/
Text
Grundlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
115
Edwin Hewitt
Kenneth A. Ross
Abstract
Harmonic Analysis I
Springer-Verlag
Grundlehren der
mathematischen Wissenschaften 115
A Series of Comprehensive Studies in Mathematics
Editors
M. Artin S.S. Chern J.M. Fr6hlich E. Heinz H. Hironaka
F. Hirzebruch L. H6rmander S. Mac Lane C.C. Moore
J.K. Moser M. Nagata W. Schmidt D.S. Scott Ya.G. Sinai
J. Tits B.L. van der Waerden M. Waldschmidt S. Watanabe
Managing Editors
M. Berger B. Eckmann S.R.S. Varadhan
Grundlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
2. Knopp: Theorie und Anwendung der unendlichen Reihe
3. Hurwitz: Vorlesungen tiber allgemeine Funktionentheorie und elliptische Funktionen
4. Madelung: Die mathematischen Hilfsmittel des Physikers
10. Schouten: Ricci-Calculus
14. Klein: Elementarmathematik vom h6heren Standpunkte aus. 1. Band: Arithmetik, Algebra,
Analysis
15. Klein: Elementarmathematik vom h6heren Standpunkte aus. 2. Band: Geometrie
16. Klein: Elementarmathematik vom h6heren Standpunkte aus. 3. Band: Prizisions- und
Approximationsmathematik
20. P61ya/Szeg: Aufgaben und Lehrsitze aus der Analysis II: Funktionentheorie, Nullstellen,
Polynome, Determinauten, Zahlentheorie
22. Klein: Vorlesungen tiber h6here Geometrie
26. Klein: Vorlesungen tiber nicht-euklidische Geometrie
27. Hilbert/Ackermann: Grundztige der theoretischen Logik
30. Lichtenstein: Grundlagen der Hydromechanik
31. Kellogg: Foundations of Potential Theory
32. Reidemeister: Vorlesungen fiber Grundlagen der Geometrie
38. Neumann: Mathematische Grundlagen der Quantenmechanik
40. Hilbert/Bernays: Grundlagen der Mathematik I
43. Neugebauer: Vorlesungen fiber Geschichte der antiken mathematischen Wissenschaften. Band
1: Vorgriechische Mathematik
50. Hilbert/Bernays: Grundlagen der Mathematik II
52. Magnus/Oberhettinger/Soni: Formulas and Theorems for the Special Functions of
Mathematical Physics
57. Hamel: Theoretische Mechanik
58. Blaschke/Reichardt: Einffihrung in die Differentialgeometrie
59. Hasse: Vorlesungen tiber Zahlentheorie
60. Collatz: The Numerical Treatment of Differential Equations
61. Maak: Fastperiodische Funktionen
62. Sauer: Anfangswertprobleme bei partiellen Differentialgleichungen
64. Nevanlinna: Uniformisierung
66. Bieberbach: Theorie der gew6hnlichen Differentialgleichungen
68. Aumann: Reelle Funktionen
69. Schmidt: Mathematische Gesetze der Logik I
71. Meixner/Schifke: Mathieusche Funktionen und Sphiroidfunktionen mit Anwendungen auf
physikalische und technische Probleme
73. Hermes: Einffihrung in die Verbandstheorie
74. Boerner: Darstellungen von Gruppen
75. Rado/Reichelderfer: Continuous Transformations in Analysis, with an Introduction to
Algebraic Topology
76. Tricomi: Vorlesungen fiber Orthogonalreihen
77. Behnke/Sommer: Theorie der analytischen Funktionen einer komplexen Verinderlichen
78. Lorenzen: Einftihrung in die operative Logik und Mathematik
80. Pickert: Projektive Ebenen
continued after index
Edwin Hewitt
Kenneth A. Ross
Abstract Harmonic
Analysis
Volume I
Structure of Topological Groups
Integration Theory Group Representations
Second Edition
Springer-Verlag
New York Berlin Heidelberg London Paris
Tokyo Hong Kong Barcelona Budapest
Edwin Hewitt
Department of Mathematics GN-50
University of Washington
Seattle, WA 98195, USA
Kenneth A. Ross
Department of Mathematics
University of Oregon
Eugene, OR 97403, USA
Mathematics Subject Classification (1991)" 43A70, 22B05, 22A10, 43A15
Library of Congress Cataloging in Publication Data. Hewitt, Edwin, 1920-. Abstract
harmonic analysis. (Grundlehren der mathematischen Wissenschaften; 115). Bibliography:
p. Includes indexes.
ISBN 0-387-94190-8
CONTENTS: v. 1. Structure of topological groups, integration theory, group representa-
tions. 1. Harmonic analysis. I. Ross, Kenneth A., joint author. II. Title. II. Series: Die
Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen; 115.
QA403.H4. 1979. 515'.2433. 79-13097
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Preface to the Second Edition
It has not been possible to rewrite the entire book for this Second
Edition. It would have been gratifying to resurvey the theory of topo-
logical groups in the light of progress made in the period 1962--1978, to
amplify some sections and curtail others, and in general to profit from
our experience since the book was published. Market conditions and other
commitments incurred by the authors have dictated otherwise. We have
nonetheless taken advantage of the kindness of Springer-Verlag to make
a number of improvements in the text and of course to correct misprints
and mathematical blunders.
We are in debt to the readers who have written to us or spoken with
us about the text, and we have tried to follow their suggestions. We are
happy here to record our gratitude to ROBERT B. BURCKEL, W. WISTAR
COMFORT, ROBERT E. EDWARDS, ROBERT E. JAMISON, JORGE M. L6PEZ,
THEODORE W. PALMER, WILLARD A. PARKER, KARL R. STROMBERG, and
FRED THOELE, as well as to a host of others who have kindly made sug-
gestions to us.
Our thanks are due as well to Springer-Verlag for their support of
our work.
Seattle, Washington
Eugene, Oregon
EDWIN HEWITT
KENNETH A. Ross
January 1979
Preface to the First Edition VII
Preface to the First Edition
When we accepted the kind illvitation of Prof. Dr. F.K. SCIIMII)T to
write a monograph on abstract harmonic analysis for the Grundlehren
der Mathematischen Wissenscha/ten series, ve intended to vrite all that
we could find out about the subject in a text of about 600 printed pages.
We intended that our book should be accessible to beginners, and we
hoped to make it useful to specialists as well. These aims proved to be
mutually inconsistent. Hence the present volume comprises only half
of the projected work. It gives all of the structure of topological groups
needed for harmonic analysis as it is known to us; it treats integration
on locally compact groups in detail; it contains an introduction to the
theory of group representations. In the second volume we will treat
harmonic analysis on compact groups and locally compact Abelian groups,
in considerable detail.
The book is based on courses given by E. HEWlTa" at the University
of Washington and the University of Uppsala, although naturally the
material of these courses has been enormously expanded to meet the
needs of a formal monograph. Like the. other treatments of harmonic
analysis that have appeared since 940, the book is a lineal descendant
of A. WEIL'S fundamental treatise (V/EIL [4]). The debt of all workers
in the fieId to WEIL'S work is well known and enormous. We have also
borrowed freely from LOOMIS'S treatment of the subject (LooMIS [2]),
from NAMARK [, and most especially from PONTRYAGIN [7].. In our
exposition of the structure of locally compact Abelian groups and of
the PONTRYAGIN-VAN KAMPEN duality theorem, we have been strongly
influenced by PONTRYAGIN'S treatment. We hope to have justified the
writing of yet another treatise on abstract harmonic analysis by taking
up recent work, by writing out the details of every important construction
and theorem, and by including a large number of concrete examples and
facts not available in other textbooks.
The book is intended to be readable by students who have had basic
graduate courses in real analysis, set-theoretic topology, and algebra as
given in United States universities at the present day. That is, we
suppose that the reader knows elementary set theory, set-theoretic topol-
ogy, measure theory, and algebra. Our ground rule is [although this is
1 Here and throughout the book, numbers in square brackets designate works
in the bibliography found at the end of the volume. These are arranged and
numbered by authors.
not hard and fast I that facts and concepts from KEI,LEY [2], HALMOS [2],
and VAN DERWaERDEN 1] may be used without explanation or proof.
In our effort to make the book useful for specialists, we have included
a large corpus of material which a beginning reader may find it wise to
omit. The following suggestions are for the beginning student who
wishes to get to the root of the matter as quickly as possible. First read
§ § t -- 7, Subheads (8.1) -- (8.7), (9. t ) -- (9.14), and all of § I 0. The reader
who is already familiar with or wishes to take on faith the theory of
integration on locally compact Hausdorff spaces may skip §§11--14.
Section 5 is absolutely essential and should be read with careful atten-
tion. Sections 16--18 may be omitted. Sections 19 through 24 are vital,
and should be read carefully. Sections 25 and 26 are rather special,
but are so interesting that we hope all readers will work through them.
Sections 4--1 and 15--26 contain subsections entitled Miscellaneous
Theorems and Examples. Some of these are worked out in detail; for
others, proofs are sketched or omitted entirely. We refer occasionally in
the main text to a result drawn from the Miscellaneous Theorems and
Examples. All such results are easy and are supplied with proofs. The
reader is counselled at least to read the statements of the Miscellaneous
Theorems and Examples, and to use them as exercises ad libitum.
Many sections also contain historical notes. We have tried to trace
the history of the principal theorems and concepts, but we obviously
have not produced a complete history, and it would be foolish to claim
that we have produced one correct in every detail. Also, while some of
the results we give are new so far as we know,/ailure to cite a re/erence
/or a given theorem should not be construed as a claim o/ originality on
our part.
For the reader's convenience, we have assembled in three appendices
certain ancillary material not easily accessible elsewhere, which is essen-
tial for one part or another of the theory. These appendices may be
read as the reader encounters references to them in the main text.
We have been assisted by many colleagues. W. WISTAR COMFORT,
JOHN M. ERDMAN, ALBERT J. FRODERBERG, L. FUCHS, JAMES MICHELOW,
RICHARD S. PIERCE, KARL R. STROMBERG, ELMAR THOMA, VEERAVALLI
G. VARADARAJAN, and HERBERT S. ZUCKERMAN have all read various
parts of the manuscript, made valuable improvements, and saved us
on occasion from grievous error. Improvements have also been made by
NORMAN J. BLOCH, FRANCIS X. CONNOLLY, GERALD L. ITZKOWITZ, and
RICHARD T. SHANNON. For the kelp so generously given by our friends
we are sincerely grateful. Our thanks are due to Mlles. LYNNE HARPER
and JEANNE SLOPER :for preparation of the typescript. Mr. ALBERT
J. FRODERBERG has most generously assisted us in the task of proof-
reading.
VIII Preface to the First Edition
The reading and research by both of us on which the book is based
have been generously supported by the National Science Foundation,
U.S.A., and by a fellowship of the John Simon Guggenheim Memorial
Foundation granted to E. HEWlTT. We extend our sincere thanks to
the authorities of these foundations, without whose aid our work could
not have been done.
Finally, we record our gratitude to Prof. Dr. F.K. SCHMIDT for his
original suggestion that tile book be written, and to the publishers for
their dispatch, skill, and attention to our every wistl irl producing the
book.
Seattle, Washington,
Rochester, New York
EDWIN HEWlTT
KENNETrI A. Ross
October t962
Table of Contents
Page
Prefaces ............................... V
Chapter One" Preliminaries ..................... t
Section 1. Notation and terminology ................ 1
Section 2. Group theory ..................... 3
Section 3. Topology ....................... 9
Chapter Two" Elements of the theory of topological groups ........ t 5
Section 4. Basic definitions and facts ................ 16
Section 5. Subgroups and quotient groups .............. 32
Section 6. Product groups and projective limits ........... 52
Section 7. Properties of topological groups involving connectedness . . 60
Section 8. Invariant pseudo-metrics and separation axioms ..... 67
Section 9. Structure theory for compact and locally compact Abelian
groups ........................ 83
Section 10. Some special locally compact Abelian groups ....... 106
Chapter Three" Integration on locally compact spaces ........... 117
Section 11. Extension of a linear functional and construction of a measure 118
Section 12. The spaces p (X) (1 p =< oo) .............. 135
Section 13. Integration on product spaces ............. 150
Section 14. Complex measures ................... 166
Chapter Four" Invariant functionals .................. 184
Section 5. The Haar integral ................... 184
Section 16. More about Haar measure ................ 215
Section 17. Invariant means defined for all bounded functions ..... 230
Section 18. Invariant means on almost periodic functions ...... 245
Chapter Five" Convolutions and group representations .......... 261
Section 19. Introduction to convolutions ............... 262
Section 20. Convolutions of functions and measures .......... 283
Section 21. Introduction to representation theory ........... 311
Section 22. Unitary representations of locally compact groups ..... 335
Chapter Six" Characters and duality of locally compact Abelian groups . . • 355
Section 23. The character group of a locally compa,ct Abelian group • • 355
Section 24. The duality theorem .................. 376
Section 25. Special structure theorems .............. 399
Section 26. Miscellaneous consequences of the duality theorem .... 426
Appendix A" Abelian groups ................... 439
B" Topological linear spaces ................. 451
C" Introduction to normed algebras ............ 469
Bibliography ........................... 492
Index of symbols ...................
...... 506
Index of authors and terms .................... 509
Chapter One
Preliminaries
Abstract harmonic analysis has evolved over the past few decades
on the basis of several theories. First we have the classical theory of
Fourier series and integrals, set forth in many treatises, such as ZVG-
MuxI [1 and BOCHNER [I. Second, we have the algebraic theory of
groups and their representations, which is also described in many stand-
ard texts (e.g. VAN IERWAERIEN [1). Third, we have the theory of
topological spaces, by now a fundamental tool of analysis, and also the
subject of standard texts (e.g. KEIIEY [2). The latter two subjects
were combined to form the notion of a topological group. This is an
entity which is both a group and a topological space and in which the
group operations and the topology are appropriately connected. The
structure of topological groups was extensively studied in the years
t925--1940; and the subject is far from dead even today.
Using a fundamental construction published in 1933 by A. HAAR [3,
A.WEII [41 in 1940 showed that Fourier series and integrals are but
special cases of a construct which can be produced on a very wide class
of topological groups. Furthermore, several classical theorems about
Fourier series and integrals [PARSEVAL'S equality, PLANCHEREL'S theorem,
the HERGLOTZ-BOCHNER theorem, and the HAUSDORFF-YOUNG inequality,
to name a few_ could be meaningfully stated, and proved, in this general
situation. The end of the development initiated by WEIL is nowhere in
sight. Problems in one branch or another of harmonic analysis are
occupying the attention of many mathematicians at the present day,
and we cannot predict at all where future developments will lead.
At any rate, the foundations of abstract harmonic analysis now seem
to be clear. The present volume is devoted to a study of these founda-
tions. Our first task is to learn thoroughly what a topological group is.
We naturally need an array of facts about groups and topological spaces
per se, which are set down in the present chapter.
§ 1. Notation and terminology
In this section, we explain some of the notation and terminology
used throughout the text. Standard concepts and notation are used
without explanation.
Hewitt and Ross, Abstract harmonic analysis, vol. I 1
2 Chapter I. Preliminaries § 2. Group theory 3
The symbols c and D mean ordinary inclusion between sets; they
do not exclude the possibility of equality. The void set is denoted by O.
Frequently the sets we deal with are subsets of some universal set,
say E. In this case we denote by A' the complement of the set A:
A'-- {x'xE E and xCA. f
In every case it will be clear what the set E is. For sets A and B, the
symmetric di//erence A/B is defined to be the set (Afl B') U (A'fl B).
A family of sets has the/inite intersection propertyif fl {A'AE$'}=O
for all finite subfamilies o- of . In particular, no set in is void if
has the finite intersection property. A collection {A,},z of sets is said
to partition a set X if U A,--X, each A is nonvoid, and t' sets A
I '
are pairwise disjoint. A family {A}, of sets is a covering or cover of
X if U A,D X. Given a set X, (X) denotes the family of all subsets of X.
Knowledge of elementary cardinal and ordinal arithmetic is assumed.
The family of countable sets includes finite sets and the void set. Infinite
countable sets have cardinal number N 0 and the real line has cardinal
number 2 t°. We will write c for 2 0. The cardinal number of an arbitrary
set A is denoted by A.
A subset Y of a partially ordered set X is said to be co/inal in X if
for every xX, there is a y e Y such that x=< y.
The terms "mapping", "transformation", and "correspondence" are
synonymous with "function". The term "operator" has a special mean-
ing, which is explained in (B.2). A function / will often be defined by an
expression
- 1()
where x denotes a generic element of the domain of the function and ](x)
denotes its image under ]. For a function / and a subset A of its domain,
][A denotes the restriction of / to A.
If ! is a function on X into Y and g is a function on Y into Z, then
the composition of g by / is the function go/on X into Z defined by
gol(x)=g (l(x)) for xeX.
Let X be a set and A any subset of X. 'The symbol a will denote
the function defined on X such that
{t 0 for xeA,
G (x) -- for xEA'.
The function. a is called the characteristic/unction of A.
Let {X," e I} be a nonvoid family of sets. We define ,PX, to be the
set of all functions x from I into U X, such that x(e)X, for all I.
This set is called the Cartesian product of the sets X,. We will almost
invariably write the elements of ,X, as (x,) where x,= x(t). Thus (x,)
is an element of ,PX, and for each , x, is an element of X,. If the
cardinal number of I is m and if all of the sets X, are a fixed set X,
we will frequently write X m for ,PX,. If I is finite, say I= {i, 2 ..... m},
we sometimes write ,X, as ,=P1X or XxXx... xX,,.
We reserve the symbols R for the set of all real numbers, K for the
set of all complex numbers, Q for the set of all rational real numbers,
and Z for the set of all integers. The symbol exp stands for the exponen-
tial function defined on K. The symbol T represents the set {exp (2zix)"
0<x<}; it is a subset of K. The set of all numbers exp(2zik/p"),
where p is a fixed prime, k runs through all integers, and n through all
nonnegative integers, will be denoted by Z (poo). The function signum
z
or sgn on K is defined by sgn z-- ] for z=4= 0 and sgn 0= 0.
For real numbers a and b, where a__< b, we define[a, b/---- {xR :a <=x<=b},
la, b[--{xER'a<x<b}, ]a,b]={xR'a<x<_b}, and [a,b[={xeR'a<__
x< b). We also define ]a, oo[-- {x R'a< x), with analogous definitions
for [a, oo[, ]--oo, a[, ]--oo, a], and --oo,
The sets K"and R"are complex n-dimensional space and real n-dimen-
sional space, respectively (n--2, 3, 4 .... ). For a=(a 1 .... ,a. and
b--(b ..... b,,) in K " we define the inner product (a, b) as ,ab and
the norm Ilalt as (a, a)Ѕ. Let S_ denote the set of all aeR" such
that [] a [] = 1.
For a nonvoid set X and a positive real number p, we define l (X)
as the set of all complex-valued functions aon X such that [a(x)
xX
Obviously a function in l (X) vanishes outside of a countable subset of X.
The norm of a El (X) is the number
See § 12 for a complete discussion.
For an arbitrary nonvoid set X, let d denote the function on XxX
such that d=t for all xX and d=0 if x=4=y and x, yeX. This
function is cal'led Kronecker's delte Junction.
The end of each proof is indicated by the symbol ].
§2. Group theory
In this section, we establish the terminology and notation concerning
groups that will be used throughout the book. We also prove a few
theorems about groups, although most of these are quite standard 1. We
The only nonstandard theorem in §2 is (2.9).
4 Chapter I. Preliminaries §2. Group theory
do this to clarify analogous theorems appearing in subsequent sections,
where topological considerations also play a r61e. For a very detailed
account of the theory of groups, the reader is referred to KuRog [1 ; for
a shorter and more elementary version, see for example VAN DERWAER-
For the detailed analysis given in §§ 9, 23--26, we require many
refined theorems about Abelian groups. These are given in Appendix A,
with complete proofs.
In dealing with abstract groups, we will adhere to multiplicative
notation for the group operation, except in part of Appendix A, where
it is convenient to write the group operation as addition. In dealing
with R, Z, R , etc., which are obviously groups under addition, we will
write the group operation as addition. In discussions involving a multi-
plicative group G, the symbol e will be reserved for the identity element
of G.
We are principally concerned with groups in this book. For some
purposes, however, it is useful to consider semigroups. A semigroup is
a nonvoid set G and a mapping (x,y)---xy of G><G into G such that
x(yz)--(xy)z for all x, y, z in G. That is, a semigroup is any nonvoid
set with an associative multiplication. We do not assume the existence
of an identity or the validity of any cancellation law. Note too that
semigroups may contain subgroups and that groups may contain sub-
semigroups that are not groups.
Let G be a group. For a fixed acG, the mappings x--+ax and x--xa
of G onto itself are called le[t and right translation by the element a,
respectively. The mapping x-- x -1 of G onto itself is called inversion.
Mappings x--axa -1 of G onto itself are called inner automorphisms
of G. We will frequently write inner automorphisms as 0: O(xl=axa -.
The set of automorphisms of a group G is itself a group under the opera-
tion of composition. The set of all inner automorphisms of G forms a
subgroup of the group of all automorphisms of G; and the mapping
a-- is a homomorphism of G into its automorphism group.
Let A and B be subsets of a group G. The symbol A B denotes the
set {ab'acA, bB}, and A - denotes {a-'aA}. We write aB for {a}B
and B a for B {a). We write AA as A 2, AAA as A a, etc. Also we write
A-A - as A -2, etc. A subgroup H of G such that H=# G and H=# {e) is
called a proper subgroup of G.
For a subgroup H of a group G, the symbol G/H will always denote
the space of le/t cosets of H in G. Thus points of G/H are sets xH, and
G/H={xH:xG). Let H be a normal subgroup of the group G. The
mapping x-- x H of G onto G/H is called the natural mapping o/ G onto
G/H. It is obviously a homomorphism. The group G/H is called the
quotient group of G by H. For an element a of G, let a be the mapping
of G/H onto itself defined by av/(xH)=(ax)H, for all x HG/H. It is
clear that o is well defined, that a is a one-to-one mapping of G/H onto
itself, and that the set {a: a c G} forms a group under composition that
is a homomorphic image of G under the mapping a--. Also, given
cosets xH and yH, the mapping y,-l obviously carries xH onto yH.
Two elements a and b in G are said to be conjugate if some inner
automorphism maps a onto b, i.e. if a=xbx - for some xG. There
exists a partition {A,),z of G such that tvo elements are conjugate if
and only if they belong to the same A,; the sets A, are called the con-
ugacy classes of G.
As already noted, R, Z, Q, R ", K", ... are Abelian groups under addi-
tion. Observe also that T and z(p°) are Abelian groups under multi-
plication. The groups R, T, Q, and Z(p) are of special importance in
the structure theory developed in §§ 9, 24--26. The group T in addition
is of vital importance in harmonic analysis: this is explained in §§23
and 24.
The symbol Z(m) denotes the finite cyclic group of m elements
(m=2, 3, ...): we will frequently represent this group as the set of
integers {0, , 2 .... , m--1 }, with addition modulo m.
(2.1) First isomorphism theorem. Let G be a group, H a normal
subgroup o/ G, and A an arbitrary subgroup o/G. Then AH=HA is a
subgroup o/ G, H is a normal subgroup o/ A H, ad HC?A is a normal
subgroup o/ A. The groups (AH)/H and A/(HC?A) are isomorphic. In
/act, the mapping de/ined by a H-- (a H) C? A = a (H C? A ), a c. A, is an
isomorphism o/(AH)/H oto A/(HC?A).
Proof. It is obvious that AH=HA, that HA is a subgroup of G,
that H is a normal subgroup of HA, and that H A is a normal subgroup
of A. Consider the mapping of A onto (AH)/H defined by a-aH. It
is easy to see that this mapping is a homomorphism and that its kernel
is H A. By the fundamental homomorphism theorem for groups, AH/H
and A/(HC?A) are isomorphic and the isomorphism can be given by
a(HC?A)--aH; let be the inverse of this mapping. [
(2.2) Second isomorphism theorem. Let G and be groups with
identity elements e and , respectively, and let q be a homomorphism o/G
onto . Let be any normal subgroup o/, H=q - (), and N=q - ().
Then the groups G/H, /, and (G/N)/(H/N) are isomorphic.
Proof. The mapping x--q)(x)/ is a homomorphism of G onto
with kernel H. Hence G/H is isomorphic with /r. Since is iso-
morphic with GIN and is isomorphic with H/N, ,/ is isomorphic
with (G/N)/(H/N).
6 Chapter I. Preliminaries § 2. Group theory 7
(2.3) Direct products. Let {G,',I} be a nonvoid family of groups
and let ,G, be the Cartesian product of the sets G,. For (x,)and (y,)
in P G,, let (x,) (y,) be the element (x,y,) in ,PzG,. Under this multi-
plication, ,G, is a group; it is called the direct product of the groups G,.
The groups G, are called/actors. The identity of , G, is (e,), where each
• P* G, be the set of all (x,) P G, such that
e, is the identity of G, Let ,
x,--e, for all but a finite set of indices this set varying with (x,)]. Then
P* G, is a subgroup of ,G," it is called the weak direct product, of the
I ' .-"
groups G,. If the cardinal number of I is m and if each G,=G,,then we
will write G r" for P G, and G m* for P* G,.
(2.4) Theorem. Let G be a group with identity e, and let N 1, N 2 ..... N,,,
be a collection o/normal subgroups o/G satis/ying:
(i) NI N2 ... N,,, -- G ;
(ii) (N Nz... N)f'lN+ = {e} /or k -- t, 2,..., m-- t.
Then G is isomorphic with the direct product P=xN"
Proof. Every element x of G can be written as xxz ... x where
each x belongs to N. By virtue of (ii), this representation is unique.
Hence for xG, we define (x)--(x)P__lN where xxz.., x,,,=x. It
is then simple to show that is an isomorphism of G onto =PxN"
(2.5) Theorem. Let G be a group with identity e, and let {N,:tI}
be a nonvoid /amily o/normal subgroups o/G. For each I, let M, be the
smallest subgroup o] G containing all N /or . Suppose that:
(i) the smallest subgroup o/G containing all N, is G itsel/;
(ii) /or every I, the relation M, fqN,--(e} obtains.
Then G is isomorphic with the weak direct product N,.
Proof. For x] N, (/'-- 1, 2), we have xl xz x x = (x x2 x ) x N,,,
and similarly xl xz x x x N,,. Hence (ii) shows that x xz----- xz x. By (i)
and the foregoing, every element of G can be written in the form x x..., x
where xiN, ('--t, 2, ..., k). This representation is unique. For, if
xl x2 ... x-- y y ... y (x i, yi N,), then yX x ... y_ x_x-- y x 1, and
(ii) implies that y----x. It isnow obvious that the mapping x xz... x-->(y,)
where y,-- x i for/'----t, 2, ..., k and y,--e otherwise] is an isomorphism
of G onto ,*N,.
(2.6) Semidirect products. Let L be a group, and suppose that L
contains a normal subgroup G and a subgroup H such that GH--L and
G f3H= (e}. That is, suppose that one can select exactly one element h
from each coset of G so that {h} forms a subgroup, H. If H is also normal,
then L is isomorphic with the direct product GxH. If H fails to be
normal, we can still reconstruct L if we know how the inner auto-
morphisms h behave on G. Namely, for xi G and hi H (/'-- I, 2), we have
(x ) (x ) : x : (x , (x)) .
The construction just given can be cast in abstract form. Let G and H
be groups and suppose that there is a homomorphism h--> zh which carries
H onto a group of automorphisms of G. That is, Zhoz,---Zh, for h,h'H.
Let G@H denote the Cartesian product of G and H. For (x, h) and
(x', h') in G@H, define
(x, = (x Ix'l),
Then G@H is a group; it is called a semidirect poduct of G and H. Its
identity is (el, e.), where e and e. are the identities of G and H, respec-
tively. The inverse of (x,h)is (zh-,(x-),h-). Let G--{(x,e.)'xG}
and H--{(e, h):hH). Then G1 is a normal subgroup of G @H, and H x
is a subgroup. Since (e, h). (x, ez). (e, h)---(z(x), e.), the inner auto-
morphism p(,,h) for (e, h)H reproduces the action of h on G. Thus
every semidirect product is obtained by the process described in the
preceding paragraph.
(2.7) Linear groups. (a) Let A-- (a],)i,= be an nxn matrix, where
the coefficients a1 are complex numbers. The transposetA of the matrixA
is the matrix (a.),=l and the coniugate of A is the matrix
where . is the complex conjugate of the number ai,. We define A* to
be the matrix *()----(). The trace of the matrix A is aii and is
i=1
denoted by trA; the determinant of a matrix A is denoted by detA.
(b) An nxn matrix A is said to be orthogonal if A--,/i and 5t--A -,
unitary if A---A *, symmetric if A =5t, skew-symmetric if A----A,
Hermitian if A = A*, skew-Hermitian if A =--A*, and real if A--
Let F be any subfield of K, such as K or R. The set of all nxn
matrices whose coefficients belong to F is denoted by 9X (n, F). The set
of all nonsingular matrices in 9X (n, F) forms a group under multiplication,
is called the general linear group over F, and is denoted by @ (n,F). The
subgroup of ®(n, F) consisting of the matrices of determinant t is the
special linear group over F and is denoted by ®(n,F). The unitary
group 1I (n) and the orthogonal group © (n) consist of all unitary matrices
and all orthogonal matrices, respectively; they are subgroups of'® (n, K).
Finally, we define the special unitary group and the special orthogonal
group as ®ll (n) -- ® (n,K) (q ll (n) and ®© (n)-- ® (n, K) f-l© (n), respec-
tively. The set 9X (n, F), the group ® (n, F), and all its subgroups can
be regarded as subsets of F ' in an obvious way; we will frequently do so.
8 Chapter I. Preliminaries 3. Topology 9
(2.8) Free groups and symmetric groups. (a) Let X be any non-
void set. A word is either void (written e) or a finite formal product
xlx. .... x n of elements of X where e--q-1 the elements x need not
be distinct. A word is reduced if it is void or if ek----e+l whenever
x--x+l. The length of the reduced word av--xlx. .... x n is n and the
length of e is 0. Let F denote the set of all reduced words of X. If
- . .... - ana ,-/'y .... y lon to
is defined as follows. Consider x lx. 2 x
...... y,. If this.word is
reduced, we define it to be avid. If it is not reduced, then x--yl and
= • ' -Y" Y,7' and if this is redtmed we
e, --bl Then consider x'x. "",-1 ""
define it to be avg. If it is not reduced, we continue in the indicated way
until a reduced word is obtained and define it to be avg. With this
multiplication, F is a group and is called the/ree group generated by X.
..... x is x -n --... xi-.
The identity of F is e and the inverse of xx2
The proof that av (d z) = (avd)z for av, d, z F is a straightforvard induction
argument on the length of the word d.
(b) Let X be any nonvoid set. For each x X, let F, be the free group
generated by x; i.e., F= {G, x, x -, x 2, x-Z, ...}. Then the weak direct
product P*F is the jree Abelian group generated by X.
xX
(c) If N is any nonvoid set, then the set of all one-to-one mappings
of N onto N forms a group under the operation of composition. A set {/}
of these mappings is called transitive if for every x,yN, there is an
/0 d {/} such that/0 (x) -- y.
If N is finite and has n elements, then the group of all one-to-one
mappings of N onto N is called the symmetric grop , on n letters. The
elements of ®.are called persutations and any subgroup of ®, is called
a permutation group on n letters.
(2.9) Theorem. Let X be any nonvoid set and let xIx. .... x (n >= 1)
be any reduced word o/ the /ree group generated by X. Let t denote
• .... x does not
identity permutation o/ ,+ or ,,+ Suppose hat xx.
have the /orm y- y-.., y-= y-' and tha n 1. Then there is a mapping
x --Px o/X into , + such that."
(i) P= t i/x is different/rom x, x., ..., x,
(ii) P:: oP;: o...o P;2 =';
(iii) P2; o P o...o P;2 P /or = , 2,..., n .
I/ x x .... x= y-, here is a mapping o/ X into + satis/ying (i), (ii),
and (iii). I/n = 1 and q = t, hen there is a mapping o/ X into satis/ying
(i) and (ii).
Proof. We here regard @nq-1 as the set of all one-to-one mappings
of {1, 2, ..., n+ 1} onto itself. For x{x, ,% .... , x,,}, define P, as .
For xX appearing in the sequence {x, x. ..... x,}, suppose that
x=xi= .... xi,. Then if %=1, let P,(l'+l)=f; if %=--1, let
P, (f) =f+ t. Make these definitions for k= 1, 2 .... ,1. Since ei=
whenever xi= xi+ , there is no contradiction inherent in this definition.
If x4 =x. and x appears in {x, x,. .... , x}, let P,(u+I)=n+t. If
G=--t, let t be the least integer such that x,= x,+l ..... x. Then
define P, (n+ t)=t. Extend the P,'s in any way to be permutations of
{t, 2, ..., + t}. It follows at once that PoP:o ...oP2(n+ 1)=1. This
proves (ii). Since P2 o P,: o... oP,] (n + 1) = 1 and P,, (n + 1) = n + 1 for
xix,, (iii) is obvious for x.4=x. Since P,,(n+ 1)=n if e=l and
P (n+ t)=t > 1 if e=--I, (iii) also holds for x.
The case xIx. .... x= y- is easily dealt with. For xq= y, let
Let Py(/e)=/e+ 1 (k=t, 2 .... , n+ 1) and Py(n+2)=l. Then (i), (ii),
and (iii) are immediate.
The case =t and q=l is trivial: set P,(t)=2 and P,(2)=I.
§ 3. Topology
In this section, we review briefly the parts of set-theoretic topology
needed in the book. We give definitions only where terminology or
notation is not universally agreed on, and we give proofs only where
no correct proof as short as ours is available in standard texts. For a
detailed account of set-theoretic topology, we refer the reader to
KELLEY [2].
In dealing with topological spaces, we require neighborhoods to be
open. We frequently define a topology by its family of open sets, written
as (ў. We denote the interior of a set A in a topological space by A °
and the closure of A by A-. Let (ў and (ў,. be two topologies on a set X.
If (.01 C (.02, then (.01 is said to be weaker than (G. and (G. is said to be stronger
than (91 .
Let X be a topological space and suppose that for every pair of
points x, yX, there is a homeomorphism / of X onto itself such that
/(x)--y. Then X is called a homogeneous topological space.
In agreement with many writers on set-theoretic topology, although
not with KELLEY 2, we make T O separation part of the axioms of
regularity and complete regularity. Thus in our terminology, regular
and completely regular spaces satisfy Hausdorff's separation axiom.
For a topological space X, let W (X) denote the least cardinal number
of an open basis and let b(X) denote the least cardinal number of a
dense subset.
(3.1) Partitions of unity (DIEUDONNI [1]). Let X be a normal space,
F a closed subset o/ X, and U, U. ..... U open sets such that U UDF.
/=1
Chapter I. lreliminaries §3. Topology
Then there exist continuous/unctions
such that
(i) h(x)=t /or all x F ;
/=1
(ii) h(U)= 0 [or k= t, ..., n.
Proof. (I) Suppose that U U=X; we will prove that there are
/=1 n
closed sets A 1 .... ,A, such that U A=X andAcU, for k--t, ..., n.
/=1
The proof is by induction andis obvious for n=t. Suppose that UIUU,.=X.
Then U and U are disjoint closed sets and there exist disjoint open sets
V and V,. such that U C V ancl U c V,.. Setting A= V' and A,.= V',
we obtain the present assertion for n= 2.
Suppose that the assertion is true for n--t and that U__IU=X.
Since X=\ U U., there are closed sets A and A. such that
--1 / 1
A c U U, A, C U,, and A UA,= X. For k = t .... , n-- t, define V = UfA.
/=1
Then U V = A and by the induction hypothesis applied to A, there are
/=1 n--1
sets A, A 2 .... , A,_I, closed in A, such that U A=A and AcV
/=1
(k=t, ..., n--t). Clearly the A's are also closed in X, each A is con-
tained in U, and U A--X.
/=1
(II) We now prove the theorem in the case that F--X. Suppose
that U U=X and that A1,..., A, are determined as in (I). By URY-
/=1
soN's lemma, there are continuous functions [ on X such that
/(X)c0,t, /(A)=t, and /,(U)=0 (k=t ..... n). For k=t, ...,n,
let
h(x)=---- for xX.
Since f.() t or all , it is clear that each h is continuous ad (i)
and (ii) are obvious.
(III) Finally, suppose thatF is a closed subset of X and that__UUDF.
Define Uo=F' and note that U U--X. Thus by (II), there are con-
/=0
tinuous functions h 0, h, ..., h, on X with values in [0,t] such that
. h(x)=t for all xX and h(U)=0 (k--0, t ..... n). Since h0(F)--0,
/=0 n
we have . h (x)--t for xF. That is, h ..... h satisfy the conclusions
of the theorem.
(3.2) Compact and locally compact spaces. A topological space X
is compact counlably compacl] if every covering [every countable covering
of X by open sets admits a finite subcovering. A topological space X is
locally compact locally counlably compact if every point of X has a
neighborhood U such that U-is compact countably compact_ as a
subspace of X. A topological space X is called a-compact if it is a
countable union of compact subspaces. A topological space X has the
Lindel6/properly if every covering of X by open sets admits a countable
subcovering.
(3.3) Theorem. Let X be a topological space having the property that
/or each x EX and neighborhood U o] x, there is a neighborhood V o[ x
such lhat V-C U. Then lhe closure o[ a compact subset o[ X is again
compacl . Also, i[ [ is a continuous mapping o[ a topological space Y
into X, and i[ A is a subset o[ Y with compact closure, then [(A) also has
compacl closure.
Proof. Let A be a compact subset of X. To show that A- is compact,
it suffices to show that every open cover of A also covers A-. For then
if is an open cover of A-, there is a finite subfamily of q./covering A,
and hence coveting A-.
Suppose that q/is an open cover of A and let W= U {U:Uq./}. For
each x A, there is a neighborhood V of x such that V-C W. Since A is
compact, there exist x, ..., x,A such that A CUV,.__ This implies
that A-CU=V,- c W. That is, A- is covered by q./.
Suppose now that [ is a continuous mapping of Y into X and that
the subset A of Y has compact closure. Then /(A-) is compact and
hence/(A-)- is also compact. This implies that [(A)- is compact.
(3.4) Connectedness. A topological space X is connected if and only
if it is not the disjoint union of two nonvoid sets that are both open and
closed. A topological space X is locally connecled if the connected open
subsets of X form an open basis for the topology of X. A component of
a topological space is a connected subset that is properly contained in no
other connected subset. A topological space is tolally disconnected if all
of its components are points. A topological space X is O-dimensional
if the family of all sets that are both open and closed is an open basis
for the topology.
A topological space X is arcwise connected if for every pair of points
x and y in X, there exists a continuous mapping [ of 0,t into X such
1 The closure of a compact set need not be compact. Consider Z><Z and let a
subset F of Z><Z be closed if F= Z><Z or if the set {m Z:(m, n) F} is finite for
every nZ. Then A = {(m, O):mZ} is compact and A--= Z><Z is not.
Chapter I. Preliminaries § 3. Topology t 3
that [(0)= x and/(t) =y. A topological space X is locally arcwise con-
nected if for every x X and neighborhood U of x, there is a neighborhood
V of x such that for every pair of points y and z in V, there exists a
continuous mapping [ of [0,t into U such that/(0) =y and/(t) = z.
(3.5) Theorem. Let X be locally compact, Hausdor[[, and totally dis-
connected. Then X is O-dimensional.
Proof. Let a be any point of X and U any neighborhood of a such
that U- is compact. We need to show that there is an open and closed
set V such that a V c U. We give the proof in three steps.
(I) Let F be a closed subset of U- and suppose that every point x
of F can be separated from a fixed point b U-by an open and closed
subset of U-. Then there is an open and closed subset C of U-such
that b C' A U- and F c C. To see this, let C be open and closed in U-,
xC,, bC,, for each xF. A finite number C1, C,, .... C,m of these
sets cover F. Hence if C=C,IUC,,U ...UC,,, C satisfies the conditions
stated.
(II) Let M be the set of all points of U - that cannot be separated
from a by an open and closed subset of U-. Then M is connected. To
prove this, note first that M is closed in U - and hence closed in X,
and that a M. Assume that M is disconnected. Then there are nonvoid
closed subsets E and F of M/and hence of X that partition M. We may
suppose that a E. Since U- is compact and Hausdorff and hence normal,
there is an open subset W of X such that E G W and W-C?F= f3. Thus
we have (W-C? W') N M = (W-C? W') C? (EUF) -- 3. By the definition of M,
every point of W-C? W'C? U- is separated from a by an open and closed
subset of U -. It follows from (I) that there is an open and closed subset
C of U- such that a C and W-C? W'A U-- C. Since C is closed in U--,
it is closed in X, and hence WC? C'C? U- is an open subset of U--. We also
have WC? C'C? U-= W-C? C' U-. Thus WC? C'C? U- is an open and closed
subset of U- containing a and disjoint from F. This contradicts the
definition of M, and shows that M is connected.
(III) Since X is totally disconnected, the connected component of a
is {a}. Hence (II) shows that for every x U-, xa, there is an open and
closed subset C, of U - such that aC, and xўC,. Applying (I) /with
F= U-C? U', we see that there is an open and closed subset C of U-
such that aC and C C? U- U'=(3. Plainly C is an open and closed
subset of X.
(3.6) Theorem. Let X be a completely regular space and let Y be an
arcwise connected space. Let Xl, x2, ..., Xm be distinct points o/ X and
Yl, Y., ..., Y, arbitrary points o! Y. The, there is a cortinuous mapping
o! X into Y such that 5v(x)= y (k= I, 2 ..... m).
Proof. Let U1, U2 ..... U m be pairwise disjoint neighborhoods of
x, x2, ..., x, respectively, and let V be a neighborhood of x such that
V/-C U (k=t, 2 .... , m). Let be a continuous mapping of X into [0,t
such that/ (x)= t and/ (V/)=0. Let Y0 be any point of Y and let
be a continuous mapping of [0,t } into Y such that z (0) = Y0 and z (t) =y
(k=t, 2, ..., m). For xV, let o(x)=zo/(x), k=t, 2 ..... m. For
X 1 , let o(x)--y o. It is easy to see that o has the properties
claimed.
(3.7) Let X be a metric space with metric d. For a positive real
number e, an e-mesh in X is a finite set {Xl .... ,x,}GX such that for
every xX, min[d(x,x), d(x2,x), ..., d(x,,x)<o. A metric space X
is said to be totally bounded if it admits an e-mesh for every e >0. The
following facts are occasionally useful. A metric space X is compact if
and only if it is complete and totally bounded. A subset A of a com-
plete metric space has compact closure if and only if A is totally lound-
ed. For proofs, see for example KELLEY [2, p. t98, Theorem 32.
A topological space X is called a locally Euclidean space if there is a
positive integer n such that every x X has a neighborhood U such that U
is homeomorphic to the open unit ball of R"" {(x, ...,x,)R"" Y..
(3.8) Paracompact spaces. A cover N of a set X is a re/inement of
a cover a/if each member of is a subset of a member of ag. A family
of subsets of a topological space X is locally/inite if each point of X has
a neighborhood which intersects only finitely many members of
A family of subsets of a topological space is -locally /inite if it is the
countable union of locally finite subfamilies. A topological space is
paracompact if it is regular and each open cover has an open locally
finite refinement.
A regular topological space X is paracompact if and only if every
open cover has an open -locally finite refinement. Every metrizable
space is paracompact. A paracompact space is normal. [See KELLEY [2,
pp. t 56--t60._
(3.9) Theorem. Let {X,},ez be a /amily o/ topological spaces. The
product space P X, is -compact i/and only i/ all o/ the /actors X, are
-compact and all but a/inite number o/ them are compact.
Proof. Let X and Y be -compact topological spaces. Then X = U X,
and Y U Y where X and Y are compact (n= t 2, . .). Then Xx Y=
U U=l(X,,xy,,) and hence Xx Y is -compact. By finite induction we
infer that XxX. x... xX is -compact whenever each X is -compact
Chapter I. Preliminaries § 3. Topology 15
(k----1, ..., n). Now suppose that {X,}, is a family of a-compact spaces
and all but a finite number, say X1, ..., X, are compact. Then
P X, is compact by TIHONOV'S theorem and hence.
P X,= XxX,x...xX,,,x P X,
I ў{I .....
is a-compact.
Suppose that ,x P X, -- sUl= A s, where each As is compact. Let , denote
the natural projection of,P X, onto X,. Since X,=sI,(As)= and each
, (As) is compact, each X, is a-compact. Assume that X, is noncompact
for ,=,, '2 ..... Then (A,)q=X, n for n--t, 2,.... Let (x,) be any
pointof P X, such thatx,nўz,(As) foralln=t 2 Then (x,)ў U
this contradiction shows that X, is compact except for a finite number
of indices ,.
(3.10) Nets. (a) Let D be a nonvoid set. A transitive partial order-
ing 1 >" directs D if for every ,/D, there is a D such that
and >-/.
A net x--x() is any function on a directed set. A net Ya, with
domain E, is a subnet of a net x, with domain D, if there is a function N
with domain E and range contained in D such that"
(i) Ya = XN (/) for each fl E;
(ii) for each D there is a/ E such that N(y) whenever
(b) Let x= be a net with domain D and range contained in a topo-
logical space X. The net x= is said to converge to an element x X if for
every neighborhood U of x there exists an element/D such that if
D and >'/, then x U. A point xX is a cluster point of the net
if for every neighborhood U of x and every D, there exists an D
such that >" and x= U.
A net x=, with domain D, is in a set A if x= A for all D.
(c) We now quote some useful facts about nets. Let X be a topo-
logical space. If A is a subset of X, then x A- if and only if there is a
net in A converging to x. The space X is compact if and only if each net
in X has a subnet converging to some point of X. The space X is Haus-
dorff if and only if each net in X converges to at most one point. A point
x X is a cluster point of a net x= if and only if some subnet of x= con-
verges to x. Let / be a function on X to a topological space Y. Then
is continuous if and only if for each net x= in X that converges to a
point xX, the net/(x=) in Y converges to/(x). See KELLEY 2, pp. 66,
t36, 67, 7t, and 86.
Here we use the notation >" although we do not assume that ___ is reflexive.
(3.11) Dimension. There are several definitions of dimension of
a topological space. We choose one adapted to our later needs (24.28).
Let X be a set and ag a finite family of subsets of X. For x X, let
re(x) be the cardinal number of the subfamily {A a/: x A}. The multi-
plicity n(ag)is defined as max{rn(x):xX}.
Let X be a compact Hausdorff space, and let n be a nonnegative
integer. Then X is said to have dimension n if the two following conditions
are satisfied:
(i) every finite open coveting of X admits a finite closed refinement
" for which n (')_<_ n + t;
(ii) there is some finite open covering °R of X such that if # is a
finite closed refinement of OR, then m (') => n + t.
We write dim (X) = n. If dim (X) = n for no nonnegative integer n,
then X is said to have in/inite dimension, and we write dim(X)=oo.
(3.12) Theorem. For a compact Hausdor// space X, the de/inition
o/ o-dimensionality given in (3.4) agrees with the de/inition o/dimension 0
given in (3-t 1).
Proof. Suppose that open and closed sets form an open basis for X
and that OR is a finite open covering of X. Then OR admits a finite refine-
ment ------ {F, F=, ..., F} consisting of open and closed sets. Let E=
and E-- F gl (F U... U F_)' (k = 2 ..... n). Then
is a refinement of OR consisting of closed sets such that m ( = t. Hence
dim (X)--0.
Conversely, suppose that dim (X)--0. Let x be any point of X, U any
neighborhood of x, and V a neighborhood of x such that V-C U. The
covering {U,V-'} of X admits a closed refinement '--{FI, F ..... F}
for which m(')--1. The F i containing x is an open and closed set
contained in U, and so the open and closed subsets of X form an open
basis.
Let X be a compact Hausdorff space and Y a closed subspace of
Then dim(Y)=<dim(X). This is easy to show, and we omit the proof.
Chapter Two
Elements of the theory of topological groups
In this chapter we initiate the program described at the beginning of
Chapter One" namely, we define and study the structure of topological
groups. Certain refined results in structure theory must be postponed to
Chapter Five, where the powerful analytical tool of representation theory
will have been made available. It seems appropriate to do at the outset
Chapter II. Elements of the theory of topological groups § 4. Basic definitions and facts 17
what can be done with one's bare hands, so to say; and this we will now do.
Section t0 contains the description of a few special topological groups
which are important for later use; the remainder of the hapter deals
with structural properties of one or another class of topological groups.
§ 4. Basic definitions and facts
A topological group is a set endowed with two structures: that of a
group and that of a topological space. These structures are connected
in such a way that algebraic properties of the group affect topological
properties of the space, and vice versa. The present chapter is an intro-
duction to the theory of the structure of topological groups. We begin
with the exact definition of a topological group.
(4.1) Definition. Let G be a set that is a group and also a topological
space. Suppose that:
(i) the mapping (x, y)---> x y of GЧG onto G is a continuous mapping
of the Cartesian product GxG onto G;
(ii) the mapping x--- x -1 of G onto G is continuous.
Then G is called a topological group .
In terms of open sets, condition (i) asserts that for every neighbor-
hood U of x y, there are neighborhoods V and W of x and y, respectively,
such that V W c U. Condition (ii) asserts that for every neighborhood U
of x -, there is a neighborhood V of x such that V - c U.
We make the convention that whenever we write "connected group",
"T O group", ..., we mean a topological group which as a topological
space is connected, T 0, ....
(4.2) Theorem. Let G be a topological group. For a cG, le[t and
right translations by a are homeomorphisms o/ G. Inversion is also a
homeomorphism o/G.
Proof. These facts are immediate consequences of (4.t).
(4.3) Theorem. Let G be a topological group, and let be an open
basis at e. Then the [amilies {xU} and {U x}, where x runs through all
elements o[ G and U runs through all elements o[ l, are open bases [or G.
1 Observe that we here assume no separation axiom for topological groups.
Throughout § § 4--8, the increased generality obtained by this convention seems
worth while. However, beginning with § 9, the term "topological group" will mean
"topological group that is a T o topological space", unless the contrary is specifically
stated.
Our sole violation of this excellent rule is in our use of "normal subgroup"
to mean a subgroup of a group invariant under inner automorphisms, and not a
normal topological space. "Normal" is simply too popular a term.
Proof. Let W be any nonvoid open subset of G, and a any element of
W. The mapping x-->a-lx carries W onto the open set a -1W, which
contains e. Since is an open basis at e, there is a U C such that
eUC a-lW. Hence aўaUG W. Thus W is a union of sets aU; i. e.,
{x U} is an open basis for G. The proof for the family {Ux} is similar.
We next prove a theorem about product sets which will be frequently
applied in the sequel.
(4.4) Theorem. Let G be a topological group and A and B subsets
o/G. I/A is opeў and B is arbitrary, then A B and B A are open. I/A
and B are compact, then A B is compact. I/A is closed and B is compact,
then A B and B A are closed. I[ A and B are closed, A B need not be
closed.
Proof. The first assertion is simple. We have A B= U {A b: b B} ;
if A is open, so is A b for all b B. Thus A B is the union of open sets and
hence is open; similarly for the set B A.
Suppose now that A and B are compact. Then AxB is a compact
subset of GxG TIHoNov's theorem and A B is the image of AxB under
the continuous mapping (x, y)-->xy. Since a continuous image of a
compact space is compact, A B is compact.
Next suppose that A is closed and B is compact. We will use nets to
prove that A B is closed. The argument is typical of those used in apply-
ing nets and so, for the sake of the reader unfamiliar with nets, we will
give a detailed argument; subsequent explanations will be briefer. Let D
be a directed set and let x, eD, be a net in AB that converges to
xoG. To show that AB is closed, it suffices to prove that xoAB
(3.t0.c). For each eD, we have x=yz, where yA and zў B. Since
B is compact, there are a z0 B and a subnet za,/ў E, of z such thatza
converges to z 0 (3.t0.c.). Clearly the net xa,E, converges to x 0. It
is easy to see that (xa, za) is a net in GxG that converges to (x 0, z0).
Thus the net ya= xaz converges to XoZ 1 since it is the composition of
the net (xa, ya) with the continuous function (x, y)-->xy -1 (3.t0.c).
Since A is closed and each ya belongs to A, we have XoZlA. Thus
Xo(XoZ )zoўAB. Similarly, BA is closed.
The last assertion of the theorem is easily proved by an example.
In the additive group R, consider the closed sets Z and eZ, where e is
any irrational number. The set Z+oZ consists of all numbers m+no,
where m and n are integers. This is obviously a dense and nonclosed
subset of R lit is actually a subgroupS.
We now return to the analysis of the topology of a topological group.
It can be completely described by properties of an open basis at e, as
follows.
Hewitt and Ross, Abstract harmonic analysis, vol. I 2
18 Chapter II. Elements of the theory of topoldgical groups §4. Basic definitions and facts 19
(4.5) Theorem. Let G be a topological group, and l an open basis at e.
Then"
(i)/or every U C l, there is a V ! such that V c U;
(ii)/or every U, there is a V such that V-IC U;
(iii)/or every U and every x U, there is a V such that x V c U;
(iv) /or every Ul and xG, there is a Vl such that x Vx-lC U.
Conversely, let G be a group, and let l be a/amily o/subsets o/G with
the/inite intersection property/or which (i), (ii), (iii), and (iv) hold. The
/amily o/sets {x U}, where U runs through and x runs through G, is an
open subbasis /or a topology on G. With this topology, G is a topological
group. [The /amily o/ sets {Ux} is a subbasis /or the same topology.
I/ the /amily also satis/ies
(v)/or U, V l, there is a W l such that W U Cl V,
then {x U} and { U x} are open bases/or the topology o/G.
Proof. If G is a topological group, then properties (i) and (ii) assert
that the mappings (x, y)---> x y and x-+x -1 are continuous at e. Property
(iii) asserts that U is open. Property (iv) follows from the fact that
x-+a x--->a xa -1 is a homeomorphism of G (4.2).
To prove the converse, let ' satisfy conditions (i), (ii), (iii), and (iv)
in the group G and have the finite intersection property. Then for U/,
there are V and W in /such that VC U and W-1C V. Since V f W@O,
we have e VW- V U. Thus all elements of /contain e. Let /
be the family of all sets U whereU ... . ForU ... ,
=I ' ' ' ' '
there exist .... , V such that , k=t .... , n. Then we have
CnC.= Thus property (i)holds for. Since ----
=x -, property (ii) holds for . Since x V =(x ) and x V x -
= (xx -) properties (iii) and (iv) hold for .
=1 '
By definition of a subbasis, the nonvoid sets (xU) where xG
=1 '
and , form an open basis for the topology of G. Let y be any
element of (x) and let V have the property that xycU
=1 '
(k=t,2,...,n) [see (iii)?. Then y V =Q(yV)(x). Thus the
sets y U, as U runs through , form an open basis at y, for each yG.
To prove that G is a topological group, let a and b be any elements
of G and U any set in . By (i) and (iv) for , there are sets V and W in
such that (b-Wb) VcU. Thus (a W) (b V) c a b U, so that (4.1.i) is
verified. [Note that V and W depend only upon U and b, not upon a.
Property (4.t.ii) follows similarly from (ii) and (iv). The equivalence of
the topologies generated by {x U} and {U x} follows also from (iv).
The last statement of the theorem is now obvious.
(4.6) Theorem. Every topological group G has an open basis at e
consisting o/ neighborhoods U such that U= U - [sets having this property
are called symmetric.
Proof. For an arbitrary neighborhood U of e, let V-- U Cl U -1. Then
plainly V= V -, V is a neighborhood of e, and V C U.
(4.7) Corollary. Let G be atopological group. For every neighborhood U
o/e, there is a neighborhood V o/e such that V-C U.
Proof. Let V be a symmetric neighborhood of e such that VC U.
Then if x V-, we have (xV)( V=O. Hence xv--v 2 for some vl, v2 V,
and thus x = v v C V V-l= V C U.
(4.8) Theorem. Let G be a T o topological group. Then G is regular
and hence Hausdor//.
Proof. By (4.7), G satisfies the axiom of regularity at e. By (4.2),
G accordingly satisfies the axiom of regularity at every point. It is
obvious that a regular space is Hausdorff.
(4.9) Theorem. Let G be a topological group, let U be any neighborhood
o/ e, and let F be any compact subset o/G. Then there is a neighborhood V o/
e such that x V x - C U/or all x F.
Proof. Let W be a symmetric neighborhood of e such that W 3 C U.
Since F C U Wx and F is compact, there exist xl x, F such that
xEF ' "'" '
F Wx. Let V- (? x Wx,. Clearly V is a neighborhood of e, and
k=l =I
X,t Vx 1 C W for k = 1 .... , n. If x F, then x Wx for some k, k = t .... , n.
Thus x---wx k for some w W, and hence
xVx-=wxVxXw-XCwWw-C W3C U.
Theorem (4.9) shows in particular that property (4.5.iv) can be
radically improved for compact groups" a single V can be found that
works for all x G. This fact will be useful in several places in the sequel.
x We will show in (8.4) that a T O group is actually completely regular.
2*
:20 Chapter II. Elements of the theory of topological groups §4. Basic definitions and facts 21
(4.10) Theorem. Let G be a topological group with identity e, F a
compact subset o/G, and U an open subset o[ G such that F C U. Then
there is a neighborhood V o/e such that (FV) U (VF) U. I/G is locally
compact, then V can be chosen so that ((FV)U (VF))- is compact.
Proof. For each xcF, there are a neighborhood W x of e such that
-x I/V, C U and a neighborhood V, of e such that V VV. Since F U xVx,
xEF
there exist xl,.., xcF such that FC xkV,,. Let V1-- V,,. Then
' k----I
Similarly, there is a neighborhood V2 of e such that V.Fc U. Letting
V-- V fl V2, we obtain ((F V) U (VF)) c U. I G is ocany compact, then V
can be chosen so that V- is compact. It follows that F(V-) is closed and
compact (4.4). Since FVF(V-) and F(V-) is closed, ve have
(FV)-cF(V-), and hence (FV)- is compact. Similarly, (VF)- is
compact, so that ((F V) U (VF))- is compact.
The fact that left and right translations are homeomorphisms of a
topological group G makes it possible to introduce a notion of "uniform
nearness" of two points in G, and also to define uniform continuity of
real and complex functions on G, as well as other mappings. Consider the
first notion. Given two points x and y in G, translate x [say on the left 1
by x -, and translate y by the same element : x-+ x -1 x = e, y --> x -1 y. If
x-ly is in a certain symmetric neighborhood U of e in G, then we may
say that x and y are "U-near in the sense of left translation". Similarly
if yx-l U, we may say that x and y are "U-near in the sense of right
translation". Both of these notions are u,i/orm concepts: they may be
applied to points x and y anywhere in the group G. If 99 is a complex
function on G, then we can say that is left [right I uniformly continuous
if for every e>O, there is a neighborhood U of e in G such that
I whenever x and y are U-near in the sense of left [right?
translation. Thus for left [right I uniform continuity, we must have
Iq(x)--q(xu)l<e [Iq(x)--q(ux)l <e for all xG and all uU.
The notions of lett and right uniform continuity of complex functions
on G are natural generalizations of the notion of uniform continuity of a
real or complex function defined, let us say, on R: for every e> 0, there
is a d>0 such that Iq(x)--q(x+t)l<e whenever Itl< d. Instead of a
single b> 0 that works for all xE R, we have a single neighborhood U of e
that works tor all x EG. An important difference between the case of a
general topological group and the case of the line R is that a general
topological group may be noncommutative, so that the notions of left
and right uniform continuity may be quite different.
It may not be inappropriate at this point to remark that topological
groups, which are generalizations of and abstractions from the familiar
elementary groups -- R, T, ® (n, K), and so on -- admit many struc-
tures, mappings, etc., that are direct generalizations of corresponding
notions that have been exhaustively investigated for the elementary
classical groups. The similarities as well as the differences between
general topological groups and the elementary models from which their
theory springs are a source of much of the fascination ot the theory of
topological groups and of analysis on topological groups. Furthermore,
a very fruitful source of theorems about topological groups is the obser-
vation of a fact for some particular group. Does a corresponding phenom-
enon then hold for a whole class of topological groups ? We shall be
concerned with this sort of question throughout the entire work.
We now give a precise definition of uniform structures on a topological
group 1.
(4.11) Definition. Let G be a topological group. For every
neighborhood U of e in G, let Lv be the set of all pairs (x, y) Gx G such
that x-yC U, and let Rv be the set of all pairs (x, y)GxG such that
y x-l U. The family of all sets Lv [Rv, as U runs through all neighbor-
hoods of e in G, is written as (G) [ў, (G), and is cMled the le/t right
uni/orm structure on G.
(4.12) Definition. Let G and H be topological groups, and let
be a mapping of G into H. Let and /" be open bases at the identities
of G and H, respectively. Suppose that for every V /', there is a U
such that (9(x), 9(y))cL v for all (x, y)L v. Then 9 is said to be a
uni/ormly continuous mapping for the pair of uniform structures
((G), (H)). lit is evident that the uniform continuity of 9 is
independent of the choices of the open bases ' and /'.1 Uniform con-
tinuity for the pairs of uniform structures (9 (G), 9 °, (H)), (9 °, (G), ,(H)),
and (,(G), 9,(H)) is defined similarly.
(4.13) Definition. Let G be a topological group, and let be the
identity mapping of G onto itself. If, is uniformly continuous for the
pairs ( (G), (G)) and (9 °, (G), (G)), then the uniform structures
9 (G) and (G) are said to be equivalent.
There is an extensive theory of uniform spaces, quite independent of the theorv
of topological groups. We shall not reproduce any part of this general theor3 ,
confining ourselves merely to the description of and facts about uniform structures
on groups, needed in the sequel. For the general theory of uniform spaces, see
KELLEY
22 Chapter II. Elements of the theory of topological groups §4. Basic definitions and facts 23
(4.14) We now list some facts about uniform continuity. The
proofs are extremely simple, where they are not trivial, and are left to
the reader.
(a) If the identity mapping of G onto itself is uniformly continuous for
one of the pairs (5(G), 9a,(G)) or (ga,(G), 5(G)), then it is uniformly
continuous for the other, and (G) and 9 a, (G) are equivalent.
(b) If G is an Abelian topological group, the structures 5(G) and
(G) are equivalent in fact, they are identical.
(c) Every left or right translation of a topological group G is a uni-
formly continuous mapping of G onto itself for the pairs ((G), (G))
and (9,(G), ,(G)).
(d) Let a, b be any elements of G. The mapping x-->a xb of G onto
itself is uniformly continuous for the pais (.(G), (G)) and
( (c), (c)).
(e) Inversion in G is uniformly continuous for the pairs ( (G),
and (5,(G), 5(G)).
(f) The structures (G) and 5,(G) are equivalent if and only if
inversion in G is uniformly continuous for the pair ( (G), (G)) or
for the pair (ga,(G), o',(G)).
(g) The structures 5z (G) and 5, (G) are equivalent if and only if for
every neighborhood U of e, there is a neighborhood V of e such that
xVx -1CU for all xG.
Our main theorem about uniform continuity is the following.
(4.15) Theorem. Let G and H be topological groups and let q be a
continuous mapping o/G into H such that [or every neighborhood W o[ the
identity in H, there is a compact subset A w o/G [or which q(A'w) W.
Then 9 is uni/ormly continuous [or each o/the [ollowing pairs o[ uni/orm
structures:
(5(G), 5(H))" (5(G), 5,(H)). (5,(G), 5(H)) ; (9°,(G),5,(H)).
Proof. We will prove the theorem for the pairs of uniform structures
(5(G), (H)) and (9(G), 5,(H))" the pairs of uniform structures
(5, (G), 5 (H)) and (9 a, (G), 9 a, (H)) present nothing new. All neigh-
borhoods of the identity in G and in H used in the present proof will be
taken as symmetric.
Let W be any neighborhood of the identity in H and let Y be another
neighborhood of the identity in H such that Y2cW. Let x be an
arbitrary point of G. By the continuity of % there is a neighborhood
of the identity in G such that q(xU)c (q(x) Y)N (Yq(x)). For each
x cG, let V X be a neighborhood of the identity in G such that V
Next let Ay be a compact subset of G such that 9(A)C Y. The
family of open sets {xV},.r is an open covering of Ay. Since Ay is
compact, there exist x 1 .... ,xAy such that UxV,kAy. Let
V= NV.k.
Now let x and y be elements of G such that x-y V. Suppose that
xAy. Then x xV** for some k, k-- ..... m, and so
y x V x V** V x V2, x U**.
It is obvious that xxU,. Hence we have 9(y)9(x)Y and
q(x)q(x) Y. That is, 9(x)-lq(y) Y and g(x)-9(x) Y--- Y. There-
fore q(x)-9(y)=q(x)-9(x)9(x)-9(y)Y2cW. If yAy, then the
same argument shows that 9(y)-9(x) W, and since W is symmetric,
we have 9 (x)-19 (Y) W. Suppose finally that neither x nor y is in A y.
Then 9(x) and 9(Y) lie in Y, so that 9(x)-q(y) YC W. This proves
that q is uniformly continuous for the pair of uniform structures
( (G), 5 (H)).
Again let x, y G be such that x - y V. If x A y, then x, y x U
and hence q(y) Yq(x) and q(x)c Yq(x). It follows that q(y)9(x) --
9(y)9(x)-(x)9(x)- Y w. I x, yA'y, then q(y)q(x)- YC W.
Hence q is uniformly continuous for the pair of uniform structures
(5(G), 9a,(H)). [
(4.16) Corollary. Every continuous mapping o[ a compact group G
into a topological group H is uni[ormly continuous /or all [our pairs
uni/orm structures listed in (4.15).
Proof. For every neighborhood W of e in H, let A w--G, and use
(4. 5). [2
(4.17) Corollary. Let G be a compact group. Then the structures
(G) and ,(G) are equivalent.
Proof. The identity mapping of G onto itself is continuous and
hence uniformly continuous for all pairs of uniform structures listed in
(4.5). Thus by (4.3), the structures 9(G) and 5,(G) are equivalent.
The corollary also follows immediately from (4.14.g) and (4.9).
Miscellaneous theorems and examples
We now list a number of examples of topological groups and give
other illustrations of the definition of a topological group, uniform
structures, etc. Some of the following assertions are proved only sketch-
ily, and some are not proved at all. Verification in such cases is left to
the reader.
24 Chapter II. Elements of the theory of topological groups §4. Basic definitions and facts 25
(4.18) Elementary examples. (a) Let G be an arbitrary group,
and let g) be the family of all subsets of G the discrete topology. With
this topology, G is a topological group. We shall often refer to such a G
as a discrete group.
(b) Let G be an arbitrary group, and let g) consist of 3 and G alone.
Then G is a topological group. We shall have little use for this topology
[except in the trivial case in which G= {e}.
(c) Let G be an arbitrary infinite group and let g) consist of G and the
subsets of G having finite complements; this is the weakest possible T 1
topology on the set G. Then G is not a topological group. [In fact,
G satisfies the T 1 separation axiom but not Hausdorff's separation axiom.
(d) The additive group R of all real numbers with its usual topology
is a locally compact, noncompact, Abelian group.
(e) The multiplicative group T with its topology as a subset of the
complex number field K is a compact Abelian group.
(f) Let G be an arbitrary subgroup of the group ® (n, K) with its
topology as a subspace of n2-dimensional complex Euclidean space.
Then G is a topological group. [To see this, it is necessary only to note
that the formula for multiplying two matrices and the formula for
inverting a matrix employ only continuous functions of the entries of the
matrices.
(g) Let G be any group and let {},x be any collection of topologies
for G, each of which makes G into a topological group. Let 9 be the
weakest topology stronger than all of the topologies . Then G is a
topological group under the topology d).
(h) Let H be any topological Abelian group and let G be an Abelian
group containing H as a subgroup. Let be an open basis at e in the
group H. Then the same sets can be taken as an open basis at e in G.
Properties (4. 5 .i) -- (4. 5 .v) obviously hold, and G becomes a topological
group with open sets as defined in (4.5). The subgroup H is open.
(i) Let E be a topological linear space as defined in (B.5). Then
regarded as an additive group, E is clearly a topological group.
(4.19) Ordered groups. (a) Let G be a group with more than
one element that is linearly ordered by a relation < [write x< y or
y>x 1. Suppose also that x<y and acG implyax<ay and xa<ya.
Fora, bcGsuchthata<b, leta,b[={xG:a<x<b}. Let the family
of all sets a, b [ be an open basis for a topology on G; note that G has
no greatest or least element, so that G=U {a, b:a< b}. Then G is a
normal T o group.
[Show that a<b implies b-l<a -1. This proves that inversion is
continuous. To prove multiplication continuous, it is sufficient to prove
that if a> e and there is some x for which e< x< a, then there is a b> e
such that b 2< a. If x<= a, set b= x. If x> a, set b--a x -1. Then again
ba, because otherwise b--a x-la x-l> a implies that x-la x-l> e and
hence a> x 2, which is a contradiction.
We now outline a proof that G is normal'it can easily be modified to
apply to any linearly ordered set with a topology having sets of the form
la, b, {xG'x<a}, and {xG'x> b} as an open basis. A subset C of G
is convex if x, yC and x<z<y always imply zcC. Let A and B be
disjoint closed subsets of G and let {Ca}aA be the family of all nonvoid
maximal convex subsets of (A U B)'. For each .A, choose xaCa. Let
xA. If IY, x cA for some y< x, let xU= y, x}. Otherwise, for some
o A, we have y B, y < x, and c C;.o imply y < c; that is, Cao lies beween
{y B" y< x} and x. In this case, let ,U= xa,, x. Define in an analogous
manner sets U, for xA and sets ,V and V for xB. Then U=
U{,UUU'xcA} and V=U{xVUV,'xB } are disjoint open sets con-
taining A and B, respectively.
(b) (DIEUDONN [2). Consider a well-ordered set S. Let G be the
set of all real-valued functions defined on S. For/, gG and e S, let
(/+g) (e)--/(e)+g(e). Then G is plainly an [additive Abelian group.
We write [> g if for some o S, we have / (o) > g (o) and [ () = g () for
all .<eo. Then G is an ordered group. With the topology described in
(a), G is a nondiscrete topological group. If S has a last element 2, then
there is an open basis at 0 [the function identically 0 consisting of open
intervals --g, g[ where g ()--0 if < and g ()- t, t > 0. If S has
no last element but has a countable cofinal subset, then there is a count-
able open basis at 0. If S has no cofinal countable subset, then there is
no countable open basis at 0. In this case, the intersection of every
countable family of open sets is open, every countable set is closed, and
all compact subsets of G are finite.
(4.20) Independence of the axioms (4.1). (a) Properties (4.t.i)
and (4.t.ii) are independent of each other, as this and the following
example show. Let R have ordinary addition as its group operation, and
let the sets [a, b[, for all a, b c R such that a< b, be a basis for open sets.
The mapping (x, y)-->x+ y is continuous, but the mapping x---> -- xis not.
(b) Let R have ordinary addition as its group operation, and let the
open sets be all sets of the torm AC', where A is open in the usual
topology of R and C is countable. Then the mapping x- -- x is continuous,
but the mapping (x, y)-->x+ y is not. It is easy to see that x---x is
continuous. Since there are neighborhoods V ot 0 such that U-C V for
no neighborhood U of 0, addition cannot be continuous.
(4.21) Topologies defined by subgroups. (a) Let G be any group
and let tF= {N} be a family of normal subgroups of G closed under the
formation of finite intersections. Let the family of all sets of the form x N,
9.6 Chapter II. Elements of the theory of topological groups § 4. Basic definitions and facts
as x runs through G and N runs through /fr, be an open basis for G.
Then G is a 0-dimensional topological group; in fact, every N C /fr is both
open and closed.
(b) In (a), let V consist of a single normal subgroup N.
G fails to satisfy the T O separation axiom. If A is any nonvoid finite
subset of G that is not the union of cosets of N, then A is compact [like
every finite space but is nonclosed. Clearly the set N is the closure of {e).
Every topology on a finite group making it into a topological group
is obtained from a single normal subgroup N in this way.
(c) Let G be a non-Abelian group, let N be a nonnormal subgroup of G,
and let 9 consist of O and all unions of left cosets xN of N(xcG). Then G
is not a topological group. [The mapping x-x -1 is continuous at aG
if and only if aNa-Ic N.
(d) (MARSHALL HALL t .) As another example of (a), consider any
group G and the family 3'= {H) of subgroups of G having finite index
G" H. First, let H and H. be in 'd. If x, y H and x H.--yH., then
y- x H N H. and x (H N H.) -- y (H1VI H.). Hence if x (H H) and
y (H VI H2) are distinct cosets of H VIH. in H, x H. and y H. are distinct
cosets of H2 in G. That is, [HI'HH._[G'H.. It is easy to see that
[G'HNH.--[G'H] • H'HIH., and so HH. has finite index in G.
Next, every H 3' contains a normal subgroup N in 3f'. For, consider
the coset space G]H--{xH, x.H ..... x,H), where [G'H]=r and x=e.
For xG, let P(x) be the permutation of G]H defined by P(x)(xH)
--x xH. It is clear that P is a homomorphism of G onto a transitive]
subgroup of the group of all permutations of G]H. It is also clear that
P(H) is exactly the subgroup of P(G) leaving H fixed. Let N be P- (),
where is the identity permutation of G]H. Then we have N cH,
[G" N ---- (GIN) n l, and N is normal. The family 3' thus satisfies
conditions (4. 5 .i) -- (4. .v) . Using the sets xH (xcG, H a') as a basis for
open sets, we see that G is a topological group; G is a Hausdorff group if
and onlyif for every x= e in G, there is a subgroup H 3' not containing x.
(e) (IWASAWA [t]; proof adapted from KURO [1], pp. 236--237. See
also voN NEUMANN and WIGNER [I].)
A free group F provides an example of the construction given in (d)
resulting in a Hausdorff topology for F. For, if gF and g&e, there is a
normal subgroup of F of finite index not containing g. Let X be a set
of free generators for F. Consider a reduced word g= x, .-. x, where
• i is --t or t and xiX ('---- t, ..., n). By (2.9), there is a positive integer
and a mapping x-P,, of X into .1 or +. such that --P**;o ...
is not the identity permutation. Now extend the mapping x-+P,, over all
of F in the obvious way, sending products into products. We obtain a
homomorphism of F into the finite group . x or ... The image of g
is o and therefore g is not in the kernel of the homomorphism just
defined. This kernel is plainly a normal subgroup of F with finite index.
(4.22) Other topologies for / and Z. (a) Consider once again
the additive group R, and let H be any Hamel basis for R over the ratio-
nal number field Q. Let J be a subset of H whose complement relative
to H is countable. Let A j be the set of all real numbers of the form
rlhl+ro.h+ ... +r,,hm, where the ri's are in Q and the hi's are in J.
Plainly A j is a subgroup of R. Let the family of all subgroups A j be
used as the family ,4:--{N} of (4.2.a). Then R is a topological group,
and the smallest cardinal number of an open basis at 0 is c.
(b) Consider the weakest topology for R that is stronger than the
topology defined in (a) and the usual topology. Then R is a topological
group. There is a countable family of open sets whose intersection is 0,
but the smallest cardinal number of an open basis at 0 is c. Furthermore,
if {xn}°°: is a sequence in R converging to 0, then x is equal to 0 for all
sufficiently large n.
(c) Consider all of the functions x-exp (i x) on R, where the param-
eter runs through all real numbers. Let R have the weakest topology
making all of these functions continuous. Then R is a topological group
with topology strictly weaker than the usual topology. [Consider the sets
{xcR:texp(iox)--t ]<e} for e>0 and R and use KRONECKER'S theo-
rem (26.t 9.b).]
(d) The additive group Z can be given a topology under which it is a
topological group and for which there is no countable open basis at 0.
Thus a countably infinite topological group need not have a countable
open basis at the identity. [Let H be a Hamel basis for R over Q con-
taining t. For , ..... nf-){t}' ande>0, let U(,
{nZ:]on--p[< for some pZ, k----t, 2 ..... m}. It is easy to verify
that the family of sets U(, ..... m;e) satisfies conditions (4.5.i)--
(4.5.v) and hence defines a topology onZ making Z into a topological group.
KROCKER'S theorem (26.t9.d) shows that every set U (1,
is infinite. The cardinal number of every open basis at 0 is c.]
(4.23) Infinite Abelian groups. (a) Every infinite Abelian group
G admits a nondiscrete Hausdorff topology under which it is a topological
group. Since T is divisible, it follows from (A.7) that there are enough
homomorphisms of G into T [i.e., characters of G] to distinguish an arbi-
trary x = e from e. The weakest topology for G under which all characters
are continuous is then a nondiscrete Hausdorff topology for G under
which G is a topological group. We will verify this in (26.t4).
(b) (KERT]SZ and SZELE [I].) Every infinite Abelian group G admits
a nondiscrete Hausdorff topology under which there is a countable
open basis at e. [First, if G has an element a of infinite order, let
28 Chapter II. Elements of the theory of topological groups § 4. Basic definitions and facts 29
A k be the subgroup generated by a2k(k--O, t, 2 .... ). Then we have
AIDA2D... DAk ... and FI A--{e}. Hence the A's can be taken
'
as an open basis at e, in accordance with (4.2t.a). Now suppose that every
element of G has finite order. Then G is the weak direct product of
p-primary groups G (A.3). If there are an infinite number of these @,
say @,, @, .... @,, .... different from {e}, let B be the set of
x= (x, x .... , x,, .... )G for which Xl, ..., x are the identity elements
of @, .... @, respectively (k=t, 2 .... ). Then {B}= is a decreasing
sequence of subgroups of G whose intersection is {e}, and so the B's can
be used as an open basis at e, as in (4.2t.a). If there are only a finite
number of p-primary subgroups @ different from {e}, at least one of
them, say Gў, is infinite. There is no loss of generality in supposing that
Gў is countably infinite. Suppose that Gq contains a divisible subgroup
different from {e}. Then Gq contains a subgroup A isomorphic to Z(q)
(A.t4). The subgroup A can be given the topology of Z(q) as a sub-
space of T. Let an open basis at e in G be neighborhoods of e in A obtained
in this way. Then G is a topological group, with a countable basis at e.
If Gq has no divisible subgroup different from {e}, form the transfinite se-
quence of subgroups S 1 --{xq" x Gq}, Sz = {xq" x Gq}, . . . , S n -- {x" x Gq},
..., So-- S, So+i={x q" x So}, ... . The transfinite sequence {S}
is obviously decreasing, and must be ultimately constant, S=S+
for some countable ordinal e. Since Gq has no divisible subgroup except
for {e}, we have S={e}. If e is finite, then every element of Gq has
order q,and so Gq is the weak direct product of an infinite number of
finite cyclic groups (A.25). Thus {e} is the intersection of a decreasing
sequence of proper subgroups. If SD S,I for all positive integers n,
write the set {Sa" Sa@ {e}} in any way as a sequence J1, J2, Ja,..., J,...,
and write H-- J ... J, (n =t, 2 .... ). Once again, {e} is the intersection
of a decreasing sequence of proper subgroups. As above, G then admits
a nondiscrete topology with a countable open basis at e under which it is a
topological group.
(4.24) Examples on uniform structures. Let G be any T o group.
If there are sequences {x}n=l and {Y}n=l in G such that lim x y,--e and
,,lirnooyx--z=e, then the left and right uniform structures of G are
inequivalent. [Let U and W be disjoint neighborhoods of e and z,
respectively. Let V be any neighborhood of e. If n is chosen so that
x, y, V and y, x, W, then (xl) -1Yn 1'7 and y, (x)-l I/V; i.e., y, (x-Ј)-ўU.
Hence the uniform structures are inequivalent.
(a) The groups L (n, F) and ® (ў, F), where ў>_ 2 and F is any
subfield of K, are topological groups as subspaces of K "'. In none of
them are right and left uniform structures equivalent. To see this, let E
be the identity matrix in 63 (n, F)" for e,/ c:F and e4= 0, let A (e,/) be
the matrix (a.).,;x in ®(n, F)such that al=e , a,,----,e aii=l for
I < n, ax,--t, and a.=0 for all other pairs ', k. It is easy to see that
2 ..... ,et
A(,
and Ym=A(m,-). Then
lim XY-- lim A t,+ --A (t, 0)--E,
and
lim grnXrn-- lim A(t, t +t)=A(t, 2).
Since A(t, 2)4=E, the left and right uniform structures of ®(n,F)
and ® (n, F) are inequivalent.
(b) Consider the subgroup G of 63 (2,F) consisting of all matrices 0
where x, yF and x4=0. Then G has inequivalent uniform structures.
and
We have lim
m--,oo
lim t -- hrnoo -m- + -m- = .
(4.25) Linear groups. (a) The groups 1I (n), © (n), (n), and
O(n) are compact. [Since the mappings A, AN, AA -,
AA*, and A det A are continuous, the groups (n), (n), (n),
and (n) are closed subsets of K"'. Since (n), (n), and (n)
are closed subsets of H(n), we need only show that (n) is compact.
If A = (ai)i= is unitary, then =E, the identity matrix, and hence
2 %i/=t for i=t, ..., n. Thus a for all i and k, so that
m=l
is closed and bounded. That is, (n) is compact.
(b) The groups (n, K), (n, K), U (n), U (n), (n, R),
(n, R), (n), and (n) are all locally Euclidean" they possess
neighborhoods of the identity E that are hmeomorphic to open balls
in real Euclidean spaces of dimensions 2 n , 2 n - 2, n , --t, , --t,
2 and respectively. [To prove this, we introduce the
exponential of a matrix. For A(n, K), let exp(A) be the matrix
1
A', where AO=E and A'=AA ... A (l times) [If e=max
l0 "
30 Chapter II. Elements of the theory of topological groups §4. Basic definitions and facts 3t
]aa. l ..... l a,l), then it is easy to see that no entry of the matrix A' has
t AZ
absolute value exceeding (n) z. Hence each entry of the matrix , .,
l=0
is the mth term of a convergent sequence, so that exp (A) is well defined;
o t A z is uniform on compact subsets of (n,/).
the convergence of 7[.,
It is easy to show the following.
() If B is anymatrix in % (n,K), thenexp(B-1AB)= B-l- exp (A) • B.
(2) If x .... , x are the eigenvalues of A, then exp (xl) ..... exp (x)
are the eigenvalues of exp (A). Prove by induction on n.
(3) The determinant of exp (A) is exp (tr A).
(4) For every A, exp (A) ® (n, K).
() I A B = BA, then exp(A + B)--exp(A) • exp (B).
(6) exp (!4) ----'(exp (A)).
(7) exp (A) -- exp (A).
We can now show that there is a neighborhood of 0 in X(n, K)
such that exp is a homeomorphism of onto a neighborhood of E in
®(n, I). The entries of the matrix exp(A) are obviously entire
functions of the entries (a11, al., .. a,) of A say exp(A)--(
Since exp (A) =E + A + + ..., it is plain that F.k (aal, aa. .... , a)--
.+ a. plus terms of higher order in aa .... , a. Now regard exp as a
mapping of K "' into K'. The Jacobian of exp at 0 is equal to , and
hence, by the inverse function theorem (APoSTOL , p. 44), exp is one-
to-one in some neighborhood of 0. As exp is continuous, if we take
compact, it follows that exp is a homeomorphism on , which carries
onto a neighborhood of E in ® (n, K).
We now prove in detail that lt(n) is locally Euclidean with dimen-
sion n . Let 93 be a neighborhood of 0 in (n, K) such that" exp is a
homeomorphism on " A implies A ,/ , and --A . ESuch a
3 exists by the continuity of the mappings A- A ,A-+., and A-+-- A. 1
Let consist of all the skew-Hermitian matrices in , that is, matrices
such that !4----/. For A--(ai)i",=l to be skew-Hermitian, it is
necessary and sufficient that aii be pure imaginary and that aki------a-.
(l-<_-<kn). Thus the set of all skew-Hermitian matrices is a real
Euclidean space of dimension n", and 3 can be taken as an open ball in
R ' with center at 0. For A in , we have (exp(A))*='(exp(A))--
exp (A) ---- exp (-- A) -- (exp (A))-l. Thus exp (A) is unitary. On the other
hand, if exp (A) is unitary and lies in exp (3), then A is skew-Hermitian.
The mapping exp is therefore a homeomorphism of onto a neighbor-
hood of E in It (n).
Analogous mappings are obtained for the other linear groups men-
tioned above. We indicate them as follows.
For ® (n, K), take A X (n, K) f-I "
for ® (n, K), take A , where tr A -- 0"
for ®lI(n, K/, take A , where tr A--0 and ai---- --."
for ® (n, R), take A X (n, R) f-I "
for ® (n, R), take A X (n, R) f-I , where tr A -- 0"
for © (n), take A (n, R) f'l , where a. ---- -- ak ."
for ®©(n), take A(n,R)fl, where ai----ai.
For ® (n, K) and ® 1I (n, K), the neighborhood 3 of 0 in (n, K)
must also have the property that ]tr A I < 2 for all A " this is plainly
possible to secure.]
(4.26) Every nondiscrete, locally countably compact, T o group G
has cardinal number greater than or equal to c. iIn fact, every nonvoid
open subset U of G has cardinal number greater than or equal to c.
With no loss of generality, we may suppose that U- is countably compact.
Since G is nondiscrete and T O hence Ha.sdorff, U is infinite. Let x o
and x be distinct points of U and let U 0 and U be neighborhoods of x o
and x, respectively, such that U-U Uf- U and U- f'l Uf---. Let Xoo
and x0 be distinct points of U0, and U00 and U0 be neighborhoods of x00
and x0 , respectively, such that UU U U 0 and Uof'l U0--" define
x0, x, U10 , and U analogously. Continuing this process by finite
induction, we construct nonvoid open sets Ui.... for all finite sequences
', '. .... , 'm of O'S and 's, with the following properties"
all sets U.,...i are contained in U.
Now let Am:U{V/..i.-- 0 or for k: .... ,m) for each positive
integer m. Let D-- F/ A The countable compactness of U- implies
m=l m •
that Dў. In fact, for each sequence {'m)°°__l of 0's and 's, the set
mQ1U...iD is nonvoid and the family of all such sets is pairwise
disjoint.]
Notes
The notion of topological group in the form used today was appar-
ently first introduced by F. LEJA [t] in t927" in t925, O. SCHEIE []
had given axioms for and studied groups that are Fr6chet L-spaces in
which the group operations are continuous. Another axiomatic treat-
ment, inequivalent to LEJA'S, was published by R. BAER in t929 [].
32 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 33
Locally Euclidean groups were discussed per se by 1. CARTAN in 1930 [1 ].
By the early t930's topological groups were being treated with facility
and precision by many writers. See tor example D. van DANTZlG
and [3], A. HAAR r3], J. vo NEUMANN [3]"
Some, although by no means all, of the early writers on topological
groups restricted themselves to topological groups having countable open
bases: this is the case for example in PONTRYaGIN 4 and _6, FRFUDFN-
THAL It], and HAAR [3]- The nugatory character of this restriction was
stated emphatically by WEIL in [3] and [4]. WEIL'S monograph [4] has
thoroughly expunged countability restrictions from the theory of topo-
logical groups. It is interesting to note that the second edition of PONTRY-
AClN'S monograph [7] removes the countability restriction. Throughout
the present book, we scrupulously avoid unneeded countability restric-
tions.
Many of the theorems of §§ 4--7 are found in one or more of the
monographs BOURBAKI Ill, LOOMIS [2], MONTGOMERY and ZIPPIN
PONTRYAGIN [71, WEIL [4 I. It is pointless to trace every avatar of every
theorem, and we will not attempt to do so. The following primary sources
of results in § 4 may be of interest. Theorems (4.4), (4.7), and
(4.8) are largely due to A. MaRIOV . Special cases of (4.6), (4.7), and
(4.8) appear in D. van DANTZIG 31" The earliest appearance of the
axioms (4.5.i)-- (4.5.v) known to us is in A. WEIL E3, P" " See also
VAN DANTZIG 31, TG. 22. Theorem (4.9) must have been long known;
the first reference we have found is a special case proved in MONT-
GOMERY and ZIPPIN , p. 55. Theorem (4. 0) is due to VAN IAMPEN
Uniform spaces are the creation of A. WEIL E3], who also described the
uniform structures (G) and ,(G). Theorems (4. 5) -- (4. 7) are essen-
tially due to WEIL [lOC. cit.. See also VAN KAMPEN ll, P" 4.
§ 5. Subgroups and quotient groups
In this section, we take up some methods of forming new topological
groups from a given topological group. We begin with a study of sub-
groups.
(5.1) Theorem. Let G be a topological group and H a subgroup o/G.
With its relative topology as a sub@ace o/G, H is a topological group.
Proof. The mapping (x, y)--->xy of HxH onto H and the mapping
x-- x -1 of H onto H are continuous, since they are restrictions of the cor-
responding mappings of G xG and G. [
(5.2) Theorem. Let A and B be subsets o] a topological group G.
Then we have:
(i) (A-)(B-) (A B)-;
(ii) (A-) -1-- (A-l)-;
(iii) xA-y=(xA y)- /or all x, ycG.
G is a T O topological group, then we also have:
(iv) i/ a b = b a/or all a A and b B, then a b = ba /or all a A- and
bB--.
Proof. To prove (i), suppose that xA-, y cB-, and that U is any
neighborhood of e. Then there is a neighborhood V of e such that
(x V)(y V) xy U. Since xA- and yB-, there are points a A and
bB such that axV and byV. Hence ab(AB) fl(xyU), so that
xyc(AB)-.
Assertion (ii) follows from the fact that the mapping z--z - is a
homeomorphism of G onto G, and (iii) follows from the fact that z--> x z y
is a homeomorphism of G onto G.
We now prove (iv). Suppose that ab--ba for all aA, bB. The
mapping (a, b)--aba-b - of GxG into G is continuous. Since {e} is
closed by (4.8), we infer that g={(a, b)GxG:aba-b-=e} is closed.
Clearly Ax B C H and (Ax B)- = A-x B-. Thus, we have A-x B- c H.
That is, a b =b a for all a
(5.3) Corollary. I/ H is a subsemigroup, subgroup, or normal
subgroup o! a topological group G, then H- is also a subsemigroup, sub-
group, or normal subgroup, respectively, o! G. If G is a T o topological
group and H is an A belian subsemigroup or subgroup o/ G, then H- is
also an A belian subsemigroup or subgroup, respectively, o] G.
Proof. If H is a subsemigroup of G, then H"C H and by (5.2.i), we
have (H-)" (H.)- H-" that is, H- is a subsemigroup of G. If H is a
subgroup of G, then also H- H. Hence (H-)-= (H-)- H-, so that
H- is a subgroup of G. The remainder of the corollary is proved in like
manner.
(5.4) Theorem. Let G be a topological group. The closure {e}-
o! {e} in G is a closed normal subgroup o/G and is the smallest closed sub-
group o! G. The closure o/a point a G is the coset a {e}-= {e} a.
Proof. The set {e}- is a closed normal subgroup of G by (5.3). It is
clearly the smallest closed subgroup of G. Left translation is a homeo-
morphism, and therefore the closure of {a} is a{e}- for eversz aG.
Moreover, since {e}-is a normal subgroup, we have a {e}-= {}-a. [
(5.5) Theorem. A subgroup H o/ a topological group G is open i/
and only i! its interior is nonvoid. Every open subgroup H o] G is closed.
Hewitt arid Ross, Abstract harmonic analysis, vol. I 3
34 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 35
Proof. Suppose that H has an interior point x. Then there is a
neighborhood U of e in G such that x Uc H. For every y H, we then
have y U-- y x -1 x U yx-IH-- H. Hence H is open. If H is open, then
by definition every point of H is an interior point.
If H is an open subgroup of G, then H'= U {xH:xў H}. Each set xH
is open, and so H' is open; that is, H is closed. [
(5.6) Theorem. Let be a/amily o/ neighborhoods o/ e in a topo-
logical group G such that:
(i) [or each U , there is a V such that VC U;
(ii) [or each U ,Y, there is a V ,Y such that V - U;
(iii) [or each U, V ,, there is a IV ,Y such that W U r V.
Let H--r (u: u a'}. Then H is a closed subgroup o[ G. I[ in addition,
(iv) [or every U and xG, there is a V such that x V x-c U,
then H is a normal subgroup o[ G.
Proof. Suppose that x, y H and that U a'. Let V a' be such that
V" c U. Then x, y V, so that x y V c U. Hence x y H. Similarly
x-H if xH. To see that H is closed, let a be any element of G that is
not in H. Then aў U for some Ua'. Let V, V2, Va' be such that
V c U, Vi-I V, and V v r v2 ; then V V- U. Hence if (av) r V = ,
we have a VV - c U, a contradiction. Hence we have a a V H', and H'
is accordingly open. That is, H is closed.
Suppose that (iv) holds and let a H and x G. For U I, let V
be such that x V x - U. Plainly x a x - x V x - U, and since U a' is
arbitrary, we have xa x- H. Hence H is a normal subgroup.
As a sort of complement to (5.6), we show next how to generate open
and closed subgroups from neighborhoods of e.
(5.7) Theorem. Let U be any symmetric neighborhood o[ e in a
topological group G. Then the set L-- U U n is an open and closed subgroup
ol G.
Proof. If x U k and y U , then x y U + and x-I (U-)= U.
Hence L is a subgroup of G. By Theorem (5.5), L is open and closed. I
(5.8) Theorem. A subgroup H o[ a topological group G is discrete i[
and only i[ it has an isolated point.
Proof. Suppose that xH and that x is isolated in the relative
topology of H c G. That is, there is a neighborhood U of e in G such that
(x U) r H= (x}. Then for arbitrary yH, we have (yU)rH=(yU)r
(yx-H)--yx-l((xU)rH)={y}. Thus every point of H is isolated, so
that H is indeed discrete. If H is discrete, then by definition all of its
points are isolated.
(5.9) Theorem. Let G be a topological group and H a .bgroup o/ G
such that U-r H is closed in G [or some neighborhood U o/ e in G. Then H
is closed.
Proof. Let U be a neighborhood of e in G such that U-f)H is closed
in G. Let V be a symmetric neighborhood of e in G such that Vc U.
Now let x be any point in H-, and let x, D, be a net in H such that x
converges to x. Since x-H - (5.3), there is an element y in Vx-rH.
There is an 0D such that xxV for all 0. Thus, if c 0, we
have yx(Vx-)(xV)--V" U and hence yxU-rH. Since the net
y x, c a0, converges to y x and U- f-) H is closed, we have y x 6 U- f-) H.
Hence x--y-lyxH and therefore H-cH, i.e., H is closed.
(5.10) Theorem. Every discrete subgroup H o/ a T O group G is
closed.
Proof. Let U be a neighborhood of e in G such that U rH={e}.
By (4.7), there is a neighborhood V of e such that V-c U. Then V-
{e), which is closed since G is Hausdorff (4.8). Theorem (5.9) now
implies that H is closed. [
The hypothesis in (5.0) that G be T O is essential, as Example (5.37.g)
shows. The following generalization of (5.0) also obtains.
(5.11) Theorem. Let G be a T O group and H a subgroup o/ G that is
locally compact in its relative topology. Then H is closed.
Proof. Let U be a neighborhood of e in G such that U-f-)H is com-
pact as a subset of H, and therefore as a subset of G. Since G is Haus-
dorff, U--f-) H is closed. Thus by (5.9), H is closed.
(5.12) Definition. A topological group G is said to be compactly
generated if it contains a compact subset F for which the subgroup
generated by F is G" that is, G--{e} U nU= (FUF-) ".
(5.13) Theorem. Let G be a locally compact topological group. Then
the /ollowing assertions are equivalent:
(i) G is compactly generated;
(ii) there is an open subset U o[ G such that U-is compact and U
generates G;
(iii) there is a neighborhood U o[ e in G such that U- is compact and U
generates G.
Proof. It is obvious that (iii) implies (ii) and that (ii) implies (i).
Suppose then that G is compactly generated" we will prove that (iii)
holds. Let F be a compact subset of G generating G. Then F U{e) is
3*
36 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 37
compact, and by (4.t0) there is an open set U containing FU{e} such
that U- is compact. Clearly U generates G.
(5.14) Theorem. Let G be a locally compact group with identity e
and let F be any compact subset o! G. Then there is an open and closed
compactly generated subgroup o! G containing F.
Proof. Since FU {e} is compact, (4.10)implies that there is an open set
U containing F U {e} such that U- is compact. Therefore U (U U U-l)" --
,=UI(U-U(U-)-I)"is a compactly generated set, and it is an open and
closed subgroup of G by (5.7).
Just as in the purely algebraic theory of groups, subgroups of topo-
logical groups play an essential r61e in the formation of homomorphic
images. Topological properties of the subgroups in question will also
play an important part in our constructions. We now take up the con-
sideration of continuous homomorphisms of topological groups and how
they are obtained from normal subgroups. We begin by discussing a
somewhat more general construction.
(5.15) Definition. Let G be a topological group, and let H be a
subgroup of G. Let 9 be the natural mapping x--->xH of G onto G/H.
Define a topology (9(G/H) for G/H by the following rule: a subset
{xH:xX} of G/H is in (ў(G/H) if and only if q-l({xH:xX}) is open
inG. Inotherwords, {x H : x _X} is open if and only if U (x H : x _X} = X H
is open in G.
Since {xH:xX}={xH:xXH}, it follows that every open set
in G/H has the form {uH: u U} where U is an open subset of G. Thus
(ў(G/H) consists of all sets of the form {uH:u U} where U is open. It is
also easy to see that sets of the form {uH: u U}, where U is open in G
and a U, form an open basisat each point a H of G/H.
(5.16) Theorem. The /amily (ў(G/H) is a topology/or G/H. Under
this topology, the natural mapping 9 o/ G onto G/H is continuous, and
(G/H) is the strongest topology on G/H under which the mapping 9 is
continuous.
Proof. Let {uH:uU}A be a family of open subsets of G/H
where each U is open in G. Then U {uH'uU}={uH'u U U} is
open in G/H since U U is open in G. Similarly, the intersection of any
finite number of sets in (G/H) is again in (ў(G/H). Clearly and G/H
are in (G/H) and therefore (ў(G/H) is a topology on G/H. The remaining
statements of the theorem are easy to verify: we omit the argument.
The topological space G/H is called the quotient space of G by H.
We now list a number of useful facts about G/H.
(5.17) Theorem. The natural mapping 9 o/ G onto G/H is an open
mapping.
Proof. If U is an open subset of G, then UH is open (4.4) and hence
q; ( U) = {, It : u U} is open in G/H.
It is easy to see that the natural mapping q)of G onto G/H need not be a
closed mapping: (A) may be nonclosed in G/H for closed subsets A of G.
A simple example is provided by the additive group R and R/Z. Every
coset x+Z in R contains the number x-- Ix I I[xl is the integral part of x l
and no other number in [0, [. Thus [0,[ can be taken as the space R/Z.
It is not hard to see that the topology imposed on [0, [ as a model of the
space R/Z is the following. An open basis consists of all sets lc, fl[
where 0<</5<t andof all sets [0, c[U ItS, 1[ where 0<<rS<t. Now
let A ={, -,..., n+2 -" .... }. The set A is closed in R, but (A) c [0, t[
1
is the set {, ј, ..., ,...}, which is nonclosed.
In view of this example, the following theorem is of interest.
(5.18) Theorem. I! G is a topological group and H is a compact
subgroup o/G, then the natural mapping 9 o/ G onto G/H is a closed map-
ping.
Proof. Suppose that A is a closed subset of G; we will show that
(A)' is open in G/H. Consider x in G such that xH does not belong to
q) (A); then x does not belong to A H. Since A H is closed (4.4), there is a
neighborhood U of x such that UVI(AH)=(3. Clearly {uH:uўU} is an
open set in G/H containing xH, and {uH:uўU} is disjoint from (A).
Therefore q(A)' is open.
(5.19) Theorem. Let G be a topological group, H a subgroup o/ G,
and U, V neighborhoods o/e in G such that V -1V c U. Let q; be the natral
mapping o/ G onto G/H. Then (9(V))-C 9(U).
Proof. Let xHG/I-I be in (9(V)) . Then {vxH:vV} is a neighbor-
hood of xH and hence contains some point of 9(V). That is, there are
points v 1, v2 Vsuch that vlxH=v2H , i.e., xH=vl-lv,2H{wH:w V -1 V}
{:uU}=(u).
(5.20) Theorem. Let G be a topological group and H a subgroup o/ G.
For aG, let ? be the mapping defined in §2:
v(xH)=(ax)H /or xHG/H.
Then v is a homeomorphism o/G/H. Thus G/H is a homogeneous space.
Proof. Since p is a one-to-one mapping of G/H onto itself and
()-1 ,_v, we need only show that v is an open mapping. Let {uH: u U}
be an open subset of G/H, where U is an open subset of G. Then
38 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 39
ap({uH:uU})={auH:uU}={vH:vaU} is open, since aU is open
inG. [3
(5.21) Theorem. Let G be a topological group and H a subgroup o/G.
Then G/H is a discrete space i/ and only i/H is open in G. I/ H is closed,
then G/H is a regular and hence Hausdor[/ space. I! G/H is a T o space,
then H is closed and G/H is a regular space.
Proof. If H is open in G, then aH is open in G for all aG, and so
q)-I({aH})=aH is open in G for every point aHG/H. That is, every
point of G/H is an open set, and hence every subset of G/H is open.
Conversely, if G/H is discrete, then the set {H} is open in G/H, and thus
0-1({H}) =H is open in G.
Now suppose that H is closed in G. Then a H is closed in G for all
aG, and (aH)'=U{xH:xH4=aH} is open in G. This implies that the
complement of each set {all} in G/H is open in G/H. Hence each point
in G/H is a closed set. This property is equivalent to the T separation
axiom. This and (5.19) imply that G/H is regular.
Finally, suppose that G/H is a T o space. Then by (5.19), it is also a T
space, so that the set {xH:xH@H} is open in G/H. This implies that
H=(U{xH:xH@H})' is closed in G.
(5.22) Theorem. Let G be a compact [locally compact group, and
let H be a subgroup o/ G. Then the space G/H is compact [locally compact].
Proof. If G is compact, then G/H, being a continuous image of G
(5.16), is also compact. Suppose that G is locally compact, and that U
is a neighborhood of e in G such that U- is compact. Then by (3.3)
and (5.19), 9(U) has compact closure in G/H. Since G/H is homo-
geneous (5.20), G/H is locally compact. [3
(5.23) Lemma. Let G be a topological group, H a subgroup o/ G,
and 9 the natural mapping o/G onto G/H. Suppose that U is a symmetric
neighborhood o/e such that ((U-)3)-f3H is compact and that {xH: xX} is
a closed compact subset o/G/H such that {xn:xX}c {uH:uU}. Then
U-f3 (XH) is closed and compact iў G.
Proof. Evidently XH=9-1({xH:xX}) is closed so that U-f) (XH)
is closed. Assume that U-f)(XH) is not compact. Then there is a net
x, D, in U-N (XH) such that no subnet converges to a point of
U-f3 (XH). Since U-f3 (XH) is closed, no subnet of x converges to any
point of G. Since each xH is in {xH:xX}, there is a subnet x, flE,
of x and an xoX such that xaH converges to xoH in the topology
P(G/H). By hypothesis, we have xoH=uo H for some uoU. Write
A=((U-)a) -. Evidently xa has no subnet converging to a point of
uo(Af3H); that is, no point of uo(Af3H ) is a cluster point of xa (3.10.c).
Hence for each XUo(Af3H), there is a neighborhood U, of e and a flE
such that
fl>___fl, implies xўU.x. (1)
For each xu o (AVIH), let V. be a neighborhood of e such that V U..
By hypothesis, A Cl H is compact and therefore u o (A Cl H) is also compact.
Hence there are points xl .... , x. in u o (A f3 H) such that
U V,, x D u 0 (A f3 H). (2)
Let V be any symmetric neighborhood of e such that V(f=
Clearly {vuoH:v V} is a neighborhood of uoH=xoH in G/H, and thus
there is a floE such that flflo implies xaH{vuoH.v V}. Let E be
such that ,/50 and ,/5, for k=l,..., n. Then x,H=vuoH for some
v V; that is, x./=vuoh for some hH. Now
h--uv-lx, UVU- (U--)3(Z A
and, consequently, uohuo(AH ). Applying (2), wc have uohV,,x for
some k (k=l, ..., n), and hence
x=vuo vv x c v v x c u x.
Now () shows that this is impossible since Vfl,. It follows that
U- (XH) is compact.
(5.24) Note. (a) In (5.23), suppose that H is compact. Then the
hypotheses of (5.23) hold if we take U=G. In this case, Lamina (5.23)
remains valid if the hypothesis that {xH'xX} be closed in G/H is
dropped.. Thus we find" if H is a compact subgroup of G and if {xH" xX}
AS a compact subset of G/H, then XH is compact in G. The details of the
argument are very like those of (5.23), and are omitted.
(b) If G is locally compact, n is a subgroup of G, and {xH" x E} is a
closed compact subset of G/H, then there is a closed compact subset F
of G such that {xH" xF}= {xH" xE}. In fact, if U is any neighborhood
of e in G such that U- is compact and 9 denotes the natural mapping of G
onto G/H, then { (s U)'sG} is an open covering of GIn. A finite sub-
family, say 9(sU),9(sU) .... ,9(sU), covers {xH'xE}. Then we
can take F(s U-UsU ... UsU-)-({xH'xE}).
(5.25) Theorem. Let G be a topological group and H a subgroup o/G.
I/ H and G/H are compact, then G itsel[ is compact. I/H and G/H are
locally compact, then G is also locally compact.
Proof. Suppose that H and G/H are compact. Then letting U = X =G
and applying Lemma (5.23), we see that G=G-O(GH) is compact .
ms can also be proved directly by using (5.18) and the fact that if a family
of closed subsets of G has the finite intersection property, then so does {9 (F)" F }.
40 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 4t
Suppose that H and G/H are locally compact. Let V be a neighbor-
hood of e in G such that V-f3H is compact. Let U be a symmetric
neighborhood of e such that (U-)3c V [use (4.5.i), (4.6), and 4.7).
Then obviously ((U-)3)-f "1H is compact. Let W be a neighborhood of e
such that {xH:x W}- is compact in G/H. By (5.t9), there is a neighbor-
hood W 0 of e in G for which 9 (W0)- c 9(U), that is,
0}-c
Choose a subset X of G for which (x H:x _ X} = (x H:x W f3 W0}-. Then
(xH:xX} is clearly compact in G/H and
By (5.23), U- f') (XH) is closed and compact in G. Moreover,
ecUf3((Wf3Wo)H)cU-N(XH ). Therefore e has a neighborhood
U f3 ((W f-) W0)H) with compact closure ; i.e., G is locally compact. 0
We now specialize the construction of and topology in G/H to the case
in which H is a normal subgroup, so that G/H is a group.
(5.26) Theorem. Let G be a topological group and let H be a normal
subgroup o] G. Let the quotient group G/H be given a topology as in ( . 5).
Then G/H is a topological group. Also the natural mapping q o[ G onto G/H
is an open, continuous homomorphism o] G onto G/H. The group G/H is
discrete i] and only i[ H is open and is a T o and hence regular group i!
and only i[ H is closed.
Proof. Let us show that G/H is a topological group by proving that
the family of all neighborhoods of H the identity! in G/H satisfies
conditions (4. 5 .i) -- (4. 5.v). Let {uH: u U} be an arbitrary neighborhood
of H in G/H, where U is a neighborhood of e. By (4.5.i), there is a neigh-
borhood V of e in G such that VC U. Clearly (vH:v V} is open in G/H
and, moreover, {vH:v V}= (yH: y_ V)c{uH:u U}. Therefore (4.5.i)
holds for the group G/H and the topology (9(G/H). The verification of
(4.5.ii) for G/H is similar.
We verify (4.5.iii) for G/H. Again let {un:u U} be an arbitrary
neighborhood of H, where U is a neighborhood of e in G, and let uoH be
any element of {uH:ucU}. If V is a neighborhood of e in G such that
u 0VC U (4.5.iii), then (vH: v V} is a neighborhood of H in G/H, and
uoH. {vH:vV}=(uovH:vV}c(uH:u U}. This is just (4.5.iii) for
G/H and the topology (9(G/H).
To verify (4.5.iv) for G/H and the topology (9 (G/H), we need only note
that xH. {vH:vV}. (xH)-l--{xvx-ln:vV} and apply (4.5.iv) for
the group G and its topology. Property (4.5.v) for G/H is virtually
obvious: we omit the verification.
The other statements of the present theorem follow at once from
(.7), (.6), (.21), and (.2t) combined with (4.8), in that order. [3
Theorem (5.26) admits a converse to the effect that all open, con-
tinuous homomorphic images of a topological group can be realized as
quotient groups G/H with the topology (9(G/H).
(5.27) Theorem. Let G and G be topological groups with identities e
and , respectively, and let ] be an open, continuous homomorphism o G
onto G. Then the kernel [- () o] ] is a normal subgroup H o[ G, and the
sets [- (:), G, are exactly the distinct cosets o[ H in G. The mapping
:__>]-1(;;) = b(;;) is a homeomorphism and an isomorphism o/ onto the
group G/H with the topology (9 (G/H).
Proof. In view of (2.2), all that we need to do here is to show that
the mapping q9 is a homeomorphism of onto G/H. Let be an open
subset of G. The set b(U) is the family of cosets {-1(). }, which is
open in G/H since U {/-1 (). }=/-1 () is open in G recall that / is
continuous I. Thus # is an open mapping, i.e., -1 is continuous. Let
{u H: u; U} be an open subset of G/H, where U is open in G. Then we have
q-({uH'ucV})=f'/-(;;)=uH for some uV}= {/(u)'uU}--
I(U). Since ! is by hypothesis an open mapping, the set [ (U) is open in .
Thus q9-1 and # are continuous, i.e., b is a homeomorphism. [
We shall have frequent occasion throughout the book to discuss
topological groups G and G for which there is a mapping [ with domain G
and range G which is both a group isomorphism and a homeomorphism.
We permit ourselves the slight solecism of calling such groups topologically
isomorphic. [Under a strict interpretation of the last phrase, two groups
mightbe called topologically isomorphic if there were two mappings of G
onto G, one a group isomorphism and the other a homeomorphism. The
content~ of (5.26) and (5.27) is that G/H is topologically isomorphic to
G if and only if the homomorphism / carrying G onto is continuous and
open [H=/- (). Thus can be reconstructed not only as a group but
also as a topological space from G and the kernel 1-1(?) provided that !
is an open, continuous homomorphism. Simple examples show that these
restrictions on / are needed. Consider the additive group R with its discrete
topology (R), and let ! be the identity mapping of R onto itself.
As we have already seen in (4.18.b), (4.18.d), and (4.22), R has many
distinct topologies under which it is a topological group. The mapping
is a group homomorphism in fact an isomorphism and is continuous as
a mapping of R with the topology (R) onto R with any other topology (9.
It will be open, however, only if (9=(R). Thus there are many
42 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 43
continuous homomorphisms in fact automorphisms] of R that cannot be
reconstructed simply from the given topology on R and the kernel of the
homomorphism.
Nevertheless, a reasonably large class of continuous homomorphisms
are automatically open, as Theorem (5.29)in/ra shows. The following
lemma is a trifling improvement on the classical BAIRE category theorem
(B.19).
(5.28) Lemma. A locally countably compact regular space X is not
the union o/a countable number of closed sets all having void interior.
Proof. Suppose that X-- U A,, where each A, is closed and A, °--.
For each n, let D--A. Then each D is open and dense in X. We will
show that f D,O, contradicting the equality X-- U
n=l n=l
Let U 0 be a nonvoid open subset of X such that U0-is countably
compact. Using regularity and the fact that D1 is open and dense, we
choose a nonvoid open set U1 such that U-c U0f D. Having chosen a
nonvoid open set U such that U-c U,_I f D,, we choose a nonvoid open
set U,+ 1 such that UI c U, f D,+ 1. Since U-is countably compact and
each U,-is nonvoid, the intersection f U,-is nonvoid. Points in this
intersection must lie in f D,. 1 [
(5.29) Theorem. Let G be a locally compact group that is a-compact:
G= U A,,, where A 1, A., ... are compact. Let ] be a continuous homomor-
phism o/G onto a locally countably compact T O group G. Then ] is an open
mapping.
Proof. Let q/be the family of all symmetric neighborhoods of the
identity e in G and the family of all neighborhoods of the identity
in G. Suppose that
for every U q/, there is a such that c [ (U). (I)
Then consider an arbitrary open subset B of G. For each xE B, there is a
UEq/such that xUc B. If Lis asin (t), we have/(x)E/(x)Tcf(x)/(U)---
/(xU)c/(B). Thus/(B) is open in. To prove the present theorem,
therefore, it suffices to establish (1).
To do this, let U be any set in q/, and let V q/have the property that
V- is compact and
(V-) -x (V-) C U. (2)
1 We have proved in fact that f D n is dense in X.
Such a V exists in view of (4.5.i), (4.7), and (5.2.ii). The family of sets
{x V'xG} is an open covering of G and hence of each A,. Since A is
compact, a finite number of the sets x V cover A, and therefore a count-
able number of the sets xV cover G itself" thus G---- U 1 (x V)---- U (xV-)
n= n=l '
where each x, G. 1 Plainly we have G =,=IU / (x V-) =,__U 1 / (x) / (V-).
We now assert that the set [ (V-) has nonvoid interior in . The sets
x, V-are compact subsets of G, and so their continuous images [ (x,)[ (V-)
are compact subsets of G. Since G is a T O group, it is also Hausdorff
(4.8), and its compact subsets /(x,)[ (V-) are closed. Assume that [ (V-)
has void interior. Since left translation is a homeomorphism, all of the
sets /(x,)j(V-)then have void interior. Then G-- U [(x,,)[(V-) is the
countable union of closed sets having void interior. This is impossible
by (5.28), and accordingly [ (V-) contains a nonvoid open subset of ft.
To complete the proof, select any point and any point
x/-(c)f V-. Then by (2), we have x -1V-C U and hence
/(U) )/(x -1 r--)____/(X)_I/(l/r-- ) -1 .
The set - is a member of , and so the present proof is completed. [
(5.30) Note. Two special cases of topological groups satisfying the
hypotheses on G in (5.29) are worth noting. First, if G is a locally compact
group with a countable open basis, then it is obviously Lindel6f and
therefore a-compact. Second, if G is a compactly generated locally
compact group, then G is clearly a-compact.
(5.31) Theorem. Let G be a topological group, H a normal subgroup
o! G, and L any subgroup o[ G. Let 9 be the natural mapping o[ G onto G/H.
Then the quotient group L H/H is topologically isomorphic with the subgroup
9 (L) o/ G/H.
Proof. We have pointed out in (2.1) that LH is a subgroup of G.
Plainly. H is normal in L H, and so the quotient group L H/H exists and
AS a topological group with its topology (9 (L H/H). Considered simply as
families of subsets of G, L H/H and 9 (L) are identical" the elements of
LH/H are the cosets x H with x LH, and the elements of 9 (L) are the
cosets xH with x L. These are obviously the same families of sets, and
the identity mapping, of xH" x L H) onto {xH" x L) is an isomorphism
of L H/H onto 9(L).
We actually use here the Lindelf property for G. A locally compact space is
e-compact if and only if it has the Lindelf property, so that we would not strengthen
the present theorem by supposing that G has the Lindelf property.
44 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups
We will prove that is a homeomorphism. Let X be a subset of LH.
Then {xH:x X} is open in LH/H if and only if X H is relatively open
in L H; that is, if and only if XH-- U ('1 (L H), where U is open in G. The
subset {xH: x X} is open in (L) if and only if it is the intersection with
9 (L) of an open subset of 9(G); that is, if and only if there is an open
subset V of G such that {xH:xX}--(L)(I{vH:vV}.
Now if {xH: x _ X} is open in L H/H, so that XH= U f-I (L H) for an
open subset U of G, we have
{xn. x)= {xn. u a (t n)) = {xn-x u)a {xn. t n)
I-I n e x X)i (Z:). t at x X)
(Z:). for V x X):
XH=(Vn( H))H=
(VH) (1 (LH.) Since VH is open, it follows that {xH" xX} is open in
L H/H. Thus the two topologies are identical, so that is a homeo-
morphism. t
We now take up the analogue for topological groups of the first
isomorphism theorem (2.1). If G is a topological group, A is a subgroup
of G, and H is a normal subgroup of G, then (2.t) shows that (AH)/H
and A/(A(IH) are isomorphic, the mapping 3 defined by z(aH)--
a(A (IH) being an isomorphism. The mapping 3 does not need to be a
homeomorphism, however, and in fact simple examples show that
(AH)/H and A/(A (IH) may be nonhomeomorphic. See (5.39.d) below.]
The isomorphism 3 is always an open mapping, however, and under
some circumstances it is a homeomorphism. We describe the situation
precisely in the following two theorems.
(5.32) Theorem. Let G be a topological group, A a subgroup o[ G,
and H a normal subgroup o/ G. Let 3 be the isomorphism (a H) -- a (A (1 H)
(aA) described in the/irst isomorphism theorem, carrying (AH)/H onto
A/(A(IH). Then 3 carries open subsets o/ (AH)/H onto open subsets o[
A/(A(I H).
Proof. An open subset of (AH)/H is a subset {xH'xX}, where
X c A, such that XH is open in A H regarded as a subspace of G. The
set 3({xH" xX}) is theset {x(AlH)" xX}. Since X(A(IH)=(XH) (lA,
it follows that X (A ('IH) is an open subset of A in its relative topology as
a subspace of G, and so, by the definition of the topology in A/(A(IH),
(x(A(lH)" xX) is an open subset of A/(A(IH).
(5.33) Theorem. Notation is the same as i (5.32). Suppose that A
is locally compact and -compact, that H is closed, and that A H is locally
compact. Then 3 is a homeomorphism, and so (A H)/H atd A/(A(IH) are
topologically isomorphic.
Proof. In view of (2.t) and (.32), we need only show that 3 -1
carries open subsets of A/(A(IH) onto open subsets of (AH)/H.
Consider first the natural mapping 9 of G onto G/H. Restricted to the
subgroup A, 9 carries A onto the subgroup (AH)/H of G/H. Since AH
is locally compact and H is closed, the quotient group (AH)/H is locally
compact (5.22) and T O (.21). Thus 9 is a continuous homomorphism of
the locally compact a-compact group A onto the locally compact T O
group (AH)/H. By (.29), 9 is an open mapping on A.
Now let {y(AClH):yY},YcA, be an open subset of A/(A(ln);
that is, Y(A (IH) is an open subset of A. Computing 9(Y(A (IH)), we
see at once that it is the subset {yH: y Y} of (AH)/H. Since 9 is an
open mapping on A, the set {yH:y Y} is open in (An)/H. We also
have, by the definition of 3, that {yH: y Y} = 3 -l{y (A('IH): y Y}.
That is, 3 -1 carries open subsets of A/(A(IH) onto open subsets of
(AH)/H.
The second isomorphism theorem (2.2) has a complete analogue for
topological groups, as follows.
(5.34) Theorem. Let G and G be topological groups with identity
elements e and , respectively" let / be an open continuous homomorphism o/
G onto . Let be any normal subgroup o/, H:/-1 (I), and N: [-1 ().
Then the groups G/H, G/H, and (G/N)/(H/N) are topologically isomorphic.
Proof. Let W be the natural mapping of onto GH. By (.26), is
an open continuous homomorphism and hence W o / is an open continuous
homomorphism of G onto / with kernel H. Hence / is topologically
isomorphic with G/H by (.27). Since the mapping __>/-1() is a topo-
logical isomorphism of onto GIN by (.27) and the image of under
this mapping is H/N, / is topologically isomorphic with (G/N)/(H/N).
Theorem (5.34) can be estated as follows.
(5.35) Theorem. Let G be a topological group and H and N normal
subgroups o/ G such that N C H. Then G/H is topologically isomorphic
with (G/N)/(H/N).
We can now show to what extent the study of topological groups
can be reduced to the study of Hausdorff groups.
(5.36) Theorem. Let G be a topological group. Then G/{e}- is a
Hausdor]/... group. I! ! is any continuous homomorphism o/ G onto a T O
group G with identity , then/-1 ()D {e}--. I[ / is also open, then is an
open continuous homomorphic image o/ G/{e}-.
Proof. Since {e}- is a closed normal subgroup of G (5.4), G/{e}-is a T
group (5.2t) and hence a Hausdorff group (4.8). Now ]-1() is a normal
46 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 47
subgroup of G and is closed since [ is continuous. Hence we have
-1() e)-. Finally, if [ is an open continuous homomorphism, G is
topologically isomorphic with G/]-I(). By (5.35), G/[ -1 () is topologically
isomorphic with (G/e)-)/(]-l()/{e)-), which is an open continuous
homomorphic image of G/(e)- (5.26). [
Miscellaneous theorems and examples
(5.37) Subgroups. (a) (BOURBAKI _, p. 0.) A subgroup H of a
topological group G is closed if and only if there is an open subset U of
G such that UVIH= UV)H-and UV)H=4=(3.
(b) If H is a nonclosed subgroup of a topological group, then H-V)H'
is dense in H-.
(c) Let H be a dense subgroup of a topological group G, and let ] be a
normal subgroup of H. Then ]- is a normal subgroup of G.
(d) The intersection of all open subgroups and the intersection of all
closed subgroups of a topological group G are closed normal subgroups of G.
(e) The center of a T O group G is a closed normal subgroup of G.
(f) Let G be a topological group such that U V n-- G for every neigh-
borhood V of e. Let H be a discrete normal subgroup of G. Then H is
contained in the center of G. [There is a neighborhood U of e such that
(a U)V)H--(a} for every a6H. By continuity of multiplication, there is
a neighborhood V of e such that V-la VC a U. Hence x-la x-a for all
aH and xV.
(g) Let G be any group and N a normal subgroup of G having at
least two elements. Topologize G as indicated in (4.21.b). Then G is a
topological group and (e) is a discrete subgroup that is not closed" com-
pare with (5. 0).
(h) Let G be any topological group, with identity e. The subset (e)-
of G has the weakest topology: and (e)- are its only open sets.
(i) The additive group R contains open and nonclosed subsemigroups,
nondiscrete closed subsemigroups having isolated points, and discrete
nonclosed subsemigroups. An example of a discrete nonclosed sub-
semigroup of R is the semigroup generated by the set (1 + r)°°=l where
(r.°°= 1 is a sequence of positive real numbers converging to 0 for which the
set 1, r 1, r. .... ) is rationally independent.]
(5.38) Quotient spaces. (a) If G is a compact group and H is a
closed subgroup of G, then the natural mapping of G onto G/H is a
closed mapping. [This follows from (5.8). To see that H needs to be
closed, consider G-- T and H--(exp(2ir):rQ).
(b) Let G be any topological group and H a subgroup of G. Let F
be a closed subset of G that is the union of cosets x H. Then 9(F) is
closed in G/H.
(C) (ALEKSANDROV and HOPF [1], p. 66.) Let G be a topological
group and H a subgroup of G. Introduce a topology . into the set G/H
of left cosets x H by the following rule. Given a coset all, consider any
open set U such that a H C U c G. Consider the family of all cosets x H
such that x H c U. Let the collection of all such families of cosets be an
open basis for the open sets of .. Then the topology . is stronger than
@(G/H), andS---O(G/H)if and only if for every coset aH and every open
set U such that aH C U, there is an open set V such that aH c V and
such that xG and xHV)V=4=(3 imply that xHc U. If H is compact,
then .P. =- (G/H).
(d) Let G be a topological group and H a subgroup of G. Then 0 (G/H)
is the weakest topology, {, G/H}, if and only if H---G.
(e) (Adapted from GRAEV [4].) Let G be a topological group and H
a subgroup of G. If H and G/H have countable open bases at each point,
so does G. [Let {W,}_ 1 be a sequence of symmetric neighborhoods of e
in G such that W+I c W, for each n and such that {W,V)H}= 1 is an open
basis ate for the group H. Let {{uH:uU,,}}=l be an open basis at the
point H of G/H, where the U, are neighborhoods of e in G. For n = 1, 2 .... ,
let
P, -- (W,+ 1 • (H V) W2))-' VI (U, H)
and Q.- P V) p. V) ... V) p,. It is easy to see that every Q, is a neighborhood
of e. Also (Q,,}=I is an open basis at e in G. To prove this, let Y be any
neighborhood of e in G and V a neighborhood of e such that V
denote, as usual, the natural mapping of G onto G/H. There is an m
such that W V) H c V V) H, and a k > m such that q (U) c q (V V) W+ 1).
Then we have
QnP,. (UH)n (w+. (H n w.:))-'
Q ((V V) Wm+I) H) V) ((V V) Wm+I) . (H V) W/,,))' c (V V) W,,+I) • (H V) Wm)
C (V V) W,,+I ) . (V V) H) c Vc
(f) Let G be a topological group and H a subgroup of G such that H
and G/H contain dense subsets of cardinal numbers m and n, respectively.
Then G contains a dense subset of cardinal number less than or equal
to ma. The cardinal numbers m and a may be finite or infinite. [Let A
be dense in H and let {bH" b B} be dense in G/H. If U is open in G and
U, then there is a bH in {uH:uU}. Thus we have
and HV)(b-IU)=4=(3. Hence AV)(b-IU)=4= and (BA)V)U=:4:=(3. Con-
sequently, BA is dense in G.]
48 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups 49
(g) An instructive example of a topological group, a closed subgroup,
and the corresponding coset space is the following. Consider the group
(xY) for which
®(2, R), which consists of all 2x2 real matrices z w
A=xw--yz=t=O. Consider the subgroup consisting of all matrices (g IY )
with x0. Consider the mapping T of ®(2, R) onto RN {(0, 0)}'=E
defined by
Given matrices A and B in ® (2, R), it is not hard to prove that T(A)=
T(B) if and only if B-1Ag. Hence, if we define T*(Ag)=T(A) for
A 06® (2, R)/g, then T* is a one-to-one mapping of ® (2, R)/ onto E.
It is easy to see that T* is continuous. To show that T* is open, use the
U 2 -- V 2
fact that for any (u, v). E, T --(u, v) and the fact that u, v,
U '
U2-- V 2
u.+v2, and u2+v 2 are continuous functions of u and v. Thus
the topology of ®(2, R)/g) is the ordinary topology of E. For
A=(ca)®g(2, R), with A=ad--bc, the homeomorphism A of
® (2, R)/g) (5.20) corresponds to the homeomorphism ]a of E defined by
(5.39) Quotient groups. (a) The group R/Z is topologically iso-
morphic with the group T.
(b) Let G be a topological group and H a normal subgroup of G.
The correspondence L-->9(L ) is a one-to-one correspondence carrying
the family of all subgroups of G containing H onto the family of all
subgroups of G/H. Furthermore, a subgroup L is closed if and only if
0 (L) is closed.
(c) In (5.31), suppose that L is an arbitrary nonvoid subset of G,
and let M be the smallest subgroup of G containing L H. Then 9(L)
is homeomorphic with the image of L H in M/H.
(d) Let 0 be any irrational real number. Then (0Z)/((0Z) CI Z) is
discrete and ((OZ)+Z)/Z is not. [Describe the group ((OZ)+Z)/Z as a
subgroup of the compact group R/Z. Why does this example not contra-
dict (5-33) ?
(e) The results obtained in (5.32) and (5.33) on the second isomor-
phism theorem can be extended. Notation is the same as in (5.32). In
proving (5.33), we showed that if 9 is open as a mapping of A onto AH[H,
then z is a homeomorphism. The converse is also true. [The condition
that z be a homeomorphism can be recast as follows: if Y is a subset of A
such that Y.(Af-IH)=Af-IU for some open subset U of G, then
YH = V fl (AH) for some other open subset V of G. Now if W is open in G,
it is easy to see that (A fl W). (Afl H) = A fl (W (Afl H)). Thus if z is a
homeomorphism, we have (A fl W) • H= V fl (AH) for some open subset V
of G. Hence 9 is an open mapping of A onto AH/H.
(f) (FREUDENTHAL [1.) Notation is as in (5.32). Suppose that z is a
homeomorphism, that A is locally compact, and that H is closed. Then
AH is closed in G. [By (S.22), A/(A VIH) is locally compact. Hence AH/H
is locally compact and therefore closed in G/H, by (5.t I). This obviously
implies that AH is closed in G.
(g) Let G be a topological group and H a normal subgroup of G.
Then G/H-is topologically isomorphic with (G/H)/{H}-, where {H}- is
the closure in G/H of the identity element H. [Note that {H}-is H-/H,
and apply (5.35).1
(h) Let G be a topological group with a compact normal subgroup H
such that G/H is compactly generated. Then G itself is compactly
generated. [Let be the natural mapping of G onto G/H and suppose
that (xH:xA} is a compact subset of G/H that generates G/H. By
(5.24.a), AH is compact. Therefore (AH)UH is compact; plainly
(AH)UH generates G.
(i) Let G be a locally compact group and H a normal subgroup of G.
If both G/H and H are compactly generated,then G is also compactly
generated. [Let {xH:xX} be a closed compact subset of G/H that
generates G/H. [Such a set exists by (3.3) and (4.7). By (5.24.b), we
may suppose that X is compact. If A is a compact subset of H that
generates H, then A U X is a compact subset of G that generates G.
(j) Theorem (5.27) has the following analogue. Let /be a continuous
homomorphism of a compact group G onto a Hausdorff group
is an open mapping, and G/H and are topologically isomorphic, where H
denotes the kernel of [. [For , let b()=[-l(). Then b is an iso-
morphism of onto G/H. As in the proof of (5.27), the continuity of
implies that 05-1 is a continuous mapping of G/H onto . Since G/H is
compact (5.22), 05-1 is a homeomorphism. That is, and G/H are topo-
logically isomorphic- ! is now an open mapping by (5.26) since/(x) =
q)-I (xH) for x G.
(5.40) Homomorphisms. (a) Let G and be topological groups
with identity elements e and , respectively, and let ! be a homomorphism
of G into . If ! is continuous at some point of G, then / is uniformly
continuous for the pairs of uniform structures ((G), o()) and
(°,(G),9°,()). [Suppose that [ is continuous at xoG. Then if
Hewitt and Ross, Abstract harmordc analysis, vol. I
50 Chapter II. Elements of the theory of topological groups § 5. Subgroups and quotient groups
is a neighborhood of , there is a neighborhood U of e such that
[(xoU)C [(Xo)7. Thus if y-lxcU, we have ](xoy-lx)c[(xo)U. That is,
[ (y)-l[ (x) . Consequently, [ is uniformly continuous for the pair of
uniform structures (oo (G), oo ()); a similar argument applies to the
pair of uniform structures (oo r (G), oor ()).
(b) Let G and be topological groups, and let ] be a homomorphism
of G onto . If ! (U) has nonvoid interior for all U in an open basis at e,
then ] is an open mapping.
(c) Let G, , and be topological groups. Let [ and be continuous
homomorphisms of G onto and of onto G, respectively. If o ! is a
topological isomorphism, then ] and are also topological isomorphisms.
(d) Let G be a locally compact group, a topological group, and 0 a
continuous homomorphism of G into such that 0 is an open mapping
of G onto (G) . Then 0 (G) is a locally compact subgroup of " if
is a T o group ,then (G) is closed in . By (5.27), (G) is topologically
isomorphic with G/-I(). By (.22), G/-() is locally compact. By
(5.tt), (G) is closed if G is a T o group.
(e) The image of a closed subgroup under an open continuous homo-
morphism need not be closed. Consider the mapping t-->exp 2rit]
of R onto T, and the image of the subgroup {n V.n z).
(f) The group R is an open continuous homomorphic image of a
totally disconnected topological group. Consider the group G of all
Cauchy sequences '----(rl, r. .... , r,...) of rational numbers, and define
addition coordinate-wise. For t> 0, let A t consist of all ,G such that
[rl __<t for all n. For >0, let V--tU<A t. Then {V+">0, ,G) is
an open basis for a topology on G under which G is a topological group.
Let H be the subgroup of G consisting of all , such that lim r= 0. Then
G]H with the topology #(G/H) is topologically isomorphic with R.
Suppose that s, where s, cG. Then So=to for some positive
integer n 0" say s0 < t0. Let 0 s,,0, t0 be irrational. Then
{'G'ro< 0) is open and closed, sU, and *U'. Thus G is totally
disconnected. The preceding is simply the usual construction of R
from Q using sequences ,with the topology of R defined in a novel way.]
(g) (GIiAEV 4.) Let G be a group and let ] be a homomorphism of G
onto a topological group . Define a topology # on G by" UC # if and
only if :7--] -1 (), where is open in . Then G with the topology
is a topological group.
(h) (GIiAEV 4.) Let G be a topological group with topology #, and
let X be a subset of G. Let {#)A be the family of all topologies under
which G is a topological group and which induce a topology on X weaker
:;1
than the topology of X induced by #. Let #o be the weakest topology
on G stronger than all of the #a. Then if ] is a homomorphism of G into a
topological groupG and ] is continuous when restricted to the domain X,
is continuous on G in the topology #o.
(i) Let ] be a homomorphism of the topological group H into the
topological group G and let 0 be the natural mapping of G onto G/{e)-,
where e is the identity in G. If o ! is continuous, then [ is continuous.
IIf :7 is an open subset of G, then U{e)-=U, and hence
o 1)-: x
(j) Let [ be a homomorphism of the topological group H onto the
topological group G and let 0 be the natural mapping of G onto G/{e}-,
where e is the identity in G. I: o! is an open mapping and ]-1 ((e}-)=
(]-: (e))-, then ] is an open mapping. :Let U be an open set in n. Clearly
-loo/(U) is open in G; we will show that/(U)---loo/(U). The
inclusion [ (U) : 0 -1 o 0 o ] (U) always holds. Let y: 0 -1 o 0 o ] (U). Then
for some z:U, 0(y)--0(/(z)) and clearly y--/(x) for some x:n. This
implies that
1 = 1 (z) = (z)
and hence x-lz1-1 ({e}-)= (1-1 (e))-. Now x -1U is a neighborhood of x-lz
and hence intersects ]-1 (e). That is, for some uc U we have ] (x-lu)--e
and hence y----](x)--](u):](U).]
Notes
Like the contents of §4, many of the results of § go back
to the early t930's: D. VAN I:)ANTZIG [2] and [3], H. FREUDENTHAL
[t], D. VAN KAMPEN [t] and A. IIARKOV [t] have all contributed to
the theory of subgroups and quotient groups. The topology described in
(.t ) was introduced by VAN DANTZlG [2], and independently [also in a
more general framework] by R. BAER and F. LEVI [t]. ]:ARKOV [t]
proved (. t 7), and also showed that not every continuous homomorphism
is open. In view of (5.29), this phenomenon could not be recognized so
long as attention was restricted to locally compact groups with countable
open bases. VAN DANTZlG [3] did, however, state an incorrect result
[TG. 20, p. 609]. FREUDENTHAL [t ] also emphasized the distinction
between continuous homomorphisms and open continuous homomor-
phisms. Theorem (.22) appears in VAN KAMPEN [t], M:ARKOV [t], and
FIEUDENTHAL [t]. The first assertion in (.2) is due to Fi:EUI)EN-
TIAL [t]. The second appears in MONTGOMEiY and ZIPPIN [t], p. 2;
our proof is modelled on theirs. Theorems (.26) and (.27) are first
explicitly stated in FREUDENTHAL [t], although MAIKOV [t] comes
fairly close. Theorem (.29) is a modification of PONTI:YAGIN [7],
4*
Chapter II. Elements of the theory of topological groups § 6. Product groups and projective limits 53
Theorem 12. Theorem (5.31) is taken from ]OURBAKI [I], p. 14.
FREUDENTHAL [1 proved a special case of (5.33), as well as (5.34).
Theorem (5.36) appears in BOURBAKI [t, p. t3.
1. YA. VILENKIN [t4 has made a detailed study of topological
groups not satisfying the T O separation axiom, and in particular of
quotient groups G/H where H is a nonclosed normal subgroup of G.
§ 6. Product groups and projective limits
The principal method of constructing topological groups to be
considered in this section is the formation of Cartesian products. This is
an extremely important process, which frequently makes it possible
to reduce problems concerning complicated topological groups to prob-
lems concerning simpler ones.
(6.1) Definition. Let {G,: tcI} be a nonvoid family of topological
groups, and let ,P G, be the direct product group (2.3). Then the group
P G with the Cartesian product topology, is called the direct product of
EI '
the topological groups G,.
Whenever we refer to the direct product of topological groups G,,
we will always mean the group ,P G, with the Cartesian product topology.
The following theorem is just what one ought to conjecture about direct
products of topological groups.
(6.2) Theorem. The direct product ,P G, o[ the topological groups G,
is a topological group and P'G, is a dense subgroup o/,P G
' tEI "
Proof. To show that P G is a topological group, we need only to
EI
verify axioms (4.t.i) and (4.t.ii). Let us verify (4.t.i). Let (a,) and (b,)
be points of P G and let A be any neighborhood of (a,) (b,). Then, by the
definition of the Cartesian product topology, there are open subsets U,
of G, such that a, b,c U, for all t I, such that only a finite number of the U,
are different from G,, and such that ,EP U, C A. Let V, and W, be neighbor-
hoods of a, and b,, respectively, in G, such that V,W, cU,, for all ,I,
and such that V, and W, are equal to G, if U, is equal to G,. Let V=,EPV,
and W =,EPW,. Then it is obvious that V and W are neighborhoods of (a,)
and (b,), respectively, and that VW ,EPU, A. Axiom (4.t .ii) is verified
similarly. Hence ,EPI G, is a topological group.
It is obvious that P* G is a subgroup of ,IG To prove that it is
EI "
dense, we must show that every nonvoid open subset of ,IG' contains a
point of P*G Let ,PxU' where each U, is open in G and only a finite
t6I t • ' t
number of the U, are different from G,, be any nonvoid basic open set
in ,PG,. Let x, be any point of U, for all , such that U,4= G,, and let
x,= e, [the identity in G, for all, such that U,= G,. Then
(x,)(,PIU,)A(,Pz*G,). fl
(6.3) Theorem [Associativity of direct products]. Let {G,: 1} be a
nouvoid /amily o] topological groups, and let {Ia}aA be a partitio o/I. LetA a
be the topological group P G, ]or each A. Then P G, is topologically
I ' EI
isomorphic with PAA"
This theorem is obvious upon a little reflection, although to write
out the proof in all of its details is somewhat cumbersome. We leave this
task to the interested reader.
As is to be expected, special properties of the groups G, are reflected
in special properties of the direct product P G Throughout (6.4) to
(6.9), to avoid tedious repetition of notation, we take {G,: I} to be a
nonvoid family of topological groups; we write e, for the identity in G,,
and G for the direct product P G
tl
(6.4) Theorem. The group G is a gro i] and only i/ each G,
is a T o gro. The gro G is coeced i/ ad oly i/ every group G is
comced, ad G is compac i/ ad oly i/each G, is compact. Finally,
he gro G is locally compac i/ and oly i/all o/ he groups G, are locally
comac ad all b a lipide mber o/ hem are compact.
Proof. These assertions follow immediately from the corresponding
assertions for arbitrary topological spaces. [
(6.5) Note. It is also well known that G--P G is completely
tel t
regular if and only if every G, is completely regular. The corresponding
assertion for normal groups fails. We vill show in (8.t t) and (8.t2) that
there are many nonnormal products of normal groups.
(6.6) Theorem. The group G is Abelian i/ and only i/ all o/ the
groups G, are A belian.
Proof. Obvious. 1
We shall now take up subgroups and quotient groups of G, and their
relation to the corresponding entities in the groups G,.
(6.7) Theorem. Let A, be a subset o/ G,/or all I. Then i/ A, is a
subsemigroup, subgroup, or normal subgroup o/ G,/or all I, it [ollows
o/thatG. 'Exp A, is a subsemigroup, subgroup, or normal subgroup, respectively,
Proof. Obvious.
54 Chapter II. Elements of the theory of topological groups § 6. Product groups and projective limits 55
(6.8) Theorem. Let J be any nonvoid subclass o[ I. Let the mapping
o! G o, to P G
EJ
is called a projection. Then I is an open continuous homomorphism o[ G
onto P G Its kernel consists o[ all (x,) G such that x,= e,/or all tE J.
j "
Proof. This result too is obvious upon a little reflection, and we leave
the verification to the reader.
(6.9) Theorem. For each I, let H, be a subgroup o/G,. Consider
mapping o/the [irst space onto the second defined by (x,H,) = (x,),H,=
P x,H,. Then the so-called aatural mapping o/the first space onto
the second is a homeomorphism. I[ each H, is a normal subgroup, so that
both spaces are groups, • is an isomorphism. Thus, in this case, , (G,/H,)
and (,IG')/(,H') are topologically isomorphic.
Proof. It is obvious that • is one-to-one and onto. Consider any
open set in ,(G,/H,) of the form ,{x,H," x, U,}, where each U, is open
in G, and only finitely many U, are different from G,. The image under #
of this set is the set {,IX,H,'(x,),U,}. The union in G=,IG , of the
sets in this collection is the set ,U,H,, which is obviously open in G.
Thus # is an open mapping. The same argument shows that #- is
open" i.e., • is a homeomorphism.
Suppose next that H, is a normal subgroup of G, for all I. Let (x,H,)
and (y,H,) be any elements of ,i (G,/H,). Then we have • (x,H,) (y,H,) =
=
Thus • is a one-to-one homomorphism, and is therefore an isomorphism.
(6.10) Suppose that we are given a topological group X and a collec-
tion of subgroups, {},i, of X. Under what conditions can we assert
that X is topologically isomorphic with the direct product of the sub-
groups N, ? Let us first consider some obvious necessary conditions. Let
G=,G,, and for each 06 I, let ,o= G,ox,,o{e,}" Clearly each ,. is a
normal suboup of G, an6 is closed if and only if all of the G, ( 0)
are T 0 groups. If 0 and are distinct indices in I, then every element of
,o commutes with every element of G,. Each product set G, ,, ...,, is a
normal subgroup of G; (G, G,.-. ,,) 0 G= {(e,)} if is different from
..... ,; and the union of all of the subgroups ,,G, ... G,is dense in G.
If I is finite, say I= {, 2 .... , m}, then G=GG... G,. Furthermore, if
I= {t, 2,..., m}, and if , is a neighborhood of (e,) in , relative topol-
ogy for =, 2 ..... m, then the product set -.. , contains a
neighborhood of (e,) in G.
We can now attempt to answer the question posed above. For the
general case, where X has an infinite number of subgroups that satisfy
the conditions observed for the groups G,, there are serious difficulties,
and we know of no way to construct X as a/ull direct product. For the
case of a finite number of subgroups a satisfactory answer to our question
can be provided. We give it in the following two theorems.
(6.11) Theorem. Let X be a topological group, and let N1, N. .... ,
be normal subgroups o/X with the/ollowing properties:
(i) NI N ... N,n-- X',
(ii) (NN2 ... N)N+I--(e} (k=t, 2, ..., m--t);
(iii) i/ Si is a neighborhood o/ e in N [relative topologyJ /or i--, 2 .... , m,
then S 1S ''" S,, contains a neighborhood o/ e in X.
Then X is topologically isomorphic with the direct product
N,ЧN,Ч ...
Proof. Conditions (i) and (ii) imply that X is isomorphic with the
direct product NixN2x... xNm: every x in X can be written in precisely
one way as a product YlY2 "'" Ym, where yi N/ (=t, 2 .... , m). Every
element of commutes vith every element of N, (/" 4= k;/'--t, 2 .... ,
k--t, 2 .... , rn). Thus (YlY2 "'" Y/) (ZlZ2 "'" Zm)=YlZlY.Z. "'" YmZm • It
follows that the mapping : of N1xN,.x ... xN m onto X defined by
lfl (Yl, Y2 .... , Ym)= Yl Y2"'" Y,, is an isomorphism. The continuity of p is
assured by the continuity of multiplication in X: if Yl, Y.,-.., Y, are
all sufficiently close to e, then YlY.'"Ym is arbitrarily close to e. The
continuity of :-1 is guaranteed by hypothesis (iii). [
(6.12) Theorem. Let X be a locally countably compact T o group, and
let N1, N 2 .... , N,,, be locally compact, a-compact normal subgroups o/ X
satis/ying conditions (i) and (ii) o/ (6.t t). Then X is topologically iso-
morphic wit/, N xN2x ... xN m .
Proof. The mapping : of (6.tt) is a continuous isomorphism of
NixN2x... xN,, onto X. The group NxN2x... xN m is locally compact
(6.4) and a-compact (3.9). Theorem (.29) implies that : is also an open
mapping.
(6.13) Definition. Let A be a set directed by a partial ordering
[\Ve will write < fl to indicate that fl and 4=fl. For every A, let
there be given a topological group G. Suppose that for every a, fiEA
such that .<fl, there is an open continuous homomorphism/ of G
into G. Suppose finally that if fl-<y, then/--/a o/. The object
consisting of A, the groups G, and the mappings/as is called an inverse
mapping system. Consider next the group G= P G and the subset of G
Chapter II. Elements of the theory of topological groups § 6. Product groups and projective limits
56
consisting of all (x) such that if -</5, then x----/(x). This subset is
called the proective limit of the given inverse mapping system.
(6.14) Theorem. Let there be given an inverse mapping system as in
(6.t 3). The pro]ective limit is a subgroup o[ the direct product G. I[ all
o] the groups G are T o groups, then the proective limit is a closed subgroup
o] the direct product G.
Proof. It is easy to verify that the projective limit is a subgroup
of G. If (x)G and (xv) is not in the projective limit, then for some
and/5 in A, we have -</5 and x :4: ] (x). If G is To, then there are
disjoint open subsets U and V of G such that xc U and [(x)cV
see (4.8). Thn the set of all (y)G such that y U and y[(V) is an
open set in G containing (x) and disjoint from the projective limit.
Miscellaneous theorems and examples
(6.15) Different topologies for product groups. (a) Let {G,:,I}
be a nonvoid family of topological groups. Let W, be an open subset of G,
for each ,c I, and let the family of all sets of the form ,PzW, be a basis for
open sets. With the usual operation (x,) (y,)---- (x,y,), and this topology,
the Cartesian product ,PzG, is a topological group.
(b) Let G----,PzG, be a direct product of topological groups, where e,
is the identity of G, for all , I. Let I 1 and I. be complementary nonvoid
subsets of I. We introduce another topology . on the set G, as follows.
Let an open basis at (e,) consist of all sets ,, {e,}x,,U,, where each U,
is a neighborhood of e, in G, and only a finite number are distinct from G,.
Then G, with the topology 0, is a topological group. If all G, are compact
for , I9., then G is locally compact in the topology 0, and is compact if
and only if ,,G, is finite. The ordinary product topology of G agrees
with the topology 0 if and only if ,Pt,G, is a discrete group in its Cartesian
product topology.
(c) (BRACONNIER [t .) Let {G,}, be a nonvoid family of topological
groups and let H, be a normal subgroup of G for each t I. Let H---- P H,"
EI
plainly H is a normal subgroup of G---- ,Pz G,. Let A be a finite subset of I,
and let U, be a neighborhood of e, in the subgroup H, for each ,A; in
other words, U, is the intersection of H, with an open subset of G,. Let
the sets, L U'X,A H, be taken as an open basis at e ---- (e ,) in G. It follows
from (4.5) that G is a topological group. The group H is an open subgroup
of G.
(6.16) Local direct products (BRACONNIER [t) 1. Let {G,},,
G, and H be as in (6.t5.c). Suppose further that each H, is open. Let
1 This construction is a fruitful source of useful examples. See in particular
§§ 24 and 25.
Go be the subgroup of G consisting of all (x,) for which x,C H, for all but
a finite number of indices ,. As a subspace of G topologized as in (6.t 5.c),
Go is a topological group and is called the local direct product o/ the G,'s
relative to the open normal subgroups H,.
(a) The identity {e}---,Pz (e,}- obtains; thus G O is a T o group if and
only if each G, is a T O group.
(b) The group G O is compact if and only if all G, are compact and
H, = G, for all but finitely many ds. In this case, G o is the entire product
,zG,. [If Go is compact, then each G,, as a continuous image of G 0, is
also compact. The group Go/H is compact and discrete and therefore is
finite. It follows that G,=H, except for finitely many t's, and that
(c) The group G o is locally compact if and only if all G, are locally
compact and all but a finite number of the H, are compact. [This
follows at once from (6.4).]
(d) The group G O is discrete if and only if each G, is discrete and
H,--{e,} for all but a finite number of indices t. [If G O is discrete, then
the direct product ,PzH, must be discrete. This implies the present
assertion.
(6.17) Examples and counterexamples involving B. (a) Let m
be any nonzero cardinal number. Then Rm/Z TM is topologically iso-
morphic with Tm.
(b) The group R/Q is isomorphic with R and has the weakest topology
{(, R/Q}. [The second statement follows from (5.38.d). To prove the
first, consider a Hamel basis B for R over Q such that t B. Plainly R
is isomorphic with P* Qb) where each Q<b)is Q. The quotient group R/Q
bB '
is isomorphic with *
eB 0}" Q)' which is isomorphic with P* Q). Both
groups are in fact isomorphic with Qc..
(c) Consider the group R , written as all pairs (x, y) of real numbers,
with its usual topology. Let H={(x, 0)R:xQ}. Then R/H - is
.topologically isomorphic with R and also with (R/H)/(H-/H). Thus R
s an open continuous homomorphic image of a non T o group; the kernel
is isomorphic with R and has the weakest topology. This construction
can be generalized to any topological group G containing a nonclosed
and nondense normal subgroup H.
(d) (CARTAN and DIEUDONN [_t .) Let B be a Hamel basis for R over Q
and let B1, B. .... ,/m be a partition of B, m > t. For 7"--t, 2 ..... m,
let be the set of all numbers rlXl+ r.x. +... + rXs, where the r's
are in Q and the x's are in B.. Topologize each with its relative topol-
ogy as a subspace of R. Then the subgroups N1, Nz ..... N m of R satisfy
58 Chapter II. Elements of the theory of topological groups § 6. Product groups and proective limits 59
(6.tt.i) and (6.tt.ii). Thus R is isomorphic with N=N1xN.x ... xNm.
However, R is not homeomorphic with N. Note also that N has a count-
able basis of open sets. [R is connected, while N has dimension 0.]
(e) Consider the group R and the closed subgroup H consisting of all
elements (t, t) for tcR and some fixed cR, :#0. Let q) be the natural
mapping of R 2 onto R2/Z . Then q)(H) is a closed subgroup of R/Z
if and only if is rational, and q)(H) is a dense [proper] subgroup of
R/Z if is irrational. Furthermore, q) is one-to-one on H if and only
if is irrational, and q) is an open mapping of H onto q)(H) if and only
if c is rational. Nevertheless, H, q (H), and q) do not contradict (5.29) for
irrational e.
(6.18) Let m be any cardinal number greater than Vl 0. Consider
the multiplicative group (--t, t} with the discrete topology and the
direct product G= {--t, t}m. Let H be the subgroup of G consisting of
all elements (x,) such that x,= t except for a countable set of indices
Then H is a subgroup of G, H is countably compact, and H is not com-
pact.
(6.19) (KASUGA [t].) Let G be any group and suppose that
and 2 are topologies for G that make G into a e-compact locally com-
pact T O group. If there is any Hausdorff topology g)3 on G weaker than
and g)2, then 1=g)2. Note that G need not be a priori a topological
group under g)a. lit is convenient to write 0x * for the topology on GxG
obtained when the first factor G has the topology and the second
factor G has the topology *. Let A= {(x, x)GxG:xG}. Since )3 is a
Hausdorff topology, A is a closed subset of GxG under the topology
g)axg)a. Since this topology is weaker than g)lxg)2, it follows that A is
also closed under Ч g)2. Give A the relative topology of g)x
Evidently GxG, with the topology g)xg)., is e-compact and locally
compact, and hence the closed subset A is also e-compact and locally
compact. The mapping (x, x)-->x of A onto G is a continuous isomor-
phism when G has the topology g) and when G has the topology g). By
(5.29), the mapping (x, x)-->x is also open when G has either of these
topologies. Hence (x, x) --> x is a topological isomorphism when G has the
topology g) or 2. It follows that the topologies g)l and g). are the same.]
One can now easily deduce the following corollary. Let G be a To
group with topology ) and H a subgroup of G. Suppose that H admits
another topology 2 under which it is a e-compact locally compact To
group and such that the identity mapping from H with the g). topology
onto H with the relative g)l topology is continuous. Then g)2 is uniquely
determined.
(6.20) Semidirect products. Let G and H be topological groups
and let G @H be a semidirect product of G and H as in (2.6). Suppose
also that the mapping (x, h)-->zh(x) is a continuous mapping of GxH
onto G. In particular, each zh is a homeomorphism of G onto itself.
Then the semidirect product G@H with the product topology is a
topological group. This fact follows readily from the continuity of
multiplication and inversion in G and H.
(6.21) The following result is related to (6.t2). Let X be a locally
compact To group, and let N,..., N m be closed normal subgroups of X
satisfying (6.t t.i) and (6.t t.ii). If N ..... N,_ are compactly generated,
then X is topologically isomorphic with N xN.x.., xN m. [An induction
argument shows that it suffices to consider the case m = 2. Let F denote a
compact subset of N that generates N. By (5.t4), there is a compactly
generated open subgroup H of G that contains F. It follows that N1 c H.
As in (6.t t), there is an isomorphism of NxN2 onto G; it is defined by
o ( x, y) = x y where ( x, y) N xN . We next show that v/ (N x (N f H) ) = H.
It is obvious that v/(Nx(N2C?H))c n. Moreover, if hn, then h=xy
where x c N 1 and y N.. Since x is also in H, we infer that yH, so that
h=xyN 1 • (N.H)=v/(Nx(N2ClH)). Since n is locally compact and
N 1 and N2 H are e-compact and locally compact, (6.t 2) implies that is
a topological isomorphism of the open subgroup N x (N2 H) of N xN2
onto the open subgroup H of G. By (5.40.a), is a topological isomorphism
of NxN onto G.]
(6.22) (WIL I41, PP. 20 and 95.) (a) Let G be a T O group, H a nor-
mal subgroup of G, and suppose that there is a continuous homomor-
phism 9 of G onto H such that 9 (x) = x for x H. Then H is closed, G is
topologically isomorphic with Hxq-(e), and q)-l(e) is topologically
isomorphic with G/H. [Let L = q)- (e). Plainly L is a closed normal sub-
group of G. For xcG, we have x=q(x)q(x)-xHL since
q(q(x-))9(x)=e. Thus G=HL=LH. It is obvious that HC?L={e},
since 9 is the identity mapping on H. Let UI, U. be arbitrary neighbor-
hoods of e in G, and let V and W be symmetric neighborhoods of e in G
such that V UI (? Us, W V, and 0 (W) c H V. Then it is easy to see
that WC (UC?H). (UeClL). Thus (6.tt) implies that G is topologically
isomorphic with H x L, and so also L is topologically isomorphic with G/H.
(b) Let G be a T O Abelian group, and let H be an open divisible (A.
subgroup of G. Then G is topologically isomorphic with H x(G/H).
[By (A. 7), the identity mapping of H onto H admits an extension over G
to a homomorphism 9. Since 9 is the identity on the open subgroup H,
9 is continuous. Now apply (a). l
Notes
Direct products of topological groups have been used since the
inception of the theory of topological groups. As will appear in § § 9,
24, and 25, direct products shed much light on the structure of locally
60 Chapter II. Elements of the theory of topological groups
compact Abelian groups. PONTRYAGIN 4 exploited countable direct
products heavily, and in 6, §§ 20 and 2t, treated special cases of
finite and infinite direct products. The construction of arbitrary direct
products appears in WElL 4, § 4. Theorems (6.tt) and (6.t2) are
from PONTRYAGIN 7, § 2t. Projective limits are defined in WEIL 4,
§ 5, where one will also find remarks on the development of the concept.
Example (6.t 7.d) corrects a mis-statement in FREUDENTHAL t, 7.
Local direct products of topological groups play an important r61e
in the constructions of ]3RACONNIER t . They have also been treated by
GRAEV t I and very extensively by VILENKIN 7. For further references
to VILENKIN'S use of local direct products, see the notes to §25.
§ 7. Properties of topological groups involving connectedness
In this section, we take up some elementary properties of topological
groups that depend upon connectedness or disconnectedness of the group
considered as a topological space. Since connectedness is a purely
topological property, the results of this section have no analogue in the
theory of groups without topology.
We begin with three theorems dealing with the component of the
identity element.
(7.1) Theorem. Let G be a topological group and let C be the com-
ponent o/ the identity e. Then C is a closed normal subgroup o/G.
Proof. Since inversion is a homeomorphism of G, C -1 is a connected
set containing e, and hence C-1C C. If a is any point of C, then also
a-cC. Therefore a C is a connected set containing e, and so a Cc C.
Thus C2C C, so that C is a subgroup of G. If a cG, then a Ca - is a
connected set containing e _x-->axa - is a homeomorphism of G, so
that a Ca-c C. Thus C is a normal subgroup of G. Like all components
in topological spaces, C is closed.
(7.2) Theorem. Let G be a topological group and let C be the com-
ponent o/ the identity in G. Then/or all a G, a C ---- C a is the component o[ a.
Proof. The mapping x--*ax is a homeomorphism of G, and C is a
normal subgroup of G.
(7.3) Theorem. Let G be a topological group and let C be the compo-
nent o[ the identity in G. Then G/C is a totally disconnected Hausdor[/ group.
Proof. In view of (7.2) and (5.26), we need only show that the com-
ponent of the identity C of G/C is C itself. Let {xC:x X} be any subset
of G/C properly containing {C}. We will shov that {xC:xX} is dis-
connected in G/C. Let 9 be the natural mapping of G onto G/C, and let A be
any subset of G. It is easyto verify that 9(A fl (XC))--q(A)fl {xC: xX}.
The set X C properly contains C, and so is disconnected:
§ 7. Properties of topological groups involving connectedness 61
(v n (x c)) u (v n (x c)), (u n (x c)) N (V N (X C)) =0, neither set
is void, and U and V are open in G. Thus
{xC" x;X}--(q(U) n {xC" xcX})U (9(V)n{xc.xX}),
where (U) and (V) are open in G/C, as is an open mapping. For
x X, we have x C-- (U N (x C)) U (V N (x C)). Thus since x C is connected,
either x C c U N (X C) or x C c V N (X C). Consequently, U fl (X C) and
VN (X C) are unions of cosets of C, and so they have disjoint images
under q. Thus (q(U)N{xC'xX})N(q(V)N{xC'xX})--_O, so that
{xC:xX} is disconnected.
The following is a simple but frequently useful fact.
(7.4) Theorem. Let G be a topological group, C the component o/e,
and U any neighborhood o/ e. Then C C U U'*; in particular, i[ G is
connected, then G-- U U '.
Proof. Let V be a symmetric neighborhood of e such that VC U.
Then U V" is open and closed, by (.7). Since C is connected, we have
Cc U Vc U U". F1
n=l n=l
The two following theorems show how to construct compact, open
subgroups in certain topological groups.
(7.5) Theorem. Let G be a topological group and let U be a compact
neighborhood o/ the identity e. Then U contains a subgroup H o] G that is
compact, open, and closed.
Proof. Since UC U, U is compact, and U is open, we may apply
(4.0) and (4.6) to obtain a symmetric neighborhood V of e such that
UVCU. We have V----eVCUVCU and hence V CUVCU. By
finite induction, we have V _-- V -1VZ U V c U for n = 3,4 ..... We
now set H= U V and apply (5.7).
(7.6) Theorem. Let G be a compact group and let U be a closed
neighborhood o[ the identity e. Then U contains open and closed normal
subgroup N o/G. The group GIN is finite.
Proof. By (7.5), U contains a compact, open, and closed subgroup H.
Let N xH x -. It is obvious that N is a closed normal subgroup of G
contained in U. By (4.9), there is a neighborhood V of e such that
x-Vx C H for all x G. That is, V C xH x - for all x< G, so that V C N.
This shows that N is also open (5.5). Since N is open, GIN is discrete
Recall our convention that neighborhoods are open sets (page 9).
62 Chapter II. Elements of the theory of topological groups § 7. Properties of topological groups involving connectedness 63
(5.2t), and since G is compact, GIN is compact (5.22). Hence GIN is
finite.
Theorems (7.5) and (7.6) show that certain locally compact groups
contain arbitrarily small compact and open subgroups 1.
(7.7) Theorem. Let G be a totally disconnected or o-dimensional
group that is locally compact Vcompact. Then every neighborhood o] the
identity e contains a compact open compact open normal subgroup.
Proof. If G is totally disconnected, it is T 1 and hence Hausdorff
(4.8). Therefore G is 0-dimensional, by (3.5). If U is any neighborhood
of e, then U contains an open and closed neighborhood, which is obviously
compact if U-is compact. Now apply (7.5) and (7.6).
Theorem (7.7) yields a new description of the component of the identity
in locally compact groups.
(7.8) Theorem. Let G be a locally compact group and let C be the
component o] the identity in G. Then C is the intersection o] all open subgroups
o! G.
Proof. Every open subgroup is dosed. Since it also contains e,
it must contain the connected component C of e. Thus C is contained in
the intersection of all open subgroups of G. Now let x be any dement of G
not in C. Consider the quotient group G/C, which is totally disconnected
(7.3) and locally compact (5.22). By (7.7), there is a compact open
subgroup {uC" uc U} of G/C that does not contain the element xC of G/C.
We may take U to be a neighborhood of e in G. Then UC is an open
subgroup of G not containing x.
(7.9) Corollary. The ]ollowing assertions about a locally compact
group G are equivalent:
(i) G is connected;
(ii) G has no proper open subgroups;
(iii) ]or every neighborhood U o] e, we have U U = G.
Proof. This is obvious from (7.8) and (7.4).
(7.10) Theorem. Let G be a locally compact group and let C be the
component o] e in G. I] G/C is compactly generated, then G is compactly
generated.
Proof. Since C is closed in G, C is locally compact. It follows
immediately from (7.4) that any locally compact connected group is
compactly generated. In particular, C is compactly generated" hence G
is compactly generated by (5.39.i).
x Ve say that a topological group contains arbitrarily small subsets of a given
type if every neighborhood of the identity contains such a subset.
(7.11) Theorem. Let G be a locally compact O-dimensional group
and let H be a closed subgroup o] G. Then G/H is O-dimensional.
Proof. Let U be a neighborhood of e in G. By (7.7), U contains a
compact open subgroup L of G. The set {xH" xcL) is open in G/H by
(5.t 5) and closed in G/H because it is a compact subset of the Hausdorff
space G/H (.2t). Thus H in G/H has arbitrarily small open and closed
neighborhoods. Since G/H is homogeneous, the assertion follows.
(7.12) Theorem. Let G be a locally compact group and let C be the
component o] the identity e in G. Let ] be an open, continuous homomorphism
o] G onto a T o group G, and let C be the component o] the identity in G.
Then ] (C)--- C.
Proof. Since ](C)- is connected, we have ](C)-cC. Clearly ](C)-
is a closed normal subgroup of , and n----]-1 (] (C)-) is a closed normal
subgroup of G containing C. By (.34), G/](C) is topologically iso-
morphic with G/H. By (.3), G/H is also topologically isomorphic
with (G/C)/(H/C). Since G/C is totally disconnected (7.3), Hausdorff
(7.3), and locally compact (.22), G/c is 0-dimensional (3.). It is easily
verified that H/C is a closed normal subgroup of G/C. Hence by
(G/C)/(H/C) is 0-dimensional and Hausdorff, hence totally disconnected.
That is, G/](C)-is totally disconnected. As a continuous image of the
connected set C, ] (C)-" ) is connected in /] (C)- Hence we have
([(C)-'--(](C)-), and thusc/(C)- ]
(7.13) Corollary. Let G be a locally compact group and let C be the com-
ponent o] the identity e in G. Let ] be an open, continuous homomorphism o[ G
onto a T O group G and let C be the component o] the identity in G.
1-1 () C C or C c/-1 (), then /(C)---- .
Proof. If /-1()C or C/-1(), then /-1().C is equal to C or
/-(), respectively, and hence is a closed subgroup of G in either case.
Since /- (/(C)) =/- (). C, we have / (C)'= / ((/- (). C)') so that / (C)'
is open in . Thus/(C) is closed in , and (7.t2) implies that/(C)---- .
In (t0.t4) we will give an example showing that the hypotheses of
(7.t3) are needed. We next give a result on connectedness reminiscent
of (5.25).
(7.14) Theorem. Let G be a topological group and let H be a subgroup
o/ G. I/H and G/H are connected, then G is connected.
Proof. Assume that G= UU V where U and V are disjoint nonvoid
open sets. Since H is connected, each coset of H is either a subset of U
or a subset of V. Thus the relation G/H= {xH" x H C U} U {x tt" x H
64 Chapter II. Elements of the theory of topological groups § 7. Properties of topological groups involving connectedness 65
{xH: x U} U {xH: x C V} expresses G/H as the union of disjoint nonvoid
open sets. This contradicts the hypothesis that G/H is connected.
The following fact about unitary groups 1I (n) will be useful later on.
(7.15) Theorem. For every positive integer , the unitary group 1I (n)
is arcwise connected.
Proof. As pointed out in (4.25), 1I (n) is locally Euclidean and hence
obviously locally arcwise connected. Since any connected, locally arcwise
connected space is arcwise connected, we need only prove that ll (n) is
connected. Let consist of all the diagonal matrices in la(n); thus
(ajk)i k;, belongs to if and only if I a, I --d. for all ] and k. It is evident
that is homeomorphic with T" and so is connected. Thus we have
C 62, where 62 denotes the connected component of the identity in 11 (n).
It is well known that every unitary matrix A can be diagonalized by a
unitary matrix P; see for example S. MACLANE and G. BIRKHOFF: Alge-
bra (New York: The Macmillan Co., t967), page 4t3, Theorem 2C. That
is, for any A in 1I (n), there exists an element P in 11 (n) such that p-lAp
belongs to , and hence to . Since 62 is a normal subgroup (7.t), it fol-
lows that A belongs to 62. We have shown that lI(n)= and therefore
Lt (n) is connected.
Miscellaneous theorems and examples
(7.16) (a) Let G be a connected [locally connected] group and H a
subgroup of G. Then G/H is connected [locally connected 1.
(b) (SCHREIE [t].) The component of the identity in a normal
subgroup is itself a normal subgroup. [Let C be the component of the
identity e in the normal subgroup H of G. For x G, xC x - is connected
and xC x - xH x - H since H is normal in G. Thus xC x - C.]
(7.17) The center of a connected group. (a) Every discrete
normal subgroup of a connected topological group G is contained in the
center of G. [See (7.4) and (5.37.f).]
(b) (HOFMANN Ill.) Let G be a connected T O group and H a totally
disconnected normal subgroup. Then H- is contained in the center of G.
[Fix an element a H. Then the mapping x-- xa x - carries G continuously
into H. The image of G is connected and hence is a single point. Thus
A related but much less trivial result along these lines appears in
(26.0).
(7.18) Dimension zero and total disconnectedness. (a) (ERI)6S
t.) In the space l,. with its usual metric topology, consider the set E of
all points (x, x2 ..... x,, .... ) all of whose coordinates are real and
rational. This is evidently a subgroup of the additive group 12. The
group E is obviously totally disconnected. [In fact, for every pair of
distinct points e and ?/in E, there is an open and closed set U such that
eU and ?/6 U'.] However, E is not 0-dimensional. [Let V be any
neighborhood of 0= (0, 0, ..., 0,...)in E such that ]1]= x, < t for
all V. For vl and a subset B of l 2, we write d(v, B) =in{ll--Vl
B}. Consider the set AI= {x 1 Q'(Xl, o ..... o,...) v}. Clearly A 1 is
nonvoid, open, and bounded, and contains 0. There is therefore an alA1
such that p(1)= (al, 0 ..... 0 .... ) V and d (1), V') < el, where el is a fixed
but arbitrary positive number. Let An= {x2 Q" (al, x2, 0, ..., 0, ...) V}.
Clearly A is nonvoid, open, and bounded, and contains 0. Thus there is
an a2 A 2 such that p(=)= (fi, a, 0 .... ,0 .... ) V and d (2), V') <
being a fixed but arbitrary positive number. Continuing by induc-
tion, we find a sequence (a,,l of rational numbers such that
(al, an, ..., an, 0, 0, ...) V for all n and d(p(", V')< e. Now set
=< t, that belongs to
(a, a= .... , a,.... ). It is obvious that
and that if e,0, belongs to V'--- V'. Thus V is not closed.
(b) For a topological oup G with component of the identity C, G/C
need not be 0-dimensional. [This is shown by the oup E of (a).]
(c) A continuous isomohic image of a non 0-dimensional group may
be 0-dimensional. [The group E of (a) can be imbedded in Q. with its
direct product topology and addition. Plainly Q. is 0"dimensional and
hence E C Q' is also 0-dimensional. The topolo of E as a subspace of
Qe0 is obviously strictly weaker than its topolo as a subspace of
(d) A finite oup with a nondiscrete topology under which it is a
topological oup is 0-dimensional and compact but not totally dis-
connected. There are compact groups of arbitrarily large cardinal
number having the same property.
(7.19) Comments on theorem (7.7). (a) Let G be locally com-
pact and totally disconnected, and suppose that the uniform structure
9°,(G) is equivalent to the uniform structure 9° (G). Then every neigh-
borhood of e contains a compact open normal subgroup. In view of
(4.t 7), this result generalizes (7.7). [Let U be an arbitrary neighborhood
of e and let H be an open compact subgroup of G contained in U. Let V
be a neighborhood of e such that x - VxH for all xG (4.t4.g). Then
V x Hx-, so that CI x Hx - is a compact open normal subgroup
xG xG
of G contained in H.]
Hewitt and Ross, Abstract harmonic analysis, vol. I 5
66 Chapter II. Elements of the theory of topological groups § 8. Invariant pseudo-metrics and separation axioms 67
(b) Not all 0-dimensional locally compact groups have arbitrarily
small open normal subgroups; such a group may indeed have no compact
open normal subgroups. The following example is adapted from MOnT-
GOMERY and ZIPII [t ], p. 57. Let G be any group containing an element
a not equal to e, and consider the direct product PczG, where each G
is G. Write the elements of this product as m----m (n), where m is a function
on Z with values in G. Let H be the subgroup of the product group
consisting of all m such that m (n)= e for all n_<_ n o cZ [n o depends upon m].
Let T be the operator on H defined by (Tin)(n)--m(n + t). Now let S
be the set of all ordered pairs (m, m), where incH and mZ. For (m, m)
and (/, n) in S, let (m, m)@/, n)=((T't),rn+n), where T°m=m. It is
easy to verify that S is a group under this operation. [Actually S is a
semidirect product of H and Z, as defined in (2.6).] The identity of S
is (e, 0), where e(n)----e for all nZ. For nonnegative integers m, let U m be
the set of all (m, 0) in S such that m (n)----e for n__< m. The family of all
sets Um satisfies (4.5.i)--(4.5.v) and hence defines a topology on S making
S into a T o topological group.
Each U is topologically isomorphic with P G where each G= G
--tn+l '
has the discrete topology. Therefore S is locally compact if G is finite.
If G is 0-dimensional, then S is 0-dimensional. Suppose that N is an
open normal subgroup of S. Then there is a neighborhood U m contained
in N. Denoting by /that point of H for which y (n)----e for n m+t
and t(m+t)=a:e, ve see that @/, 0)EUcN. Since N is normal, all
elements (e, k) @/, 0) (e, k)-l-- -(TB, 0) lie in N for kEZ. For k=
m + 2, m + 3, ..., the neighborhoods (T*/, 0) • U 0 are pairwise disjoint.
0 oo
It follows that the net {(T'y, )},=,+. has no cluster point. Thus N is
noncompact.]
(c) Local compactness is necessary for the validity of (7.7). The
group Q is 0-dimensional but contains no bounded open subgroups.
The group Q also shows that (7.8) and (7.9) may fail in the absence of
local compactness.
(7.20) (HARTMAN and MYClELSKI [I].) Every T O group G is a
closed subgroup of an arcwise connected and locally arcwise connected
group G. [Consider the set G of all functions ] on [0, t [ with values in G
for which there is a sequence 0:a0< a< a.< ... < a:t such that [ is
constant on [a, a+l[ (k: 0, t .... , n-- t). With ]g (t) : ] (t) g (t) and
- (t) : (! (t))- (0_--< t < ), U is obviously a group. The function identically
equal to e is the identity of G. For e> 0 and a neighborhood V of e in G,
let U(V,e) be all /cG such that ({t[0, t['/(t)ўV})<e. Here . is
Lebesgue measure on [0, t [. Properties (4.5.i) -- (4.5.v) are easy to verify.
Hence G is a topological group. The constant functions form a closed
subgroup of G topologically isomorphic with G. If ! and g are elements
of G , and 0 _--< t, let h (t) :/' (t) for 0 t < and h (t) : g (t) for t < t.
Then ho:g, hi:l, and the mapping -+h is continuous. Moreover, if
[ and g lie in (V,-), then all of the h lie in (V, e).]
!
(7.21) (D. VAN DANTZlG [3].) Let G be a topological group andA
a connected subset of G such that eA. Then the smallest subgroup of G
containing A is also connected. [For each aA, Aa -1 is connected and
contains e. Hence the set AA-I: U Aa -1 is connected. Similarly the
sets (AA-1) and U (AA-1) are connected.]
Notes
The special properties of connected groups were noticed by the first
writers on topological groups" SCHREIER [t] proved (7.t) and LEJA [t]
(7.4). VAN DANTZlG [3] also proved (7.t). The first statement of (7.7)
appears to be in VAN DANTZlG [2]. It also is proved by VAN KAMPEN [t ].
Our proofs of (7.t4)and (7.t 5)are taken from CHEVALLEY [t], pp. 36--37.
§ 8. Invariant pseudo-metrics and separation axioms
In this section, we show that the topology of a topological group can
be completely described by means of a family of left invariant pseudo-
metrics and that every T O topological group is completely regular. We
also explore other separation properties for topological groups.
(8.1) Definition. A metric or pseudo-metric d on a group G is
said to be le]t invariant if d(ax, ay)=d(x, y) for all a, x, yG. If
d(xa, ya)=d(x, y) for all a, x, ycG, then d is said to be right invariant.
If d is both left and right invariant, it is said to be two-sided invariant.
Similarly, if H is any subgroup of G, then a metric or pseudo-metric d
on G/H is said to be le]t invariant if d(axH, ayH)=d(xH, yH) for all
a, x, yG.
Our first theorem is rather technical, and its proof may not be
obvious at first glance. It is, however, the kev to the metrizability
.theorems for topological groups [(8.3), (8.5), and (86)], and it immediately
Implies complete regularity for T o groups (8.4).
5*
68 Chapter II. Elements of the theory of topological groups
(8.2) Theorem. Let (U/}°__l be a sequence o/symmetric neighborhoods
o/ e in a topological group G such that U+I c U /or k:t, 2 ..... Let
H = ( U. Then there is a left invariant pseudo-metric a on G such that"
(i) a is uni/ormly continuous/or the left uni/orm structure o/GxG;
(ii) a(x, y):0 i/and only i/y-lxH;
(iii) (x, y)__< 2 -+ whenever y-lxU;
(iv) 2 - <= a (x, y) whenever y-1 x ў U.
I/, in addition, x Ux-l= U/or all xG and k= t, 2 ..... then a is also
right invariant and
() (x-l, y-):(x, y)/or x, y.
Proof. It is convenient to rename the sets U. Let V,_,--U for
k--t, 2 ..... We now define sets V for all dyadic rational numbers
r, 0 < r < t, as follows. For
r-=2-h +2-'+ "'" +2 -', 0</1< "" <ln, (!)
let
V, -- V-,1V-,, • • • V.-. (2)
We also define V,-- G for all dyadic rational numbers r => t. We first show
that
r < s implies V, c V. (3)
We may suppose that s < t, since the inclusion is obvious for s>=t. Let r
be given by (l) and s by 2 -=,+... +2-"%0<m<m 2<... <rn#.
There exists a unique integer k such that li--n for f<k and l,>m.
Letting W= V._,V,_, ... V__, we then have
v, = w v_, v_,+ v_,+. . . . v_,
c w v,_,,v,_,,_ v,_,,_.. ...
wv,_,,v,._,, wv,_,,+ wv,_.,-- v,_,.v,_,....
c v,_.. v,_.. . . . v,_.,_ , v,_., v,_.,+ , ...
We next show that for every r of the form (I) and every positive
integer I, we have
v,v,.,cv,+,.,+,.. (4)
Since (4) is obvious if r-+- 2-*+__> t, we suppose that r-+- 2-*+<t. If
l> l,,, then
v,v,_,=V,+,_, (5)
§ 8. Invariant pseudo-metrics and separation axioms
69
and (4) is again obvious. If l <= l, let k be the positive integer such that
l_l<l<=l [define/0=0. Let rl-2-*+-2-*-2-*+ ...... 2 - and
r.-r + rl. Then it is evident that r < r2< r + 2 -*+1. Applying (5) and (3)
to r2, we obtain
This proves (4).
For xG, let 0(x)=inf {r:xV,}. It is obvious that 0(x)=0 if and
only if xH. Finally, we define the function a on GxG by the following
rule:
a(x, y)=sup {1 q(zx)--q(zy)[ : zG}.
It is obvious that a(x, y)=a(y, x) and that a(x, x)---O for x, yG.
It is easyto verify that a(x, z)<=a(x, y)+a(y, z) and that a(ax, ay)---
a (x, y) for a, x, y, z G. Therefore a is a left invariant pseudo-metric on G.
To establish (iii), suppose that 1 is any positive integer, that u V._,
and that zG. If zV,, then by (4), we have zuV,+_ч.. Hence
9 (z u) 9 (z) + 2 -+. Similarly, if z u V,, then z V, V- V,+ 2_ч and
hence 9 (z) q (z u) + 2 -+. Thus [ q (z) -- 9 (z u)] 2 -+ for u V2_, an d
zG. Consequently, we have a(u,e)<=2 -+ for uV._. Since a is left
invariant, we conclude that a(x, y)_<2 -*+ whenever y-xV._----U.
Thus (iii) holds.
We now prove (i). Consider (x, y) and (x, y) in G x G. Using the left
invariance of a and the triangle inequality, we obtain
] a ( x,Y) --a ( xl,Yl) [ -- [ a( x-lx,x- l y) -- a ( xl y,e)-j-a ( e, y-lxl) -- o' (y-11;1,y-1yl)[
[ a(xi 1 x, x-Xy) --a(x-Xy, e)] -J- [ o'(e, y-X Xl ) _a(y-X Xx ' y-Xyx) [
< [ cr(x -1 x, e)[ -J-] o'(e, y-Xyx) [ .
This inequality and (iii) show that [ ў(x, y) --O'(Xl, y)[ _2 -+3 whenever
x;lx and y7ly are in U. This proves (i).
We next prove (iv). Suppose that y-1 x ў U= V._. Then 0 (y- x) _>_ 2 -
and so ў(x, y)=a(y-lx, e)>=[9(ey-lx)--(ee)[=9(y-x)_>_2 -. This
proves (iv). Assertion (ii) follows immediately from (iii) and (iv).
Finally, suppose that x U x -1= U for x G and k =t, 2 ..... Then
xV, x-=V, for all dyadic rational numbers r>O, and hence (xyx-1)=
(y) for all x, y G. Thus for a, x,. yG, we have
-.up {I q,(,zx)-q,(,zy)l .z}
=up {[(zx)-(zy)l.z}=(x, y);
70 Chapter II. Elements of the theory of topological groups §8. Invariant pseudo-metrics and separation axioms 7t
and so a is right invariant. Also for x, y C G, we have
(x-, -)--(, -x)-(, x)--(x, ).
(8.3) Theorem. Let G be a T o topological group. Then G is metrizable
i] and only i] there is a countable open basis at e. In this case, the metric can
be taken to be le]t invariant.
Proof. If G is metrizable, then plainly there is a countable open
basis at e. Suppose now that {V}°__I is a countable open basis at e. Let
U--V Vf . If U, ..., U_ have been defined, let U be a symmetric
neighborhood of e such that U c U ffl ... ffl U_ 1 ffl V and U c U_I.
Such a set U exists by (4.5). The family {U}°__I satisfies the hypotheses
of (8.2) and H-- if) U----{e} by (4.8). Let a be the left invariant pseudo-
' /=1
metric as in (8.2) for this family {U}°=I. Clearly a is a true metric; that
is, a (x, y) --0 if and only if x---- y. From (8.2.iii) and (8.2.iv), it follows that
{xcG:a(x, e)<2-k}cUc {xG:a(x, e)_<_2 -k+} for k----t, :2 ..... This
implies immediately that the topology defined by a coincides with the
given topology of G.
(8.4) Theorem. Let G be a topological group, a an element o/G, and
F a closed subset o] G not containing a. Then there is a continuous real
[unction v on G such that (a)--0 and (x)----t ]or each xF. Thus a T o
group is completely regular.
Proof. Let U be a symmetric neighborhood of e such that (a Ua)
Choose neighborhoods U., U 3, ... of e such that {U}°__I satisfies the
hypotheses of (8.2). Such a sequence exists by (4.5). Let a be a left
invariant pseudo-metric constructed for {U}°__ as in (8.2). For x6G, let
(x) = min {t, 2a (a, x)}. Then is continuous by (8.2.i) and clearly
V2(a)=0. If xF, we have a-xCU, and hence a(a, x)_>--2 - by (8.2.iv).
Consequently, (x)--t for all x6F.
(8.5) Theorem. A locally countably compact group G admits a left
invariant metric compatible with its topology i/and only i/{e} is the inter-
section o/ a countable ]amily o/ open sets.
Proof. If G admits a metric compatible with its topology, then
clearly {e} is the intersection of a countable family of open sets. Suppose
that {e}--,71__ 1U,, where each U, is open in G. Let V-- U1. By finite induc-
tion, we choose a sequence {V}__. of neighborhoods of e such that
V,- c V_I N U and V,- is countably compact, n-- 2, 3, ..- • To show that
{V}°°__ is a basis at e, let W be an arbitrary neighborhood of e. If the
inclusion V c W fails for all n, then the sequence of countably corn-
We) oo
pact sets {V-CI .t=. has the finite intersection property. Hence
f (V VI W') is nonvoid, contradicting the relation
n (v.-nw') = v.- nw'c u. nw'=(}nw'=e.
n----2 2
Consequently, G has a countable open basis at e and hence by (8.3)
admits a left invariant metric compatible with its topology.
(8.6) Theorem. Let G be a compact group such that {e} is the inter-
section o] a countable ]amily o] open sets. Then G admits a two-sided
invariant metric yielding the topology o] G.
Proof. By (8.5), G admits a left invariant metric compatible with
its topology. For x, y G, let
e(x, y)=p ((, y)).
Since G is compact, a is bounded and hence O(x, y) is finite for all
x, yc G. It is routine to verify that 0 is a two-sided invariant metric on G.
Let e be an rbitrary positive number. By virtue of (4.9), there is a
> 0 such that
for all z6G. Thus if xG and a(x, e)< , then a(z-xz, e)--a(xz, ez)< e
for all z6G and therefore 0(x, e)_<_ e. That is, {xG'a(x, e)< O)c {xcG"
(x, e)_<_e). This and the obvious inclusion {x6G'o(x, e)< e)c {xG"
a (x, e)< e} show that the topologies defined by a and 0 coincide.
(8.7) Theorem. Let G be a a-compact locally compact group with iden-
tity e. Then ]or every countable ]amily {U..°__ o] neighborhoods o] e, there
is a compact normal subgroup N o] G such that N
metrizable and has a countable basis ]or its open sets.
Proof. We can write G = U F,, where {F.,°°__l is an increasing se-
quence of compact subsets of G. Let V be a neighborhood of e such
that is compact. Using (4.5) and (4.9), we construct a sequence
{V},= of symmetric neighborhoods of e such that V V,_ U,, and
such that xVx-1 V,_ for all F, ( =t,2 .... ). As in the proof of (4.7),
it follows that V-V_ (=t, 2 .... ). Let N= V. Obviously
N ,;C U, and thus, by Theorem (5.6) applied to the family {V'n =
0,t,2 .... }, N is a closed normal subgroup of G. Since NC V-, N is
also compact.
72 Chapter II. Elements of the theory of topological groups § 8. Invariant pseudo-metrics and separation axioms 73
Let 9 denote the natural mapping of G onto GIN. We now show that
{9(V,))=1 is a basis at N in GIN. Let wN'w6W) be an arbitrary
neighborhood of N in GIN. For some no, we have V, o C W N. For in the
contrary case, the family {V-N (WN)'),=I of compact sets has the finite
intersection property and hence (V (WN)') is nonvoid. This is
n=l
impossible since (V n (WN)') = V n (WN)'= V n (WN)'
=1 --1 n=l
= N (WN)'= . Since V0 C WN, we have 9 (V0) C {w N'w W).
Since GIN has a countable open basis at N, GIN is metrizable (8.3).
Since GIN = 0 9(F), the group GIN is -compact, and hence Linde-
16f. As a metric space satisfying the Lindel6f property, GIN has a
countable basis for its open sets.
While every T O group is completely regular (8.4), there are many
nonnormal T O groups. We shall present two constructions of such
groups. The first of them (8.8) and (8.t0) is very general. The second,
Theorem (8.12), is rather special, but shows that nonnormal groups
exist in profusion. Our first construction, given in (8.8), bears a certain
family resemblance to the well-known Stone-Cech compactification of a
completely regular space.
(8.8) Theorem. For every completely regular space X, there exists a
topological group F with the/ollowing properties:
(i) X is a closed subspace o/F;
(ii) algebraically, F is the/ree group generated by X;
(iii)/or any continuous mapping 9 o/X into any topological group G,
there exists a continuous homomorphism o/F into G such that
for all x X. 1
Proof. Let f be a family of topological groups such that
:<max(X, c) for G6 f, distinct members of f are not topologically
isomorphic, and every topological group H for which H _<_ max (X, c)
is topologically isomorphic with some G in f. Let {(G,, %)}m consist of
all pairs (G,, 9,) where G, and 9, is a continuous mapping of X into G,.
If H is a topological group, H = max (X, c), and 9 is a continuous mapping
of X into H, then there is an *o I and a topological isomorphism z of G,o
onto H such that z o %0:9" In such a case, we will identify the pair
(H, 9) with the pair (G,o, %0)"
1 The group F is called the/ree topological group generated by X.
Let Go:,PIG ,, and let e denote the identity of G o. For xX, let
v (x) Go be defined by v (x),: 9, (x). The function v is easily shown to be
a homeomorphism of X onto v (X). [Note that X can be mapped homeo-
morphically into a product of real lines or apply KELLEY 2], p. 1 t6,
Lemma 5.] Henceforth we identify X with v(X); that is, X is regarded
as a subset of G o. Now let F be the subgroup of G o generated by X.
Let 1 be any positive integer. For each permutation P6®, let
Up Lt(l) be the matrix (U.k),k= 1 such that ujk:t if/': P(k) and u.:0
if/" P (k). It is obvious that the mapping P-- U is an isomorphism of
® into Lt (1).
We next verify (ii). Let xx .... x: be any reduced word formed
from elements of X. By (2.9) and the preceding paragraph, there is a
mapping A of the set not sequence l] {Xl ..... x, into Lt(1), where
1: n + or n + 2, such that A (Xl)tt A (x.) ..... A (x,) * E; here E is the
lxl identity matrix. By (7.t 5) and (3.6), there is a continuous mapping
v of X into Lt (1) such that 9 (x) : A (x) for k:t .... , n. For some *o I,
the pair (Lt (l), 9)is (G,0, %°). Hence
(xx. .... x),o:A(Xl)*A(x.) * .... A(x,)* E,
so that x x. .... x, e. That is, F is the free group generated by X.
A similar argument proves (i). Consider any element of F NX'. It
* .... x, where Jў> t or 1 :-- t 8'-- 4- t and
can be written as xx.- , , ,
e:e+l whenever x.: x.+. Applying (2.9) note in particular (2.9.iii)]
and the isomorphism of ® into Lt (1) described in the last paragraph but
one [l:n+ or n+2], we obtain matrices A(Xl), ...,A(x,)6Lt(1) such
that B:A(Xl)*IA(x.)* .... A(x) * is distinct from each A(x). Consider
a neighborhood of B in Lt (1) such that -f ' is homeomorphic with
a sphere S,_ 1(4.25.b) and such that A(Xl) ..... A(x,) lie in -'. It is
easy to see that 1I (l) f ' is arcwise connected. Thus we may apply (3.6)
to find a continuous mapping of X into 1I (1) f ' such that (xk) : A (x)
for k-- .... ,n. The pair (1I (l), ) is a pair (G,0, %°). The neighborhood
in F of xx. .... x: consisting of all (y,)F for which y,o plainly
excludes all points of X. Hence FX' is open, i.e., X is closed in F.
Property (iii) is almost trivial. If 9 is a continuous mapping of X
into any topological group H, the image 9 (X) in H is contained in a sub-
group j of H such that =< max (, c). Thus (J, 9) is a pair (G,o, %0)"
Then 9 is merely projection onto the ,o-th axis in Fc,PIG,. As such,
9 can obviously be extended to a continuous homomorphism still a
projection] of F into J:H. [
74 Chapter II. Elements of the theory of topological groups § 8. Invariant pseudo-metrics and separation axioms 75
The theory of free topological groups has been extensively developed
see for example GRAEV 4. We cannot set forth all of it here. We
mention, however, the following fact.
(8.9) Theorem. Let X be a completely regular space, F the topological
group constructed in (8.8), and F any topological group such that:
(i) X is a subspace o[ F;
(ii) the smallest closed subgroup o] F that contains X is F itsel];
(iii) ]or every continuous mapping 9 o] X into a topological group G,
there is a continuous homomorphism q5 carrying into G such that
#(x)-9(x ) ]or aII xX.
Then there is a topological isomorphism z o] F onto F such that z(x)--x [or
all x X.
Proof. In view of (8.8.iii) and (iii), there are a continuous homo-
morphism # of F into and a continuous homomorphism of into F
such that qS(x)=#(x)--x for all xcX. The mapping #o# is a con-
tinuous homomorphism of F into itself that is the identity mapping on X.
In view of (8.8.ii), o # is the identity mapping of F onto itself. Similarly,
# o is the identity mapping on the subgroup of generated algebrai-
cally by X. Since this subgroup is dense in and # o is continuous, we
see that qgo is the identity mapping on . Thus _#-1 and we may
take z= #.
(8.10) Theorem. There exist nonnormal T o groups.
Proof. Consider any nonnormal completely regular space X. As X
is closed in its free topological group F, and every closed subspace of a
normal space is normal, F cannot be a normal space. 0
We will now show that some fairly simple groups are nonnormal.
(8.11) Theorem. I] m is any uncountable cardinal number, then Z "t
is a nonnormal completely regular topological group.
Proof. Since Z r is a T o group, it is completely regular (8.4). Write
Z m as ,P Z,, vhere --m and each Z, is Z. To show that Z m is not
normal, let
A={(x,)cZ m" for n:0, there is at most one index, for which x,--n,
and
B--{(x,)Z m" for nl, there is at most one index, for which x,--n}.
If (x,) CA, then we have x,0 -- x,1 = n for some n cZ, n 4: 0, and some
*o, '1 I, *o 4:,1. Hence {(y,) cZ m" y,0 ---- y,, = n} is an open set containing
(x,) that is disjoint from A. Thus A is a closed set; similarly B is closed.
Obviously A N B-- . Let U and V be open sets in Z m such that U A
and V B. To prove that Z m is nonnormal, it suffices to prove that
UVd:.
Let (xl))Z m be defined by xl)=0 for all I. Since (xl))A c U,
there exist distinct '1 .... , , I such that
(xl))C{(x,)Zm'x,,=O for k----t, ..., ma)C U.
Let (xl))Z m be defined by x*)=k if =, (t<=k<=mx) and x*)=0
otherwise. Since (xl))6A c U, there exist *,+1, ..., ,,6I, distinct from
each other and from '1,-.., *., such that
(x *)) C {(x,)Z m" x,,-- k for k--t ..... m a and
x,=0 for k=m a + t, ..., m2} U.
Continuing in this manner, we define by finite induction a sequence
((xl")))=l of elements in Z m, a sequence (t}°=l of indices from I, and a
strictly increasing sequence of integers (m,,},,°°= 1 such that xl ")- k if t--t
(t _<_ k_<_ m,_l) and xl ")--0 otherwise and such that
(xl"))6{(x,)6Zn'x,,=k for k----t ..... m,,_ 1 and
x,----0 for k=m, 1 + I ..... m,a U.
We now define (y,)Z m as follows. Let y,=k if t--t (k--t, 2 .... ) and
y,--t otherwise. Clearly (y,)c B. Hence for some finite subset J of I,
we have
, J)c v.
Clearly there exists an n o such that ў J whenever k> m,,o. Finally, we
define (z,)6Z m by the rule"
z,=k if -- and
z,=O if -- and m,o+
z,-- t otherwise.
Then (z,)((x,)Zm.x,=y, for J)c V and
(z,)c((x,)Zm'x,--k for k--t .... , m0 and
x,,=O for k=m,,+ t ..... m,,o+l}c U.
That is, UV3.
(8.12) Theorem. I/ the product P G o/ T o topological groups G,
is normal, then all but at most a countable number o/ the groups G, are
countably compact.
76 Chapter II. Elements of the theory of topological groups §8. Invariant pseudo-metrics and separation axioms 77
Proof. Suppose that G, is not countably compact for
where J is uncountable. Then for ,E J, there exists a countable closed
discrete subset Z, of G,. _Choose a countable open covering {U 1, U., ...}
of G, with no finite subcovering, let x,6(U1O ... U U,)', and Z,--
{xl, x., .. }.] Then P Z x Pj{e,} is a closed subset of,PxG homeomorphic
with Z:. By (8.1t), it is nonnormal. It follows that ,PzG,is nonnormal.
The reader has undoubtedly already observed that (8.11) and (8. t 2)
are purely topological" no group structure is needed. Theorem (8.12) is
true for products of TI spaces.]
In view of Theorems (8.t0) and (8.t2), it is interesting to note that a
large class of topological groups are normal.
(8.13) Theorem. Every locally compact T O group is paracompact
and hence normal.
Proof. Let G be a locally compact T o group, and let U be a symmetric
neighborhood of the identity in G such that U-is compact. Let L-- tA U ""
t=l
then L is an open and closed subgroup of G, by (.7). Since U-C U , L
is also a-compact and is hence Lindel6f. Let / be any open covering
fLr(n)oo
of G. For each coset x L of L, there is a countable subfamily , xr s--1
of / such that xLc 0 V ). For n:t,2 ..... let #",:{V(fq(xL) •
n=l
x L G/L). Then : tA ў/', is clearly an open -locally finite refinement
n=l
of the covering /. Thus G is paracompact and hence normal (3.8).
Miscellaneous theorems and examples
(8.14) Invariant pseudo-metrics for coset spaces. (a) Theorem
(8.2) shows that the topology of a topological group can be described in
terms of continuous left invariant pseudo-metrics: if consists of all
continuous left invariant pseudo-metrics on G, then the family of all
sets {xG:r(x, e) }, where r runs through and through the positive
real numbers, forms a basis at e for the topology of G.
If G is a topological group and H is any subgroup, then the topology
of G/H can be described by continuous, but not necessarily invariant,
pseudo-metrics. Our construction follows that of MOnTGOmERY and
ZIIII [I , p. 36. Let r be a right invariant pseudo-metric on G. Define
r* on G/H by
r*(xH, yH)= inf {r(a, b):axH, byH}.
Let e>0 and choose axH, b, cyH, and dzH such that
r(a, b)<=r*(xH, yH) + e and r(c, d)<=r*(yH, zH) + e.
Since d c -l b zH, we have
r* (xH, zH) (r (a, dc-b) <:r (a, b) + r(b, dc-b)
-- r(a, b) +r(c, d)<:r* (xH, yn) -ka* (yH, zH) + 2.
This proves the triangle inequality; the remaining axioms for a pseudo-
metric are obvious. Let {xH:xVy} be a neighborhood of yH in G/H,
where V is a neighborhood of e. If a continuous right invariant pseudo-
metric and a positive real number e are chosen such that {xG:
a(x, e) < }C V, then xH:a* (xH, yH) < ) {xH:x Vy. It is routine
to verify that a* is continuous on (G/H) x (G/H). Thus the topology of G/H
is describable by continuous pseudo-metrics. Also, as in (8.4), every T o
coset space is completely regular.
(b) If G is a metric group and H is a closed subgroup of G, then G/H is
metrizable. Let r be a right invariant metric on G compatible with the
topology of G (8.3), and let a* be as in (a). Then a* is a metric and the
topology of G/H coincides with the a*-topology. In fact, for a fixed
y G, we have
(x, y)<
for all e > 0.
(c) In (a) and (b), a* is not necessarily invariant. If H is a compact
subgroup of G, then the topology of G/H can be described using only
continuous left invariant pseudo-metrics. To see this, we first remark
that if the hypothesis in (8.2) that U be symmetric is replaced by the
condition that U-I C U for all k, then we obtain a left invariant pseudo-
metric a such that (8.2.i) and (8.2.iv) hold and such that a(x, y)_< 2 -k+a
whenever y-x U. Simply repeat the proof of (8.2) with slight modi-
fications.]
Let {xH:xV,} be a family of neighborhoods of H, where V, is a
neighborhood of e in G, n=t, 2 ..... Let U and W be neighborhoods of e
such that U C V, W C U1, W 1 is symmetric, and h W h - C U 1 for all h H
(4.9). By finite induction, we choose U,, such that UcW,_ and find
symmetric neighborhoods W of e such that W, C U.f V.and h W.h - C U,
for all hH. Then (W,+IH)2cW, H and (W,+IH)-cW, H for n--t, 2, ....
Now let a be the left invariant pseudo-metric which is guaranteed by the
remark of the last paragraph for U take WH, k--l, 2 .... ]. Define
(r* (xH, yH)--(r(x, y) for xH, yHG/H. It is easy to verify that a* is a
continuous left invariant pseudo-metric on G/H.
(d) Using (c), we can prove an, analogue of (8.3) due to KRISTESEN t .
Let G be a topological group ano let H be a compact closed subgroup of G.
Then G/H is metrizable if and only if there is a countable open basis at H.
Moreover, in this case the metric can be taken to be left invariant. To
78 Chapter II. Elements of the theory of topological groups § 8. Invariant pseudo-metrics and separation axioms 79
see this, let the sets (xH" xrV} in (c) be a basis at H. Note that the sets
(xH" x rW} constructed in (c) also form a basis at H. The pseudo-
metric o* of (c) is then a metric since f'l WH--H, and the topology of
G/H coincides with the a*-topology, since
{xH'a*(xH, yH) < 2-}C {xH'xyW,,}C {xH'a*(xH, yH) =<2-"+a},
for yH G/H and n-- t, 2 .....
(8.15) Examples concerning metrizability. (a) The construction
of (8.2) applied to the additive real numbers R, with U,,=V._,,--
]--2 -, 2-[, gives a(x, y)-- min (Ix--yl, t) for all x, yR.
(b) (KAKUTANI [I], VAN DANTZIG [3]') For X=(xik)in, k=l in the
matrix group @ (n, K), let []X][ be the norm of X regarded as a linear
transformation of n-dimensional complex space onto itself"
]IX]I--max ]xikv] ']vl]+]v.]+...+]v,]= .
Then it is easy to verify that [IX+ Y]] __< ]lXllчllYtl, ]lXYll I]xliilYil, and
]iEIi= EE is the identity nn matrix]. The function a(X, Y)=
log 0 +[ix- Y-Ell+]IY-X--EII is a left invafiant metric on (n, K)
compatible th the ordinary topology of (n, K). However, the
oup @ (n, K) admits no metric that is two-sided invafiant. This
follows readily from (4.24); see (8.t8) in]ra.]
(C) (ALEXANDER and COHE It].) Let G be any discrete group with
identity e. Consider the group G ,, realized as the group of aft sequences
=(x, x, ..., x, ...) with xG. For G ,, let ()=0 if
(e, e,..., e,...), and 6() =/n if x ..... x_=e and xe, n=, 2, ....
For , y6 G ., let a (, y) = (y-). Then a is a two-sided invafiant
metric on G , compatible with the topolo of the product oup G ,.
Let , ..... , ... be a fixed finite or infinite sequence of elements of
G . such that () > () > ... > () > .-.. Then for every sequence
(= of integers, lim ,_z .... exists. The set of all such limits
is a closed suboup of G ,. Every closed suboup H of Z , has this
form. [To prove the last assertion, consider for every positive integer
the set of all yH such that 6(y)=i]n. Let y be one of these such
that y, is positive d is as small as possible. If there is no element y
of H for which 6(y)= t In, let y= e. Then H is the set of all limits
lim , ,... .
Removing the y's that are e, we obtain the desired representation.]
(d) There exist nonmetrizable groups for which the identity is the
intersection of a countable number of open sets. Thus the requirement of
local countable compactness, or something like it, is needed in (8.5).
IFor example, see (4.22.b) and (4.22.d).]
(e) Every infinite Abelian group G admits a nondiscrete Hausdorff
topology under which G is metrizable. [Combine (4.23.b) with (8.3).]
(8.16) (JONES [t] and HULANICKI [t].) Every locally compact T O
group of cardinal number V: 1 is metrizable. [By (8.5), it suffices to prove
that some point of G is the intersection of a countable family of open sets.
Assume that this is not the case: no point of G is the intersection of a
countable family of open sets. Well order G: G={x 1, x., ..., x, ...},
where runs through all ordinal numbers less than , the first uncount-
able ordinal. Let U1 be a nonvoid open set such that U- is compact and
xl c U-. Suppose that is some ordinal less than/2 and that nonvoid
open sets U have been defined for all/5< such that
U a is nonvoid for , < ; (I)
xўU- for fl<; (2)
U-C Ua_ whenever fl is a nonlimit ordinal and <fl< . (3)
By assumption we have f') Ua= (x}. If is not a limit ordinal, let U
be a nonvoid open subset of G such that U- U_ 1 , U f')(2-xU) ,
and xў U-. If is a limit ordinal, then f'l b--aU-, and we let
U be a nonvoid open set such that Ufl (2Ua) and xў U-. We
then have f')U---[x, x., ..., x .... }'----, but also f')U 3 by
the compactness of .]
(8.17) (GRA.V [4.) Let G be a T o group admitting an open basis
at the identity e consisting of sets U such that xUx---U for all xrG.
Then G is topologically isomorphic with a subgroup of a direct product of
topological groups each of which has a two-sided invariant metric. [The
construction of a in (8.2) can be carried out so that H is a normal sub-
group, and so that the first neighborhood U l is contained in an arbitrarily
preassigned neighborhood of e; recall the modification of (8.2) indicated
in (8.t4.c). Define a metric aH in the group G/H by all(all, bH)--a(a, b).
Then a H is a metric on G/H making G/H into a metric group [although its
topology may be different from the quotient group topology of G/H].
Forming the direct product of all of the groups G/H with their metric
topologies, and mapping G into this direct product in the natural way,
we obtain the result asserted.]
(8.18) Let G be any T o group having a countable open basis at the
identity e. The following conditions on G are equivalent"
80 Chapter II. Elements of the theory of topological groups § 8. Invariant pseudo-metrics and separation axioms 81
(i) G admits a two-sided invariant metric compatible with its
topology"
(ii) the uniform structures (G) and , (G) are equivalent;
(iii) there is a countable open basis {U}°°__ I at e such that x U, x-l= U
for n=t, 2 .... and all x6G;
X oo oo
(iv) if { n},=l is a sequence of elements of G with limit e and {Y,,},--1
is any sequence of elements of G, then {y,x, yT,1}=l has limit e.
(8.19) Let G be a T O group with a subgroup H such that H and G]H
have countable dense subsets and countable open bases at every point.
Then G is metrizable and has a countable open basis for all open sets.
By (5.38. e), G has a countable open basis at e and is metrizable by (8.3).
By (5.38. f), G has a countable dense subset. A metric space with a
countable dense subset has a countable basis for all open sets.
(8.20) (a) Let G be a compact T0 grou p, and let {U,},°°__ 1 be any countable
family of neighborhoods of e in G. Then there is a closed normal sub-
group N of G such that N C fl U, and such that GIN admits a two-
sided invariant metric compatible with its topology. This is immediate
from (8.7) and (8.6). However, we shall indicate a completely different
and interesting proof due to L. GRABAR' (see PONTRYAGIN 7, § 20,
Example 37, P. t26). For each n (n--t, 2, ...), let /, be a real con-
tinuous function on G such that [ (e) -- t, [ (U,) = 0, and [ (G) c 0, I .
Let [--Y, 2-"/,; [ is a continuous function on G into 0, t such that
n=l
!
/(e)=t and [/n=l U/,)=0. For a, bG,
let
•
Then 0 is a two-sided invariant pseudo-metric on G. Let N--
{a6G'(a, e)--0}. Then N is a closed normal subgroup of G contained
in fl U,. If for aN and bN in G/N, we define *(aN, bN)=(a,b)
n=l
we obtain a two-sided invariant metric for GIN.
The foregoing construction can be carried out in any T o group.
However, in general the metric * on GIN will not yield the quotient
group topology of GIN. If G is compact, the metric 0" does yield the
topology of GIN. Let s be any positive number. Since [ is uniformly
continuous on G (4.t6), there is a neighborhood V of e such that
][(u)--[(v)l<s whenever u-lvV. Let W be a neighborhood of e such
that y-lWycV for all yG (4.9). Then aW implies that (a, e)-----
O*(aN, N)e. Hence the 0*-topology on GIN is weaker than the
quotient space topology of GIN. Since GIN is compact in the quotient
space topology and Hausdorff in the *-topology, the two topologies
coincide.l
(b) The function of (a) may be a metric on G incompatible with
the topology of G. For example, let G= ® (n, K), and choose / so that
[ (E) I, / (X) < t for X 4= E, and / (® (n, K)) C 0, t 1. Then the con-
structed from this function / as in (a) cannot be compatible with the
topology of ®(n, K), since ®(n, K) admits no two-sided invariant
metric.
(8.21) (H. H. CORSON, oral communication.) If X is a completely
regular space, then there is a complex topological linear space (B.5)
such that X is a closed subset of L. Let (X) denote the family of all
continuous complex-valued functions on X, define addition and scalar
multiplication pointwise, and endow (X) with the topology of point-
wise convergence; then a net [ of elements in ff (X) converges to/ (X)
if and only if [ (x) -->[ (x) for each x X. Define ( (X)) in the same
manner and let L=((X)). For xX, let v(x)L be defined by
v(x) (/)--/(x). Then v is a homeomorphism of X onto v(X). To show
that v(X) is closed, let 9v(X)-VI (v(X)'). Then there is a net x, D,
such that v(x)--. For each D, let N--{xaX:fl;>}. For each
finite set F C X such that F N--- , choose /,F (X) such that
/,F(F)-----0 and /,F(N--)=t. Nov /,F is a net whose domain is
{(, F): D and F N-= } and where (, F) ;> (0, F0) if and only if
a>--a0 and FF 0. It is easy to check that/,F-+0, the function identi-
cally zero. We also have for each and F,
(l,) = lin u(x)(l,) -- lin/,r (xz) = t.
Hence
0 ---- lim (x) (0) = 7)(0) -- lim (p (1 F) -- t.
a (t, F) '
This contradiction shows that f(X)-f(f(X)')=; that is, (X) is
closed.
(8.22) (a) Theorem (8.8) can be modified as follows. If property (iii)
is required to hold only for T o groups G, then the free topological group F
can also be taken to be T o . [Repeat the proof of (8.8) verbatim, replacing
the term "topological group" by "T o group". 1
(b) (KAKUTANI [5.) Many imbeddings similar to that described in
(8.8) can be produced. We cite two examples. Let X be any completely
regular space. Then there is a topological group A such that:
(i) X is a closed subspace of A;
(ii) algebraically, A is the free Abelian group generated by X;
(iii) every continuous mapping of X into a topological Abelian group
G can be extended to a continuous homomorphism of A into G.
Hewitt and Ross, Abstract harmonic analysis, vol. I 6
82 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 83
There is also a topological group E such that the left and right
uniform structures of E are equivalent and such that:
(i') X is a closed subspace of E;
(ii') algebraically, E is the free group generated by X;
(iii') every continuous mapping of X into a topological group G
whose structures are equivalent can be extended to a continuous homo-
morphism of E into G.
[The proofs follow that of (8.8). In the Abelian case, only continuous
mappings of X into T need be used to prove (i) and (ii). In the case of E,
one needs only to note that all of the groups 1I (n) have equivalent left
and right uniform structures.
(8.23) (GRAEV [4].) (a) Let X be any nonvoid completely regular
space and let be an open continuous mapping of X into a topological
group G. Then the extension of to a homomorphism carrying the
free topological group F of X into G is an open mapping. [Let V be a
neighborhood of the identity in F and x a point of X. Then (x V)X
is open in X, and so ((x V) f X) -- ((x V) X) is open in G. Hence
(x -1) ((xV)NX) =v(VN (x-IX)) is a neighborhood of the identity
in G, and T(V) (V (x - X)). This implies that is an open map-
ping.]
(b) Every T o group G is the quotient group F/H of a free topological
group F by a closed normal subgroup H. [Consider the identity mapping
of G onto itself and apply (a).]
(8.24) (VILENKIN [2], [6], and GRAEV [4].) Every non T o group G
is a quotient group of a certain T o group with respect to a nonclosed
normal subgroup. Let e be the identity of G, and let e- denote the set
{e}-. Let A be a completely regular space with a dense subset D such
that there exists a one-to-one correspondence o of A f')D' onto
Form the Cartesian product space X=(G/e-)xA. In each coset xe--
in G, select a point, denoted by (xe-), and let (e-)--e. Let 9 be the
natural homomorphism of G onto G/e-. Then obviously 9o(xe--) =xe-"
for all xe-G/e-. Let [ be the mapping of X onto G defined by:
-)o)(a) if aAf'lD',
I(xe-, a) = (xe-) if aD.
Let F be the free T o group generated by X; see (8.22.a). Since F is
the free group generated by X, [ can be extended to' a homomorphism
also written as [, of F onto G. Since 9o[(xe-, a)=xe- for
90[ is a continuous open mapping of X onto G/e-. By (S.40.i), [ is
also continuous. Moreover, (5.40.j) implies that [ is an open mapping
provided that ]-(e-)--(/-(e))-. Then (5.27) will show that G is the
quotient group of the T o group F by the nonclosed normal subgroup
]-x (e). The continuity of [ yields at once the inclusion [-1 (e-) ([-1 (e))-.
For aD, we have [(e-, a)=e so that {(e-, a)'aD}c/-(e). Since D
is dense in A, it follows that
/-1(--) ={(--, a).aA)={(g-, a).aD}-c(/-l(e))-.]
Notes
Invariant metrics on groups were introduced by VAN I)ANTZIG [3]"
Theorems (8.2) and (8.3) are due to KAKUTANI [t and G. BIRKHOFF[t];
BIRKHOFF did not point out that his metric is left invariant. A closely
related construction, which also proves (8.4), is attributed to PONTR-
YAGIN by WEIL[3 ], p. t 3. Theorem (8.7) is due to KAKUTANI and
KODAIRA [t]; for our version see also MONTGOMERY and ZIPPIN
p. 58.
Free topological groups have been studied by many writers. The
concept was invented by A. 1V[ARKOV [2]; a complete exposition appeared
in MARKOV [4. M.ARKOV proved (8.8) by very complicated means;
simpler constructions were given by NAKAYAMA [t ], KAKUTANI [5J, and
GRAEV [31- Our treatment follows KAKUTANI'S. 1V[ARKOV [3] used his
method to construct a connected group of arbitrary cardinal _ c all of
whose elements have order 2. See GRAEV [4], Theorem 28, for a sim-
plified proof.
Theorems (8.tt) and (8.t2) are due to A.H. STOlE[t. The
normality of locally compact T o groups was pointed out to us by E. A.
MICHAEL.
Throughout the remainder of this book, ALL TOPOLOGICAL
GROUPS WILL BE TAKEN TO BE 0--and hence completely reg-
ular-unless the contrary is explicitly specified.
§ 9. Structure theory for compact and locally
compact Abelian groups
The principal result of this section (9.8) is that every locally compact,
compactly generated Abelian group is topologically isomorphic with a
product of a compact Abelian group and a finite number of copies of
R and Z. This famous theorem is the continuous analogue of the structure
theorem for finitely generated Abelian groups. A discrete Abelian group
is compactly generated if and only if it is finitely generated; and every
such group has the form ZxF, where a is a nonnegative integer and F
is a finite group (A.27).
6*
84 Chapter II. Elements of the theory of topological groups §9. Structure theory for compact and locally compact Abelian groups 85
Theorem (9.1) is of independent interest; it implies, for example,
that if x is any element in a locally compact group, then the smallest
closed group containing x is either compact or topologically isomorphic
with Z.
In Theorem (9.t 5) we show that all infinite 0-dimensional compact
groups are homeomorphic with products of two-element spaces. We
close the section with a brief introduction to compact topological semi-
groups.
(9.1) Theorem. Let H be any subgroup o] the additive group R,
with its relative topology [ormed [rom the usual topology o/R. Let [ be a
continuous homomorphism o/ H into a locally compact group G. Then
either [ is a topological isomorphism o/ H onto [(H), or ([(H))- is a
compact A belian subgroup o/G. I[ [ is not a topological isomorphism, then
/or every neighborhood U o/e in G, there is a positive number #H such
that/or every hH, the set/(h, h + #NH)N U is nonvoid.
Proof. In steps (I), (II), and (III), we suppose that H----R or H=Z.
(I) Suppose first that there is a neighborhood U of e in G such that
{t H: ! (t) U} is a bounded subset of H:
/ (t) U implies
Then / is one-to-one, since otherwise the set/-1 (e) would be a subgroup
of H different from {0}" such a subgroup is necessarily unbounded. If
V is a neighborhood of e in G such that V-C U and V- is compact,
then we have V-f/(H)--V-C/([--,INH). Since H is R or Z,
[-- , a H is a compact subset of H" therefore V- / (-- ., H) is
compact. Hence the closure of V l/(H) in the relative topology of [(H)
is compact. That is, /(H) is locally compact. By (.29), / is a topological
isomorphism.
(II) Next suppose that / is not a topological isomorphism. With no
loss of generality we may suppose that G -- (/ (H) )-. Then G is Abelian,
by (5.3). Consider any a in H and let A be any nonvoid open subset of G.
Then there are an element tH and a symmetric neighborhood U of e
in G such that /(t)U cA. By (I), there is an element vH such that
v > ,a, ч I tl and / (v) U. Thus we have [ (v + t) = / (t) / (v) / (t) U C A,
so that the set /(a, o[ H) is dense in G.
Next let V be any symmetric neighborhood of e in G such that V-
is compact. Let a be any nonnegative element in H. We have just
proved that G=/(a, o[H) V. Since V- is compact, there are
elements tl, t 2 .... t mla, o[H such that V-C U !(t/) V. Let
' i=1
#--max(t1, t 2, ...,tm). Let x be any element of G, and let
inf ([-(xV--)(3 0, oI). Since x V--is closed, we have rx H and
/(v,)6xV-, and hence /(z,)x/(ti)V for some j=t, ..., m. Thus
/ (z,-- t]) x V c x V-. Since t i is positive, we find that r, -- t i < 0. Thus
z,</, and consequently xq/([O,t[CH)V-c/([O, #](H)V-. That is,
G--/([O, #ClH) V-.
Now [0, #] N H is compact since H is R or Z, and so /([0, F]NH) is
compact. By (4.4), [(0, #] NH)V-is compact. We have thus proved
that G is compact.
(III) The last assertion of the theorem is now easy to verify if H= R
or H=Z. For a given neighborhood U of e in G, choose a symmetric
neighborhood V of e in G such that V-C U and V- is compact. By (II)
if hH there is a r[0,#ENH such that /(--h)/(r)V-. Thus
1( + ) (v-)- = v-c g.
(IV) Now suppose that H is any subgroup of R. If H={0}, the
theorem is trivial. If H contains a smallest positive element, H is
topologically isomorphic with Z and so comes under (I) -- (III). If H
is not {0} and contains no least positive element, then H is obviously
dense in R. If we can extend the continuous homomorphism / to be a
continuous homomorphism ] of R into G, then we can apply (I) and (II)
to infer that /is a topological isomorphism of H or (/(H))-is compact.
The last assertion is also obvious in this case" since ] is continuous, we
have ([h, h +#1)C(h, h +#[ ('IH)-, so that if an open set Uintersects
([h, h + #[), then it intersects/([h, h + #[ H).
To extend / continuously over R, we recall that ] is uniformly con-
tinuous (5.40. a). Let x be any element of R('IH' and let {h}°°__l be any
sequence of elements of H such that li_m h--x. Then for every neigh-
borhood U of e in G, there is a positive integer m 0 such that/(hm) ](hn)-lg
if m, n_m o. If U- is compact, then we have ](hm) U/(hmo) for all
m > too, and there is a cluster point in U-/(hmo ) of the net {1 (h)}m > m0"
Let this cluster point be (x). It is easy to see that ](x) is unique, both
130 t 130
for the sequence {h,,l and also for any other sequence {h,,= with
limit x. Furthermore, is a continuous homomorphism of R into G
agreeing with [ on H. We omit the details.
(9.2) Definition. Let G be a topological group. If G contains a
dense cyclic subgroup, then G is said to be monothetic. Let be a
continuous homomorphism of R into G. Then r(R) is called a one-
Parameter subgroup of G. If 6 contains a dense one-parameter subgroup,
then G is said to be solenoidal.
Theorem (9.1) shows that a locally compact monothetic [solenoidal 1
group is topologically isomorphic with Z [R or is compact. A detailed
86 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 87
analysis of the structure of compact monothetic and compact solenoidal
groups appears in § 25.
We now establish the structure of locally compact, compactly
generated Abelian groups. In (9.3)--(9.7), we give preliminaries to the
main theorem, (9.8).
(9.3) Theorem. Let G be a locally compact, compactly generated
A belian group. Then G contains a discrete subgroup N with a finite
number o] linearly independent generators such that GIN is a compact group.
Proof. Let U be a symmetric neighborhood of e in G such that U-
is compact and 0 U"--G (5.13). Since (U-) is compact (4.4) and
n=l
GU--G, there is a finite subset {a,a.,...,a} of G such that
Uc(U-)CiUaiU. Let A be the smallest subgroup of G containing
(a, a., ..., a). It is plain that UcAU and hence AUAU--AU.
Then we have U 3-- U U A U A U, U 4-- U U A UU A U, and so
on. Hence G-- O U" = A U.
n=.
Let C i be the smallest subgroup of G containing a i ('----1, 2, ..., m).
We have G--C-C... C- U-, so that if C-, C-, ..., C are compact,
then G is compact, and we can take N= (e). If some C- is noncompact,
then by (9.t), the group C i is an infinite cyclic discrete subgroup of G.
Let b 1 be any one of the a i for which (b'nEZ} is a discrete infinite
subgroup of G. We define elements b., b .... by induction. Suppose
that b 1, b., ..., b k have been selected from among the elements
a, a., ..., a m in such a way that bl, b., ..., bk generate a discrete sub-
group N of G and bl, b.,.. ., b are linearly independent" if 11 ."... b=
where the i s are any integers, then 1--. ..... --0. Consider the
quotient group GINs. If it is compact, then our process terminates and
we take N--N. If G/N is noncompact, then, writing 9 for the natural
homomorphism of G onto GINs, we have G]N-- 9 (A) 9 (U-), and since
9 (U-) is compact, 9 (A)- cannot be compact. Clearly we have 9 (A) =
9 (C) ... 9 (C). If each 9 (Ci)- is compact, then 9 (C1)-'." 9 (C)- is
compact and closed so that 9(A)-9(C1)-... 9(C,)-. But 9(A)- is
not compact, and thus there must be some " for which 9(Ci)- is not
compact. Since G/N is locally compact (5.22), we infer from (9.1) that
9(Ci) is a discrete, infinite cyclic subgroup of G]N. Let
It is clear that no power b+ (n:0) can be equal to any b .... b ,
because all of the cosets b
+lNk are distinct. Thus b, b., ..., b+l are
linearly independent. Let N+I be the subgroup of G generated by
bl, b. .... , b+l. Let V be a neighborhood of e in G such that V
and such that the open subset {vN:vE V} of G/Nk intersects {b+N: nZ}
only in N. Then VN+I= {e), so that N+ 1 is a discrete subgroup of G.
As the set {a, a.,..., a,) is finite, our process must terminate with
a group N for which G/N is compact.
(9.4) Theorem. Let G be a locally compact Abelian group having a
discrete subgroup N with a finite number o] generators such that GIN is
topologically isomorphic with T'xF o, where r is a nonnegative integer and
Fo is a finite Abelian group. Then G is topologically isomorphic with
TxRxZxF, where a, b, c are nonnegative integers and F is a finite
A belian group.
Proof. (I) Suppose first that G is connected. Then GIN is also
connected, as the continuous image of a connected space, and so GIN
has the form T . Let 9 be the natural mapping of G onto T . We may
also consider T as R]Z; let be the natural mapping of R onto T .
Since Z and N are discrete subgroups of R and G, respectively, there
are a neighborhood U of 0 in R and a neighborhood V of the identity
in G such that is one-to-one on U, 9 is one-to-one on V, and 9 (V) -- (U).
In fact, choose a symmetric neighborhood U0 of 0 in R such that
UZ - 0). Then choose a symmetric neighborhood V0 of e in G
such that V0 f N---- (e) and 9 (V0) c (U0). Finally, choose UC
U0fV-l(9(V0)) and let V--Vo(9-1(v(U)). With no loss of generality,
we may suppose that U-- (R: II ll < ) for some positive number
[I I] denotes the norm described in § I. For U, let #() be
the unique element in V c e such that 9 ( # ()) ---- ()- It is obvious
that if , /, and + /are in U, then # ( + /) -- # () # (/), and that
(-) - ()-.
We extend b to be a mapping, also denoted by b, of R into G
by the following rule. For every 6R, there are an integer n and an
element /U such that n/--. Then we set #()----(b(/)) ". To see
that # is still single-valued, suppose that nl, m. are integers, that
Y, y. U, and n 1 /1 =m. /.. If n or m. is 0, then plainly #(n 1 /)=
#(m. V.)--e. If neither n x nor m. is 0, then YE U for k--0, +t,
2 .... ,-+-m., and we have #(/1)=(#(v/)'; similarly we have
\ \-Iii
It is likewise easy to see that is an open continuous homomorphism
of R into G. Since (R ) contains a neighborhood of the identity
element of G and G is connected, it also follows that (R ) --G. See
(7.4).]
Since o q()--() for U and U generates the group R , we
have o =. It ollows that the kernel, say H, o q is contained in
the kernel Z of . I H= {0), G is topologically isomorphic with R .
Suppose that H{0. By Theorem (A.26); we can choose a basis
88 Chapter II. Elements of the theory of topologica! groups § 9. Structure theory for compact and locally compact Abelian groups 89
el, e. .... , ep for Z p and positive integers dl, d,. ..... dk (t <= k<= p) such
that de, d,.e2, ...,dke are independent generators for H. Since
e, e,., ..., e are independent generators for Z , the elements e, e,.,..., e
are a basis for R as a vector space over R. Thus every coset /+ H
contains an element xlel+ x,.e.+ ... + xpe, where 0=< x< d .... ,
0_<__ x< d, and x+l, ..., x are arbitrary real numbers. Furthermore,
distinct elements of this type lie in distinct cosets of R with respect
to H. It follows at once that G is topologically isomorphic with
TxR-k.
(II) Now let G be any group satisfying the hypotheses of the theorem.
With no loss of generality, we may take GIN equal to T'xF o, where F 0
is finite. Let 9 be the natural homomorphism of G onto GIN and let
be the projection of T'xF o onto T r. Then o 9 is an open continuous
homomorphism of G onto T'. The kernel, say M, of this homomorphism
is a finite union of distinct cosets: M-- N 0 xN U x,.N 0... 0 xzN. Since
N is closed (5.10), each xjN is also closed and does not contain the
identity of G. Hence M is a discrete subgroup of G with a finite number
of generators.
Consider then the group G/M= T'. If r--O, i.e., M=G, then G is
a discrete group with a finite number of generators, and by (A.27)
G has the form Z, where c is a nonnegative integer and is a
finite Abelian group.
Suppose finally that G/M= T" with r > 0. Since M is discrete, o 9
is one-to-one on some neighborhood U of the identity in G. Thus o 9
is a homeomorphism on U, since it is continuous and open. As T is
locally connected, we may suppose that U is connected and symmetric.
The set U is a continuous image of the connected product U U... U
n times for n--2 3 .... Hence U U is also connected and is an
' ' n=l
open subgroup of G (5.7). Since U U is also closed, it is the connected
component of the identity in G; we write C= U U . Since T' is a
connected group, it is generated by the neighborhood og(U ). Thus
we have T'og(C)og(G ), and T'=C/(CAM) (.32). Since C
is connected, we may apply Step (I) of the present proof to infer that
C is topologically isomorphic with TxR "-, where k is a nonnegative
integer.
Since o 9 (C) = o 9 (G), we clearly have G CM. By the purely
algebraic form of the first isomorphism theorem (2.), we know that
G/C=CM/C is algebraically isomorphic vith M/MAC. Since CM/C
and M/M A C are discrete, this algebraic isomorphism is automatically
1 Observe that step (I) of the present proof holds for any discrete subgroup N
of G: /V need not have a finite number of generators.
a honaeomorphism. It follows that G/C has a finite number of generators.
133 (A.27), G/C is topologically isomorphic with ZCxF, where c is a
nonnegative integer and F is a finite Abelian group. Since C is open
and divisible, (6.22.b) shows that G is topologically isomorphic with
Cx(G/C) and hence with ZCxF1xTkxR "-.
In order to complete the description of the structure of compactly
generated, locally compact Abelian groups, we must borrow a theorem
that is proved in Chapter Six: see (24.7). The proof of (24.7) is of course
independent of the present section.
(9.5) Theorem. Let G be a compact Abelian group and let U be a
neighborhood o/e in G. There is a closed subgroup H o/G such that H C U
and G/H is topologically isomorphic with TxF, where a is a nonnegative
integer and F is a/inite A belian group.
This is Theorem (24.7).
(9.6) Theorem. Let G be a locally compact, compactly generated
A belian group. Every neighborhood U o/ e in G contains a compact
subgroup H such that G/H is topologically isomorphic with TxRbxZCxF1,
where a, b, and c are nonnegative integers and F 1 is a/inite A belian group.
Proof. By (9.3), there is a discrete subgroup N of G with finitely
many generators such that GIN is compact. Let 9 denote the natural
mapping of G onto GIN. Let W be a symmetric neighborhood of e in G
with compact closure such that W C U and WAN=e). By (5.t7),
9(W) is open in GIN. Thus (9.5) implies that there is a closed subgroup
H c ? (W) of the group GIN such that (G/N)/ is topologically isomorphic
with TxF: here a is a nonnegative integer and F is a finite Abelian
group. Let H0--9-1(/ ) and H--HoA W. It is obvious that 9(H0) --/;
we now prove that 9 (H) -----/. Clearly we have 9 (H) C/; if /, then
-----9(w) for some wW. Since 9(w)/, we have wH o and therefore
---- ? (w) 9 (H0 A W) -- 9 (H). Since 9 is one-to-one on W- and W- is
compact, 9 is a homeomorphism of W-onto 9(W-). In particular,
9 is a homeomorphism of H onto 9(H)-----.. Since / is compact, it
follows that H is compact. We now show that H is a subgroup of G.
If x, yH, then xy-Ho and for some hH, we have 9(h)----9(xy-).
Hence xy-h-N and plainly xy-h-HH -H -CW , so that
x y-1 h- __ e. That is, x y- -- h H.
Let denote the natural mapping of G onto G/H. To prove the present
theorem it suffices by (9.4) to show that (N) is a discrete subgroup of
G/H and that (G/H)/(N) is topologically isomorphic with TxF • i.e.,
with (G/N)/. Note that (N) is finitely generated since N is. We
neЧt show that Ho=HN. If hH and xN, then 9(xh) --9(h)?(H)--
90 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 91
and so x hcH o. Conversely, if x cH o, then there is an h in H such that
9(x)=9(h); thus we have x-----xh-lhNH. By (5.34), we see that
(G/H)/v (N) is topologically isomorphic with G/y - (y (N) ) and likewise
(G/N)/ is topologically isomorphic vith G/q-l()=G/no . Since
v-I(v(N))---:HN=Ho, (G/H)/v (N) is topologically isomorphic with
(GIN)In.
Finally, (N) is discrete in G/H. Note first that NfilH C NfilWc Nfil W a
= {e}. Clearly N'U {e} is an open set containing H so that by (4.10),
there is a neighborhood V of e in G such that HVcN'U {e}. That is,
we have
(HV) filN-- (e}. (t)
This implies that p(V)fily(N)----(H}. In fact, if v H--xH, where v V
and xN, then vh=xforsomehH andso by (1), we havevh--x:e.
Before proceeding to the main theorem, we set down an obvious
lemma.
(9.7) Lemma. Suppose that G is a topological A belian group, that H
is a subgroup o/G, and that G/H=RxZbxF, where a and b are non-
negative integers and F is a compact A belian group. I/ 9 denotes the
,atural mapping o/ G onto G/H, then every compact subgroup o/ G is
Proof. If E is a compact subgroup of G, then 9(E) is a compact
subgroup of GIg. It follows that 9 (E) C (0} >< (0} xF, so that
z c v (v (z)) c v
(9.8) Theorem. Every locally compact, compactly generated Abelian
group G is topologically isomorphic with R'><Zb><F /or some nonnegative
integers a and b and some compact A belian group F.
Proof. We will apply ZORN'S lemma in conjunction with the preceding
theorems.
By (9.6), there is a compact subgroup H cG such that G/H is
topologically isomorphic with RxZxF, where/r is a compact Abelian
group. Here, and in similar situations later on, we will regard G/H as
equal to RxZxF; we will also identify R'xZ with the subgroup
RxZx {e'}, and F with the subgroup {0}x{0}xF [e' is the identity
of F]. Let 9 denote the natural mapping of G onto G/H. Let M = 9 -1 (F) ;
by (5.24. a), M is compact. In view of (9.7), M is the largest compact
subgroup of G. By (5.34), G/M is topologically isomorphic with
(RxZbxF)/F and hence with RxZ .
Let .Z' denote the family of all closed subgroups L of G such that
G --LM. Partially order .L by the rule"
L'<L whenever LL.
Obviously G belongs to .Z', so that .Z' is nonvoid. Suppose next that
{L}ea is a linearly ordered subset of .W; we will show that {L}ea has
an upper bound in .Z'. Let L--ill L. Clearly L is a closed subgroup of
G and L L for all A. To show that L .v, consider an arbitrary
xG. We direct the set A by writing :' whenever L>'L,. For each
A, we have G -- LM and therefore x -- y m for some y L and
nM. Now {m,'ocA} is a net in M and has a convergent subnet
qr, 7E; let m 0 denote the limit of qr. Let N denote the function from
E into A that relates the net with its subnet [see (3.t0)]. That is,
qr=mN() for E. Consider any 0 in A. There exists a 70e E such that
N(7) >" 0 whenever >" 0- Thus for >" 0, we have
xq-
-- x m N (,} -- YN(r)
It follows that the limit xm z of the net xqT*, vE, belongs to Lo.
Since 0 is arbitrary, we have xm -z R L-- L. Consequently, x --
x mmoLM. Thus G=LM and L is an upper bound for {L}a.
By ZoRN's lemma, there is a maximal element L 0 in .Z'; obviously
we have G ----LoM. Next we show that LofilM= {e}. Assume that there
is an element z in L 0 R M such that z 4= e. Let U be a neighborhood of
e such that zў U. Being compactly generated, G is a-compact, and its
closed subgroup L 0 is also a-compact. By (5.33), Lo/(Lo filM) is topo-
logically isomorphic with LoM/M--G/M and hence with RxZ . In
particular, Lo/(LofilM ) is compactly generated. Since LofilM is compact,
(.39.h) shows that L 0 itself is compactly generated. By (9.6), there is
a compact subgroup H 0 of L 0 such that H 0 c L0fil U and Lo/H o is topo-
logically isomorphic with R'oxZoxFo, where F 0 is compact. Let 9o
denote the natural mapping of L 0 onto Lo/Ho; let L=9Z(R'oxZ o)
and M0___ 9 - (F0). It is trivial that L1 c L 0 , L fil M 0-- H 0, and L 0 = L M 0 .
By (9.7), every compact subgroup of L 0 is contained in M 0. In particular,
we have
L filM --L fil (L filM) L filM 0 =H0 U.
Thus z does not belong to LfilM and since zM, we see that zCL.
In other words, we have L L 0. By (5.24.a), M0 is compact and so
MoM is a compact s'ubgroup of G. Therefore MoM= M since M contains
all compact subgroups of G. Using this, we see that
G =LoM --LMoM =LM.
Thus L1 belongs to .Z'.; and L. L 0 since L L 0. This contradiction
shows that L0fil M---
By (5.33), we see that L o is topologically isomorphic with G/M and
hence with R'xZ . Finally, by (6.t2), G is topologically isomorphic
with Lox M and hence with R'xZbxM.
92 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 93
We now give some applications of (9.8).
(9.9) Definition. Let G be a topological group and a an element of G.
The element a is said to be compact if the smallest closed subgroup of G
containing a is compact.
(9.10) Theorem. Let G be a locally compact Abelian group. The
set B o/ all compact elements oWG is a closed subgroup oWG.
Proof. Let M--{e, a, a -1, a 2, /g -2, ..., a n, a -n, ...--, for all acG.
Since Mb C M. Mb and M_I--M, (4.4) shows that B is a subgroup of G.
Now suppose that x c B-. There is an open compactly generated sub-
group H of G that contains x (5.t4). The group H is topologically iso-
morphic with RxZbxF by (9.8), where a and b are nonnegative integers
and F is a compact Abelian group. We may regard RxZbxF as an
open subgroup of G. If the R'-coordinate of x or the Z-coordinate of x
were different from zero, there would be an entire neighborhood of x
in H, and hence in G, disjoint from B, since R and Z contain no
nonzero compact elements. It follows that x is in F, and so x is in B. 1
The following technical result is useful in proving some consequences
of (9.8). It is also needed in the proof of Theorem (24.36).
(9.11) Theorem. Let H be a closed subgroup oWR c di//erent /rom {0}.
There are vectors x, ..., xa, y, ..., y in H that are linearly independent
in R c, such that H is exactly the set o/ vectors that can be written in the/orm
(i) o xi +... + o x + ml y + ... + m y,
where o ..... Oa R and m, ..., m Z. Thus H is topologically isomorphic
with RxZ and is also compactly generated.
Proof. The proof is by induction on the number c. For c = I, the
theorem is quite elementary. We may thus suppose that c >=I, that the
theorem has been proved for R , and that H C R +. There are two cases.
Case I. The subgroup H is discrete. Let Y0 be a vector in H such
that t=IyoI =min{lu]'uH, u0}. Plainly t is positive. Let L=
{ay0"aR}. Let uHL', let a be any real number, and let m be an
integer such that la--m I g }. Then ,ve have
t
tIlu-myoli lu-yoII+ Y°--myo] Iu-y°H+ 2'
so that
t
2 "
Consider the quotient group R+/L, and the natural mapping of R +
onto R+/L. It is obvious that R+/L is topologically isomorphic to R
1 In non-Abelian locally compact groups, the set B of all compact elements
may fail to be a group or even a closed set" see (9.26.c) and (9.26.d).
and that y is a linear mapping. In view of (t), y(H) is a discrete and
hence closed subgroup of Rc+/L (5.t0).
By our inductive hypothesis, there are vectors (yl), ..., (yb) in
(H) that are linearly independent in Rc+/L and such that q(H)
consists precisely of the vectors
(mZ). (2)
Plainly we can choose y,H. Now for ucH, we can find ma,..., mZ
such that y(u) --q(my +...+ my). This implies that u--mly ....
-- m yb -- m0 Y0 for some m 0 Z, or
u=m oyo+... + my.
If v0, v, ..., v R and v y = 0, then also 0 = v y = v (y)
b k=0 k=0 k=0
v,(y,). Hence v=v2 ..... v=0, and finally v0=0. Thus
k=l
Y0, Y, ..., Y are linearly independent in R +, and the induction is
complete for discrete subgroups.
Case II. The subgroup H is nondiscrete. In this case the construction
of L must be carried out differently. We will show that H contains a
i-dimensional linear subspace of R +. Let {u,}, be a sequence of
vectors in R + and {2,}, a sequence of positive real numbers such
that lu ll=a and 2,u,H for all n and such that lira 2--0. Let 0
e .
be any vector in R + such that a subsequence of {u,}, converg s to 0
Let a be any real number and e any positive number. Choose a positive
e Let p be an integer
integer m such that 2me and Ix o -Itml[< t [ + e"
such that p 2,,-- I < 2m e. Then
Since p 2m Um H and H is closed, it follows that a w0 H and hence that
Consider now the quotient group R+/L and the natural mapping
of R+ onto R+/L • clearly is linear and R+/L is topologically
isomorphic with R . By (5.3t), (H) is topologically isomorphic with
H/L and is hence locally compact (5.22) and closed in R+/L (5.11).
By our inductive hypothesis, there are linearly independent vectors
9(a) .... , (), (y) ..... (y) in R+/Lsuch that (H) is just the
set of all vectors
o (x) + ... + o (x) + . (y) + ...+ m (y)
94 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 95
where %.R and mk<Z. We may obviously suppose that the vectors
and yk lie in H.
All vectors 2aide i (eiR)lie in H. For, if vH and q)(v)=
then v-- aiai--a0a0 for somea0R, andso ciej=v--%eoH. The
i=I i=I
proofs that H consists exactly of the vectors in R c+l of the form
b i=o
my (.iR, mZ) and that a o, a 1, ..., a, Yl, ..., Yb are linearly
/----- 1
independent in R c+1 are like the corresponding proofs for Case I.
(9.12) Theorem. Let 3 be a topological isomorphism o/ R'xZbxF
into RcxZxE, zvhere a, b, c, d are nonnegative integers and F and E are
compact groups [not necessarily Abelian. Then a<c m,d a+b<=c @d. 1
Proof. Let r] be a topological isomorphism of Rx Z b onto RxZ x {e},
where e is the identity of _F. Then 3 o 1 is a topological isomorphism of
RxZ into RxZxE. Hence in this proof, we may disregard the
factor F and suppose that the domain of 3 is R><Z .
(I) Assume that a> c. Since 3(Rx{0}) is a connected subgroup of
RcxZxE, it follows that
Let ] be a topological isomorphism of R onto Rx{0}, and let 0 be the
projection of RxZxE onto RxE. Then 30=0 o 3or] is a topological
isomorphism of R into RxE.
Let be the projection of RxE onto R . Then the mapping
is a continuous homomorphism of R into R . It is easy to see that is
actually a linear mapping of R into R , and since a>c, the linear
subspace S={eR" p(e)=0} is different from {0}. For eR , write
3o (e) = (j, z), with j R and z E. It is obvious that 30 (e) = (0, z) for
every e;S; that is, the inclusion 30(S)c {0}xE obtains. Since 30 is a
topological isomorphism, 30(S) is closed in {0}xE (5.11). Since E is
compact, S is also compact. This is a contradiction, since S is a Euclidean
space of positive dimension. Therefore the inequality a< c obtains.
(II) We return to the mapping 3 of RxZ into RxZxE. Let
be the projection of RxZxE onto RxZ . Plainly q)o 3 is a continuous
homomorphism of RxZ into RxZ . Suppose that
where eR><Z . Then 3(e) belongs to {O}xEcRxZxE, and hence
3(e) is a compact element of RxZxE. This means that e itself is a
compact element in RxZ , and so e--0. Thus f o 3 is a one-to-one
mapping and q) is one-to-one on 3(RxZ). Since R><Z is locally
1 We are indebted to ROGER W. RICHARDSON for the inequality a c and its
proof.
compact, 3 (RxZ ) is closed in RўxZxE (5.1 t). Now (5.18) implies
that fo 3(RxZ ) is closed in RўxZ . It follows that q)o 3(RxZ ) is a
locally compact group. By (5.29), f o3 is a topological isomorphism.
We may obviously regard q)o 3(RxZ ) as a closed subgroup of R ў+.
By (9.1t), q)o3(RxZ ) contains a+b vectors that are linearly inde-
pendent in R +a. That is, a+ b=< c+ d.
(9.13) Corollary. Suppose that RxZbxF and RўxZxE are topo-
logically isomorphic, where a, b, c, d are nonnegative integers and F and E
are compact groups. Then a=c, b=d, and F and E are topologically
isomorphic.
Proof. The isomorphism 3 must carry compact elements onto
compact elements; therefore 3({0}x{0}xF)--{0}x{0}xE. By (9.12),
we have a=c and b--d.
We now present a useful consequence of (9.8).
(9.14) Theorem. Let G be a locally compact Abelian group, and let C
be the component o/ e in G. Then C is topologically isomorphic with
R"xE, where n is a nonnegative integer and E is a compact connected
A belian group. The number n is the largest possible dimension o/a sub-
group o/ G topologically isomorphic with R /or a nonnegative integer a.
I! H is a compactly generated open subgroup o/ G, then H is topologically
isomorphic with R"xZxF, where b is a nonnegative integer and F is a
compact A belian group.
Proof. Let H be a compactly generated subgroup of G. By (9.8),
H is topologically isomorphic with R><Z><F for nonnegative integers
a and b and a compact Abelian group F. The subgroup C is compactly
generated (7.4) and connected, so that C is topologically isomorphic
with RxE, where n is a nonnegative integer and E is a compact
connected Abelian group. [No factor Z appears because C is connected
and Z is not. The set C is defined as the union of all connected subsets
of G containing e, and so there is a topological isomorphism carrying
R into R"xE. From (9.t2) we infer that a=<.
Let H now be any open compactly generated subgroup of G: let H
be topologically isomorphic with RxZxP, where c and d are non-
negative integers and P is a compact Abelian group. The group C is
the intersection of all open subgroups of G (7.8). Hence C is contained
in a topological isomorph of RxP [which is plainly open in H; hence
its isomorph is open in G. Thus there is a topological isomorphism
carrying R"into RxP, so that (9.12) implies that n<_c._ Since we can
take a=c in the preceding paragraph, it follows that c=n.
(9.15) Theorem. Let G be a O-dimensional, in/inite, compact group,
and let m be the least cardinal number o/ an open basis at the identity e o/ G.
96 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 97
Then G, regarded only as a topological space, is homeomorphic with the
space 0, I) m, where 0, I) is discrete 1.
Proof. We give the proof in a number of steps.
(I) Let (U,}, be an open basis at e having cardinal number In.
Using (7.7), choose a compact open normal subgroup V, of G such that
V, c U,, for each ,I. Now well order the family V,},I, and rewrite it
as {V, V 2 ..... V,...}, where runs through all ordinals less than, say,
the first ordinal ,a with cardinal number In. Note that In must be
infinite. With no loss of generality, we may suppose that V--G. For
every ordinal/5, I <fl<#, let 2V--fl<aV , and let NI=-G. It is clear that
each N is a compact normal subgroup of G. Fix an ordinal fl<#, and
let X be any subset of G that is the intersection of sets of the form
alVIUa2VU...Ua, V, where ajG and fl.</5 (/'--1,2 .... ,s). It is
obvious that 2VX- X.
(II) We next define certain open and closed subsets of G. Let B be
the set of ordinals/5 < # for which V. 2V V. Both V a and N V a are open
and closed normal subgroups of G. The quotient groups NV/V a and
G/NV are compact (.22) and discrete (.21) and hence finite. Let
b(IlNV .... , b(a"alNV be the distinct cosets of 2VV in G, and let
a(llV .... , a(alVa be the distinct cosets of V in NV. For each integer ',
i=,2,...,k a, let
A )= b(fl 1) a]) V e b( ) a') V U''" 1.3 b( re.a) a ") Vfl.
Thus each A "/is a finite union of cosets of V, and each A( i/intersects
every coset of 2VV in a coset of Vў. It is obvious that
if '=t=l and that UAI=G.
i=
We next show that if X is a nonvoid subset of G and ]" = I, 2, ..., k
then (NX)A(jl4=(3. Since U b(JlNV=G, there is some b(J / for which
(NX/n (bgNV/4= e. Sie i in NV, we ave V(}NV =
bg I a/iV V = 2V b / a/V. Thus there are elements a 1, ae ivy, x X, and
yV such that alx--a 2b la/y. Hence a lalx=b/a/y. Since
alalXNX and b(JlalyA I, we have (NX)A(I4=(3.
Now let ill,/52, ..., fls be any distinct elements of B; we may suppose
that/51<fl< -." < fls. We will show that
()
if t =< i<=ka,(l=1, 2, ..., s). It follows from (I) that A:>=N.,.A()2 ). By
the preceding paragraph, we have
For a related result, see (25.35) inlra.
Applying (I) and the preceding paragraph again, we get
(N,(A(/IfA(I))A(: 14=e. Finite induction now shows that (I) holds.
To complete our construction, we need one more fact, viz." if x, y
__ __
are in G and x=t=y, then there is a/5B and a ', I <'<k a, such that A
contains one and only one of the points x and y. Since x-ly 4=e, there
is an e<# such that x - y cV. Let/5 be the least of all these e's. Then
x-lyN, so that x-lyNV and x-lycV. Hence we have VNV,
so that fl B. In the notation used above to define the sets A/, we have
x, y b)N Va for some t, I <- t <- m a . Since 2 -1 y V a, we have x
and yb(la(JlV, where '=t=l. Hence xA and yA(J I, so that yCA.
(III) To finish the proof, we form the Cartesian product Y--
P {t 2 ka}, each finite space {I 2, ka} being given the discrete
flB ' ' "''' ' "'''
topology. Define a mapping q of G into Y by q(x)= (')B, where '
is the integer such that xA 1, for each fl B. Since all of the sets A /
are open and closed, is a continuous mapping. Since the sets A 1
separate points of G, is one-to-one. Hence is a homeomorphism,
and (G) is a compact and hence closed subspace of Y. In view of (I),
(/) (G) is dense in Y, and thus (G)--Y.
The constructions in (I) and (II) above show that B<--in. Since G
and Y are homeomorphic, we must have B--in, inasmuch as B is the
smallest cardinal number of an open basis at an arbitrary point of Y.
To show that Y is homeomorphic to {0, 1)m, write B as the union of a
family of pairwise disjoint countably infinite sets.
The proof will be complete when we have shown that a countably
infinite Cartesian product X of finite discrete spaces is homeomorphic
with {0, t} 0. Write this product X as the set of all sequences
(x o, x, x2 .... ), where x {0, I .... , a,-- I } and the a,'s are integers greater
thanl (n=0, I,2 .... ). ForxX, letz(x)=A. (do+ )(a+ )...(ai+ ) '
]=0
oo a i -- 1
where A -1 is the number (a ° + )(a + t)'." (ai + 1) " It is easy to see that
i=0
z is continuous" if x, X and xi--y i for '=0, l,...,ra, then
+ a i -- I
The
mapping
is
also
one-to-one. To see this, consider x, yX and suppose that x.--yi for
]" 0 .... , m- I, and that x> y,. Then we have
(x) - (y)>__ (o+ )'" (.,+ )
Ч[_ +- +- ]
ara+l+ I -- (am+ 1+ l)(am+ 2-+- 1) .... > (a 0+1) "- (a re+l)
am+l -- I am+ 1 -- I (am+l + t) (am+.-Jv 1)
(a o+1) - (am+l) am++ I t 3 9 -7 .... >0.
Hewitt alld Ross, Abstract harmollic allalysis, vol. I 7
98 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups 99
Thus z is a one-to-one continuous mapping of X into [0, 1 and accord-
ingly is a homeomorphism. Hence z(X) is a compact, 0-dimensional
subspace of [0, 1 having no isolated points. In particular, z (X) contains
no interval, and there is a point tclЅ,§[ such that fez(X). Let
A0-- {ecX' (e)< t} and AI= {eX' (e) > t}. Then A 0 and A1 are
complementary nonvoid open and closed subsets of X, and I () - (')1 =< §
if e, e' lie in A0 or if e, e' lie in A 1. In like fashion we partition A0 and
A 1" A 0 =A00UA01, A 1 --A10UA11, where the Ai. are open and closed in
X and ]z(e)- (e')I =< (§)s if e and e' lie in the same A,i. Proceeding
by finite induction, suppose that sets Ai,,s... i have already been defined
for l = 1 ..... k and {i 1, is, ..., ik} any sequence of O's and l's. Suppose
too that {Ai, is... h-10, Ai, is... h-11} is a partition of A, 1is... i_1 into open
and closed sets and that I(e)- (e')l <= (§)' if e and e' lie in the same
set Ai,is... h, for all relevant choices of i's and l's. Then (Ai, is...ik) is
contained in an interval of length (§)k, and the middle third of this
interval contains a point t' ў (X). Define A,l,s ... ,k0 as {e Ah, s ... , " (e) < t'}
and Ai,is...l as {aAi,s...'z(a)>t' }. This defines the sets A,,...i,
for all positive integers k and sequences {i 1, is .... , i} of O's and 's.
Now for an arbitrary infinite sequence of O's and 's, say f--
(i 1, i s, i .... , i, ...), define ў(i) =Ai,Ai,sAi, isi.... For every i,
the set ў(i) is a single point because [z(e)--z(e')] =<(§) whenever a
and e' lie in a single set Aili, ... i. The set of all points (i) is X itself,
because for each positive integer k, the sets Ai, is...i k partition X. For
the same reason, is one-to-one. Finally, to see that is continuous,
let e--ў(i) be any point of X and let U be any neighborhood of a.
Then the compactness of X guarantees that there is a positive integer k
such that A,is... U. Hence if i' is a sequence of O's and l's such that
• I .!
il = h, i2-- '2, "'', i =,, we have #(i') AiA ,... i U. Thus # is a one-
to-one continuous mapping of {0, }° onto X and is thus a homeo-
morphism 1.
We now make some observations about compact topological semi-
groups. A topological semigroup S is a semigroup endowed with a Haus-
dorff topology for which the mapping (x, y)--xy of S xS into S is
continuous. If z x =zy always implies that x =y, then S is said to
With only a little more effort we can prove that every totally disconnected
compact metric space Y with no isolated points is homeomorphic with {0,1} t%.
It is easy to see that Y can be partitioned into a finite number of open and closed
sets B ..... Ba each of diameter not exceeding 1. Each B i can be partitioned into
a finite number of open and closed sets Bil , Bi ..... Bia [a is the same positive
integer for each i], where each Bii has diameter not exceeding Ѕ, and so on. This
leads by an argument like that involving to the fact that Y is homeomorphic
with the product of a countably infinite number of finite discrete spaces. Then
using the result proved in the theorem, we find that Y is homeomorphic with
{o, }o.
satisfy the le/t cancellation law; if x z = y z implies that x = y, then S
satisfies the right cancellation law. An element / in a semigroup such
that/2--/is called an idempotent.
There are many differences between topological groups and arbitrary
topological semigroups. One striking difference is that topological semi-
groups need not be homogeneous. For example, let S= {0, I, , ,...}.
Define xy =max(x, y) in S and give S the topology it inherits from R.
Then S is a nonhomogeneous compact topological semigroup.
(9.16) Theorem. A compact semigroup S satis/ying the le/t and right
cacellation laws is a topological group.
Proof. Suppose first that S also satisfies
aS=Sa=S for all acS. ()
Fixa 0in SandchooseeSsuchthat ea o=a 0. Then for anyaS, we
havea 0x=aforsomexSandhenceea=ea 0x=a 0x=a. Thuseisa
left identity for S. The existence of a left inverse is immediate. As is
well known, S is therefore a group. To show that S is a topological
group, we need only to verify that inversion is continuous. Let U be
an open subset of S; we have to show that V = U - is open. Let x, e D,
be a net of elements in V' converging to x0 S. Each x 1 is in U' and
thus by the compactness of U', there is a subnet x 1,/hcE, of xg , and
an element y U' such that lira x=y. By the continuity of multi-
plication, x 0 y = lim xax t = e. That is, x o- y- V'. Therefore V' is
closed and V is open.
The previous paragraph shows that the theorem will be proved if
we prove (). Let a S be fixed and let A be any nonvoid closed subset
of S such that aA cA. [For example, S itself is such a set. 1 Evidently
ADaA DaeA D... and each aA is a closed set. The sequence
is a net in S and therefore there is a subnet a , oD, and a y S such
that lim a--y. For x cA, we have yx=lim akx. For each positive
integer k, there is an e0D such that ec 0 implies k>=k. That is,
axaA for e-->50. Consequently, yxaA for all k" i.e., yx aA.
We thus have
yA aA. (2)
Now consider xc aA. Then for each integer k> there is an xkA
k=l -- '
for which x =a x. There is a subnet xa ,/E, of x= and an element
zA such that lim =z. Then
x = lim (a kaxa ) = (lin aa (liam xa ) = y z y A
•
7*
100 Chapter II. Elements of the theory of topological groups § 9. Structure theory for compact and locally compact Abelian groups t01
This and (2) yield
yA= El aA. (3)
k=l
Setting A equal to S and then to aS in (3), we obtain y S =
and yaS = aaS. It follows that y S=yaS. Since S satisfies the
=1
left cancellation law, we obtain S=aS. The equaliўy S = Sa is proved
in like manner.
(9.17) Lemma. Let S be a compact semigroup and let a be any
element o/ S. Then {a, a , a a .... }- contains a closed A belian group G,
namely, G {a , a +1, .. .}-.
Proof. For k =, 2,..., let A={a , a + .... }. These sets have the
finite intersection property and therefore G= A is nonvoid. It is
=1
easy to verify that G is a closed Abelian subsemigroup of S.
Therefore, as shown in the proof of (9.6), it suffices to establish the
equality
yG=G, for all yG. ()
Assume that yGG for some yG. Let zG(yG)'; then yxz
for all xG. Therefore, for each xG, there exist neighborhoods U,, V,,
and W, of y, x, and z, respectively, in S, such that W, (U,V,)=.
Since U VDG and G is compact, there exist x .... x,v G such that
xEG '
U GG. Let U-- G, V= U V,, and W= . Clearly U,V,
= = = =
and W are open sets, y4 U, z W, VD G, and
wn(uv) =. (2)
Since yG, there is an integer q such that aqE U, and since zeG,
there are positive integers r, re, r a .... such that q<r<r=<.., and
--q
a'W(i-- 2 3 .... ). The net {a' }= has a cluster point u in S
which is necessarily also in G. For some k, k = .... , n, we have uV,
and thus for some m, re=l, 2 ..... we must have a'=-qI. Now
a r=a qa r=-q UV.UV.
Since a '= is also in W, this contradicts (2); therefore () holds.
(9.18) Theorem. Every compact semigroup contains at least one
idemotent.
Proof. This is immediate from (9.7).
(9.19) Definition. A nonvoid subset M of a semigroup S is called a
left ideal rigt ideal if SMcM MScM]. If M is a left ideal and a
right ideal, M is called a two-sided ideal. A semigroup S with no ideals
different from S itself is called simple. A semigroup S is called com-
pletely simple if it is simple and satisfies:
(i) if e and / are idempotents in S such that e/--/--/e, then e--/;
(ii) for every xS, there are idempotents e and / in S such that
eXXzX/.
(9.20) Lemma. A compact simple semigroup S is completely simple.
Proof. We first verify (9.19. ii). Let xS. Since S is simple, we
have S = S x S. Thus x--a x b for some a, b S, and evidently x--ax b
for k -- 2, 3,.-. • By (9.17), there is an idempotent e in the set (a, a 2, ...}-.
Let a , D, be a net in {a, a2,...} converging to e. There is a subnet
bka, fl E, of b , and a point b 0 (b, b 2 .... }- such that lim b a- b 0. Then
e x b 0-- lim a a x b a -- x. Therefore e x -- e (e x b0) = e x b 0-- x. Similarly
x/--x for some idempotent/ (b, b 2 .... }-.
Suppose now that e and / are idempotents in S such that e/=/-/e.
Since S is simple, there are elements a and b of S such that e a/b.
Let y-- ea/ and z z/b. Then y/z --e and yez--e, and therefore ya/z ----e
for k--2, 3,-... There is an idempotent g(y, y2 .... }-- and a net yk,
aD, converging to it. Further, there is a subnet z a,/5E, of z con-
verging to a point Zo(Z , z 2, ...}-. Therefore g/zo--e. Since y--ea/=
ea/e eSe and eSe is a closed subsemigroup of S, we have geSe.
Therefore ge--g so that g---ge----gg/zo--g/zo=e; i.e., e--e/zo--/z o.
Consequently,/--/e=//Zo--/zo--e.
(9.21) Theorem. A compact semigroup S contains a smallest two-
sided ideal K. The ideal K is closed and completely simple.
Proof. Let E be the set of idempotents in S; by (9.8) E is nonvoid.
Let K-- f-) SeS. If e, ... eE then each set SGS, k -- m,
eE ' ' ' " " " '
contains the element e e2...em---e I (e I e2... ea_l) ea (e+l ... era) e m and
hence the family (SeS'eE} of closed sets has the finite intersection
property. Therefore K is nonvoid. It is easy to see that K is a two-
sided ideal of S.
Let M be any, two-sided ideal of S and let xM. Then SxS is a
closed ideal and S x S c M. By (9. 8), S x S contains an idempotent e 0.
It follows that K C S eoSCM. Therefore K is the intersection of all
two-sided ideals of S, and is therefore the smallest two-sided ideal of S.
If xK, then KxK is a closed ideal of S and KxKcK. Hence
KxK =K. It follows that K is simple. By (9.20), K is completely
simple. ]
The smallest two-sided ideal of a conpact semigroup S, whose
existence was established in (9.2t), is called the kernel sometimes the
Sukevi5 kernel] of S.
102 Chapter II. Elements of the theory of topological groups §9. Structure theory for compact and locally compact Abelian groups 103
(9.22) Lemma. A completely simple semigroup S with identity e is a
group.
Proof. Let x be any element of S. Since S is simple, there are
elements a and b in S such that axb=e. Then bax and xba are idem-
potents. Thus by (9.19. i), we have b a x ---- e and x b a -- e. That is, b a
is an inverse for x. Therefore S is a group.
(9.23) Theorem. A simple compact semigroup S is a pairwise dis-
oint union o[ closed subgroups.
Proof. Let E be the set of all idempotents in S. We first prove that
if e and ] are in E, then e S] is a group. By (9.t6), it suffices to verify
the cancellation laws in e S[. The semigroup e Se is simple; indeed if
eye, exeeSe, then y--aexeb for some a,bS and hence eye=
(eae)(exe)(ebe). This implies that eSe is simple. By (9.20), eSe is
completely simple. Plainly e is an identity for e Se and thus by (9.22),
e Se is a group. Suppose now that
(z/) (x/) = (z/) (y/),
where x, y,zS. Let (ezfe) -1 be the inverse of ezfe in eSe. Then
exf--(ezfe)-(ezfe) xf--(ezfe)-(ezfe)yf--eyf. The right cancellation
law for e S/is proved in like manner.
Since S is completely simple (9.20), we have
s=u{sl:,l};
see (9.19.ii). We now show that the family of sets {eSl'e, IE} is
pairwise disjoint. Consider e Sf and let e 0 be its identity. Clearly we
have eSfce0Se 0. Thus e/ is in the group e0Se 0 so that e 0Se 0
e f (e 0 S e0) e f -- e (f e 0 S e0e ) f c e S f. Hence
eoSeo--eS f. (t)
Suppose now that G and H are in {e S f" e, fE} and have a point a in
common. Let eg and eh be the identities of G and H, respectively. Then
ega--a--eha. Let bG be such that ab--Q. Then e--eab
ee. Similarly en--ene and so eh----e. By (I), we have G--egSeg=
e S e-- H.
(9.24) Corollary. Let S be a compact semigroup, with kernel K. Then
K is the union of pairwise disjoint closed subgroups.
Miscellaneous theorems and examples
(9.25). Let L be a subgroup of R with a topology strictly stronger
than its topology as a subspace of R, under which it is a topological
group. Suppose that H 0 is a topological group and that there is a con-
tinuous homomorphism of H 0 onto L. Then Theorem (9.1) does not
hold if H is replaced by H 0. Let H 1 be any continuous homomorphic
image of a subgroup of R with its usual topology. Then Theorem (9.1)
holds for H. [The second assertion uses (.40.c).]
(9.26) Concerning compact elements. (a) (PONTRYAGIN [7], § 40,
Example 74.) Let G be a locally compact Abelian group, let C be
the component of e in G, and let B be the subgroup of all compact
elements of G. Then BC is an open subgroup of G. [Since B and C are
subgroups [(9.10) and (7.1)], BC is clearly a subgroup of G. Let H be
any compactly generated open subgroup of G; H is topologically iso-
morphic with R"xZ'xF. Let be a topological isomorphism carrying
R"xZ'xF onto H. Then z" (R") . "c (F) is an open subgroup of G, and
r (F) -- B H is the largest compact subgroup of H. Thus (R) • (F). CB
is open in G. Since (R ) is a connected subgroup of G, we have (R ) C C,
and so -c(R").z'(F).CB-CB.]
(b) If G is a locally compact, compactly generated Abelian group,
then the subgroup B of all compact elements is itself compact; it is the
largest compact subgroup of G. For an arbitrary locally compact
Abelian group, the subgroup B of all compact elements need not be
compact. [The first assertion is immediate from (9.8). The second
assertion follows from the fact that there are infinite torsion groups.
Simply endow such a group with the discrete topology. As a less trivial
example, consider any group ,Q [(10.5) in/ra].
(C) (E. THOMA, letter to the writers.) Consider the discrete group G
generated by elements a and b and satisfying the relations a=b=e.
Then a and b are compact elements, but a b is not a compact element.
Thus the compact elements of G do not form a group.
(d) Consider the general linear group 63(2, K). For n =2, 3 .... ,
let z--exp [2i (I/n). An easy induction argument shows that
In particular, we see that
z, +z, +..-+z,+1
t '
since the sum of all n-th roots of unity is zero for n = 2, 3 ..... Therefore
t generates a finite subgroup of ®g (2, K) and is a compact element.
To show that the set of all compact elements of ®g (2, K) is not closed,
104
Chapter II. Elements of the theory of topological groups
we need only observe that 1 o 1
This is obvious since I, -- for k--I, 2,....
(e) (MARKOV [I .) A topological Abelian group G is topologically iso-
morphic withR for some positive integer n if and onlyif: (I) G 4= {e}; (2) G
is locally compact; (3) G is connected; and (4) G contains no compact
subgroup except {e}. [This follows readily from (9.14).
(f) (vA KAp [I, p. 460.) Let G be a locally compact Abelian
group and C the component of the identity. If G/C is compact, then G
is topologically isomorphic with RxF for a nonnegative integer n and
a compact Abelian group F. [By (7.4), C is compactly generated. By
(5.39.h), G is compactly generated and hence Theorem (9.8) tells us
that G is topologically isomorphic with R xZxF, where F is a compact
Abelian group. If C 0 is the component of the identity e in F, then C is
topologically isomorphic with RxCo . Since Rx{O}xCoCRx{O}xF
cRxZxF, Theorem (5.35)implies that (RxZ)/(R'{O}xF)
is topologically isomorphic with (G/C)/(F/Co). That is, Z is topologically
isomorphic with a continuous image of the compact group G/C. Con-
sequently, Z is compact and m=0.
(9.27) Topological semigroups. (a) The group G given in (9.17) is
the largest subgroup of {a, a,...}-. [Use the fact that {a, a 2, ...}-=
(b) The partition of a simple compact semigroup into subgroups is
unique;see (9.23).
(c) (GELBAUM, KALISCH, and OLMSTED [I].) Unlike groups, semi-
groups need not be homogeneous. In particular, the following implica-
tion may fail:
U open in S and x in S imply x U open in S. (I)
However, given any commutative topological semigroup S with identity
e, we can find a stronger topology under which S is a topological semi-
group satisfying (I) and such that the neighborhoods at e are the same
under these two topologies. F A topological semigroup not satisfying (I)
is [0, I with the usual multiplication and topology, since 0. 0, I- {0}.
To prove the second assertion, let 0 denote the given topology of S
and let ' be a basis of open sets at e. We define the new topology 01
on S by requiring that {xU: U[} be a neighborhood basis at x, for
each xS. Denote this open basis by 1; we must verify that N1 is
really a basis for a topology. This will be the case if given aU, b V,l
where U, V and caUf'lbV, there exists a W?[ such that
cWCaUf'lbV. We have c=au=bv where uU and vV. By con-
tinuity of the mapping x--ux at e, there is a U1 c ' such that u[J C U.
§ 9. Structure theory for compact and locally compact Abelian groups 105
Similarly, v V c V for some V '. Choosing W in ' such that W U1 n v1,
we find that
cW c (c n (c = u n vx) c u) n v) .
To show that multiplication is continuous in the )x-topology, let
a, b(S and U(@'. If V(/ and VC U, then a Vb V=ab VScab U.
Obviously the Ox-topology of S satisfies (I). Finally, we see that 1
is stronger than , because if W(D and a(W, then the continuity of
x-+ax at e implies that aUc W for some U(@'.
(9.28) Monothetic semigroups. A topological semigroup S is said
to be monothetic if it contains an element a such that {a, a s .... } is dense
in S. In this context only, a will be called a generator of S.
(a) (HEWlTT [4.) Let S be a compact monothetic semigroup, and
let a denote a generator of S. Let G denote the closed subgroup
Cl {a *, a*+l, ...}-, and let e be the identity of G. One of the following
situations must hold"
(i) S =G, in which case S is a compact monothetic group"
(ii) there is a positive integer m such that SIG'={a, a s, ..., a"}
and all of the elements a, a .... , a are isolated points of S;
(iii) S gl G'= {a, a s .... } and all of the elements a, as, ... are isolated
points of S.
[By (9.17), G is a closed subset of S that is a group" thus by (9.16),
G is a topological group. We next prove that if a "G and p> n, then
a G. Let k be an arbitrary positive integer and U a neighborhood of a .
By continuity of multiplication, there is a neighborhood V of a"such
that Va -' C U. For some integer l>= k, a is in V since a {a , a+l,...}-.
Then a+t -( U and l + p -- n> k, so that U intersects {a , a *+x .... }.
+1, ..}-; as k is arbitrary, a t is in G.
Thus a {a , a .
The last paragraph shows that one of the following equalities holds"
.... } .... ),
Gf-I {a, a, .... } ={a '+1, a '+e .... ) where m=> I, (2)
,r. .... }=e.
Any element of S that is not an isolated point must lie in every set
{ a, a+ .... }- and hence in G. In particular, we have S gl {a, a s .... }'c G
and S f) G'= {a, a .... } gl G'. Also, all points in S f) G' are isolated. Using
these remarks, we see that () implies (i), (2) implies (ii), and (3) implies
(iii).]
(b) A compact monothetic semigroup S with two distinct generator.
a and b is a topological group. [Assume that S is not a topological groups
106 Chapter II. Elements of the theory of topological groups § t0. Some special locally compact Abelian groups 107
Then a and b are isolated points by (a), and hence ac{b, b .... } and
bc{a, a s .... }. Therefore a=b" and b--a"for some integers m and n
greater than I. Thus a=a"" and a is in G = fl {a , a +1, ...}-. Con-
sequently, S =G and S is a topological group.] ---1
Notes
At the present time, it is difficult to assign exact priorities to all of
the results (9.1)--(9.14). Theorem (9.3) for connected locally compact
Abelian groups was proved by PONTRYAGIN [3 and [4], VAN KAMPEN [1 ],
and MARKOV [I. The earliest statement known to us of (9.t) [in a
trivially more special form is in WEIL [4, p. 96. The fact that (9.3)
holds for compactly generated groups and not merely connected groups
was apparently first noticed by WEIL E4, p. 97. Theorem (9.5) is due to
VAN KAMPEN [t], p. 458. The first explicit statement of (9.8) known
to us is in WEIL [4, p. 99, where a proof attributed to VAN KAMPEN [t]
is given. A second proof of (9.8) based on the PONTRYAGIN-VAN KAMPEI
duality theorem appears in WEIL [4, p. t I0. Theorem (9.t 4) for groups
with countable open bases is due to PONTRYAGIN [2] and [4]. A general-
ization of (9.14) appears in VAN KAMPEN [, p. 460.
Theorem (9.15) is due to IVANOVSKIi [I] and KUZ'MINOV [I. For
a commutative G, see VILENKIN [I 7. The proof given here is based on
HULANICKI [4]. KUZ'MINOV proves a different result: every compact.
group is a continuous image of some space {0, 1}m. In this connection,
see (25.3 5).
In (9.1) -- (9.t 5), we merely scratch the surface of what is known
about the structure of locally compact groups. For non-Abelian groups,
in particular Lie groups, we refer the reader to the monographs of
MONTGOMERY and ZIPPIN [I], CHEVALLEY [I], and PONTRYAGIN [7],
Chaps. VII--XI. Locally compact Abelian groups are examined in some
detail in §§ 24--26 in/ra. Even for this case there is an enormous litera-
ture: for a partial bibliography, see the notes to § 25.
Theorems (9.16)--(9.22) are due to NUMAKURA [I. Again, we have
given only a glimpse of the known structure of topological semigroups.
For a useful survey of the field, see A.D. WALLACE
§ 10. Some special locally compact Abelian groups
We now describe a class of locally compact Abelian groups which are
of importance in the structure theory of locally compact Abelian groups
and are also interesting on their own account. To define these groups,
we begin with some quite elementary facts about integers.
(10.1) Let a =(a 0, aa, a. ..... a, .... ) be any sequence of integers
each of which is greater than 1. If M is any positive integer, we obtain
by successive divisions"
M= x 0 + M 0 a o , where x o {0, 1 .... , a 0-1}
and M0 is a nonnegative integer;
Mo=xl+Mlax, where xl {0, t ..... aa--l}
and M is a nonnegative integer;
M=x.+M.a., "where x.{0, t, ..., a.--t} (1)
and Ms is a nonnegative integer;
Mk_ = x+ M a, where x {0, 1 ..... a-- t }
and Mk is a nonnegative integer;
•
The Mk's decrease strictly as long as they are positive, and so there is
a least integer m such that M,,,=O; defining M_ as x0, we have in all
cases 0 =t= M_t = x, and
M-- x0 + x a0 + x a0 + "-" + x a0 a... %_. (2)
It is also easy to see that every positive integer can be written in exactly
one way as a sum of the form (2). Thus we have a one-to-one correspond-
ence between the set of all nonnegative integers and the set of all
sequences (Xo, X,X .... ,xt,... ) such that xt{O,l ..... at--l } for
l =0, I, 2 .... , and only a finite number of the xt's are different from 0.
For a positive integer N, let us carry out the divisions (t), obtaining
X=yo+oao, No= y + N , N= y+ N, a, ..... N_ = y. (3)
where the 's and yi's satisfy conditions as in (). Then we get
N= Yo+ Y ao+ Y, ao + "'" + Y.ao "" a_t. (4)
Similarly, we write
M + = Zo+ ao, = z + , = z + a, ..... g_ = z,, (5)
and
M+ N=zo+ z ao+ z ao + ... + z ao a ... %_. (6)
Obviously the z's are deteined from the x's and y's. To make this
COmputation directly, we proceed as follows. From (), (3), and (5), we
have M+N=z o+a o=xo+yo+(Mo+No) ao, so that xo+Yo=
108 Chapter II. Elements of the theory of topological groups § t0. Some special locally compact &belian groups t09
(Po--Mo--No) ao+Zo--toao+Zo. Plainly z o is the least nonnegative
residue of x o + Yo modulo ao, and this also determines t o --Po- Mo--No
directly from x o and Yo. Using (I), (3), and () again, we have Po
z 1 + P a 1- t o + M o + N o- t o + xl + y + (M + N) a. That is, xl + y + t o
z+ (P-- M-- N) al-- z+ t al. Plainly z is the least nonnegative
residue of x + yl + t o modulo a, and this fact also defines t--P--M--N
in terms of x o, x, Yo, and yl.
To complete the construction by finite induction, suppose that we have"
Xo+ Yo-- to ao + Zo, x + y + t o-- t a + zl, ..., x k + yk + t_ -- tl a k + z,
where z i is the least nonnegative residue of xj+ yj+tj_ modulo a i
('-- 0, I,..., k; t_--0)" and t.---- P.-- M.-- (]'-- 0, .... , k). Then we have
z+ + P+I a+-- P-- t + Mk + N= t + x+ + y+ + (M+ + Nk+I) ak+;
thus x+ + y+ + t--z+l+ (P+--M+I--N+) a+--zk+ + t+ a+.
Since z+l 0, .... , a+--}, the induction is complete.
Thus addition in the semigroup of nonnegative integers can be trans-
ferred to the set of sequences (x o, x, x 2 .... ) 0<= xn< a n for all n, and
xn--0 for n sufficiently large] without reference to the correspondence
indicated in (2). If (x o, x, x. ..... x m, 0, 0 .... ) and (Yo, Y, Y2 .... , Yn,
0, 0 .... ) are two such sequences, we define their sum by the following
rule. Write Xo+Yo--toao+Zo, where Zo0, , ...,ao--I ) and t o is an
integer. Write xl+ y+ t o- t a + z, where zl 0, , ..., a 1- ) and tl
is an integer. When z o, z, ..., z_ and t o, t, ..., t_ have been defined,
write x+ y+ tk_l-- t a+ z, where z 0, , ..., ak-- ) and t is an
integer. This defines sequences (t o, t, t 2 .... ) and (Zo, z, z2,...)" note
that each t is either 0 or . It is clear that the largest integer p such that
z =t= 0 is max (m, n) or max (m, n) + . We define (x o , x, x 2 .... , x, 0, 0, ...)
-- (Yo, Yl, Y2 .... , y, 0, 0,...) to be (z o, z, z2, ..., z, 0, 0 .... ).
A moment's reflection shows that in the definition of addition just
given, there is no need for the added sequences to have only finitely
many nonzero entries. Another moment's reflection shows that there
is no need for the sequences to begin with x o" sequences ( .... x_,,
xl_, ..., x_2, x_, x o, x, x 2, ..., x .... ) can be added, provided only
that x is 0 [or k < n o, n o being some integer depending on the sequence.
We proceed to a formal definition.
(10.2) Definition. Let be a fixed but arbitrary doubly infinite
sequence of positive integers:
a 2 .... , a n .... ), where each a n is greater than . Consider the Cartesian
product Pz0, I, ..., a n- ). Let Qu be the set of all e--(x) in this
product space such that xn is 0 for all n<n o, where n o is an integer
that depends upon e. For e and t--(Yn) in Q, let z--(zn) be defined
as follows. Suppose that Xmo 4 = 0 and x,,=0 for n < rn o, and that yo=t=0
and Yn--0 for n< no. Then let zn--0 for n<Po=min (rno, no). Write
-- where ў {0, 1, , -- 1 } and is an integer.
,'po + Ypo = l0 apo Zo , Zo'- "'" apo
Suppose that Zo, zfo,..., z and to, tfo,,..., t have been defined.
Then write x+ + y_ + tk=t+ a+ + z+, where z+l{0, , ..., a+-- }
and t ,1 is an integer. This defines by induction a sequence z = (z,)
We define the sum a+ to be the sequence z. To complete the definition
of addition in ,O,,, we define 0+ e=e+ 0=e for all e,O,,, where 0 is
the sequence in 2, that is identically zero. The subset of 2, consisting
of all e for which x=0 for n< 0 will be denoted by
The set D,, with the addition defined above will be called the et-adic
umbers. The set A, with the addition defined above will be called the
-adic integers. If all of the integers a, are equal to some fixed integer
r> , we write 2, and A, for 2,, and A,,, and call these objects the
r-adic numbers and r-adic integers respectively. When A,, is considered
by itself, there is obviously no need to consider negative indices n, and
we will take the liberty of writing elements of A,, as a= (x 0, x, xe,...)
and of ignoring a for n < 0.
(10.3) Theorem. With the operation+ de/ined in (0.2), the set
is an A belian group, containing A, as a subgroup. The group A, contains
a subgroup isomorphic to Z.
Proof. It is obvious that 0+e=e for all eQ, and
for all e, ,. We now find the inverse of an element eD, different
from 0. Suppose that x,, is different from 0 and xn= 0 for n< m. Define
the element in , by the following rule : y = 0 for n < m; Ym = am-- Xm
y = a,-- x-- 1 for n > m. It is plain that e+ = O.
To complete the proof that ,O, is an Abelian group, we need only to
show that the associative law holds. Let e, V, z be any elements of
and let m be the least integer for which x or Ym or z,, is different from O.
It is then trivial that the values of (e+ )+ z and e+ (+ z) at the
index n are equal, and equal to O, for all n< m. Consider now the set
X of all elements u,O, such that u--O for n<rn and only a finite
number of the values u n are different from O. It follows from (I0.I)
that the mapping U--+Um+U,n+ am+U,+ea,nam+ + ... is an isomor-
phism of Z' onto the additive semigroup of nonnegative integers.
Consequently. the associative law holds in Z'm, since addition of integers
s associative. Return now to our arbitrary e, , and z. Consider any
fixed, index prn and the "truncated" elements e', ', ' such that
x x for n_<__p and X'n=O for n> p, with ' and z' defined similarly.
Then (a'+ /')+ z'--a'+ ('+ z') since e', ', and z' are in m" The
Value of (a'+ t')+ z' at the index p is obviously the same as the value
of (a+ /)+ z at the index p" a similar remark applies to e'+ ('+
and a+ (_+_ z). This proves the associative law in Ј2, and completes
the proof that Ј2, is an Abelian group.
Chapter II. Elements of the theory of topological groups § t0. Some special locally compact Abelian groups
It is obvious from the foregoing that Ala is a subgroup of Qa. If
and y are in ,, and x= y----0 for n< 0, then x+ y has the value 0 at
all indices n < 0, and -- x also has the value 0 at all indices n < 0. Let u
be the element of Al,, such that u----0 for n<0, u0----t, and u----0 for
n:>0. Then 0, u, 2u ..... ku, ...} is a subsemigroup of/l isomorphic
to the additive nonnegative integers, and so Al,, contains the subgroup
(0, + u, -+- 2 u ..... -+- k u .... ), which is isomorphic to Z./Of course every el-
ement of 2a of infinite order generates a subgroup isomorphic with Z.]
As is to be expected, both Q and Al are topological groups under a
suitable topology.
(10.4) Definition. For each integer k, let Ah be the set of all xQ a
such that x----0 for all n< k. For distinct elements x, y, let a(x, y)
be the number 2 -m, where m is the least integer for which Xm:4: Ym"
For all x, let a(x, x)=0.
(10.5) Theorem. The sets .... A_h, ..., A_., A_I, Ao, At, A., A3,
• .., Ah, ... satis/y conditions (4.5. i)-- (4.5. v). Hence they define a topology
on 2, as in (4.5) under which [2, is a topological group. Under this topol-
ogy, 2 is Hausdor//, locally compact, a-compact, and O-dimensional. The
sets Ah are compact subgroups o/[2,. [Note that A o = A,.] The/unction a
is an invariant metric on [2, compatible with the topology o/[2.
Proof. It is evident that the sets A are subgroups of Qa, so that
(4.5. i), (4.5. ii), and (4.5. iii) are trivially satisfied. Since . is Abelian,
(4.5.iv) is also trivially satisfied, and (4.5. v) is obvious. Plainly 0 is in
allah, so that (4.5) may be applied, anda is a topological group in which
the sets x+ Ah are a basis for open sets. [Note that this is a special case
of (4.2t.a).] Since CI Ah= {0), ,, is a Hausdorff group (4.8).
k=0
Considered as a set, Ah may be regarded as the complete Cartesian
product ._P,{0, t ..... a--t). For xAh and l>k, the set x+A, is the
set of all elements (..., 0, ..., 0(h-), xh, xh+ ..... x,_, z,, z,+ .... ) of
Q., where z i is an arbitrary number in {0, t, ..., a i- t } for/'----l, l+ t, ... •
This shows that the relative topology of Ah as a subspace of a is the
Cartesian product topology of {0, t a--t}. This is a compact
' ...,
topology, and so the open subgroupsA are compact and closed. Therefore
. is locally compact and 0-dimensional.
The set A of all x 2,, such that x,=0 for n:> 0 is plainly a countably
infinite set. Since/2.-- A + A 0 ==U a (x + A0), . is a-compact.
It remains to verify our assertions about the function a. It is obvious
that a(x, y)--a(y, ) and that a(x, y) is positive unless x=y. To
verify the identity
( + z, u + z)=(, u), ()
it suffices to consider the case in which xm = Ym and x= y for all n < m.
Applying (t0.2), we write xm+ Zm+ tin-l--tin am+ Vm and
t a=+ v,, where v m, Vm{0, t, ..., a m- t} and tin_l, t m, tm are integers.
If v,=v,, we would have x m = ym(mod am), which is impossible. It
follows that is invariant.
The function satisfies an inequality stronger than the usual triangle
inequality, viz."
(x + y, 0) _ max { (x, 0), (y, 0)}. (2)
To prove (2), let x and y be different from 0 [if either is 0, (2) is trivial,
and let p and q be the least indices for which we have xf=0 and yf0,
respectively. If p 4:q, then the least index at which x+ y is different
from 0 is min(p, q), so that in this case (x+y, 0):2-m"{f'q}=
max (2 -f, 2-q). If p=q, then the least index r at which x + y is different
from 0 may not exist or may be any integer greater than or equal to p.
Thus we have a (x + y, 0) = 0 or a (x + y, 0) : 2-' =< 2 -. Thus (2) is
verified. The relations (t) and (2) obviously imply that the triangle
inequality holds for a.
Finally, to see that a is compatible with the topology of Q,,, we need
only note that A={/2'a(, 0)<2 -+} for all kZ.
(10.6) Note. All of the groups Ah are compact and monothetic.
Let u be the element of A such that u,=0 if n :4: k and uh--t. Then it
is clear that the set {l u}x is dense in A. The set {l u},°°___oo is a [ortiori
dense in A.
(10.7) Our success in introducing addition in in such a way as to
make it an Abelian group leads at once to the thought that perhaps a
natural multiplication can also be introduced in , which can be inter-
preted as ordinary multiplication of rational numbers for elements of
having only a finite number of nonzero values.
It is simple to describe this multiplication in 4 for an arbitrary
sequence a=(a0, a, a, ...). In this paragraph, we write the elements
of /l as sequences x=(x0, x, x, ...). Let u denote the sequence
(t, 0, 0 .... )/l. For l,l'Z, let (lu)(l'u) be defined as (ll')u. It is plain
that in this way we reproduce multiplication of integers in the subset
{l u" lZ} of 4a. Thus the mapping l-->l u is an isomorphism under both
addition and multiplication. To define xy for arbitrary x, y4, we
cc ,,
truncate x and y. For every nonnegative integer m, let x (m) be the
element (x(m)) of /I. such that xm)=x,, if n=<m and x(ml=0 if n>m.
Define y() similarly. Then a(m)y (m) is already defined for all m=
0, t, 2 ..... We define x y as lim x(m)y Ira). One can prove that this
limit exists and defines an associative, commutative, and distributive
112 Chapter II. Elements of the theory of topological groups § 10. Some special locally compact Abelian groups
product in Aa. Furthermore, the mapping (oe, y)-oey of AaxA. onto
A. is continuous. Thus A. is a compact topological ring 1.
(10.8) We now take up the problem of defining a reasonable multi-
plication in the larger groups Q,,. Here it is convenient to restrict
ourselves to the case in which all a are equal: that is, we consider only
the groups O, for an arbitrary integer r:> t. As in the case of addition
in 2a, we define multiplication in 2, as an extension of ordinary multi-
plication. Consider any positive rational number x whose denominator
is a power of r:
x= x r + x+l /+1 + ... + Xm r', (t)
where l<=m, l and m are integers, and the x]'s are integers in the set
{0, ..... r-- }. If we require further that x:4=0 and x m :4=0, then this
representation is unique. Given another rational number of the same sort,
,s+l
y ---= y, r' + Ys+l + "'" + Yt
,/+s+l
their product xy has the form Z+s
[Note that z+, may be 0. As in (t0.t), we can compute the numbers z
directly from the xi's and y/s by the following rule. Write xy=
v+,r+z+,, where z+,{0, t, ..., r--t} and v+, is an integer. Next
write x+ly,+ xy,+l +v+,=v+,+l r +Z+s+l, where Zl+s+l{O , t .... ,
r-- t} and v+,+l is an integer. Proceeding in an obvious way by induc-
tion, we see that the product of two rational numbers of the form (t)
can be computed directly from the x] s and Yi s. This leads to a definition
of multiplication in
(10.9) Definition. Let oe and y be any elements of D,. If =0 or
y--0, let oey be defined as 0. If neither oe nor y is 0, suppose that
x0 and x=0 for n<l and y:4=0 and y=0 for n<s. Define an
element zD, by the following rules. For n<l+s, set z=0. Write
xy=v+r +z+s, where Z+s{0, t, ..., r-- I} and v+ s is an integer.
Suppose that the numbers v+, v++l, ..., V+s+k_l and z+, z++l, ...,
z+,+_ 1 have been defined. Then we write x+kys+X+_ly,+l+
x+_y+ + ... + xy+ + V+s+_l=v++r + z+s+k, where z+,+
{0, t, ..., r--t} and V+s+ is an integer. This defines the numbers
by induction for all n>=l+s. Let the product oey be the element z.
(10.10) Theorem. With the addition o/ (t0.2) and the multiplication
o/(t0.9), D, is a commutative ring with unit. The ring D, has divisors o/0
i/r is not a prime power. It is a/ield i/and only i/r is a prime power.
1 A topological ring A is a ring endowed with a topology such that, as an additive
group, A is a topological group and such that the mapping (x, y) --> x y is continuous.
A topological field F is a'field that is a topological ring for which the mapping x -> x -x
is a homeomorphism on F {0}'.
The field Qp is usually called the p-adic number field.
113
Proof. In view of (10.3), where we proved inter alia that c2,, is an
Abelian group, Ј2, will be proved to be a commutative ring as soon as
the identities oey=yoe, oe(yz)--(oey)z, and oe(y+z)--oey+ez
(x, y, z-Q,) are established. These are obvious for oe, y, z having only
a finite number of nonzero entries, since in this case our multiplication
is nothing but multiplication of nonnegative rational numbers. These
identities are extended to arbitrary elements of Ј2, by the argument
used in the proof of (10.3) to prove the associative law for addition. We
omit the details.
The element uЈ), such that u,=do,, [Kronecker's delta symbol is
plainly a multiplicative unit for Ј2,.
Let r be a positive integer that is not a prime power" then r has the
form ab where a and b are integers greater than I and (a, b)=t. The ringЈ),
abounds in divisors of 0. We give one example. Let x--0 for <0,
x0 =a, and y--0 for n< 0, yo=b. Let Xl, Yl, and u 1 be chosen so that
0<xl<r, 0--<_yl<r, Ul is an integer, and Xl b+yla+t =ulr. Such
xl, Yl, and u I can be found, since (a, b) = I. When x 1 .... , x, Yl, ..., Y,
and Ul, ..., u have been chosen, let X+l, Y+I, and u+ 1 be determined
so that xn+ 1 b+ Yn+l a+ x Yl @ "'" @ Xl Yn-- gn--U,,+l r. This defines
e and y by induction, and it is obvious that oe y = 0.
Now let p be a prime and a positive integer. To prove that D
is a field, we need only to show that every nonzero element oe of Ј2
has a multiplicative inverse. Suppose that x@0 and that x--0 for
n< l. If (x, p)= , we define an element y Ј2 by the following rules.
For n<--l, let y.0. Let y_ be the [unique I integer such that
y_x=l-/v opt, where y_{l, 2, ..., p-- I} and v 0 is a nonnegative
integer. Suppose that y_, Y-+I .... , Y-+-I and v0, Vl, ... , v_ 1 have
been defined. Let y_+ be the unique integer such that y_+ x+
Y-l+k--1Xll - "'" - Y--l Xl+k -Vk-I--Vk pCZ, where y_+{0, I, 2 .... ,
P--I} and v is a nonnegative integer. This description defines y by
induction for all n>=--l. It is obvious from (10.9) that oey----u. Suppose
next that (x,p)>, so that x--px, where (x,p)--1 and 1_<_fl<.
Let ',v6Q be defined by w,,--p - bo,, (nZ) and let z--woe. Then
plainly z----0 for n <2 l. We also have p-x--px--t p+ z, so that
z0 and t-- x. Furthermore, to compute z 1, we have p- X+l + t--
P-ZX+l+X--t+lp+z+l . This implies that z+14=0 and that
(z+, P)1. Hence the element zЈ) satisfies the conditions of the
previous case and so admits a multiplicative inverse z -. That is, we
have u zz-l= oe (iv z-l), so that oe too has a multiplicative inverse. ° [
1 The symbol (a, b) as usual means the greatest common divisor of a and b.
e The last statement of this theorem for :> 1, and its proof, were kindly pointed
out to us by JAMES MICHELOW.
Hewitt and Ross, Abstract harlnonic analysis, vol. I N
114 Chapter II. Elements of the theory of topological groups § 10. Some special locally compact Abelian groups 115
(10.11) Theorem. The mapping (x, t) -oet is a continuous mapping
o[ D, xD, onto ,. Thus , is a locally compact ring. I] r is a prime
power p, the mapping --oe -1 is a homeomorphism o] DN{0}' onto
itsel]. Thus Ј21a is a locally compact [ield.
Proof. Let A be any neighborhood of 0, as defined in (t0.4). Let
e and f be arbitrary elements of ,. If e 4= 0, let s be the least integer
such that cs0; if e=0, let s=0. If f4=0, let t be the least integer
such that d0; if f=0, let t=0. Let l=max(k--s, k--t, s). Then if
x--cA and y--dA, we have xy--cd=x(y--d) + (x--c)dA.
Thus the mapping (x, y)--xy is continuous.
Suppose now that r is a prime power, say p. Let c be a nonzero
element of 2, and let s be the least integer such that cs 0. Let A
be any neighborhood of 0 as in (t0.4) such that k> Is[. I (c,, p)--, the
inverse d=c - has the form ( .... 0, ..., 0, d_s, d_s+, ...), where d_s4:0,
as was shown in the proof of (t0.t0). If x--cA+2s, the rule given in
the proof of (t0.t0) for computing inverses shows that oe---c-A.
If (c,, p)=pa with t _--</<, then as in the proof of (t0.t0), we define
w by w-=p -a 0, and write =wc. The element z has the form
(..., 0, ..., 0, zs+, zs+2, ...), where zs+l=t=0 and (zs+,/5)=t. Let k be
an integer greater than Is+ t [. Suppose that x--cA+2(s+. Then we
have woe--wcw • A+2(,+cA+,.(s+. The previous case shows that
(W ) -I -- (W C) -I /k . Hence we have
w • A :A. This proves that inversion is continuous. Since (e-1)-1=,
the mapping e-1-- is also continuous.
The groups A. are useful in constructing certain examples of compact
connected groups.
(10.12) Definition. Consider the locally compact Abelian group
RxA,,, written additively, where = (do, aa, a.,...) is any sequence of
integers all greater than t. Let u=(u) be (t, 0, 0, ..., 0, ...). Let B
be the subgroup {(n, n u)}=_oo of RxA,,. Let Z'. be the group (RxA,)[B.
We call . the a-adic solenoid.
(10.13) Theorem. The group 27. is a compact, connected Abelia
group containing a continuous homomorph o/ R as a dense subgroup.
Proof. It is easy to see that B is an infinite cyclic discrete subgroup
of RxA. and is hence closed (5.t0). Thus Z'. is a T o group (5.2t) and
is locally compact. Since RxA. is Abelian, so is Z'..
Let 9 denote the natural homomorphism of RxAa onto Z'a. Then
we have 9(Rx{0})={(, 0)+ B'R}, which is a subgroup of Z'.. If
?R and m is any integer, then (?,mu)+B=(?--m, 0)+B is in
9(Rx{0}). Thus {(, mu)+ B'R, mZ}=9(Rx{O})" this equality
and the fact that {mu'mZ} is dense in A. show that 9(Rx{0}) is
dense in .. If =(t, 0, t, 0, t, 0, ...), then (0, e)+B is not in
0(Rx{0}), so that 9(Rx{0}) is a proper subgroup of &. Since Rx{0}
is homeomorphic with R, its continuous image 9(Rx{0}) is connected
and its closure Z'. is also connected.
To show that Z'. is compact, we will apply Theorem (9.t). Assume
that the continuous homomorphism established above of Rx {0} into Z'.
is a topological isomorphism. Then by (5.tt), 9(Rx{0}) is a closed
subgroup of Z'.. This contradicts the fact that 9(Rx{0}) is a proper
dense subset of Z'.. Hence (9.t) shows that 9(Rx{0})-=Z'. is com-
pact. [
(10.14) The groups RxA. and Z'. show that the hypotheses of
Corollary (7.t3) are essential. Since Z'. is connected, the component of
the identity in Z'. is Z'.. The component of the identity in RxA,, is
obviously Rx{0}. Since 9(Rx{0})is not the entire group Z', the
conclusion of (7.t3) is false for RxA., , and the mapping 9. Note
that the kernel of 9 is B, which is not contained in Rx{0}, nor is
Rx(0} contained in B. Thus the hypotheses of (7.t3) are violated in the
present example. Note too that (7.t2) holds in the present example.
The group X. can be realized in a quite concrete way.
(10.15) Theorem. Consider the set [0, t[xAa, with the /ollowing
binary operation +:
(i) (, e) + (, y) = ( + -- [ + r, e + y- [ + r
is as de[ined in (t0.t2). For n=t, 2, ..., let
1
(ii) U -- {(, e) [0, t[ xM,,'O <= < n and x o ..... x_l-- 0}
u {(, ) [0, F_ Ч z.- - ± < <
..... .
With the operation + and the neighborhoods U,taken as an open basis at
(0, 0), [0, t [xA. is a topological group topologically isomorphic with ,Y,..
Proof. Every coset (, e)+BZ'. contains the element (--[],
a-- [] u). In other words, (, e) + B contains an element of [0, t [xA..
Suppose that the elements (, e) and (, /) of [0, t [xA,, belong to the
same coset of B. Then (--r/, e--/) = (n, n ) for some integer n. This
implies that n=0, so that (, e)=(, /). Thus each coset (, e)+ B
contains exactly one element of [0, t[xA., and we have an obvious
one-to-one mapping of Z'. onto [0, t [xA.. The definition (i) of addition
in [0, t [xA,, is obviously the same as for ((, e)+ B) +((, /)+ B) =
(+, + )+ B.
An open basis at the identity B in Z',, is obtained as {9(]-- e, e[xA)}
for 0< e< t, and k=t, 2 .... • this is pointed out in (5.t 5). It follows
Chapter II. Elements of the theory of topological groups Chapter III. Integration on locally compact spaces 117
---- xA, is an open basis at B. The set 0,
maps onto U,[ 1) and the set 1-- --n , O[ x A,, onto U. (2). Thus
witl the structure defined by (i) and (ii) is a replica of Z',.
Miscellaneous theorems and examples
(10.16) (a) The only proper closed subgroups of Qp, where p is a
prime, are the subgroups Ak defined in (10.4). Let a= (..., a_l, a 0, a .... )
be a doubly infinite sequence of integers that are not all equal to the
same prime number. Then Qa contains a compact open subgroup H
distinct from all the subgroups A.
Let H denote a proper closed subgroup of p. We first prove that
if H contains an element ac such that xn--0 for n<k and xk0, then
AcH. It suffices to prove that ac generates a dense subgroup of A.
Consider yA and a neighborhood y+A z of y where l> k. The group
A,/Az is a finite discrete group" it is isomorphic to the cyclic group
Z (p-) under the mapping ae +A-x ч x+ p + • • • ч x_ p- -.
From this we see that ae+A generates the entire group A,/A. In
particular, ma +A=y+A, for some integer m, and hence maey+A,.
Therefore {mac)=_oo is dense in A and A, c H. Since H ,(2, there is
a least integer ]" such that AicH. The foregoing implies at once that
HcA/.
Now consider ,(2,, where a (..., p, p, p .... ) for all primes p. Sup-
pose that a, is a composite number for some integer k" a--r s for positive
integers r and s greater than I. Consider the element ac such that xk--r
and x,--0 for n k, and the smallest closed subgroup H of ,(2 contain-
ing a. Then we have H--]=0U (1ac+ A, 1), so that A+I H A, . Suppose
finally that a, is prime for all nZ. Choose k so that a,a,+. The
numbers ak and ae are relatively prime. Hence there is an integer m
such that 0_<m< a,+ and a,m + I = 0 (rood a,+l). Let ae be the element
such that x--1, x,+--m, and x,0 for all other n in Z. It is easy to
see that a, ae belongs to Ak+.. It follows that the group F in A,/A,+.
generated by ac+A,+, has at most a elements. Plainly A,/A+. itself
has a ak+ elements, and therefore F is a proper subgroup of A,/A,+..
ak--1
It follows that H--iU ° (jae +A+.) is a compact open subgroup of
and it is a proper subgroup of A. Since a belongs to H and not to A+,
we infer that H is equal to no
(b) Some groups ,Q contain proper closed subgroups that are not
open. [For example, let a be defined by a--2 for n=< I and a--3 for
n:>2. Let a be defined by x,--0 forn=<0and xn--t for n_>_1. Then
{0, ae} is a closed subgroup of -Qa since ae+ a--0.
(c) No group ,(2a is compactly generated, although every element of
every -(2a is a compact element" see Definition (9.9).
Notes
The construction of ,(2a and AI a goes back in all essentials to K. HEN-
SEL Ill, who explicitly constructed ,(2, and ZI,. A construction more
general than that of a and ZI a was given by H. PR0ER , in a form
different from that found in the text. J. yON NEUMANN I recast
PROFER'S construction in a form that closely resembles our special case.
D. VAN DANWZlG 41 has explored generalizations of tle construction of
Ll, and in [_5 has given a wealth of details about/1 for a= (2, 3, 4 .... ).
VaN DANWZlG [I 1 contains the definition of 'a for a= (n, n, n .... ), and
the construction of Za in general is given in VAN DANWZlG [_6, p. 4t9.
Chapter Three
Integration on locally compact spaces
Harmonic analysis on a locally compact group is the study of certain
spaces and algebras of functions and measures defined on the group.
The function spaces in question are defined by measurability and inte-
grability conditions; manipulation of the relevant measures requires
delicate techniques from measure and integration theory. Also, the
fundamental theorems of the theory of representations of locally compact
groups [_see §221 depend wholly upon the study of a certain space of
square integrable functions. Thus the theories of measure and integration
play a vital r61e in the present book.
There are many treatments of integration theory in the extant
literature. None of those available to us is precisely what we need.
Integration theory per se is not our main theme; but we must be sure
of the validity of the theorems from integration theory on which our
subsequent analysis rests, as applied in the very general situation pre-
sented by a locally compact group. Therefore we have set down the
constructions and proved the theorems from integration theory that are
needed in the sequel. We also entertain a hope that the present chapter
may prove useful to students who wish a rapid introduction to integration
theory at our level.
We shall assume as known the rudiments of abstract measure and
integration theory. See for example SAKS E{, Chapters I-III, or
HALMOS [2, Chapters I--V.
Chapter III. Integration on locally compact spaces § 11. Extension of a linear functional and construction of a measure
§ 11. Extension of a linear functional
and construction of a measure
For the sake of completeness, we first repeat a number of definitions
and review a few technicalities.
(11.1) Notation and terminology. Let Y denote an arbitrary non-
void set. A nonvoid family S of subsets of Y is called a ring or ring
o/sets] if A U B S and A Q B' S whenever A, B S. If the set Y is
in the ring S, then is called an algebra o] sets. A a-ring a-algebra
o] sets is a ring [algebra such that U AkS whenever (A 1, A. .... ,
A,, ...)C .
If Y is a topological space, and d) is the family of all open subsets
of Y, then the family of Borel sets in Y is defined s the smallest a-algebra
of sets containing d). Let be the family of all subsets of Y of the form
{y Y'/(y)> 0) for some real-valued continuous function ! on Y. The
smallest a-algebra of sets containing is called the family of Baire sets
in Y. If Y is a normal space, then is exactly the family of open subsets
of Y that are unions of countably many closed sets 1.
A finitely additive measure ,u is a nonnegative extended real-valued
set function defined on a ring S of subsets of a nonvoid set Y such that"
(i) l, (3) -- O"
(ii) # ( A = /(A) if each A and theA'sarepairwise
disjoint, k=
If property (ii) can be replaced by the stronger property
A = /=1
(iii) # =1ў(A) if each A 9 , U A , and the A's are
pairwise disjoint,
then # is called a counably additive measure or simply a measure. A
complex-valued set function # satisfying (i) and (iii) [in (iii), the series
is required to be absolutely convergent is called a complex measure.
Given a nonnegative set function # on a ring 9 , a subset A of Y is
said to be a-/inie if it can be written as U A k where each A c and
/----1
(A )< o.
Let c be a ring of subsets of Y. An extended real-valued function
on Y is 9-measurable if the set {x Y'/(x)>o} belongs to 9 for every
cR. A complex-valued function / on Y is -measurable if its real and
imaginary parts are 9-measurable. If Y is a topological space and
is the a-algebra of Borel [Baire I sets of Y, an 9%measurable function is
called Borel [Baire measurable.
Observe that our definitions of Borel and Baire sets differ from the defini-
tions in HALMOS [2], pp. 219,.220. Our definitions are appropriate to our construction
of measures, which differs in various points from HALMOS'S.
Let # be a measure on S. Let [ be any function on Y with values
in [0, . Then the Lebesgue integral f ](x) d# (x) [or simply f [ dў,
is defined as Y Y
sup/:1 [inf (/(x)'x A)/, (Ak) • {A 1 .....
is a partition of Y with each A
If / is any extended real-valued function, if /+=max(/, 0) and /-=
-min(/, 0), and if either f l+ d# or f I-d# is finite, then f/d# is
Y Y Y
defined to be f/+ d#-- f/- d. * If /=/1+i/. is a complex-valued
Y Y
function on Y, where/ and/, are real-valued, and if f/. d is defined
Y
and finite ('= I, 2), then f / d# is defined to be f/ d# + i f [. d#.
Y Y Y
(11.2) Definition. Let X be any topological space. Let ff (X) denote
the set of all bounded complex-valued continuous functions on X. Let
(0(X) denote the set of all complex-valued continuous functions / on X
such that for every e > 0, there exists a compact subset F of X [depending
upon / and e-] such that [/(x)]< e for all xF'. Let oo(X) denote the
set of all complex-valued continuous functions [ on X such that there
exists a compact subset F of X [_depending upon ]] such that /(x)=O
for all xF'. For/(X), let Illll.- up{ll(x)l.x X }.
(11.3) Remarks. It is obvious that (X) is a complex normed linear
space, ,vith pointwise linear operations" (]+g)(x)=/(x)+g(x), (e/) (x) =
e(/(x)). It is also obvious that 620(X) is a linear subspace of 62(X) and
that if00 (X) is a linear subspace of 2 0 (X). It is elementary to show that
if(X) and 0 (X) are complete in the metric induced by the norm
that is, if(X) and 20(X) are complex Banach spaces. Also ff00(X) is a
dense linear subspace of 0 (X). Finally, we note that 0 (X) may consist
of 0 alone, even for very common spaces. However, if X is a nonvoid
locally compact Hausdorff space, then there are always nonzero functions
in if0 (X). If X is a compact topological space, then 620 (X) coincides with
the space 62(X). If X is a locally compact, noncompact, Hausdorff
space, then we have oo (X)Co (X) .(X) • the equality oo (X) = o(X)
may obtain even for locally compact, noncompact, normal spaces.
However, if G is a locally compact, noncompact group, then we have
ў00(G)C. ff0(G). [_See (.43.e).]
We adopt the usual conventions, here and throughout this book, in dealing
With the extended real numbers oo [sometimes written + oo] and -- oo. That is"
too if tR or t=--c" --cx<t if tR or t=oe" t+oo--oe+t=oo if tR or
too. (oc)+oo and oe+(--oo) are undefined" t--oe=--oe+t=--oe if tR
or too. t.oc=oo.t=oo if tR and t>O" 0-oo= oe.0=0.(-- oe)= (-- o,) . 0= 0-
t'oe-c.t=_oc if tR and t<0" similar formulas for t.(--oe)" (oe)(cx)=
(--- oe) ( oc)= oo and (o) (-- oe) = (-- oe) (o o)= -- o.
Chapter III. Integration on locally compact spaces § 11. Extension of a linear functional and construction of a measure 121
120
(11.4) Definition. Let Y be any nonvoid set, and let (Y) be any
linear space of complex-valued functions defined on Y. The symbol
'(Y) will denote tile set of all real-valued functions in (Y); and the
symbol ч (Y) will denote the set of all nonnegative functions in ' (Y).
For example, if X is a topological space, 0 (X) denotes the set of all
nonnegative continuous functions on X each of wilicil vanishes outside
of a compact subset of X. Wilere no confusion can arise, we will write
, ', and + for (Y), '(Y), and ч (Y), respectively.
In our discussion of integration theory, it is advantageous to consider
an arbitrary nonvoid locally compact Hausdorff space X, the linear
space 00(X), and an arbitrary complex linear functional I defined on
00 (X) that is nonnegative : I(/)>= 0 if/0 (X). To avoid an occasional
solecism, we will also suppose that I is not the zero functional. These
notations will be fixed throughout Chapter Three.
(11.5) Theorem. For 1oo, we have
Proof. Write I(/)=qexp(iO), O<=q<o, 00<2:z, and write
exp(--i0)/--gl+ig, with gl, g2E0. Then q--exp(--i0)
I(exp (-- i 0)/) -- I(gl)+ iI(g2) = I(gl), since I is obviously real for real-
valued functions in E00. We have gl [g[ =< [gl+ ig2]= [/[ ,1 and thus
II(/) [ = 1 = I(gl) " (lll).
(11.6) Theorem. Let A be any compact subset o/ X. There is a
nonnegative number fl depending only upon A such that II(l)l-<-flllll[/or
all/oo such that/(A') :0.
Proof. Consider any o0 such that o(A)= and apply (.5).
The next result expresses a sort of continuity property enjoyed by I.
(11.7) Theorem. Let 3 be any nonvoid subset o/ o such that /or
/, 1 78, there is an / 78 /or which / <= min (/1,/.), and such that
inf{l(x):l}--o /or every xX. Then/or every e>O, there is an
such that /,(x)<e /or all xX; /urthermore we have inf{I(/):/}--O.
Proof. Fix e>0, and let A/--{xX:/(x)>=e} for each [. All of
the sets A t are compact, and their intersection is void. Hence
AhAAI, V1...V1AI,----(3 for some/1,/2, ...,/n). If [,=< min (/1, /2, ...,/n),
then /,(x)<e for all xX. To show that inf{I(/):/}--O, consider
any fixed g, any function h such that h_< min (g, /,), and apply
(tt.6) with the set {xX:g(x)>O}- as A.
For extending the functional I, two classes of functions are useful.
We now define them.
1 For real-valued functions / and g defined on an arbitrary set Y, the expression
/g means/(x) g(x) for all x Y. Also the expression 1:# 0 means that / is not
equal to the function identically zero: that is, / assumes somewhere on Y a value
different from 0.
(11.8) Definition. A function / on X with values in [0, oo [that is,
o is a possible value for/} is said to be lower semicontinuous if the follow-
ing conditions hold at every point x 0 in X. If/(Xo) < oo, then for every
e;>0, there is a neighborhood U of x 0 in X such that/(x)>/(Xo)--e for
all xU. If /(x0)=oo , then for every positive number A, there is a
neighborhood U of x 0 such that/(x) > A for all x U. This set of functions
is denoted by J+.
(11.9) Definition. A function [ on X with values in 0, oo is said
to be upper semicontinuous if for every x0ўX and every > 0, there is a
neighborhood U of x 0 such that [(x)/(Xo)+a for all xўU. Tile set
of all such functions is denoted by +.
We next state without proof some simple properties of functions in
(11.10) Theorem. (i) I//X + and o is a nonnegative real number,
then z / +.
(ii) I//1,/2 ..... /m ч, then rnin (/1,/2, ...,/m) ч"
(iii) I/ C ч and is nonvoid, then sup {/:/)} ч.
m
(iv) I//1,/2 .... ,/m ч, then /i ч.
i--1
(v) I// ч, then/-- sup {g'g o, g-
We now make our first extension of the functional I.
(11.11) Definition. For /Jч, let (/)--sup{I(g)'go,g<=/.
[Note that ](/) may be + o.
The following assertions are obvious.
(11.12) Theorem. (i) I//o, then/+ and I(/)--(/).
(ii) I//1<=/2 and/1,/2 jч, then
(iii) I//ч and o is a nonnegative real number, then (/)=(/).
We next state and prove a continuity property of .
(11.13) Theorem. Let be a nonvoid subset o/ ч such that /or
/, /278, there is an/78 such that/>max(/1,/2). Then ?(sup{/:/})--
sup{(/):l)}. In particular, il 11,19 .... ,In .... J+ gbd 11/9.""
< In <=..., then 7(,,lirn In) -- li,_m
Proof. Let /o=SUp{/']7}. If {/o}Uco, tile present theorem
follows from (II.7), since {/o--/'/} satisfies tile hypotheses of (t.7),
and 0= inf {I(/o--/)'/C )}= inf {I(/o)- I(/) • / )} = I(/o)- sup {I(/) • / }.
In the general case, it obviously suffices to prove that
sup{7(/)./ Let ={go.g<=/ for some /2}}. Let go be any
function in such that go/o. By (ll.10. v), we have
lo= sup {sup [ggo: g </]:/ } = sup {g: g
122
Chapter III. Integration on locally compact spaces
and accordingly
go = min (go, [o) = sup {min (go, g)'g }"
By the previous case, we have
I(go)=Sup{I(min (go, g))"g }<= sup {I(g)"g } = sup {(/)"[}.
(11.14) Corollary. I//:,/.:gJ+, then (/1--/) =(/1)+ (/.)"
Proof. Note that ]:+]e=sup{g:+ge'gi:o, gi]i for i=:,2};
then apply (t t.t 3) and the additivity of I.
(11.15) Corollary. For c +, we have 7(2/)=27(/).:
Proof. For finite , this follows from (1t.t4)" for infinite , use
(.3).
We now extend [ to all nonnegative functions defined on X.
(11.16) Definition. Let h be any function defined on X and assuming
values in [0, . [In the present discussion, such functions be caed
nonnegative, and the set of all such functions will be denoted by .]
Then the [perhaps infinite] number f(h) is defined as inf{i(g)'g*,
(11.17) Theorem. (i) I[ g+, thenI(g)=I(g).
(ii) I[ h , he and h , then i (h 0 I (he).
(iii) I/h and 0< , then f(h)=I(h).
(iv) I/hx, h , then (h+ h) I (h) + I (h) .
We omit the proofs of these simple assertions.
(11.18) Theorem. I/ {h.}.c and h...h,..., then
i(lim h,) = lim I (h,).
Proof. It suffices to prove that (2 h, I(h,), and we may
suppose that lim(h)<. For an arbitrary positive number e, let
g, h, be a function in + such that
f(g)_,.2--<f(h) (=, 2, 3, ...). ()
* II Y is a nonvoid set and a is a function defined on Y with values in [0, ],
then a (y) is defined as the supremum of all sums a (y), as Y1 runs through all
yy
finite subsets of Y.
* The functional is not in general additive. For example, let X = [0, t ] an'd
x
I (1) = f ] (t) dr. II h is the characteristic function of a set A [0, 1 ] that is not
0
Lebesgue measurable and h, is the characteristic function of [0, 1] A', rhea
(h + hz) = I <(h) +(hz). In some cases, of course,
§ 11. Extension of a linear functional and construction of a measure t23
Let g'.max (gl, g ..... g. (n= t, 2, 3,...)- Then we clearly have
g.l+ min (g',, g,+) =g',, + gn+,
and from (tt.t4), (tt.t2.ii), and (t) we obtain
I(gn+O (g) + (g+) - (min (g', g,+))
_ , _ = _ , = = _. j (2)
<=I(g,+I(g+O--I(h)<I(g,+I (h+)--I(h)+. 2- .
Adding (2) over n=t, 2, ..., m--t, and taking account of (t), we have
(g) <(h)+ e (m=l, 2, 3 .... ). (3)
The present theorem follows from (3) and (tt.t3).
(11.19) Corollary. I! h, /or n = , 2, 3,..., then i < , i (h).
This follows at once from (tt.t8).
(11.20) Definition. For each subset A of X, let (A)=I(a).
(11.21) Theorem. The set [unction is a Carathdodory outer measure
on (X). That is:
(i) 0g ,(A)g/or all A cX;
(ii) ,(A) g,(B) i/ A cBcX;
(iii) ,() = 0;
(iv) G (A) i[ A, A,, ..., A. ...(X).
=1
Proof. Assertions (i) and (iii) are obvious; (ii) is a special case of
(tt.17. ii) and (iv)of (.9).
(11.22) Theorem. For every A cX, (A)=inf{(" U is open and
UDA}.
Proof. The assertion is obvious if (A)=. If (A)< and
0< e< t, there is a function g* such that ga and i(g)-- e < ,(A).
Let U={xX'g(x)> t--e}. Since gis lower secontinuous, U is open;
and we have ( _ e) g v. Thus (U) = I (v) i ( _ e) g = (t - e I (g)
< [,(A)+e. Since e is arbitrary, the proof is complete.
(11.23) Theorem. Let U be an open subset o] X. Then (U)=
p{,()- is om#ct Fc V}=p{(V)" is o, - is ompt,
and V-C U}.
Proof. Suppose that <,(U). Let ]0 be such that [ and
I(/)>. For 0<< t, let ={xX'J(x)} and V={xX'/(x)>}.
Let V={xX'J(x)>O}. Plainly we have VcU, ,(V)(U), and
I(/)<(V) By (1t.20)and (ti.18) we have <I(])(V)=i(limF)=
t24
Chapter III. Integration on locally compact spaces
.T(lim v) = lira t(F) -- lira t(V) so that ,(F)
ўo o o
ufficientl small.
(11.24) Theorem. I[ A X and A is contained in a compact set F,
then ,(A)< .
Proof. There is a function /0C + such that /. Thus
,(A),(F)I(/) =I(/)<.
(11.25) Theorem. Let be any/amily o/ pairwise disjoint open sets.
raen ,(o{g" gў})= Z ,(g). AZso, taere is a unique dosed suset E
u
o/ X such that ,(E')=0 and such that ,(E U)> 0 i/ U is open in X and
UE@. [The set E is called the support o/ and is denoted by the
symbol S(,).?
Proof. The first assertion follows at once from (ti.15), since a is
lower semicontinuous if A is open. To prove the second, let W be the
union of all open sets of t-measure 0. Corollary (tt.15) implies that
,(W)=0. Let E=W'. Then if U is open and UE, we have
(U) > 0, and (U) (E U) + (W U) = (E U). The uniqueness of
E is easy to show" we omit the argument.
(11.26) Definition. If A cX and ,(A)=0, we call A an t-null set.
If A F is t-null for every compact subset F of X, then A is said to be a
locally t-null set. The family of locally t-null subsets of X will be denoted
by ,. A property that holds on X except for an t-null set is said to
hold t-almost everywhere. A property that holds on X except for a
locally t-null set is said to hold locally t-almost everywhere. A complex-
or extended real-valued function / on X that vanishes t-almost every-
where [locally t-almost everywhere is called an t-null [locally ,-null]
function. In using these terms we shall often drop the prefix ,- where no
confusion can occur.
(11.27) Theorem. Le h be a nonnegative extended real-valued/unction
on X. Then I(h)=0 i/ and oy i/ the set A={xX'h(x)>O} is a null
set. I/ (h)<, then B={xX'h(x)=} is
{h},% is a sequence o/ nonnegative /unctions on X such that lim I (h,)=0,
then there is a subsequence {h} o/ {h}% such that lim h,(x)=0
almost everywhere on X.
Proof. Suppose that I(h)=0. Then we have a lim n h and so
by (t t.t8),
,(A) =I(A)I(lim nh) = limI(nh) =lim hi(h)=0.
If , (A) = 0, then h lim na and I (h) = 0 by the same computation.
1 Observe that is closed under the formation of countable unions and arbitrary
subsets.
§ 11. Extension of a linear functional and construction of a measure 125
Suppose now that I(h)<. Then for e>0, we have B<__eh so
ttat ,(B)=I(B)GcI(h). That is, ,(B)=0.
Finally, suppose that limI(h) =0. Choose n 1< n2<'" such that
[(h)< . Then by (11.19), we have I h, < . The previous
paragraph shows that h,(x)< almost everywhere and hence
lira h, (x) = 0 almost everywhere.
We adopt the classical definition of measurability for subsets of X,
due to CARATHODORY.
(11.28) Definition. A subset A of X is said to be t-measurable if,
for every subset S of X, the inequality
(i) ,(S),(SnA)+,(SnA')
obtains. The family of all t-measurable subsets of X is denoted by ,.
A function on X is said to be t-measurable if it is ,-measurable. In
using these terms we shall often drop the prefix t- where no confusion
can arise.
(11.29) Theorem. The/amily , is a a-algebra o/ subsets o/ X, and
the set/unction t is countably additive on ,.
All statements of this theorem hold for every Carath6odory outer
measure. Proofs may be found, for example, in HALMOS [2, pp. 44--47,
and in SAI<S I, pp. 43--46. We shall not repeat these proofs.
(11.30) Theorem. Every Borel subset o/ X is measurable, and every
locally null set is measurable.
Proof. (I) We first show that a subset B of X is measurable provided
that
,(U) ,(Un B)+ ,(Un B') ()
for all open subsets U of X such that t(U)<. Let S be any subset
ofX. Ift(S)=, then clearly t(S)t(SB)+,(SB'). If
then for open sets UDS such that t(U)< , we have t(U)t(UB)+
*(UB'),(SB)+t(SB'). Henceby (ti.22), wehave
*(S)=inf{,(U)'U is open and UDS}t(SB)+ ,(SB'),
and B is measurable.
(II) Let V be an open set. To show that V is measurable, let U be an
arbitrary open set such that ,(U)< . Let e be an arbitrary positive
number. Use (t .22) to choose an open subset H of X such that H D U V'
and
(H) ,<,(U fl V') + 2 " (2)
t26 Chapter III. Integration on locally compact spaces § 11. Extension of a linear functional and construction of a measure
Applying (tt.23), choose an open set W such that W-C U f'/V and
(w) + > (v c v). (3)
Let W0----U(-1HFI(W-)'; then W and W0 are disjoint open sets. Since
U(1V' cWocn, (2) shows that I,(W0)--,(U(-1V')]< -. Combining this
with (3), we obtain I,(W)+,(Wo)--(,(VF1V>+,(vF1e'))l<. Using
this and (tt.25), we have
,(UF1V')--e. As e is arbitrary, (t) holds with B--V, so that V is
measurable.
Tile foregoing and (tt.29) show that all Borel sets are measurable.
(III) Finally, let A be a locally null set and U any open set for which
,(U)< oo. For e>0, let F be a compact subset of X such that FCU
and ,(F) + e> ,(U) (tt.23). Then ,(U)----,(F)+ ,(UF1F') U and F are
measurable, so that, ( U (1F') < e. Thus, ( U (1 A) , (F (1 A) +, ( U (1F' (1 A)
<_O+ ,(UF1F') < e. Hence ,(U(-1A)=O, so that
:,(UnA')__<,(U). 1
(11.31) Theorem. Let A be a subset o/X. The ]ollowing statements
are equivalent."
(i) A is measurable"
(ii) , (U) >:, (U (1 A) +, (U (1A') /or all open subsets U o/X such that
(V)< oo"
(iii) A (1 U is measurable/or all open subsets U o/X such that, ( U) < oo ;
(iv) A (1F is measurable/or all compact subsets F o/X.
Proof. Clearly (i) implies (iv). Suppose that (iv) holds and let U be
an open subset of X such that ,(U)<oo. By (tt.23), we see that
U= ,UF, UN where each F, is compact and (N)=O. It follows that
AU=(,(F,A))U(NA),= is measurable. Thus (iii)holds.
It is trivial that (iii) implies (ii). The fact that (ii) implies (i) is proved
in Part (I) of (1t.30).
(11.32) Theorem. Let A be a measurable set o] finite measure. The
(i) t(A):sup{t(F)'F is compact and FcA).
Proof. For e >0, choose first an open set V D A such that t (V f'/A') < -.
Then apply (t t .23) to select a compact subset E o f V such that ,(V f'/E') < -.
Finally choose an open set W such that V W V (1A' and , (W) < -.
Then define F--E (-1W'. Obviously F is compact and F A. Also we
e * and
have ,(A F1F'=) ,((A (-1E') LJ (A F1W)) <: t(VfIE') + ,(W)<- + ,
hence t(F) :> ,(A) -- e.
(11.33) Note. Theorem (tt.32) holds if A (.d, and A is contained in
the union of a countable family of sets of finite measure [apply (tt.t8)
and (t t.32)]. However, (t t.32.i) does not universally hold for measurable
sets. Consider for example tile space R with tile topology in which sets
{(x0, Y)"< Y<fl} Ix0 fixed are an open basis. Plainly this is the topol-
ogy of RdxR, where Rd denotes tile discrete reals 1. Thus RxR is a
locally compact Abelian group. For /(00(RxR), let {xl ..... xm) be
m oo
the set of numbers x such that/({x)xR) 0. Let I(/): . f/(x i, y) dy,
where f ... dy denotes ordinary Riemann integration. The closed and
discrete subgroup Rx{0) has curious measure-theoretic structure. It is
easy to see that for A cRux(0}, t(A):0 if A is countable and t(A) :
if A is uncountable. The set Rx{0} is measurable since it is closed.
Since the only compact subsets of Rx{0} are finite, (.32) fails for
Rx(0). Note also that Rx{0} is locally null but not null.
This 0-- phenomenon persists for all locally null sets and arbitrary
X and I. That is, every locally null set A X has measure 0 or . The
set A is measurable (.30). If ,(A)<, then t(A):sup(t(F):F is
compact and FA) (tt.32). Hence t(A):0. Note that a measurable
set A is locally null if and only if t (F):0 for all compact sets F A.
(11.34) Definition. Let be a a-algebra of subsets of X containing
all open sets and let # be a finitely additive measure defined on .
Suppose that" for every open set V, we have (V):sup{(F)'F is
compact and FC V}; for all A , we have (A)=inf{(V)" V is open
and V D A}. Then is said to be regular.
By repeating verbatim the proof of (.32) witll t replaced by , we
see that if A and , (A) < , then (A) = sup { (F)" F is compact and
F cA). Theorems ( .29), ( .22), ( .23), and ( .30) can be combined
in the statement that t is a countably additive, regular measure on a
a-algebra of subsets of X containing all open sets and all sets of t-measure 0.
Our principal reason for constructing t from I is to represent I as
the integral f/d . We now show that this can be done.
X
(11.35) Lemma. Let A and B be disyoint measurable subsets o/ X
and and fl nonnegative real numbers. Then
( + ) = (G)+ ().
Proof. In view of (.7. iv) and (.7.iii), it suffices to show that
(G + fl) ;()+ fl() • ()
1 Given a topological group G, G d will always denote this group with the discrete
topology.
128 Chapter III. Integration on locally compact spaces
Tile cases t(A)--o, t(A)--0, ,(B)--o, ,(B)--0, --0 and /--0 are
trivial. Thus we may suppose that 0 <, (A) < o, 0 <, (B) < o, :> 0,
and/:> 0.
Suppose that A and B are compact. Tilen A and B are contained in
disjoint open sets U 0 and V0, respectively, which we may obviously take
to be of finite measure. It is also clear that I (SeA +/SeB) is finite. Let e
be an arbitrary positive number. Then there is an [Jч sucil that
[>= SeA -+-/seB and
E --
i(i)- - <r(G + ). (2)
Let d be a number such that 0 < d < min 3(g.) ' 3() ' ' . Since [
is lower semicontinuous, there are open sets U and V such that A C U C U,
BcVcV o, /(x)>--d for xU, and /(x)>--d for xV. Thus we
have/ (--d)v + (--d) [note that U V=, and the functions
(--d) v and (-- d) are in *. Applying (11.14) and other properties
already established, we obtain
(/ ((- + (- ) = ( ( + (- f( ]
= (- ) ( + (- () ( + ( (3)
- - 2e
Combining (2) and (3) and recalling that e is arbitrary, we obtain (t).
Consider now arbitrary A and B and e>0. By (11.32), there are
compact sets E and F such that E C A, F c B, (E) > (A ) , and
(>(_
. Naking use of (I) for compact sets, we have
( +) z( +)-z()+z()>z( +z(-. a
(11.36) Theorem. Le / be a omegaive measumbZe /cion o X.
x
Proof. Let A--{xX'/(x)=}. For every positive integer , and
for h=l,2 .... ,-2 -I, let A,= xX'/(x)< a j. Define
n 2n--1
k
the function h by h: $A.+n$A. Then we have h,
k=l
h...h.., and lim h(x):](x)for all xX" also f]d,=lim rhode
n X X
by the theorem on integrating monotone increain sequence, and
'(/) = li (h) by (.8). Finally, by (.5), we have
k=l X
§ 11. Extension of a linear functional and construction of a measure 129
(11.37) Corollary [F. RIESZ'S representation theorem]. For every
1o, w h X(/)=f ld.
x
(11.38) Note. Corollary (]].37) asserts nothing about the uniqueness
of the measure t that gives the integral representation for I. There are
pathological cases in which two different measures yield the same integral
for all functions in if00 .1 However, we have the following.
Let/ and v be measures defined on a-algebras and , respectively,
of subsets of X. Suppose that and are regular (]].34), and that
t(F) and (F) are finite for compact F. Note that and J4 both
contain all Borel sets.] If f d = f d for all 0, then (A)=, (A)
x x
for all A,. To see this, suppose Јirst that F is compact. Let
{U,},I be a decreasing sequence of open sets such that
n=X
lira ,u (U,)= (F), and lim (U,)=(F). Let , be a function in ff0 such
that , (F) = , .(U) -- 0, and (X) c 0, . Let %-- rain (1 .... , ,).
Then we have lira %--F -almost everywhere and -almost everywhere.
Hence
(F)=fFd=ff%d=lim f%d--lim f%dv= fFdv=v(F).
X X X X
Now for any open set U, we have
(U) = sup { (F)" F is compact and F c U)
= sup { (F)" F is compact and F C U) -- (U),
and likewise for A , we have
(A) = inf{(U)" U is open and UDA}
=inf{v(U)'U is open and UDA}=v(A).
We next establish a technical but useful property of .
(11.39) Theorem. There exists a/amily o/ nonvoid compact sub-
sets o/X such that."
(i) the sets in are pairwise disjoint"
(ii) (F U) > 0 i/ U is open, F , a,d F U "
(iii) every open set o/finite positive measure has nonvoid intersection
with only countably many F, and every point o/ X has a neighborhood
meeting only countably many F "
(iv) the set N= (U {F })' is locally null.
Proof. The void family of compact sets trivially satisfies (i) and (ii).
A _ simple well-ordering argument shows that there is a maximal family
See for example HALMOS [2], p. 23, Example (IO).
Hewitt and Ross, Abstract hoaic aaalysis, vol. I 9
30
Chapter III. Integration on locally compact spaces
of nonvoid compact subsets of X satisfying (i) and (ii)1. Property (iii)
is an immediate consequence of (ii).
If U is an open set with finite measure and F, F, ..., F, ... are tile
members of having nonvoid intersection with U, then t(UnN')+
,(VnN)--,(vn(fiU&U...UF, U...)) + ,(un(fiUFU...UF, U...)')
t(U), since U and (FUFU..-UF,U...) are measurable. This implies
that N and N' are measurable.
Finally, assume that N is not locally null. Since N is measurable,
tllere is a compact set E c N suck that t (E)> 0; tllis is pointed out
the last sentence oi (tt.33). Consider the family of all open sets
U in X such that (E n u) - 0. Let B -- U {E n u: u E n (v u. u).
Then t(B)--0, since in the contrary case there would be a compact set
D B such that t(D)> 0, and D would be contained in the union of a
finite number of sets DR U, each of measure 0. Let Sz--ERB'. Then
we have t(Sz)-- t(E)>0, andif V is open in X and SRV--SR(ErV)
:4:, then t(snv)--t(Env)>o. Hence we could add S to the
family - and obtain a strictly larger family of compact sets satisfying
(i) and (ii). This proves (iv).
The following result will also be useful in the sequel.
(11.40) Theorem. Let ] be a nonnegative, measurable/unction on X
such that f ] tit< oo. Then there is a/unction ]' defined on X such that
X
]or every o>=0, the set {xCX:/'(x)>o) is -compact, 0<__/'<__], and
f/'dt--f/dt.
X x
ProoL For every positive integer n, and for k--t, 2, ..., let A,,-----
tt k + It is plain that A is measurable and that
x.X'- _<_/(x)< 2------j. ,
t(A,,,,) < oo. By (I 1.32), there is a -compact set B,, such that B,, c
and (A,,n B,,) = O. Let /',= max -r e." =t, 2, ..., . It is
plain that
oo k
2-,(A,,,)<--_f/'.d*<----f/d*. ()
k=l x x
Now let ]'= lira ]',,. Since f ] d t= lim 1 k
oo x -oo = 2 -- t(A,), inequality (t) and
the theorem on integration of monotone increasing sequences imply
that f ]' dr--f ] dr. It is also easy to see that for _-->0, the set
x j, is equal to U B, --> . and hence is c-compact.
{x X. (x)> •
1 Recall that the union of a void family is . Thus if " were void, we would
define N to be X.
§ 11. Extension of a linear functional and construction of a measure
(11.41) Corollary. I] ] is any t-measurable /unction such that
f Ill d t < oo, then there is a Borel measurable/unction ]' equal to ] t-almost
x
everywhere.
We now state and prove an elementary theorem about measurable
functions which is analogous to (t t.3 t).
(11.42) Theorem. Let ] be a real- or complex-valued/unction on X.
Then the ]ollowing statements are equivalent."
(i) ] is measurable;
(ii) ]v is measurable/or all open subsets o/X such that t(U)<
(iii) ] is measurable/or all compact subsets F o/X;
(iv) ]q is measurable/or all/unctions q in o.
Proof. Clearly (i) implies (iv). Suppose that (iv) holds and let F
be a compact subset of X. Then, as ill (tt.38), there is a sequence
{},o__ of functions in 0 such that = lim ,almost everywhere.
Therefore/y= lira/,almost everywhere, so that ] is measurable.
This shows that (iii) holds. Statement (iii) implies (ii) in the same way
[/#v = li,om/#, almost everywhere for a suitably chosen sequence
of compact sets].
Suppose that (ii) holds. To show (i) it suffices to consider nonnegative
functions ]. Let m be a positive number. Then for open sets U such that
(U) < oo, we have
V= X. l (x)
by (ii), this set is measurable. By (tt.3t), the set {xcX'l(x)>oў } is
measurable. Hence ] is measurable.
Miscellaneous theorems and examples
(11.43) Examples concerning {0 (X) and {00(X). (a) Let X be
any Hausdorff space. Then /(X) is in 0(X) if and only if {xX:
I/(x)l> }- is compact for every >0, and /00(X) if and only if
{xX. l/(x)]> 0}-- is compact.
(b) Consider the space Q of rational numbers with its usual topology
as a subspace of R. Then ( (Q) is a large space, but (0 (Q) consists of 0
alone. /If IC(Q) and ](a)=O, then the set {xQ" II(x)l>Ѕ11(a)l}-is
noncompact, so that/ў(0 (Q).]
(c) If X is a locally compact Hausdorff space, select any point aX
and any neighborhood U of a such that U- is compact. By complete
regularity, there is a continuous function ] with domain X and range
contained in [0, 1 ] such that ] (a) ----- 1 and ] (U') -- O. Thus / =4= 0 and
loo(X).
9*
132 Chapter III. Integration on locally compact spaces
(d) Let .Q be the smallest uncountable ordinal, and let X be a well-
ordered set of order type .Q. With the order topology [see (4.19. a)],
X is a countably compact, locally compact, noncompact, normal space,
on which every continuous complex-valued function is bounded. [Assume
that /is continuous and unbounded on X. For each positive integer n,
there is an n X such that 1] (n)] >= n. Since .Q is the least uncountable
ordinal, there is an element vX such that v_>_,for all n=l, 2 .....
Thus/is unbounded on the compact set {fl X'fl_<_v}, which is impossible.]
Every compact subset F of X is bounded" there is an X such that
/<__ for all/F. Also, we have0(X) =00(X). [Suppose that/o(X).
t
Then for every positive integer n, there is an X such that [/(/) ] < -
for all/__>. As before, there is an element vX such that v>=,for all
1 for all positive integers
n--1,2,3,.... Iffl_>_v, thenwehave]/(/5)]<-
n, and so ] (fl)--0. Since the set {fl X'fl v} is compact, it follows that
/oo(X).
(e) Let G be a locally compact, noncompact group. Then we have
ff00(G).Cff0(G ). Let U be a neighborhood of e in G such that U- is
compact. Then the family of sets {xU'xG} is an open covering of G
that admits no finite subcovering. If it did, G would be the union of a finite
number of compact sets and hence would be compact. Thus there is an
infinite subset {xl, x2, x3 ..... x, .... } of G such that x,ўU=lXkU for
n--2, 3,.... Now let V be a symmetric neighborhood of e such that
V2c U. Then it is plain that the sets {x.V}n°=l are pairwise disjoint.
Let ! be a function in ff (G) such that ! (G) c 0, 1 , ] (e) -- 1, and ! (V') =0.
Let g be the function such that g(x) = 2-"(xlx) for xG. It is then
easy to see that go(G) and gўoo(G).
(11.44) The values of a continuous measure. (a) JR. E. JAMISON,
oral communication. 1 Let X be a locally compact Hausdorff space, and
suppose that I is a nonnegative linear functional on ff00(X) whose cor-
responding measure t is continuous, i.e., ,({x})=0 for all xX. Suppose
that B is an t-measurable subset of X such that
, (B) = sup{, (F) : F compact and F C B},
and that is any real number such that 0<<,(B). Then there is a
compact subset E o of B such that t(Eo)
[Let F be a compact subset of B such that ,(F) > fl, and let - be the
family of all compact subsets E of F such that t (E) >=/5. Let {ET}7r be a
subfamily of - that is linearly ordered by inclusion, and write E1 for
the set f3 ET. Clearly E 1 is nonvoid. Assume that, (El) <ft. Then there
§ 1 t. Extension of a linear functional and construction of a measure 133
is an open set U containing E such that ,(U)<. However, we have
U'VI E = U'VI ( fl E,) and since the E,'s are compact, there are
,EF m
7'"" 7m in/' for which f3 ET c U. Since {E7}7 r is linearly ordered,
/=1
we have some ETC U, a contradiction. It follows by ZORN'S lemma
that there is a minimal compact subset E 0 of E such that t(Eo)fl.
Assume that t(E0)>ft. Let x be any point of E 0. Since t({x})=0, we
can find a neighborhood U of x such that t(U) < t(E0) --ft. We find
,(Eo U')=,(Eo)--,(Eo U) >,(Eo)--,(U) >fl,
and this contradicts the minimality of E0.1
(b) The hypotheses of (a) obviously hold for every open subset U of
X and also for every a-finite t-measurable subset of X. The conclusion
of (a) must fail for all locally null nonnull sets Isee (tl.33)1.
(11.45) Measures on subspaces. Throughout this example, let X
be a locally compact Hausdorff space and X* a nonvoid closed subset
of X. Let I* be a nonnegative linear functional on Soo(X*) and let *
be the set function defined on all subsets of X*as in (11.20). We will
show how to construct from I*a functional on (Soo (X) and will compute
the corresponding measure.
(a) For a function / on X, let/* denote / with its domain restricted
to X °. If / is in ffoo (X), then/* is in ffoo (X*). Also, every function g in
ff0o(X* has the form/*for some / in (So0 (X). IIt is easy to prove that
/* is in ffoo(X*) whenever/is in ffoo (X). The second statement is equiv-
alent to the assertion that every function g in (oo(X* admits an ex-
tension / over X such that / is in (Soo (X). If X is compact, / exists by the
TIETZE extension theorem IKELLEY I2, p. 242. If X is noncompact,
let Xoo be the l-point compactification of X, and let be the adjoined
point. Let E be a compact subset of X*such that g vanishes on X*VIE'.
Then E is also compact as a subset of X, and there is an open subset
U of X such that E C U and U- is compact. It is obvious that
Y X* (U' fiX)U{} is a closed subset of X. Let/1 be the function
on Y such that/(x)=g(x) for xX" and/(x) =0 for x(U'VIX)U{}.
Then [ is continuous on Y, and by the TIETZE extension theorem there
is a continuous complex-valued function /2 on X such that /2 is an
extension of [. Let /be the restriction of [2 to the space X. Note that
if g(X* [0, 1-], then the inclusion/(X) I0, 11 can be realized.
(b) For [200 (X), let I(/)= I*(/*. Then I is evidently a nonnegative
linear functional on if00 (X). Let , denote the set function on all subsets
of X defined for I as in (I 1.20). Then for every subset A of X, we have
(A)(AVIX*)=*(AVIX*). lit is easy to see that (X*')=0, and
134 Chapter III. Integration on locally compact spaces § 12. The spaces ,p (X) (1 ___p ___<oo) 135
since X* is ,-measurable, the equality ,(A)=,(Af3X*) holds for all
A cX. Now consider an open subset U of X. By (11.11) and part (a),
we have
,(U) = sp {S(I)"IGo(X), I<}
= sup {I* (/*)"l(o(X), /<u}
__ • +
p{'() 00(x') .}
- ," (UX').
To show that ,(U),'(UX'), consider any function g in
such that gGvnx.. Let h be the function on the closed set
X'U(U'X) such that h(x)=g(x) for xX" and h(x)=0 for xU'X.
Then h belongs to G0 (Y) and by part (a), there is a function / in ў0 (X)
such that [(X)c0, t and [ is an extension of h. Clearly /'--g and
[Gv. Thus ,(U)I(/)=I'(/')=I'(g), and, as g is arbitrary, we have
• (u) ,.(ux.). Ths ,(U)=,.(UnX').
Finally, let A be any subset of X *. By (1t.22) and the preceding
paragraph, we have
," (A) = inf {," (U X')" U is open in X, U X" D A }
-- inf{,(U)'U is open in X, UX'DA}
= inf{,(U)" U is open in X, UDA}
,(A).
This completes the proof.
Notes
The theory of integration as formulated in this section is the work
of many hands. An extended discussion of the historical development
of the subject would be out of place here" we mention only a few high-
lights . Corollary (11.37) for X--0, 1 and the integral a Riemann-
Stieltjes integral is of course due to FReDeRIC RIESZ 1 . It is fascinating
to see how RIESZ chose the right way to represent bounded linear func-
tionals on G (0, t ), in contrast to earlier representations which he refers
to loc. cir., p. 974.
The basic idea of extending a functional and so producing a measure
is due to P. J. DANIELL 1 . The construction of a measure directly from
a nonnegative functional on G(X) and Corollary (11.37) for a compact
Hausdorff space X appear in yON NEUMANN 5, which is the earliest
statement known to us of this technique and result. The same con-
struction, with more details, appears in KAKUTANI [3" It is remarked
with no proofs or constructions in WEIL [4, p. 32, that "il y a donc
An interesting historical discussion appears in OURBAKI [3], PP. 113126.
correspondance biunivoque entre les mesures de Radon et les fonction-
nelles I([)". Unquestionably WElL was in possession of a construction
similar to that of the present section, although he leaves everything to
the reader, referring only to the fascicule [2 of BOURBAKI, which appeared
in print eleven years after WElL 4. A construction similar to but not
identical with our construction of f and [ is sketched in H. tARTAN [2.
CARTAN'S process is developed with full details by EDWARDS [1.
Measures, whether countably or finitely additive, which yield integrals
reproducing given linear functionals in various abstract settings, have
been considered by several writers within recent years. One such treat-
ment is in HEWlTT 1, where references to earlier literature also appear.
ITheorem 2.17 of HEWlTT [1 is false, as was pointed out by I. GLICKS-
BERO: an additional separation property of sets in by functions in
is required. An alternative treatment is given by LooMIs [3. All of
these constructions lead to essentially the same result: the choice of one
or another set of axioms and mode of procedure seems to be largely a
matter of taste. Countable additivity is obviously essential for the
application of the most powerful tools of integration theory, viz.,
LEBESGUE'S theorem on dominated convergence and FUBINI'S theorem.
The appearance of this phenomenon is discussed in GLICKSBERG [1.
The construction of , r, and , given in the present section is very
like [although not quite the same as that found in NMMARK [1, §6.
We have also borrowed freely from BOURBaKI [2. Locally null sets
were apparently first introduced by BOURBAKI [2, with a definition
formally different from but actually equivalent to ours.
§ 12. The spaces p (X) (1 <=p <_ oo)
Throughout this section, X, I, ,, t',, and q, are as in § t 1. We take
the elementary properties of measurable functions as known. See for
example HALMOS [2, pp. 73--86 or SAKs [1, pp. t2--t
(12.1) Definition. Let p be a positive real number. Let ] be an
extended real- or complex-valued measurable function on X such that
f l/lPd,<oo. Then we say that ]Cp(X, ,)" where no confusion can
x
arise we will write p for p (X, ,). The norm ]l/lip is defined by
Given two functions/, gCp, we have if and only if
almost everywhere on X. Two functions such that I[/-gll =0 r
taken to be identical in p. That is, p actually consists of equivalence
classes of functions. We will frequently refer to a function ] in p, etc.
It will be clear from the context in every case whether we mean the
t 36 Chapter III. Integration on locally compact spaces § 12. The spaces p (X) (t =_< p < oo) 137
fixed function ] or the equivalence class in p containing /. The real
functions in p will be denoted by ;. We shall be concerned with
for values of p >_ 1 almost exclusively.
The pathological character of t on sets that are locally null but not
null see (.33)1 permits us to improve a little upon (12.1).
(12.2) Theorem. Let / be a complex-valued/unction or an extended
real-valued/unction on X that is locally null but not null. Then f Ill d,= oo
x
(1 =<p<). Hence i/ /, g and /(x)=g(x) locally almost everywhere,
t f I-1 d-o. Th.s ! d are equal almost everywhere and define
x
the same element o/.
Proof. As pointed out in (11.33), if N is a locally null set but not a
null set, then ,(N)--. For some > 0, we have ,({xX" I/(x)[2> })=t=0,
and hence , ({x X" l/(x)IP> }) -- oo. It follows that f Il d,-- oo. For
x
],gf, we have ]/-gi<=(i/l+igi)<=[2max(I/[, igl)l_<_2p(l/l+lgl ).
Hence J I[-g] d t is finite and is therefore 0. r
x
It is obvious that if [=/' locally almost everywhere and g=g'
locally almost everywhere, then [ + g =/' + g' locally almost everywhere.
Thus addition is defined unambiguously in . Similarly /is defined
unambiguously if [ p and K. Consequently, is a complex linear
space. The norm II/b ha the usual properties ascribed to a norm [see
(B.7)I. That is" I]/]f is nonnegative and is zero if and only if/--0 [we
repeat, in the sense of " l[/+glf<=ll/ip+ilgll, and l[/[l=ll" II/[[ for
K. Note that these properties are obvious for the case p= 1. A close
examination of certain inequalities implying [[/чII< II/llч I111 is
required for the detailed study of , and we therefore digress to set
them down.
Throughout (12.3) to (12.6) inclusive, p will be any number such that
P
1 < p < o, and p' will be the number p-1 ' so that - + )7-1. Note
that 2'-- 2.
(12.3) Lemma. Let a and b be nonnegative real numbers. Then
a p b p"
(i) ab<_-p- + p, ,
and equality holds i/ and only i/aP= b '.
1
•
Proof. For u>_ 0, the function u -1 and its inverse u -1 are strictly
increasing and continuous. From this it is obvious upon inspecting the
1
areas under the curves v=u - and u=v P- that
f / 1 ap bp,
ab < u#-l du @ v #-1 dr---p- -- p, ,
0 0
unless b =a-l, which is equivalent to a =b p'. Plainly equality holds
in (i) if a = b '.
(12.4) Theorem [HOLDER'S inequality]. Let/ and g ,. Then
/g " in/act
(i) I[ / g" d t I fl/l ,
and
(ii f I/J II/Jl
x
Equality holds in (i) i/ and only i/ sgn [/ (x) g (x)] is constant almost
everywhere on the set where/g 0. Equality holds in (ii) i/ and only i/
there are nonnegative numbers A and B not both 0 such that A ]/(x)l=
Big(x) [" almost everywhere. Thus
and equality holds in (iii) i/and only i/ sgn [] (x) g (x)] is constant almost
everywhere on the set where/g 0 and A ]/(x)1= Big(x)1" almost every-
where .
Proof. Inequality (i) holds for any function h t"
x
Thi inequMity i proved ]ut as (11.5) wa proved. Let
h(z)0}. If sn k()--exp(i)()Mmot everywhere on E, then we
hve
- xp ()Il - xp () I1
x x
o that equality obtain in (1). Suppose conversely that equality obtains
in (I), so that exp () [h t =] [kl t, lor some . Write exp ()
x x
=+i, where and are reM-vMued unctions. Then we have
f 9dr= f91 dt + i f 92dt = f Eg + 93 dr, so that f dt < f
X X X X X
f [9 + 9t d t = f 9 d . This implies that (x)> 0 almost everywhere
X X
a (x)=0 amot evwhr. Hence (x)=l(x)[=lh(x)] amot
h, an h()=xp(-- )]h(x) ] amot eeywhe.
We turn now to (ii). If / or g vanishes almost everywhere, then
equality trivially holds in (ii). If neither / nor g vanishes almost every-
where, then (2.3) shows that
II11, gl,,-; I/ +' IIg: ' ()
InequMity (iii) is Hozg's ieqliy" sometimes (ii) is also reerred to as
6LDE'S inequality. For = p'= 2, (iii) is called Cvez's ieqliy or
138 Chapter III. Integration on locally compact spaces § 12. The spaces p(X)(1 <p <oo) t39
with equality holding if and only if
IIg IIN: [/(x)I' -11/11 , Ig(x)I".
We integrate (2) over X, obtaining (ii), and note that if (3) fails on a set
of positive measure, then strict inequality obtains in (ii).
(12.5) Corollary. Let ]1, ..., ], be nonnegative ]unctions in 1 and let
oq , ..., o be positive numbers such that + + . . . + = . Then
X
Proof. Use (t2.4) and induction on n.
(12.6) Theorem [INKOWSKI'S inequality]. Let [ and g be [unctions
in (1 < p < ). Then
and equality obtains i[ and only i[ A [= B g [or nonnegative numbers A
and B not both O.
Proof. We have I/+gl =<2 (I/1+ let ) as in (12.2), so that
Applying (t2.4), we have
x x
f Ill" tl+ 1 '-1 + f Igl" I/+ gl -' d*
This clearly implies (i).
The condition stated for equality in (i) is obviously sufficient.
Suppose conversely that equality holds in (i). Then the inequalities
in (t) are both equalities. From the second of these and the condition
for equality in (12.4.ii), we have
A' i/<x)l - B' I/<x> ч g(x)lP and A" Ig(x)l - B" I/<x> ч g(x)lP (2)
almost everywhere, where A', B', A", B" are nonnegative real numbers
and (A' + B' ) (A" + B" ) > 0. If !+ g= 0 or 1= 0 or g= 0 [in we
plainly have A[=Bg. If not, (2) implies that Ig(x) l=v[[(x)[ =
l] (x) + g (x) l almost everywhere, where , > 0. Since the first ine-
quality in (1) is also an equality, we have
almost everywhere on the set E = {x X'[ (x) + g (x) 0}. We thus have
almost everywhere on E, so that also g(x)=??[(x) almost everywhere
on E. Thus g(x)=??J(x) almost everywhere.
(2.7) Theorem. Let [ and g be [unctions in 1. Then llZчgIll<=
II/1[ч Ilgll:, nd equality holds i[ and only i/there is a positive measurable
[unction e on X such that g(x)=q(x) ](x) almost everywhere on the set
where [ (x) g (x) : O.
Proof. The inequality is obvious:
X
Equality obtains in (1) if and only if
I/<x>+g<x>[: I/<x>l + Ig
(2)
almost everywhere on X. The equality (2) is no restriction unless
/(x) g(x):o. If this occurs, then (2) is equivalent to the condition that
g()
0 (x)= be real and positive.
(12.8) Theorem. For t <=p< oo, the spaces are complex Banach
spaces, and . is a complex Hilbert space with inner product
(i) <[, g) -- f [ d .1
X
Proof. Theorems (t2.6) and (:2.7) show that IIl[ is a norm for
Isee (B.7). To show that g is complete, let {/=}=t be a Cauchy sequence
in p and let nt< m.<-..< nk<.., be a sequence of integers such that
(k--t, 2 .... ). From (12.7) and (t2.6), we have
Writing g =lg, we apply the theorem on monotone convergence
and find f g a,_ I11/., I1,ч .
X
2 -- [, (x)) converges to a
This implies that the series [,, (x) +=x(/-,+l (x)
complex number, say [(x) for almost all xX. Thus lim [,, ( x) -- J ( x)
almost everywhere. For e>0, choose l so large that for m_n, and
k>=x, we ave I1--.1< . Then by FATOU'S lemma on integrating
sequences of nonnegative functions, we have
X X -*oo --o X
1 Horizontal bars denote complex conjugates throughout this section, as they
do everywhere in the text excep for (9.2) s., where certain functions are
written with horizontal bars.
t40 Chapter III. Integration on locally compact spaces § t2. The spaces p (X) (t p __<_ oo) t4 t
To prove that 2 is a Hilbert space, we need only note that
f ! { d t is a legitimate inner product [see (B.39)
x
The concepts and results of (12.1) and (12.3)--(t2.8) hold for the
spaces p defined for any measure space" plainly (12.2) requires the
special X and discussed in § 11. Another special result follows.
(12.9) Definition. Given X, I, , and t', as in § 11, let ® denote the
class of functions on X of the form ai Aj, where the %.'s are complex
i=l
numbers and the sets A i are measurable and have finite measure. In
other words, ® is the class of complex-valued measurable functions on X
that assume only finitely many values and vanish outside of sets of
finite measure.
(12.10) Theorem. For 1 <=p< c, the linear space oo is a dense
sub@ace o/, and o is a dense subspace o[
Proof. We carry out the proof for 00 and f" the spaces 0 and
are dealt with similarly. Since 0 _<_ I(]/ [P) = f ]/l d, for/0o (11.36),
x
it is obvious that 00C2. To show that 00 is dense in p, we first
prove that ® Iwhich is obviously a linear subspace of is dense in
.1 Suppose that ] p and that ] is real and nonnegative. For every
positive integer n, and for k=t, 2, ..., n.2" 1, let Ak, n= xX" 2--
k+,}
/(x)< 2" . Define the function % by or,= A,- Then we
have cr ®, cq=<cr 2=<... =<%=<..., and lim %--] everywhere on X.
Furthermore, we have
I/-o,,1'_-< (I-+-o,,)*=<
Since (2 [)# 1, we apply LEBESGUE'S theorem on dominated convergence
to infer that lira f [[--crn[ d,-- f lira I/--cr] d,=0. This obviously
n-eo X X n--eo
implies that ,irn II]--cr, lp--O. Every function in p can be written as
]1--/.+i(/--/), where ]1,/2,/,/ are real, nonnegative, and in p.
Hence ® is dense in
To prove that G00 is dense in 2p, it thus obviously suffices to show
the following. Let A be a measurable subset of X such that 0< (A)< oo.
Then there is an/G0 such that f I/--A 1 d t < e, where e is an arbitrary
x
positive number. This assertion follows at once from (tt.32). Indeed,
choose a compact set F and an open set U such that F cA c U and
(gc)f') < e. Then select any function/I00 such that/(f)- 1,/(g') =0,
and/(X)c[O, ]. It is obvious that f ]/--AI
x
x This assertion is valid for p defined for an arbitrary measure space.
In connection with Theorem (12.10), note that Illll may well be zero
for a nonzero / in G00. It is the equivalence classes containing elements
of S00 that are dense in 2.
We now define the space of "essentially bounded" measurable func-
tions. Our definition is motivated by Theorem (t2.2) and differs some-
what from the classical one.
(12.11) Definition. Let g), denote the space of all bounded, com-
plex-valued, measurable functions on X. For/ g),, let [I/I,=sup{[/(x) [ "
x.X}, as usual. Let , denote the space of all locally null functions in
93,. Plainly g), is a Banach space under the norm /, and pointwise
linear operations, and it is easy to see that , is a closed linear subspace
of ,. \Ve define 200(2, ',) [or as the quotient space
Thus we may consider 2 as the space of all bounded, measurable,
complex-valued functions on X, two functions being regarded as equal
if they differ only on a locally null set. The norm of/ is defined as
inf{/--/' ,'/'9,}" we write this number as
It is easy to show that [[/[[oo--inf{R'>=0 and
is locally null}. Furthermore, this infimum is attained, i.e., there is a
/3>_0 such that {xX'[/(x)[>fl} is locally null, and such that if 0--<y<fl,
then {x(: X" /(x) l> } fails to be locally null.
(12.12) Theorem. The space 200 is a Banach space.
Proof. The only property to be verified is completeness. This is a
property of all quotient spaces of Banach spaces by closed linear sub-
spaces (B.I 7).
We now prove a theorem which will be used in identifying the con-
jugate spaces of
As before, if 1 < p < oo, then p' denotes -. p '
oo and oo' as 1. p_'we also define I as
(12.13) Theorem. For/ 2p (1 <= p <= ), we have
(i) /=sup{[xf /q) d].q)oo ' ,=<1}.
A similar assertion holds/or ' and 'oo.
Proof. (I) Suppose that 1 < p< oo. Then for ] 2f, HOLI)ER'S inequal-
ity (12.4) shows that the right side of (i) does not exceed the left side.
To prove equality for / 2f, we may plainly restrict ourselves to the
case /Ip:>0 First let h= /}-P I/1 sgn U. Then it is obvious that
h<,, [recall that p'=/-] and that h ,,=1. Furthermore,
_ X
/ -P f/I/I p-1 sgn/d= 1/- f I11' d,-=ll ;. Now, using (12.10) choose
X X '
% (Ј00 such that h-- 9a Jlp' < 2 I]l-' where e is an arbitrary positive
142 Chapter III. Integration on locally compact spaces § 12. The spaces p(X) (1 p <= o0) 143
number less than 2l[/1[#, Then we have II111'-- a l = i1111'--Ilhll'l
IIw-- hll' < m I-tll--; Let = ; note that I111.> o. Then IIll.:
applying H6LDER'S inequality and the foregoing, we obtain I / dt--
(II) Suppose that p=]. For 1 9 and h 9, we obviously have
Consider any function 00. Let F be a compact set such that (F') =0,
a or >0 t =:{<X'l()l4. Lt =00 b suc that
o (E) : ], w (E'I) : 0, and w (X) C E0, ]. Let -- sgn . w. Ten
it is clear that 00 and I1=11 fo, . xt i y to e
Xim r V'= : I1 ': I111. (z)
0 F
Now consider an arbitrary 19, and let be an arbitrary positive
number. By (]2.]0), there is a 00 such that I1-111< . o,
sufficiently small, (2) and (]) and elementary norm inequalities give
(III) Suppose finally that = . In considering (i), we may ignore
t t,ivi c II111-0. zt b y ,mb, ,ch that
Then (]2.i]) and (]].26) imply that there is a set , such that
f
x
sic I1-11 c b ma arbitrarily small with 00, ()) and
wit short comp,tation eld (i) for = .
The proof for 9 is similar an is omitted.
We now prove a partial converse of (i2.i)), also used in fining
conjugate spaces of
(12.14) Theorem. L ] rel nmbr. I/I is
(i) sup {# lP d t'F is compact, F c X} = oo, 1
then
(it) sup {fxlgdt" 9 o, [19llp • _--<l}--
For p= , (it) holds/or any measurable, nonnegative, extended real-valued
/unction on X that is not in Ehere p'--i .
Proof. Suppose that ]p< and that (i) holds. Let fl be an
arbitrary positive number and let FcX be a compact set such that
f/dt>. For n=],2 .... , let /,=min(l,n); then lim fldt--
f l d t>. Let n 0 be chosen so that f 1o d t>. Since (F) < , we
F F
have /,o. and obviously ]]Lo.vll>fl. Hence by (2.3), there
is a 00, I111' a, sc that ] 1' dt]>fl. Moreover,
x
Sinc I1 a, a III i I1' , thi prov (it).
Suppose now that / is nonnegative and measurable and that 1 .
For every fl>0, (.33) implies that there is a compact setF such that
• ({xF'/(x)>2fl})>O. Let /0=rain(l, 2fl) ; clearly 10 and
]1o [] = 2ft. By (2. 3), there is a oo such that [[[] t and ]/odt [ > ft.
Then we have
x
(1.1) Discussion. Or next aim is to show that the conjugate
space (B.2]) of 9 (]p<) can be Mentified with ,.
inequality (]g.4.iii) and its trivial analogue (]2.]3.i) for
show that every element g of , efines a bounded linear functional
on .
(i) lf 1,= (1).
x
eorem (11.3) implies as a special case that I1%1I Eee (.8)a is eqa to
I111,. We wi to how that every bounded linear functional on has
the form for some g 9, (] p < ). To do this, we require another
result, known as the LEBESGUE-RADON-NIKODM teorem, which is of
fundamental importance in its ovn right.
(12.16) Suppose that I, , ,, and are as in 1 t. ConsMer another
nonnegative linear functional J on 0o(X), and construct for it just as
for I the extended functionals ] an ] an the Carath6odory outer
1 This condition holds, for example, if 1 # and I vanishes outside of a set which
is the union of countably many sets of finite measure. It lails for N if N is locally
null, although N may not be in #.
t44 Chapter III. Integration on locally compact spaces § 12. The spaces p(X) (1 < p <=oo) t45
measure f($a), which we now write as (A). One way to form a functional
J from I is to select a nonnegative t-measurable function g such that
g" F is in 9 (X, t) for every compact set F c X" call such a function g
a weight /unction. We can then define J(qg)=f 99g d t for all 99cI00.
x
We shall show that this description of J is equivalent vith several rela-
tions between I and J and t and 7-
{12.17) Theorem [LEBESGUE-RADOII-NIKODqVl]. Let I, J,, and 7 b,
as in (12. t6) and let /[,, /[, M/',, be the classes o] sets defined ]or t and
7 as in (t t.28) and (t t.26). The ]ollowing statements are equivalent.
(i) There is a weight ]unction g on X such that
I () - f g a, /or aU
x
(ii) Let A C X have the [m A = U A,, where each A,is t-measurabl
g is a weight ]unction depending only on and
(iii) I[ [ is an -measurable complex- or extended red-valu [umio
x
X such that -aZost everywhere, and f/d=f/'gd,.
Again g is a weigh /unction depeing o on and t. x x
(il
(v) Fo ever ad ]o eve positive
Prowl We ben by prong that (i) implies (ii). Suppose first
that F is a compact subset of X. Using (1 t.22), we find open sets
and a compact set E such that
FC...CUn+ICUC...cUicE,
lim t(U)--t(F), and lim 7(U):7(F). For n=l, 2, ..., let 99,(Y0 be
such that qgn(F)=t, qg.(U)=0, and qg.(X)c[0, 1]. Then lissom qg.(x)
F(X) r-almost everywhere and the functions p,, are dominated by E,
which is in g(X,). Also lim ,(x)g(x)=F(X)g(x) t-almost every-
where and the functions p,,g are dominated by . gc I(X, O. Hence
we may apply (i) and LF.BF.SGUF-'S theorem on dominated convergence to
obtain
--- f ed-- lim f qg, d-- lim f ,p, gdt----fegdt--- f gdt. ()
7 (F) x ,,-, x ,,-, x x F
Now let A -- A,, where each A, is t-measurable and 7-measurable
and t(An)< , 7(A,3< . By virtue of (tt.32), A= F UN where
each Fn is compact and t (N) -- (N) -- 0. We may obviously suppose
that F c F.+I for all n----t, 2 ..... Writing D= U F. and using (t), we
see that
?(A)-----?(D)----li,m (F,)- lim y gdt-- y gd,---- y gdt. (2)
n'-°°F D A
This completes the proof that (i) implies (ii).
We now show that (ii) implies (iii). We may obviously restrict our-
selves to the case ]>__ 0. Apply (t t .40) to / and the measure 7 to obtain
For every positive integer n and k--t,2,...,n2"--t, let A,--
xX'--=<['(x) < 2, . Each A,,, is plainly -and 7-measurable and
is the union of a countable family of sets for which t and 7 are finite.
By (ii), we have
k=l k=l Ak, n X k=l
n2--I .
The functions ,- Sa,.. increase monotonically to /', and so (3)
k=l
implies that f / d = f/' d = f 1' g d,. Thus (iii) holds.
x x X
We next show that (iii) implies (iv). Let A be a set that is locally
t-null" t(AF)=0 for every compact set F cX. For each such F,
there is a sequence UI D U. D--. D U D--- D A RF of open sets Such that
Let E = U" plainly ED A RF and
is compact and t(U)< -. =1 '
,(E) -- 0. The function Sv. is t-measurable and (U) =< (U-) < oo. By
(iii) there is a nonnegative function/,--$v,, that is t-measurable and for
which (U)--f/,g d,. The functions /,g are dominated by
x
which is in (X, ,), and so, by LEBESGUE'S theorem on dominated con-
vergence, we have 0 = f lim/ g d t >= lim f/, g d t -- lim (U). Thus
lira (U) =0, and so (ARF)=0. This shows that A is in and (iv)
,
is established.
We next prove that (iv) implies (v). Suppose in fact that (v) fails.
Then there are a 0ffo and a positive number e0 such that for
n t 2, , there is a function p o0 for which 0 :< :< 90, I(n)< 2-"
and j()>=e0. Now set h:sup{p, P+x ..... p+t, ...} (n--l, 2, ...),
and h lim h,. Then we have f h, d t < f d t= , j' dr< 2-2-",
X ---X = X
and % f,,d f hd (=1, 2 .... ). Since h,,,, Lsu's
X X
Hewitt d Ross, Nbsaet oe ysis, vol. I t O
146 Chapter III. Integration on locally compact spaces § 12. The spaces p(X) (1 p =< ) 147
theorem on dominated convergence implies that
f h d t -- lim f h.d t -- 0, (4)
X n-->a X
and
f hd-- lim f h,,d l >= to> O. (5)
X n-oo X
From (4), (11.27), (5), and (11.32), we infer that there is a compact set
Ec{xcX'h(x)>O} such that (E)>0 and t(E)=0. The set E is
plainly in and not in . Thus (iv) fails.
It remains to prove that (v) implies (i). This proof requires several
steps. Suppose first that J GI and that (X)<. Then for every
G;0, CAUCHY'S inequality (12.4.iii) implies that
These inequalities show that the mapping f d is a linear Iunc-
x
tional on ;0 that is continuous in the norm lle of (X, ). Since
is dense in (X, e) (12.10), this mapping can be extended in one and
only one way so as to be a bounded linear [real-valued functional
on (X, e) (B.II). By (B.45) , there is a function gE(X, ) such that
ў(]) = f ] g de for all ]e (X, e). For E;0, we thus have
x
x x
For ffg0, the inequalities 0 ] I imply that
0 f gd, f d,.
x x
It is easy to see from these inequalities that Og(x) t for all xeX
except for an t-null set. We may thus suppose that 0g(x) 1 for
xX. In particular, g.$z[(X, t) for all compact subsets E of X.
This proves (i) in the very special case ] I and (X)< .
Second, suppose that ] I and that (X)=. Let be a family
of compact sets constructed for the measure t as in (1.9). For each
F if, let I F and ]F be the linear functionals on 00 defined by Iy (T) =f
F
and ]F(T)- f d, 00- We now show that ]Iy for Fff. As
in the first paragraph of the proof, there is a dominated sequence
{..1 of functions in ff0 such that 2i.x)=$F(x)t-almost every-
Since (B.45) applies to complex Hilbert spaces, a little explanation is called
for. Extend ў to a complex linear functional ўc on (X, t) (B.38). Then apply
(B.45) to obtain a function g(X, ) such that ўc(/)= f/gdt for
It is easy to see that g must be real-valued, x
where and 7-almost everywhere. Hence if CGo, then
JF(P)-- f pd -- f SFd = lim f
F X n-eo X
lim fq,,v/d=f$pd,=I(p).
n-oo x x
Write tF and TF for the measures corresponding to I F and JF, respectively.
If p0 and p=< 1, then JF(P) = f d<(F). Hence by (11.11), TF(X)
F
is finite. Consequently the preceding case applies to the functionals JF
and IF. Thus for every FE-, we find a function gFE.(X, tF) such that
o<gF(x) =< 1 everywhere on X and for all 00,
f cpd-- f cpgFd,.
F F
We may obviously suppose that gF(X)---O for xF'. Now define the
function g as gF. That is, g (x) = gF (X) for x F (F o), and g (x) = 0
F
for x (U {FEo-})'. It is easy to see that g is an t-measurable function.
Infact, let A = {xX:g(x)>}whereo>=O. Then A F1F= {xX:gF(X)> }
is t-measurable for each Fo-, and A= U (ARF). If U is an open
F
set of finite positive t-measure, we have URF4:J for only countably
many Fco-, say F, F .... (t 1.39.iii). Hence
- .(A +, un( _I(A =,(u).
,(UnA)+,(UnA') , un .
By (11.31), A is t-measurable. It is obvious that g. $E 1 (X, t) for all
compact sets E c X.
Now let q) be any function in if00 and let E be a compact subset of X
such that (E')--0. It follows at once from (11.39.iii) that E has
nonvoid intersection with only countably many F, say F, .....
F ..... Write N=(U{Fo'})'. As noted in (11.39), N is locally t-null,
and since J<= I, it follows that N is locally -null. Hence we have
X E NnE n=l
UF,,
=1
= Z f qgd,= f qgd,.
n=lF X
Thus we have proved (i) if J =< I.
Consider finally the general case. For every positive integer n, let
J rrfin (hi, J) where min (n I, J) is the linear functional on 00 defined
in (B.34)" We first show that J()= lim J,, () for each ffg0 .1 That is,
we assert that for an arbitrary > 0,
inf{nI(y)+J(cp--y)'yo and y=<0}>_--7()-- (6)
1 The reader will note that (v) is used in the proof only to establish this equality.
10"
148 Chapter III. Integration on locally compact spaces § 12. The spaces p(X) (l =<p oo) 149
if n is sufficiently large. We restate (6) as"
x()>=j()- ()
if 0=<=<q, c--0, and n is sufficiently large. For J(q)_<_, (7) is no
restriction. For J(q)>, we use (v) to assert the existence of a 6>0
such that if 0_<__<q, c0, and J()__>, then I()__>& For every
integer n >
, we find that if J()->=o, then nI(v)> J(q)-o>
J()-. Thus (7) and hence (6) hold.
Plainly we have J.<= n I for n= t, 2, ..., and so by the preceding
case there is an ,-measurable function g. such that O<--g.<=n and
J(q)= qg.d, for all q(00- Since Jl<=J.<=..., the sequence of
x
functions gl, g. .... is nondecreasing ignoring as usual a locally t-null
set], and hence we have
() = im . () = f [2 -] '',
X
for (-o- Thus we set g-- lim g. [locally ,-almost everywhere], and the
proof is complete.
Our first application of the LEBESGUE-RADON-XlIKODM theorem
follows.
(12.18) Theorem. For t <--p< oo, the conjugate space o/p= p(X, 0
is ., in the sense that/or every bounded linear/unctional
is a/unction g . such ,ha
(i) (/) = f/g d,/or all
x
and 11 • [[=Jig lip'- [Recall that we take t'--oo.]
Proof. We begin with the space 3. Let be any nonnegative
bounded linear functional on . Since . is a [real linear] subspace of
, we may regard as a linear functional on . that is nonnegative
on _." hence (t2.17) may be applied, if relevant, to and the original
functional I in terms of which (X, ,) was defined. Consider any
function q9 in _.. Then, if 0__<p=<q9 and pc_., we have
This implies that and ! satisfy condition (t2.t7.v), and hence by
Theorem (t2.t 7), there is an t-measurable function g>= 0 on X such that
g" F is in 9a (X, ,) for all compact F C X and
'() = f 9og a, fo u o.
x
Eo 4o nd I111__< a, we thus have
f I1 g a, _<_ II '11.
x
If F is a compact subset of X, then
IY e a,] __< YII e a,__< II ll-
By (t2.t4), therefore, we have gFJ3, and by (t2.t3), IIllrllll.
Let =P{IIII,:F i compact and FcX}. Let CC---Cc---
be a sequence of compact sets such that 1 I111,=- Let
Then g is locy almost eye,here 0 on D'. If this were not the ce,
there wod be a compact subset E of D' such that, (E)> 0 and g(x)> 0
for xE [see (-33)]- Then we have f g#" dr>0 and F, UE is compact
for n=t, 2, .... Since lim f g" d,= we then obtain
y e'a,=ye'a,+yg'a,>"
UE
ior n sufficiently large. Redefining g to be 0 on D', we obtn a fction
, o wi 0) o]. om 02.>, we ve IIll:ll&. Since
is dense in (2.0), the representation () holds for a
For an arbitrary bounded ne function on , we use (.7)
to te -- , where both and e noegative. If
e as in () for and , respectively, then we plnly obtn (i) th
g:g--g, for IreS] fctions on . The representation (i) d
Theorem ( 2. ) show that I[ [I : II &-
We now consider arbitrary bounded ne functional on the
complex anach space . Then for each/ , we have (
i(/), where (1)d (1)e re. It is easy to see that" (1+1'):
• (/):- (/)- a p {I (/)1 Illll } II ll (i: , =). Considered
on the [real] e space , therefore, is a bounded ne fctional,
d by the preous scussion ats a reprentation
X
for . Sily ats inteal reprentation on the IreS]
subspace of consisting of a functions i th : hence we have
(i) : f d, vhere h,. Thus for/:+i, we have
x
( = (1) + ( = ( - (1)
= () + () - i [- () + (i)] = f [ +
x
Sce g and h e ,, g--ih is in ,, so that (i) is estabshed. A f
appeal to Theorem (2.) shows that II-l,:llll.
Hotes
Ts section necessy deps somewhat from clsic treatments
of spaces fod for example in H, Lwoo, d P
and Duo d Scwz []. Ts of cose ses because of the
150 Chapter III. Integration on locally compact spaces § 13. Integration on product spaces
possible existence of locally null, nonnull sets; (12.2), (12.11), (12.12),
(12.14), (12.17), and (12.18) all differ from the corresponding classical
assertions. The very useful Theorem (12.t 0) obviously has no analogue
for gp spaces defined for abstract measure spaces.
Lemma (12.3) is due to W. H. YOUNG [11, and the proof of (12.6)
to F. RIESZ [31, P. 45. Theorems (12.13) and (12.14) are adapted from
BOURBAKI [2}, Oh. IV, §6, N ° 4, Remarque 2). The classical case of
(12.13) and (12.14) is due to F. RIESZ [2. Theorem (12.17) is of vital
importance in harmonic analysis. For X= R and ,= Lebesgue measure,
it is due to LEBESC, UE [1 ; for X an interval in R"and general set func-
tions, to RAI)O [11; for abstract measure spaces, to NIKODM [11. Our
treatment is based on BOURBAKI [31, Oh. V, § 5, N ° 5, Th6orme 2. For
p> 1, Theorem (12.18) is the translation to our setting of a classical
result of F. RIESZ [21, p. 475. For p=l, the classical case of (12.18) is
due to H. STEIIAIJS
§ 13. Integration on product spaces
(13.1) Discussion. Throughout (13.1)--(13.14), X and Y will denote
arbitrary nonvoid locally compact Hausdorff spaces, and I and J will
denote arbitrary nonnegative, nonzero, linear functionals on 00 (X) and
00 (Y) respectively. For I and J, we construct the extended functionals
I,, , and of, and the corresponding outer measures, and . The families
of sets ',, 4 , and /n, n are then defined as in §11. Note that
//,, f,C (X) and
We are concerned with the product space Xx Y and with functionals,
measures, and integration on Xx Y defined by means of I, J and
Let / be a complex- or extended real-valued function on Xx Y such that
for some xoX, the function " y--/(x o, y) is in 00(Y). We then write
J(/(Xo, y)) for J() ; the expressions I,(/(x, Yo)), L ([(x, Yo)), (/(Xo, Y)),
(/(x, Yo)), and Ј(/(x o, y)) are defined analogously. For functions [
on X and g on Y, let /g denote the function (x, y)--/(x) g(y) defined
on Xx Y. [The ranges of / and g will always be such that the definition
is meaningful. 1
(13.2) Theorem. For every/oo(XxY), we have J(/(x, y)) 00(X)
and I([(x, y))00(Y). Also the equality
(i) L[J(/(x, y)) =J[L(/(x, y))
obtains. The /unctional de/ined by (i) is a nonnegative, nonzero, linear
/unctional on oo (Xx Y).
Proof. We may suppose that ] is real-valued. For /o(XxY)'
there are obviously open sets UcX and Vc Y such that U- and V--
are compact and ! (x, y) =0 for (x, y) (Ux V)'. [Consider the projections
of the set {(x, y)XxY:j(x, y)@0} onto X and Y. The open subset
UxV of XxY is evidently a locally compact Hausdorff space in its
relative topology. Consider the space g) of all functions h on Ux V of the
form %.oi wllere , ..., G0(U) and o ..... G0(V). It is
j=l
clear that is a subalgebra of (UxV) that separates points" if
(u, v) and (u, v) are in UxV and uu, there is a such that
(u) = 1, (u) = 0 and a such that (v) = 1. The STOE-WEIERSTRASS
theorem implies that is uniformly dense in G(UxV). Now the
function ], with its domain restricted to Ux V, is plainly in G (Ux V).
Hence ] can be arbitrarily uniformly approximated on Ux V by functions
in . We extend the functions ii in ) by letting i(x)--O for
i=1
xXU' and i(y)=0 for yYV' (=l, ..., m). Then for every
e>0, there are functions , ..., ,0(X) vanishing on U' and func-
tions , ..., ,0 (Y) vanishing on V' such that
- < (x, Y.
By Theorem (11.6), there are positive numbers and fl such that
tJ( )t II ll,
oo(Y) and (V-') =0.
Now consider a ]ixed xX. The function
/ (x, y) - X (y),
defined for yY, is obviously in 0(Y) and vanishes on V' V-'.
Hence () shows that
There are several forms of the SOE-WEIERSRASS theorem. A set of func-
tions on a set X is said to separate points o/X if for x, y X, x y, there is an
such that [(x) [(y). Several versions of the SOE-WEIERSTRASS theorem follow.
(a) For a compact Hausdorff space X, a subalgebra of r(X) that separates
points of X and vanishes identically at no point of X is uniformly dense in r (X).
(b) For a compact Hausdorff space X, a subalgebra of (X) that separates
points, vanishes identically at no point of X, and is closed under complex con-
jugation, is uniformly dense in (X).
(c) For a locally compact Hausdorff space X, a subalgebra of (X) that sepa-
rates points and vanishes identically at no point of X is uniformly dense in (X).
(d) For a locally compact Hausdorff space X, a subalgebra of 0 (X) that sepa-
rates points, vanishes identically at no point of X, and is closed under complex
conjugation, is uniformly dense in 0 (X).
For a detailed proof of (a) and sketches of proofs of (b)--(d), see for example
HEWlTT and STROEe [2], (7.30) and (7.37).
t 52 Chapter III. Integration on locally compact spaces § t 3. Integration on product spaces 153
The estimate (2) is independent of xX. Thus the function Jy(/(x, y))
is the uniform limit on X of functions Y. J(Pi) Pio (X), each vanishing
on U'D U-' It follows that J, (! (x, y)) is in -o (X) and vanishes on U'.
From (2) we also infer that
[;. [:, - ;%) <
i=1
Precisely the same argument shows that I.(/(x, y)), qua function of y,
is in [o (Y) and vanishes on V'; also
]Jy [I,, (] (x, y))]- I(Pi) J(pi)] < fl0te. (4)
Since e is arbitrary and 0t and fl are fixed as soon as ] is fixed, (3) and (4)
establish (i). The linearity and nonnegativity of the functional defined
by (i) are obvious. If I@)4=0 and J(W)4=o, then
I(p) I(P) :! = 0. [i
{ 13.3) Definition. The linear functional defined by (t 3.2. i) is called
the product of the functionals I and J and is written IxJ. The outer
measure on #(Xx Y) defined as in §t t from IxJ is called the produa
of the measures e and /and is denoted by
13.41 aorm. lo, g + (XЧ '), , y, (g(,,
I (g (x, y)) R ч (I o. Furlherrcun, e
Proof. Let 11 be the set of functions/4o(XЧY) such that Ot<=g.
As shown in 03-2), each function ](/(x, y)) is in 4o(X). By 0t.t0.v),
w w g-u{I:lu}. Itnc fo ch nxa ox, w ve g(o, y)=
sup{/(x0, y)-/II}, for a yY. Each function /(xo, y) is
and Theorem (it-t3) implies that y,(g(xo, y) ) = su {L (/ (x o, y)):/tt}.
This being true for all xoX, (tt.t0.iii)implies that
function of x is in all ч (X). The argument for i,(g(x, y)) is the same.
Using (it.it), (t3-3), and (it-t3) twice, we have
IxJ(g) = sup {Ix J(/} "[11} = sup {I. [J (/(x, y))] "[11}
• h equat IЧI()= J,[I,(g(, y))] i ova iny. n
{ 13.5) CorollatT. IIh is any o..gative extended real-valued [unaio
o Xx Y, then
(i) IxJ(h)>__max{i [j(h(x, y))], [i.(h(x, y))]}.
Proof. If gR+(XxY) and gh, then we have
The monotonicity of i and f (tt.t7.ii) implies at once that
Now take the infimum over all g.
For functions of the form /(x)g(y), we can improve upon (13.).
(13.6) Theorem. Let / and g be arbitrary nonnegative extended real-
valued/unctions de/ined on X and Y respectively. Then
(i) Ixl(/g)--i(/) y-(g)
unless one o/i (/), [ (g) is oo and the other is O.
Proof. The inequality
IxI (I) <i,(l) i() ( )
is a]] that needs to be proved, in view of (13.). The only case to be
verified is that in which_ i(1)<_oo, 1(ў)< oo. Let fl and , be any real
numbers greater than i (1) and f(g), respectively. Then there are func-
tions_ ]'R+(X) and g'R+(Y) such that 1<__1', g_<g', /(/')<fl, and
I (g') <,. The function ]'g' is evidently in X + (Xx Y) [recall that
0.oo-----0] and so ve have fl,>i(l')f(g')-i,[(l'(x)g'(y))]--
IЧ](/' g') > IЧ](/g). This establishes
We now take up the problem of computing integrals with respect to
,Чr/in terms of integrals with respect to
(13.7) Theorem. Let A be a subset o/ XЧY. I/ A is ,Чpnull,
then the set A={yY'(x, y)A} is 7-null /or ,-almost all xX and the
Proof. We consider only A. If ,x r/ (A) -- 0, (t3.5) implies that
0=/(ea) [(e a (x, y))]--(r/(A))0. Theorem (tt.27) shows
that (A,) vanishes for ,-almost all xX.
(13.8) Theorem [FLIBINI'S theorem/. Let I be in (Xx Y, ,x). Then
l(x, y) qua /unction o/x is in (X, ) /or 7-almost all y Y, and l(x, y)
qua /unction o/ y is in (Y, 7) /or ,-almost all x X. Furthermore the
/unction o/ x defined by x---> f l (x, y) d 7 (y) where the integral exists and 0
elsewhere is in (X, ,);z similarly/or f l(x, y)d,(x). Finally, u2e have
x
1 The sets A x and A are called sections of A.
a Throughout this book, the function x-+ fl(x, y) d(y) will be defined every-
Y
where on X by this convention; similarly for y-+fl(x, y) d,(x).
Chapter III. Integration on locally compact spaces § 13. Integration on product spaces
(i) f /(x, y)dtx(x, y)= f f /(x, y)d(y)d,(x)
XXY X Y
-- f f /(x, y)dr(x)d(y) .:
YX
Proof. Consider first a ]ixed t x7-measurable function ] for which
f 1/ dtx< . There is no loss of generality in supposing throughout
xY
the present proof that / is nonnegative. We first show that the function
x/(x, y) is e-measurable for -almost all y. By (tt.40), there is a
function /' defined on XxY such that f /'(x,y) dtx(x,y)=
x Y
f /(x, y)dtx(x, y),/'/, and the set {(x, y)XxY'/'(x, y)>e}is
xY
a-compact for every e0. Thus /=['+h, where h is a nonnegative
function on Xx Y that vanishes except for a set N of tx-measure 0.
Let P be the set of yY for which t{xX'(x, y)X}--0" by (t3.7),
we have (P') =0. Now if y P and e0, we have
{xўX'/(x, y)> e}= {xўX:/'(x, y)> } U {xўX: (x, y)ўN }
and h(x, y) > -- /'(x, y)}. , (1)
The first set on the right side of (t) is a-compact and the second set is
-null, so that their union is -measurable. Similarly the function
y/(x, y) is -measurable for ,-almost all x.
Now by (t2.t0), there is a sequence , .... of functions in
ў0 (Xx Y) such that lira f /(x, y)-- (x, y) dx (x, y)=0. From
(t3.5) and (tt.36), we can then write
li ] { f [/ (x, y) -- % (x, y) [ d (y) ] = O, (2)
where we replace f l/(x, y)-- %(x, y) l d (y) by 0 if this integral is
Y
undefined, i.e., if /(x, y)- % (x, y) is not -measurable. By (t t.27), there
is a subsequence of the .s [which we again write as , .... such
that
lira f 1/(x, y) -- % (x, y)] d (y) = 0 ,-almost everywhere on X. (3)
y
Since f %(x, y)d(y)=(%(x,y)) [use (It.36) is in ў0(X) as a
Y
function of x (3.2), we infer from (3) that f/(x, y) d(y) is ,-measurable"
Y
it is the limit ,-almost everywhere of functions in ў;0 (X). It now follows
1 Here and throughout this book we use f ... d like parentheses. Thus the right
member of (i) is to be interpreted as
Y
from (2) that
lim : I: l(x, y)d (y)- f (x, y) d ()i , (x) = o, (
--->oo y
and so the function x-->f/(x, y)d(y) is in :(X, t). Finally, taking
note of (t3.3), we have Y
This implies the first equality in (i). The remaining equalities are obvious.
To complete the proof, it suffices to notice that if g(x, y)=/(x, y)
x7-almost everywhere, then for e-almost all x, the equality /(x, y)--
g(x, y) holds for 7-almost all y, and for 7-almost all y, /(x, y)=g(x, y)
for -almost all x. This follows immediately from (t3.7).
(13.9) Theorem. Let / be a nonnegative extended real-valued Junction
de/ined on XxY that is tx7-measurable and vanishes tx7-almost every-
where outside o/ a set U A n where each A is tx-measurable and
n=l n
x 7 (An) < oo. Then the/unction x-.] (x, y) is e-measurable/or 7-almost
all y Y, and the ]unction y -. f / (x, y) d t (x) is 7-measurable; similarly
x
/or y-./(x, y) and x-. f/(x, y) dw (y). Furthermore, we have
Y
(i) f /(x, y)dtx(x, y)= ff/(x, y)d(y)de(x)
XxY X Y
-- f f /(x, y)d,(x)drl(y ) .
YX
Proof. We may clearly suppose that AcAc...cAnc.... Let
/,-min (n,/A,) (n=t, 2 .... ). Then it is clear that ]<=/<=...<],,<=...,
lira/,=/tx-almost everywhere and/n (XxY, tx) for n=l 2,
Let P be the set of y Y for which li,m/nx,y ) =/(x,y) e-almost everywhere
and for which the function x--/n (x, y) is e-measurable (n= I, 2 .... ). By
(t3.7) and (t 3.8), ve have 7 (P') --0. Clearly the function x-+ li_.m/n(x, y)--
/(x, y) is e-measurable for y;,P. Also each function y--f ],,(x, y)de(x)
x
is ]-measurable and is in (Y, 7). By the theorem on monotone con-
vergence, we have f/(x, y) de(x)= lim f/n(X, y) de(x) for yP. Note
X n--oo X
too that the functions y-. f/, (x, y) de (x) are monotone increasing with
x
156 Chapter III. Integration on locally compact spaces § ! 3. Integration on product spaces
n (y P). Combining these observations with (13.8), we obtain
.f /(x, y)a,x,(, y) = im f /.(x, y) d,x,(x,
= lira ff/.(x, y)d,(x)d7(y ) = f lim f/.(x, y) d(x)d(y)
n-oo y X y n--o X
-- f f /(, y)d()d/(y).
YX
xY
(13.10) Theorem. Le / be coml- or xd m-,
measurable /unction on XY vanishing -most everywhere outside
o/a set O A, where each A, is ,x-measurable and ,xy(A,) < . The
three integrMs
(i) /(.. r)d.n(., r). ff/(., r)dn(r)d.(.).
XY X Y
YX
are ]inite and equM to each other i] and only i] one o] the integrals
(ii) f ]/(x. y)] d.xn(x, y). ff[/(x, y)[ dn(y)
XY X Y
f f ]l (x. y) l d. (x) dn (y)
YX
is ]inite.
Proof. We may write ] as /--/,+i(l--/,) where 0b[]l o
i=l, 2, 3, 4. Then f [](x, y)]dey(x, y) is finite if and only if each
f ]i (x, y) dexy (x, y) is finite (i= l, 2, 3, 4); similarly for the iterated
integrals. Thus the assertion follows immeately from (13.9).
(13.11) Theorem. Let A X a BY and suppose tha ,(A)=0
and B is the union o] an increasing sequence B, B, ..... B .... o] [not
necessarily y-measurable/ sets o/ ]inite y-measure. Then we have
,x(AxB)=0. I] B is any set, and A is locMly e-null, then A xB is
locally ,xy-null. Similar sertions hold i] the r6les o] A and B are
interchange.
Proof. By (t3.6), we have IxJ (GXB) =?xJ (G B.) =(G) ()
=,(A) y(B,)=0. Thus by (li.18), we have ,x(AxB)
= lim IxJ
Now suppose that A is locally e-null d B is bitry. If F is any
compact subset of Xx Y, then F is contMned in a set DxE, where D
is a compact subset of X and E is a compact subset of Y. Then we have
(AxB)Fc(AD)x(BE), and ,x((AD)x(BE))=O by the
first case.
(13.12) Theorem. Let / be in (X, ) and g in (Y, ). Then the
/unction (x, y)--->/(x) g (y) [written as /g] is in (Xx Y, x) and
f /(x)g(y)d,x7(x, y)--f/(x)d,(x)fg(y)d7(y ).
XXY X Y
Proof. In view of (13.6) and (11.36), we need only to prove that
is x-measurable. Using (12.10), choose a sequence 1, . .... , n,---
of functions in 00(X) and a sequence of functions pl, p., ..., Pn,---
in 00(Y) such that li.om ][/--n]--0 and li.om [] g -- pn ] -- 0. Elementary
properties oflxJ, (13.6), and (11.36) imply that
xj (I/g- . .l) < i xJ(I/ -/.1 + I1. - . .1)
<_x](l/-/.l) + Ч1 (1/.- .. l)
=([/I)([- .[)
t q {11.11} i bo, t t xp=io go to o
n-o. Hence by (1 t.27) a subsequence of {l/g--Cpn pn [}n°°__ converges to 0
txl-almost everywhere on XxY. Since each Tn Pn is in oo(XЧY),
it follows that ]g is ,x-measurable.
(13.13) Theorem. Let A c X and B Y be measurable and a-]inite.
Then A xB is ,x?-measurable and tx7 (A xB) -- (A ) • ? (B).
Proof. Let A 1, A. .... , be a monotone increasing sequence of
measurable subsets of X of finite t-measure such that 13 An--A ; simi-
larly for B1, B. ..... Then by (t3.12), #.X.=A.#. 9-1(XЧY, ,Ч),
and ,x?(AnxBn)--,(An) ?(Bn). Now take limits as
A result similar to (t3.t3) holds for arbitrary sets A and B so long
as the product ,(A) "7 (B) does not have the form 0-oo or oo. 0.
(13.14) Theorem. Let A c X and B C Y be arbitrary sets such that
t(A) . (B) is not o/the/ormO.o or oo. O. Then ,x(A><B)--,(A).7(B ).
Proof. By (13.6), we have
• x7(AxB ) =I Чy (A>) =i(A)f() =
(13.15) We now take up the theory of integration on arbitrary
products of compact Hausdorff spaces. For the remainder of the present
section, {Xv}v . will denote an arbitrary nonvoid family of nonvoid
compact Hausdorff spaces and X will denote vP, X v. For each ,F,
I will denote an arbitrary nonnegative linear functional on (Xv) for
which I v (1)--I. We will shortly consider a linear functional I on (X).
For all of these functionals, I, I v, , I=v, , and v will have tile meanings
established in § t 1.
Chapter III. Integration on locally compact spaces § 13. Integration on product spaces 159
For 70 F and a function g on X n, we can consider the function g
on the product space X. ]Recall that v0 is the projection that carries
(xv)X onto xn. 1 It is obvious that if g(Xn), then go:n(X ).
Throughout the remainder of this section, we will write g o v° as g where
no confusion can arise.
Also, consider any function ]((X). Select a 70F and a point
avX v for all 7:7o" Write a' for the point (aV)vPmvoXv. The function
xn-->/((a' , xvo)) is obviously continuous on X n. Applying the func-
tional I n to this function, we obtain a complex-valued function on v.P
Its value at a' is written as In[/(a')l. It is convenient to define
on all of X by the rule that In[[(a) for aX is equal to IvoI/(a')],
where a'=(av).vo. That is to say, In[J(a)l does not depend on the
70-th coordinate of a. Finally, we will abbreviate Iv,[Iw.[...
as I,...... , (/); similarly for [v ...... w (]) and I,...... , (/).
We will adopt similar conventions with regard to integration.
Consider any complex- or extended real-valued function h on X. Select
a 70F and a point arc X v for all 7:4: 70. Write a' for the point (av)
Suppose that the function xvo-->-h((a', Xvo) ) belongs to l(Xn). Then
integrating this function with respect to v0, we obtain a complex-valued
function defined for some or all a'. Its value at a' is written as
f h((a',Xvo)) dvo (xn) or more briefly f h d n. As in the case of functionals,
XVo XV°
we define f hdt n at (a', an)PrX as f h((a',xn)) dtn(x o) for all
XVo XVo
choices of aVo X n.
(13.16) Theorem. For [(X) and 7o1-', the [unction In(/) is also
in (X). Furthermore, i/ 71, ..., 7 are distinct elements o/ I', and
{nl, ..., n} is any permutation o/{t, ..., p}, we have
(i) I v ...... v, (/)---- Ivn, ..... w, (/)"
Proof. Let Yj denote the space of all functions on X of the form
h- /-/v], where each q?i, belongs to 62 (X0) and 1 ..... ,are
i=i =I
distinct elements of/'. Let 7 be any element of/'. Then Iv(h ) is in
In fact, if 7 is 0 for some k 0- t, ..., n, then
o )
and if 7 is distinct from all the b, then
Iv(h)--h.
These expressions for Iv(h) show that Iv(h ) is in Y). It also follows that
both sides of (i) are well defined and belong to
Next consider an arbitrary /
The function space is a subalgebra of (X) that is closed under complex
conjugation, separates points of X, and contains . The STONE-WEIER-
STASS theorem implies that there is an h. for which
For 70 -P, it is clear that I Ivo (]) -- In (h) l < e. Since Ivo (h) ,2, we see
that Iv0 (]) is the uniform limit on X of functions in 2, and thus Iv0 (]) C (X).
Therefore both sides of (i) are defined for f c(X), and both sides are
functions in ( (X).
With all coordinates x
as a Icontinuous] function on Xv, x... ><Xv, Then ( 3.2) shows that the
two sides of (i) are equal. []
(13.17) Definition. For a finite subset A:(71, ..., 7} of/' and a
function/(X), we define Ia(]) to be the function in (X) defined by
(t3.6.i).
(13.18) Theorem. Let (A) be the set o! all finite subsets o! I', directed
by inclusion. Then ]or every ](X), the net I (]) o[ ]unctions i (X)
converges uni/ormly to a constant/unction, which we denote by I(/). The
/unctional I on (X) is linear and nonnegative, and I()
Proof. Let be a positive number. As pointed out in the proof of
(3.6), there is a function h: fv], fv],(X), where
]=1/ў=1
bl .... , are distinct elements of/'] such that I[]--hll . Let 30--
( ..... b). If zI zI0, then
i=1/ў=1
Thus all of the functions I (h), ZI D ZI0, are the constant function whose
value is I0 (h). We also have
]1I (/)-- I (h)]]<, for all ZI. (2)
From (1) and (2), we obtain
]Ia(/)-Iao(h)ll,<e for all ADA 0.
Since Iao(h ) is a constant function, whose value can be regarded as
depending only upon e [once h:h, has been chosen], an elementary
argument shows that Ia (/) converges uniformly to a constant function.
The linearity and nonnegativity of I follow from the linearity and
nonnegativity of each
For the remainder of this section, I will denote the linear functional
on (X) given in (i 3.18).
160 Chapter III. Integration on locally compact spaces § 13. Integration on product spaces t61
(13.19) Corollary. Let ]=/1/. "'" ],. where//(X#) and .... ,
are distinct elements o/I'. Then
Z(l) = H +(1).
j---1
(13.20) Theorem. Let g: gx g,. "" g,, where gi6 X + (Xaj) and Ox .... ,
are distinct elements o/I'. Then
(i) i(g)=
Proof. We first observe that if At ..... A,, are nonvoid subsets of
[0, oo], then
H sup{a'a6Ai}-----sup{at a,..., a,,'ai6Ai}. (t)
i=l
We omit the proof of (1), which is elementary; the validity of (t) depends
upon our convention that 0-oo--0.
Since gi:sup{]i:]i6+(Xej)and ]i<__gi} (1t.t0.v), the equality (t)
implies that
g-- sup {/x/.""/,,,'/i 6 + (Xe) and /i <= gi' i -- 1,..., m}.
Applying (11.t3), 03.19), (1) again, and (ll.ii), we have
[(g) -- sup {I(/ /. . . . /,, " /i 6 + (Xn,) and /i <=
-- sup { f-l l+, (/i) " /i 6 ++ (X+,) and
j=l j=l
each A v is v-measurable and such that Av:X v ]or all but finitely many
coordinates 761. Then A is e-measurable and
(i) , (A) = H er (A)
[this is really a finite product].
Proof. Let A be a finite subset of/' containing {?6/':Av=Xv} and
suppose that A >__ 2.
We first prove (i) in the case that each A v is open or closed. Let
Ac: {06A'An is not open}; we prove (i) by induction on Ac. If
then each ean6 X + (X0) since A n is open, and therefore
,(A) =i()=i H :+)=Hi+(+)
(eEzt Ezt
by (13.20). Suppose now that (i) is valid and that A is e-measurable
whenever At<n, and suppose that Ac--n, n>=t. Choose o in zl,.
Writing Ao--A n {0o}' and Y=rPn.X,, we see that
Xn.xeP.A+x Y--(Ae.xPn.A+ x Y) U (A;.xPn.A+ x Y). (t)
The inductive hypothesis applies to X.xePa.AexY and also to
A.xga.Aex Y. Therefore a = Ae.xPn.ax Y is e-measurable, and as
e is finite, we use () and our inductive hypothesis to write
and
171, (A,) = ,C A) +
,(A) = [t--ee.(A.)] H q(A+)= H
This proves (i) in case each Aa is open or closed.
Consider now the general case. Suppose that ee, (A e,) ---- 0 for some
80EA. Then for > 0. there is an open set U, D Ae, for which re, (U,)< .
Hence e (A) =< e(U,><,Pe, X) = re, (U,) < . Suppose finally that q (Aa) > 0
for all 6EA and that A =m. Let be an arbitrary number such that
0< e< 1. Then there are open sets U and compact sets F such that
Fe C Ae C Ue and q (Ue) < q (Ae) <- q (Fe). Then
1 t / (2)
= (g,nЧ& x 4
Since is bitrary, we have ,(A) = H *n (An).
The inequalities (2) also show that A is the union of a a-compact
set and a set of ,-meure 0 if all q(An) e positive. Hence A itself is
*-measurable.
(13.22) Theorem. Let Av be a subset oWX v/or each ?E./", //all but
a countable number o/the A are equal to X and all A are e,-measurable,
thenPrA, is ,-measurable an e(PrAv)=# e, (A,). t I/ev(A,)<t/or an uncountable number o/t indices y, then
Proof. The fst statement follows at once from (13.21) and the
COuntable adtivity of ,. To prove the second, note that there is a
number < l such that ,(A)< for infinite number of indices
and so there e open sets in these Xv such that A
This product is dfind inf {t (Av,) .... (Av.) "{x ..... n} is a finite subset
of
z If (Av ) 1 and Av X v for an uncountabl number of th indics ?, thn
dv need not b masurabl; s (16.13.f).
Hewitt d Ross, Abstct hoc Mysis, 1. I
162 Chapter III. Integration on locally compact spaces § 13. Integration on product spaces t63
Consequently we have t(,gvA, ) z ' for n= 1, 2, 3, .-., so that t(,PvA, )
-----0o
We now take up the analogue of Theorem (13.8) for infinite products.
Several preliminaries are needed.
(13.23) Let X=,PI,X , and suppose that {0}0,o is a partition of P.
If for each 0 O xve define Yo= P X,, then X is homeomorphic with the
' 'PO
product Y--ooYo. In fact, the mapping of X onto Y given by
(Ix)) = Iol, w o = I x)o, i ovio a oooi. o
each 00, let Jo be the functional on (Yo) formed as in (3.18) from the
functionals {Iy}Vo, and let J be the functional on g(Y) formed as in
(13.18) from the functionals Jo. Let 0 be the measure on Yo associated
as in }11 with Jo. If I denotes the functional on g(X) formed as in
(13.18) from the functionals {Iy}yv, it is natural to expect I and J to be
essentially the same functional. The following theorem shows this to
be the case.
(13.24) Theorem. Le {}oo, Yo, Y, , Jo, and J be as in (13.23).
Then
(i) z(l)=J(lo,-) lor zz l(x).
Proof. Since z is a homeomorphism, it is clear that / belongs to
g (X) if and only if / o z- belongs to g (. It is also easy to see that the
mapping /J(/oz -) is a nonnegative [and hence continuous line
functional on g (X). We omit the argument.
Let cg(X) be as in the proof of (13.16). Since is dense in the
uniform topology of g(X), it suffices to establish (i) for functions in .
Since I and J are linear functionals, it suffices to establish (i) for func-
tions of the form h = where each belongs to g (Xe) and ,...,
are distinct elements of F.
Let A0= { ..... ,} and O0= {OO'AoO@}" clearly O0 is finite.
For 00o, we define Ao--Ao and W0= %" we regard W0 as defined
8 a 0
on Yo-- P Xr. Finally, we observe that the function W 0oW° defined
yv o
on Y is equal to hoz -. For 00o, we have Jo(Wo)=(Jo)ao( )
da0
II(9). Therefore we infer that
da0
In further consideration of X, Y, I, and J as in (13.23), we will
identify X and Y and I and J. Theorem (13.24) shows that this is
justified.
We now state and prove the p analogue of Theorem (13.16).
(13.25) Theorem. Let 1 <--_p< co. For/p(X) and 7oI', the/unc-
tion f / dt,o is de/ined and is complex-valued /or t-almost all (x)X.
Xo
Where otherwise unde/ined, set f/do=0. The /unction f/dv, is in
(X) and x. x.
Furthermore, let ,..., be distinct elements ol F, and let {n .... , n}
be any permutation o/{1 .... , m}. Then we have
(ii) f f ... f /d,,...d,= f f ... f /d...d,,
Xm Xm_l Xyl XyN X3N_ 1 Xyn l
this equality holding e-almost everywhere in X.
Proof. Since (X)=I, we have o(X)(X). Now, to apply
(13.24), write O= {1, 2}, = {0}, =F {0}'. Then in the notation
of (13.23), we have =Xr., =Iro, and =,r.. By (13.24), we have
X=XoxY and I=Iox]. By (13.3), then, we have
We now apply FUBINI'S theorem (13.8). We have /((xv))=
l(xo, (x)G)=l(xo, y), where yY. As a function of xo, l(Xo, y)
is in g(X0 ) for all yA, where AcY and (A)=0. Thus we can
write f l(xo , y)d0(x;,0) and obtain a complex number for all yA.
X?0
The convention in (13.15) under which f I d,o is defined shows that
X0
f I d o is defined for (Xo, y)XoxA . By (13.14), this set has -meas-
X 0
ure 0. Thus the first statement of the present theorem is proved.
The function f ]/(x0 , y2)I f do (xo) also is defined everywhere on X
X0
by the same rule.
H6LDER'S inequality (12.4.iii) for p> 1 and an obvious estimate for
P t give us
• ke relations (1) koM -lmost everkere o X.
The functions f I d<o and f I11 are in , (X), as (13.8) and (13.12)
X7o Xo
show recall the definition of these Јunctions in (3.). Usin (),
164 Chapter III. Integration on locally compact spaces § t3. Integration on product spaces
(t3.t2), and 03.8), we obtain
fl f/dtv°l d, <= ff I/I a,-- f f I/I f I/I d,. (2)
X X, XX, YtX, X
Tking -th roots (2), we hve
Ts establishes (i) and of course the fact that f/dG,(X).
Both sides of (ii) e well defined and e complex numbers ,-Mmost
everhere, since at each step the inteand belongs to x (X), as we
have just sho. Now break up F into = {, ..., } and 5='.
For any fixed yzYz for wch f/(Yx, Yz) dx (Y) exists and is fite, we
can apply (t3.8) to obtain (ii). The set of such yz's is M1 yzYz except
for a set of z-mease 0, as (13.8) shows. Then (t3.tt) completes the
proof of (ii).
(13.26) Definition. For a finite subset A = {, ..., ,} of F and a
function I(X), f/11 denote the function defied by (3.25.ii).
We now state d prove the analoe of (t 3.8) for infite products.
(13.27) Theorem. Let {A} be the sa o/l finite subsets o/F, direa
by inclusion, and let ] # (X), t # < . Tn
lim I1-- 'll =0.
Proof. Let >0 and choose g(X) such that II/-gl<T 02.o).
By (3.t8), there is a finite subset Ao of F such that
IIz(g) - z (g)II. < for
Repeated application of (t3.25.i) shows that
fite sets A. Using mOWSK'S inequMity (t2.6) and (t2.7), we have,
if A D
< + II1() - I()1+ d-
x
[The inequality IIg- 11 IIg- 11 onow fom
Theorem (13.27) shows that the integral f/dt is the limit in the
x
p metric of the functions f], for ]E3p(X). If/ is countably infinite,
this result can be sharpened considerably: the functions f] converge
valmost everywhere to f ] dr. To show this, we first prove a lemma.
x
(13.28) Lemma. Suppose that I'--{t,2 .... ); /or each positive
integer n, let A,,--
A0
Proof. Let
B. = {xX
and C. BFIB' = , .. o
= .+l, for n t 2, .. Then {C,},=1 partitions A e into
measurable sets. [Theorem (t3.25) shows that the functions f I are
t-measurable.] Therefore to establish (i) it suffices by virtue of countable
additivity to show that
0,(C,) <= f 111 dt ()
for n=t, 2 ..... In order to apply (t3.24), let O= {t, 2}, --A,,, and
--/'flA,. Then Y --Pn.Xv and Y,.=P;,Xv. For k > n, the function
f/does not depend upon the coordinates Xl, ..., xk, and therefore
does not depend upon the coordinates xa .... , x.. It follows that c.
does not depend upon the coordinates xa ..... x,. Equivalently, c,,(Yl,
does not depend upon the yl-coordinate. Using this fact along with
(t3.8) and (t3.24), we obtain
C,,, X
Recalling that */1 (Y)--t and that C, c {x6 X: l( ])(x)l > 6},we see that
C-m Y,Y, X
This proves (t). []
(13.29) Theorem. Let 1 and A. be as in 03.28). Then/or/6(X),
lim f / -- f / d t t-amosA,
n-to A X
166 Chapter III. Integration on locally compact spaces § 14. Complex measures 167
Proof. For xX, let r (x) be defined by
r (x)= -oolim [sup {l(/) (x)- (L/) (x)l'm'n >= P}]"
Let 6 and e be positive real numbers. By (12.10), there is a g(X) such
that H/- gl]l -ў and, as shown in the proof of (t3.16), there is a function
h such that [[g--h[l< - and f h-- f h dt for all sufficiently large n.
An X
Evidently [[/--h[[l< e ;writing for/--h, we have
r(x)--p-oolim [sup {]( )(x)- (j,,p)(x)l'm,n >= p}]
This inequality implies that
{xX'r (x) >2 6)C {xX" sup {1()(x)]} >
Applying (13.28) to the function % we therefore have
(5. t({xX:r(x)>2(5}) f ]] d,--I[lll< ,.
x
Since e is arbitrary, ve have t({xX:r(x)>2b})--O, and since b is
arbitrary, it follows that r(x)--O t-almost everywhere. This implies
that lim f ] exists t-almost everywhere. By FATOfJ'S lemlna and (13.27),
we have
f lim f/--f/d,[d,<:limk;] / f/d,[d,--O.
X n'-°°An X n,--c X
Consequently, lira f/-- f /d t-almost everywhere (11.27). [
n--oo A n X
Notes
For the early history of FUBN'S theorem, see SAs I11, p. 77 and
tOURBAKI ), pp. 119--120. In (1.1)--(1.14), we have followed in
part ]3OURBAKI 31, Ch. V, §8 and also in part NAiMARK I11, §6, N ° 48.
For the history of (13.15)--(13.29), see DUNORD and SCHWARTZ
p. 235. A different treatment of integration on product spaces, and a
number of sharper results for countable products, are given in HEWlTT and
STROMBERG I2], Ch. VI.
§ 14. Complex measures
Throughout this section, X will denote as usual an arbitrary nonvoid
locally compact Hausdorff space. Where no confusion is to be feared,
we will write if0 for 0 (X)" similarly for fro0, ff, etc. We are concerned
with the conjugate space ' of if0. That is, we are concerned with
complex-valued linear functionals q9 on 0 such that for
all [(:(So, where/5 is a constant depending upon qg. As noted in (B.8),
there is a least nonnegative number/5 for which this inequality holds:
it is the same as sup{]qg(/)l:][/l[<:l,/o}, is written as ][qgll, and is
called the norm of
Since (o0 is a dense linear subspace of 0, it is obvious that each
09' is completely determined by its values on if00. Thus we can
attempt to carry over the techniques and results of §§11--43 to the
present case. There are two important differences. First, functionals
in ' can assume arbitrary complex values for functions in . This
apparent drawback is offset by the second difference, which is that the
functionals in ' are bounded, while nonnegative functionals on 00
may well be unbounded.
(14.1) Theorem. Let I be a nonnegative linear /unctional on oo,
and let be constructed/rom I as in § 11. The/unctional I is bounded i/
and only i/ (X)< oo, and then [[I][--(X).
Proof. Since (X)--sup(I(/)'O<__/<=t, /0), it is plain that I is
unbounded if (X)-- oo. Since [I(/)1I([/[ ) (1t.5), it is also clear that I
is bounded if t(X)<
(14.2) Note. It is obvious from (14.1) that every nonnegative linear
functional on ff is bounded if X is compact. If X is not countably
compact, then X contains a closed, infinite, discrete subset A. Every
compact subset of X meets A in a finite set. For/00, therefore, the
sum /(x) has only a finite number of nonzero terms, and the func-
xA
tional I(/)--/(x) is a nonnegative, unbounded linear functional on
00 .1 xA
We next show how to represent functionals in ' as integrals with
respect to bounded complex measures.
(14.3) Theorem. Let q9 be a/unctional in . Then there are non-
negative boundedl linear/unctionals I1, I, I, I in such that
(i) q 11 _ I + i (I- I).
The /unctionals I, I and I, I can be chosen so that min(I, I.)--
rnin (I, I)--0. Under this added restriction, the resolution (i) is unique.
Proof. Suppose that
(/) = I (1) -- Z (1) + i [I (1) -- I (/)] (1)
We have no reasonable characterization of the locally compact Hausdorff
Spaces A" for which (00(X) admits an unbounded, nonnegative, linear functional.
2 For the definition of min (I, J), see (B.34).
Chapter III. Integration on locally compact spaces § 14. Complex measures t69
for all/ffg. Then the same holds for all/ffo, as (B.38) shows. Hence
we may confine ourselves to functions in _ or lg. Clearly we have
q--ql+i q., where ql and #z e real-valued, adtive, and reM-
homogeneous. It is obous that # and # e bounded. Now restrict
# to the space fig (= 4, 2). Clely g is a vector space over R such that
I[l Eff if [Eff. Also, if [E, gE, and Ig[/, we have
and so IOi(g)]lIOI]. II/l[- Hence , #, and Oz satisfy the hothes
of Theorem (B.37). Therefore, there are unique nonnegative bounded
line functionals I, I, on such that # = I--I, and m (I, I,)=0;
similly #, = I- I,.
(14.4) Theorem. Let be a unctiond in . Tre is a comple
measure de]in on a a-algebra o] subsets o] X containing t Bel
sets in X and such that
x
Proof. For the nonnegative bounded functionals I i of (i4.3), con-
struct the meur i as in i t (i= t, 2, 3, 4). By (t t.36), we have
x
d by definition of the inteM, also for all [0- Since I i is bounded,
*i (X) is finite and 0 ,i (A) ,i (X) for all A cX. Now xte
--,,+i(--q). Clely (A) is a complex number for all A cX, d
[(A)],i(A ). On the a-algebra =,,.,,,,, is
obously countably adtive. The representation (i) now follows from
(t4.3.i) and the definition of f d.
X
We next find a nonnegative fine functional J that majofizes [#(T)]
for eve ў0, in the sense that
(14.5) Theorem. Let be any [unctionM in . For every , let
Then eended to a nonnegative linear unctiond in
the least nonnegative red, orang o .
Proof. Suppose that J is any nonnegative majorant of
Then if T, Wў0, and [IT, we have Hence
to show that [[ is the let nonnegative majort of in , we need
only to show that ]] is a bounded fine functional.
Fst, it is obous that Iol ( ) lloll-I1 11 an
then I1 11,, 11 11., and Io( )l lloll. II ll llOll-II - Tins shows that
is a nonnegative real number for an and Mso that ]] is
a bounded functional. It is also obous that IOl () if a is a
nonnegative reM number and
Next let 01, T,. be any functions in . For e > 0, let Pi be a function
in o such that Ipil j and I(pi) l + >11 (i) (J=l, 2). Write
#(w)=exp(/0)l#(wi)l (00<2), d te e for exp (i (O,-- O) ).
Then we have
Thus we have I#l (T) +l#[ (T,) l#l (T +
To prove the reversed inequity, suppose that 0 and Il
Let =n(T,ll) and O,=l--- It is plain that
piTi, and pi6 (i=t, 2). Now let i=Pi s (i=t, 2). It is ey
to see that io, ]il=piTi, and l+Z=- Hence we have
Consequently I#] is adtive on . We now extend l#l the usuM
way (B.38) to be a complex line functional on 0-
(14.6) Theorem. Let be an element o] . Let be the complex
measure corresponding to and write Il /or the nonnegative measure
corresponding to I # l . Then Il( =[I #
Proof. Recall first that Il (=sup{l#l (), IIll.t}- If
I1,11. we (1"1) 1 1
that II#[lll (x). On the other hand, for T such that IIT. t, and
for e>0, there is a 0 such that IIT and
This impfies that II #[I Il (x). 0
{14.7) Theorem. Let be an dement o I , a let a Il be the
mmsures corresponding to a ] # l, resptivdy. Then
Ts follows immeately from (t4.5.i).
(14.8) Definition. If I and J e nonnegative e nctionMs on
0 and t and e the respective mees, then we te n (t, ) for
the measure coespong to (I, J). If IJ, then we te
Suppose that I, J, ,, and e above. Then it is ey to see that
the follong seions e eqvMent"
, (A) (A) for all subsets A of X, (2)
t(F)(F) for all compact subsets F of X. (3)
t 70 Chapter III. Integration on locally compact spaces § 14. Complex measures
(14.9) Definition. Let .7F/(X) denote the set of all complex measures
# obtained as in (t4.4) from linear functionals 05 in G' (X). For #M(X),
we define the norm I]# ]] of # as (x).
(14.10) Theorem. The mapping qb--# given by (t4.4) is a one-to-one,
linear, norm-preserving, order-preserving 1 mapping o/ (X) onto JFI(X).
Proof. The mapping 05--# is plainly order-preserving and onto; it
is norm-preserving by (t4.6); and it is one-to-one by formula (t4.4.i).
We need to show that the mapping 05--/ is linear.
Let be another member of G' and let v be its corresponding
measure; also let a be any complex number. Let 77 be the measure
corresponding to ў+; then we must show that ]--#+v this
implies linearity of 09--#. We have =1--.+ i(3--a) where
('--t, 2, 3, 4). We also have
= I- L.+ i (I3- I4), = J- J.+ i (J3- J4) , (t)
and
qb+ T--L1-- L.+ i (L3-- L4) , (2)
where the functionals I i, J., and L i are chosen as in (t4.3). Denote the
measures corresponding to Ij, J], and L i by ў*i, vi, and 77" respectively
(/'=t, 2, 3, 4). Substituting the identities (t) into (2) and equating real
parts, we obtain
oI + .. L. + 0ў413 -- 0ў314 @J1 @ L. -- 0ў.I 1 -- 0ў112 -- 0ў3I 3 -- ohI4 +J. +Lr
It follows from (t t.38) that the measures
0ў 1 [Al -31- 0ў 2/'2 -31- 0ў 4/'3 -31- 0ў 3/'4 -31- V1-31- 2 (3)
and
09./b 1 q- 0 1/b9. -- 3/b3 @ 04/b4 @ Vg. @ 771 (4)
agree on all Borel sets. For any finite measures and 0 constructed as
in § t t from nonnegative linear functionals on G0, and every set A c X,
the identity
inf { (U): U is open and U
+ inf{,0(u):g is open and UDA}
-- inf { (U) + '0 (U) : U is open and U D A },
is easily verified. This depends upon (t .2.ii) and the trivial fact that
if U and V are open, U DA, and V DA, then U AV is also open and
UAVDA. Since the measures (3) and (4) agree on Borel sets, (5)
together with (tt.22) implies that the outer measures (3) and (4) agree
1 That is, # is a nonnegative functional if and only if the corresponding measure
# is nonnegative.
on all subsets of X. This means that
(01 -- 02)(/ll -- }/2) -- (03 -- 04)(/23 --/24) AV (1 -- 2) = 771 -- 772,
and a similar argument shows that the imaginary parts of
are equal. Hence we have et,+v=77.
We now collect some elementary facts about 2FI(X).
(14.11) Theorem. Let ў and be in , and let and , respectively,
be lhe corresponding measures in M (X). Then we have
(ii) [+v[all+l l.
Also, i/I] air/, then every ]vt-measurabZe set is ]l-n, eas,trabZe, i.e.,
(iii) ,I c
Proof. From (t4.5.i)we have I ўI-I l. Iў1 for and this
implies that I 1=I 1
For/eg and e0, [] /, we have
lel It follows that
lel (/}+ and hence 1#+ This is equivalent to (ii).
Suppose finally that [ў1Gv[ and that A is Iv I-measurable. Then
A=BUN where B is a a-compact set, N is ]v[-measurable, and
(x>=0. Hence 1 (N)=0 and N is l-measurable. Consequently,
A is [-measurable.
(4.2) Theorem. Let be an element o/e, and let and ] be as
i,, (14.4) and (14.6), respectively. There is an []-measurable /unction g
such that."
(ii) (/) = f/(x) g(x) dll <x)/o
X
Proof. Consider the nonnegative functionals I, I2, Ia, I a of (t4.3),
the nonnegative functional [# of (14.5), and the measures , t2, t, ta,
and corresponding to these functionals. Let # and # be as in the
proof of (14.3). We prove first that IiG# ]
and J[. For 9G0, we have
t 72 Chapter III. Integration on locally compact spaces § t4. Complex easmures 173
[The last equality is just the definition of Ii=max(#l, 0)" see (t4.3)
and (B.34).] We also have I=I--. Hence for 99c, we have
() = up { ()-;, o } - ()
= up ( (- ).;, o )
= up { ()-;, - o}
upl()Io, I1 } I1 ().
The same arment shows that Ia I and I I1.
In the proof of the LEBESGUE-RADoN-NIKODM theorem (t2.t 7), the
follong is proved. The inequahty I]1 implies that there is
]gl-measurable weight function gi such that 0g(x)gt for all xўX
and such that
()=fgll for o. ()
Thus we have
• () = f [a- e, + ў (e- g,)3 a I1 for %. (2)
X
Write g=g-- g.-k i (ga-- g,). Plainly (2) is equivalent with (ii).
It remains to show that Igў*)l=a Igl -am°st everhere. Assume
that lg)l> on [mub] t F of positive Igl -measure. We may
suppose that F is compact, since I1 is finite for all subsets of X (t4.6).
There is a sequence ..- .-.. of functions in @if0 such that
.(X)c[0, ] and ,(F)=I for n=l, 2,...,and lim f.all=ll(F).
g
Now consider the function s. Ts is plainly a function in Ј1(X, I1),
and so by (2.0), there is a sequence of fctions 1, ..... ., ... in
such that lim f IW.--s el dll-o. Now let =r" m ,t* (' t) "
X
/we aee that >t]. It is obous that ,ffo0 and that
Since Is()l i o or , an elementary arment shows that
I.()-sgC)llw.C)-sg()l for all xX. Hence we have
lim f I.- e lall = o.
The ofinal definition of I1 in (t4.5.i) and (2) show that
We also have
Since Igl is bounded, we infer from (4)that lim f ..all-
f lgl 11- Taking the limit as noo on both sides of (3), we obtain
theX ]xf --flglx al,l- Since Ig(x)l>a for
relations Igl (F)> lgl algl I
xF, this last inequality is impossible. Thus we have
Ig(.)l__< [/,[-almost everywhere. (5)
Finally, assume that [g(x)l< on a compact set F such that
I/z/(F) >0. Choose a sequence 9h > 0,.>---> 0.>--- of functions in
such that lim 0. = eg, where E is an 1/, [-measurable set containing F such
that [#[ (E flF') -- 0. Then we have lim [[ (0.) = lim [ 0. d[#[ =[#1 (F).
Using the definition of [[ in (4.5.i), choose functions P.coo such that
t
[p,I =<0. and [(p.)[ + >[1 (0.) (n=l, 2, ...). Using (2) and (5) and
obvious integral inequalities, we now have
I(v,.)l--- ]xfv,,,galll = < f Iv,.I Igl all f Iv,.I all fg.all. (6)
Taking limits as n-+oo in (6), we find that
(7)
LEBESGUE'S theorem on dominated convergence and the fact that
Ig (x) l< on F give us
=] Igl all =ff Igl all < I1 (F). (8)
Relations (7) d (8) ve an obous contraction. Thus [g(x)l=a for
Igl-aost an x. We replace g by a fction equM to it [gl-Mmost
eye.here to obtNn (i).
only # h(A)=h(A)=h(A)=q(A)=O. A subset o/X is Igl--easurabze
il and oy q it is ti-measurable lot i= , 2, 3, 4, and/or 1 I-measurable
sets A, we have
(i) (A) = f g all.
Furtherre, a complex- e.ended re-valu /unction / on X is in
(ii) f I d = f I g all-
X X
174 Chapter III. Integration on locally compact spaces § 14. Complex measures 175
Proof. As shown in the proof of (14.12), we have Ii_<_lql (i=t, 2,
3, 4), and this immediately implies that ,i=<lF]. Hence if A is [/].
measurable, then by (14.1 l.iii) A is ,.-measurable for /= 1, 2, 3, 4. Tile
functional I1+ I.+ I3+ 14 is easily shown to be a nonnegative majorant
of q, and (14.5) thus implies that Iq[<=Ii+I.+I3+I4, so that
]# ] =< '1+ *.+ %+ q. Now if A is ,j-measurable for i-- 1, 2, 3, 4, then
A is clearly q+,.+%+q-measurable, and (14.11.iii) implies that A is
[#]-measurable.
Consider a complex- or extended real-valued function / on X. From
the previous paragraph, it is clear that /ei(X, I#])if and only if
[_I(X, ,) for i=1, 2, 3, 4. The linear functionals /-+ f / d# and
x
/-+ f ! g d]#] on 1 (X, [tt[) are clearly bounded in the 1 norm. By
x
(14.12.ii), they agree on 0, which is dense in 1 (X, [/A[) by (12.10). It
follows that these functionals agree on I(X, ]#[). Hence (ii) holds;
(i) is a special case of (ii).
Finally we characterize [#] in terms of .
(14.14) Theorem. Let A be an ]#]-,neasurable set. Then
(i) Il(A)--sup 1(.)1"{1, ..., ,,} s prXiXo o! .4 Xo I 1-
}
measurable sets .
Proof. For {El,..., E} as in (i), we have
Hence I#l (A) is greater than or equal to the right side of (i).
To prove the reversed inequality, let e be a positive number. By
(t2.10), there is a function h= , eiE such that the E i are pairwise
disioint I# [-measurable subsets of X and
suppose that the sets {E}I partition A. Let
[again > 1, and write h0= i ,. Then an elementary argument
i=1
@[and In particular, have
shows that Ih0 - ]h- Iflil t. we
f]hog-- [ dlўl<e. It follows that
X
x
As e is arbitrary, the proof is complete.
(14.15) Corollary. Let E be an I#]-measurable subset o/ X. Then
the/ollowing assertions are equivalent."
(i) ]/[ (E) --0;
(ii) # (A)--0/or all ]t [-measurable subsets A o/E;
(iii) # (F)--0/or all compact subsets F o/E.
This follows immediately from (14.14).
We next identify '(X) for discrete spaces X which are trivially
locally compact].
(14.16) Theorem. Let X be a discrete space, and let I be any
negative linear/unctional on oo. Then there is a nonnegative real-valued
/unction zv de/ined on X such that
(i) I(/) -- E /(x) w(x) /or all /oo
xX
Ithe sum is actually /inite. Conversely, every nonnegative real-valued
/unction w on X de/ines a nonnegative linear /unctional on oo by the
/ormula (i). Let q be a bounded complex linear/unctional on o. Then
there is a complex-valued/unction w de/ined on X such that
(ii) Y [w(x)l< 00 1
xX
and
(iii) q(/)-- y,/(x) w (x) /or all / o.
xX
Conversely, every Wll(X) defines a/unctional in by the/ormula (iii).
Proof. For every xX, let b, be the function on X defined by
-/a0 if y--x Define w by the equation w(x)--I(b,) xX. Then
(y) =
if y=x" '
the function w satisfies (i). The converse proposition is obvious.
Let q be a nonzero bounded complex linear functional on 0, let
be as above (xX), and let w(x)=q(b,) for xX. If (ii) fails, we can
find a sequence x, ..., x, of elements in X such that I (x )l
k=l
Then if/0 is equal to sgn w(xk) b,, we have q(/0) -- sgn w(xk) w(xk) --
. k=l
w(x)l>llq [ and I1/0[--1 which is impossible. Hence Z ]w(x)]<
kl ' ---
xX
qgl;< c and (ii)holds.
Linearity implies that (iii) holds for all/oo. Let/fio be arbitrary
(/0), and let e be a positive number. Choose a function/o in oo such
that l/- 011 < -Ii T Then (]) -- ,/(x) w(x) __<1(/)--(/o)1
X
That is, w is in l (X): see § 1 for the definition.
! 76 Chapter III. Integration on locally compact spaces § t4. Complex measures t 77
xEX
x
The proof of the last statement of the theorem is simple and
otted.
In the next two theorems, we investigate a special class of measles
in
(14.17) Theorem. La I be a nonnegalive linear ]unctionM on oo,
and let be the corresponding measure as in t t. suppose that w is
[unction in (X, O, and let
x
Then • is a /unctional in . Write lot the complex measure
spondng to • as n (a4.4), and writ, E lot the set {xX:w(x)O}.
A subset A o/X is ! [-measurable i] and o i/A E is t-measurable. For
such sets, we have
(ii) (A) = f w d,
and
(iii) Il (A) = f I!
z pezr, IIll=lllll d q ,(A)o, th (A)=ll(A)=0. Also
/o/ (x, I1), h,/(x, ,),
x E
a
(v) fl alal = f 111 a,.
x
Proof. It is easy to see that # is a functional in d that
I11111. To proe (ii), suppose initily that w is nonnegative. We
prove (ii) first for open sets U. By (t t. ), we have
so that (U) f w dr. The set E is clearly a countable union of
u
measurable sets having finite t-measure. Therefore U E can be tten
as (xF)U N where t (N)= 0 and {F}x is an increasing sequence of
compact sets. Choose / in o such that /(F)=t, /(U')=O, d
/ (X) c [0, ]. Then/ u for n = i, 2 .... , and lim/ w = u w t-almost
everhere. Therefore using (), we have
(U) lim f la,= f lim/a,=fa,= f
X X U UE
Thus (ii) holds for all open sets. A simple computation shows that (ii)
holds for all sets in the a-algebra of Borel sets.
We now show that
if AcX, AfqEE.Jl,, anat(AfqE)--O, thenAand#(A)--0. (2)
We note that E has the form BoUN o where B o is a Borel set, No
and t(No)=0. Also there is a Borel set B such that A fqE c B and
t (B) = 0. Then A c B U B o, and
#(BUB;)-- f wdt= f wdt--O.
(Bi3B;)CIE BCIE
It follows that A ў and # (A)=0.
Now suppose that A is any set such that A fqE,. Then A fqE=
DUN where D is a Borel set and t(N)=0. By (2), we have N
and AfE'E, so that A=DUNU(AfE').
Suppose that A is in ў. Let B and C be Borel sets such that
B C A C C and # (B)=# (A) =# (C). Then
f wdt--#(CfqB')=O,
and as w(x)>0 for xE, we see by (1t.27) that CfqB'fqE is t-null.
Thus A IqE = (BfqE) U (A fqB' fqE) is t-measurable.
Finally suppose that # (A)= f w d t for some A ў. Since
A CIE
It(A) +It(A')=It(X)-- fwdt= f wdt+ f wdt, either
X ACIE A'CIE
or
# (A) < f w at (3)
A f'IE
# (A')< f w dr (4)
A'CIE
obtains, we may suppose that (3) holds. Choose an open set U D A
such that/z (U) < f w dr. Then we have # (U) < f u d, < f w d,, which
A fie A fqE UfqE
is impossible. This completes the proof of (ii) in the case that w=>0.
For arbitrary w g (X, t), we write w = wl-- w.-t- i (w 3- w4), where
' +
the wis are functions in 9(X,t) and min(wl, w.)--mJn(w3, w4)--0.
For ]o and ?'=t, 2, 3, 4, we define ўi(])= f/widt" An elementary
x
computation shows that min (ў, ў.) = min (ў3, ў) = 01 and therefore
(14.3) implies that ў--ўt-- ў.+i(ўz-- ў) is the resolution of (t4.3).
Write E(i)--{xEX.wi(x)>O}; then Iti(A)--fwidt for all sets A such
that A fiE(i)is t-measurable (1" t 2, 3 4) a
-- , , . IIA isasubset of X, then
A fIE is t-measurable if and only if A fIE (i) is t-measurable for ?" = t, 2, 3, 4,
since E=Ea)UE(.)UE(Z)UE(). It follows that A flE is t-measurable if
1 These minima are the minima of nonnegative linear functionals, defined as in
(B.34).
Hewitt and Ross, Abstract harmonic analysis, vol. I | 2
178 Chapter III. Integration on locally compact spaces § 14. Complex measures t 79
and only if A is 1# I-measurable, and in this case/Z (A) =/Zl (A) --/Z2 (A) -
( (A)-- (A)) = f .
ANE
We now prove (iv). Let ® consist of all functions a having the form
m
aix, where aiK and A i is It-rneasurable (i-l, ..., m). Because
i=1
of (ii), (iv) obviously holds for a®. Suppose now that /i(X, t/z[)
and that w>=0. Then, as is shown in the proof of (12.10), there is an
increasing sequence {a,},--1 of nonnegative functions in ® such that
lira a,=[ everywhere on X. Using LEBESGUE'S theorem on monotone
convergence, we have
f/d/z= lim fa,dtz= lim f a,wd=f /wd.
X n-°°X n,---oo E E
Relation (iv) for arbitrary/I(X, I l) and Wl(X , t)is now proved
by writing / and w as sums of functions in (X, I 1) and
respectively, and applying the case just considered.
Next we prove (iii); it suffices to prove that [q[ (/) = f/Iw[ dt for
x
/G0, because then (ii) can be applied to I/Z]. The functional defined by
[-- f/[w[ dt for/G0 is obviously a nonnegative linear functional on Go,
x
and is a nonnegative majorant of q" Iq(/)] = Ix f/w dr]-----<x f I/[ d.
Thus by (14.5), it suffices to prove that
x
for all /ffff. Let {co}n°°__l be a sequence of functions in oo such that
I .l_<a for all n, and lim f [o%--sgn ] dl/zl=0. [This sequence is
n x
constructed just as in the proof of (14.12) for the function sgn g. For
/, we have [ff m./d#--x f sgn N / d[ [I/ll.x f Ira. sgn N[ d]#[, so
that lira f ./d = f sgn /d = f/w] d" the last equality follows
X X X
from (iv) since sgn/x(X,[] ). It follows from (14.5.i) that
] (/) f/]w[ d. This proves (5) and hence (iii).
x
Assertion (v) follows from (iv) and the fact that 1[ (/)= f/[w] d
x
for /0. Finally we observe that ][=[[(X)=f[w[ d=[w]] by
(14.6) and (iii). S x
(14.18) For Wgx(X, ) and as defined from (14.17. i), we write
d = w d, meaning that (14.17.i)--(14.17.v) hold for ў and [[.
Let be a complex measure as in (14.4), and let w be in 1 (X, [?[).
By (14.12), there is an [[-measurable function g on X such that
u=gu[l. et b the m ch that g=gUll in (4.Zl.
We then have f/d/z= f/wdv for /o, and we write d/z=wdv. A
x x
theorem analogous to (t4.17) is now valid.
(14.19) Theorem. Let I be a nonnegative linear [unctional on oo
and let t be the corresponding measure as in §11. Let # be a complex
measure as constructed in (14.4) with the property that 1#] (F)=0/or all
compact sets F c X that have t-measure O. Then there is a Borel measurable
/unction wi I (X, t) such that d#=w
Proof. Let A be any locally t-null set (11.26). Consider any compact
subset F of X. Then t (A fq F)=0 and hence there is a Borel set E such
that A fq FcE and t(E)=0 E can be taken to be a countable inter-
section of open sets. Using (tl.32), we see that I/z] (E)=0 and hence
1#[ (A F)=0. Thus A is locally ]#I-null. [Note that I/z[ (A)=0 if A is
locally ]#I-null, since ]#[ is finite on all subsets of X.] Thus we have
,Cuf,l, and we apply (t2.17) to assert the existence of a nonnegative
t-measurable function w 0 such that w 0 F is in 1 (X, t) for all compact
subsets F of X and such that
f/dl/l- f/Wo dt for all /oo. (1)
x x
Although w 0 need not be in Ј1 (X, t), we will find a function Wl
such that (1) holds with w 0 replaced by w 1. It is easy to see that there is
a sequence {U}n__ of open subsets of X such that U,- is compact,
U-C U,+ for n=l, 2, ..., and 2irn I1 (g.)=ll (x). For each n, let
Gff0 be taken so that %()-- 1 F,() 0, and
Write U U.as E, and let Wl=ew 0. Plainly {F}L1 is an increasing
sequence of functions, 2i w=* everywhere, and I1 (E3 =0. Now for
/6Gff, (1) implies that
f /d[[= f e/d[[= lim f %/d[] }
x x x
- im fw./o,=f/*o,= f/1 a,. (2)
X X X
Plainly (2) also holds for all [G0. Moreover, f w 1 d t = lim f w o dr =
n X
imdll=l(E)< so that Wll(,t ). By (11.41) we may
Suppose that w 1 is Borel measurable.
Let g be the l-measurable function for defined in (14.12); then
f/d=f/gdJ I for /G0. (3)
x x
By (11.41), we may also suppose that g is Borel measurable. Since (2)
holds for/o, we can apply (14.17.iv) to the function [gg(X,
12.
180 Chapter III. Integration on locally compact spaces § t4. Complex measures t8t
this yields
X X X
Now let w--gwl. [l
(14.20) Definition. Let # and t be set functions defined on a sub-
family of (X) satisfying the hypotheses of Theorem (t4.t9). Then
is said to be absolutdy continuous with respect to . [We also define
absolute continuity for measures and r/of the kind considered in
saying that r/ is absolutdy continuous with respect to if the conditions
(t2.t7.i)--(t2.t7.v) obtain. It is clear that the second definition is
consistent with the first.]
We now define and investigate a property of measures that is in a
sense the antithesis of absolute continuity.
(14.21) Definition. Let I be a nonnegative linear functional on if,0
and let be the corresponding measure as in § t t. Let # be a complex
measure as constructed in (t4.4), and suppose that there is a Borel set
BoX such that t(B)--0 and [# [ (B') -- 0. [We say that I#l is con-
centrated on B.] Then # is said to be singular with respect to .
(14.22) Theorem. Let I and be as in (t4.2t) and let t* be any
complex measure as constructed in (t 4.4). Then there are complex, measures
Pa and t*. as in (t4.4) such that
(i)
(ii) / is absolugdy continuous with respect to ,,
and
(iii) /, is singular with respect to t.
I1 vl and v are any complex measures [as in (t4.4)] with properties (i)-- (iii),
then Pa--vx and #.--v. Also
(iv) I/*l = 11 ч I,. 1, a,a II II - I1111 ч II,. II-
Proof. Let # be the complex linear functional on 0 in terms of
which /, is defined and let Il be as in (t4.5). It is obvious that
I ql <i+ I#l and that the [nonnegative] measure corresponding to
I+ I#] is ,+ Igl- It is trivial that I/*l (F) =0 if F is compact and
(*+l/*l) (F)--0. Thus applying (t4.t9) to/,, we find a complex-valued
Borel measurable function w in (X, t+ I/*l) for which
X X X X
0)
for all lo- Let E={xcX'w(x)#O}. Then E is a Borel set and by
(t 4. t 7) we have
/,(A) = f u, d, + f u, all
A flE d flE
for subsets A of X such that AE is (,+ll)-meable. Let
c(x:l()l > } a B=(X:[() I =}. Then from (3) we have
I1 (c) = f I1 , + f Il 11,
that Igl (c)=0 =d,(C)=0. W may therefore supse that I(x) I
for xў X. Setting A = B (), we have
Ig[ (B) = ,(B) + Il (B),
at
,(B) = 0. (4)
Now let a d be e cfion such that
X X X
for ]o, d let & d g,, rpectively, be the cong complex
me. It foaows kom (t), (4), ad (5) that =x+,
g=&+g,. It foHows om (t4.t7.) that I,](B')=0; ts equity
d (4) show at g, is s th rpect to t. Supse that F is
compact d that t (F)=0. Setg A =F (), we have
=0, sce ]] on FB'. AI (4.7.), e have
Fn Fn
Fy, let x d be e statement of e rem. en
we have --v=vz--,. For eve compact set F such at ,(F)=0,
we hve I'.-- a. I (F) = Im--'l (F) Z I1 ( + I'1
abmlutely continuous rpect to ,. e other hd, if
I.1 (c')=0..here .(B)=.(C)=0. me I'.--ml (B'OC')I,.I (B')+
I,] (C')=0, d ,(BUC)=0. Hence v,--, is th s d ab
Mtely conuo th rpect to , d is erefore 0. opey (iv)
oows om (), 0), =d 04.t7).
(14.23) Rem. Let # be d let be e mee coe-
ng to # (t 4.4). LEBESGOE'S theorem on donated convergence
vd for t th rpect to . More precely, if
ost evehere t of a sequence {],}k%x of ] ]-mable ctions
on X d if ere a cfion s(X, I[) such at ll,(x)ls(x ) for
! 82 Chapter III. Integration on locally compact spaces § 14. Complex measures 183
[[-almost all xX, then lim f/, d# exists and is equal to f/d. To
n--oo X X
see this, let g be as defined in (14.12). Then the sequence {[/,g[}= of
functions is dominated by s, lira/,g=/g, and the usual dominated con-
vergence theorem, together with (14.13.ii), gives us
f /d#=f /gd[#[= f /,d[#]-- lira f /,d.
X X X n X
We now discuss product measures #v where the measures and v
are complex.
Let X and Y be locally compact Hausdorff spaces. Suppose that I
and J are bounded, nonnegative, linear functionals on 0o (X) and oo (Y),
respectively, and that and are their respective measures. Then
IxJ is a bounded, nonnegative, linear functional on oo(XxY) and
can be extended uniquely to a nonnegative [and hence bounded] line
functional on o (Xx Y).
Let # and be in (X) and E (Y), respectively, and let and v
be the corresponding measures. Let #= ili and = fl be the
resolutions of # and , respectively, given in (t4.3). We then define
#x to be the functional i&Iix; #x belongs to
i=1
g (Xx Y). If #i and vk are the measures corresponding to Ii and ,
respectively (f, k=l, 2, 3, 4), then it is easy to see that the measure
corresponding to #x is i flki xvk' and we define #xv to be
this measure, i= k=l
(14.24) Theorem. Let #, v, and #xv be as in (14.23). Then we have
Proof. Let gl and g2 be measurable functions on X and Y, respec-
tively, such that ]gl[ = t, [g[ = t,
f/d=f/gd[#[ for all /o(X),
X x
and
f/dvf/gd[v for all /0(Y);
Y Y
see (t4.12). By (tl.41), we may suppose that g and g are Borel measur-
able. We first show that
XY
for all/o(XxY). It suffices to verify () for all functions / havg
the form/(x, y)= (x)(y) where o(X) and 0(Y), since line
combinations of functions of this sort are uniformly dense in o(X Y)
[see the proof of (13.2). For such /, we use (t3.t2) twice to obtain
f / (X, y) gl (X) gg. (y) d I lx [ (, y)
xxY
f q(x)g(x)p(y)g(y)d[#lxlv[ (x, y)
XY
-- f gxd][ f g2d]v[ = f d#f dv
X Y X Y
4 4
i=1 = x
4 4
- E Z fl f (x) (y) dx (x, y)
/=1=1 XXY
4 4
= Z Z & Zx j (/) - x (/).
In view of (1), we can apply (14.t7.v) and see that
XXY
for all /G0(XxY).
Therefore we have I#xg*l = Ibl x I g*l and
We now prove FUBINI'S theorem (13.8) for complex measures.
(14.25) Theorem. Let , v, and #xv be as in (14.23), and let / be in
9n(XxY, [#xv[>. Then /(x, y) qua /unction o/ x is in l(X, [#[> /o'
[v]-aZmost aZl y Y, so that the integral f [(x, y)d#(x) exists/or [vl-almost
x
all y Y. Define this integral as 0 where it does not already exist; then the
[unction o/ y defined by y--> f /(x, y)d#(x) belongs to 9n(Y, I l). Similar
x
remarks apply to/(x, y) qua/unction o/ y and the/unction x-->f /(x,y)dv(y).
Also we have v
(i) f /d#xv-- f f /(x, y)dv(y)d#(x)-- f f /(x, y)d#(x)dv(y).
XXY XY YX
Proof. Let gl and g. be the Bore1 measurable functions used in the
proof of (t 4.24). The function (x, y)-+ /(x, y) gl (x) g (y) clearly belongs
to 9q(XxY, [,xv[)= l(XX Y, I lx I 1) (14.24). All statements of the
present theorem except the last follow at once from (13.8). By (14.24.1)
[for/go(XxY), and (14.17.iv), we have
f /d#xv- f /(x, y)gl(X)g2(y)d[/]xlv[ (x, y).
XXY XXY
Hence by (13.8), we have
f / d#x, = f f i(x, y) gl (x) g2(Y) d[V[ (y) d //I (X)
XXY X Y
-- f f 1 (x, y) dr (y) d/u (x).
XY
The remaining equality of (i) is shown similarly.
t 84 Chapter IV. Invariant functionals § 15. The Haar integral 185
Notes
The contents of this section are well known to many functional
analysts; for example, see W. RUDIN [2], p. 228 and P. J. COHEN [t],
p. t93, for assertions related to (t4.6) and 04.2), respectively. Curi-
ously enough, no coherent treatment seems to exist in print. Some of the
results appear in HEWlTT [3]- Complex measures are also defined in
BOURBAKI [4], Ch. VI, §2, N ° 8, but are treated from an altogether
different point of view.
Chapter Four
Invariant functionals
Invariant flmctionals, measures, and integrals are a vital tool in
studying representations of locally compact groups and in establishing
the detailed structure of locally compact Abelian groups. They also
provide the function algebras and function spaces that are studied in
harmonic analysis. The subject of invariant functionals is large, and
we cannot treat it with any completeness. In §t5, we construct the
Haar integral, which is essential for all of our subsequent work. In
§ t6, we give some technical but interesting facts about Haar measure,
and in §§ t 7 and t 8, we follow some interesting byways.
§ 15. The Haar integral
We begin with some definitions.
(15.1) Definition. Let G be any group, E any nonvoid set, and
any function with domain G and range contained in E. For a fixed
element aG, let / ,] be the function on G such that
[ (x):/(xa)] for all x G. Then [/] is called the left translate [right
translate] o//by a. Let/* be the function on G such that/*(x):/(x
for all xG.
(15.2) Definition. Let G be any group and let be a set of functions
on G. Suppose that / and a G imply / [, ]. Let I be any func-
tion on such that I(]):I(d ) [I(t):I(/,)] for all/ and aG. Then
I is said to be left invariant or invariant under left translations [right
invariant or invariant under right translations]. If I is both left and
right invariant, it is said to be two-sided invariant. Suppose that ]J
implies/*. If 1(D=1(1") for all/, then I is said to be inversion
invariant, or to be invariant under inversion.
Of course every function space of the sort described in (t5.2)
admits a function that is two-sided invariant and inversion invariant:
namely, any constant function. This trivial example is of no value to us.
We shall be concerned throughout the present chapter with the con-
struction and description of nonzero linear/unctionals on a variety of
function spaces which are invariant under left or right translations or
inversion.
There is an intimate relationship between certain linear functionals
and measures; this relationship is worked out for 00 and 0 in Chapter
Three. Frequently when an invariant functional can be represented as
the integral with respect to a measure, the corresponding measure will
enjoy invariance properties like those of the functional. We accordingly
make a formal definition of invariance for measures.
(15.3) Definition. Let G be a group and let ў be a family of subsets
of G. Let E be any nonvoid set, and let 2 be a function with domain ў
and range contained in E. Suppose that A Eў and xEG imply xA
[Axsl]. If 2(xA)=2(A) for all xG and A W [2(Ax)=2(A) for all
xEG and A6ў], then 2 is said to be left invariant [right invariant].
Suppose that A ў implies A-a E aў. If 2 (A-) = 2 (A) for all A aў, then
2 is said to be inversion invariant.
The first and most important function space for which we construct
an invariant functional is 00(G), where G is a locally compact group.
As in Chapter Three, we will write 00 for 00 (G), etc., when no confusion
can arise.
We first make a simple observation about 0-
(15.4) Theorem. Leg G be any topological group, and let / be any
/unction in o (G). Then/or every >0, there is a neighborhood U o/e
in G such that II()-l(y)l< lot ,at ,yG suc that x-yU [1 is Zelt
uni]ormly continuous]. Similarly, there is a neighborhood V o/e in G
uni[orml y continuous].
Proof. This follows immediately from (4.t 5), if we take H to be the
additive group of complex numbers.
We now state the fundamental theorem on invariant functionals.
On this theorem all of our subsequent analysis depends.
(15.5) Theorem. Let G be a locally compact group. Then there is a
]unctional I on o such that:
(i) I(l) is real aut positive/or /=t:0;
(ii) I(+ g)=I(/)+ I(g) ]or all l, g0;
(iii) I(ml)=mI(l)/or >= 0 and
(iv) I(d)=I(l )/or all lffo and
186 Chapter IV. Invariant functionals § 15. The Haar integral 187
Furthermore, i] J is any nonnegative ]unctional on o satis]ying condi-
tions (ii)--(iv) and not vanishing identically, then J--cI [or some positive
number c.
(15.6) Remarks. Before proceeding to the proof, we make a few
remarks. The functional I described in (1 .) is not a linear functional,
since it is not defined on a linear space. However, properties (ii) and (iii)
are as close to linearity as can be achieved on 0- We call I a le[t
Haar integral on o. It is clear that a right invariant Haar integral can
also be constructed on 0, and that it too will be unique up to a multi-
plicative constant.
Proof of (15.5). The proof, as is to be expected, requires a number
of steps.
(I) We begin with any two functions ] and in o such that :k 0,
and seek some estimate of the size of ] relative to . Consider all
finite sequences sl, s. .... , s) of elements of G and all sequences
c 1, c 2 .... , c) of positive numbers such that __< c i,j; that is,
t(x) <--_ c i q(six) for all xG. (1)
Define the expression (]" ) as the infimum of all sums c i for which
the inequality (1) obtains. We note first that such sums c. exist.
=1
Let F be a compact subset of G such that [(F')--O, and let
Let a be a point of G, U a neighborhood of e in G, and # a positive number
such that (x)# for all xaU. By compactness, there are a finite
number of sets Yl U, y. U, ..., y,0 U that cover F. It is then easy to see
that /(x) __< 0 (ay -1 x) for all x G. That is, (/" q)) exists and does not
o
exceed m,___. Next we note that if (t) holds, then I/,,<----Xc;llll, so
j=l
that we have
IIIII,, < (!" ) < -,oli/II,, (2)
We next list three obvious properties of (]" )"
(a/: q)= (l'a9) = (1" 9)
(0 1" q) = O (1"
(1,. + l.'qO =< (l,.'qO + (l:,'q)
for all a G; (3)
for all :> 0" (4)
for all /1,/2ff;0. (5)
If and are nonzero functions in ;0, then we also have
(1"') <---- (1" ) @'') •
(6)
To establish (6), we need only note that if /(x)G , ci q(six) for all
i=1
G and 0 (y) ___< d ( ) for all G, then 1 (x) c d ( s ) =
=i i=1
Z c i dk (tn s i x) for all x G. Thus (1:) d = c i dk .
=1 =1 i=1 =
Next choose once and for all a function 1o0 such that 100,
and define
(1:9)
z, (1) = ( ), (7)
for nonzero functions in ffo. Then I,(0)=0 since (0" )--0, and for
/ 0, we see from (6) that
u0:t) z, (1) (11o). (8)
From (3), (4), and (5), respectively, we also have"
I()=I(1) for all aG and
lG0
(9)
i ( l) = Z, (l) o, o
10;
(0)
I, (1 + 12) I (1)+ I (12) for all 1, 120" (] )
I(1)I(1) if 1, 12 o and ]l. (]2)
The idea of the proof is now to consider all of the functionals I,
where vanishes outside of a neighborhood of e. Then as
"decreases", it will be shown that I converges to a limit functional I,
which enjoys properties (i)--(iv). The uniqueness of I is also an imme-
diate consequence. To carry out this program, we must establish two
technical lemmas.
(II) Let 1, ..., l be nonzero functions in ffgo and let 6 and be any
positive numbers. Then there is a neighborhood U of e such that
2 i I(] i) G I( 2 i 1)+6 (3)
J=X =1
if $0, +0, (U')--0, and 0 2iA (i=, 2 .... , m).
To prove (II), let E be a compact subset of G such that/i (E')=0 for
i 1, 2 .... , m, and let V be a neighborhood of e such that V- is compact.
Choose a function g in ff0 having the property that g(E)= and
g(G) Q 0, t]. Let M denote A m. max(]/], ..., []/]]). Let e be an
arbitrary positive number less than M. The functions ],/ .... ,/m,
and g are left uniformly continuous, as pointed out in ( 5.4). Therefore
there is a symmetric neighborhood U of e in G such that U C V and
[/(s)--/i(x)]<4M iЈ s-xU (/=1,2 .... ,m) (14)
t 88 Chapter IV. Invariant functionals § t 5. The Haar integral t 89
and
I0)-()1<- i -lxV.
Now let p be in ff0 where =t= 0 and 9 (U')
be taken so that 0 2i (= , 2 .... , m). Let # denote the function
2i [i + *g, and define the functions , h, ..., h by
hi(x ) = ,(x) for
0 for xCE.
It is obvious that #ff0, that h i#=2 i/i for =t, 2, ..., m, and that
h i t. If s - x U, then by (t 4) and (t 5), we have
/=1
I()- o()1 Z lb()- 1()I +
= [ 06)
< m 4MmA + s 4M -- 2M "
It is evident that llOll It/tI+*M+ s<2M. We now show
that
* if s-x6U (]=,2, ,m) (t7)
Ih (s) - h ()1 <. ....
In fact, if x and s also belong to E, then using (t 4) and (6) we have
If E and s-U, then sE, since otherse we would have
=ss-EUE. If sE and s-U, then E, since other-
Mse s=-sEU-EV -. Therefore if either • or s fails to be in
E, the left side of (t 7) is ero.
Let us now estimate (" ). If we have
() c(s) for all G,
then, since (U') =0, we also have, for fixed G,
the summation being restricted to those indices h for which s U.
From (t 7) and (t 8), we obtain
() () N' c (s ) (s
and hence
for all G. This impes that
sumng over = t, 2 .... , m and recang that h i t, we obtMn
Since c c be taken close we Msh to (" ), we have
2 (a 1 ( + ( l.
Dividing by (/,: ), we get
Z I (2i/i) (t + ) I (#). (t9)
Applfing (t0), (), the definition of #, and (8) to 09), we obtMn
a i(11 ( + 0 a + i (
i 1 + ,. I(10 + ( + 0 (
These inequaties show that (t 3) holds if e is sufficiently sma, and thus
(II) is estabshed.
O second lemma foows.
(III) Let ] be any function g, ] 0, e y positive number, and
U a neighborhood of e G such that II()-l(y)l<e if , yG d
- U. Let ] vanish outside of the compact set E G. Let g be a
function in g such that g0 and g(U')=0. Then for every
there are elements , .... , E - and nonnegative numbers c, c .....
At this point, we could shoen the prf by appeMing to Tmoo's rem,
and by selecting some pot in a lge Cesi product of closed inteaN. is
selection, narally, involves the iom of choice, and leaves the niqen f the
left Haar inteal to be proved sepately. We prefer to give e remader f the
proof in a strictly consctive form, which simultaneously demonsates e exist-
ence and uniqueness of the left H inteal and does not depend n the iom
of choice. For a sketch of the shoer, nonconsuctive, proof, see (15.25).
t 90 Chapter IV. Invariant functionals § 15. The Haar integral
such that c i>0 and
i--1
/(x)-- c ig(t ix) for all xcG. (20)
To prove (III), we note first that
/ (x) - (s- x) / () (s- x) / (x) + (s- x)
for all s,xG. If s-xU, then I/(s)-/(x)l,z if s-lxe, then
g(sqx)--O.] Now choose >0 so small that (/'g*) <--e. Let V be
a neighborhood of e in G with compact closure such that [g (u) -- g (v)[
for all u, vG such that u v- V. Notice that we use the right uniform
continuity of g.] Since / vanishes outside of the compact set E, there are
elements s,s,.., s mE such that U s iVE{xG'/(x)O}. By
(3.), there are functions h,h,...,hmo such that hi((siV)')=O
(f= , 2 ..... m) and h i (x) = if /(x) 0. Now we have
h (s) /(s) (s- x) -- h (s) /(s) (s? x) h (s) /(s) (s- x) + ] (22)
for all s, xG and f= , 2, ..., m. If s; s = (sT x) (s qx) - V, then
[g (s - x)-- g (s x)[< ; if s s i V, then h i (s)= 0.] Summing (22) over
-t, 2 ..... m and taking cognizance of (2t), we obtain
[/(x) - (s- x) - v / (s) E () /(s) (s? x)
i=I
[/(x) + , (- x) + n /().
Now consider any o, 0. Using (9), (0), (), and (2), we
infer from the last inequalities that
(23)
[/(x) + , (.) + (/).
I ( < ([. g,)= - where . Thus,
Now (7) and (6) imply that I(g*)= '
dividing (23) by I(g*) and using (t0), we have
= t z( / /(x) + . (4)
We now apply (II) with d=e--, A=(/.'g ) I11 [see (8) 2--
, -- (g) '
and/=hi/. This, together with (t0) and (tt), tells us that there is a
neighborhood W of e in G such that if our vanishes on W', then
( g(s;x) -. ()
Combining (24) and (25), we have (20), with ti--s 1 and
iv , (g.) (J"-- t, 2 ..... m). (26)
(IV) We now complete the proof of the existence of a left Haar
integral. For every neighborhood U of e in G, choose a function 99--99 v
in ;0 such that 99q:0 and qo(U')--O. Given 991--99v, and 992--99v,,
write 992991 if U1D U. The set {99) is then directed by >>. We will
show that lira I(1)--I(1) exists for every/CG0. We may suppose that
/4:0. By (II), this limit functional I will satisfy (ii). It will obviously
satisfy (i), (iii), and (iv), in view of (8), (t0), and (9), respectively. To
show that lira I(1) exists, it obviously suffices to show that for every
e>0, there is a 990 such that [ I** (l) -- I,, (l) [ < e for 91>-90 and 92>-90.
First let U 0 be any neighborhood of e in G such that U0- is compact,
and let o0 be a function in Ca0 that assumes the value t everywhere on
the compact set F={xG:l(x)+lo(x ) q:0}-. U-. Let e be any number
such that 0< e < t, and let
8
Y -- 4 [- 1. + (oa : lo) -] [- 1. + (1: lo) -] "
Let U be a neighborhood of e in C such that U C U 0, II (x)- I (y) l<
2
if y-lx U, and I/0 (x)--/0 (Y)] < if xC U. Then choose a nonzero
function g in 0 such that g(U')--O and take =--y. Applying (III),
we obtain t 1 ..... tm((xcG:/(x):#:O)-) -1 and nonnegative numbers
q .... , c m such that
/(x)--Y,c ig(t ix) <__ for all xG.
i=1
Since / and all of the functions 0g vanish on F', we have
/(x)--.cg(tix) <=yo(x) for all xG.
For every q)g, q) 0, this last inequality implies that
(27)
-- < I,(oo) < (oo-lo). (28)
i=1
[Simply write (27) as two inequalities and apply (t2) and (tt), followed
by (8). Now we apply (II). Each c i is given by (26), and since hi/<= [,
we have c i < I (1)
_/-=< (]'g*). From (II) we infer that there is a neigh-
borhood V of e in G such that if q), q)4=0, and q(V')=O, then
II (.= c i jg)- I, (g)(i= ci) I
(29)
t92 Chapter IV. Invariant funcfionals
Combining (28) and (29), and writing c= c i (c > 0), we obtain
i I, (/)- c I, (g) i< [a + (/o)] • (30)
Apply precisely the same arment to the "be" function [0, and note
that I, (]0)= - We find a neighborhood of e in G and a d > 0 such that
for nonzero eg0 such that (')=0. Combining (30) and (3t) in an
obvious way, we obtain
if (o, 0, and (V'U')=0. Inequaty (32) impes at once that
(-B + (10)2) < (1) + [ + ( Io)],
z(/)+ 2[+(1"/0)]. Ts estimate, com-
so that 2+t< t_[+(o:to)]_
bined th (32), yields
(- < [+ @'g)] e[ + (1"1.)] - •
s ft and f are any functions in g,fiO,f(V'U')=O
ff = t, 2), then we have
I.(l)-Z.(l)l<-
As explned above, ts proves that lim I (/) ests Ior all [(0, and
thus a left Ha inteal exists.
(V) It remains only to show that two left Haar inteals on
ffer only by a constant factor. Let be any nonnegative [unction
on ff0 satisIng (ii), (i), and (iv), and such that (/)0 for me
/x(ff0- For any/0, /0, we have c i /Ior appropriate re
i=l
c i and sG. Hence ]([) ci J(, and thus ]( is positive
nonzero ] in g.. For y [tg. and nonzero g in g., suppose that
[tN E ci . Then J([t) c J(g), so that we have
Jet1)
(h g) -j(e) •
Consider any nonzero [ in ff0; [ vanishes outside of a compact set E
in G. Let V be a neighborhood of e with compact and select a fixed
(0 so that = 2 on E. Now for y e > 0, apply (III) with = 2e
to obtain a neighborhood U c V of e with the following property. If g is
any nonzero Iunction in 0 vanishing outside oI U, then there are points
§ t 5. The Haar integral
t93
tl, ... , t m in G and nonnegative numbers c 1 ..... c,,, such that
/(x) <= t to (x) + c i g (t i x), (4)
and
l(x) + o,(x)> 7, c i g(tix), (35)
for all xў G. It follows at once from (34) that
(/'g) < (o: g) + c i. (36)
,kpplying J to both sides of (35), we have
J(l) + e J@)> c j(g).
Using (36) and (6), we have
J(l) + e J(o) => [- e-(l:g 1 (l"g) J(g) >-_ [- e(o'l) (l'g) J(g). (37)
Let ] be any nonzero function in g.. Divide (37) by J(]l) and use (33).
This gives us
(t) + [a- (1)-(t :e) z,() • (38)
Now take the limit in (8) over g and then as 0. This gives
(1) > Z(l)
J(h) = z(10 "
Interchanging/ and/, we have
I(/) = Z(l),
for all/(qff0.
(15.7) Note. There is a unique extension of [ from q0 to the complex
linear sace q00 that is a complex linear fuctiga] o q00. This is a
special case of (B.8). The extension is necessarily ]eft invariant. We
i]l write the extended functional as , d we will call a left Haar
nt@ral on q00, or on G.
(15.8) Remarks. We now use the construction of to associate
a set function with our ]eft Haar inteal I. We denote this set function
by 2. Thus 2 is a nonnegative, extended real-valued set function defined
for all subsets of G, and is a measure on the a-algebra ў of 2-measurable
subsets of G. Every open set is in . The set function 2 has the follow-
ing properties"
(i) 0< 2(U) for all nonvoid open sets U;
(ii) 2(U)<
(iii) 2 (a B) = 2 (B) for all B C G and a(G [2 is left invariant in the
sense of ( 5.3)].
Hewitt and Ross, Abstract haoaic alysis, vol. I
194 Chapter IV. Invariant functionals § 15. The Haar integral
To prove (i), we use (15.5.i) and the definition of 2(U) in (tt.tl)
+
and (tt.20). Given xU, there is a function q)00 such that
and q)----<v. Hence we have 2(U)I(0)>0. Property (ii) is shared by
2 with all of the set functions t constructed in § t t" 2 (A) is finite if A-
is compact. Property (t 5.5.iv) implies (iii), as follows. Since I(oj)=I(D
for all [0 and since q)<= if and only if a0____<a for any extended real-
valued functions q) and on G, Definitions (tt.tt) and (tt.t6) show
that I(ah)=I(h) for all aG and all nonnegative extended real-valued
functions h on G. We thus have 2 (B)= (B)=(-, (B))=[()= 2 (aB).
The measure . constructed from a left Haar integral on 0 will be called
a le]t Haar measure on G.
The measure 2 on J6 gives us an integral f [ d. defined for all non-
negative .-measurable functions ! on G, the integral giving rise to the
function spaces gp (G, ) (t =< p < oo); . itself gives us the function space
oo(G, 2). The left invariance of 2 shows at once that f ] d2= f
G G
for all /gl (G, 2), that (a/, og)--(/, g) if/, g are in g.(G, 2), and that
I1 /II -11 11 for (a __<p<oo), see also Theorem (20.t).
Where no confusion can arise, we will often write f / d 2 as f /(x) dx,
G G
f /(y) dy, and so on. Thus dx, dy .... will always indicate integration
G
with respect to a left Haar measure on G. The identity f a/d2 = f [ d.
G G
thus receives the suggestive form f/(x) dx = f/(ax) dx. We will call
G G
f / d2 a le/t Haar integral o! / whenever it is defined. This terminology
G
is justified by the identity I(/)= f/d2, valid for all nonnegative
G
measurable functions ! on G [see (t t.36).
It is to be expected from (t5.5) that left Haar measures, like left
Haar integrals on 00, are unique up to a multiplicative constant. This
is indeed the case, under certain restrictions on the domain of definition
of the measures. The exact result is the following. Let N denote the
family of Borel sets in G. Let/ be any measure on such that"
(iv) / (F) < oo if F is compact;
(v) / (U) > 0 for some open set U;
(vi) / (a B) =k, (B) for all B N and a G"
(vii) / is regular in the sense of (1 t.34).
Then if 2 is any left Haar measure as described above, there is a positive
number c such that tt (B) = c 2 (B) for all B N.
To prove this fact, we construct a functional J from # in the obvious
way" J(q)) = f d/ for q)oo. It is easy to verify that J enjoys all
the properties (t 5.5.i)--(t 5.5.iv). Therefore, the uniqueness of the Haar
integral implies that J = c I, where I is an arbitrary Haar integral, and c
is a positive number. From (t t.38), we infer that/ (B) =c . (B) for all
B.
(15.9) Theorem. Let G be a locally compact group with a le]t Haar
measure . Then (G) is ]inite i] and only i] G is compact. I] G is compact,
we will always normalize le]t Haar measure by the requirement that
Proof. If G is compact, then t =00. Thus I(t) exists and is a
positive real number. Since I(t)=;t(G), it follows that .(G) is finite.
If G is noncompact, let U be any neighborhood of e in G such that U-
is compact. Then no finite collection of sets xl U, x.U, ..., x m U can
cover G, since if it did, G would be the union of finitely many compact
sets and would be compact. Thus we can choose an infinite sequence
of points {x, x. .... , x,...} in G such that x+ ўiU=xi U for all positive
integers n. Let V be any symmetric neighborhood of e such that Va U.
Then the sets x V, x. V ..... x, V, ... are pairwise disjoint, and we have
(G)>= 2(xiV)=n.2(V) for all positive integers n. This implies that
'=1
(15.10) Right Haar integral and measure. The constructions and
arguments used in (t 5.5) and (t 5.6)--(t 5.9) are plainly symmetric wtih
respect to right and left translations of functions on the group G.
Therefore we can state with no further ado that there exists a func-
tional I' on 0 with properties (t5.5.i)--(t5.5.iii) and I'(/)=I'(/) for
all [g0 and aG. The functional I' can be extended over 00 as in
(15.7), and I' is unique up to a multiplicative constant. There is also
a right Haar measure .' on G with properties (t 5.8.i)--(t 5.8.ii) and
2'(Ba)=2'(B) for all aG and all sets BCG.
We shall see in (t 5.t 5) how to obtain a right Haar integral from a left
Haar integral.
We next introduce an important function.
(15.11) Theorem. Let G be a locally compact group, and let I be a
le/t Haar integral on oo. For/o, / 4=0, and/or xG, let A(x)-
(/) •
Then A depends only upon x, and not upon I or/. The /unction A is
continuous, positive throughout G, and satis/ies the/unctiozal equation
(i) A(xy)=A(x)A(y) /or all x,yG.
Proof. Consider the functional J, on g0 defined by J, (])= I (L-,).
It is obvious that J is a left Haar integral on 0o, and so by (t 5.5),
13"
196 Chapter IV. Invariant functionals § t 5. The Haar integral t97
there is for each x EG a positive number A(x) such that J (l)= I (I-,)-
A(x) I(1) for all lEff0. Thus the notation A(x) is justified. It is clear
from (t5.5) that A does not depend upon the particular choice of
We also have, for x, y, u G,
(1,,_.),-. (u) =/,,-, (u y-) = /(uy- -) = / (u (y)-9 = h,r, (").
Hence we have
I(1()-,___) _ l((l:Of') 1((1-,)-,) z(b,-') _ A(y) A(x)
A ( x y) = (1) -- (1) = (1:) • (1) •
Thus (i) holds.
It remains to show that A is continuous. Since A is a homomorphism
of G into the multiplicative group of positive real numbers, we need only
show that A is continuous at e (5.40.a). Let U be any neighborhood
e in G such that U- is compact, and let / be a nonzero function in
Let o be a function in ff0 such that o)({sG'/(s)>O}-.U-)=t, let
be a positive number, and let V be a neighborhood of e such that V C U
I(/)
and I1(.)-/(,1< z() whenever u-tvV. Then if xV, we have
I1(-')- 1()1 <
= z-- () o n < , a o I z (1:)- z (1) 1 < z (/).
Thus
I()-1__<, <. u
(15.12} Remark. The Iunction A is called the modular lunaion o
the locally compact group G. I A- t on G, then G is called unimodular.
[Note that ZI = t if and only if the class oI left Haar integrals is the same
as the class of right Haar integrals.] It is obvious that every locally
compact Abelian group is unimodular.
(1.13} Theorem. Every comlbac group is unimodular .
First proof. The set A (G) is a compact subgroup of the multiplicative
gop ]o, [. ry (} i oy bgop o ]0, [, o
A(G)=, i.., A(x)=] or all
Second proof. It is worth while to note that the present theorem can
be proved without using the modular Iunction. Let I be the normalized
leIt Haar integral on . For IE and xG, let l'(x) =I((x])*). The uni-
form continuity of I implies that 1' (. Now write (l)=Z(l'). It is
obvious that ] has properties (t5.5.i)--(t.5.iii) and that ](t)=t.
We prove first that ] is left invariat: ](,/)=](1). We hae
(d)' ()=z ((,(d))*) =z ((,d)*) =I'(,)=,(1')(). tn (d)=z (,(I')) =
(1') = ](/t). Since J(t) -- I(t) -- t, we have jr_ 1 by the uniqueness
property of L
1 See also (|9.28).
To prove that J is right invariant, we note first that (/a) * (Y) -----
[, (y-l) -- / (y-la) -- / ((a -1 y)-l) = a-' (/*) (Y)- That is,
(1,)*=,-,(1"). ()
Similarly, we have
(d)*= (l*),-l. (2)
Then we have (.f,,)' (x)--I[(,,([,,))*]--I[((d'},,)*]--I[,,_,((d'}*)]--I[(* ] --
['(x). Hence J([,)-I(([)')--I([')=]( D. Since I=J, our second proof
is complete.
(15.14) Theorem. Let I be a le[t Haar integral. For every
e have I(/) = I (]* -) .
Proof. Write J(l')---I*4). We plainly have
A.-,--A(a-I) A.
Using
(t 5.t .2), we thus have
A (.-l)i [(ll)a__l _1
J(3 -/[(3* ---]--z l(/*).-, ] - A (a-l)
-- A(a-1)I [ (lr)a-' A|a -' l-- A(a-l)Zl(a)I (/*) -- J(/).
That is, ] is a left invariant functional. Plainly ] also satisfies (t 5.5.i)-
(t 5.5.iii) [note that [ -ffo if [ffo and is zero only if [ is zero]. The
uniqueness of left Haar integrals shows that J--ci, where c is a positive
number.
To show that c-- t, let e be a positive number and let U be a neighbor-
hood of e in G such that A lx) --t < r for all xU recall that A is
continuous and that A(x -) = A () " Let g be a nolero hmction in
such that g=g" and g(U')=0. Then we have Ig (x) -- g (x) A(I I
for all xG and therefore II(g)-I(g)l<=eI(g). Thus
I-1__<,
that ct. Consequently, I([)=I ([*--) for all [I... rl
We now constrct a right Ham- integral from a left Ham- integral.
(15.15) Theorem. Let I be a le/t Haar integral on 6oo(G), where G is a
Then J1 and J. are right Haar integrals on G, and [urthermore, J1-J..
Proof. By (t 5.t4), we have
Z (J) -- I([*)--I ((/*) * --I ([ ---)-- J. (/'),
198 Chapter IV. Invariant functionals § 15. The Haar integral 199
for [00" thus J1-- J2. It is obvious that J1 satisfies conditions (t 5.5.i)---
(t 5.5.iii). To check the right invariance of J, we use (t5.t3.t) to write
J,. (1o/- s (/l/*) - s (o_,/l*)) -- s (/*/- J,./l).
(15.16) Corollary. Let G be a locally compact group and I a left
Haar integral on oo. Then G is unimodular if and only if I is inversion
invariant.
Proof. Suppose that I(/)=I(/*)= ] (/) for all/ffoo. Then by (t 5.t 5),
I is right invariant, i.e., G is unimodular. If I is right invariant, then
A--t, and again by (t$.t$), I(i*)--II[-)--I([). []
(15.17) Examples of Haar integrals and measures. (a) Let G be a fi-
=
nite group, G=r. Then it is obvious that the functional I([)-
is the left and right normalized Haar integral for the space of all com-
plex-valued functions on G. The corresponding measure on subsets of G
is given by 2 (A) = 2.
(b) Let G be any discrete infinite group. Then 0 consists of the
nonnegative functions on G that vanish outside of finite sets. Let
denote the function such that b (e) = t and (x) = 0 for x e. Then a
nonzero function [ in 0 can be written uniquely in the form
i= i' S, /(ai): i, f:, 2, ..., m, and /(x)--O other-
wise. Thus a left or right Haar integral I on oo has the form I(/)=
e I(d). The group G is plainly unimodular. It is convenient in
dealing with infinite discrete groups to take I()= t, and this we will
always do in the sequel. The measure 2 corresponding to I is countably
additive on alZ subsets of G" 2(A)= if A is a finite subset of G and
(A)= if A is an infinite subset of G.
If G is locally compact and nondiscrete, then 2 (A) -- 0 for all countable
subsets A of G. It suffices to prove that 2 ({e}) =0. Clearly 2 ({x}) =2 ({e
for xG. Let UbeanopensetinGsuchthat0<2(U)<. Since U
is infinite, we would have (U)= if ({e})0.
(c) Consider the additive group R. A Haar integral on 00(R) is
I(/) = f /(x) dx, the integral being an ordinary Riemann integral, which
for each / is in reality extended only over a finite interval. The measure
corresponding to I is ordinary Lebesgue measure on R. There is no
single normalization for the Haar integral on 0o (R) that is suitable for
all purposes. We shall find it convenient in some subsequent chapters
touse 2- /(x) dx.
(d) Consider the compact multiplicative group T. Here the normalized
Haar integral is- /(exp(it)) dt, for all ](T), where again the
integral is the Riemann integral on the interval [0,2. The correspond-
times Lebesgue measure on [0,2
ing measure is - .
(e) Let G be any topological goup with the following three properties.
First, as a topological space, G is an open subset of some real Euclidean
space R ". For --(x, ..., ,) and =( .... , ,) in G, the product
is thus a function F(x .... , x,, , ..., ,) mapping GxGR onto
G c R'. Our second hypothesis is that all of the partial derivatives
0x ' 0y '
exist and are continuous throughout GxG (i, k--t ..... n) [ is the
projection of F onto the i-th coordinate. For each aG, let a [b be
the transformation of G onto itself defined by a()
for all G. That is, a and 8 are left and right translation by the ele-
ment a. The third property that we require of G is that the Jacobian
of each of the transformations a and 6 be a constant; that is, they
depend only upon a.
We denote the Jacobian of a transformation z by the symbol J(z).
For aG, we define S(a) to be ]J(au)] and D(a) to be ]J(bu). Since
aoa=a, we use the familiar Jacobian identity J(aoa)-
J(a).J(a) to point out that S(ab)=S(a)S(b), for all a, bG.
Similarly, we have bu = 8 o 8a, and therefore D (a b) = D (a) D (b).
Write e for the identity element of G. Then a, and 8, are the identity
transformation on G, so that S (e)- D (e)= t. It follows that S and D
are continuous homomorphisms of G into the multiplicative group
We form the left and right Haar integrals on 00 with the functions
S and D, respectively, together with Riemann integration in R . Given
a continuous complex-valued function 9 defined on G and vanishing
outside of a compact set, let f 9 () d denote the ordinary n-dimensional
1 These conditions are fulfilled, for example, if each . has the form
=1 l=l
Where the real numbers ci) and d i do not depend upon
200 Chapter IV. Invariamt functionals § t 5. The Haar integral
Riemann integral of . Then we claim that
f ,
is a left Haar integral on ffoo and that
/a(/) = /(e) D()de
is a right Haar integral on o0- To prove these assertions, we make use
of the familiar formula for transformation of multiple integrals. Applied
to G and aa, this formula is:
f (a) da -- f ( o %) (y)If (a,,) ()l
aa(G)
Here 9 is, let us say, any function in oo- For 9---,-, we obtain
L(.-,/)=..-d(e) s(l de= ((.-,I)oo) () (Sool(l
Since Is obviously satisfies the properties (t5.5.i)--(t5.5.iii), we have
shown that I s is a left Haar integral. The proof that Ia is a fight Haar
integral is almost the same, and is omitted.
It is of interest to compute the modar flmction for G. We have
f f '
f ()
Hence the modular function A for G is DIS.
(f) We first apply (e) to the multiplicative group R YI {0}' of nonzero
real mtmbers. The transformation x-+ax has Jacobian a, and so a
Haar integral has the form
where for each lg..(en {o}') the integral is actually extended over
[-- fl, -- ] 13 [, fl] for real numbers and fl, 0 < < ft. Similarly a Haar
integral for flmcfions on the multiplicafive group of all nonzero complex
nmnbers is
f f '(*+'"
x*+ y* dxdy.
(g) We next apply (e) to the group G of all matrices
with x, yR and x =t=0. For convenience we write the elements of G as
(x, y), with (, )(, v)=(, v+ ). Topoloze
Then G is plaly a topolocN oup satisfng the htheses of (e).
For (a, b)G, the trsfoation a(.,b) h the fo a(.,bl(X, y)=
(ax, ay+ b). The Jacobi of a, is a*. Thus a le Ha teM on
00 (G) is prodded by
where the intern is, for each fixed [, actuNly extended over a bonded
closed subset of R that does not intersect the le {(, y)R " =0}.
The trsfoation (.,b) on G has the fo 8(.,b) (x, y) = (ax, bx+ y);
the Jacobi of <.,b is a. Thus
is a right Ha tegal on g, (G).
This oup is one of the splest exples of a locy compact
goup wNch the Nght d leR Ha teMs e eentiy Iferent.
The modular ction is A(, )= s
(h) Consider next the oup N g (2, R). For A = N g (2, R),
a simple computation shows that J()=]()=(
(det A) . Thus the ght d le Ha teMs e identicM, d e
Nven by
. (Xll X- Xl XI) * "
(i) Let G1, G, ..... G be locally compact oups, d let Ii be a
le Ha teal on G i (=t, 2 ..... m). The product ctional
I,...I is a left Ha teal on GG,...G. This follows
meately from (t3.3) d (t3.2).
(j) Let {G}r be bitr nonvoid faly of ompaa oups;
let I be the notated Ha teal on G for each [i.e., I (t) = t ] ;
d let I be the fctional on (rG) defined (t3.8). Then I is
the noed Ha te on P G
202 Chapter IV. Invariant functionals § I f;. The Haar integral 203
(k) We now describe normalized Haar measure ). on the groups
(t0.2). Notation is as in (t0.2). Topologically, A a is the Cartesian prod-
uct P 0, t .... , a--t }, where each finite factor space is discrete. The
/=0
finite space 0, t, ..., a-- t) admits the measure such that (A) =
for all A 0, t, ..., a-- ). Haar measure on is the product measure
of these measures 0, , ..., , ..., as constructed in (3.t)--(3.22).
Given subsets A of 0, t, ..., a--t) for k=0, t, ..., let C(Ao,A , ...,A)
be the set of all such that each x belongs to A for k = 0, .... , n.
For n--0, t, ..., let be the element of such that u=i
(k=0, t . ). We will compute - A). Consider first C((00 )
ao--1
(," x0=0 ). Since ,= 0 (k"° + C ({0}0)) and the translates just
written are pairwise disjoint, the invariance of implies that (C({0}0))
. A set C(Ao) is the union of 0 pairwise disjoint translates of C
a o
Hence (C(Ao))= . Similarly, there are a pairwise disjoint
a 0
translates of C(A o, (0)) whose union is C(Ao), from which we
Finite induction shows that
infer that (C(Ao, A))= H a"
A for = 2, 3, ..- • The countable additiv-
..., =
ity of 2 implies that2(P A= lim2(C(A A A,))=H
X}=0 1 0, 1, " • •, "
}=0
Characteristic functions of sets C(A o, A,..., A) span a linear subspace
of @() that is dense in E(). /This follows from the STONE-WEIER-
STRASS theorem (t3.2), since the product of two of these characteristic
functions is another of the same sort, and these characteristic functions
separate points of . The foregoing and (t3.22) show that
aees with f/d,, where , is the product measure for A0, A, ..., A,...,
for / in a dense subspace of @(). Hence A is the product measure.]
(1) Using (k), it is easy to compute Haar measure A on Q. We
normalize A by requiring that A():t. For each kZ, let A be the
measure on 0, t, ..., a--t} defined in (k). For each nZ, let denote
the product measure of these A's on the space (0, t, ..., a--};
1 It is curious that Haar measure on Aa is identical with Haar measure on
kPoC°= {0, 1, ... , a-- I ), with the topology and group operation as the direct product
of finite groups of order a}. See (j) supra.
this space is homeomorphic with the set A defined in (t0.4). If A is a
A-measurable subset of A where n < 0, then ). (A) -- a a+l ..- a_l .// (A).
For an arbitrary ).-measurable subset A of 2, we of course have ). (A)--
lira a a+ ... a_ .// (A NA).
(15.18) We will now investigate relatively invariant flmctionals on
quotient spaces. Throughout (t .18) -- (t f;.24), we will suppose that G
is a locally compact group and that H is a closed subgroup of G. Given
any function V on G/H and any acG, let aV be the function on G/H
defined by av(xH)=v(axH). The space G/H is locally compact (.22)
and Hausdorff (5.2t), so that the function space oo(G/H) is nontrivial
if H G. We are concerned here with finding nonzero functionals J on
oo (G/H) such that:
(i) o__<J()<oo for vCo(G/n);
(it) J( + 2) --J()+ J(v2) for VI,Voo(G/H);
(iii) J() =J() for K and Voo(G/n);
(iv) J()--Z(a)J() for all aG and VCoo(G/H).
For aG, the number z(a) is defined by (iv) and is to be independent of
the choice of . Such a functional J is called relatively invariant.
Let J be as above. As in part (V) of the proof of (t S.S), it is easy to
see that J() > 0 for all nonzero 0 (G/H). Property (iv) is responsible
for this. Since b---b() for 6, bG, the function Z satisfies the functional
equation
(v) z(ab) -- Z(a) z(b) for all a, bG,
and also has the property that Z (G) 0, oo[. A less trivial fact is that Z
is continuous. To prove this we need a lemma.
(15.19) Lernrna. Let v be a continuous complex-valued /unction on
G/H such that/or every (>0 there is a compact subset (xH:xcF) o/G/H
outside o/which ]v I is less than . Then/or every e > O, there is a symmetric
neighborhood V o/ e in G such that i/ x, yG and yx-V, then
[v (x H) -- v (y n) [ < e.
Proof. This lemma and its proof are similar to but not identical
with (4.t). Consider e>0 and choose (zH'z(F) to be a compact set
such that IV (wn)[ < for wile {zn" z(F}. Now for every xcF, there
is a neighborhood U of e in G such that if yH(zH'zUx}, then
Iv(yH)--v(xn)l<- . Let V be a symmetric neighborhood of e in G
such that V c U. A finite number of the open sets {zH:zV,x} cover
204 Chapter IV. Invaxiant functionals § 15. The Haar integral 205
the compact set {zH:zF}; that is,
{gH" zt 7} Ck__J l(ZH: z Wxk " X,k} ,
for some xl, ..., x,F. Let V
Now consider any x, yG such that yx-l V. If xH{zH:zF}, then
we have xV, kxkH for some k--t, ..., m. Therefore y=yx-x VVxHc
U xH, and yH {zH:z U x). Obously xH also belongs to
(z H : z U, x} and hence
I +
Similarly, if y H {z H : z F}, then
nor xH lies in {zH:zF}, then we have i (yH)- ( H)lalw(yH)l+
(15.20) Theorem. Let J be a relatively invariant /unctiond on
oo(G/H) and let g be as in ( 5.iS.iv). Then g is continuous.
Proof. Let be a nonzero function in o(G/H), and let {xH:xF}
be a compact subset of G/H outside of which vanishes. By (5.24.b),
we may suppose that F is compact. Let U be a neighborhood of e
G such that is compact. Then the set {u x H: u , x F} is compact
in G[H since F is compact in G. Let o be a function in o(G[H)
such that o({uxH:uV-,xF})=t. For >0, use (t5.t9) to choose
a symmetric neighborhood V of e in G such that V c U and
J()
for all v V and x G. Then we have
J() (xH) for all xHG/H and vV;
I __<
applying J to the last inequality, we have
IX() ](V,)--J(V,)I<__,I(V,) for a vўV.
This implies that Z is continuous at e, and since Z is a homomorphism
of G onto a topological group, is continuous throughout G.
We will ultimately establish a relationship between the function
and the modular functions for the groups G and H. We first establish
the following theorem.
(15.21) Theorem. Let L denote a left Haar integral on (00(H). Then
the correspondence/-->./' given by
(i) /'(xn)=L(,,/)
defines a linear mapping o/ 00(G) onto oo(G/H). [In the expression
L(x/), / is regarded as the/unction / restricted to
Proof. Let / 00 (G) ; then / vanishes outside of a compact set F c G.
If we restrict the domain of /to H, we see at once that ,/is continuous
on H and that /vanishes outside of the compact set (x-XF)fill. Thus
L(x/) is defined for xG and/oo(G). If hH, then we have L(,/)----
L (h(,/)) --L (/). Thus the function L(,/), defined for all xG, is constant
on the cosets xH; we can therefore define/" on G]H by the equality (i).
The uniform continuity of / guarantees that /' is continuous on G]H.
We omit the details of this argument, which should by now be familiar
to the reader. Also/' vanishes outside of the compact set {x H: x F} c G/H.
Thus (i) defines a mapping of (00(G) into oo(G]H). It is obvious that
(/+/2)'--/'1+/., (/)'--/' (K), that /'-->_0 if /_-->0, and that /'--0
if and only if/--0 for /-->_ 0]. In particular,/-/' is a linear mapping.
Now let p be any function in oo(G/H). Suppose that p vanishes
outside of the compact set {xH:xF}. By (5.24.b), we may suppose
that F is compact in G. Let g be a function in g0 (G) such that g (F)--t.
Let U--{xG:g(x)>O}. H. Then it is clear that g'(uH)>O if and only
if u U recall that L is a strictly positive functional]. Let 0 be the
natural mapping of G onto G/H. Let be the function on G defined by
g()
(x)--g'o,Cx) ocp(x) if g(x)=O,
0 otherwise.
It is clear that oo (G) is continuous since -- g
g,oPo 0 on the
open set U, --0 on the open set (FH)', and G--UU(FH)']. For
x U, we have
xr_/(y ) __ xg (Y)P (x H)
L (.g) for all y H,
and therefore L(,o)--(xH). If xtU, then clearly L(,=O--p(xH).
That is, v(xH)--'(xH) for all xHG/H. Hence the mapping /-+/'
carries 00 (G) onto oo (G/H).
(15.22) Theorem. Let J be a relatively invariant /unctional on
oo(G/H) as in (t5.t8) and let g be as in (t5.t8.iv). Let A and denote
the modular/unctions o/ G and H, respectively. Then we have
(i) Z (h) = /or all h H.
Proof. Let L be a left Haar integral on (00 (H) and let the mapping
/---/' be given by (t 5.2i.i). For/oo(G), let K (/) --J (/'). For all x, sG,
it is evident that (,/)'(xn)--L(x(,/))--L(,,/)--/'(sxn). Thus K(,/)--
J(,(/'))--Z(s) J(/')--Z(s)K(/). This tells us that K is a "relatively
x Thus, if there is a relatively invariant functional J, the homomorphism Aft5 of H
into the multiplicative group ]O, oo[ admits an extension over G which is a contin-
Uous homomorphism of G into ]O, oo [.
206 Chapter IV. Invariant functionals § 15. The Haar integral :207
invariant" in an obvious sense], strictly positive, linear functional on
)00(G). Consider the functional I(D--K(z/) on 00(G)- We have
% (s) I([) -- % (s) K(%]) -- K (,(%/)) -- K (% (s) %.
= y, (s) K (%. (s])) -- Y, (s) I(s[),
for ]c00(G) and s cG. Since I is plainly linear, it follows that I is a
left Haar integral on 00(G). Thus we have
J(/') -- K(/) = I(Z -/) . (1)
Now take any hH, and consider the function ]h-,. For xG, we have
(/,_,)' (xH) -- L (x(/,-,)) -- L ((x/),-,) ---- ((h)L(x/) -- ((h) /'(xH).
Thus
((/-0') - () I(/') • (2)
We also have
z- (x)/(x-)= z (-)z-(x -)/(x-),
so that
l(z-(/,-,)) -- z(h -) I((;Z,- f),-) -- z(h -) A(h) I(;Z, -/). (3)
Applying (2), (t), (3), and then (t) again, we find
(:) j(/')= z(: -) /t(:) :(/,).
Hence the equality (i) holds.
(15.23) Remark. Suppose that H is a closed normal subgroup of G,
so that G/H is a locally compact group. For J, let us take a left Haar
integral on oo(G/H). Since (aH)(xH)--axH for all a, xG, we have
(4 u)!P -- a!P for all functions !P on G/H. Thus J satisfies (t . t 8.i) - (t 5. t 8.iv)
with %(a)--t for all aG. From (.22), we infer that 6(h)--A(h) for
h H, where A and are the modular functions for G and H, respectively.
The converse of (t 5.22) also holds.
(15.24) Theorem. Let A and be the modular/unctions/or G and H
respectively, and suppose that there is a continuous homomorphism %, o/
G into ]0, oo[ such that
(i) Z (h) A (h) /or all hcH.
()
Then there is a/unctional J on (oo(G/H) satis/ying (t5.t8.i)--(t5.t8.iii)
and (t 5.t 8.iv) with the given homomorphism Z,.
Proof. Let I and L denote left Haar integrals on G and H, respec-
tively, and for ]oo(G), let/' be defined by (t5.2t.i). Define the func-
tional K on 00 (G) by
K(/)-- I(Z-1/). (t)
1 Here Z -1 designates the function l/Z " • it is not to be confused with the inverse
mapping.
Ve will show that K can be regarded as a functional J on oo(G/H).
Since the mapping/--->/' is linear and onto (t 5.2t), this will be accom-
plished if we show that /'--0 implies /<(/)--0. [We will then define
j(/')K(/).] Suppose then that L(/)--O for all xG. By (t5.t4), we
then have L ((]) -) -- 0 for all x G. Now let g be any function in 0 (G)
such that g'(xH)=t for all x{yG:/(y)@0}-. That is,
L(,g)=t whenever /(x)0. (2)
Let us use the notation I gL] to indicate that the functional I ILl is
being applied to a function of x G [yH]. We then have
1.
Applying Future's theosem to this equality, we obtain
By (15. t t), we have
/)=z(y) (g, .
Substituting this into (3) and using (i), we find
Ly [I x (g (x y) %-1 (x) 1 (x))] -- O.
Applying FUBINI'S theorem (t3.2) again, we have
L [z(,g) z-' (x) 1 (x)] = 0. (4)
In view of (2), (4) implies that I(Z,-I])--O. That is, K([)--O. Thus
J(/')=K(]) defines a nonzero functional on oo(G/H) satisfying (t 5.t8.i)
to (1 .18.iii). Furthermore,
and thus J is relatively invariant on goo(G/H) with the given homo-
morphism Z appearing in (15.8.iv).
Since /Eg00(G) and gEgoo(G), the function (x, y) z- (x) g (x)
is in 00 (GxH). Thus we need only the simplest version of FUBINI'S theorem (13.2)
to obtain (3).
:208 Chapter IV. Invariant functionals § 15. The Haar integral :209
Miscellaneous theorems and examples
(15.25) We sketch here a proof of the existence of the Haar integral
using steps (I) and (II) of (.5) and TIHONOV'S theorem. For each
nonzero /,,, let X 1- (/,'1) ' (/'/")' and let X--PXI, where [ runs
through all nonzero / in 0. For nonzero q)0, the functional I, can
be regarded as the point (xl) of X where xl=I(/). Let ' consist of all
neighborhoods of e in G. For each U', choose exactly one
such that q)v (U') = 0. The set ' is directed by inclusion C, i.e., U_;>V
if U c V. Thus I, U , is a net in the compact space X. Consequently,
there is a subnet I E, of I and an I X such that lim Iў--I. Then
'
I is a Haar integral; the linearity follows from step (II) of (t 5.5).
(15.26) Haar integrals and automorphisms of (BRACONIE
[t). Let G be a locally compact group and let be a topological auto-
morphism of G oto G; i.e. let "r be an automorphism and a homeo-
morphism. Let I be a left Haar integral on 00. The mapping/---/.
is plainly a one-to-one linear transformation of 00 onto itself that
preserves nonnegativity and also preserves the norm II/11 . The linear
functional I(/)=I(/o-r) thus obviously satisfies conditions (t
(t5.5.iii). For aG, we have ([)o=_,()([o), as is easy to verify.
Hence we have I(/)=I(-,l)(/oz)), and as I is a left Haar integral,
we obtain I, (a/)= I, (/). That is, I, is also a left Haar integral, and by
the uniqueness of left Haar integrals (t5.5), there is a positive number
A(z) such that I([)=A(z)I([) for all [00. Given z and z,., it is
easy to see that A(zoz)=A(z) A(z). That is, the mapping
is a homomorphism of the group of all z's into the multiplicative group
0, oo[. We obtain the function x-+A(x) of (t5.tt)by considering the
inner automorphisms k): y-+xyx -. Then plainly A(0,) in our present
notation is A (x) in the notation of (t 5. t t) .
Similarly, if J is a right Haar integral on 00, we define J ([)
and find J (/)=J (([) o z) = J (([ o z)-(l) =J([ o z) =J (/) and hence
J ([) = (z) J(]), where d (z) > 0 and (z o zt) = d (z) d (z).
(15.27) Comments on right Haar integrals. (a) Let G be a locally
compact group, J a right Haar integral on 00, and A the modular func-
tion on G. Then the mappings [-+J(/A) and [--J(/) are identical and
are left Haar integrals on 00. I This follows from (t 5.t ). Given a left
Haar integral I, [-->I ([ )is a right Haar integral. Thus we can write
1
J(/)-I (/-) and J(/A)--I (/A -A ) =I([ ). Moreover, we have by (t5.t5)
that
t For more facts about Zl (r), see (26.21).
(b) Let G, A, and J be as in (a). Then J(a/)=A(a) j(/) for all [00
aG. [We have J(])-I([-), where I is a left Haar integral. Thus
and
(15.28) More examples of Haar integrals. (a) The construction of
(15.17.h) can be generalized to (bg(n,R). For A(bg(n,R), we have
j(b)=J(q)=(detA): this is shown by an elementary computation.
Thus ®g(n, R)is unimodular, and the Haar integral on 00((bg(n, R))
has the form /" l (x)
Idet (X)I n dX,
the integral in each case being extended over a compact subset of R"'
that is disjoint from the set {(x .... , x,,): det (X) =0}.
(b) Consider the group G of all triangular n xў matrices
such that xiR (t <=f<=k<=n) and x x9.9. ... Xnn4=0. Plainly G can be
n(n+l)
regarded as an open subset of R 9. , and satisfies the hypotheses of
(t 5.t 7.e). A left Haar integral on 00(G) is the integral
• .. !(x, x ..... Xnn) I '"" nnl dx dx,..., dxnn.
A right Haar integral on 00 (G) is the integral
ff
• .. l(x, x ..... Xnn) I ."'nnnl dx dx2.., dxnn.
(c) Let G be the group ofmatrices ( Y)
x- [x,yR, x0, with its
topology as an open subset of R For A = G, we have ](e)- a
and J(d)=l. [We use x and as parameters to describe G. Thus
Hewitt aad Ross, Abstract haonic analysis, vol. I 14
Chapter IV. Invariant functionals § 15. The Haar integral
210
is a left Haar integral on 00 (G) and
is a right Haarintegral on 00(G). The modular function isA(x, y)--
(d) Consider the group of nonzero quaternions. We describe the qua-
ternions informally as the set of all linear combinations x+ i y+jz+ ku,
where x, y, z, and w are arbitrary real numbers and the quaternion
units t, --t, i, --i,, --, k, --k form a group" t is the identity,
(--1)2=t, i2=j2=k2=--t, ij=--ji=k, jk=--kj--i, ki=--ik=j.
The product (x+iy+jz+ kw)(x'+ iy'+jz'+ kw') is formed by multi-
plying out, assuming the validity of the usual axioms for an algebra over
a field, and applying the above multiplication table for the quaternion
units. Thus
(x+ iy+jz+ kw)(x'+ iy' +jz'+
=(xx'--yy'--zz'--ww')+i(xy'+ yx'+zw'--wz')
+(xz'--yw'+ zx'+ wy')+ k(xw'+ yz'--zy'+ wx').
The quaternion t-t-i0+0+ k0 is an identity for multiplication, and
0+i0+0+ k0 is a zero for multiplication. The nonzero quaternions
form a [non-Abelian group under multiplication. [The inverse of
x+ iy+z+ kw is Z (x--iy--z-- kw), where =x2+y2+ z2+ w. Let
us denote this group by G. We can evidently regard G topologically as
R 4 without the origin (0, 0, 0, 0), and with this topology G is a locally
compact group. Left and right Haar integrals on 00 (G) are the same,
and are given by
[X22r_ y2_ Z2 - W22 dx dy dz dw.
(15.29) Haar integrals for semidirect products. Let G and H be
locally compact oups, and let h z be a homomorphism of H into the
group of automorphisms of G. Suppose also that the mapping (x, h)z(x)
is a continuous mapping of GxH onto G. Then the semidirect product
G @H, vith the Cartesian product topology, is a locally compact oup
[see (2.6) and (6.20).
(a) Let J and K be right Haar integrals for G and H respectively.
Then the product functional JxK is a right Haar integral for G@H.
[Let [ be any .function in 00(G@H). Then, using FUBINI'S theorem
(t3.2), we have
= =
=L (/(x, = L (/(x, = j
(b) We next compute the left Haar integrals for G@H. While there
is no necessity for doing so, we find it convenient to use the right Haar
integrals J and K of (a). Let A and A n be the modular functions for
the groups G and H, respectively. For fixed hcH, the functional
j(/o z) on 00 (G) has the form J(/o z)- d(h) J(/) • moreover, d(hl h2)--
5(hl) d(h2), (h)>0, and is continuous on H. [All of these properties
of except for continuity follow at once from (t 5.26). It suffices to
check continuity at the identity e' of H. Let U be any neighborhood
of the identity e in G such that U- is compact, and let / be any nonzero
function in 0 (G). Let A denote the set {y c G: / (y) > 0}-, and let
o) 0 (G) be such that o) (A. U-) = t. For e > 0, let U 0 be a neighborhood
of e such that U 0 c U and
e J (/)
1/(u) -- / (v)] < - whenever u -1 v U 0 .
Finally, let U1 be a symmetric neighborhood of e such that U c U 0.
For each xA, joint continuity at %,(x)=x implies that there exist
symmetric neighborhoods V and W of e' and e, respectively, such that
WcU1 and
z (y) x U1 whenever h c V and y x W.
Let xlWxl ... xmW be a finite covering of A and define V as fl Vxi.
J (/)
If hV and yA, then y-1 z(y)U ° and hence I/(Y)--/° z(y) l<ej( .
It follows that
J (1)
I/(y) - 1 o (y)[ __< o, (y)
for h c V and all y G. Applying J to this inequality, we obtain
l J(/) - (h) ](/)l * ](/) for h< V.]
We now have"
That is, the modular function for the semidirect product G@H is
Zl(x, h)= d(h) A(h) A(x). Comment (15.27.a) now gives us the left
Haar integrals on oo(G.@H)" they all have the form/--JxK([dAnA).
(c) The group of matrices (; ) discussed in (t 5.17.g) is a semidirect
product of the multiplicative group --o, 0[ U 0, o[ [which we take
Here we use the identities ((Xoq)Oho)(X)=q(Xoo(x)) and
Zl (a) J (9) [see (15.27.b)].
212 Chapter IV. Invariant functionals § 15. The Haar integral 213
as H] and the additive group R which we take as G. For real x:#0
and real y, let zx(Y)--xY. Then (y,x)(y',x')--(y+xy',xx')" this
corresponds precisely to the product of the matrices ( )and(' ').
Applying(15 t7.f) and (b) above wesee once again that - ! ]_(x, y) dxdy
is a right Haar integral for this group. It is also easyto see that ((x)- ix I ,
which gives us from (b)another verification that] -- -- ] [(---Y) dx dy is a
left Haar integral for our group. -oo
(d) Let H be any locally compact group with modular function
and let G be the additive real numbers R. For h cH and x CR, let
zh (x) -- /l (h) x. Plainly we define in this manner a semidirect product
of R and H. To compute (, we write J([o,J-f [(/l(h)x)dx=
A(h) /(x) dx--A(h)J(/)= (h)J(/). Thus (b) gives the modular
function for R@H as --- A (h). t = t. Therefore every locally com-
pact group admits a semidirect product with R that is unimodular.
(15.30) Relatively invariant integrals. (a) A simple example of a
relatively invariant but not invariant integral is the following. Consider
the additive group R 2 and its closed subgroup H= {(x, 0)'xR}. Let
be any nonzero real number• Then f f exp(y)/(x, y) dx dy defines
as in (t 5.24) a relatively invariant integral on oo(R/H), for which the
proportionality factor Z in (t 5. t 8.iv) is 2: (x, y) = exp (-- y). One needs
only to note that R is Abelian so that A/d= t on H, and that 2: is a
continuous homomorphism of R onto ]0, for which 2: (H)= t .
(b) Consider the special linear group ®g(2, R), and its subgroup
/_.
consisting of all matrices{ ') with x,,R and x@0. The group
/
®g(2, R) is unimodular, as (t5.t7.h) and (t5.23) show. There is no
relatively invariant functional en 00(®(2, R)/g) in the sense
(t5t8). The modular function d of g has the form d ( ')
• X -1 --- -- as
(.28.c) shows. If there were a relatively invariant functional on
00(®(2, R)/), there would be a continuous positive function Z on
®(2, R) such that x(XY)--(X)x(Y) for X,Y®(2, R) and
z(X)_ A (x) for X (t 5 22) It clearly suffices to show that
(X) -- (x) • •
the only positive function Z on (2, R) such that Z (XY) -- Z (X) Z (Y)
is the function I. On the subgroup of all matrices (t0 ),writez(t0
[(x). Then/(x+y):/(x)/(y). Similarly ifwewriteg(y)=Z(ty ), we
g(x+ y)=g(x)g(y). On the subgroup of all matrices (z
/
\
get
z -1/, writ e
Z z- = h (z). Since
t at 0) l(x) = I x) (z), t at l(x) - I x) x
and a]] real z0. Hence [(x) is constant for x>0, and since/(2)--/() ,
we get [(x)--1 for x>0. Since /(--x):and ](0):I, it follows
z- . If ad--bc-- and b0, then
nd he e ore (:
so that Z a- = 1. Hence we have Z--t. That is, the group g (2, R)
admits o nonconstant homomorphism into the multiplicative group
Notes
Invariant integration on one or another special class of oups has
long been known and used. A detailed computation of the invariant
integral on () was given in t897 by Huwz [1. Scu and
Fous in the years t900--t920 made frequent use of averages over
finite groups" for references, see the notes in WY
computed and applied intensively the invariant inteals for ()
and (). WY in [t computed the invariant integals for (),
(n), the unitary suboup of the symplectic group, and [more or less
explicitly for certain other compact Lie groups. WYg and P in
showed the existence of an invariant inteal for any compact Lie group.
:214 Chapter IV. Invariant functionals § 16. More about Haar measure :215
The decisive step in founding modern harmonic analysis was taken by
A. HAAR 3 in t933. He proved directly the existence but not the
uniqueness of left Haar measure on a locally compact group with a
countable open basis. His construction was reformulated in terms of
linear functionals and extended to arbitrary locally compact groups by
A. WEIL [t3, 23, and 4], pp. 33--38. KAKUTANI 2] pointed out also
that HAAR'S construction can be extended to all locally compact groups.
Theorem (t5.5) as stated is thus due to WEIL. The proof we present
is due to H. CARTAN [t .
For an arbitrary compact group G, yON NEUMA?;?; [5] proved the
existence and uniqueness of the Haar integral, as well as its two-sided
and inversion invariance. In [6], vo N-u=a proved the uniqueness
of left Haar measure for locally compact G with a countable open basis;
a special case was also established by Sz.-NAGY [t . WEIL [43, pp. 3 7--38,
proved the uniqueness of the left Haar integral for all locally compact
groups. A generalization was proved by RAKOV 2; see also AUBERT t].
All of these uniqueness proofs function only ex post/acto, when the axiom
of choice has already been invoked to prove existence. The virtue of
CARTAN'S existence proof is that it demonstrates uniqueness and existence
at one and the same time.
Theorem (t 5.9) is due to WEIL 4, p. 38; VON NEUMAN 3] pointed
out that 2(G) is finite if G is compact. It is interesting to note that
WEYL [t ], Kap. I, § 5, considered art invariant measure of total meas-
ure t ort tile group ® (n,R), and concluded that the existence of such
a measure was doubtful, because ® (n, R) is noncompact.
Theorem (t 5.t t) appears in vo?; NEUMAIVN 6 and is attributed by
VON NEUMANN to A. HAAR. Theorem (t 5.t3) is due to vo?; NEUMANN 53,
and Theorem(t5.14) to WEIL[4, pp. 39--40. Example (t5.t7.g)is
immensely popular; so far as we know, it was first computed by vo?; NEU-
MANN 63. The other examples of (t 5.t7) are widely known. It is diffi-
cult, as well as of little interest, to trace their history.
Explicit computation of the Haar integral for groups like ® (n, R),
® (n, K), 1I (n), and Ј), (n), which are not given as open subsets of some
R [in which case the simple process of (t5.t7.e) does not apply]
requires some sort of parametrization of the group. The computation,
still in abstract form, is given for all Lie groups by CHEVALLEY t3,
pp. t67--t70. The Haar integral for lI(n) and other compact classical
groups is described in WEҐI. 3], PP. t94--t98, 2t8, 22S. A description
of the Haar integral for lI(,), the symplectic unitary group, and (n)
is also given By T6YAMA t]. An illuminating discussion of the r61e of
invariant integration in general appears in WEYL 3, PP- t85--t94.
All of (15.t8)--(t5.24) is taken from WEIL 43, pp. 42--45; see also
WEIL [1 "].
We now mention some further results which the reader interested
in invariant measures may wish to look into. A.D. ALEKSa?VI)ROV
has sketched the construction and properties of a/izitely additive ana-
logue of Haar measure for groups that are not necessarily locally compact
but only locally bounded. A topological group G is locally bounded if
there is a neighborhood U of e such that for every neighborhood V of
e, a finite set {x, ..., x}c G exists such that U x V D U.] OXTOBY
/=1
has made a careful examination of the existence and strange properties
of countably additive invariant Borel measures on complete metric
groups. H.F. Davis [t] has characterized Haar measure as a measure-
theoretic product of n-dimensional Lebesgue measure, Haar measure on
a compact group, and "counting" measure on a discrete set. LooMIs
has studied invariant measures in uniform structures. RosE?; [t] and
ScIWaRZ [2] have examined invariant means on G(S) for a compact
semigroup S.
§ 16. More about Haar measure
In this section, we present a remarkable theorem of KAKUTANI and
OXTOBY on extensions of Haar measure and also some facts about
nonmeasurable sets.
(16.1) Let G be any infinite compact metric group. Let 2 be nor-
malized Haar measure on G, and let //' be the C-algebra of 2-measurable
subsets of G. By (4.26), we have G_> c. Since G contains a countable
dense subset D and every element of G is the limit of a sequence of
elements of D, we see that G=c. Throughout (t6.t)--(t6.tt), G,
and.///' will be as defined here. We will have occasion to use the notation
E", where e is either t or ': this will denote E or E', respectively.
It is known that there is no countably additive measure defined for
all subsets of G, agreeing with 2 on /ў', and invariant under, say, left
translations. [For a discussion, see (t6.t3).] Nevertheless, 2 can be
extended to a very much larger measure space.
(16.2) Definition. Let X be any set, 9 ° any C-algebra of subsets
of X, and / any measure defined on 9 °. Then the character of the
measure space (X, cM',/) is the smallest cardinal number In for which
there is a subfamily aў' of 9 ° such that 7= m and such that for every
S and every e>0, there exists a set Aaў' satisfying #(S/A)<
Any such subfamily aў' of cM' will be called a basis for (X, 9 °, #).
We first note that the measure space (G,K/, 2) has character N0.
There is a countable open basis for the topology of G" let ad consist of
finite unions of sets from this countable open basis. If M Z and > 0,
then there is a compact set F and an open set U such that FCMC U
216 Chapter IV. Invariant functionals § 16. More about Haar measure 217
and 2(UVIF')< e. Since F is compact, there is a set A c such that
F c A c U. Then clearly 2 (M/ A) < e. Since -- 0, (G, t', 2) has
character 0. lit is easy to see that (G, t', 2) cannot have finite character.]
Given a set X and two measure spaces (X, 't' 1, 1) and (X, t'2,
we say that the second is an extension of the first if t'2Dt' 1 and
#2 (A)--#l (A) for all A ct' 1.
We will prove the following assertion.
(16.3) Theorem. There is an extension (G,[*, *) o[ (G,[, ) such
that the character o/(G, /[*, *) is 2 c and * is invariant under left trans-
lations, right translations, and inversion.
The proof of this theorem is long, and is broken up into a sequence
of lemmas. We shall repeatedly use well ordering and transfinite induc-
tion, but we do not use the continuum hypothesis. In order to avoid
using the continuum hypothesis, we first extend (G, t', 2) to a measure
space (G, t 't, 2t) such that every set A c G for which < c belongs to
t 't and has 2t-measure zero. We will then extend (G, 't, 2t) to the
desired measure space (G, ?t'*, 2").
Let consist of the transformations x--ax, x-->xa, and x--x -1 of G
onto itself (aG). To check that a measure is [left and right] translation
and inversion invariant, it suffices to check that if M is measurable, then
each (M) is measurable () and that the measures of M and
are equal.
(16.4) Lemma. The measure space (G,, ) can be extended to a
measure space (G, 't, 2t) such that every set A c G [or which <
belongs to //[ and has 2t-measure zero. Furthermore, 2t is translation and
inversion invariant and (G, l t, ) is complete; that is, i/N t'*, 2t (N) =0,
and N 1 c N, then N 1
Proof. Let be the family of subsets P of G such that P<
It is not known whether c'; however, the continuum hypothesis
obviously implies this inclusion, since ' contains all countable sets.
At any rate, if Pcflt', then 2 (P) ---- 0 .1 We also have
P, P,., ..., P, .... implies U P, . (t)
ў--1
1 If Pt' and 2.(P) > 0, then P contains an uncountable compact set. Every
compact subset F of G is the union of a countable set C and a perfect set F 0" let C
be the set of all xF such that U f'l F is countable, for some neighborhood U of x.
All nonvoid perfect subsets of compact metric spaces have cardinal number
This is shown by the fact that every nonvoid perfect set F 0 contains a homeo-
morphic image of {0, l} eo" such a subset of F 0 can be constructed by trivially
modifying the proof of (4.26). The set F 0 plays the rdle of U, and each Uiv..i, is
chosen to have diameter less than l/m. Then for any sequence {jm}m__i of O's and t%
the set Uf...im consists of exactly one point Xh...i,,,... and this point belongs to
The set consisting of all of the points xit...i,.., is homeomorphic with {0, 1}
For, KONIa'S theorem [see HAUSDORFF [t, p. 34 states that if {mr}vr
and {nr}rCr are sets of cardinal numbers such that m r < n for all
then m,< H nr- Therefore, since .< c for all n, we have U P. F yF n=l
c=c. Thatis, UP..
n=l
Let consist of all subsets M of G such that the symmetric
difference MtM is in for some M[. In other words, a set lies
in t if and only if it has the form (MA')UB where M and
A, B. We first verify that is a a-algebra. If Mt t and
M tM where M, then (Mt)'M'--M tM so that
(M*) d*. Also, if {M.. c and Md G , then
(M. C. (MG M. by (t). Thus U M *.
n : n=l
For M dt and M & M , M , we define
2t (M t) -- 2 (M).
If M / M and M t/N M , where M 1 and M 2 are in /, then we
have (Mx & M2) c (M M ) U (M & M t) , so that 2 (M & M2) = 0 and
2()=2(M2). Therefore 2 t is well defined on the a-algebra t. To
is a pairwise
prove that 2 is countably additive, suppose that {M,},=
disjoint sequence in and that M & M. where M. . Let Q = M
n--1
and Q.M.( Q ') for n=2, 3,.... Then {Q.= is a pairwise
h 1
disjoint sequence in , M Qc (MM), and ( M)
k=l n
Q . Thus 2 * .l .1 = =1 '
,1 M : Q, : (Q,):E (M) andwe
have proved that 2' is a measure on
We next show that (G,*, 2*) is complete. Suppose that
i*(N)=0, and CN. Then NM for some M such that
i(M) =0. HenceN,M,2(NM)=O, and&(NM) cN&M ,
so that N .
Finally, we check that 2' is translation and inversion invariant on
Clearly p implies that z(P) for all z. Thus if M*
M* M and M <, then z (M*) z (M) = z (M* M) . Conse-
quently, z (M*) * and 2 (z (M*)) = a (z (M)) = a (M) = 2 (M*) for all
1 To verify this inclusion, suppose that x Qn and xMn; then xM n, so that
xMMn" Suppose that xMd and x 0,- If zўM, then clearly xMntM,.
Otherwise we have xMk for some k < n. Since xM n and _Mkr CI Mn : , we have
xC (Mkt), ' so that xMkt/XMk.
218 Chapter IV. Invariant functionals
(16.5) Lemma. The set G contains exactly c distinct compact subsets
having cardinal number c. Let oc denote the smallest ordinal having cardinal
number c, and let {F'I <<oc} be any well ordering o/ all compact
subsets o/G having cardinal number c. Then/or every M* C * such that
* (M*)>O and /or every ordinal number <oc, there is an ordinal
such that > and F c
Proof. Every open subset of G is a union of sets in a countable open
basis, and thus the number of compact subsets of G having cardinal
number c does not exceed c. In the next paragraph, we will see that G
contains exactly c compact sets F such that = c.
To prove the lemma, it suffices to show that every M* Ct 'f with
;*(M*)>0 contains c compact sets having cardinal number c. Let
M ' have the property that M*/X M . Then we have ] (M) > 0 and
M*/X M< c. It thus suffices to prove that M contains c pairwise disjoint
compact subsets each of cardinal number c, because then c of these
compact sets are necessarily disjoint from M*/XM and are hence con-
tained in M*. The set M contains an uncountable compact set, and
therefore M contains a nonvoid perfect set. As shown in the proof of
(t6.4), this set contains a subset homeomorphic with (0, )°. This
space is plainly homeomorphic with 0, }°x(0, }°=U(0,
(x}" x {0, t}°} • this is a compact space which is the union of c pairwise
disjoint compact subsets each having cardinal number c.
(16.6) Definitions. For every subset A of G, we define the outer
*-measure # (A) of A to be
inf 2' (M*)" M* * and A C
A subset A of G is said to be absolutely invariant with respect to * and ,]
if for every zc%, z(A)/NA6* and 2' (z(A)/XA)--0.
It is easy to verify that the family of absolutely invariant subsets of
G is a a-algebra of subsets of G recall that (G,'*, ]*) is complete].
(16.7) Lemma. There exists a /amily (X,}, N o/ subsets X, o] G
with the/ollowing properties"
(i) N----c;
(ii) the sets X, v, N, are pairwise disjoint"
(iii) # (X,)= t /or v N"
(iv) /or every subset No o/N, the union U X, is absolutely invariant.
yEN0
Proof. Let (F'-<=<oc} be as in (t6.5), and let
be any well ordering of with order type oc. For each xEG and each
ordinal +< oc, let C (x) be the set of points of G that can be written as
vj:o...oz(x) where t G____< and e is t or --t, k-- ..... n, and
§ 16. More about Haar measure
219
n=l, 2, 3 ..... Clearly we have xEC(x) and for xcG and
we have z(C(x))=C(x). We also have C(x)__<max(, 0)< c and
therefore C (x) E
We shall show that there exists a transfinite double sequence
{x't G fl G < c} of elements of G such that"
the sets {C(x)" GflG<c} are pairwise disjoint. (2)
If we a'ee that (, )< (, fl) whenever < or whenever = and
< fl [lexicographic ordering], then {(, fl)'l G fl G < ў} is a well-
ordered set. We shall define the sequence {x'tGflG<c} by trans-
finite induction. Let x be an arbitrary point of . Suppose that
t G flG<c and that x has already been defined for all pairs (y,
< (, fl), G G y. Consider the union D (, fl) of the sets C (x) as (y,
runs over all pairs (, )< (, fl). Then D (, fl) G (). max (, 0)= max
(, 0)< c. Since = c, the set (D (, fl))' is nonvoid. Let x be an
arbitrary point of G (D (, fl))'. Then C (xj) is disjoint from every
C(x), (, ) < (, fl). Otherwise we would have
t 0 • 0
..
where , Oiy, e is or --, and i is or --, k= .... , n,
]= .... , m. Hence
- - ,, o .. o G c D
=, d o...o a:,o •
and this contradicts the choice of x.
Now let N= {v: v is an ordinal and G v< c} and let
X,=U{C(x):v<o}, vN. ()
Property (i) is obvious and (ii) follows immediately from (2). To prove
(iii), assume that (X)< for some v N. Then there is an Mt d* such
that X,M* and 2t(Mt)< . Therefore Lemma (6.5) applies to the
set (M*)' and there exists an ordinal , v<<c, such that (Mt) '.
This is impossible since x by () and xC=(x)XM* by (3).
This proves (iii).
To prove (iv), it suffices to show that r(0X)G(0X) has
cardinal number less than c for all y< mc and N 0 N. Since
U X=U{C(x)-V<No, v<<m,}
vNo
1 If m is an ordinal, we write for the cardinal number of the set of ordinals
220 Chapter IV. Invariant functionals § 16. More about Haar measure 221
is a disjoint union by (2), and since r,(C(x))-- C(x) whenever
we have
(4)
Since C(x) U r(C(x)) max(, 0) for v = and since
{(, v)'vN0, vG<y} __< (y)2, the right side of (4) has cardinal number
less than or equal to max (y, 0)" (y)--max (y, 0)< c.
The following purely set-theoretical lemma is a special case of a
lemma of A. TARSKI ].
(16.8) Lemma. Let N={v} be an arbitrary set having cardinal num-
ber c. Then there exists a family {N0}0 e of distinct subsets of N such that"
(i) O=2 c-
(ii) the set N; is nonvoid for every sequence {0} of distinct elemenls
=1
from 0 and every sequence {e}, with e,,= t or ', n= t, 2,....
Proof. It suffices to prove the lemma for some particular set N
such that N=c. Let O be the family of all subsets of [0, t[" clearly
O=2 c. For each 0O, let B(O)=OU(t,2[R(O+t)')" here
{x+t'x0}. Wehave
0,0O and 00 imply B(O)B(O)'. (t)
In fact, if x00' ' .
, then xB(O)B(O2)', and if xOO, then
,+ B (0) n B (G)'.
Let be the family of all countable subsets of [0, 2[, and let N be
the family of all countable subsets of . Then = N=c. For each
00, define % as the family of all countable subsets of B (0)c[0, 2[,
and N0 as the family of all countable subsets of (%)'. Then we
obviously have % c and N 0 N for 0 O.
Let {0} and {e}, be given sequences as described in assertion
(ii). Choose any element 000 such that 00@0, for n=t, 2,..., and
put s0 equalto t or' sothat s 0@s. Let Ibe the set ofnonnegative
integers i such that si= t and let ] be the set of nonnegative integers
such that s i--'. Then I and ] are disjoint nonvoid sets whose union
is {0, t, 2 .... }. [The reason for adjoining 00 is to ensure that both I
and ] are nonvoid. For every iI and ], we have Oi@Oiand therefore
B(O)B(Oi)'@ by (t). For every such pair (i,)Ix], choose an
element x in the set B(Oi)B(O)'. For ], let Ci={xi'iI} and let
Then obviously and and have
xB(O) and so CcB(Oi). Hence each C i belongs to 0 and thus
u N0. That is,
N (;)
j 0.
Likewise, for iI and '], we have xiiB(Oi) and so CjcI:B(Oi). There-
fore Ciўoi for all 'J and thus vcf(o,)'. This implies that
and, as i is arbitrary, it follows that v, N. From this and (2) we
infer that if) N;= .
=1
From (ii) it follows that the members of N are distinct.
We are now in a position to state our crucial lemma, which gives us
a family {Eo}o o of 2 ў subsets of G satisfying certain properties. Later
in the construction we will find that the sets E o are all 2*-measurable
and that 2*(E0,/Eo, ) =Ѕ whenever 0@02. This property implies
immediately that the character of (G, '*, 2*) is 2 ў.
(16.9) Lemma. There exists a/amily {Eo}o o o/distinct subsets o/G
such that
(i) 0-- 2 c"
(ii) ( Eo is absolutely invariant [or every sequence {0..__ o] [not
necessarily distinct elements [rom 6) and every sequence ,}oo= where
is1 or',n--t,2 .... •
(iii) ff (.E) --- t [or every sequence {0},% o[ distinct elements [rom
and every sequence {}.__ where G is or ', n--, 2 .....
Proof. Let {X} be the family of subsets of G obtained in (6.7),
and let {N0}0o be the family of subsets of N obtained in (6.8). For
06), we define Eo to be U {X'v No}. The sets Eo are all distinct because
the sets X,are pairwise disjoint. Property (i) is exactly (6.8.i).
Each set Eo is absolutely invariant by (t6.7.iv). Property (ii) follows
from this and the fact that the family of absolutely invariant subsets of
G is a -algebra.
To prove (iii), we first choose any 0 from ffl N (t6.8.ii). By defini-
tion we have Eo=U {X'v No}, and since the sets {X} are pairwise dis-
joint (t6.7.ii), w,e also have E; D U {X:v N;}. That is, E; D
when e is t or . Hence we have the relations
n (- 1
(u n No,,
n:l = n:l "
- '0n .
Since ff(Xo) t (t6.7.iii), it follows that
(16.10) Let {Eo}oo be the family of subsets of G obtained in (t6.9).
Consider the family of all subsets E of G having the form
(i) E U ......... } 1
222 Chapter IV. Invariant functionals § 16. More about Haar measure 223
where n is a positive integer, 01,..., 0,) is a finite subset of 6), and for
each sequence {el .... , e,) with e= t or ', k= t, ..., n, M...=, is an ele-
ment of'1.. Here U(, ...... ,,} means the union over the 2 distinct sequences
, ..., e} where e= or '. Similarly ....... , will denote the sum over
all of the distinct sequences e, ..., e) n is fixed.
(16.11) Lemma. The ]amily is an algebra o] subsets o[ G and
. Let
t (M. )
(i) 2" (E)= ( ....... , ..
whenever E has the/orm (t6.t0.i). Then * is well defined and finitely
additive on # and *(Mў)= ў(M ў) /or all Mў . Finally, i/ E
,, t () # * (,())=* ().
Proof. We first observe that if E has the form
E=U( ...... ) ...
and (0,+1 .... ,0) is a finite subset of 0 disjoint from (0, ..., 0,}, then
we M...,. i i, to M...,, o a - (, ..... ) we
ei= t or '. Moreover, we have
* * (M. ,,) - * .... * (M .... )
These observations imply that whenever in the present proof we have
two sets E and F in #, we may suppose that E has the form () and F
has the form
That is, the sets (0, ..., 0,) in(t) an (2) are the same.
If E # has the form (t), then
Also if I and W have ehe foms (1) and (2), especўively, then
E UF = U(= ...... ==) Q ...
Hence is an algebra of subsets of G. It is obvious that ў .
Suppose that E has the fors () and (2)" i.., E=I. To prove
that i* is well defined on , need only to show that
* (MI... =.) = * (MI]. =.) ()
for all sequences {/1 .... n}" From (t)and (2) we see that Q EI n
' ' k 1
;I ...... Eo N Consequently, we have
k = 1 e... en "
= (4)
Sinceff(lE;)=t (t6.9.iii), every 2*-measurable subset
has 2*-measure 0. Thus (4) implies (3).
We now show that 2" is finitely additive on N. Suppose that E
and F are disjoint sets having the forms (t) and (2). Then
(=1 E;: N M... == N ff=,... =, = e so that M ..... N M.== =t E;: . As in
the previous paragraph, it follows that 2 ў (M...,NM..,,) =0. Then we
have 2 ў (M ..... U M.,)= 2 ў (M...,)+ 2 t (M.,,), and it follows at once
from (i) that 2*(EUF)=2*(E)+ 2*(F). The fact that 2"(Mў)=2 ў (M ў)
for all M ў ў is obvious.
Finally, let E have the form (t) and let z be in . Then
If we let
we find that
T(E)-- U{ ........ n} ( (k=lE;)O(M"'=n)) "
e o = U( .........} M=E; N (M...=,) ,
"'" k--i k i
Each term of this union belongs to J* and has 2Kmeasure 0 by (16.9.ii).
Thus Z (E)/ E 0 C dt'1. C 6* and obviously we have E 0 N. Hence T (E) E d
[recall that (G, ..'f,/1.1) is complete]" then using (t6.4) we obtain
Proof of Theorem (16.3). Let //* be the a-algebra of subsets of G
generated by 5 . To show that 2* can be extended to a countably
additive I measure on //t'*, it suffices to prove that if {Ep}°= is a decreasing
sequence of sets in 5 and 2*(Ep)_>_>0 for some positive number e
and p= t, 2 ..... then N Ep 4=. [See Theorem t3.A, HALMOS [2]" the
p=l '
above property implies immediately that 2* is a measure on the algebra
5 ° in HALMOS'S sense.] The observations of the first paragraph in the
proof of (t6.tt) show that we lose no generality if we suppose that we
have a sequence {0,},=°° of distinct elements of O, a strictly increasing
224 Chapter IV. Invariant functionals § 16. More about Haar measure 225
sequence {n#}°_l of positive integers, and elements M...n" in for
each sequence {el .... , en,} such that
, were replaced by the set N M.. the equality
If in (t) each set M .....
would not be destroyed Irecall that {E}°_I is a decreasing sequence of
sets 1. Hence we may also suppose that
M.t cM T...
8t"" np + l np
for /5=t, 2,... and every sequence {el .... , enp+,}. By hypothesis, we
( ).E. = 2--'-1 ,t,, .... n,} t (M...)=> for each p and therefore
have
2"
2t(M,,,...,,) , ()
..., exi t ome , ...,
where d--t or ' for which the set
of positive integers is infinite. Having defined {81 .... , dn_,} and an
infinite set Zp_l of positive integers, we choose {d,_1+1, ..., d,} in
such a manner that
Zp={qZ_l"q>= p and e(#l=d for all k=n_+t, ..., n}
is infinite. For p = t, 2,..., there is a q>= p such that
2 (Md,...d,)= 2t(M,!e,...,@ t (M,,,...d#2) > O, (4)
the first inequality following from (2) and the second from (3). Since
{M...o}_ is a decreasing sequence of sets, we have
2 t N ... >0.
Xp = 1
M... is not disjoint from Eoe •
Since # = E = t, it follows that =
Thus we have
p=l p 1 k=l p 1
and we have shown that * can be extended to a measure, also denoted
by 2", on *.
We now show that (G,*, 2") has character 2 c. Let {Eo}oo be as
in (t6.9). From (t6.tt.i) we have *(EoE;,)= and therefore
*(Eo& Eo,) ={ for 0, 00 and 0 02. Suppose that * is a
basis for (G,/*, 2"). Then for every 00, there is an Ao such that
2*(Eo2XAo)<l-. If 01=4 =02 and Ao,--Ao,, then Ѕ=2*(E0, A
2,(Eo,&Ao,)+*(Ao&Eo)<, which is impossible. Thus {Ao'OO}
consists of distinct elements and has cardinal number 2 c.
It remains to prove that M** implies that (M*)6#/* and
2*((M*)) =2*(M*) for all z. Let consist of those sets B#/*
such that (B)* and *((B))=*(B) for all . Since
by (t6.tt), Theorem 6.B of HALMOS 2] shows that =* if is a
monotone class; that is, if is closed under the formation of unions of
increasing sequences of sets and of intersections of decreasing sequences
of sets. If {B,}, is an increasing sequence of sets in and <, then
B = (B,)* and
n 1 =
* ( ( B))= * (1 (B))= li * ( (B))
= lim *(B.= * (;. B),
so that U B.. Similarly BN if {B}, is a decreasing sequence
n=l n=l
of sets in .
Miscellaneous theorems and examples
(16.12) Character of a measure space and dimension of ,2- Let
(X, ', #) be a measure space such that # (X) = 1. Let b be the dimension
of 2(X,/[, #)1 we will write 2 for 2(X,', #) and let m be the
character of (X, /, #) in the sense of (6.2). If b is finite, then m= 2 b.
If b is infinite, then m=b. I Suppose first that # assumes only finitely
many values; let 1 be the least positive value of #. Let Et be a set in '
such that #(E1)= 1. Then for every set A '/ such that A E 1, we
have #(A)=1 or #(A)=0. Let be the smallest positive value assumed
by # on subsets of E', if there are any such values. Note that 2>__1.
Choosing E2 E' such that # (E2)=cz2, we see that E 2 is like El" induction
gives us a partition of X into sets E 1, ..., E, where ,a(E)=, and
every subset A of E, A ', has measure or 0 (k=t, 2 ..... n). It
is then easy to see that {}=1 is an orthogonal basis in 0, so that
2 has dimension . It is also easy to see that the character of the measure
space (X,t', #) is 2". In fact, the sets and all unions EU...UE
for nonvoid subsets {kl, ..., k,} of {t, 2 .... , n} form a basis" and no
basis can have fewer elements. That is, m= 2b= 2"in this case.
The space . (X, t', #) is defined as usual as the normed linear space of all
t'-measurable complex-valued functions i on X such that Illllg=flll
X
see §12 for a special case. With the inner product (, g)= f] d#, .(X,#, #)
is a Hilbert space, x
Itewitt and Ross, Abstract harmonic aralysis, vol. I
226 Chapter IV. Invariant functionals § 16. More about Haar measure 227
Suppose next that # assumes infinitely many values. Then there is
an infinite family {An}_-i c g of pairwise disjoint sets of positive meas-
ure: this is shown by an elementary argument, which we omit. The
functions An ( n= t, 2 .... ) are in 2 and are plainly orthogonal, so that b
is infinite 1.
It remains to show that m--b if b is infinite. Let p be the least
cardinal number of a dense subset of 2" it is easy to see that 3=.
Let 5' be a basis for (X, g, #) of cardinal number m. Functions of
the form (i+ii)M (i,jcQ, Mcg) are dense in 2, and every
i=l
function can be approximated arbitrarily in the 2 metric by a
function a with A', since HA--BII2=/Z(A/B). This implies that
m: p--3. To show that m 3, select a dense subset of 2 such that
--3. For every Mg and positive integer n, there is an [ such
1 For every [ and n for which an M g exists satisfying
that I][-- M ]12 < -"
the last inequality, choose one such M" let 5' be the family of all the
sets M so selected. It is obvious that 5'_< 0" 3--b. Now, given M C'
I and
and a positive integer n, there is an [ such that
I Therefore, using MINKOWSKI'Sinequal-
an A ' such that I1 a 112 --"
2
ity (t2.6), we have /z(A/\M)Ѕ--HA--MH2:HA--/II.+II/--MI]2<-.
That is, ' is a basis for (X, ', #).
(16.13) Nonmeasurable sets. (a) Every nondiscrete locally compact
group G contains a subset E that is nonmeasurable under every left
translation invariant measure/Z that is an extension of left Haar measure
Let U be a neighborhood of e such that (U)<oo and let V be a
symmetric neighborhood of e such that Vac U. Let D be any countably
infinite subset of V and let H be the subgroup of G generated by
D. Clearly H is countable. Denote the set of distinct right cosets of
n by {nxo}o A ; let A0= {z A." (nx) [ V :: }. For z A0, choose
exactly one element y in (Hx)( V and define E to be the set {y:sA0}.
Assume that E is/z-measurable. Since H f3 V is countably infinite, the
set U {xE: x c H fl V 2}-- (H fl V 2) E is a disjoint union of a countably
infinite number of left translates of E. It follows that (H f3 V 2) E must
have either/z-measure 0 or/z-measure oo. Since (V)> 0 and
this assertion will be contradicted when we have established the following
inclusions:
V c (H f V 2) E c U.
For every cardinal number rt > 0 there is a compact group G, with Haar
measure ., such that dim (z(G, ))=It. See (24.16) in/ra.
If v V, then vHx--Hy for some A 0" thus v--hy for some hH.
Since h--vy, VV-Ic V 2, we have v--hy (H ( V ) E. The inclusions
(H f3 V 2) E c V 2 E c V 2 V c U are obvious.]
(b) Let G be a group, and let H be a subgroup of G not necessarily
normal such that the Ileft coset space G/H is countably infinite"
G= U a,, where the sets a, are pairwise disjoint. Assume that
r=l
there is a left invariant measure/Z on an algebra 5' of subsets of G such
that H 5' and/z (G) = t. If/, (H) is 0, then we have/z (G) -- /z (a,H) --
E # (H)= 0. If/, (H) is positive, then we have /z (G)-- # (H)- c.
:1
It is therefore impossible for H to belong to 5'. In particular, if G is a
compact group with normalized Haar measure , then H is not -meas-
urable, and there is no left invariant extension of Haar measure under
which H is measurable.
(c) Let G be an infinite Abelian group. Then G contains a subgroup H
such that G/H is countably infinite. Case I" G is a torsion group.
Since G is the weak direct product of a countable number of primary
groups (A.3), it suffices to prove our assertion for the case in which G
is p-primary for some prime p. By (A.24), G contains a subgroup B
such that G/B is divisible and B is isomorphic with a weak direct product
of cyclic groups. If B--G, then our assertion clearly holds. If B@G,
then G/B is a weak direct product of copies of z(p°) (A.t4). It is clear
that G/B admits a subgroup J such that (G/B)/J is countably infinite,
and hence G does also.
Case II" G is not a torsion group. Let aG have infinite order, let
L be the group a)___o, and let M be a subgroup of G such that
Mf3L: (e) and M is maximal with respect to the property M f3L : e).
Then LM is a subgroup of G, and G/(LM) is a torsion group. If G/(LM)
is countable, then G/M is countably infinite. If G/(LM) is uncountable,
we can apply Case I.]
(d) Every infinite compact Abelian group contains a subgroup H
that is nonmeasurable under every translation invariant measure that
is an extension of Haar measure. Simply apply (b) and (c).
(e) Every infinite compact metric group G contains c pairwise disjoint
subsets that are not measurable with respect to Haar measure. Consider
the c sets X, constructed in (6.7), let # be the set function of (6.6),
and let denote the set function defined from the Haar integral as in
( .20). It is obvious from ( .22) and (6.6) that (X,) __>/z (X,) for all v.
By (t6.7), we have 2(X,): for all v. If any X,o were 2-measurable,
we would have 2 (X,,) 2(X,o) :0 for all VVo.
t5"
228 Chapter IV. Invariant functionals § 16. More about Haar measure :229
(f) Relatively simple nonmeasurable sets can be obtained by resorting
to large product spaces. Let /' be an uncountable index set, and for
each VCF, let G v be an infinite compact group. Let ,,v denote normalized
Haar measure on @ and let denote normalized Haar measure on
G= P @. Then is the product measure (t3.t8) of the measures )v
(t 5.t7.j). For each V c/', let A v be a proper <measurable subset of @
such that ;(Av)=t. Then A =vPrA is nonmeasurable. [Assume that
A is measurable. Let F be a compact subset of A, and for y/', let Ev
be the projection of F into @. Then E is a compact subset of A and
,(E,) <t. Thus (,PrE,)=0 (t3.22), and since Fc,PrE ,, we have
2(F)=0. Thus 2(A)=0 by (tt.32).
The Baire sets of G are defined in (tt.t). According to (t9.30.b)
in/ra [which is of course proved without recourse to the present discus-
sion, there is a Baire set B such that BDA and 2(B)=2(A)=0.
Since G is normal, the Baire sets in G are the smallest a-algebra of subsets
of G containing every open set that is the union of a countable family
of closed sets. Now consider the subsets D of G for which there is a
countable subset of F such that
(x)D, (yv)G, and xv=y for 7.F imply (y)cD.
This family of sets is a a-algebra containing every open subset of G
that is a union of a countable family of closed sets, since such a set is
a countable union of basic open sets. It follows that there is a countable
subset of F such that
(xv)B, (Yv) G, and xv=yv for V imply (Yv) B-
We infer from this and the inclusion B D A that B D vg.r'.AxvPro' @"
Theorem (t3.22) now implies that 2(B)=t, which is a contradiction.]
(16.14) Locally null sets. Every nondiscrete locally compact topo-
logical group G that is not a-compact contains a locally 2-null subset
that is not R-null; compare (tt.33). [By (5.7), G contains an open and
closed a-compact subgroup H. Plainly the quotient space G[H is un-
countable. Choose exactly one element from each left coset of H, and
let A denote the set of elements so chosen. The set A is locally 2-null
because the only compact subsets of A are finite and hence have 2-measure
zero. We now show that 2(A)= oo. If U is an open set containing A,
then ,((xH)NU)>0 for all left cosets xH of H. Since G/H is un-
countable, this implies that (U)--oo (tt.25). Hence by (.22), we
have
Notes
The Theorem (t6.3) of KAKUTANI and OXTOBY [1 is remarkable
in showing that Haar measure does not, so to say, tell the whole story
regarding invariant integration on compact metric groups. This theorem
opens the door to possible extensions of harmonic analysis; we will
return to this matter at an appropriate place in the sequel.
There are many other important and interesting facts about invariant
measures. Some of these are taken up in § § t 7 and t8. Both § t 7 and § t 8
deal with what may be called "finitely additive" situations. In §§t5
and t6, we have treated only countably additive invariant measures,
which are by far the most important invariant measures for harmonic
analysis in its present form. In these sections, we have by no means
exhausted the useful known facts about invariant measures 1. Limita-
tions of both space and time prevent us from treating them all.
Several topics are important enough for certain specialized applica-
tions to warrant at least mention. The first of these is A. WEII.'S con,
verse to the construction of Haar measure, proved in WEIL [4, pp. t 40---
t46. A slightly stronger version appears in HALOS [2, pp. 266--276.
The basic idea is simple. As is shown in (20.t7) in/ra, every Borel set A
of finite positive measure in a locally compact group G has the property
that A(A -) contains a neighborhood of e. Now let G be a group with
no topology but with a left invariant measure # on a left invariant
a-ring s' of subsets such that the mapping (x, y)-(x, xy) of GxG onto
itself preserves measurability for the product measure #x/, ,2 and such that
every A in s' is a-finite with respect to . Suppose also that for xG
and x 4=e, there is an A s' such that 0 </ (A) < oo and/ ((xA)/ A ) > O.
Then taking sets {xG:#((xA)/A)<e} as an open basis at e, one
obtains a topology on G under which it is a topological group. There
is a locally compact group G O in which G is a dense subgroup, and in
which Haar measure is closely related to/. For the details, we refer the
reader to HALMOS, loc. cir. For groups in which the a-ring ў' is a
a-algebra of a special sort, this result has been greatly sharpened by
IVACKEY [3" The details are rather technical and will not be repro-
duced here.
There is an extensive literature on the subject of (AB) for subsets
A and B of a locally compact group. See for example KEMPERMAN t,
I(NESER [t, MACBEATH t. Isolated but interesting theorems on Haar
measure appear in gWlERCZKOWSKI It and 2, and URBANIK t.
1 Our selection of topics for § § 6--18 was governed partly by sheer personal
preference.
2 Since G need not be locally compact and/ need not be as constructed in § 11,
we cannot define/x/ by the procedure of § 3. \Ve must use the abstract con-
Struction found, for example, in HALMOS [2], ch. VII.
230 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions 231
§ 17. Invariant means defined for all bounded functions
As is pointed out in Definition (t 5.2), the notion of invariance for a
linear functional defined on a space of real or complex functions on a
group need not be limited to the case of a locally compact group G
and the Haar integrals defined on 00(G). In this section, we take up
the interesting land so far only partially solved I problem of finding the
groups for which there is an invariant mean defined for all bounded
functions on the group.
Many of our previous definitions given for groups also have meaning
for semigroups. For example, if S is a semigroup, if a is in S, and
is a function on S with values in any nonvoid set, then a/and/a have
exactly the same meanings as in Definition (t 5.1). In the cases where
definitions are carried over verbatim from groups to semigroups, we will
usually not spell out the semigroup definitions in detail. A number of
the theorems of the present section can be stated and proved for semi-
groups just as easily as for groups. We shall make this extension where
it is possible.
We proceed to exact definitions.
(17.1) Definition. Let X be any nonvoid set. Let (X) denote the
set of all bounded complex-valued functions on X; '(X) and ч(X)
are defined as in (tt.4). For/(X), let II/llu:sup{I/(x)l:xX }.
Clearly (X) and ' (X) are complex and real linear spaces, respec-
tively, under pointwise addition and scalar multiplication. With the
norm n II-, (X) and '(X) are obviously complex and real Banach
spaces, respectively.
(17.2) Definition. Let X be any nonvoid set and let be a real
linear subspace of ' (X). A mean M on is a real linear functional on
having the property that
(i) inf {/ (x) : x X} <= M(/) _<__ sup {/ (x) : x X} for all/.
Note that if M is a real linear functional on and if contains the
constant functions, then property (i) holds if and only if
M(/)>_O whenever / and />0 (1)
and
M()=. (2)
(17.3) Definition. Let S be a semigroup and let be a linear
subspace of 3'(S) such that x S and / imply
[right invariant mean M on is a mean such that M(I)=M.(/)
[X(/,)=X(/) for all/ and x S. A mean X that satisfies M(,/)
M(/x)=M(/) for/ and xS is said to be a two-sided invariant mean
or simply an invariant mean 1.
1 Note the close connection with Definition (15.2).
(17.4) Theorem. Let S be a semigroup and let be a linear sub@ace
o/3"(S) such that x S and/ imply / . Let consist o/all/unctions
h having the/orm
/or some/1,...,/, and a ..... a, S. Then there exists a left invariant
mean/or i/ and only i/
(i) sup{h(x)'xS}>=O /or all hz.
Suppose/urther that x S and/ imply/. Let consist o/ all
/unctions h having the/orm
k=l i=l
/or some/1, ...,/,, gl, ..., gr and a .... , a n, b I .... , br S. The there
exists a two-sided invariant mean/or i/and only i/
(ii) sup{h(x):xS}>O /or all hЈ).
Proof. We will prove the second assertion; the proof of the first
one is the same. If M is a two-sided invariant mean for , then M(h)=0
for all h g and thus
up { (x): x s} > M() =0.
Suppose now that (ii) holds. Clearly Ј is a linear subspace of .
For h 5), we define
Mo (h) = O .
Then clearly we have
i o () = 0 =< ( (x):x S
for h,. Letting p(/)--sup(/(x):x S) for/ and applying (B.13), we
see that M o can be extended to a linear functional M on satisfying
M(/) p {I (): x s}
for/ . Also we have
-- M(/) = M(--I) < sup {-- /(x): x S}=-- inf {1 (x): x S}
so that
inf {/(x): xc S} =_< M(/).
Finally if aS and /, then /--/ and /--/ belong to so that
M (/) = ў (a/) = X (la) .
(17.5) Theorem. Let S be it commutative semigroup. Then there is
an invariant mean M on '(S). [In particular, every linear sub@ace
o/ qd" (S) such that / whenever x S and/ has an invariant ,mean.
232 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions 233
Proof. Let/1 ..... / be elements of 93' (S), let a ..... a be elements
of S, and let
/=I.
By virtue of (t7.4), it suffices to prove that sup{h(x)'xS}>=O. Assume
then that for some e> 0, we have
sup {Ca (x)'x< S}=-- e. (I)
Let p be any positive integer. Let A consist of all functions 2 with
domain {t ..... n} and range contained in {t, ..., p}. Evidently A con-
tains exactly pn elements. Let r be the mapping of A into S defined by
7(2)--a (1/a (2/... a (n). For a fixed k, k=t .... , n, we wish to estimate
the sum
22 (2)
It is easy to see that all of the terms in (2) cancel each other except
possibly those/k (7(2)) such that 2(k) =1 and those ](a k z(2)) such that
; (k)= p. The number of these terms is 2p *. Therefore
Applying (t), we have
-,p"> Y, )2
EA EA /=1
/=1 AEA
>= -- , 2 p,-t A = -- 2 n p"- A,
/=1
where A -- max {I/ ]l'k-- t ..... ,,}. Consequently we have ep <= 2hA.
Since p can be chosen arbitrarily, this leads to a contradiction. Hence (t)
cannot hold and there is an invariant mean on
We now state an elementary but useful theorem.
(17.6) Theorem. Let S be a semigroup and let be a linear sub@ace
of " (S). Suppose that - is a family of subsemigroups o] S such that
(i) i/T 1 , Te -, then there is a T q- such that T TI U Te;
(ii) U{T" T-}=S.
For each T -, let r be the linear space o/all/unctions in with their
domains o/definition restricted to T. I/J whenever x S and ] and
i/ there exists a left invariant mean on each r, then there exists a left
invariaut mean on 3. If also f whenever x S and f and if there
exists a two-sided invariant mean on each r, then there exists a two-sided
invariat mean on .
Proof. We prove the first assertion' the proof of the second is
similar. Let/1 ..... /,, , a ..... a,, S, and consider h= , I/k--/. For
some To-, we have a 1 ..... aT o. By (t7.4) applied to r0, we see
that sup{h(x)'xTo}>_O. Then obviously sup{h(x)'x S}=>0. By (t7.4)
again, we see that there is a left invariant mean on
Suppose that S is a topological 9emigroup, i.e., a semigroup that is
a topological space in which the mapping (x, y)-->xy is continuous on
S x S into S. Then " (S) is plainly a translation invariant linear subspace
of '(S), so that (t7.4)--(t7.6) hold for --'(S).
We now list three corollaries of (t 7.6).
(17.7) Corollary. Let S be a semigroup. There exists a [left, right,
two-sided] invariant mean on 3" (S) i/ there exists a [left, right, two-sided]
invariant mean on " (T) /or all/initely gef, erated subsemigroups T o/S.
(17.8) Corollary. Let G be a group. There exists a two-sided invariant
mea oft " (G) i/there exists a two-sided invariant mean on " (H) /or all
/iuitdy generated subgroups H o! G.
(17.9) Corollary. Let G be a group such that every/inite subset o/G
geuerates a/iuite group. Theft " (G) admits a two-sided invariant mean.
Corollary (t 7.9) follows from the fact that if H is a finite group, then
' (H) has a two-sided invariant mean, namely the Haar integral.
(17.10) Theorem. Let S be a semigroup and suppose that there exist
lc/ dud right invariant means /or 93" (S). Then there exists a two-sided
iuvariat mean for ,r (S).
Proof. Let M and M' be left and right invariant means, respectively,
for '(S). For /r(S) and xS, let /'(x)=M'(J)" clearly /''(S).
Now let
Mo (I) = M (/') .
Since the mapping /-+/' is linear on 93"(S), we see that M 0 is linear.
Moreover M0(t) =t ' and Mo(/)>_O whenever/:>0. For a, xS, we have
(/)' (x): M' ((/)) = M'(,J)=/'(ax) --(/') (x). Therefore
Mo (1) = M( (al)') = M(a(l') ) = M(I') -- Mo (1).
For a, x S, we have (la)'(x)--M'(x(la)) =M'((xl)a) =M'(,cl)=l'(x). It
follows that
Mo (la) = M( (la)') = M(I') = M o (1).
Consequently M 0 is a two-sided invariant mean for
\Ve omit hcrc the result for left and right invariant means, because of (17.11)
[q.v.].
234 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions :235
(17.11) Theorem. Let G be a group and suppose that there exists a
le/t invariant mean M/or F (G). Then there exists a two-sided invariant
mean/or " (G).
Proof. As in Definition (t5.t), [* denotes the function on G such
that ]*(x)=/(x-1). For/q,F(G), let M' (/) -- M (/*). Then M' is a right
invariant mean for '(G); now apply (t7.t0).
(17.12) Theorem. Let G be a group and let H be a subgroup o/ G.
I/3" (G) admits a two-sided invariant memў M, then so does " (H).
Proof. For each right coset Hx of H, let 7(Hx) be an arbitrary but
fixed element of Hx. Every element x of G then has a unique representa-
tion x--hxT(Hx), where hxH. For/'(H) and xG, we define
Clearly 1--+]' is a linear mapping of r(H) into r(G). For /,'(H),
let M o (/) = M(/') ; obviously M 0 is linear, M o (t) = t, and M o (/) => 0 when-
ever/>---0. We will prove that M o is left invariant on ,'(H). For xG
and aH, we have ax=ah,-c(Hx) and also ax=h,-c(Hax)=hx'c(Hx).
Thus ah,=h,,, and for any/' (H), the equalities
(al)' (X) =al N) = l = l = f (aX) =a(f) (X)
hold. Thus we have
(al) = (al)') = ) = M(I') = (1).
Since M o is a left invariant mean for '(H), Theorem (t7.tt) shows
that there is a two-sided invariant mean for '(H). V
(17.13) Corollary. Let G be a group. Then F(G) admits a two-sided
invariant mean i/ and only i/ (H) admits a two-sided invariant mean
/or all/initely generated subgroups H o/G.
Proof. This follows from (t7.8) and (7.2). ]
(17.14) Theorem. Let G be a group and let H be a normal subgroup
o/ G. Then " (G) admits an invariaut mean i/ and only i/ both " (H) and
" (G/H) admit invariant neans 1.
Proof. Suppose that M is an invariant mean for ,F(G). By (t7.t2),
' (H) admits an invariant mean. Let denote the natural mapping of
G onto G/H. For /'(G/H), let Mo(/)=M(/o q)). It is easy to show
that M 0 is linear on ' (G/H), that M0(l ) -- 1, and that M 0 (/) > 0 whenever
/=>0. For/'(G/H) and x, yG, we see that ((,u/)oq))(y)--/(xyH)
1 Note the analogy between this theorem and Theorems (5.25) and (7.14).
,(/o q)(y) and consequently
M o (, ,1) = M ((,,/) o qg) = M (x(/o )) = M(/o qg) = M o (/).
Similarly Mo is right invariant on ' (G/H).
Next suppose that M and M' are invariant means for '(H) and
"(G/H), respectively. For/!3"(G) and xG, let/(x)=M(,/)" here the
domain of J is restricted to H. We first prove that if x and y are in the
same coset of H, then ](x)=J(y). Thus suppose that x--yh o where
hoH. For hH, we have J(h)=/(xh)=/(yhoh)=o(,/)(h ). Therefore
(x)=M(,/)=M(°(/)) =M(/)=](y). Since is constant on the cosets_
of H, we may define the function/' on G/H by the rule/'(xH)--/(x).
Finally, we set Mo(/)=M'(/' ) for /'(G). The linearity of M 0, the
equality M0(l)=t, and the nonnegativity of M 0 are obvious. For
a, xG and/'(G), we obtain
(al)' ( x H) = M (x(al) ) = M (axl) = I'
which implies that
Mo (a/) = M' ( (a/)') = M' (/') = M o (I).
Thus M o is left invariant and, by (17.t t), ,' (G) has a two-sided invariant
mean.
(17.15) Theorem. Let S be a semigroup and let consist o] all
h 3" (S) having the/orm
/or/1,...,/, " (S) ad a,..., a S. The he /ollowig salemes are
equivalent"
(i) there is a le/t invariant mean M on " (S);
(ii) inf{][t-- h[l,'h
I] S satis/ies the left cancellation law xy=xz implies y=z], then (i)
and (ii) are equivalent to
(iii) the uniJorm closure o] is not equal to " (S).
Proof. Suppose that (i) holds. Then by (17.4), we have
sup{h(x):xS}>=O for all hg This implies that ]lt-h][>t for all
hg, and hence that inf{ll-ll.:hs)z}>. Since Ila we infer
that (ii) holds.
Suppose now that inf {lla - h • h a. Then by (B. t 5), there is a
linear functional M on N'(S) such that M(t)=t, [IM][= t, and M(h) =0
for all h g. Recall that [IMII--sup{IM(/)I ll/ll --a and/' (S)}. Since
M(g) =0, M is left invariant. To show that M is a mean, we need only
show that />=0 implies M(/)>=O. Assume that />=0 and M(/)<0. If
236 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions 237
=11111.-I, then Ilell.<__lllll. and I (e)I---IIIII.- (I)>IIIII.>_=IIeL. This
implies that IIMII> contrary to the choice of M. Therefore M is a
left invariant mean on 23' (S).
Obviously (ii) implies (iii). Suppose finally that (iii) holds and that S
satisfies the left cancellation law. Choose a function/0 in ' (S)71 (g-)'.
By (B.I 5), there is a bounded linear functional L on 23'(S) such that
L(/0)=l and L(5)-)=0. By (B.37), L has the form L+--L_, where
L+ = max (L, 0), and L+ and L_ are nonnegative linear functionals. Note
that by definition,
(1)= 0__<g__< 1}
for nonnegative/. We now prove that L+ is left invariant; we already
have L(x/)=L(/) for all / since L(g&)=0. Suppose that/>0 and xES.
Then g<=/implies xg=< x/. Moreover, if h_<_ /, then h=g for some gEm'(S)
such that O<---g</. [To see this, note that every y xS has the form xs,
where s is unique by the left cancellation law. Then let g(y)=h(s) for
y=xs and let g(y)=0 for ySf(xS)'.] It follows from (t) that
L+ (,I)=L+ (1). Since every function in '(S) is the difference of two
nonnegative functions, we have L+ (/)=L+ (/) for all/'(S). Since L
and L+ are left invariant, so is L_. We have either L+(/0)=t =0 or
L_(Io) 4=0, say L+ (lo)4 =0. Since (lo)l =< +(IIIoIL)-II1o L (a), ,ve have
L+(I)
L+ (t)>0. Defining M(/)- L.(1)' we obtain a left invariant mean for
We now prove that for some groups G, ' (G) admits no left invariant
mean.
(17.16) Theorem. Let G be a group containing a/ree subgroup having
two generators. Then there is no le/t invariant mean on " (G).
Proof. In view of (t7.tt) and (17.t2), it suffices to show that 3'(F)
admits no left invariant mean, where F is the free group having two
generators a and b. Each element of F has a unique representation as a
reduced word (2.8). Let A be the set of elements of F that begin with
a or a -1 when written as reduced words, and let /(x)=t for xA and
/(x)=0 for xcA. By (t7.4), it suffices to show that the function
= + ((- 1) - I))
has the property that sup {h (x): x F} < 0.
We have
(x) = I x) + / x) -- 1 -- I (x),
for xF. For all xF, either x or ax belongs to A. Therefore h(x)
except possibly in the cases that b a -ix or b -la - x belongs to A. For one
of these situations to hold, x must have the form a b - ay or a b ay where
s t and a * y is reduced. In each of these cases, both x and ax belong
to A whereas only one of the elements b a-lx and b-ta-tx belongs to A.
Thus in these cases we have h(x) =-- 1. Hence sup {h (x) : x F} =< -- t .
Miscellaneous theorems and examples
(17.17) (ROBISON [t].) Theorem (t7.4) can be generalized as follows.
Let X be any nonvoid set, any linear subspace of ' (X) containing
some function /=4=0, and " any family of linear transformations of
into . A necessary and sufficient condition that there exist a nonnegative
linear functional M on satisfying
(i) M(T/)=M(]) for each TE" and i,
and
(ii) inf{/(x):xX}<=M(/)<=sup{/(x):xX},
is that for every finite set of pairs (], T), ], TkEgT"(k=t, ..., n), we
have
(iii) sup t, (/-- T/)(x)"xX1 >--O.
k=l
(17.18) Invariant means for@(S). (a) Let S and T be semi-
groups and let q0 be a homomorphism of S onto T. If there is a left
[two-sided] invariant mean for ' (S), there is also a left [two-sided]
invariant mean for '(T). [If M is the mean for '(S), and/'(r),
define M o (/) = M(/o qo) and show that M 0 is the desired mean for ' (T).]
(b) Let S and T be semigroups and suppose that M and M' are left
[two-sided] invariant means for ' (S) and ' (T), respectively. Then
3' (Sx T) also admits a left [two-sided] invariant mean . [For / '(SxT),
let
Mo (l) = Ms (M' (l (s, t)))
where Ms IMp' ] indicates that the variable is s Site T]. It is evident that
M 0 is well defined, linear, and nonnegative. Also we have M0(t)=t.
Suppose that M and M' are left invariant. Then if (a, b) Sx T, we have
Mo(ў,,o)l) = M(M/ (/(as, bt))) = M(M' (l(s, t))) -= M(I)
by applying the left invariance of M' and then that of M.]
(c) Suppose that {S,},t is a family of semigroups each having an
identity e, and let S= P* S, If each ' (S,) admits a left [two-sided]
lI "
invariant mean, then so does ' (S). Each S, may be regarded as a
1 The definition of the direct product of a family of semigroups is exactly like
that of groups (2.3). If we choose a fixed element in each of a family of semigroups
[the identity if there is one], then the weak direct product of semigroups can also
be defined by analogy with (2.3).
238 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions 239
subsemigroup of S in a natural way. In fact, x c S,o corresponds to the
element s c S, where
x if *--*0,
St --
e, if ,4=0.
By (b), all finite products ,°S, have the property that ',,EI0 ( P S,) admits
an invariant mean. If we let 3- consist of all such ,oS,, then (t7.6)
shows that ' (S) admits an invariant mean.]
(d) The result of (c) does not necessarily hold if we take S to be
P'St The analogous statement for groups does not
P S rather than E z "
even hold. [Let F be the free group generated by a and b with identity e,
and let F0=FA{e}'. Let x=x'x .... x" be any reduced word in F0;
each x is a or b and each e is t or -- t. By (2.9), there exist an integer l
and elements P and P in the permutation group ® such that
P2:o P:o... o P2; is different from the identity permutation. We define
Gx as ®, ax as P,, and b as Pb. Having made these definitions for each
x F0 we define G to be P G. Since each G is finite, ' (G) admits a
' xEo
two-sided invariant mean for all x F 0. The elements (ax) and (b) of
generate a free group. That is, G contains a free group on two generators.
Thus by (t7.16), '(G) does not admit a left invariant mean. Note the
analogy between this result and Theorem (8.8).]
(e) Let S be any semigroup whatever; ' (S) may or may not admit
invariant means. Let S* = S U {0} where s 0-- 0 s = 0 for all s S*. Then
' (S*) admits a two-sided invariant mean. [For ]' (S*), let
(f) (DAY [5].) In view of (e), the following result is interesting. Let S
be a semigroup and let M be a left invariant mean for ' (S). Suppose that
T is a subsemigroup of S such that M(r)>0, where r is as usual the
characteristic function of T. Then ' (T) admits a left invariant mean.
[For /c'(r), let /''(S) be defined by /'(x)=/(x) for xT and
M(Z') Then M 0 is a mean on ' (T).
/' (x)=0 for x T', and let M0(/)-- M(r) "
To show that M 0 is left invariant, let x T, [C'(T), and consider
g=(,j)'--,(]'). It is easy to see that g=ge E where E= {y S'yў T and
xy T}. For any s S, one can show easily that at most one of the elements
{x k s}_-i belongs to E. Hence for every positive integer n, we have
.k(E)(S)__<--t for all sS.
ThusnM(v.)=M(;,())<=M(t)<oo, so that M() =0. Now we have
IM( )I- <M(llgll and so M(g):0. We have thus shown
that M((,/)')--M(,(/')) for all xC T and [' (r). This implies immediately
that Mo is left invariant.]
(17.19) Invariant rneaas for !B (G). (a) The two-sided invariant
mean for ' (G) in (t 7.t t) can be taken to be inversion invariant. ILet
M be the two-sided invariant mean and for /'(G), define M0(/)=
M(I*)].]
(b) Let G be the group generated by a and b and satisfying the
relations a--b=e and no others. Then a---a, b---b, and G= {e, a, ab,
aba, ..., b, ba, bab, ...}. Let H consist of all of the elements of even
length" that is, H= {e, ab, abab, ..., ba, baba, ...}. Then H is an infinite
cyclic normal subgroup of G and G/H consists of two elements. Thus
'(H) and '(G/H) admit invariant means. By (t7.t4), '(G) also
admits invariant means.
There is a left invariant mean on ' (G) that is not right invariant.
ILet A consist of all elements in G ending in a" that is, A = {a, ba, aba,
baba, ...}. For p<q, let A ff''q) be the set of elements xA such that
p=< length x=<q. Consider an arbitrary element y of G whose length is
ra and an xA whose length is f>ra. Then yx is also in A and it is
either the element of length f--m or the element of length f+ m. Thus
if p and q satisfy ra< p andp+ 2ra< q, we see that y-IA (+''q-') cA ('q),
so that yA I'ql contains the set A (+','q-rl. Thus if /C3'(G) and we
consider the sum I/(x)--/(yx), we see that all except at most 4m
xEA (p, q)
terms cancel. It follows that
E [l(x)-l(yx)] <=4 lllll,,. (t)
xA(P, q)
Consider h--- , [/k--y:/k] where /1, ..., /, t () and YI .... , Yn G.
We assert that =
inf{h(x)'xA}<_O. (2)
Assume that inf{h(x)'xA}-->O. Let m 0 be the maximum of the
lengths of Yl .... , y, and let B-max{ll/lll ..... II/ ll }. Choose p and q
so that rao< p and p+ 2m0< q. Now apply (t)" this yields the relations
2 F,
xE A(P,q)
-- , 2 [/ (x) -- 1 (y x)] 4 mo B = 4 n m o B.
k=l xEA(P,q) k=l
Since q_p can be taken arbitrarily large, we have a contradiction. Thus
(2) holds.
The function --(), is clearly equal to t on the set A. Hence
by (2), it is not in the uniform closure of all functions of the form
h [/--y/3. An argument like that used in proving that (7.t 5.iii)
implies (7. 5.i) nmv shows that there is a left invariant mean M on
240 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions 241
3,(G) such that M((A)--(A)a)4 =0. This mean is obviously not right
invariant.]
(c) Let G be a noncompact locally compact topological group and
suppose that M is a mean for '(G). Then M(/)--0 for all functions
that are arbitrarily small outside of compact sets. [As in (t 5.9), there is
an open set V in G and a sequence x 1 .... , x .... in G such that V- is
compact and {xV}__l is a pairwise disjoint sequence of sets. Then
xkv= xil(v)=<t for all n so that nM(v)M(t ). Thus M(v) =0.
k =1
Now suppose that I/(x)t < for xcF', where F is compact. Then for some
yt .... , ym C G we have iU yi V D F. Thus we have
and hence IM(I)! <M(,)=,.]
(17.20) Invariant means for r(R). (a) Let M be an invariant
mean for ' (R). Suppose that the limits lim ] (t) and lim /(t) exist.
t-+oo
Then the number M(]) lies between these two limits. In particular, if
the above limits are equal, then M(/) is equal to the common limit.
[Let g(x)=,m/(t) for x=>0 and g(x)= lim /(t)for x<0. Then
lim (g (t) -- J (t) ) = lim (g (t) -- J (t) ) = O, and so (t7.t9.c) implies that
M(g--/)=0. Since M(/)=M(g), it is clear that M(/) lies between
lim /(t) and lim ] (t).
(b) Theorems (t7.5) and (t7.4) show us that sup{h(x):xR}>=O for
all h= [/-- ak[ where/k ' (R) and a R (k = t, ..., n). By modifying
the proof of (t7.5), one can prove that sup{h(x):xN}>=O for any
positive number N. [We may suppose that each a is positive; otherwise
replace ]--k[ by (--/)--_(--k/). Let p be any positive integer
and let A consist of all functions 2 with domain {t, ..., n} and range
contained in {t .... , p}. Let z be the mapping of A into IN, o[ defined
by ().)=N+ ,).(k)a. Assume that sup{h(x)'xN)=.<O. Using
this and taking into account the terms that cancel, we see that
> 22
EA
-- 2 p" max {IIA II. .... , II1.lid = - 2 p-* max {IIA II. ..... III: liD.
k=l
Since p is arbitrary, the inequality e< 0 is impossible.
(c) There is an invariant mean M for ' (R) such that M(/) = lim ](t)
whenever this limit exists. [Let g0 consist of those functions h in 3'(R)
of the form
= +
where g, 1 .... , i, " (R), a 1 ..... a,, R, and lim g (t) -- 0. Clearly 5)0 is a
linear subspace of '(R). Let h be as in (t) and choose N so large that
By virtue of (b), we have
[1,, (t) - A + => m => o.
Combining (t), (2), and (3), we have sup {h (t) " t >= N} >= -- Ѕ. Thus
]lh--(--t) ]I,>= Ѕ- for all hE 5)0. As in (t 7.t 5), there is a mean M on N' (R)
such that M(0)--0. Then clearly M is invariant. Also if lim/(t) =e,
then i--g)0, so that M(i)=M(i--)+M()--.
(d) Part (c) remains valid if R is replaced by any additive subsemi-
group S of R having the property that x S implies Ixl S. In particular,
it is valid if S isZ, Q, {xcR:xN}, {xZ:xN}, or {xQ:xN} where
N=> 0. [Repeat the proofs of (b) and (c) verbatim, using S instead of R.
(17.21) Llniqueness of means (DaY [51, §7). If G is a finite group,
it is trivial that ' (G) has exactly one invariant mean. We will show in
(c) that fr infinite Abelian groups, 3' (G) always admits more than one
invariant mean.
(a) Let G be a group such that ' (G) admits an invariant mean, and
suppose that H is a normal subgroup of G such that ' (H) admits more
than one left invariant mean. Then '(G) admits more than one left
invariant mean. [By (17.t4) and (t7.tt), '(G/H) has an invariant mean
M'. Let M and M be distinct left invariant means for ' (H)" we
construct M0 (t/ and M0 (1 from M and M' and M and M', respectively,
exactly as in the last paragraph of the proof of (t 7. t 4), using the notation
of this paragraph with obvious modifications. We will show that
M0 (14 = M0 (1. Choose g ' (H) such that M (g) 4= M (g). Write the distinct
elements of G/H as {xH}A , where the x are fixed elements of G.
Thus each y G can be uniquely written as xh for e c A and h H" define
]'(G)_ by ] (y) -- g (h). For all eA, we have /-(x)=Ml(,/)=Ml(g )
and /(x)=M(g). Hence /; ( xH) -- M (g) and /'(xH)--M.2(g) for all
eA. It follows that Mo (1 (/)=M'(M(g)) =M (g) and M0 ('1 (/)=M.(g). 1
(b) Suppose that G is a group and that G= U G, where {G,,}°=I is a
strictly increasing sequence of finite subgroups of G. Then ' (G) admits
more than one left invariant mean. [Let k denote the number of
elements in G, n= 1, 2 ..... By taking a subsequence if necessary, we
may suppose that k>= 3 k_ for n--2, 3 ..... For i C' (G), let p (i)
li-
,-oo ] (x). Let N be the closed linear subspace of " (G) consisting
xEGn
Hewitt and Ross, Abstract harmonic analysis, vol. I 16
24:2 Chapter IV. Invariant functionals § 17. Invariant means defined for all bounded functions :243
of all functions [ such that
lim [ -- an/(x) ] = O.
Consider / in ' (G) and a in G. Then aUG. for some m. Clearly we have
a G.-- G.for n > m, and therefore
&- Ed(x)-l()]=o
for n m. It follows that contains all linear combinations of functions
of the form J--].
Let ]0 be defined on G by ]o(X)= if xGU(GG)U...
(G,+ G,) ... and ]0 (x)=0 otherwise. Since
p(/o)=li m 1 t
. 10 (x) > 0 () >
l0 does not belong to . Note also that
-- P (--/0) = lim - ]0 (x) < lim ]0 (x) < 2
-- n 2n 3 "
Let M(/)=0 for 1 . It is easy to show that
sp {-- p (-- 1-- 1o)--M(I)'I } ---- P (-- 1o)
and
{P (1 + 1o) - (1) 1 } = g (1o) .
Theorems (B.I 2) and (B.I 3) show that for any real number a such that
Ѕ=<=<§, there is a linear functional M s on !3'(G) with the following
properties" M s ()--0; M s (/0)-- a; andM (l)=<P (l) for all lc!3' (G). Since
-p(-l)<=M(l)<=p(l) for all Ice'(G), it follows that M(t)=t and
that M s is nonnegative. Since M s (l)--0 for all l , M is also left in-
variant. Therefore ' (G) admits at least c distinct left invariant means.]
(c) Let G be an infinite Abelian group. Then ' (G) admits more than
one invariant mean. Elf G contains an element of infinite order, then G
contains a subgroup H isomorphic with Z. By (t7.20.c) and (t7.20.d),
there are distinct invariant means for ' (Z); such means M 1 and Ms
may be chosen so that M1 (l) -- lira l (n) and M. (l) -- lim / (-- n) whenever
these limits exist. Now apply (a). If all elements of G have finite order,
there is a strictly increasing sequence {H,)n°°= 1 of finite subgroups of G.
Let H-- U H and apply (b) and (a).]
(17.22) Invariant finitely additive measures. (a) Let G be any
group such that ' (G) admits an invariant mean M Ewhich in view of
(t 7.t t) may be taken to be two-sided invariant]. For any subset A of G,
define v(A)--M(A). Then v is a finitely additive two-sided invariant
measure.
(b) For any subset A of R, let v (A)=M(A) where M is an invariant
mean for J' (R) as in (17.20.c). Then v is a finitely additive set function
satisfying
(i) O<=v(A)<_ t for all A
(ii) v (A) = 1 if , o[ C A for some R;
(iii) v(A)=0 if A is bounded above;
(iv) v (A + a) --v (A ) for all A cR and aR.
(c) (Adapted from YON NEUANN 21.) Let G be a locally compact
group such that ' (G) admits a two-sided invariant mean M. Let
denote left Haar measure on G. Then there is a finitely additive set
function if, defined for all subsets of G, having the following properties:
(i) 0_--< ff (A) + o for all A C G;
(ii) #(xA)=#(A) for all xG and A
(iii) # (A)--2 (A) for all Haar measurable subsets A of G.
That is, we obtain a finitely additive measure defined for all subsets of
G that is invariant under left translations, and which agrees with 2 on
measurable subsets of G.
IThe proof is carried out in two steps.
(I) There is a finitely additive set function v, defined for all subsets
of G, satisfying (i) and (iii) .
This assertion depends upon the axiom of choice. Let g be the
family of all Haar measurable subsets of G. Since the theorem is
trivial when ', consists of all subsets of G, we suppose that g is
different from (G), the family of all subsets of G. Well order all of the
subsets of G, putting all of the sets in g first. Thus the family (G)
is written as
A 1, A .... , A, ...,
where A is [say the smallest ordinal with corresponding cardinal 2,
and AC' for e</; here/<A. Our purpose is to define v in such a
way that if a 1 .... , a,, b .... , b, are positive real numbers and e ..... e,,,,
.... , , are ordinals (%.< A, fl< A), then
m
Eai,<EbA implies Eaiv(A,)<__Y,bv(A&). (t)
= = = =
Condition (t) implies that is finitely additive" in fact, if A B=,
then + e =ue. For e<F, let (A) = 2 (A). Let us verify (t) in
the
case
that
< F and fl < F, /= t .... , m, k = t .... , n. Integrating
For a different proof of this fact, see IRKHOFF [2], pp. 185--186.
16"
244 Chapter IV. Invariant functionals § 18. Invariant means on almost periodic functions 245
the left inequality of (t) with respect to 2 gives
We now extend the definition of v by transfinite induction. Suppose
that v (As) has been defined for all e< in such a way that (t) holds,
where F<=]< A. If
inf
h=l
we define v(An)=+ . Then it is simple to verify (t) for all ordinals
ei and fl less than or equal to . Suppose now that inf b v(A)"
b n} < + " Then there exist positive real numbers a i and ba
and ordinals ei and fl less than such that
=1
b v (Aa)< . (3)
-- aiv(A,)'(2)and (3)hold. To
We define
show that (t) holds for all ordinals e, e G, it suffices to consider the
following two cases"
ai + Gb &a,- (4)
i=I
i=1 =1
If (4) holds, then clearly
Ј a i v (A:j) + v (A n <= b v (A&).
i=i =i
Suppose finally that (5) holds. Consider any a i, b, ei, and fl for which
(2) and (3) hold. Adding (2) and (5) and applying (t) and the induction
hypothesis, we obtain
Z (A) + E (A) E b (X) + E b; (X;).
i=1 i=i =1 =1
Taking the infimum over all a i, b, ei, and fl satisfying (2) and (3),
' 0 ' t
we get a i v (A;) G v (An) + b , (A;).
i=1 =1
This completes the proof of the existence of v.
(II) We now use the measure v of (I) and the mean M to construct
a finitely additive measure # satisfying (i) -- (iii). For each subset A
of G, we define a function , on G by (x)=v(xA). The function is
nonnegative, but may be unbounded and may also assume the value
ч o. Now let ,u (A) -- lim M (min (A, n)). Conditions (i) and (iii) are
obvious. To show that # is additive, let A and B be disjoint subsets of G.
Then clearly A U B = A + B. The inequalities
min ( + , n) =< min (, n) + min (/, u) _G< min (5 +/, 2 n)
then imply that # (A U B) = ,irn M (min ( +/, n)) = lim M (min (, n))
+ lim M(min(B, n)) =# (A) +# (B). To verify (ii), let xcG and A cG.
Then we have xA = (A),, and therefore
#(xA)- lim M(min(x, n)) -- lim M(min((,) n))
-- lim M((min (, n)),) = lim M(min(X, n)) =#(A).]
Notes
The theory of invariant means on (G) was founded by VON NEU-
5ANN 21, who established for groups the main results of the present
section. Invariant means have exerted a strange fascination for mathe-
maticians in recent years, and a startling number of writers on the subject
have published previously published results. We make no effort to trace
all of these repetitions.
Theorem (t7.4) is due to DIXMIER [t. Theorem (t7.5) for groups
appears in VON NEUMANN [2], and for semigroups in DAY [2]. Our
proof of (t7.5) is taken from DIXMIER [t]. DAY proved (17.6), (t7.7),
and (t7.9) 4], as well as (t7.tt) in slightly disguised form [41, Lemma 7.
Theorem (t7.t2) was announced without proof by DAY [31" the first
published proof was given by FOLNER [3J. For left invariant means,
(t7.t4) was proved by VON NEUMA'N [2, and for invariant means by
Day [41, Theorem 6. Theorems similar to (t7.t 5) appear in DAY [41,
FOLNER [4], and RAIMI It]. Theorem (17.16)is dueto VON NEUMANN [2].
The reader wishing to pursue the subject further should consult"
DAy [51 for a useful survey" FOLNER 3 and KESTEN Ill for other
conditions under which invariant means exist; LORENTZ 1 and Lu-
TIAR [t for uniqueness questions" SILVERMAN t], 2], [3-
§ 18. Invariant means on almost periodic functions
In § t 7, we obtained invariant means on (G) and (G) in various
cases where the HAHN-BANACH theorem can be applied. In the present
section, we prove the existence of invariant means in other function
246 Chapter IV. Invariant functionals § 18. Invariant means on almost periodic functions :247
classes by the use of compactness arguments. This seems altogether
fitting, since compactness and well ordering are two of the most powerful
concepts used for existence proofs in analysis. As we shall show in
Vol. II, some of the results presented here can be subsumed under the
construction of Haar measure on compact groups. Others cannot, and
in any case it seems worth while to give a completely elementary con-
struction, as we will do.
We begin with a description of the class of almost periodic functions
on a group G. For a complex function / on G and acG, let D/be the
function on GxG=G 2 such that DJ(x, y)=J(xay).
(18.1) Theorem. Let G be a group and let ] be a Junction in
Let- denote closure in the uni/orm topology ]or (G). The [ollowing
properties o] ] are equivalent"
(i) {]'acG}- is compact in (G);
(it) {[" aG}- is compact in 3(G) ;
(iii) {b]'a, bG}- is compact in (G);
(iv) {D]" aG}- is compact in (G2).
Proof. The space (G) under the metric II]-gll, is a complete metric
space; a subset 9I of (G) is thus compact if and only if it is closed and
totally bounded, and 9I has compact closure if and only if it is totally
bounded /see (3.7). Hence we have only to prove the equivalence of
total boundedness for the sets {]a" aG}, {" ac G}, {]" a, bG}, {Da]" aG}.
It is obvious that (iii) implies (i), (iii) implies (it), (iv) implies (i),
and (iv) implies (it). We will prove that (i) implies (iv), (it) implies (iv),
(i) implies (iii), and (it) implies (iii). This will complete the proof.
Suppose, then, that {]'aG} is totally bounded. Given e>O, there
is a finite subset {al, a,..., %} of G such that {/,}=1 is an e/4-mesh in
{['aG}. 1 For '--t, 2, ..., m, let A-{aG'll/-/,lI,<e/4 }. Consider
the family of all sets (A,arl)A(A,a[I)N...A(A,,a), where /.C{t,2,
.... m} for '=1, 2, ..., m. Write the nonvoid sets in this family as
BI, B.,..., B,, and choose bk B for k= t, 2, ..., n. It is obvious that
U Bk=G. Now consider any cG; let B, be a set B such that c B0.
Let (x, y) be an arbitrary point in G; select '0 (/0=t, 2 .... , m) such
that yAio. Then we have
l/(xcy)-/(xb0y)l < l/(xcy)-/(xc..)[ + ]/(xc.)-/(Xb,oo) I } ()
The first and third summands on the right side of (t) are less than
because y CAio. The second summand is less than e/2 because c and bk0
For the definition of -mesh, see (3.7).
are both in Aoao for sonde l.0. Since (x, y) is arbitrary, the functions
Do[, Dv/ .... , Dv/ form an e-mesh in {D/:cG}. Thus (iv) holds.
The proof that (it) implies (iv) is very like the proof just given.
Without writing it out, therefore, we may assert that (i), (it), and (iv)
are equivalent.
Finally, suppose that (i) holds. Then (it) holds also. Let {/}.__ and
{/}= be e/2-meshes in {/:aG} and {/:bG}, respectively. Then for
all x, a, b in G, we have
l! (bx=)- ! (b x =;) l I! (Z)x =)-- I (Z) x =)I + [! (Z) x =)- I (b,cx I
for some k and '. That is, {l,}il, : is an e-mesh in {l" a, bG}.
(18.2) Definition. Let G be a group. A function / in (G) satisfying
one [and hence all of the conditions (t 8.t.i)--(t 8.t.iv) is said to be
almost periodic. The set of all almost periodic functions on G is denoted
by N(G). For a topological group G, the set of all continuous functions
in N(G)is denoted by (G).
(18.3) Theorem. Let G be a group.
(i) Every constant/unction is in N(G).
(it) I//is in (G), then also Re/, Im/, and are in (G).
(iii) I//, g are in N(G), then ]+ g and/g are iў N(G).
(iv) I//(,/(,/( .... are in N(G) and 2i ]]/(--/]--0, then/N(G).
(v) I//is in N(G) and a, b are in G, then /, ], and / are in [(G).
Similar assertions hold/or (G).
Proof. Assertions (i), (it), and (v) are obvious. To prove (iii), con-
sider first / + g. Let e be any positive number" let {/}iQ be an e/4-mesh
for {].aG} and {gv}=t be an e/4-mesh for {gvbG}. Let Ai--
{aG']/--/,<e/4} (=t,2,...,m) and B--{bG"
(kt, 2 .... , n). Let thenonvoid sets AiB be written as C, C2, ..., Cf,
and choose cCz (/=, 2, ..., p). Then it is easy to see that
is an e-mesh for {(/+g)'cG}. To show that /gN(G), suppose that
/>0 and [g],,>0, and form an }]-mesh for {/'aG} and an
)--mesh for {g'bG}. Then, as above, the sets Cz and points czC
yield an e-mesh for {(/g)-c G}.
To prove (iv), note that if {/ .... ,/} is an e/3-mesh for {]" a(G},
and l/--/()[< e/3, then {/, ...,/} is an e-mesh for {/" aG}.
We will show that (G) admits a unique, strictly positive, two-sided
invariant mean. We first prove a combinatorial lemma /in somewhat
greater generality than we actually need.
248 Chapter IV. Invariant functionals § 18. Invarant means on almost periodic functions 249
(18.4) Lemma. Let P and Q be nonvoid sets. Let be a ]unction
that maps P into the [amily o[ nonvoid subsets o[ Q. Suppose that
(i) U { (p) " p C P) >= P
[or all finite subsets P o[ P. Suppose also that P is finite or that e (P) is
[inite [or all p C P. Then there is a one-to-one [unction mapping P into
Q with the property that (P)e(#) or all p P.
Proof. Suppose that P is finite, P--n. If n--, the result is trivial.
For n> 1, there are two cases to consider. Suppose first that the left
side of (i) is greater than P if PcP and 0<P<n. Choose any PoP
and let (Po) be any element of 0 (Po). Then P fpo' and the function
p-+(p)A((po) )' satisfy the inductive hypothesis, and PfPo)' is n--I.
Hence can be constructed in this case.
Second, suppose that there is a PoCP such that 0<Po<n and
U (p)--o. Then Po satisfies the inductive hypothesis and so can be
PPo
defined on Po so as to be one-to-one, and such that (p)c(p) for PPo.
Note that (Po)-- U (p). Now look at the set PPo'. If there were a
P Po
subset P2 of Pf-IPd such that U{(p)f"la(Po)"pcPe}<Pe, then we would
have U (p) -- (a (Po) (P)) -- Po + U (p) ffl (Po)' < Po P. That
PEPoUP PE P, PEP,
is, (i) would fail for Po U P. Hence we can apply the inductive hypothesis
to Pf-IP and the mapping *(P) =(P) a(P0)', and define a on PfGP.
The case in which P is infinite and all (p) are finite is handled by
a compactness argument. With the discrete topology, each 0(p) is
compact, and so the Cartesian product X=Ppo (p) is compact. For a
finite nonvoid subset F of P, let H be the set of all (qf)C X such that all
qf are distinct for p CF. By the first case, H is nonvoid. It is also
closed. Since Hs,f-1...f-lHsDHscj...cjs, the intersection f-l{Hs'F is a
finite nonvoid subset of P} is nonvoid. Choose any point (q) in this
intersection, and let a (p)--qf for all pc P. Plainly this a has all of the
required properties. []
Lemma (t8.4) can be used to establish a useful fact about metric
spaces.
(18.5) Lemma. Let X be a metric space with metric d, and suppose
that {x , x ..... x} is an e-mesh in X/or some e > O, such that the number
o/elements o/the mesh is as small as possible [or the given e. Let Y be any
subset o/X such that/or each x c X there is a y c Y such that d (x, y) < e.
Then there are a one-to-one mapping o[ {x 1, x,..., x} into Y and a
sequence {z, z2, ..., z,} o/ elements o/ X such that d(x, z)<e and
This assertion is called "the marriage lemma" for an obvious reason.
Thus Y is an e-mesh if it is finite.
Proof. For every k-- t, 2 .... , n, let ў (x) -- {yc Y'd (x, z) < e and
d (y, z)< e for some zc X}. Note that all x ..... x are distinct. Consider
an arbitrary nonvoid subset {xh, ..., xi, } of {x, ..., x} and write the
remaining elements if any I as x],+, ..., x]. We will prove that
e (xh) U . . . U (x,) r. ( )
If the set (xA)U... U 0 (x#) is infinite, there is nothing to prove. Suppose
then that it is finite" we write (xA)...O(xi)={yl, y2 ..... Ys}, where
the yz's are distinct. Form the set A={yl, y2 .... , Ys, xi+, ..., xi}.
Some yz's may be equal to xi's in A, but at any rate we have A s + n-- r.
Also A is an t-mesh. For, if zX and d(z, xi, )e for p--r+ t,..., n,
then d (z, xi ) < e for some p = t, 2, ..., r. There is also a y Y such that
d(z, y)< e, and by the definition of , we have y(xi,)" i.e., y=y for
some l--t, 2,..., s. By the choice of n, we have s+n--rAn, so
that s r. This establishes (t).
Now apply (t8.4) with P={x, ..., x} and Q-Y.
With the aid of (t8.5) it is easy to construct an invariant mean on
().
(18.6) Theorem. Let G be a group, / a /unction in (G), and s
positive number. Let (D,], ..., D]) and (D,], ..., D,,]) be e-meshes
in {D]" aG} both having the least cardinal number n among all e-meshes.
Then we have
k=l k=l u
Proof. We apply Lemma (t8.5) to the e-meshes {DI} and {DI}
and the metric space {D/'aG}. Thus we can find q, c= .... , c,,<G and
a one-to-one mapping of {t, 2,..., n} onto itself such that
IID/-- D/II < e, IID/- Dbo(,/l < e (k = l, 2 .... , n).
Thus we have
IIDI--D:<,III.<2e (k=, 2, ..., n). (t)
Adding the inequalities (t) from t to n, dividing by n, using elementary
properties of the norm, and noting that a carries {t, 2 ..... n} onto itself,
we obtain (i).
(18.7) Theorem. Let G,/, e, and {D,/ .... , D/} be as in ( 8.6). Then
(i) _ I(D-w =
k=l k =1 u
1 Observe that only the elementary par of (18.4) is used in this proof: TIONOv's
theorem is not required.
250 Chapter IV. Invariant functionals {} t8. II}variant means on almost periodic functions 25t
Proof. Let u, v be arbitrary elements of G. Then
is an e-mesh in {D/:acG}. For, given acG, we
for some k=t, 2 .... , n, and so ]ID/--D,,ak/II,,<e. For {D,/} and
{Do,/} with b=uav, compute the function in (t8.6.i) at the point
(e, e) G . This gives (i).
(18.8) Theorem. Let G be a group. There is a complex linear/unc-
tional M on N(G) such that:
(i) M(/)=M(]) /or ]N(G) and a, bG;
(ii) M(/)>0 i/]+(G) and/0;
(iii) M(t) = .
Proof. Let [ N (G) and let e be a positive number. Let E, be the set
of all complex numbers z such that for some sequence (c 1 ..... c} of
elements of G, the inequality
holds. Theorem (t 8.7) shows that E, is nonvoid. Suppose that 21 and z,
are in E,. That is,
P
I ' I
1- /(xcy) <, for aii (x, y) (2)
and
q
z-- , l(xdy) < e for all (x, y)G . (3)
/e=l
Setting x--e and y=d/e in (2), adding over k= t, 2,..., q, and dividing
by q, we obtain
q P
Setting y=e and x=c i in (3), we obtain similarly
# q
Hence we have
]Zl--Z21<2. (4)
It is easy to see that if zE,, then
I1 < II/ll,,ч .
1 On g[r(G), M is thus a two-sided invariant mean in the sense of (17.3). We
could of course define M only on 91 r(G) and then extend it by (B.38), but there is
no advantage in doing this.
Let F, be the closure [in the ordinary topology of the complex plane
of E,. Since F,, fq ... fq F,m D Fminl ....... ,ml, compactness shows that the
set f3 F, is nonvoid. The relation (4) shows that fq F, contains exactly
e>0
one point. This number we take as M(]). Since we have F, c E,, for all
e, e' such that 0< e< e', it follows that M(]) is the unique point lying
in all E,. Thus we have:
M(]) is the unique complex number such that for every
there is a sequence {aa ..... a} of group elements for which
M(I)- = D 1 < .
(5)
It is obvious that M(o/)=oM(/) for all/91(G) and complex num-
bers , that M(b/)=M(/) for all a, bG, and that M(t)=t.
We now prove that M (/+ g) = M (/) + M (g) for/, g 91(G). For e > 0,
apply (5) to choose sequences {a 1 ..... a} and {bl, ..., b} of elements
of G such that
m
II
-- D, < (6)
M(I) -- m . 2
and
Db, g < --
M(g)-- ,z /e=i , 2"
From (6) we see that
m
M (/) -- --.
for all x, y in G. Summing over k and dividing by n, we get
(7)
m
J /(xab/ey) < 2
M(I) m = =
for all x, y in G. That is,
M (/) -- m- ,, 2"
/e=i /=i
Similarly we find
M (g) -- D,g <
mn o u
1=1 k=l
Adding (8) and (9), we have
• M (I) + M (g) -- m---/e=, =,
Since e is arbitrary, (5) shows that M (/) + M (g) = M (/+ g).
(s)
(9)
252 Chapter IV. Invariant functionals § 18. Invatiant means on almost periodic functions 253
We next establish (ii). It is obvious that M(/)>=O for /91ч(G).
Suppose that [91ч(G) and that ](b)>0 for some bG. Let {Da, ....
be an ](b)/2-mesh in {Da/'aG }. Then for all x, y, aG, we have
l(xaly) Av ... 2v j(xanY ) max[[(xaly ) .... ](xany)l > ](xay) -- [(b)
' 2
Setting y=e and a--x -lb, we obtain
] (X al) -Jr-'" -Jr- / (x an) - [ (b__)_) for all x G.
2
Hence we have M([a,+...+/)=nM(])>=](b)/2, since M is linear and
is nonnegative on g[ч(G). 1
(18.9) Theorem. The linear ]unctional M constructed in (t8.8) is
uniquely determined by properties (t 8.8.i)--(t8.8.iii). In/act, suppose that
G is a topological group and that M' is any complex linear ]unctional on
9 c (G) such that
(iz) M' (if) =M'([) /or all a G and ] 9 (G)
or
(i,) M' (L) =M' (]) [or all a G and ! 9 (G),
(ii) M' ([) >= 0/or ! I+ (G),
(iii) M'(t)= t.
Then M' (]) =M(/) ]or all ] 9 (G).
Proof. Since M and M' are linear on N(G), we need only prove
M'(/)=M(]) for ]N(G). By (t8.8.5), we have
-- e < M(/) ---
=I
for all x, yG, where e is an arbitrary positive number and al, ... , a
are appropriate elements of G. If M' satisfies (i), we set x=e in (),
and obtain
- < M(I) -- ,1 < . (2)
=1
Applying M' to (2), we get
-- e M(I) -- (1) e.
=1
Hence M'(])=M(/) for /91(G). The case in which (i,) holds is dealt
with similarly.
In some cases, the mean value M on [ (G) has a particularly simple
form. We begin with a general theorem.
(18.10) Theorem. Let G be a locally compact group with a left Haar
measure 2. Suppose that there is a sequence {Hn}= 1 o/subsets o/ G such
that 0< 2 (Hn)< oo/or all n and such that/or each x G,
(i) lim 2 ((xHn)rH)
n-+oo X(H,) =0.
Then/or every/ 9 (G), we have
' fl(x)dx.
(ii) M(/) --n-oolim 2 ()
Proof. We first prove that for/(G) and bG, we have
, f
lim 2(Hn ) . (b/(X)-/(X)) dx= O. (t)
H
Clearly we have
S (x) xl= I S (x) x S (x) x I }
" (2)
f ..]/(x)l dxg
(bHn) A
We also have
Substituting () in (2), dividing the result b (H,), and applying
we obtain ().
We shall use () to establish (ii). It obviousl suffices to prove (ii)
or i (G). For such l, let
, f
p (/) = ,lim -2- . / (x) dx.
It is evident that p(/+g)gp(/)+p(g) and p(/)=p(/) for real non-
negative numbers . By the HAHN-BANACH theorem (B.tb), there is a
linear functional M 0 on (G) such that
-- p (--/) M 0 (/) p (/) for all / ; (G). (4)
By (1) we have p(ff--/)=--p(--ff+/)=O for all/2(G) and all bG.
This and (4) show that M 0 (/)=M 0 (if). It is also clear that M 0 is non-
negative and M 0(t) = t. From (18.9), we infer that M 0(/) =M(/) for
/(G). Now if there were a function/(G) such that
-- p(--/) = lim /(x) dx< lim (H) /(x) dx--p(/),
Theorem (B.12) would imply the existence of two distinct functionals
M0 and M 1 both satisfying (4). They would both be equal to M on (G),
and a contradiction would result. Consequently the limit in (ii) exists
and (ii) itself is established.
Sets H1, H2 ..... H .... for which (ta.10.ii) holds need not exist.
However, if G is locally compact, a-compact, and Abelian, they do. We
give the construction in a sequence of lemmas.
254 Chapter IV. Invariant functionals § t8. Invariant means on almost periodic functions 255
(18.11) Lemma. Let G be a locally compact group with left Haar
measure , let A be a 2-measurable subset o/G such that 0< it (A)< oo, and
let x be any element o/ G. Write Ao=A, and A,=A U xA U...Ux"A
(n = t, 2, 3, ...). Then
(i) (*A'nA;'I _< t (n=0, t 2 ...).
(A,) -- n+ t ' '
Proof. Let B,= A, fl A',_I (n-- t, 2, ...). For n= 2, 3, 4, ..., we have
B, = x'A n (A U xA U ... U x"-lA) ' c x'A fq (xA U ... U x"-IA) '
- x(x'- A (A xA x'-"AI')=
It follows that
(A) (B1) >= (B,)>=... >= 2 (B,>=.... (1)
It is also clear that
2(*AnnA'n) 2 (Bn+)
= (2)
(A,,) (A) + (B) +... + (B,)
(n = t, 2, 3 .... ). If (Bn+i) is zero, (2) shows that (i) holds. If
is positive, then (2) and (t) yield
2(xAnnA'n) 2 (Bn+) 2(Bn) _
2(An) (n+ t) 2(Bn) -- (n+ t) 2(Bn) n+ t
note that 2 (B,,) < oo].
(18.12) Lemma. Let G be a locally compact Abelian group with Haar
measure . Let U and V be neighborhoods o/ e in G such that U- and V-
are compact, and let e be a positive number. Then there is an open subset
H o/G such that H- is compact, V c H, and
(i) 2((HU)AH')
2(H) < e.
Proof. Since V-- U- is compact (4.4), there are elements t I , t. ..... t, eG
such that VU c V- U- ciU=l i V. For n ---- 1,2 ..... let IV, tO (t 1 t .... t, " V},
the union being extended over all ordered r-tuples (1, ,. ..... ,) of
nonnegative integers such that %_<_ n (/'--1, 2 .... , r). It is obvious that
.... ).
Now consider any /'=t, 2 ..... r. We have W=4UoA,= _ where A
U { .... t lt+'" " "'i+1 t, =" V" 0<= k = < n, k = t, ..., r, k =4=i}. Plainly we have
0 < it (A) < oo; (t 8.1 t ) shows that
;t(tiWrl W,) < t (n -- t 2, ...). (2)
2(w,) = n+ t '
Using (1) and (2), we make the following estimates"
,.((W n U, n W) < '((](-Jl t]wn) n W)
(w,) (w,)
<= 2 2 ((t]w,,) n w/,)
= (w,)
<2 t
n+t
Since r is fixed, we obtain (i) by taking H = W with any n > __r _ t. ]
(18.13) Lemma. Let G be a locally compact, a-compact, Abelian group.
There is a sequence (Hn},°°__ o/subsets o/G with the/ollowing properties"
(i) each H, is open and is compact;
(ii) HICH,.c...cH,c... ;
(iii) U H,G;
n=l
(iv) /or each xG, lira 2((xH,fH) =0.
.-+ (H,,)
Proof. Let (F,,)= be an increasing sequence of compact subsets of
G such that eF 1 and U F,,=G. Let W be any neighborhood of e with
n=l
compact closure, and let U,= Fn W (n--t, 2, 3,...). We define the sets
H by induction. Use (t8.t2) to find an open set H with compact
closure such that H U and
. ((H U) n Hi)
. (HI)
Suppose that H I, H2, ..., H_ 1 have been defined. Use (t8.t2) to find
an open set H, with compact closure such that H.D H. 1U U.and
X ((H U) nH;,) < .
X (H,
Thus {H},°__, is defined. Properties (i), (ii), and (iii) are evident. Since
U U,=G and U1cU,.c-..cU,c..- we can choose for each xG a
nl '
positive integer n 0 such that x U for n n 0. For all such n, we have
((.H. nil,;) < ((H.U. nH;,) < ±.
. (H,,) -- . (H,,)
This implies property (iv).
(18.14) Theorem. Let G be a locally compact, r-compact, Abelian
group. There is an increasing sequence {n},°=, o/open sets having compact
closure such that
256 Chapter IV. Invariant functionals § t 8. Invariant means on almost periodic functions 257
(i) M(/) : 11111 --- f/(x) dx
n--eo 2 (Hn)
H
/or al! / 91c (G).
Proof. Let {mn)nco__ 1 be as in (t8.13) and apply
Miscellaneous theorems and examples
(18.15) Applications of Theorems (18.10) and (18.14). (a) Consid-
T
er the additive group R. For / 9Ac (R), we have M(/) = t-+co 2Tlim - f/(x) dx =
T b --T
T-+co (T--a) /(x)dx= r-+colim (T + b) /(x) dx, whereaandbarearbitrary
a --T T
real numbers. [Consider the subadditive functional p (/)= lim - /(x)dx,
T- co 2 T .]
--T
and argue as in (18.10); similarly for the second and third equalities.]
(b) Consider the additive group 2'. For / 91(2"), we have
m m
M(/)= lim t , /(i) lim t ,/(i)
m-co m + t .
m 2m+ t . --a
= --m =a
b
= lira /(f).
[see (s.0).]
(c) Let G be a compact oup. For /(G), we have M(/)=I(/),
where I is the Haar inteal on (G). [This is obvious from ( 5.5) and
(,.).]
(d) Let G and be locally compact groups with left Haar measures
and 20, respectively. Suppose that {H)% and (H)% are sequences of
subsets of G and G, respectively, that satisfy (8.0.i). Then for every
(x, y) G x G, we have
(n) 0 (n)
(e) Consider a topological group of the form GoxRxZ b, where Go
is a compact group and a and b are nonnegative integers. For n = t, 2, ...,
let H be the set of all (x, Yl ..... Ya, zl ..... zb)GoxRxZ b such that
yiR and [yi[<n, zZ and [z[<=n (--t, ..., a; k--1 ..... b), and x is
arbitrary in G 0. Then for/ 9(GoxRxZ), we have
M(I) -- lim (2n)-(2n + 1) - f l(w) d2(w),
n .--. co Hn
It is easy to show that ((G) = 9 c(G) if G is compact" see Vol. II, (33.26.b).
This fact, while interesting, is unimportant for our present purpose.
where 2 is an appropriately normalized Haar measure on GoxRxZ .
Apply (d)and (t8.10).]
(18.16) Miscellaneous means (adapted from MAAK 2]). (a) Let S
be an arbitrary semigroup. For a function / in (S) and aS, let D/
be the function on SxS=S such that D/(x, y)=/(xay). For a
positive number e, let E, be the set of all complex numbers z such that
z---i =.,D/]I <e for some sequence {a, ao ..... a=} of elements of S.
If Zl, Z2E,, then ]z--z=[<2e. If all sets E, are nonvoid, then E,
e>0
consists of a single complex number, which we write as M(/). [Suppose
that
Zl--- /(xaiy ) <e and z 2-- , /(xby) < e
• n
for all (x, y)S . Set x=ai, b and y=b, in the first inequality, sum
over f', k, k', and divide by mn . This yields
This relation with 21 replaced by zz is obtained by setting x=a i, and
y=aib . in the second inequality above. The argument now follows the
proof of (t8.8).]
(b) Suppose that/3(S) and that M(/) exists. Then M(/)=M(/)
for all K, and M(/)=M(,/)=M(/)=M(,/) for all a, b S. If/93+(S)
and M(/) exists, then M(/)>_O. I II/(-ll[-+0 as n--> and M(/('o)
exists for all n, then M(/) exists and is equal to lim M(/('0). If M(/) and
M(g) exist, then M(/+g)=M(/)+M(g). All assertions but the last are
immediate. We sketch a proof of the last. If --m-- /(xaiy ) < - and
w g(xby) < -g for all (x, y) S", then z---- /(xaiby ) <
n k=l mn.
and I zo_ __. g(xaiby)l < g" This implies that I(+zo)--
- (/+g) (xaibY) <-4-' and so E is nonvoid for the function/+ g.
Weals°have IM(/)--I < -i-' IM(g)--wl < ' and lM(/+g)--(+w)l < ,
so that IM(/)чM(g)_M(/ч g)l < '
(c) A ero eleen or ero of a semigroup S is an element 0 S such
that 0x=x0=0 for all xS. If S has a zero, then M(/) exists and is
equal to/(0) for all/3(S). The functional/-+/(0) is the only invariant
mean on 3(S). See also (17.18.e).
Hewitt and Ross, Abstract armonic analysis, vol. I 17
258
Chapter IV. Invariant functionals
(d) Let S be an infinite cancellation semigroup" ax=ay implies x-y
and xa--ya implies x--y, for all x, y, a S. I t () and {x S"
is finite for all e>0, then M(]) exists and is zero. In view of (b), it
suffices to show that M(${b})----0 for all bS. If aa, a2, ..., a, are
distinct elements of S and x, y are elements of S, then the equality
xaiy=b can hold for at most one f, f----t, 2 ..... m. Thus we have
(e) Let G be a locally compact, noncompact topological group for
which the uniform structures 5 (G) and 9 , (G) are equivalent. Let f be
a function in (G) such that the set {xG'lf(x)i>e }- is compact for
all e >0. Then M(]) exists and is equal to zero. [We will show that
there is a neighborhood W of e in G such that M(w) exists and is equal
to zero. Note also that O<=]<__g and M(g)----0 imply M(/)=0. These
facts together with (b) suffice to prove (e). Let V and W be neigh-
borhoods of e in G such that V- is compact, W is symmetric (4.6),
and xWx-lc V for all xG [(4.t4.g) and (4.5.i). Define a sequence
(ai)i°°=l of elements of G by induction. Let a be arbitrary. When
a 1, a2 ..... a_l have been defined, let a be any element not in
(al V)U (a2 V) U--. U (a,,_l V). Now consider the function h----- ] D o (w).
Its value at every (x, y)G is either or 0. In fact, if xajyW and
xakyW where 'k, then we have (xaky)-l(xaiy)--y-la;1,aiyW ,
a;la] yW2y-lc V, so that aia V. This implies that f<=k; and so f--k.
It follows that h =-- and that M()=0.
Let G be a locally compact, noncompact topological group and suppose
that there is a continuous homomorphism of G onto some noncompact
Abelian group H. 1 If/is a function in (G) and the sets {xG" I[()1
are compact for e > 0, then M(/) exists and is equal to zero. lit suffices
to prove that M(-)=0 for all nonvoid compact subsets/r of G. Let
al=e. Since FF -1 is compact, '(FF -1) is a compact subset of H and
since HI. is Abelian, we have (U xF:;-lx -1) =z'(;'F-1). It follows that
U xFF-lx-l= G; let a2 ў U xFFqx -1. Having defined aa .... , a_, we find
xEG xEG
--I (.--I ),
that ,=1,U (x6xFF-lx-1)al:C and choose a in Gf"l ,U=l. (x6 xFF-lx-1) a' "
11-: 2 II
We then have D (-) -- ,
This property holds whenever G is not unimodular [use the modular function
( 5.t t)] and also holds for some unimodular groups such as I (,/g) [use the
function A -- det A ].
We do not know if the result of (t8.16.e) holds for all locally compact groups
and all/E(G) such that {1/() I > } is compact for all e > 0. The question
is open even for/E0 (G).
§ t 8. Invarian't-means on almost periodic functions
259
(f) Let G be a group and H a normal subgroup of G such that G/H
is infinite. Then M($H)--0. [If a, a ..... a lie in distinct cosets of H,
then . D (eH) --- -.]
=1 u
(g) Let G be a group and H an infinite normal subgroup of G. Let B
be a subset of G such that B (xH) is less than or equal to p (a positive
integer) for all xG. Then M($)=0. [If , a2 ..... a are distinct
elements of H, then D () .
=1 u
(h) Let G be an infinite Abelian oup. There is a function ] in +(G)
such that /(x)>0 for all xG and M(/)=0. [By (6.3.c), G has a
subgroup H such that G/H is countably infinite. Let A, A2 .... , A, ...
be the distinct cosets of H. Then M($A ) =M($H)--0 as (b) and (f) show.
Now set ]= 2-$A and apply (b) once more .]
(18.17) o-Functions (KEINER [] and MAAK 2]). (a) We now
describe a class of functions ] in (S) IS is an arbitrary semioup] for
which the mean value defined in (8.6.a) exists. Suppose that for
every e>0, there are subsets A, Ae ..... A and E of S with the
following properties"
E=S(AU...UA)'; ()
M($)=0; (2)
if a, b S and the set B,,= ((a, b)}U ((xay, xby): x, y
nonvoid intersection with some AxA, then ]](z)--/(w)]<e (3)
for all (z, w) B,, (E'
Then [ is called an w-/unction. We will prove that M(]) [in the sense
of ( 8. 6)] exists for w-functions ]. We use a sequence of lemmas.
(b) Let [ be an w-function; let A, A, ..., A and E be as in (a)
for a given e > 0; and suppose that n has the smallest possible value for
this e. Let x, y be any fixed elements of S. Then there is a permutation
z of (, 2 .... , n} such that A(xA,ўy) for k=, 2, ..., n. [Like
(t8.7), this result depends upon (t8.4). For k=t, 2 ...., n, let B=xAy
and let e(k)=(i'AB, in). Given a set
where k<k2<...<k,n, let (i,i, ...,i) denote the set
(k) U... U e (k,), where < 2 <... < L n. It is obvious that
1 This construction shows spectacularly that there is no analogue of LEBESGUE'S
theorem on dominated convergence for the functional M. In fact, writing
hn=A,+ "'" + A, we have lim h=l, h lh... h n..., M(I)=t, and
M(hn)=0forn=l,2,3 .....
17"
260 Chapter IV. Invariant functionals Chapter V. Convolutions and group representations 26t
Now let Ct=(zS'xzyAi, } for 1--1, 2, ..., s, andletF--{zS'xzyE}.
Clearly we have M(F)=M(x(E)y ) =0 and hence M(EuF)=0. Write
the sets A different from AI, .... A, as A,+ ..... A, and consider
the family of sets
c n (EUF)', .... csn (EUF)', A,+, n (EUF)', ..., A, n (EUF)',EUF.
It is easy to see that this family of sets satisfies (t) and (3), and so
s+--r, sr. Thus (t8.4) can be applied, and (b) follows.?
(c) Let / be an m-function; let A1, A .... , A and E be as in (a)
for a given e>0; and suppose that is as small as possible for this e.
For every d > 0, there is a positive integer N and in each set A there are
elements V,l, v, ..... v, such that
N
....
=1
P
[Let =inf {11 ' *'11" ' ""' is a finite sequence of elements
/=1
of S}, for k=t, 2, ..., n. Since n is as small as possible, all 's are
positive. Let --min (1, ,..., ). Choose elements 1, .... , in
L
II
S so that D,() <d. For h=t, 2,..., , there are elements
/=1
and in S such that
L
/=1
while at the same time
L
L (xx,yy)<0 for all (x, y)S . (6)
/=1
Let N be the smallest integer such that NL. Because of (5), at least
N of the elements Xlk .... , x lie in A" take N of these elements
as V,l, v, ..... v,. On account of (6), we have (4).?
(d) Let / be an m-function on S. Then M(/) exists. Nore precisely,
for every e > 0 there are elements Wl,..., w in S such that
i=1 i=1
[Let e be a positive number, and let A1, A, ..., A. E be as in (a),
with e replaced by el3, and with the smallest possible value of . L'et
d-- e , and choose N and the elements v, (h=t, 2 ..... ; l
t, 2 ..... N) as in (c), for this value of d. Let x, be arbitrary elements
of S, let be as in (b), and let c be an element of A(xA() (
I, 2,..., n). We make the following estimates:
- y,/(v,)- _ ,/(xv,y)
k, l k, l
x (l(v,)- l(x,(>,y)) <
= = I1 (,,)-/(c) (3)
, k, l
+ /(-/xv,,l = + .
, l
Since v,A, cA, (3) shows that
2 < .
To estimate 2, consider the values of k and l such that xv;),zyE.
We have c x sy, where s A,(), and again (3) shows that
2'i/()-/(xv.,,)l (o)
nN 3'
the sum being taken over all k and l such that x v,,zyE. Let "
denote the sum over all k and l such that xv),yE. From (4), we see
that
Combining (), (0), (9), and (8), we get (7) with the wi's equal to the
U, l'S.
Notes
The literature on almost periodic functions is enormous: for a survey
see MAAK , pp. 222--23S. Lemma (8.4) is due to HALMOS and
VAUAN ; our language is regrettably less colorful than theirs. The
construction of M and the proof of its uniqueness ( 8.6) -- ( 8.9) are
taken from MAAK , pp. 3--4. Theorems (8.0)--(8.4) are based
on LUBARSKI and an unpublished manuscript of A. BIG and
E. HEwing. See also I(AWADA and STRULE for related results.
One-parameter families of sets resembling the sets (H) of (8.3)
were used by CALER6N to prove ergodic theorems.
Chapter Five
Convolutions and group representations
In the present chapter, we initiate our study of harmonic analysis
proper. The basic operation in harmonic analysis is convolution; in § t9,
we give a reasonably general definition of convolutions and develop
262 Chapter V. Convolutions and group representations § t9. Introduction to convolutions 263
with some care the fundamental properties of convolutions of measures.
In §20, we examine explicit formulas for convolutions of measures and
functions. In § 2, we present some facts about representations of groups
and algebras. In §22, we prove the existence of irreducible representa-
tions of locally compact groups.
Our main goal in this chapter is to prove the GEL'FAND-RAKOV
theorem (22.2)- this theorem is one of the most important in the entire
book and incidentally is needed at once in the theory of locally compact
Abelian groups, which we treat in Chapter Six. It would be perfectly
reasonable to postpone a detailed discussion of convolutions until we
have treated representations of topological groups and the theory of
locally compact Abelian groups. However, we wish to treat representa-
tions of locally compact groups by means of the algebra 1 (G, ). To
define multiplication in I(G, 2) [which is convolutionS, we need to
define convolutions in general. Our discussion of convolutions in the
present chapter is only introductory: several subsequent chapters are
devoted to a detailed study of various convolution algebras.
§ 19. Introduction to convolutions
As usual, we begin with a definition.
(19.1) Definition. Let S be a semigroup [not necessarily topological]
and let be a linear space of real- or complex-valued functions on S.
We suppose throughout that is left invariant" x[ if [ and xS.
For a linear functional M defined on and ], let r ] be the function
on S such that M](x)=M(x]) for all xS. Suppose further that the
linear functional M is such that r] for all ]. Then if L is any
linear functional on , the functional whose value at ] is L(])----
(LoM) ] is well defined. It is called the convolution of L and M and is
written L, M. For a right invariant space of functions , a linear func-
tional L on , and ], we define _L] as the function L_.] (x) = L(],,).
(19.2) Theorem. (i) The convolution L, M is a linear junctional on .
(it) For a [real or complex I number o, the identities o(L,M)-=
(oL) • M-- L • (oM) hold.
(iii) I/L • M and L • N exist, then L * (M+ N) exists and L • (M+ N) -
L,M+L,N" similarly (M+N),L--M,L+N,L. 1
(iv) I/ L, M, N are linear /unctionals on and and carry
into itsel/, then L • (M , N) -- (L • M) • N.
1 Properly we should write (L* M) + (L • N)" but we regard convolution as a
sort of multiplication and use the universal convention that ax + by = (ax)+ (by)
in any ring.
Proof. Assertions (i), (it), and (iii) are immediate consequences of
(19.t)" we omit their proofs. To prove (iv), note first that (L,M),N
obviously exists._ For x, yS,_ we have (,j)(y)=N(y(,/))=N(,y/)--
/(xy)--x(N/) (y).l Thus N(,/)--,(/). From this we infer that
(M---g) /(x)--(M,N) (,/)--M((,/)) --M(,(/))--M(N/) (x). That is,
M--g--2ro 2V. This shows that L,(M,N) exists, and we can write
L,(M,N) --Lo (M,N) --Lo (Mo2V)- (L o )oN=(L,M),N.
(19.3) Definition. Let be as in (19.t) and let L be a linear space
Ireal or complex according as is a real or complex linear space] of
linear functionals on such that L,M exists and is an element of L
for all L, M L. Then L is called a convolution algebra with • as multi-
plication and addition defined as usual]. It follows at once from (t9.2)
that L is an associative algebra over the real or complex field.
Convolution algebras abound in both analysis and algebra. Many
have been exhaustively studied; others, of equal interest and possibly
of equal importance, have hardly been noticed. We first give an example.
(19.4) The ll-algebra of a semigroup. Let S be any semigroup and
let be the linear space (S) of all bounded complex-valued functions
on S. For each a S, let E a be the linear functional Ea (/)----/(a). Then
E,/--/, as is easy to see, and E,, E b (/)-- E, (/b)----/ (a) -- /(ab)= Eb (/).
Thus the set of linear functionals Ea forms a semigroup under convolution
that is isomorphic with S. By forming the set of all complex linear
combinations E,k , we obtain an algebra containing an isomorph
k=l
of S as a multiplicative subsemigroup. One can go a step further and
consider all of the linear functionals . Ek such that
k=l k=i
and a, a ..... a, .... are distinct elements of S. The zero functional
must be treated separately, but is trivial.] For an obvious reason, we
call this the ll-algebra o! S, and denote it although this is not really
accurate] by ll (S). It is clear that every E is a bounded linear
functional on (S)" E (/)= /(a). It is obvious [take
k=l k=l
](a) =sgn that ]l eEall = I1 • To verify that l I(s) is a con-
. k=l [I k=l "
Volution algebra, and so to justify the expression "/i-algebra", we compute
as follows. For ](S), A=,E,ll(S), and B=,flEo,ll(S), we
oo k=l /=1
have N/= ./5/, which is a function in (S), and hence
()
1 The associativity of S is needed only to prove the identity y(x])
264 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 265
Since
(2)
we may rearrange the left side of (2) and the right side of (t) in any way
we like without changing their values. Let (Xl, x2 .... , x,,...} be the
finite or countably infinite set of all distinct products akb . Then we
have
n=l akbl=xn J
To show that A * B c 11 (S), we must show that
n=l akbl= n
Since Y. Ikl. I/51 = IlAll. IIBII <°°, this sum can be rearranged
k=l /=1
arbitrarily. For each pair of positive integers (k, l), we have ab=x,
for
precisely one n.
Therefore
=1 akbl=xa n=l akbl=xn k=l /=1
That is, A • B is in l, (S) and IIA* BII IIA I1 IIBII •
Various other examples of convolution algebras, of only ancillary
interest, are given in (t9.23)--(t9.24).
We now take up the most important of the convolution algebras to
be studied in this book, namely, the algebra of all bounded linear func-
tionals on 0(G) for a locally compact group G. Throughout (t9.5)---
(t9.28) inclusive, G will denote a locally compact group. From this
point on, the symbol p (G) (t =<p__=<_oo) will denote the Banach space
p (G, 1), where I is a fixed but arbitrary le/t Haar measure on G.
(19.5) Lemma. Let / be a/unction in go (G), and let L be any element
o/ (G). Then [,/and L_/ are in o (G).
Proof. We carry out the proof only for ,/. If Z=0 or 1-0, the
lemma is trivial" (/)=0 and ,(0)--0. First of all, it is obvious that
/and/, are in g0 (G) for every /c g0 (G) and x G. Let e be any positive
number, and choose a neighborhood V of the identity in G such that
8
]/(u)--/(v)]< I for all u, v G such that uv -1 V. This can be done
by (t5.4) /is right uniformly continuous. Then if x and y in G are
such that xy-lV, we have I[ ,/ -- / [, < i and so ]/(x)---/(y)[<e.
This shows that/,/is continuous indeed, right uniformly continuousl.
To show that ],1[ is arbitrarily small outside of a suitably chosen
compact set, we first write /=/1--/,,+ i (/a--/), where the /i's are in
CS(G). Thus we may suppose that / itself is in g(G) and not zero, since
is plainly a linear operator. Let ,u be the complex measure that
represents L as in (t4.4) and let ]/*1 be the nonnegative measure defined
in (14.6). Then as noted in (t4.6), I/*1 (G) is finite. Let e be an arbitrary
positive number. By (t.23), there is a compact set E cG such that
. There is also a compact set F C G such that ]/(z)! <
[if[ (E')< 21llll 2IffI(GW
for z F'. We now have
lZl(x)l = Iz(J)l ILl (J) = f l(xy) Iffl (y) )
()
- fl(xy)dlffl (y)+ fl(xy)lffl (y).
It is clear that e e'
fl(xy) lffl (y) II111. Iffl (E')< .
E' 2
(2)
Also, if 1 (xy) >
= 2lff I (G) ' we have xy cF; and if in addition y E, we have
x(y-') oF(E-l). Thus if x (F(E-1)) ', we have
f I (xy) d I, I(y) < I I (E)
E
Combining (3) and (2) with (t) and noting that F(E -) is compact E(4.2)
and (4.4), we see that [,/o(G).
(19.6) Theorem. The confugate space (G) is a convolution algebra
under Definition (t9.3). With the norm [ILI[, it is a Banach algebra in the
sense o/ (C.1). The lunctionals E delined by E (1) = 1 (a) /or 1 o (G)
satisly the relation
(i) E,E= E /or a, bcG.
The/unctional E is the unit ol (G)"
(ii) E, • L -- L • E,- L/or all L (G).
The algebra (G) is commutative i/ and only i/ G is A belian.
Proof. To show that ' (G) is a convolution algebra, we need only
to prove that L, M is a bounded linear functional on 0 (G) if L and M
are see (19.5). Let ff and v be the complex measures that represent
L and M, respectively, as in (t4.4). Applying (t4.7), we have [M(/)I--
[Gf l dvJ J [I[ dlv [ . Hence, for every xG, we have
G
This implies that II/ll.----< I/I" IIMII, ad ths at t,n anow
us to write
IL, M(/)I- IL(/)I _--< IILII" I1/11. _--< IILI" I1'11" II/11. ()
266 Chapter V. Convolutions and group representations § t9. Introduction to convolutions 267
The relations (t) obviously show that L • M is a bounded linear functional
on 0 (G) and also that
By (B.22), the space '(G) is complete in the metric HL--M[I. This fact
and (2) show that ' (G) is a Banach algebra.
It is obvious that the functionals E a are in '(G) and that IIEa]l =t,
for all a cG. To verify (i), we write
(E* Eb) (/) -- E (E--o/) = E (/) = /b (a) = /(ab) = Eb (/).
To verify (ii), we note that E-/(x) -- E, (,/) -- /(xe) -- /(x). That is,
E/--/, and so (L* E,)(/)=L(/). Also we have
(Ee, L) (/) = E, (/) -- Y_./ (e) = L(J) -- L(/) .
This proves (ii).
If G is non-Abelian, then (i) shows that '(G) is noncommutative.
To prove the converse, suppose that G is Abelian, and that L and M
are elements of ' (G). Let ,u and v be the measures I representing L and
M, respectively. Let / be in 0(G). The function (x, y)-+/ (xy) is con-
tinuous and bounded on GxG, and so belongs to EI(GxG, [#xv]).
Hence by (t4.25), we have
L,M(/) ---- if/(xy) dv(y)dff(x) = f f/(xy) dff(x) dr(y) ]
o G °° / (3)
-- f f ! (3'x) d# (x) d (3') -- M* L(/). ]
(19.7) Note. FUBIN'S theorem plays the essential r61e in proving
(19.6.3), and hence in showing that (G) is commutative if G is. In
situations where FtmNI's theorem fails, convolution algebras can be
noncommutative even though the underlying group or semigroup is
commutative (t 9.24).
(19.8) Definition. For L, M in (G), let # and be the complex
measures that represent L and M, respectively, as in (t4.4). Then
will denote the complex measure that represents L,M as in (t4.4).
We call ff,v the convolution of the two measures ff and v.
We now make some computations involving if, v, and ff,v.
(19.9) Theorem. Let L, M, if, and v be as in (19.8). Then
[*[<l[*[l.
x Where there is no danger o conusion, we use "measure" to mean "complex
measure as constructed in (14.4)".
Proof. Let 9cff and suppose that /0 and ]/[_<9. Then using
(14.7) twice, we have
d ffll(xy) I dl I (y)alffl (x)
f f 9(xy)all () alffl (x) = Izl • IMI ().
by (4.S.i), w a IZ*MI() IZI*IMI() o that
Iffl*ll" U
(19.10) Theorem. Suppose that L and M are functionals in
and that ff and , respectively, are the complex measures representing them.
Let z denote the mapping (x,y)xy of GxG onto G. I] [ is a ]unction
o tt vў Zmot vryr lor Iffl* I I, t fo vў
wr o,x 1o Iffl. u # /<1(, Iff* I1), t
[o is defined ]ffxv]-almost everywhere on GxG. Furthermore, [oz is
ў I(GX, Iffxl), d
(i) f I aff. - f (1 o ) dffxv = f f 1 (x) a () aft (x)
G GXG G G
GG
PooL Let be compact subset o G. 7he - (?) is closed
(.22), t i o t s tt ?c a I, II ()<
I • I (?) + . t I< E;0 () b t o tt I (?) = , I (') = o,
GXG GG
GG G
Thus we have
f #od]]x]v I [],]v](F) ()
for compact sets F c G. It follows immediately that () holds for any
a-compact subset of G. Next consider an ]], ]v]-null set A in G. Let
D be a countable intersection of open sets in G such that A c D and
[[*[v] (D)=0. We will show that aoz is an ]]x]v]-null function, by
howing that #ooz=#_,(o)is an Ill]-nuaa function. Since -X(D)
as a Bore set and hence Imllvi-measurabIe, it suffices to show that
[[[v] (E)=0 for all compact subsets E of ((x, y)GxG" xyD}. Clearly
268 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 269
r(E) is a compact subset of D and $E<=$(E)or. Hence by (t), we have
G >< G G >< G
Consequently, eDo and eAo r are I/lxlv[-null functions. Thus if A is
an Il,l[-nu]] set in G, then r-l(A)is an [lЧll-nu] set in GxG.
This implies the first statement of the theorem.
Next consider any I I* Iv l-measurable subset B of G. Then B--F U A
where FrlA =O, F is a-compact, and A is an I/1. Ivl-nu]] set. Then
e B o r= eAo r -{- # o r is [ IЧll-measurab]e and
GXG
It now follows that if /in 1(, I1"11) has the form 2keB, for
=1
[l,]l-measurable sets B and positive numbers k, then
e,<Ч, I1Ч11) and
f/odll Ч11--<_fdl[ * Il.
G><G
Now consider an arbitrary nonnegative function /in I(G,
we may suppose that / is real-valued. For all positive integers n and
let A, ,---- xG'<=/(x)< 2" . Let/,-- $Ak.," Thenwehave
/1<=/2<=...<=/,<=...<=/, and lim/,(x)--/(x) for all xG. Similarly, we
have/1 o r <=/2 o r <=... <=/, o r =<... and lira/ o r (x, y) -- / o r (x, y) for all
(x, y)GxG. Since each /,or is ]/,]x]vl-measurable, lot is
measurable. The monotone convergence theorem and (2) now imply that
f/ozdl[Чlvl- lim f /,ordltz[xlv ]
"'-°° G
/<, I#l[l>. o/. <, I#[11), II/ll o F I/I [111;
is obvious from (t9.9) that
If ld. * ] <= II/111. (4)
Moreover, using (3), we see that
Now consider the linear functionals lf I d,v and l-+f(lor)dlxv
G
on 01(, I1.11). These functionais agree on Eo(G ) since f/dlt,v--
L * M(/) = L o M(/) = f f /(xy) dv (y) d/, (x) = f / o r d/zxv, and they are
G G GXG
bounded, by (4) and (5). By (t2.t0) and (B.tt), these functionals agree
on all of 1 (G, I1" IV])" This proves the first equality in (i). The remaining
equalities are immediate consequences of (t4.25).
(19.11) Theorem. Let L, M, , v, and r be as in (t9.t0). Then ]or
every I#l,lvl-measurabZe set A, r-l(/)is ]tzxvl-measurable , and
(i) #,v(A) =/zxv (r-X(A)) = f v(x-XA) dtz(x ) = f tz(Ay -1) dr(y).
Proof. This follows immediately from (t9.t0).
(19.12) Let M(G) denote the set of all complex measures,u obtained
as in (t4.4) for linear functionals L in '(G). By (t4.t0), the mapping
L-# of '(G) onto M(G) is one-to-one, linear, norm-preserving, and
order-preserving. We have defined /,.v [where L-->/, and M-->v in
(19.8) so that the mapping preserves convolution. In our further study
of '(G), we will replace it by the equivalent entity M(G), and will
usually state our results in terms of M(G) alone. Where convenient,
of course, we will make use of the fact that M(G) is a concrete represen-
tation of ' (G), so that we can use the HAHN-BANACH theorem and other
devices of functional analysis.
The word "closed" in reference to M(G) means "closed in the topology
for M(G) defined by the metric I]/,--vll".
We now describe, and characterize so far as possible, some important
subsets of M(G).
(19.13) Definition. A measure # in M(G) is said to be purely
discontinuous if there is a countable subset E of G such that ]/,1 (E')=0.
The measure /, is said to be continuous if /,({x})=0 for all xG. A
continuous measure # is said to be singular if ,u is singular with respect
to left Haar measure 2 in the sense of (t 4.2). A measure/, is said to be
absolutely continuous if/, is absolutely continuous with respect to left
Haar measure 2 in the sense of (t 4.20). The sets of purely discontinuous,
continuous, singular, and absolutely continuous measures in M(G) are
denoted by M (G), M (G), M (G), and M (G), respectively. The symbols
Mч(G), M(G), etc., will designate the sets of nonnegative measures in
M(G), M (G), etc., respectively.
In the last expression, we mean that #(Ay-) exists for [v[-almost all y and
that # (A y-) is v-measurable as a function of y [let # (A y-) be 0, say, where it is
not already defined]. Similar considerations apply to v(x-A).
270 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 271
(19.14) Lemma. Let A be a subset o/G that is measurable/or all
#M(G). The set VA---{#M(G): Ii (A)-o) is cZosed linear sub@ace
ol re(G).
Proof. For tt, v VA, we have l# (A) + v (A)I -<- I#l (A) + Iv I(A) -- 0
(t4.tt.ii), so that #+vVA; also [a#l (A)= lal I#l (A)-- 0 for gIt7 and
#V A (t4.tt.i). Thus V A is a linear subspace of M(G). If #,VA
t 2, 3 .... ) if #M(G) and lira IIm-l1-0, then for every Borel subset
B of A, we have
l# (B)I- [#. (B)- #(B) I --I(#--- #)(B)I I#.- #1 (B) lira-- 11.
Hence #(B)=0, and so I1 <1)=0 <a4.as>.
(19.15) Theorem. For aўG, let as be the measure such that as(P)=p(a)
/or all P c G. All subsets o/ G are as-measurable, and as is the measure in
M(G) corresponding to the [unctional Eg(G) [see (t9.6). For a sequence
o[ complex numbers {}% such that I < and a sequence {a}=
o[ distinct points o[ G, the measure
(i) Ea.s
and
All measures o[ the [orm (i) are in M (G), and every nonzero measure in
M(G) has a unique representation (i) in which all e,,'s are nonzero [the
sum may be [inite. The set M(G) is a closed subalgebra o/M(G). The
equality Ma(G)=M(G) obtains i/and only i/G is discrete.
Proof. The first two assertions are obvious upon a little reflection;
we omit their proofs. To prove (ii), write # = a. e.. Then for E C G,
it is clear that # (E)=Ee., the sum being taken over all n such that
a.E. By (t4.t4), we have I1 (E)=.p ](E)i'{E, ..... E.} i
ti=l
of E} Nle. I , the sum again over all n such that a. E. Hence
paition
we have I1 i1 . Theorem (14.t4) shows that tel i the least
nonnegative majorant of #, and hence I1 Ј I.i . The equality
(iii) follows at once from (ii) and the definition of I1 II.
The definition of this measure is obvious" see also (19.4).
It is also obvious that every measure of the form (i) is in M(G).
For #M (G), let E--{al, a., a s .... } be a set [all a,,'s are distinct such
that 1# I(E')--0. For every [# I-measurable set X C G, we have
(x) = (x E) + (X E') = (X E) = )2 ({.}) •
anX
Hence #= Z # ({a.}) e, as we wished to show. For {a 1, a. ..... au}CE,
aEE
we have Y, I<{}>l<ll<G>-IIll<oo, so that 2 [#({a.})]<oo. It is
n=l
also obvious that the representation (i) is unique if the a.'s are distinct
and the % s are different from zero.
To show that M(G) is norm-closed, let {#..oo__t be a sequence of
elements of M (G) and # an element of M(G)such that lira II-m II = 0,
Let E.be a countable set such that I.1 (E'.)=-0 (--a, 2, .... ) and
let E = U E. For a compact subset X of E', we have
for each n. It follows that #(X) =0 and so I1 <E') -0 by (t4.15).
We next show that M(G) is a subalgebra of M(G). It is obvious
that M(G) is a linear subspace of M(G). If #.-# and .- in M(G),
then it is clear that
I1.1-11.-11ч I11111.-11 (19.6.2), so that #...--#.. Thus if
Z %ea. and Z tmeb,, are in M (G), a simple computation based on the
n=l m=l
identity e. e -- ў shows that
-- \m=l =1 m=l
The set of all distinct a,b, is countable, and so M (G) is closed under
convolution.
Suppose that G is discrete. Theorem (t4.t6) shows at once that every
functional in @'(G) has the form . eE, with levi<o o" thus
n=l n=l
M(G) -211 (G) if G is discrete. If G is nondiscrete, let w be any nonnega-
tire, nonzero function in i (G), and let # be the measure in M(G) such
that d,u=wd2, as in (14.17). By (15.17.b), we have 2(A)=0 for all
Countable sets AcG, and so #(A)=fwd2=0 if A is countable.
A
Since # is not the zero measure, we have M
(19.16) Theorem. The set M(G) is a closed two-sided ideal in the
algebra M(G).
272 Chapter V. Convolutions and group representations § 19. Introduction to convolutions
Proof. By (19.13) Me(G) is the set (3 Vx }, which by (19.14) is the
' xEG
intersection of closed linear subspaces. This implies that Mc(G ) is a
closed linear subspace of M(G). For #Mc (G), vM(G), and aG, (19.1 l.i)
gives us
[z,v({a))-- f lz((ay-1)) dr(y)-- 0
G
and
G
Thus Mc (G) is a two-sided ideal in M(G).
(19.17) Lemma. For all subsets A o/ G and all xG, we have
(i) (Ax) =z(x) (A).
In particular, i/A is 2-null [locally 2-null 1, then Ax is 2-null [locally
-null I .
Proof. Consider first the case of an open set U. If/0(6) and
[Gv, then plainly /x-lG(v)-l=v,. It follows immediately from
(15.11) and (11.11) that A (x) 2 (U) <= 2 (Ux). Replacing x by x -1 and
U by Ux, we also have A(x -1) 2(Ux)G2(U), and as A(x)A(x-1)=t,
(i) follows for open sets. Given any set A c G, we have A C U if and
only if Axc Ux, so that (i) follows for all subsets of G from (11.22).
Suppose that A is locally 2-null. If F is a compact subsetof A x,
then Fx -1 is a compact subset of A, and so 2(Fx -1) =0. Thus 2 (F)=
A(x) 2(Fx-1)=0; since Ax is 2-measurable, it is locally 2-null. [
(19.18) Theorem. The set M,(G) is a closed two-sided ideal in the
algebra M(G). The mapping l,--w carrying M a (G) into 1 (G) -- 1 (G, )
as in (t4.17) and (t4.t9) is a norm-preserving linear isomorphism o/M,(G)
onto (G).
Proof. An element # of M(G) is in M,(G) if and only if I#] (F)--0
for every compact set F such that (F)--0. Lemma (19.14) shows that
M, (G) is a closed linear subspace of M(G). Let # be a measure in M, (G)
and v a measure in M(G). If F is a compact set such that (F)--0,
then we have (Fy-1)--O for all yG (19.17), and hence ]#[ (Fy-1)--0
for all yG. Using (19.9) and (19.11), we obtain
[/**vl (F) [/1 *[vl (F) = f[/,[ (Fy -) dl l (y) = 0,
so that #,vM,(G). We evidently have ]#[(x-lF)=0 for all xG,
and thus
G
This shows that v,# is in M, (G)" that is, M, (G) is a two-sided ideal in
m().
273
For #M,(G), Theorem (14.19) shows that there is a function w I(G )
such that d# = w d it. The mapping of M a (G) into 1 (G) engendered in
this way is obviously linear. Theorem (14.17) shows that this mapping
is onto and norm-preserving; since it is norm-preserving, it is also
one-to-one.
(19.19) Theorem. The set M s (G) is a closed linear sub@ace o/ M(G).
Proof. Let #, be elements of M s(G). Then # + is continuous,
clearly enough, and if B and C are Borel sets such that it (B)= it (C)--0
and ]#[ I (C')--0, then I#+v[ (B'f C')___< 1# I (B')-+-[f[ (C')=0, and
. (B U C) =0. Thus #-+- f M s (G) and M, (G) is a linear subspace. Suppose
that #1, #. .... , #..... are in Ms(G ) and that lira [[# -- # [1= 0. Let
be a Borel set such that ;(B)=0 and [#[ (B)=0. Write B= U B;
then A(B)=0. If F is a compact subset of B', then =1
for all , so that/z (F)=0. Hence [#[ (B')=0; since/z is clearly contin-
uous,/z is in M s (G).
(19.20) Theorem. I/ G is nondiscree, he algebra JFI(G) is a direc
sum as a linear space:
(i) M(G) =M (G) M s (G) M, (G).
Writing #M(G) in the ]orm #=/z+#s+#,, we have
I 1= + I sl +
and in particular
(iii) 11# [1 = 11# 11 + [[#s 11 + 1[#, ]1.
I/G is discrete, then M(G)=M (G)=M, (G) and M s (G)= {0}.
Proof. Suppose that G is nondiscrete. The equality (i) asserts that
every/z in M(G) can be written as a sum of measures in M (G), M s (G),
and 3//, (G), and that this decomposition is unique. Given #M(G), the
set {xG:#({x})4=0} is clearly finite or countabmy infinite. If it is void,
set/z=0. If it is nonvoid, enumerate it in any fashion" {a, a, a .... },
where the as are distinct. Then write #= .# ({a})e. It is clear
that #M, and that/6=/z--# is a continuous measure. Now apply
Theorem (t4.22) to/z and left Haar measure . We obtain
where /z s M, (G) and /z,M, (G) ; this decomposition is unique, and
1#1 [/Zs[ + 1#,1. The equality [#1 =1#[ + 1#1 is also easy to verify" we
omit the details. This proves (ii) and (iii).
If/,=#+#'c, where/,'M(G) and #'M(G), then
so that 0=/zg--#=/,--#. Hence the decomposition
is unique, and (i) is proved.
Hewitt and Ross, Abstract harmonic analysis, vol. I 1
274 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 271;
The last statement of the theorem follows from (15.17.b), (14.16),
and (t9.t3).
(19.21) Note. The subspace M (G) is a subalgebra of M(G); M c(G)
and M(G) are two-sided ideals in M(G). However, 2Is(G ) is not in
general a subalgebra of M(G) [see (9.26)]. If G is nondiscrete, then
Mc(G)--Ms(G)@M(G).
Miscellaneous theorems and examples
(19.22) (a) We made an arbitrary choice in Definition
as a right invariant linear space of functions on S, and define _M[ as the
function whose value at x S is M([,,). If _M carries into itself and L
is a linear functional on , define LV M as the functional L o _M. Then
we have _N(lx) (y) --N(
(_N[), for all xS. Also (MVN)[(x)=(MVN)(/)--M(N_([,))=
M((_N/),) = (_Mo_N) ](x). Thus LV(MVN)=Lo(MVN)--Lo(M_o_N)=
(L o_M) o _N-- (LV M) V N. That is, the operation V, like the operation.,
is associative. It is clear that for every theorem about., there is a
corresponding theorem about V.
(b) Let =3(S) and E (aS) be as in (t9.4). Then E_2[--d[ and
(EbVE) [=[(ab)--Eb[. Thus the mapping a--E is an anti-iso-
morphism of S onto a semigroup under V of linear functionals an 3(S).
This fact makes our choice of • for the definition of convolution
somewhat more convenient than the choice of V, which as pointed out
in (a) would have been perfectly possible.
(c) We can apply the operation V to the conjugate space {(G)
identified as usual with M(G), where G is a locally compact group.
For L and M in (G) and [o(G), (19.5) shows that M_/o(G ). Thus
L V M is in {(G).
For #, vM(G) and [1 (G, ]#]V]v]), we have
f / d# v -- f f/(yx) dv(y) d#(x). ()
G GG
For I:,lv I I -measurable sets A, we have
#V v(A) = fv(Ax-1)d#(x) = f #(y-IA)dv(y). (2)
(19.23) Other examples of convolution algebras. The choice of
0(G) and '(G) in (t9.5) et seq. is not accidental. Considerations of
accessibility, availability of analytic machinery, and even personal taste
influence one's choice of function spaces and spaces of linear functionals
in which to study convolutions. Some natural candidates, however,
suffer from serious defects, as the following examples show.
(a) Let G be a noncompact, locally compact, nondiscrete group with
a countable open basis of neighborhoods at the identity. As in (t 5.9),
select a countably infinite set {xz , x. ..... x, .... } c G and a neighborhood
V of e such that V- is compact and (Vx,)A(Vx,,)=O if m=n. Let
{V},= be a decreasing sequence of neighborhoods of e such that V z c V
and f3 V,={e}. Let [ be a function in (G) such that [(G)c[0
/(x)=t for n=t,2 ..... and/((,__, Vx,)')=0 [,,.= {x} is closed and G
normal topological space[. Let L be a functional in *(G) such
is
&
that
J
z() =2i m 9(x )whenever this limit exists [apply (B.t4)]. If y V and
ycV, then y/(x)=0 for all n>m. It follows that L(/)--t and L(y/)=0
for all y Vf?{e}'. That is, the function [ is discontinuous, and *(G)
is not a convolution algebra in our sense.
(b) (Suggested by Buck [1.) Let G be any locally compact group
and let ,, (G) [,(G)] be the linear space of all right [left uniformly
continuous, bounded, complex-valued functions on G. Let
,(G) ,, (G). Under the uniform norm 1111,, these spaces are Banach
spaces [closed subspaces of (G)I. For 9,,(G) and e>O, there is a
neighborhood g of e such that Iq(ux)--9(vx)l< e for all xG provided
that uv-=(ux)(vx)- U. For M,*,(G), therefore, we have
and so L • M ,*, (G) for L, M *,, (G). That is, ,*, (G) is a convolution
algebra in the sense of (19.3). Similarly LVM [see (19.22.a)] is in
, (G) if L, M are in *
, (G). Both of these algebras are highly legitimate
objects of study. They present grave difficulties, however, because of
the great complexity of their structure. There is no known analogue for
,,(G) of Theorem (14.4), which represents linear functionals in '(G)
as integrals with respect to countably additive complex measures. It
seems unlikely, indeed, that all functionals in ,*, (G) can be represented
as integrals with respect to [initely additive measures.
(c) Apparently less tractable even than ,*,(G) is the convolution
algebra 3"(S), where S is a semigroup [no topology 1, 3(S) is as defined
in (17.1), and (S) is made into a Banach space with the uniform norm,
as in (17.1). It is trivial that _M9 and 2r0 are in 3(S) if 93(S) and
M3*(S). Thus *(S) is a convolution algebra under both • and V.
It is known that a functional L 3"(S) can be represented as an integral"
L(9) _ f q) (x) d# (x), where # is a finitely additive, complex measure
s
defined for all subsets of S. However, in no nontrivial case has the
analysis of 3"(S) been carried out. Even for the group Z, the complexity
and sheer size of 3*(Z) are such that a detailed study seems very difficult
indeed. [The representation theorem for 3"(S) is simple" let # (A)=L(A)
for all A S.
(19.24) (a) The algebras ,*,(G) and 3"(S) of (19.23) need not be
°ornmutative even if G and S are commutative. We consider ,*,,(G),
18"
276 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 277
the argument for *(S) being similar. Suppose that G,,(G) contains a
closed linear subspace 9 that is left invariant a]c9 if ]c9 and
that 1 , and that 9 admits two distinct bounded linear functionals
N 1 and N2 with the properties . () = A. ([) and . () = (y= , 2).
By (B.4), there are extensions - of . that are bounded linear func-
tionals on G, (G). Now if [ and N () N2 (), we have M* M2
M o 2 (/) -- M () M2 (/) -- M2 (/), and M2* M 1 (/) = M (/) M2 (/). Thus
, (G) is noncommutative.
(b) Let G = R. In ў (R), consider the subspace consisting of all
for which N()=i(x) and N2(9)=li9(x) exist. Obviously
2, . is invariant, and .()= . Hence G (R) and *(R) are
noncommutative.
(c) Let G be an infinite Abelian group. Then by (7.2t.c) and (a),
*(G) is noncommutative.
(19.25) Measurability with respect to [], I1, I*1, nd I[*l.
(a) Let G be a locally compact group. It is possible for a subset A of G
to be measurable with respect to measures ]#l and Iv i, where # and
are elements of M(G), while at the same time A is not measurable with
respect to ]#.v] or ]#[ .Iv I. As a simple example, take G=R, let 2 be
ordinary Lebese measure on R, and let # be the measure such that
d=[_,ld. An elementary computation using (9.) shows that
d#.#=/d2, where /(x)=max(2-Ix l, 0) for all xR" note that it
suffices to show that f f l_,l(x+ y) dx dy = f/(x) dx for all aR.
If A is a subset of , 2 that is not Lebesgue measurable, then A is
I# ]-measurable since Iml(A)0, but A is not ]# . ]-measurable or
]#l.l#l-measurable, as (14.7) shows. Note that ]#l=# and I#.#]
(b) A subset of G can be I# .v [-measurable and yet not ],[.[v
measurable. Consider the group T with the normalized Haar measure
described in (t5.17.d). Let # and v be defined by d#(x)=x d2(x) and
dr(x) = x - d2(x). Then for every integer n, we have
T TT T T
2 2
- ' t - o
o o
Since the set of all linear combinations of functions x (n=0,
2 .... ) is a dense subspace of G(T), 1 we infer that #.v=0. From
The group T is compact and G is a subalgebra of G(T) closed under complex
conjugation and separating points of T. See the footnote to (13.2). This is a special
case of a phenomenon occurring in all compact groups. For the Abelian case, seo
(23.20). The non-Abelian case will be taken up in Vol. II, (27.39).
(14.17) we see that l#l -- Iv[ -- )., and from (19.11.i) that ).,).--)..
Hence every subset of T is l#,vl-measurable, while (16.13.d) shows that
there are subgroups of T that are not )-measurable. 1
(c) The hypothesis in (9.) that A be I#] ,]v]-measurable is needed
in order to ensure that -I(A)is ]/xvl-measurable. [Take G--T and
and v the measures described in (b). Let H be a subgroup of T such
that T/H is countably infinite (16.13.c). Then H is l#,vl-measurable,
since/*v--0. Theorem (2.2) shows that (Tx T)/ -I (H) is isomorphic
with T/H. Thus -I(H) comes under (16.13.b) and is not measurable with
respect to Haar measure on TxT. If ). is normalized Haar measure
on T, then ).x). is normalized Haar measure on TxT (15.17.j). Also
by (14.24) and (b) above we have l#xvl--I#I x Ivl--). x ).. That is,
r-1(H) is not I/xv]-measurable.l
(19.26) Properties of M 8 (G).
(a) Let G be a nondiscrete locally compact group. Then M(G) is
different from {0}. In fact, every nonvoid open set in G contains a
compact set which is the support of a nonnegative measure # in M (G)
for which/, (G)----1. I Suppose first that G is metrizable, i.e., that {e} is
the intersection of a countable family of open sets (8.5). Let U be any
neighborhood of e. Then the construction used in (4.26) can be applied
to G and U to produce a subset F of U such that ).(F)--0 and F is
homeomorphic with (0, 1 }°. Thus F is homeomorphic with the product
of W 0 2-element groups. Normalized Haar measure on this product
group can be transferred to a measure on F and thence by (11.45) to a
measure # on G which is certainly continuous and singular. If G is not
metrizable, (5.14) and (8.7) show that every neighborhood of e in G
contains a compact infinite subgroup N such that ). (N)--0. Normalized
Haar measure on N can be used with (11.45) to produce once again a
continuous singular measure # on G as required. The invariance of
Haar measure on G shows that an arbitrary nonvoid open subset of G
can be used instead of a neighborhood of e. 1
(b) The subspace M(G) need not be a subalgebra of M(G). Let
G and G. be infinite compact groups, let )-1 and ).. be normalized Haar
measures on G1 and G2, respectively, and let G--GlxG.. Denote by e i
the identity element in G.. If /G(G), then the function x-->-/(x, e2)
belongs to G(G1) and /-- f/(x, e2)d2 l(x) defines an element of G*(G).
Let )-1' be the measure in M(G) such that f/d2*l--f/(x, e2)d2 (x) for
/ G(G)" similarly let 2*2 M(G) be taken so that .f / d ;t - f/(el, y) d ;2(y).
Then it is clear that 2 and ). are in M (G) and also that 2ў*v=2.2. ,. =2,
Where 2 is normalized Haar measure on G.
278 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 279
(19.27) Let # be a measure in M(G). Suppose that for every compact
subset F of G for which (F)--0, either the function x-->#(xF) or the
function x-->#(Fx) is continuous at e. Then # is absolutely continuous.
IIf the function x--># (xF) is continuous at e, so is the function x-->#(x-lF).
Let U be a neighborhood of e such that 2(U)< oo and let vMa(G ) be
defined by dv=ud. By (19.18), I#l,v and v,[#[ are in Ma(G). By
(19.11), we have 0- ,l l (F)- f f dx>__
G U
f 1# (x-lF)] dx. Since 1# (x-F)] is nonnegative, we have # (x-F)=0 for
u
2-almost all xU" and hence #(e-F)=#(F)=O. The argument if
x->ff (Fx) is continuous at e is similar.]
(19.28) (a) (BRACONNIER [11. ) Using (t9.17), we can generalize (15.t2)
and (15.13). Every locally compact group G having equivalent left and
right uniform structures is unimodular. [Assume that G is not uni-
modular and that the left and right uniform structures are equivalent.
Let U be a neighborhood of e such that 2 (U)< oo. By (4.t4.g), there is
a neighborhood V of e such that xVx-C U for all xG. Choose x 0 in G
such that A(x 1) ). (V) > ). (U). Then x o Vx c U and ). (x 0 Vx ) =
(x0 v) = (v) > (v).l
(b) The converse of (a) is false, as the groups ® (n, R) show. By
(15.28.a), these groups are unimodular; by (4.24), their left and right
uniform structures are inequivalent.
(19.29) Convolutions involving unbounded functionals. (a) Let G
be a locally compact group" let L and M be nonnegative linear functionals
on 00(G) such that the measure corresponding to M has compact-
support E (11.25) ; let ff and be the measures corresponding to L and M
respectively as in § 11. Thus v is in Mч(G) and # is arbitrary. For every
/00(G), Lemma (19.5) shows that the function M/defined by
! (x) = M(,I) = f ! (xy) dv (y)
is continuous" observe that it vanishes outside of the compact set
F= {uG'/(u)=O}-. E -, and is nonnegative if ] is nonnegative. Hence
the mapping
l-->Z(l) = f f l(xy) dv(y) d#(x)
GE
is a nonnegative iand obviously linear functional on the linear space
00(G). We write L*M for this functional, and #.v for the measure
corresponding to L*M as in §11. Thus L*M exists if L and M are
nonnegative linear functionals on 00(G) and the measure corresponding
to M has compact support.
(b) Consider a left Haar integral I on 00 (G) and any functional M
as described in (a). Let E, v, and F [defined for ]00(G) be as in (a).
Let ; as usual denote the Haar measure corresponding to I, and let dx
stand for d(x). Then we have I,M=cI, where c= f A(y -) dr(y).
E
Iltis easy to see that -(o]) =(-]) ; hence I, M(]) =I ((r])) =
I(M/)=I,M(/). By (15.5), therefore, there is a nonnegative number c
such that I*M=cI. To evaluate c, we note that
I, M(I) = f f ! (xy) dv (y) dx
GE
= f f I
FE
Hence (t 3.8) can be applied, and we have
I, M(]) = f f l (xy) dx dv (y)
EF
= f f l(xy) dxdv(y)
= f Z(l) a (y-,)
E
(c) Let if, v, and ff,v be as in (a), and suppose further that the support
of ff is d-compact. Let ] be a function in (G, if,v) and let z be as
defined in (t9.t0). Then we have ffdff,v= f (]oz) dffxv=
G GXG
f f ] (xy) dv (y) d (x) = f f ] (xy) d (x) dv (y). [The proof is carried out
GE EG
by repeating mutatis mutandis the proof of (t9.t0). An important dif-
ference occurs only in showing that ffxv(z-(D))=0 [notation is as in
the proof of (t9.10)]. Here it is needful to recall that the support of
ffxv is d-compact so that z-(D) is xv-null if every compact subset
of - (D) is xv-null.
(d) Let if, v, and ff,v be as in (a), and suppose again that the support
of ff is -compact. Let A be a if,v-measurable subset of G. Then we have
ff, (A) = f f
GXG G E
[If ff,v(A)< , this is a special case of (c). An elementary arment
using the monotone convergence theorem proves the result for arbi-
trary A.]
(e) Let G be a locally compact, d-compact group, let H be a compact
subgroup of G, let v be any nonzero measure in M+(G) whose support
is contained in H, and let 2 be a left Haar measure on G. Then for
every 2-measurable set A cG, we have 2(A)- v6) v((x-lA)OH) dx.
[This follows from (d), (b), and the obvious fact that A(y-)= 1 for all
280 Chapter V. Convolutions and group representations § 19. Introduction to convolutions 281
(19.30) Baire sets and Haar measure (KAKUTANI and KODAIRA [J "
proof based on HALMOS [21, pp. 282--289). For the definition of Baire
sets, see (J.l). Throughout this example, ). will denote a fixed left
Haar measure on whatever group G is under consideration.
(a) Let G be a compactly generated, locally compact group, and let
U be an open subset of G such that 0<2(U)< o. Then U contains a
set D such that,:
(i) D is a Baire set;
(ii) ;t (U VI D') -- 0"
(iii) there is a compact normal subgroup N of G such that UN--
{xcG'[(x)>O} for some [C'(G), and (UN)--(U).
[Define by finite induction a sequence of compact sets F,, each of the
form {xG'](x)--O} for some ]' (G), and a sequence of neighborhoods
V of e such that" ). (Fn)-+- -- >). (U)" Fn. V, is a compact subset of U"
and F.IF. V, for n=,2, .... Theorems (4.t0)and (.23) and
obvious properties of continuous real-valued functions show that this
construction is possible.
By (8.7), there is a compact normal subgroup N of G such that
NC,f=IV,and GIN has a countable basis for open sets" plainly GIN is
metrizable. Let 0 denote the natural mapping of G onto G/N, and let
D--U F. It is easy to see that D--DN, and it is obvious that (i)
and (ii) hold for D.
We now note that 9(U) is open in GIN (5.17). Since GIN is a metric
space, there is a function g'(G/N) such that g(xN)>O for xN9(U )
and g (xN) -- 0 for xN ў 9 (U). The function g o 9 is continuous on G
and positive exactly on UN.
Next let v denote normalized Haar measure on the compact group N.
From (J.45) and (19.29.e) we infer that
O--(UVID')-- f v((x-(UVID')) NN) dx }
()
> f v((x-(UOO'))ON)dx.
(UN) riD'
If xc(UN)ND', then we have
(x-](U D')) Y -- (x - U) N
as is easy to see, and thus the integrand v((x-(VD'))NY) in ()
is positive throughout (UN)D'. It follows from (1.27) that
.((UN) D')--O. Since DC UC UN, we have proved that .(UN)--.(U).
Thus (iii) is established.]
(b) Let G be as in (a) and let A be any 2-measurable subset of G.
Then there are Baire sets C and D such that D CA C C and (C D') --0.
ISuppose first that I(A)< oo. Let {U,},°°__ 1 be a decreasing sequence of
open sets such that U, D A and lira ). (U, -- ). (A). Using (a) choose for
each n a set V,of the form {xG'/(x)>O} for some ['(G) such that
V,D U, and ). (V, -- ). ( U) . Let C--C V,. It is obvious that C is a
n=l
Baire set, that CDA, and that A(C(A')--O. Let {W,)n°°__l be a decreasing
sequence of open sets such that W, C A' and lira ). (IF,:0; let Y be
an open Baire set containing W such that (,)-- (W,); let Y--
and finally let D:C Y'. Then D is a Baire set, and the relations
DcA cC and A(CD')--O are evident.
Suppose now that A is any ).-measurable subset of G. Being com-
pactly generated, G is a-compact, and so A -- U A where {A)°°= is an
n----1 n,
increasing sequence of ).-measurable sets of finite ).-measure. For each n,
choose Baire sets Dn and C such that D cA c C,, and ).(C D)--0.
Then set D-- U D and C-- U C.
(c) Let G be an arbitrary locally compact group and let A be any
).-measurable subset of G. Then there are Baire sets C and D such that
DcA cC and C D' is a locally ).-null set. [Let H be a compactly
generated open subgroup of G (5.14) and let {xH: xX} be the family
of all distinct left cosets of H in G. We obtain a left Haar measure on
H by restricting ). to subsets of H" and each coset xH is a topological
and ).-measure-theoretic replica of H. Thus for each x one can use (b)
to choose Baire sets D, and C, as in (b) such that D, cA (xH) C C, xH
and .(CD')--0. Now set D-- U D and C-- U C It is clear that
xX xX "
C 0 D' is locally ).-null" and it is easy to see from the construction in (b)
that C and D are Baire sets.
Notes
The notion of convolution [Fr. Produit de composition, Ger. Faltung,
Russ. CBelTICA is a venerable one in both analysis and algebra. The
convolution
I f sin ((n + )(t- x)) dt
9 (t) 2 sin } (t -- x)
is found in DIRICHLET'S original memoir Ill. WEIERSTRASS'S original
proof [1 of his famous approximation theorem uses the convolution
282 Chapter V. Convolutions and group representations §20. Convolutions of functions and measures 283
For I(R), the limit of this expression as 0 is I (x); the convergence
is uniform on compact sets; and this expression is also evidently the
uniform limit of polynomials, on each compact subset of R. This proof
of WEIERSTRASS'S approximation theorem retains a certain charm, de-
spite the many other proofs that have been given.
Fractional integration and differentiation are defined by means of
convolutions: see for example ZYGMUND [1, Vol. II, pp. 133--142. The
central r61e played by convolutions in classical harmonic analysis is in
evidence throughout ZYGMUND'S treatise [1. Convolutions of functions
of finite variation on R [equivalently, convolutions of complex Borel
measures on R were discussed by BOCHNER [11, §§18--21. An algebraic
viewpoint concerning convolutions of measures on R was manifested in
the papers of WIENER and PITT [1 and BEURLING [t. GEL'FAND in [t]
pointed out that gl (R) and M(R) are Banach algebras and initiated the
study of their structure that has continued up to the present day.
It is worth pointing out also that the WlENER-HOPF equation
f K(x--y) (y)dy-- (x)
0
concerns a convolution on the additive semigroup I0, oo. For a very
detailed survey of results on this equation, see M. G. KREiN 1. Con-
volutions on [0, oo[ were also considered by VOLTERRA and P:RS [t].
Convolutions have also been used for many years in algebra and
number theory. For a finite group G, the classical group algebra [over K/
consists of all "formal complex linear combinations" .,x, with
xEG
componentwise addition and scalar multiplication and with the product
It is plain that this algebra is isomorphic with M(G) [and obviously
also with gl (G). The group algebra of a finite group G is an important
tool in studying the representations of G, as is brought out for locally
compact G in §22. For the history of this technique, see the notes to
Chapter III of WEYL [3. Convolutions of functions on semigroups that
are not groups also have a long history. DAUBLEBSKY VON STERNECK
wrote down and used for number-theoretic purposes convolutions of
functions on {1, 2, 3, ...} under multiplication. Many years later,
D. H. LEHMER [1 ] used the same formulas and also convolution formulas
based on other semigroup operations on {1, 2, 3 .... }. The/-algebras
various semigroups have been studied by W. D. MUNN II and HEWlTT
and ZUCKERMAN E2,
The decisive step in uniting the algebraic and analytic lines of
development in the theory of convolutions was taken by H. WEYL.
In [2], he defined the convolution of two continuous almost periodic
functions on R and used this operation together with the theory of
integral equations to establish the principal facts about continuous
almost periodic functions on R. WEYL and PETER in [1 ] defined con-
volution of two continuous functions on a compact Lie group G and
used exactly the same technique as in WEYL [2] to study the representa-
tions of G. In [4], VON NEUMAN exploitedWEY's convolution of almost
periodic functions to establish the theory of almost periodic functions on
an arbitrary group.
Convolutions of measures in ' (G) as formulated in (19.6) --(19.12) were
defined [in a different form] by WElL [4], pp. 46--48; a definition almost
identical with ours for @(G) appeared in H. CARTA [2]. The general
definition of convolutions in (19.1)--(19.3) is due to HEWlTT and
ZUCRMA [2] and [3]. An independent and somewhat different
formulation has been presented by R. C. Buck [1] and [2].
The construction in (19.4) is due to HEWlTT and ZUCKERA [4].
The notions introduced in (19.13) go back to VITALI and LEBESGUE
for the real line: see for example LEBESGUE [1 ]. Their adaptation to the
case under study is widely known. Theorems (t 9.t 6) -- (19.20) are an-
nounced in HEWlTT [5], p. 141.
§ 20. Convolutions of functions and measures
In this section we present a number of definitions and formulas
dealing with convolutions. These are of some interest per se and are
also essential in our study of representations of locally compact groups.
Throughout this section, G will always denote a locally compact group.
We begin with some technicalities [(20.)-(20.4).
(20.1) Theorem. Let ] be a nonnegative -measurable /unction on G
and let a be any element o] G. Then ] and ] are -measurable,
(i) f d.--f ld.,
G G
and
(ii) f 1 d A -- A (a -) f I d
G G
I//is a/unction in (G), then ] and/ are also in (G) and (i) and (if)
hold/or/.
Proof. Let / be nonnegative and .-measurable, and let
(t - 5n:= l be
an increasing sequence of .-measurable functions, each assuming only
finitely many values, such that lira o"(x)=/(x) everywhere on G. We
plainly have lira ("(x)=(x) for all xG, and so lim fd)d2=
d 2, lim d f d 2. Since f d f d for all
n G G G
284 Chapter V. Convolutions and group representations § 20. Convolutions of functions and measures 985
holds. In view of (19.17), we have f a "1 d--A(a-)f d"> d3 for all n.
G G
Formula (ii) is proved from this equality just as (i) was proved.
To prove the last assertion, note first that if /(x)--g(x) ).-almost
everywhere, then af(x)--ag(x) and /a(x)-----g(x) ).-almost everyvhere.
Now write [ as [--[.+i(/--[) where each /;(G), and apply the
previous case to these functions.
(20.2) Theorem. Let [ be a nonnegative ,-measurable /unction on G.
Then ] is also -measurable,
, (G), (ii) holds,
?oof. Conside first a -nul] set A. will pove that A -1 is -null.
.:,, ..., {x C" m.
-null, it suffices to prove that A - D, is -nul] for each . Choose, > 0
and choose an open set U containing A sck that ;(U) f-. It is
obvious that U - D, is open and that A - D, U - D,,. No con-
sider an 7 function l;0() such that iv-,,. Then I*= and
(l $. t ) iplies that
sm ' f'
G UDn
Since I is arbitrary, we infer that (U - D,) , and brace also
(A - D,) ,. As is arbitrary, (A-1
-nu]l set.
Consider next a -measurable set E sch that (E)< . Then E
is the union of a 7orel set B and a -null set A. Since B -1 is also a 7ore1
set and A - is -nu]l, E-:B-UA - is -measurable. Finally, consider
an arbitrary -measurable set C. IЈ ? is compact, then ?-1C is
-measurable. 3y the preceding case, 3 C -1 is also -measurable. As
F is arbitrary, (ll.)l) implies that C -1 is -measurable. Since C -1 is
-measurable whenever C is, it follows immediately that 1" is -meaurable
whenever I is.
e now prove (ii). For functions 1 in 0 (G), Theorem (] $. ] 4) and Corol-
lary (]t 37)imply that f / d: f 1" '
. 3-d. Suppose that ge9+(G)" i.e.,
G G
g is a nonnegative lower semicontinuous function on G. Then by (11.1 t),
.lgd = sup {! ld'io(G) and/<: g}
G
-d2" [oo(G) and /g
Replacing g by g
in (1), we obtain
= - •
Now let [ be an arbitrary nonnegative 2-measurable function on G.
Then by (11.36) and (11.16), we have
f 'd2= inf ?gd2 " g+(G) and g [}
=inf g d2 " g*(G) andg/ / d2,
-
and replacing / by/
in this inequality, we also have
f /d2 > .
The remaining statements of the theorem follow at once from (ii).
(20.3) Note. Theorem (20.2) does o imply that/ (G) if/ (G)"
see (0.9.a).
(20.4) Theorem. Let p be a mber sch ha 1 p <
be a/ncion in (G). For ever e > O, here is a mighborhood U o/ e i G
s,ch
hat is, he maig x,/ o/ G io (G) is righ i/orml coHos.
For ever /ixed G ad ever e > O, here is a mighborhood V o/ e sch
ha is, he aig x/ o/ G io (G) is coHos.
Proof. Let be a function in ,, (G) such that I/--n< (12.10).
Let F be a compact set such that (F')=0 and let W be a symmetric
neighborhood of e such that W- is compact. Let U be a neighborhood
of e such that U W and such that ab- U implies that ](a)--(b) <
286 Chapter V. Convolutions and group representations
--2 (WF)--- ( .4). If st-l U, we then have by (20.1)
3
f l(xl - (xlldx- f l(s-xl - (x/i dx
that is, ]ls--olp< -. Using (20.1), (12.6), and (12.7), we now can
write
- 2 I1!- 1, + IIs-, II, < 2 + - ,
This proves (i). t
To prove (ii), choose @oo(G) such that II-/ll< 7 A(t)--. Let W
be a symmetric neighborhood of e such that W- is compact and
W{x6-A(x)<2P}. Let E be a compact set such that (E')=0 and
let V be a symmetric neighborhood of e in G such that V W and such
1 1
that b-a V implies [(b)--(a)[ < (e/4) 2(EW)--- A(t)>-. Then if s=tv,
where v V, we have from (20.1) that
= (-/ I (/- (/I < 7"
E
As above, we obtain
IlL - I, II = II1,o- I, II -<- IIl, - , II, ч II, - % II, ч I1,- I, II
- (v-,)-- (,-,) III- 11 ч I1,=- ,11 ч A ([-1) -p- [['P -- lll
<2.7+-+7=e. [
Our basic definition follows.
(20.5) Definition. Let # be a measure in M(G) and v a measure in
Ms(G ). Let / be the function in 1 (G) as in (19.18) such that dv -- / d.
By (19.18), the measures #,v and v,# are in Ma(G ). Let #,/and/,#,
respectively, denote the functions in I(G) such that d(#,v)--(#*/)d2
and d(v,#)--(/,#)d2. If # is also in Ma(G), so that d#= g d2 for
g (G), then we write/, g for the function in (G) such that d (v,#) =
(/, g) d 2. We call # •/,/, #, and/, g convolutions.
We wish to obtain explicit representations for #,/, /,#, and /,g
as defined in (20.5), and also to extend the definition of these entities
to classes of functions other than I(G). We begin with two measure-
theoretic lemmas, which are of little intrinsic interest but are essential.
"Function" means "extended real- or complex-valued function" through-
out this section.
§2o. Convolutions of functions and measures
287
(20.6) Lemma. Let X and Y be locally compact Hausdor// spaces. Let t
an be measures on X and Y, respectively, as constructed in § 11. Suppose
that z is a continuous mapping o/ X x Y into X such that
/or every t-null F, set B C X and every y Y,
Then / o is an t x -measurable /unction on X x Y/or every t-measurable
/unction / on X.
Proof. (I) We will first show that if E is an t-null subset of X, then
z -1 (E) is t x -measurable. There is a Borel subset B of X [in fact a G setJ
such that E C B and t(B) --0. Write B 0 for z-l(B). Observe that B 0 is a
Borel set in X x Y Iin fact a G setJ and hence is t x -measurable, since z
is continuous. Let F be any compact subset of X xY. By FUBINI's
theorem (13.9), we have
XXY
d,x r -- f f o (x, y)d,(x)d(y). ()
Y x
For every y Y, the set 2V,e is ,-null by (i). Since
for all (x, y) in XxY, (1) implies that
YX y
Since t x is a complete measure and z- (E)C Bo, we have proved that
F fq z-l(E) is t x -null for all compact sets F C X x Y. Therefore z-1 (E)
is tx -locally null and hence tx -measurable by (11.30).
(II) Next we show that if E is an t-measurable subset of X and if
t(E) < oo, then z-l(E) is t x -measurable. There is a Borel subset B of X
Iin fact an F, setJ such that BC E and t (B) -- t (E). Thus we can write
E -- B U A where (A) -- 0. Then z-x (E) = z "-1 (B) U z "-1 (A) is x -
measurable by (I) and the fact that "C-1 (B) is an F set and hence is x-
measurable.
(III) Now we show that if E is any -measurable subset of X, then
z - (E) isx -measurable. Let F be a compact subset of X x Y. Since z
is continuous, the set C = z (F) is compact in X. By (II), the set z-x (C r3 E)
is t x -measurable, since (C r3 E) =< t (C) < oo. The obvious equality
z-x (E) r3 F = z- (C r3 E) r3 F shows that z-1 (E) r3 F is x -measurable.
This holds for all compact sets F C X x Y and so z-(E) is t x-mea-
surable by (11.31).
(IV) Finally, consider any -measurable extended real-valued func-
tion ! on X. If B is a Borel set of real numbers,/- (B) is an -measurable
subset of X, and (III) shows that z-1 (/- (B))is an, x]-measurable subset
of XxY. Since z - (/q(B))=(/o z)-(B) and B is an arbitrary Borel set,
we conclude that /o z is ,x ]-measurable. Complex functions/are dealt
With by considering their real and imaginary parts. [
9-88 Chapter V. Convolutions and group representations § 20. Convolutions of functions and measures 9-89
(20.7) Lernrna. Let G be a locally compact group, let 2 be any le/t Haar
measure on G, and let 7 be a measure on G constructed as in § 11. Let ] be a
2-measurable [unction on G. The [ollowing [unctions are x7-rneasurable
on GxG"
(x, y) + l(xy),
(i) (x,y)-/(yx)
(x, y) l(yxy-),
(x, y) - /(xy-),
(x, y) / (y-ix),
(x, y) -/(x),
The/ollowing/unctions are x2-measurable"
(ii) (x, y) -+/(y),
(x, y) - /(y-).
(X, y) "-+ 1 (X -ly),
(X, y) -+ /(y x-l),
(X, y) -->" /(x-l).
Proof. We will check two cases of (i), leaving the rest of the lemma
to the reader. The function (x, y)--/(xy -1) has the form /o where
z(x, y)=xy -1 for (x,y)GxG. For a subset B of G and yG, we have
{xG : (x,y) B} = By and so the condition (20.6.i) fallows from (9.7).
Hence (x,y) --/(xy -1) is 2x 7-measurable by (20.6). If z is the mapping
(x,y)=x - for (x,y)GxG, then {xG: (x,y)B}=B -1 for all yG,
and the condition (20.6.i) follows in this case from (20.2). Hence the
function (x,y) --/(x -1) is 2x7-measurable, again by (20.6). ]
We can now prove the general theorem that shows the integrability
of convolutions. It too is rather technical.
(20.8) Theorem. Suppose that I <=p <= oo. Let / be a 2-measurable
/unction on G, and let be either the/unction (x, y) -+ x-ly or ( x, y)-+ y x -x
mapping G x G onto G. Suppose that q9 is an 171 -measurable/unction on G,
where either (a) 7 is a measure in M (G), or (b) 7 is a nonnegative measure
as in § 11 and q9 vanishes outside o/ a a-compact set. Finally suppose that
there is a positive number such that
(i) f f Iw(y)q(x)/o(x,y)ldydll(x)<lwl,,
GG
/or all v/6(oo (G) . I/I <= p < and / vanishes outside o/a a-compact set S,
then
(ii) f q(x)/ o w(x, y)d7(x ) --h(y)
G
exists and is finite/or 2-almost all y_G" h belongs to p(G) and [h
I/ p= oo and /6(G), then the integral in (ii) exists and is /inite /or
locally 2-almost all y-G" h is in (G)" and h <= 7.
Proof. (I) First we show that the function h of (ii) exists and is
2-measurable. In case (a) we select an ]l-measurable function g such
1 Recall our convention that 1' -- oo and oo" -- 1" see page 141.
that d] =Ndl I and = (4.2). Since there exists a a-compact subset
D of G such that I rll(D') = O, we may suppose that g vanishes outside of
a a-compact set. In case (b) we define g(x) = 1 for all xG. In this case,
vanishes outside of a a-compact set. Thus in either case we have
d=gdlrl[ and gq9 is an [[-measurable function that vanishes outside
of a a-compact set E.
Let W be in if00 (G)" then W vanishes outside of a compact set At. The
foregoing and (20.7) show that H(x,y)=(y)/o(x,y)(x)g(x) is an
x2-measurable function on GG that vanishes outside of the a-com-
pact set EA. Thus we may apply (13.10), citing (i), to obtain
f f H(x,y)dydl(x)=f f H(x,y)dl](x)dy. (1)
GG GG
By (3.0), (13.8), and again (i), the integral
f H(x,y)dl I (x)=W(Y) f / o w(x,y)(x)g(x)dl [ (x) (2)
G G
exists and is finite for 2-almost M1 yG, and defines a 2-measurable
function in El(G). For yG, we define
h(y) = f / o w(x, y) (x)g(x)d][ (x) (3)
G
provided that the inteal exists" ve define h(y)=0 otherwise, as sug-
gested by footnote 2 to page 153. We see that w h is 2-measurable for all
00(G). By ( a.42), h itself is 2-measurable.
(II) We next show that h belongs to E (G) if a p < and / vanishes
outside of a a-compact set S. Let B =ES if z(x,y) =x-y and let B =SE
if z(x,y) =yx -" plainly B is a-compact. Hence there is a sequence
{} in 0 (G) such that = iim % 2-almost everhere and W (G) c
0, for all n. From (2) we infer that
h(y) = lira W(Y) f / o z(x, y) (x)g(x)dll (x)
exists and is finite for 2-almost all y B. It is easy to show that h(y) =0
for all y B'. Since B is a-compact, we thus have
(y) 1' dy = sup {/ l h (y)]'dy'F is compact, F G G}.
For in 00 (G), we have by (1) and (i) that
ana thus (2.4) and (12.13) show that h<(G) and that Halibut.
(III) Finally, suppose that / is in (G). Let N denote the set of y<G
for which the integral in (3) does not exist as a finite number. Assume
Hewitt and Ross, Abstract haonie analysis, vol. I 9
§20. Convolutions of functions and measures 291
290
Chapter V. Convolutions and group representations
that there is a compact set F such that it (N ( F) > 0. Choose pc0 (G)
such that p(F)= t. Then the integrals in (2) do not exist as finite num-
bers for this p and for y c N N F, a contradiction. Thus N is locally it-null,
and so (3) holds locally it-almost everywhere.
If {ycG: Ih(y)] >y} is not locally it-null, then {ycG: Ih(y)l >/3} is
not locally it-null for some/3 > y. Thus {yG : ]h(y)] >/3} contains a com-
pact set F for which it(F)> 0. Choose an open set U such that F C U
(U ( F')< - (fl- x - )it (F). Choose p in ff0 (G) such that p (F)= t
and
y '
p(U') =0, and p(G)c I0, tl. Thus by (i) and (t3.t0), we have
u
< y 2 (F) + (fl -- y) 2 (F) = fl 2 (F) G f ]W (Y) h (y)] dy
f f [/o (x, y) (x) W (Y)] d ][ (x) dy
GO
= f f ] (y) (x) [ o z (x, y)] dy d ]] (x).
GO
This is contrary to (i).
With (20.8), we can now explicitly represent the functions ,[, [,,
and/,g as defined in (20.5).
(20.9) Theorem. For M(G) and M(G) so that dr=[ d2 with
[ (G), we have
(i) # , / (x) = f/(y-Xx) d (y) /or 2-almost all
and
(ii) /,(x) = f d(y-X) /(xy -) d(y) /or 2-almost all xG.
Proof. For YG00(G), (9.0) shows that
f d,v f f W (xy) / (y) dy d (x).
O GO
Using (20.), we can write
f y(xy)/(y) dy = f,[w(y):,/(y)] dy = f y(y)/(x-Xy) dy,
for all x G. Thus
f d,v f f W(y)/(x-xy) dy d(x). ()
O GO
We will now apply (20.8) with p--t. Note that / must vanish outside
of a a-compact set, in fact outside of an open a-compact subgroup. Let
r(x, y)=x-ly, qb=t, 7=ff, and r=ll llll ll. For pўoo(G), we have
dd IV: (Y)/(x-lY)l dy dl/z I (x) < Jig: lld ix_/]ll dl I () /
-- I w Iloo Ill fix II:' II = lloo II"11 I1:' II.
J
(2)
Therefore by (20.8), the function h(y)= fl(x-ly) d#(x) belongs to
G
1 (G). Moreover, using (t), (2), and (t3.8), we see that
f v, hd2---- f f W(Y)/(x-lY) dff(x) dy
G GG
--- f f W(Y)/(x-ly) dy dff(x) -- f d/z,?., -- f ff,/dit
GG G G
for all Pўo0(G). Theorem (t2.t3) shows that II,./-all.=o. and this
implies (i).
For (ii) we let "c(x, y)=yx -1, q=/1-1, 7=#, and -II llll ll. Then
for P00(G), we have
Now argue as above.
(20.10) Theorem. For ], gI (G), we have
(i) /,g(x) -- f l(y) g(y-lx) dy,
(ii) [, g (x) = f ! (xy) g (y-) dy,
6:
(iii) [,g(x) = f A(y -)/(xy -) g(y) dy,
6:
(iv) /, g (x) = f A (y-I) / (y-l) g (yx) dy,
G
each equality holding [or 2-almost all x c G.
Proof. Equalities (i) and (iii) follow at once from (20.9) and the
definition of/,g. To prove (ii), we write
f 1 (y) g (y-1 X) dy --- f / (y) g (X -1 y) dy
G G
--- f ,,-,[d (Y) g* (y)] dy -- f ! (xy) g (y-i) dy.
G G
19"
292 Chapter V. Convolutions and group representations §20. Convolutions of functions and measures 293
To prove (iv), we write
f -1) f /,(yx-1) g(y)dy
A (y) [ (xy g (y) dy -- A (y)
1
(y)
dy.
A(x -1) f[
One can also obtain (ii)--(iv) from (i) alone by applying (20.) and
(20.2).
(20.11) Remark. Convolutions of functions and translations are
connected by the following identities, in which / and g are -measurable
functions and a d x are elements of G:
(ii) [.(g) (x)=[.g(xa)=([.g)(x),
(iii) (&) • g (x) = (a -t) [. (d) (x).
Each of these identities means that whenever one of the integrals exists
and is finite, the others also exist and all are equal. These identities
follow at once from (20.t).
The convolution formulas (20.9.i) and (20.9.ii) for M(G)and/ (G)
can be extended to [(G) (t<p). The following two theorems
show how this happens.
(20.12) Theorem. Let [ be a [unction onG that is in (G) (t p ),
and let be a measure in M (G). Then the integral
exists and is finite/or all xGN', where N is -null i/ t p< and N
is locally -null i/ p =. De[ining #.[(x) as 0 where it is not defined
by (i), we obtain a/unction in (G), [or which
Proof. We apply Theorem (20.8). If p is finite, the function [ vanishes
outside of a a-compact set. Let z(x, y)=x-y,
II It little- I 00 (G), then H6LDER'S inequality yields
SIl(x-y) ()1 y II,-,111. I111, = I111-I111,.
e therefore have
ii I()I(-')1 1,1 () IIII1-I11, 11 ()= IIlt,. IIIII,I111.
The present theorem follows from (0.).
For the convolution i*, with i (), our results are less elegant
than for #*i. This difference arises because of the aisfinction Detween
left and rilh Haar measure.
1
(20.13) Theorem. Let,u b 7n M(G) and suppos that f J(y)-Y-dll(y )
' P [t'--o, o'--t, and-- --0].
is /inite, where <= p <= o and p -- p_ t co
let / be a/unction in p (G). Then the integral
(i) fA(y -1)/(xy-1)d/z(y)--/./z(x)
G
exists and is finite/or all xGCN', where N is 2-null i/ t <=p < o and N
is locally C-null i/ p--o. Relation (i) de/inns a /unction in p(G) /or
which
(ii) II/./z 1 < II/11 f A(y)-- dl/z I (y).
G
Proof. In order to apply Theorem (20.8), let (x, y) =y 37 -1, ! =/,[-1,
1
=/, and - I1 I f A(Y) -- dl/l (Y). Let be in oo(G). By H6LER'S
G
inequality (t2.4.ii) and Theorem (2b.t), we have
1
f ]/(Y x-l) w(y)l dy < I/,,-'l]s, I[ 1]' = A(x)k-IIlll IIw I,.
G
Therefore we have
f f I(y) -1 () /(y-l)i ay dl[ () -<- f ()- II/ll IIwll, a I I ()
GG G
1
(20.14) Corollary. Let/1 (G) and g (G) (t <= p <= ). The integral
(i) l,g(x) = f l(xy) g(y-1) dy-- f l(y) g(y-1.) dy
G G
exists and is finite /or all xGCN', where N is C-null i/ p < oo and is
locally C-null i/ p- o. The/unction/.g is in (G), and
(ii) ]l/*g ], <= [Jill [g [P"
II / (G) and A -- g 1 (G), the integral
(iii) l.g(x) = f A(y -1)/(xy -1) g(y) dy = f/(xy) g (y-l)dy
G
- fl (y)
G
exists and is /inite /or all x.GfN', where N is as above. The /unction
!. g is in (G), and
(iv)
Proof. The statement for/ 1 (G) and g p (G) follows at once from
(20.'12).
For /p(G) and A--ў;-gI(G), we apply Theorem (20.8). [Note
that if g happens to be in I(G), then we can simply apply (20.t3) with
1
d,u=g d/t. Let z(x, y)--yx -1, ---A-lg, --/t, and y= 1 p A--g 1"
Note that q5 is a/t-measurable function that vanishes outside of a a-com-
pact set. A computation similar to that in (20.t3) shows that
294 Chapter V. Convolutions and group representations § 20. Convolutions of functions and measures 295
1
for P00(G). l
Theorems (20.t2) and (20.t3) lead naturally to the following result,
which expresses a continuity property of the mappings /-+#,/ and
-,, for/(G).
(20.15) Theorem. Let [ be a /unction in p(G) (t <=p < co) and e
a positive number. There is a mighborhood U o/e in G such that
(i/II * !- ! I1 <
[or all # c M + (G) such that # (G) -- t and # ( U') = O. There is also a neigh-
borhood V o/e such that
(ii) ]]/,#--/]]p<
]or all #Mч(G) such that #(G)--t and (V')--0.
Proof. By (20.t 2), we have/, [ (G). Let U be a neighborhood
of e in G such that Ily-'/--/11 < if y V (20.4). Then if 00 (G) and
#(V')=0, the function (x, y)-- ][(y-lx)--/(x)l. IV(x)/ satisfies the con-
ditions of Theorem (t3.t0). Thus we have
Theorem (t 2. t 3) implies that IIt • 1-- / If < e.
To prove (ii), let V be a neighborhood of e in G such that
] f IA(y-1)--l]< -2 - for yV and/[(y-I) I],_,_[ Ip<__ for yV. Then
if M + (G), t (G) = t, and t (V') = O, we have
= (:)
I/, [a- f A(y-:)d/, (y)[ < f / ,. t- A(y-I)I d(y)<2"
v v
For 00 (G), we have
fll*" (x)- (,f A (y-')m, (y))1() I Iw()l d
= fl f (1 (xy-:)- 1())A(y-:)d (y) I I (x>l x
G V
GV
VG
i ,. f 1,_. _ i i A (-1) d,. () [p,.
v
Theorem (I 2. t 3) implies that
v 2
Combining (:) and (), we obtain (ii).
(20.16) Theorem. Le p be real mbe sch ka t < < , ad
p-l" '
(i) f l(y) (y-1) y = 1.()
exists 1o all x:G and elies
, (G). I 1 I : (G) ad
Proof. Suppose first that 1 () and *: , (), < p < . By
HO::'s inequality (7.4.ii) we have
fll(.) :(-1)1 .l, I:* I,, = III1 I1:*11,
for all :. Thus l.g(s) exists and is finite for all s, and also
I,: I1 I I:*11,. Applying HO:::'s inequality again, we have
I1.:(*)-f.:()1 Isl-,lll, :*11,
for all s, : . It follows from (20.4.i) that l*g is right uniformly contin-
uous. If 1 (:')--0 and g (f')=0, where and are subsets of 6, then
1.(()') =o. Thus if
1. is in 00(:). Otherwise, let {1,},1 and {,},: be sequences of
functions in 00() such that lim III=-11=o nd lim :-:* ,=0
(t 2.10). Then we have
Therefore
lim I=,-I, I=o,
ana i* aocoraingl7 is a function in 0 ().
: 7or the definitions oi () and ,(), see (19.13.b).
296 Chapter V. Convolutions and group representations §20. Convolutions of functions and measures 297
If/ (G) and g o (G), then f/(xy) g (y-) dy plainly exists L and
is finite for all xG. For s, tG, we have
By (20.4.i) we see that f.g is right uniformly continuous.
If/eoo(C) and g*9..l(G), then we have from (20.t) that
1, g (x) -- f 1 (xy) g (y-L) dy ----- f I (y) g
G G G
As in the preceding paragraph we see that [/, g (s)-/, g (t)] is arbitrarily
srnall if s-Lt lies in a suitable neighborhood of the identity. Thus/,g
is left uniformly continuous.
(20.17) Corollary. Let A and B be ;t-measurable subsets o[ G such
that 0< ;t(A)< co and O<;t(B -L) < oo. The/unction x-+;t(A ( (xB-L))
is in (G) and is positive somewhere. The set A B contains a nonvoid
open set, and the set A (A -L) contains a neighborhood o/the identity.
Proof. Both of the functions A and ---B-, are in p(G) for
t=<p<=oo. Applying (20.6) with p=p'=2, say, we find that A*B
is in 0(G). The value A.B(X) is just ;t(AC(xB-L)). It is also clear
that A *B (X)---0 if X (AB)'. Finally, using (t3.9), we have
f A *B (x) dx -- f f A (xy) -, (y) dy dx
G GG
--- f f A (Xy) dXB-,(y ) dy = ;t(A) f z (y-l) B-'(Y) dy = ;t(A) ;t(B) > 0.
GG G
Hence A *B is positive on a set of positive ;t-measure, and is accordingly
f,
positive on an open subset of A B. Since ;t (A -1) = -d 2> 0, it is easy
A
to see that A contains a compact set E such that ;t(E) and ;t(E -1) are
positive. We have E,-,(e)=;t(E)>0, so that E(E -L) and also
A (A -L) contain a neighborhood of e. l
Still another result along the same lines as (20.14) and (20.t6) can
be obtained.
(20.18) Theorem. Let p and q be real numbers such that
1 1 Pq so that 1
t<q<o, -+-->t" and let r-- p+q_pq , -p- +-q---
' Note that if goo (G), then g*oo (G) and I1*11 = IIllo. This is because
;t (F)= 0 if and only if ;t (F -1) = 0, for compact subsets F of G [use (20.2)].
The result in (20.17) may be false if X (A) -- oo. To see this, consider G = R X R d
and A = {0} ><Rd. Then A is a locally null non-null X-measurable subset of G, so
that X (A) = oo. However, A A -1= A does not contain a neighborhood of (0, 0).
Let / and g be ;t-measurable /unctions on G such that leap(G), gee(G),
g*ee(G), and ]gle= g*le" Then/or ;t-almost all xeG, the integral
(i) l,g(x)= f l(xy) g(y-L) dy
exists and is finite. The/unction/, g is in E,(G), L and the inequality
obtains.
Proof. Again the functions / and g must vanish outside of -compact
sets. We write
f I/(xy) g(y-1)l dy -- f (l/(xy)l g(y-1)le)-I/(xy)l- ig(y-)lt- dy. ()
By (20.t4), the function y- I/(xy)l [g(y-1)e is in I(G) for 2-almost all
xeG. For all such x, we apply (t2.5) to the right side of (t), with n= 3,
1 1 1
= r' = t -- , and = t -- . This gives us
f [1 (xy) g (y-) ] dy
( [/(xy)[p [g(y-1)[q dy) (df [/(xy)[p dy) ( ]g(y_l)[q dy) ' } (2)
q (t )}:2=q. Hence/,g(x) existsand
since (t-- )q-I =p and -- P
is finite for X-almost all x e G. We now show that/,g is actually X-mea-
surable. We may, as usual, suppose that / and g are nonnegative. Then
there is an increasing sequence {/}L of nonnegative functions such
that lim ],x)=/(x) for all xeG and such that each ] is a finite linear
combination of characteristic functions of 2-measurable sets having
finite measure. Likewise g is the limit of an increasing sequence {g}.
of such functions. Each function /,g belongs to (G) by (20.t6).
The monotone convergence theorem implies that/, g (x) = lim ], g (x)
for all xeG, and hence ],g is X-measurable.
We can rewrite (2) as
. .
It is simple to verify that the hypotheses of (t3.9) are satisfied by the
function (x, y) II(xy)I lg(y-,)If, see (20.7). Thus we have
=
1 Note that r is always greater than 1.
298 Chapter V. Convolutions and group representations §20. Convolutions of functions and measures 299
The equalities (4) show that we can integrate both sides of the inequality
(3) with respect to x, over the group G, to obtain the inequalities
Taking r th roots of both sides, we find
(20.19) Summary of convolution formulas. For convenience in
writing relations involving convolutions, we introduce some abbreviated
notation. If is a set of functions on G, then * denotes the set
{/*"/ I}. If 9X and 3 are sets of functions on G, then . 3 denotes the
set {/.g'/9, g}. I and E are subsets om(G), then D., D.,
and . D are defined analogously.
We summarize (20.t2), (20.4), and (20.t6) as follows"
(i) 3I(G) • E (G) c E (G) for t p
(ii) (G) • (G) c (G) for t p
(iii) E(G),(E,(G))*Co(G) for
(iv) 1 (G) * (G) c ru (G)"
(v) () • ( ())* c ,. ().
The proo o (20.8) shows that the hypothesis gq= [g*[ is un-
necessary in order to conclude that [, g E, (G). In view o this, we have
(vi) E (G) • [E (G) (E (G))*] c E, (G) whenever + > and
p q
As seen in (20.3) and (20.4), we cannot state that E(G)*M(G)C
E (G) and E (G) • E (G) c E (G). I the group G is unimodular, however,
these relations hold. Moreover, in this case, (iii) and (vi) may be written
as f (G) • f, (G) 0 (G) and f (G) • e (G) , (G) + -- r "
The inclusions (iv), (v), and (ii) for t p < are actually equalities.
See Vol. II, (32.45) and (32.30).
(20.20) Discussion. The algebra M(G) admits an adoint operation
[see (2t.6) and (C.27) which is very useful in establishing properties of
M(G) and (G). To see what this operation should be like, let us
consider the mapping/ў,/of 2(G) into itself, for a fixed ўM(G).
We write Tv/=,u,/. Then as (20.t2) shows, L is a bounded linear
operator, for which Tv , carrying 2(G) into 2(G). Like all
bounded linear operators on Banach spaces, T admits an adjoint operator,
(Tv)= T see (B.53). Since 2(G) is a Hilbert space (t2.8), we have
for all/, g 2 (G)"
i.g.
f ,/(x) ( x= f/( T2g(x ,
or
f /(x) Tg (x) dx -- f f /(y- x) d, (y) g (x) dx.
G GG
Using (t3.t0) and (20.t), which is legitimate [although at the moment we
are concerned only with a heuristic argument to give us a reasonable
guess as to the form of T, we find
f f /(y-1 x) dig (y) g (x) dx -- f f /(y-1 x) g (x) dx d,lg (y)
GG GG
- f f /(x) g (yx) dx d/, (y)
GG
- f /(x) f g (yx) d (y) dx
G G
G G
Combining these computations, we see that T ought to be the trans-
formation of . (G) into itself such that
(i) T g (x) = f g (yx) d# (y).
G
We will show that T is in fact a convolution operator; that is,
T g = #~, g for a measure/~ M(G). We proceed to a formal definition,
which it is convenient to state in terms of linear functionals.
(20.21) Definition. For L(G), let L~ be the functional on 0(G)
defined by L ~ ()= L @-).
(20.22) Theorem. The/unctional L~ has the/ollowing properties [/or
all L, M(G)]"
(i) L~ is bounded and linear on o (G);
(ii) (oL+flM)~=gL~+M~ /or
(iii) L~~ =L;
(iv) (L • M)~= M~, L~;
(vi) IlL II = IIL
Proof. Assertions (i), (ii), (iii), and (vi) are very simple, and we omit
their proofs. To prove (iv), let,u and v be the measures in 2t/(G) represent-
ing L and M respectively. Then for q)0(G), we have
(L , M)~ (c?) --= ff ff t(Y -lx-1)*(xy) d (Y)dtt (x)du (x) --d (y) f f t (Y-1 x-l)d(y)du(x) } (t)
G G
[the last equality follows from (t 4.25). Now it is plain from (20.2t) that
L~(W) = f w*(x) d#(x) = f )(x -1) d(x) for o(G).
G G
300 Chapter V, Convolutions and group representations § 20. Convolutions of functions and measures 301
Thus we have
*~ ()= (/))* () = f (:)()
= f f (- -) ff () u (y).
Tlle identity (iv) follows at once from (t) and (2).
(2)
We now prove (v). By (t4.5) and (20.2t), if 0 (G) we have
ILI ()- sup{IL(-)[ "Y(o (G), Il }.
For 0(G) such that I[--<_0, we have ] = [*] *, and hence
[L()] [L[(*) = L() note that is nonnegative and ILl (*)
is a nonnegative number. It follows that Igl ILI on g (G). Thus
ILl : ILI ILI= ILI,
and so [L [ : [L[ on (G); this implies that ]L [ : ]L[ on 0 (G).
If Lg(G) and is the measure in M(G) representing L, we will
write for the measure representing L.
(20.23) Theorem. Let L(G) and let be the measure in M(G)
corresponding to L. Then"
(i) a subset A o[ G is [l-measurable i/ and only i/ A - is I-meas-
urable, and in this case we have (A):(A -) and ]ff[ (A) : [ (A-)"
(ii) a/unction / is in (G, ff]) i/ and only i/ /* is in (G, [), and
in this case we have f/d: f/ d ;
G G
(iii) lot every/(G) ( p ) we have
, / () = f / (y
G
the integral being finite lot x G N', where N is A-null i/ p < and
N is locally A-null i/- , and representing a/unction in p (G)"
(iv) i/ M (G), so that d : g d 2 with g x (G), then is also in
M (G) and d : g d 2.
Proof. To prove (i), consider first an open subset U of G. It is easy
to see that there is an increasing sequence (%). of functions in 0 (G)
such that lim .=a, where A cU and Iffl(UA'):O. Similarly
there is an increasing sequence {.). of functions in 0 (G) such that
lira . :., where B c U - and I ( U-x B') : 0. For n: , 2, ..., let
.:max(%,W); then lira .=v ]l -almost everywhere and
We write g for the function g* .
lim co =u-, I#l -almost everywhere. Thus by dominated convergence
(t 4.23), we have
--- * d/l -- (U -1)
~ * dff = lim f o. .
ff (U) = lim fco. d lim ( o. The additivity of # and #~ now shows that ff'(E)--ff (E -x) for all closed
subsets E of G. From (4. 5) we infer that if ]ff~] (B)=0, then Iff[ (B-l) --0.
An ]/,~[-measurable set A can be written in the form A--(,IF,) U B,
where {F,}°°__ 1 is an increasing sequence of compact sets and B is a set
of ]#'I-measure 0 disjoint from F,. Then we have
#~(A)-- lim ff~(F,) -- limff(F
- u :
Hne -1 is Iffl-me.s=.be, .nd 0) hod. If -' i Iff I-me.r.be, we
use the identities (A-1)-I=A and ff~~=ff to infer that A is lff l-me" -
urable.
To prove (ii) 1, consider first a function
where %. K and A: t'l._ I . Using (i), we have
f a dff ~ -- Y, %. if~ (A i) -- Y, :/z (Ay ) -- f a* dff.
G i=1 /=1
If/(G, Izl), there is a sequence of functions {a,},°°__l each having
the form (t)such that lim a,, (y) = / (y) and
for all y G. Dominated convergence gives us
lim f d#~-----f / d# ~ ,
n-- G G
and the previous case shows that
Furthermore we have
*d#
f.uz= fo. .
G G
1 Relation (ii) is obvious for/ 0(G): it reduces to the definition of/z~. Like
Theorem (19.10), however, (ii) is not completely obvious for the larger class of
functions for which it is asserted.
9. Write/=/1--/9. + i (/--/4) where the/j are real and min (/1,/9.) -- min (/, h) -- 0.
Then apply the construction of (11.36) to each/i"
302 Chapter V. Convolutions and group representations § 20. Convolutions of functions and measures 303
The theorem on monotone convergence implies that lira la,*]-----I['1 is in
1 (G, [#]), and thus dominated convergence implies that
lim f a- d# = f/d#.
n-- oo G G
Thus (it) holds for I 1), then /--/** in
9(G, I#[), since #--#. Thus (it) is proved.
Assertion (iii) follows at once from (it) and (20.t2).
To prove (iv), we use (t 4.t 7). As usual, g is taken to be Borel meas-
urable. For/ 0 (G), we have
G G G G G
By (t4.t 7), this proves (iv). ]
It is convenient to put down here some further algebraic properties
of the algebras M(G) and M, (G).
(20.24) Theorem. The/ollowing statements are equivalent"
(i) G is A belian ;
(it) M(G) is commutative"
(iii) there exists a commutative ,-subsemigroup A o/ M(G) such that
/or every nonvoid open subset U o/G, there is a # A + such that/z (U)--t
and # (U') --0.
Proof. The equivalence of (i) and (it) follows from (t9.6). Obviously
(it) implies (iii).
Next suppose that G is non-Abelian, i.e., acca for some a, cG.
An elementary continuity argument shows that there is a symmetric
neighborhood V of e such that
(VcaV) n (VaVc)= .
Let A be any ,-subsemigroup of M(G) such that for every nonvoid open
subset U of G there is a # A + such that # (U) -- t and # (U') = 0. Choose
a neighborhood V of e such that V-c V. Next choose A + such that
v (c - V)= t and v ((c -1V)')=0. Let S(v) denote the support of v (t t.25).
Obviously we have S(v)cc -IV" let b be any element of S(v)-cVc •
Now let W be a symmetric neighborhood of e such that W cV and
a-WacV. Finally, let #ca + be taken so that #(Wab)--t and
.
We will show that #,v(aV)v,#(aV); this will violate (iii) and
show that (iii) implies (i). By (t9.tt), we have
/z,(aV)-- f v(x-laV) dz(x) -- f )(-laV)
G Wa b
t For example, _/i_ might be the algebra M a (G).
If xWab, then xWab=aa-lWabcaVb, so that b-lx-laV. In other
words, the open set x-laV intersects S(v), so that v(x-aV)>O. It
follows that /,v(aV)>O. We complete the proof by showing that
v,#(aV)=O. We have v,bt(aV)= f/z(x-laV) dr(x). If xc-lV, then
X -1 Vc and -, v
(x-laV) n (Wab) C (VcaV) n (VaVc)--(
by (t). Consequently, t(x-laV)--0 for all xc -11 "7 and v,#(aV)--O.
If G is discrete, the algebra Ma(G ) coincides with M(G) and hence
has a unit, viz., the measure ee. If G is nondiscrete, then Ma (G) and M c (G)
both lack even one-sided units. A little more is true, as follows.
(20.25) Theorem. Suppose that G is nondiscrete. Then there is no
measure in Mc(G) such that/,v=v/or all vM(G), and there is no
measure/' in M(G) such that v,/'--v /or all vM,(G).
Proof. Assume that / M (G) and that ,u • v-- v for all v M, (G).
Since I/l ({e})--0, there is a neighborhood U of e such that I/1 (U)
Let V be a symmetric neighborhood of e such that VzC U and 2 (V)
then (G). The relation (20.9.i) implies that for 2-almost all x V,
we have
a -- v (x) --/z *v (x) -- fv (3/-1 X) d/tt (3/) -- f v (3/-1 x) d/tt (3/)
G U
U
This contradiction proves the first statement. The second is proved in
like manner, starting from (20.9.ii).
The algebra M, (G) always contains an approximate unit, however,
as we will now prove.
(20.26) Definition. Let S be a topological semigroup. A net (xz}ot a
of elements in S is called an approximate unit for S if the relations
y -- lira x y -- lira y x hold for every y S.
(20.27) Theorem. The algebra Ma(G) contains an approximate unit.
Proof. Let q/denote the family of neighborhoods of e and regard
as a directed set in the usual way" U>'V if UcV. For each
choose a measure/v M+, (G) such that/v (U) -- t and/v (U') = 0. Then
the net {/u}u6 is an approximate unit for M,(G) by (20.t5)applied to
(G), which is isomorphic with M, (G).
(20.28) Theorem. A closed linear sub@ace A o/ M,(G) is a le/t
right ideal o/M(G) i/ and only i/ / A and xG imply that
I/,s, A. In other words, i/ 9X is the sub@ace o/ (G) corresponding
to A, then / 9X and x G imply ,/ 9X / 9X] . 1
1Note that if dtt=/d2 and xG, then dex,tt=x-x/d2 and dtz,ex=
z] (X -1) /--. d 2.
304 Chapter V. Convolutions and group representations § 20. Convolutions of functions and measures 305
Proof. Let A be a closed left ideal in M,(G) and let {%} be an
approximate unit for M, (G). Then for # A and x G we have
e. •/ = lime. • (% • },) = lim (e. • %) • }, < A
since each e,% belongs to ll/I,(G) (t9.t8) and A is a closed left ideal.
Suppose conversely that #A and xcG imply e,,,ucA. Let v be in
M, (G) and 0 in A. If every bounded linear functional • on M,(G)
for which ў(A) =0 also satisfies )(v*0) =0, then by the HAHN-BANACH
theorem (B.t 5), v*0 also belongs to A. Let ) be as above. By (t2.t8),
there is a function g oo (G) such that
We then have
q)(#) = fg d# for M (G).
(,0) = f g g,0 = f f g(*y) g0 (y)
- f(**0) () = 0,
since e**#0<A for xG and #(A)=0.
Miscellaneous theorems and examples
(20.29) Measures of inverse-sets. (a) Let G be a locally compact
group that is not unimodular. Then there is an open subset W of G
such that it(W)< o and it(W -1) =o. Let aG be such that A(a-)>=2.
Let U be a symmetric neighborhood of e in G such that Uc{xG"
< A (x) < 2}. It is easy to see that the sets aU, a" U, ..., a" U, ... are
pairwise disjoint. Since G is nondiscrete, there are symmetric neighbor-
hoods of e such that it(l/V)_<_2 - and cU (k=,2,...). Let
{n}= be an increasing sequence of positive integers such that
2-"+() for all k. If xa", then x--a"w for some w and
A (X -1) = A (W -1) A (a-l) nk ) 2 - 2 ()-. Now let W= U a . We
have2(W)=Z()tandZ(W-)= dZ= A(x -1) dx=.]
k=l W k =1 an Wk
(b) A simple and instructive special case of (a) is provided by the
group G described in (t5.t7.g). Let W={(x, y)G'x>t and lyl<t}.
Then as remarked in (t 5.t 7.g), we have
()-2 -2.
1
The modular function A(x, y) is Ix1-1,
(20.2) shows that
as noted in (t 5.t7.g). Hence
ool oo 1
: f w- -d')" : f f (Ix, hi-fi ' -
7ў dy dx = f f l dy dx =°x
G 1 --1 1 --1
Also we have W-l={(x, y)G'0< x< t, lyl< x}, and
1 x 1
ff'
it ( W - ) = -ff d y d x = -- d x = oo .
0 --x 0
(20.30) Notes on Theorem (20.4). (a) The mapping x--->,/ of G
into (G) is left uniformly continuous for all/f (G) if and only if G
has equivalent uniform structures (1 =<p < o). [Suppose that G has
inequivalent uniform structures. By (4.14.g), there is a neighborhood
U of e such that given a neighborhood V of e there is an x c G for which
xVx -1 cI-U. Choose W to be a symmetric neighborhood of e such that
IҐc U and it (W) < oo.
Now consider any neighborhood V of e. Let x in G have the property
that xVx-l(l: U, and choose z(xVx -) CI U'. It is easy to see that the
sets x-zW and x -1W are disjoint. Therefore we have
1
I[ ,-'x (ew) --x (ew)lip -- (2 it (W)) ,
and, moreover, (Z -1 X) -1 X= x-lzx< 17. Since V is arbitrary, the mapping
x-+,() is not left uniformly continuous.]
(b) The mapping x-->/, of G into Јf(G) is left uniformly continuous
for all/ Јf (G) if and only if G is unimodular. [Suppose that G is uni-
modular. By (20.2), /* also belongs to Ј(G). Given e>O, use (20.4)
to choose a neighborhood U of e such that IIs(/*)--(/*)[It,< e whenever
st-l U. Then x -y U implies that I/.--/yl[f -- II (/. -- /,) * [If =
L-, (I*)-,-. (1") < .
Suppose next that G is not unimodular. Let U be a symmetric
neighborhood of e such that U- is compact. Let x l-e" having chosen
x,_l, choose x in (=U U)'- Let W1 be a symmetric neighborhood of e
k 1 xk
such that WsC U and it(W1)<2 -1. When Wu_ 1 has been chosen, let
be a symmetric neighborhood of e such that W W_I and it (W,,) <2 -".
Finally, let W= x,l/V. Clearly we have $w;(G). To show that
x- ($) is not left uniformly continuous, let V be a symmetric neighbor-
hood of e such that V W. Choose a positive integer mso that
Then VCIWЈ'_, must be nonvoid; pick any element v of VCIW,,_I. We
next show that x m W m v is disjoint from W. Assume that x,,, W m v
for some n. Then x- x m V - W= -1 Wll a U and hence n = m. Thus
v I/V -11/V=_ Wm Win-l, which is a contradiction. Since x m W m v Wv and
z, W,v is disjoint from W, we infer that ]l($w)- ($w)-* Ill> 0. For every
Y c G, we have
1
II (),- (),- I[ = (y-l)ll ()- ()-1[- ()
Hewitt and Ross, Abstract harmonic analysis, vol. I 20
306
Chapter V. Convolutions and group representations
Since G is not unimodular, the right side of (t) can be made arbitrarily
large. Since (yv-1)-ly=v V, this shows that x-+(w)x is not left uni-
formly continuous.
(c) Suppose that G is unimodular. The mapping x-/x of G into
gp (G) is right uniformly continuous for all ] p (G) if and only if G has
equivalent uniform structures (t =< p < o). If G has equivalent uniform
structures, then (b) shows that x-/, is left [and hence right uniformly
continuous. If G has inequivalent uniform structures, choose U and W
as in part (a). For a neighborhood V of e, choose x and z so that
1
zc(xVx-1)f-lU '. Then I(w),-l,.-,--(w),-, If= (2).(W))-P- and x-l(x-tz-1)-t
_ x-lzxcV.
(20.31) Converse of (19.27). (a) Let # be a measure in M(G), and
let A be a Borel set in G. Then the function x--># (xA) is right uniformly
continuous on G and the function x-->/ (Ax) is left uniformly continuous
on G. By (t9.t8) we have d#=/d2 with /I(G), and by (t4.t7)
/ (x A) = f xA (Y) / (Y) dy = f x-,(gA) (y) / (y) dy = f /(y) dy.
G G A
Thus #(xA)--#(yA) I <= fill-I] dA< 1[./--/. Now apply (20.4.0.
A
To deal with the function x--+#(Ax), note that #(Ax)=#'(x-lA -) and
use (20.23.iv).
(b) Let # be in M(G) and suppose that for all compact sets such that
2(F) =0, the function x----/(xF) or the function x-->#(Fx) is continuous
at e. Then for every Borel set A, the function x----#(xA) x----#(Ax)
is right [left uniformly continuous throughout G. [Apply (19.27)and (a).]
(20.32) Convolutions using right Haar measure. (a) Unlike some
of the choices made in developing the theory of convolutions, our choice
of left Haar measure instead of right Haar measure was purely arbitrary.
Left Haar measure can be replaced by right Haar measure everywhere
with no loss whatever. Some formulas look different, but this is the only
f
change. Given a left Haar measure ;t on G, we write J() = f* d2 = p-dA
•
G G
for all functions on G for which the integrals have meaning [see (t 5.t
and (20.2). The identity ()*=-,(*) shows that J is right invariant.
Now restrict J to 6200(G) and construct the measure for J by the
process of § t I. For an open subset U of G, we have
q(U -) = sup{J(/)/o(G), / <=
-- sup/J l*d2"i
-- sup { f g dR'g<go(G), g < v} -- ;t (g).
G
§20. Convolutions of functions and measures
307
A similar argument shows that (A -) =2(A) for all A c G. We infer
from (20.2) that /0= and that (A)=f-d2 for all A/ 0. This
A
1
implies that = f/* = f/X *or (G, ). Thus
G G G
j(]) = f/do whenever either side is finite. Throughout (20.32), 2 is a
fixed left Haar measure and is as just defined. In (20.32), but not
elsewhere, II/I means the norm of /in p (G, ).
(b) For t--<p =< oo, the mapping/-+/* carries p (G, 2) onto (G, )
[as well as (G, ) onto (G, 2). It is linear, norm-preserving, reality-
preserving, and order-preserving. The mapping/--+/A has all of these
properties for p--t, but loses them for t < p =< oo if G is not unimodular.
(c) It is easy to see that (A) =0 if and only if 2(A)--O, for all A cG.
Hence a measure v in Ms(G ) has the property that dv = ] d for some
]gn(G, ) [see (t4.t9) and (t9.t3). Since M(G) is a two-sided ideal in
M(G), we can define/,,/, /,,/, and/,,g for ,ttM(G) and/,gP.(G,o )
by analogy with (20.5). The functions #,,/,/,,/, and/,,g can be written
as integrals involving instead of , and are of course functions in
I(G, ). For -almost all xG, we have:
(i) # ,, 1 (x) = f / (y- x) A (y) d# (y);
G
(ii) / ,, # (x) = f /(xy -) all, (y);
G
(iii) / ,, g (x) = f ! (xy -) g (y) d (y)
G
= f /(y-l) (y) (y)
= f / (y) g (y- x) A (y) d (y)
= f/(xy) g(y-)A(y)de(y ).
Note that ў / is the function such that (ў /) d 0 = /) d 2 =. ] d;
that is, /=A(.(/)); similarly for ] and ]g. [This can be
checked by comparing (i)--(iii) with (20.9) and (20.t0).
(d) The analogues of (20.t2) and (20.t3) are as follows. Suppose that
[(G, O) (t p) and that ,uM(G). Then the integral in (ii) defines
1
a function in f(G,)and l, ll, llll, l ll. If fA(y)dll(y)=
G
is finite, then the integral in (i) defines a function in (G, ) and
(e) Theorem (20.t6) has the following analogue. Suppose that
20*
308 Chapter V. Convolutions and group representations §20. Convolutions of functions and measures 309
t << p << oo. If [* p (G, ) and gc , (G, ), then the integrals in (iii)
define a function in 0 (G) and I[*, g ,, <= I/* [g]]'" I * (G, ) and
g(G, )=(G, ), then (iii) defines a function in ,,(G), and
I[gII. /*l gL. I [(G, a) and g(G,), then (iii) defines a
function in ,,(G), and [ g ][ ]gIJ.
(f) Finally we state the analogue of (20.t8). Let p, q, be as in (20.8).
Let [ (G, ) G (G, 9) and let g q (G, ). Suppose also that
[/]l = ]/*[. Then the integrals (iii) define a function in ,(G, ), and
(20.33) If G is noncompact, then neither (G),(G) nor
(G) • Ef (G) is contained in 0 (G).
1
(20.34) The hypothesis in (20.]3) that f (y)-'-d[(y) be finite
is essential. For groups G that are not unimodular and for ,
one can find a measure in M (G) and a function / in E (G) such that
1, does not belong to E, (G).
Choose a in G such that (a -x) 2 and let fl : (aq). Let U, V be
symmetric neighborhoods of e such that U xG'<A(x)<,
(g) < , and V g. Note that the sets {ag}L_ are pairwise
disjoint. For h- I, 2,..., choose symmetric neighborhoods of e such
that () 2 - and V. Let {n}L1 be an increasing sequence of
positive integers such that
(-)1/' >
: A (w) •
Define W to be U a "Wh" then ). (W) = it (W) 2 -= I. Define # in
k =1 k=l k=l
2t/ (G) by the relation d = d. Finally, define A to be a Ua
k=l
1
and / to be A-V -. To show that/ (G), we use (20.2.i) and (t9.tZ)
to obtain
f f' f
/Pd: XA-d: AdA:A(A) Ј(annUal)
G G G
= E (( () E --z() <.
=i =1
rem in how in (<. Sin e
it suffices to prove that xa-V implies that /,ў(x)t. We have
1
/ ,re(x) : f A(y -1)/(xy -1) dit (y) : f A(y -1) A (xy-1) - A-' (xY -1) w(Y)dy
G G
If yCa "*W, then yW and yx -l<a "kVa Ca '*Ua CA. Since
y-l Wa-', we have A (y-) > .A (a-) = fl-I and since x- Va ,
l =fl--}--I It follows that for xa-*V,
we have A (x -) > . A (a) .
1,() f (y-1)l/P' (-I)I/P dJ (fln,-l)Up" (fl--l)l/p () .
an k Wk
(20.35) An alternative approach to , (G) *M(G). Instead of
restricting the/,M(G) for which we define/.# for/(G)
as was done in (20.t 3), we can change the definition of convolution. For
/ p (G) and # M(G), let
1
/ [] () -- f/I (y)-T / (y-1) d/.z (y)
G
whenever the integral exists . Then /D,u (x) exists for it-almost all x G
and defines a function in (G) for which II/D/, lip -< II/ll I/,[I. For p= t,
we have /[] #=/.#. [To apply Theorem (20.8), let z(x,y)=yx -1,
1
qg--A-, =#, and y = /[ . For Wg00(G), we have
1 1
= f I[wll , a : lwl ,
G
(20.36) Invariant measures in M(G). (a) Let G be a compact
group, and let it be normalized Haar measure on G. Then for arbitrary
#cM(G) and all Borel sets A c G, we have 2,lZ (A) =/, it (A)--/ (G) it(A).
Thus it,/=/,k=#(G) it, and it generates a t-dimensional two-sided
ideal in M(G).
(b) To obtain all t-dimensional one- and two-sided ideals in M(G),
we need the following lemma. Suppose that g is a k-measurable function
on any locally compact group G and that for all a in some dense subset
D of G, we have g(x)=g (ax) for locally k-almost all xG. Then there is
a constant fl such that g(x)=fl for locally it-almost all xG. We may
suppose that g is real-valued. If g is not a constant locally it-almost
everywhere, there exists a real number such that the sets {x G: g (x) > }
and {x G: g (x)< } are not locally it-null. Hence there are compact sets
E and F such that E C {x G'g (x) > o}, F C {x G'g (x)< }, it (E) > 0,
and it(F)>0. Then it(F-l) >0 and by (20.t7), there is an aD such that
2(E(aqF))>O. If xEf"l(a-F), then g(x)>, and g(ax)<o" this is
a contradiction.J
(c) Let G be a locally compact group. A measure/M(G) generates
a t-dimensional left ideal if v,/=/ for all vcM(G), where is a
complex number depending of course on v. Similarly / generates a
l-dimensional right ideal if/,,v---# for all vM(G): see (C.t).
1 This formula with p" instead of p appears in GODEMENT [ ], p. I ; GODEMENT'S
formula is equivalent to ours only for p--2.
3 t0 Chapter V. Convolutions and group representations § 2t. Introduction to representation theory
Suppose that there is a nonzero # in M(G) that generates either a
l-dimensional left ideal or a l-dimensional right ideal. Then G is compact,
# generates a l-dimensional two-sided ideal, and d/,--flZ d l, where
is a nonzero complex number and Z is a bounded continuous nonzero
complex function on G such that y.(xy)-----y.(x) Z.(Y) for all x, ycG.
Suppose first that # generates a l-dimensional left ideal. Then we
have sa.#=z(a-1)# for all acG. It follows that f! d#--z(a -1) f
for [o(G). Since #4=0, we also have Z4=0. Since
]z(a-1)] " I#[ -<- Ileal . # --[# l, the function Z is bounded. Since
eab .#, we have Z (a) Z (b) = Z (ab). The continuity of Z follows from the right
uniform continuity of / for [0 (G). Now let #1 be the measure in M(G)
such that d#l----- Z -1 d#. It is easy to see that f/d#l---- f [ d#l for all
[o(G) and also that #14=0 Isee (14.17)]. We infer from (14.5) that
f ! d #[- f! dl# ] for all/0(G), and of course [#1[ is not zero. Hence
]#11 is a nonnegative invariant regular measure such that ]#11 (G)<
Accordingly G is compact (l .9), and d [#11- r d2, where r is a positive
number. There is a Borel measurable function g, ]gl--l, such that
d#l-- g d[#l[. Thus we have d# -- rg z d2; it follows that g(x)--g(a-lx)
for 2-almost all x, a being any fixed element of G. Hence by (b), g can
be taken as a constant function, g(x)=exp(iO), and we infer that
d#- flZ d2 with fl-- r exp(i0).
It follows easily from (20.22.iv) that #- generates a l-dimensional
right ideal. Since d#---f12; d2 by (20.23.iv), it follows that/, and
generate the same l-dimensional ideal, which is necessarily two-sided.
Suppose now that # generates a i-dimensional right ideal. Then #~
generates a l-dimensional left ideal and the previous case applies.]
(20.37) The adjoint operation L-+L ~ of (20.21) can be definedin
many convolution algebras other than M(G). and (G) for a locally
compact group G. Making no pretense at completeness, we cite * (G)
for any locally compact group G and 3*(G) for any group G. If
9(G), then 9" is also in (G); if 9c (G), then 9" is also in 3(G).
Thus the mapping 9-L(9") is defined if L is a bounded linear functional,
and plainly is again a bounded linear functional. It is worth noting that
the important relation (2o.22.iv) may fail in ,* (G) and 3*(G). Let
denote either , (G) or (G). Let contain a two-sided invariant sub-
space 9 such that l 9, [* 9 if [ 9, 9 if [ 9, and 9 admits two
distinct two-sided invariant bounded linear functionals N 1 and N,. for
which N (l) -- N 2 (1) = i. Extend N 1 and N. to functionals M 1 and
respectively, in *. Then for ]c we have M 1, M2 ([) --M. ([) as in
(19.24.a) so that (M 1,M) ~ (]) -- M(]). On the other hand, for [
and x G, we have M (,]) ----- M 1((-) -- M 1(()._1) -- M1 ~(]). Hence we
have M'.M"(/)--M"(I)M(/)--MI~(/). Since M is distinct from
M on 9, we infer that (M 1,M)-4= M.M. Specific examples
illustrating this phenomenon can be constructed as in (19.24.b) and
(19.24.c).
Notes
Most of (20.1) -- (20.19) are copies of or marginal improvements on
corresponding results in WEIL 4]. The second assertion of (20.4) is
stated by GEL'FANI) and RAIKOV 2]. Convolutions of measures and
functions in 1 (G), treated in (2o.)-(2o.9), appear in WI 4] only
implicitly, on p. 48. Formula (20.9.ii) appears in GOIEMENT l], p. t3,
for /0(G). We are unsure of the first explicit mention of (20.9.i);
WNI l uses it. The /orm of (20.9.i) and (20.9.ii) is obvious; al-
though as we have seen a rigorous proof requires some care. Theorem
(20.10) is found in WEIL 4], p. 49. Formula (20.10.ii) and its analogue,
formula (iii) in (20.32.c), have been long used as an ad hoc definition of
multiplication in 9-a (G) or 1 (G, ): see for example WEYL and PETER l ,
GEL'FAND and RAiKOV l], SEGAL l], and GEL'FAND and RAiKOV 2.
Corollary (20.14) appears in WI 4, p. 49, and also in SGA 1.
Theorem (20.16), and also (20.17) and (20.18), are due to WEIL 4,
pp. 0, 4--.
The adjoint operation in M(G), examined in (2o.2o)-(2o.23), was
introduced at an early stage. For continuous functions on compact
Lie groups, it appears in WEYL and PETER l]. The formula g---g
for gl(G) appears in GEL'ANI and RAiKOV 2] and in SEGAL 2.
The adjoint operation #-#" for measures in M(G) appears in Y. A.
gREDER Eli, and is utilized heavily in REIDER 2].
Theorem (20.25) for the case of a two-sided unit and M a(G) is due
to SEGAL. Theorem (20.27) appears in LooMIS 2, p. 124, Th. 31E. A
more general assertion was proved by SEGAL 2, Lemma i.i. Approxi-
mate units in M a (G) have long been used: see for example HELSON i],
p. 84.
§ 21. Introduction to representation theory
A central technique in contemporary analysis is the study of topo-
logico-algebraic objects of a given class by means of their continuous
homomorphisms into the most elementary objects of the same class.
Thus the bounded linear functionals on a Banach space E, which are
just the continuous linear space homomorphisms of E into a l-dimen-
sional Banach space, are a very useful entity. Thus too, continuous
algebra homomorphisms of commutative Banach algebras into the field
K are a vital tool in analyzing the structure of such algebras. This
phenomenon persists in the study of locally compact groups G and of
M(G) and its subalgebras. By a proper choice of the groups and algebras
3t2 Chapter V. Convolutions and group representations § 2t. Introduction to representation theory 3t 3
that serve as the images under these homomorphisms, we obtain much
information about the structure of G and M(G). We begin with some
quite elementary definitions and facts.
(21.1) Definition. Let S be a semigroup, with or without a topology.
Let E be a linear space over an arbitrary field F. A representation V
o/S is a homomorphism x-+V of S into the semigroup of all operators
on E (B.2). That is, for each x S, V is an operator on E, and Vy--VV
for all x, y S. The linear space E is called the representation space o! V.
A subspace E1 of E is said to be irvariart under the representation V if
V (El) c E1 for all x S.
A first obvious specialization is appropriate for representations of
groups.
(21.2) Theorem. Let V be a representation o/a group G as in (21.1).
Then E is the direct sum o/ invariant subspaces E o and E such that:
(i) V (E0) = {0}/or all x G;
(ii) V (e)=e/or all E [e is the group idertity ;
(iii) V V_, () = V-I V () = /or all E and x G.
I/E is a topological linear space and V is a continuous operator, then E
and E1 are closed in E.
Proof. Let E 0 = { E : V = 0} and let E = { E: V = }. Since
V,E-EV,--E for all xG, it is plain that V(E0)={0} for all xG.
Properties (ii) and (iii) follow at once from the definition of El. It is
also obvious that E 0 (? E1 = {0}. For E, we have
Thus E is the direct sum E o ® E. It is immediate that both E o and
are invariant and are closed if V, is continuous.
(21.3) Note. Theorem (21.2) shows that in studying representations
of groups, we lose nothing in supposing that V is the identity operator
I on E. From this point on, therefore, we shall mean by "representation
V of a group" a representation such that V= I. The situation is quite
different for semigroups that are not groups: representations V such that
V=0 for some elements x in the semigroup play a vital r61e in many
interesting and important problems. See for example HEWlTT and
ZUCKERMAN [3], E4], E]"
(21.4) Definition. Let A be an algebra over a field F and let E be a
linear space over the same field. A representation T o/ A is a homo-
morphism x-+ T, into the algebra of operators on E (B.2). That is, for
each xA, T, is an operator on E, and for x, yA and F, we have
T+y=T+ T, Txy=TT, and T,=T,. The linear space E is called
the represertatio space o/ T. A subspace E of E is said to be irvariant
uder the representation T if T, (E)c E for all x A.
(21.5) Note. (a) Let A be an algebra with a multiplicative unit u,
and let T be a representation of A as in (21.4). Then E is the direct sum
of subspaces E 0 and E1 such that T, (E0)= {0} for all x A and
for all e E. If E is a topological linear space and T, is a continuous
operator, then E 0 and E are closed subspaces. This fact is proved by
the same argument as used in (2.2). From this point on, we will always
suppose that T,=I if T is a representation of an algebra with unit.
(b) Suppose that T is a representation of an algebra A or of a semi-
group S by continuous operators on a topological linear space E. If
E is a subspace of E that is invariant under T, then E- is also invariant
under T.
The algebras M(G), M(G), M(G), and M,(G), defined in (19.13) for
a locally compact group G, all admit the adjoint operation
described in (2o.2)-(2o.23). This adjoint operation is intimately
connected with representations of G by unitary operators on Hilbert
spaces, and is an essential tool in studying these algebras. Many facts
about this adjoint operation can be obtained with no extra effort
for arbitrary algebras over K that admit an adjoint operation. We
shall therefore define and study such algebras. The term "algebra"
means "algebra over K", unless the contrary is specified.
(21.6) Definition. Let A be an algebra, and let there be a mapping
x-x " of A onto A such that for all x, yA and K the following
identities hold:
(i) (x + y) = x + y;
(ii) (x) =x;
(iii) (xy)"-- y'x';
(iv) x= x.
Then A is called an algebra with adfoirt operation or a .-algebra. If A
is also a normed algebra (C.I) and
(v) I['[I- l[ II for xA,
then A is called a rormed ...-algebra. A Banach algebra that is a normed
~-algebra is called a Barach .-.-algebra. An element xA such that
x ~ x is said to be Hermitian . Throughout Chapter Five, if a normed
algebra has a unit u, then we suppose that u I= 1.
(21.7) Remarks. Let A be a -algebra. If A has a unit u, then
u~-u • for u"--uu" and so u=u"'--(uu~)"=uu". Every xA can
be written in exactly one way as the sum x+ i x, where x and x are
1 In the literature, the symbol "*" is most commonly used for what we write
as ",, \Ґe choose the latter notation to avoid confusion with ",", already used
to denote convolution.
3 t4 Chapter V. Convolutions and group representations
Hermitian. Namely, we have x-- 2 + i 2" Every element of
the form xNx is Hermitian. If A has a unit and x has an inverse, then
(xN) -1 exists and is equal to (x-l) ~. If x has a quasi-inverse y (C.),
then yN is the quasi-inverse of x N.
Our present concern with ~-algebras stems from the fact that every
subalgebra of M(G) [where G is a locally compact group containing
along with # is a normed ~-algebra.
Certain special representations of --algebras are of great importance
to us.
(21.8) Definition. Let A be a ~-algebra and T a representation of
A by bounded operators on a Hilbert space H. Suppose that T~--T
for all xA T is the adjoint operator of T," see (B.3) ]. Then the
representation T is called a ~-representation.
Let T and T' be representations of an algebra A by bounded operators
on Hilbert spaces H and H', respectively. Suppose that there is a linear
isometry W (B.42) carrying H onto H' such that WT x W-I= T for all
xA. Then T and T' are said to be equivalent. Equivalence of two
representations V and V' of a semigroup S by bounded operators on
H and H' is defined similarly.
(21.9) Theorem. Let T be a N-representation o[ a N-algebra A, as
in (21.8). I/ H is a linear subspace o/ H invariant under T, then the
orthogonal subspace H# is also invariant under T.
Proof. For s e H#, H, and x A, we have
(T , 5 = (, T5 = (, T-5 = o.
Thus Txs e is orthogonal to all H and is thus in H.
(21.10) Corollary. Let A be a N-algebra and T a--representation o[ A
by [bounded] operators on a Hilbert space H. Then H is the direct sum o/closed
orthogonal invariant subspaces N and H such that T,(N)--{0} [or all
x A, and/or all nonzero HI there is an x A such that T, 4=0.
Proof. Let N--{seth • T, se--0 for all xA}. Plainly N is a closed
invariant subspace of H. By (21.9), the orthogonal subspace N ± is also
invariant. Setting N±--HI, we have the present assertion.
(21.11) Theorem. Let G be a group and V a representation o[ G by
unitary operators on a Hilbert space H. I/H is a linear subspace o[ H
invariant under V, then H is also invariant under V.
Proof. We have V, N- V, -1 since V is unitary and V, -1--- Vx_, since V
is a representation. Now argue as in (21.9).
Our goal in studying ~-representations of .-algebras is to decompose
them into "elementary" components so far as this can be done. Our
first step in this direction is the following.
§ 2t. Introduction to representation theory
3t5
(21.12) Definition. A representation T of an algebra A by operators
on a topological linear space E is said to be cyclic if there is a vector
se E such that the linear subspace {T,s e: x A } is dense in E. A representa-
tion V of a semigroup S by operators on E is said to be cyclic if there is a
vector s e E such that the smallest linear subspace containing {V se:x S}
is dense in E. In both cases s e is called a cyclic vector.
(21.13) Theorem. Let T be a N-representation of a ~-algebra A by
operators on a Hilbert space H. Then H is the direct sum o/ sub-
spaces N and {H}, which are closed, pairwise orthogonal, and in-
variant. We have T, (N)-----{0} for all xA, and T, restricted to H, is
cyclic for all yl-'. For every nonzero Hv, there is an xA such that
T x 4= O. For H, let , be the pro]ection o/ in H. Then T,-- Y. T, .
Proof. First, let N be as in (21.10). If N±-- {0}, then T is the trivial
representation T,--0 for all xA. If Nz# {0}, well order the nonzero
elements of N ± in any fashion:
, , ..., , ..., , ..., ()
where runs through all ordinals less than some ordinal zl. Define H
as the closure in H of the linear subspace {se+ T, se : K, x A}. It is
easy to see that H is an invariant subspace of H under the representa-
tion T. We claim that the subspace {T,:xA} is dense in H1. Let
be a vector in H orthogonal to all Txse for xA. For all y, zA and
K, we have
Since is in H, T is also in H and since {+T:K, zA} is
dense in H, we infer that T--' for all yA. Since is in N ±, this
implies that --0. It follows that {T,I'xA} is dense in Hx, so that
the representation T restricted to the subspace H is cyclic with cyclic
vector sel.
If H=N±, our construction stops. If H .c N ±, let se, be the first
vector in (1) such that ,.H. Form the subspace {se,.+ T'K,
xA}---_H. By (21.9), H#(N ± is an invariant subspace for T, so
that H. is an invariant subspace for T contained in H f-IN ±. As in the
preceding paragraph, T is cyclic on H with cyclic vector se,. The
definition now proceeds by transfinite induction. Suppose that subspaces
H, H .... , H .... of H and elements se in N ± have been defined for
all ordinals y less than 0, such that the H are pairwise orthogonal, are
contained in N ±, and have the form {o+T'oK, xA}-
=b< b,.<...< by<-..]. If there is no nonzero vector in N ± orthog-
onal to all H (<o), the construction stops. Otherwise, let o be
the first vector in (t) that is orthogonal to all H(y<yo). Construct
316 Chapter V. Convolutions and group representations § 2t. Introduction to representation theory 3 t 7
Hro={Oro+TxoTo'K, xA}-. This defines the family {Hr}Ter,
where F can be taken as a subset of the set of ordinal numbers less than A.
Since no nonzero vector in N ± is orthogonal to every HT, we infer that H
is the direct sum of N and the subspaces H7 (B.63).
The last statement is easily verified from the boundedness of each
operator Tx: we omit the details.
(21.14) Theorem. Let G be any group and let V be a representation
o! G by unitary operators on a Hilbert space H. Then H is the direct sum
o/subspaces {H}r, which are closed, pairwise orthogonal, arid irvariant
under V. The representation V restricted to H is cyclic, [or each ,I .
The proof is similar to but even simpler than that of (2.3), and is
omitted.
(21.15) Discussion. Suppose that A is a --algebra, T is a--repre-
sentation of A by operators on H, and that is a vector in H. The
function x--->(T,, )=p (x) is evidently a linear functional on A, with
the property that p(x-x)= Tx~, ) -- (T, T) >__0 for all xA.
Functionals of this sort can be studied in abstracto. They turn out to
have a wealth of useful properties and to be intimately connected with
cyclic representations of n-algebras.
(21.16) Definition. Let A be a n-algebra. A linear functional p
on A [regarded as a complex linear space is said to be a positive/unctional
if p (x-x) >= 0 for all x A.
(21.17) Theorem. Let p be a positive /unctional on a .-algebra A.
Then ]or all x, y A, we have
(i) p (y'x) = p (x y)
and
(ii) IP (yx)l < P (xx) P (Y-Y)"
Proof. For K, (21.16) and (21.6) show that
0 <= p((x + y) (x + y)) = p (x"x + yx + xy + Ilyy)
-p(xx) +p(x) + p(x'l + I (). ()
Since p (xx) and p (y'y) are nonnegative, p (y-x) + p (x-y) is real.
Setting = 1 in (1), we infer from this that Imp (yNx) + Imp (x'y) =0.
Setting = i in (1), we also infer that Re p (yx) = Re p (x"y). This
proves (i).
We can thus rewrite (1) as
0 <= p (xx)+ 2Re p (x'y) + I1 p (> • (2)
If p(x-y)--O, then (i) shows that (ii) holds. If p(xNy)4=0, set
in (2). This yields
# (xx) IP (xy)l < # (xx) P (yy) • (3)
If p (xx) is positive, then (3) implies (ii). If p(yy) is positive, we
repeat the argument interchanging x and y and again obtain (ii). If
p(XNX)--p(yNy)--O, (2) shows that Reczp(xNy)120 for all K. This
is impossible if p (xy) O, and so (ii) is established.
(21.18) Theorem. Let A be any N-algebra and let A be the algebra
obtained [rom A by ad/oining a unit u, as in (C.3)" elements (x, o)
will be written as u+ x. For u+ x A,, let ( u+ x) N- u+ x-. Then
A, is a N-algebra. Let p be a positive [unctional on A. Then there is a
positive/unctional p* on A, that agrees with p on A i! and only
(i) p (x N) = p (x) /or x A
and
(ii) Ip(x)l<__ap(xx) /or xA,
where a is a positive number independent o/ x. I/ (ii) holds, p* (u) can be
taken as any number greater than or equal to a.
Proof. If p* exists, then (i) follows from setting y--u in (21.17.i)
and noting that u-=u. Setting y--u in (21.7.ii), we get
!#* (ux)[ - [# (x)l < # (x~x) #* (),
so that (ii) holds with a= p* (u).
If (i) and (ii) hold, let b be any nonnegative number such that
IP (x)12<= bp (xNx) for all xA. Then define
p* (u+ x)=+p(x).
We have
-i1 . +p (x) + # (x) + (xx)
= Ilb+ 2ReCap (x) + p(xx)
> I1 - 2 I1 I# (x)l + # (xx)
Since pt is plainly a linear functional on A,, the proof is complete.
It is a remarkable fact that all positive functionals on a large class
of -algebras are continuous.
(21.19) Theorem. Let A be a Banach -algebra having a unit u.
Then every positive [unctional p on A satis/ies the relation
(i) I# (x)l # (u)I111 or xA.
3t8 Chapter V. Convolutions and group representations §2t. Introduction to representation theory 3t9
Proof. The binomial series
converges absolutely for all K such that I1_<_1, and its sum is
exp (Ѕ Log (l -- ) ) [where --< Im Log (l -- ) _<_ if 1 ; its sum is 0
for --1. For xA such that x 11 and --1, 2, 3,..., let
y=+ (- x .
It is clear that lim -- =0, and since A is complete, there is a
A such that lim =. Now define
c,=(-)" (= , ....
=0 n k ' '
By ME,TENS' theorem APOSTOL l, p. 376, we have
+ E ."- (-)
if aK and ][ I. Considering only real , we see that c =--l and
c,--0 for n = 2, 3, -.. • Examining the proo/of MERTENS' theorem, we see
that
lira y = lira [u + X c x ] = u- x.
J
Since lim y: y, we infer that y=u--x. If also x is Hermitian, then
it is clear that y is Hermitian. Therefore y: yy: u-- x and
o# ()-# (u)-# (x).
Consequently p (x) is real and
#(x)#(u).
Similarly
-#(x) : # (- x) # (u),
so that (i) holds for Hermitian x [recall that p is homogeneous. For
arbitrary xA, xx is Hermitian, so that
#(xx)#(u) lxx #(u) lxll .
By (2.7.ii) with y:u, we hae
]# (x)l # (,,) # (xx).
Combining the last two inequalities, we obtain (i).
(21.20) Corollary. Let A be a Banach -algebra with unit u and le
p be a positive ]unctional o A. Then p is a bounded linear ]unctional,
d I1# I1:# (u). i A y h -Zgr potv oro
junctional on A [or which (21.8.i) and (21.8.ii) hold, then p is bounded.
Let M(p)=sup }i-) "xA, p(x~x)=t=O . Then we have IP [<=M(p),
and i/ A has a ni, M(p)= p . Frheom, i/ p and q are positive
onaZs lot which (21.18.i) and (21.18.ii) hoZd, hen M(p+ q)M(p)+ M(q).
Proof. The first statement is obvious from (21.19) and the equality
= 1 (21.6). If A is any Banach -algebra and (21.18.i) and (2t.18.ii)
hold for p, then (21.18) shows that p can be extended to a positive
linear functional p* on A,. The number M(p) is the smallest value that
can be assigned as p* ()" this follows from (21.18). Since lip* I= P* (),
p must be a bounded functional on A A,, and as the norm does not
increase on a subspace, we have IP P* ()= M(p). If A has a unit, the
inequality M(p) p (u)= IP] follows from (2 .l 7.ii). As shown in (21.1 8),
M(p+ q) is the least value of (p+ q)* (u) for a positive extension of p+ q
over A,. Since M(p) + M(q) -- (p* + q*) (u) is such a value for p + q, we
infer that M(p+q)M(p)+M(q).
(21.21) Corollary. Let A be a Banach -algebra and p a positive
[umtional on A satis[ying (2.8.i) and (2.8.ii). Let M(p) be de/ined as
in (21.20). Then/or all xA, we have
1
(i) p (xx) M(p) 2i l(xx>ll.
Proof. In view of (21.18), there is no loss of generality in supposing
that A has a unit u and that M(p)=p(u). Repeated application of
(2 t. 7.ii) gives
1 1
1 1 1
1
From (21.19), we have
1 1 1
E# ((xx)'-)] # (u) (xx) "-' .,
so that
1
I# (x)l # (u) (x)""'l,-
It follows that
1
applying (C.4), we now have
1 1
= lim xx "
Corollary (21.20) has an important consequence.
320 Chapter V. Convolutions and group representations § 2t. Introduction to representation theory 32t
(21.22) Theorem. Every -represertatio T o[ a Barach -algebra A
by !bounded operators o a Hilbert space H is a continuous mapping o! A
into the space o[ bounded operators o H supplied zvith the ,orm topology.
Indeed, we have
(i) ]L ]x 1/or all x A.
Proof. For an arbitrary H, let p be the function on A such that
p(x)= (T,, ). Plainly p is linear, and we have
Thus p is a positive functional. Furthermore,
and
Thus p satisfies conditions (21.18.i) and (21.18.ii), and M(p) does not
exceed IIll , By <2.20>, # is a bounded linear functional of norm not
exceeding IItl , and so we have
IT!I = <T,,> =#(xx> 1 • I[xx Illlxl IIl Ix[ .
Taking square roots, we obtain (i).
We now take up the connection between positive functionals on a
-algebra A and -representations of A, remarked on already in (21.15).
(21.23) Theorem. Let T and T' be cyclic -representations o/ a
-algebra A by operators on Hilbert spaces H ad H', with cyclic vectors
and ў', respectively. I/ the positive /unctionals p(x)=(T,,ў) and
I t t
p'(x) (T , ) are equal, then the representations T and T' are equivalent.
Proof. For all x, yA, we have
The mapping W defined by W(T=)= T ' is thus a linear, inner product
preserving mapping of {T,: x A } onto {T2 ': x A }. As these subspaces
are dense in H and H', respectively, W can be extended to a linear
isometry that carries H onto H'. For all x and y in A, we have
W T, W- T/ ' = T; y ' = T; T/ ' .
Thus WT, W -= T on the dense subspace {T ў'" yA} of H', and so
this identity persists throughout H'. That is, T and T' are equivalent
(2.8).
(21.24) Theorem. Let A be a Banach -algebra and let p be a positive
flmctional on A such that
(i) p (x) -- p (x)
and
(ii) ]p(x)l<=ap(xx) (a>0)
/or all xA. Then there is a cyclic -representatior T o! A by operators
on a Hilbert space H, with cyclic vector , such that
(iii) p(x)=(T,ў,ў) /or all xA.
Proof. Let A=A if A has a unit; let A=A, if A has no unit. By
(21.8), there is a positive functional p on A agreeing with p on A.
We shall construct a representation T of A such that (iii) holds for all
xA. Consider first the set B={xA'p(x~x)=O}. For yA and
x B, we have p (yx)--0, since by (21.17.ii),
IP (x)l < P (x x) p (~) o.
If x, x. B, then we have
((x + x.)~(x + x.))
= p (xx) + p (xx) + p (xx) + p (xx.) = o,
and for K and xB, p((ox)x)=ll.p(xx)=O. I yA and xB,
then P((yx)yx)=p((xyy)x)=O. Furthermore, B is closed in A
since p is continuous (21.19). Thus B is a closed left ideal in A.
Form the difference space A/B [see (B.2), denoting its elements
x+ B, y+ B .... by , r], .... If x-- x. B and y-- y B, then
p(x) -p (x) = ((x- x.)) + ((- )x.)
- 0 + p (x(y- >.)) =0.
Hence the function (, ])---Pl (y~x) is well defined on AI/B" we write
(, )--p (y~x) for x and yr]. It is obvious that
<eq_ ., r]> -- (e, r]> + (e., r]), (2)
(,>=<,>, (3)
(,)>o i, q=o, (4)
for all e, rlA/B and K. Thus AI/B is an inner product space, which
in general is incomplete.
For xA, let T,' be the operator onA/B such that T,(y+B)=xy+B.
Since B is a left ideal, each T,' is well defined and is an operator on A/B.
It is obvious that the mapping x-->.T, is a representation of A. Let us
show that each T,' is a bounded operator. For this purpose, fix yA
Hewitt and Ross, Abstract harmonic analysis, vol. I 2 t
§2t. Introduction to representation theory 32q
322
Chapter V. Convolutions and group representations
and let ql be the function on A1 defined by ql(x)--pl(y-xy) for xA x.
Plainly ql is a positive functional on A 1, and ql (u)=p (y-y) u is the
unit of AI. By (21.9), we have
Iq(x)] <=p(y-y)[[x[[ for xA.
We can now compute
(T; , T; ) = (xy + B, xy + B) = p ((xy) (xy))
- q (xx) # (yy)IIx II # (yy)Iixll = , > Iixll .
Thus T' is a bounded operator on the inner product space A1/B, and
IIT;II< IIxLI,
Furthermore, we have
(T; ,) -- (T; (y + e),z + e) -- pl (z-(xy))
-- Pl ((x-z)-y) -- (y + B, x-z + B) -- (, T;~). , (6)
Finally, let H be the completion of the inner product space A1/B.
Every operator T; admits a unique extension over H, denoted by
which is a bounded operator on H. It is easy to see that the mapping
x->T, is a representation of A1 by bounded operators on H, and the
identity (6), which is preserved upon passage to H and T,, shows that
this mapping is a -representation. The inequality llr ll llxll for xA1
follows from (5) see also Theorem (21.22).
It remains to show that the representation x->,T x is cyclic when
restricted to A. Let=u+ B. Then (T;(u+ B):xAI}--{x+B:xA}=
A1/B. By the definition of H, therefore, is a cyclic vector for the
representation x-- Tx of A1. Furthermore, we have (T, , ) -- Pl (ux)
p (x) for xA. Thus the proof is complete if A1--A. If A lacks a unit,
so that A1--A,, we alter the definition of H. Let HI={H:T,=O
for all xA}, let H.--H, and write --1+2 where iHi (/'=1, 2).
Then for all xA, we have
since T,.H,. by (21.9). It is easy to see that the linear subspace
{a.+T,,.'K, xA} is dense in H., since
[[(+.)+T(ў+,.)--(+Ј.)1 ] for all --+.H, and A1/B =
{+ T, : K, x A} is dense in H. Suppose that 2 H. and that
is orthogonal to all T,. for x A. Then for y, z A and K, we have
Hence Ty x/.=0 for all yA, and this implies that /.--0. Therefore
{T : x A} is dense in H.. Putting H. and for H and in the state-
ment of the theorem, we see that the present proof is complete. ]
(21.25) Theorem. Let A be a --algebra and T a cyclic .-representa-
tion of A by operators on a nonzero Hilbert space H. Let be a cyclic
vector in H. Let p be the positive/unctional on A defined by p (x)----- (T ,).
Then M(p)= (, ).
Proof. Since n = {0}, there is an x A such that Tx =0. For all
such x, we have
p (X"X --- ( T x ў, Tx ) 115 ў I1" "
The definition of M(p) (21.20) implies that M(p)<= (ў, ў). The reversed
inequality is proved from the cyclicity of T: if xA is chosen so that
IIT ў-ў1l sufficiently small, then [(T,ў, ў) -- (ў, ў)] --<_ IIT ў-ў11" 11ў11 is
small, and elementary continuity considerations show that [
is arbitrarily close to
The reader who wishes to read our treatment of the spectral theorem
may find it advisable at this point to read (C.36)--(C.42), since (C.42)
is needed in (21.30) and some preceding theorems of §21 are needed in
(c.36)- (c.42).
For reasons that will become apparent in the sequel, we are particu-
larly concerned with representations of groups, semigroups, and algebras
admitting no proper closed invariant subspaces.
(21.26) Definition. Let {M} be a nonvoid set of continuous operators
on a topological linear space E. If {0} and E are the only closed subspaces
of E that are invariant under all M, then {M) is said to be irreducible.
In the contrary case, {M) is said to be reducible, and a proper closed
subspace F of E invariant under all M is said to reduce {M). A representa-
tion T of an algebra A is said to be irreducible if the set of operators
(T,},n is irreducible; similarly for a representation V of a semigroup S.
The following fact is useful in studying reducibility.
(21.27) Theorem. Let T be a ~-representation o/ a .-algebra A by
operators on a Hilbert space H. Let L be a closed linear subspace of H,
and let P be the profection operator mapping H onto L. Then L reduces
T if and only if
(i) PT,--T,P /or all xA.
Proof. If L is invariant under T, x is in A, and e is a vector in H,
we have PL, T,PL, and so PTxP--T,P. That is,
TxP--PT.P. (t)
Taking adjoints, we have
21"
324 Chapter V. Convolutions and group representations § 2t. Introduction to representation theory 325
and PT.~ = PT~ P (2)
for all xA. Putting x- for x in (2), we find
PT, = PT, P,
which equality, combined with (1), gives (i).
Conversely, if (i) holds, eL, and xA, then P(T,)=T,(P)=T,,
so that T,e L. Thus L is invariant under T. [
(21.28) Theorem. Let V be a representation o/ a group G by unitary
operators on a Hilbert space H. Let L be a closed linear sub@ace o/H,
and let P be the pro]ection operator mapping H onto L. Then L reduces V
i/ and only i/
(i) By,= v, P /or x .
Proof. Since V,= V_,, the argument of (21.27) can be repeated.
(21.29) Remarks. It is trivial that every representation of a semi-
group or an algebra by operators on a l-dimensional complex linear
space is irreducible. A l-dimensional representation of an algebra A
can be regarded as a multiplicative linear functional on A, in the sense
of (C.12). A l-dimensional representation of a semigroup S can be
regarded as a complex function Z on S such that
(i) Z(xy)=Z(x)Z(Y) for x, y S.
We shall explore both of these concepts in the sequel.
(21.30) Theorem. Let T be a -representation o/ a -algebra A by
operators on a Hilbert space H, di//erent /rom tae eero representation. The
lollowiug properties are equivalent:
(i) T is irreducible;
(ii) every nonzero vector in H is a cyclic vector/or T;
(iii) the only bounded operators on H commuting with all T, (xA) are
o/ the /orm zI ( K).
Properties (i)--(iii) are also equivalent/or every unitary representation
V o/a group G.
Proof. We carry out the proof only for A and T; it is similar for
G and V. Suppose that (i) holds. Let N = { <H: T, = 0 for all x A }.
Then N is closed and invariant under T, and so N is {0} or H. The
equality N = H means that T is the zero representation, and so we have
N--{0}. Now if H and gq=0, then H--{T,'xA} is a subspace of
H that is invariant under T and {0} c. H1. Since Hi- is also invariant
under T, we have H-= H. Hence (i) implies (ii). If (i) fails, and H is
a proper closed invariant subspace of H, then no vector in H can be
cyclic for T in H. Thus (i) and (ii) are equivalent.
1 This may be viewed as a form of SCHUR'S lemma; compare with (27.9), Vol. II.
We now consider (i) and (iii). If (i) fails, if H1 is a proper closed invariant
subspace of H, and if P is the projection operator onto H1, then P is not
of the form eI, and PT,--T,P for all xA (2.27). Hence (iii) fails if (i)
fails, so that (iii) implies (i). Suppose finally that (i) holds, and let B
be a nonzero Hermitian operator on H commuting with all T,. Let
{Pt'tR} be the family of projections on H for the operator B given by
the spectral theorem (C.42). By (C.42.ii), each P commutes with all
T. and so P--0 or I (21.27). By virtue of (C.42.i), there is a real number
t o such that P--0 for t< t o and P--I for t>t o. It follows from (C.42.v)
that B--toI. If now B is any bounded operator commuting with all
we also have T,B=(BT,~)=(T,~B)N--B-T, for all xA. Writing
B+B N B--B-
B= 2 +i---2----' we see that B is a linear combination of
Hermitian operators commuting with all T,, and so B--I for a complex
number .
We are going to show that every nonzero M-representation of a Banach
-algebra A gives rise to irreducible representations of A. For this
purpose we first introduce a partial ordering in the class of positive
functionals on a --algebra.
(21.31) Definition. Let A be a --algebra and p and q positive func-
tionals on A. If there is a complex number such that p-- q is a positive
functional, we write p ;> q [or q <(p].
Note that the in (21.31) can always be taken to be positive. If
q (x-x) = 0 for all x A, let -- 1 ; if q (xg'Xo) > 0 for some x o A, then the
inequality q(XXo)<=p(xgxo) forces to be positive. It is easy to
show that p < p, that p-< q and q < r imply p < r, and that 0 < p.
(21.32) Theorem. Let A be a Banach N-algebra and let p be a positive
/unctional on A satis/ying conditions (2.24.i) and (2.24.ii). Let x-+T,,
H, and be as given in (21.24). Let q be any positive /unctional on A
satis/yin (2.24.i) and (2.24.ii) and such that q<p. Then there is a
positive-de/inite Hermitian operator B on H commuting with all T, (x A)
such that
(i) q (x) = (BT, g, ) /or all x< A.
Conversely, i/ B is a positive-definite Hermitian operator commuting with
all T,, then (i) de/ines a positive /unctional q on A satis/ying (21.24.i)
lad (21.24.ii) and such that q<(p.
Proof. It is convenient to prove the converse assertion first. By
(C.35), there is a unique positive-definite Hermitian operator C on H
such that C -- B; the operator C commutes with all operators in (H)
that commute with B, and hence C commutes with all of the operators T,.
326 Chapter V. Convolutions and group representations §2t. Introduction to representation theory 327
Writing q (x) -- (BT x , ), we have
__< T cў i . Icў = <r, Cў,T Cў>.
= q(xx). Cў .
Consequently the functional q satisfies (21.24.ii) with a = IIcў11". Prop-
erty (21.24.i), linearity, and positivity of q are all obvious. It is also
easy to see that p--q is a positive functional if >_ II B II [this is the norm
of B defined in (B.8).
The direct assertion of the present theorem is more interesting and
also more complicated. Suppose that q is a positive functional on A
satisfying (21.24.i) and (21.24.ii) such that q< p. Then there is a positive
number such that
O<=q(x'`x)<__,.p(x~x) for all xA.
()
Let H'={Tx'xA}. Then H' is a dense linear subspace of H (2.24
For all x, y A, let
09(T, , Ty ) = q (y'`x) . (2)
The function 09 is actually well defined on H'xH'. If T,=T,, and
Tyў = T, , then T_,ў = Ty_,, = 0, so that
((x- x) (x- x)) - ((- ) (- y)) -o.
Thus by (1) and (21.17.ii), we have
Ie/x)- e/;x/I < Ie(/- i/x)l + le(;/x- x/)l
< q(X'`X) } q((Y-- Yl)" (Y-- Yl)) Ѕ @ q((x-- X1)'` (X-- X1)) Ѕ
The function 09 on H'xH' is obviously a bilinear functional (B.4).
Furthermore, we have
IO(T,g, T,ў)[- lq(yx)l <__ q(xx)
<p(x-x)p(y-y)-- T,ў I •
Thus 09 is continuous and can be extended in just one way to a contin-
uous bilinear functional on HxH. As noted in (B.60), there is a
bounded operator B on H such that
09(, ) = (B e, ) for all , H. (3)
For all x, yA, we have
<BTx, r,ў) = q(y'`x) =q (x~y) = (BT,, Txў ) = (T,ў,
Hence B is Hermitian. Similarly we have (BT, , T, ) = q ( x'`x) >= O.
Thus B is positive-definite. For all x, y, z A, we have
and
(T, BT,, T,ў)= (BT,ў, T,~z ) =q((x'`z)'`y)
Since H' is dense in H, we infer that (BTx, )=(T,B, ) for all
, H. This proves that B commutes with every T,.
We infer from (2) and (3) that
q (y'`x) = (BT, ў, Ty ў) -- (BTy~, ў, ў) for all x, y A. (4)
It remains to show that (i) holds a fact requiring proof only if A lacks
a unit. To do this, let q* be the functional on A defined by
q*(x)=(BTxў,ў ) for xA. (5)
As shown above, q* is a positive functional on A satisfying (21.24.i) and
(21.24.ii). Thus there are a Hilbert space H* and a cyclic "`-representa-
tion x--> Sf of A by operators on H*, with cyclic vector *, such that
qf (x) -- (S; el, el) for x A. (6)
Similarly q can be written as
q (x) -- (st t ctt, ctt)
where x--> S t is a cyclic ---representation of A by operators on a Hilbert
space H tt with cyclic vector tt. We will define a linear isometry U
carrying H t onto Htt as follows. Let USt=Sttt for all xA. Then
U is a well-defined, linear, inner product preserving mapping of a dense
subspace of Ht onto a dense subspace of Htt, as (4), (5), and (6) show.
It is easy to see that U can be extended to a linear isometry (B.42) of
H t onto H tt. The definition of U implies that U(S;(Styt)) = St.t(UStyt)
for all x, y A" since {Styt'yA} is dense inH t, we have US#t=St
for all x A and et H t.
For all x A, we now have
-- <t.t tt, tt) __ (tt, stt tt).
This implies that Ut-- ft. In consequence we have for all xA
q(x)---- <stttt,tt> -- <Sxft ut, Ut> __ <USfx t,
-- ( S Ct, Ct ) = qt ( x) = ( BT,, ,
This is just (i).
(21.33) Theorem. Let A be a Banach "`-algebra, and let A h be the
set o/all Hermitian elements o! A. Then A h is a real Banach space. Let P
328 Chapter V. Convolutions and group representations §2t. Introduction to representation theory
denote the set o[ all positive [unctionals p on A such that
(i) p (x ") = p (x)
and
(ii) ]p (x) l <__ p (x-x)
/or all x A. Eacl element o/P, when restricted to A h, is a real, bounded
linear ]unctional on A h, o/norm less than or equal to 1. Thus P can be
regarded as a subset o/A. So regarded, P is a convex set compact in the
weak-, topology o/A (B.24).
Proof. Condition (ii) is just the condition M(p) <= 1 (p =4= 0). Thus
by (21.20) each pP is a bounded linear functional on A of norm not
exceeding 1. It is trivial that A h is a real linear space, and it is simple
to prove that A is closed in A. Thus A h is a real 13anach space. Let
be any element of P. By (i), p is real-valued on A, and is obviously
linear. The norm of p is not increased when it is restricted to A, so
that lip II <=1 on A.
I we are given a real linear functional / on A such that
/(x-x)=/(x) ", we may ask for conditions under which [ is the restric-
tion to A of a functional in P. Every element of A can be written in
just one way in the form x+iy with x, yAh (21.7). Thus the only
possible linear extension of [ has the form p(x+iy)=[(x)+i[(y).
Obviously p is a complex linear functional on A. If p P, then for all
x, y A , we have
I (x+ /I.< ((x- )(x+ )) = (x.+ .+ (x- x/)
or equivalently
F.(x)+ l"(y)<=l(x)+ 1 (y) + 1 ( (xy-
Similarly, we must have
l. (x) + l (y) <= l (x.) + l (y.) + l (i (yx- xy) ) ,
and thus
1 - (x)+ 1 . ()+ I1(i (x- x))l =< 1 (x ) + 1 () ()
is a necessary condition that the function [ in A' be the restriction to A
of an element of P. Conversely, if (1) holds for l, then p is a positive
functional on A satisfying (i) and (ii). Thus /ll_<_l for all lA satisfy-
ing (1). It follows that P, regarded as a subset of A', is the part of the
unit ball in A' defined by condition (1). As such, it is obviously closed
in the weak-, topology and is hence compact (B.25). Finally, (i) and (ii)
are equivalent to the existence of a positive extension pf of p over A,
for which pf (u)_< 1. This shows that P is convex.
(21.34) Theorem. Let A, A, and P be as in (21.33). A nonzero
element p ol P is an extreme point ol P il and only il M(p)=l and the
representation x-+T, ol A constructed as in (2t.24) is irreducible.
329
Proof. It is obvious that 0 is an extreme point of P. If P and
0<M(p)<
1,
then
p=M(p)
Ie(p) + (l -- M(p)) . O, so that p is not
an extreme point of P. Suppose that p P, p =4= 0, and that the represen-
tation x--T x of A as constructed in (21.24) is reducible. Thus there is
a closed linear subspace H 1 of H invariant under all Tx such that
{0} . H1 . n. Write H. for H#. By (21.9), H. is also invariant under
all T. Write the of (2].24.iii) as ў1+ў, where iHi (=1, 2). It is
easy to see that 0, 0. Also it is clear that
= p (x) + p (x);
furthermore, p and pz are functionals belonging to P. We claim that
PLY0, pz0, and that there is no real constant such that pz=p.
Since T is a cyclic representation, there Is an xA such that
I T -- < 1 l" Then we have
Hence p0; similarly we have p0. Now assume that p=p,
where is a [necessarily 3 positive number. Let d be a positive number
such that
Choose an element aA such that IILў-I<o. Then we hav
thus
I <L ў, ў>l > I111 - ltll. <2>
A like computation gives
From (1) we infer that
Combining (2), (3), and (4), we have
Which is a contradiction.
M(p)
Finally, write q = M(pi ) Pi (= 1 2) and t-- M(Px)
h&ve '
M(p) " By (21.25), we
]]//'() = (, )= (1, 1)-- (9., 9.) = ]]//'(1)--
330 Chapter V. Convolutions and group representations § 21. Introduction to representation theory 331
Hence we also have
P = Pl q- P,. = t ql q- (t-- t)
Therefore p is not an extreme point of P.
Conversely, suppose that M(p)=t and that the representation T is
irreducible. Assume that p is not an extreme point of P. Then we have
p = tpl+ (t -- t) P2, where 0< t< t, M(pl) <- t, M(P2) t, Pl =# P. The
inequality M(p)<=tM(pl)+ (t--t)M(p,.) shows that M(pl)
, (-)
Since Tp--p,= t p., we see that p>-p, and so by (2t.32) there
is a positive-definite Hermitian operator B on H such that
p (x) = (BT,ў, ў) for all x< A ,
and such that B commutes with all T,. Then (2t.30) implies that B =aI
for some nonnegative number . Thus p,=ap; since M(pl)=M(p)=I,
we have p,= p, and so arrive at a contradiction.
(21.35) Theorem. Let A be a Banach -algebra, let p be any/unc-
tional in P, let (Xl, ..., xm} be a/inite subset o/A, and let e be a positive
number. Then there are nomero extreme points p,, ..., p, o/ P and
positive numbers t,, ..., t, such that t <__ I /or which
/=1
(i) Ip(xi)-(tlp(xi)+t2p,.(xi)+...+t,p,(xi))l<e (=t, ...,m).
Proof. This result follows at once from the KREiN-MIL'MAN theorem
(B.30). Convex combinations of extreme points of P are dense in the
weak-, topology of P as a subset of A. This establishes (i) for Hermitian
elements x,,..., x,n of A. Since every x in A has the form yq-iz with
y, z A, (i) is immediate for arbitrary Xl, ..., x,, A.
(21.36) Corollary. Let A be a Banach ...-algebra and x a, elemen o[
A /or which there is a/unctional p P such that p (x)0. Then there is
an extreme point q P such that q (x)= O.
This follows at once from (2.35).
We now present our main result on representations of -algebras.
(21.37) Theorem. Let A be a Banach .-algebra and let x be an
element o/ A. The/ollowing conditions on x are equivalent:
(i) there is a/uctional p P such that Pl (x) = 0;
(ii) there is a/umtional p2 P such that p=(xx) > 0;
(iii) there is a -representation T o/ A such that T, 0;
(iv) there is an irreducible -representation S o/ A such that S,=O.
Proof. The inequality IP*(x)]"<=P*(xx) shows that (i) implies (ii).
If (ii) holds, then by (2t.36) there is an extreme point q of P such that
q(xx) > 0. By (2t.34), the -representation S of A for which q(x)
(S,g, ) is irreducible. Since q<x x>:lts ll .>o, it follows that S,0.
Thus (iv) holds. It is trivial that (iv) implies (iii).
Finally, to show that (iii) implies (i), let T be a -representation of A
by operators on H for which T,q=0. If the relation (T,, )=0 held
for all H, then by (B.58) we would have T,=0, a contradiction.
Choosing pl as the function p(y)=(T,) for some such that
p(x)q=O and [[I[=I, we obtain (i).
Miscellaneous theorems and examples
(21.38) The following examples show that the hypotheses of Theo-
rems (21.18) and (21.t9) are really necessary.
(a) Let X be any nonvoid compact Hausdorff space, and let A be
the set g(X). Give A the usual linear operations and the trivial multi-
plication/.g=O for all/, gA. For/A, let/=and l[/[[--max{[/(x)[:
x<X}. Then A is a Banach -algebra without a unit, and every linear
functional on A is positive. Fix a in X, and define p (/)=/(a) for/ A.
Then p is a continuous positive linear functional, liP I[= I, and p (/)= b (/)
for/A. However, Ip(/)['.<=p(/~./) holds for no real number and so p
is not extensible to
If we define q to be ip, then q is also a continuous positive functional
having norm I but q (/) need not equal } (/). If X is infinite, A admits
discontinuous linear functionals [let 3 be a Hamel basis in
such that II/[l= i for/ 3, define qo on 3 so as to be unbounded, and
extend qo to g(X)l. Hence A is a Banach --algebra with a discontinuous
positive functional.
(b) Let X be a locally compact Hausdorff space such that goo(X)
admits an unbounded nonnegative linear functional I. Such a functional
will exist, for example, if X is not countably compact; see (14.2). Let A
be g00(X); define linear operations and multiplication to be pointwise.
For/<A, let/= and [I/ll=max{I/(x)l'xX }. Then A is an incomplete
normed -algebra without unit and I is a discontinuous positive func-
tional on A. Also the inequality I(1II)<I(II) hdds *or no rea
number z.
(c) Let A consist of all polynomials/(z) -- Y, z (z K) having complex
/=o
coefficients. Let the linear operations and multiplication in A be point-
wise, and for /(z)=ez , let /'(z)=,gz and
k=0 k=0
0t< I}. Notice that/'/(t)=]/(t)] 2 for/A and tR. Clearly A is an
incomplete normed -algebra with unit. For/ A, we define p (/) = / (I)
and q(/)=/(2). Then plainly p is a continuous positive functional on
d and q is a discontinuous positive functional on A. The existence of
Such a functional q shows that (21.19) may fail if A is incomplete.
332 Chapter V. Convolutions and group representations § 21. Introduction to representation theory 333
(d) (GODEMENT [2.) Let G be a nondiscrete locally compact group.
Regard oo (G) as a subalgebra of 1 (G), which in turn is identified with
M a (G). Thus for/, g00 (G) we define
1
],g(x) =f/(y)g(y-lx)dy, /U(x) = A(x) /(x-i)' and I]/l=fl/I d..
Then 00(G) is a dense subalgebra of gl(G) [the fact that /.g is in
00(G) for/, gCoo(G) is shown in the proof of (20.6). The mapping
/-,/() =p (/)
is a positive functional on 00(G) that is unbounded. It satisfies the
condition p(/U)=p([) but not the condition ]p(/)l<p(/,/). Note
that p ([u,/) = I/I-
(21.39) If A is a Banach u-algebra and p is a positive functional on
A satisfying (2t.18.i) and (21.8.ii), then Ipl[<=M(p)< by (2.20).
We now explore this inequality.
(a) Suppose that A is a Banach ~-algebra and that for every xA
and e>0, there exists a vA such that ]Iv 1=<I and Ilx--vx <. I p
is a positive nonzero functional on A satisfying (2.8.i) and (2.8.ii),
then ]pll=M(p). [Let x be a fixed nonzero element of A, and let e>0.
Choose vA such that Uvnl and x-vx]< 2pnH. Then
I1 (l - I (vll (- (v)l = (x) + (vxl . (x -
and therefore by (21.17.ii), we have
[P ()1 < IP (vx)l + <-- P () P (vvl + < (xx) lip II Ilv vll +
-< (/11 II + .
Since e is arbitrary, we have ]p(x)l.<=p(x~x)llpH. It follows that
M(P)<=ttPtt.
(b) Let A denote all bounded complex-valued functions / defined and
analytic on the open unit disk D = (z K" Izl < I) and for which /(0)--0. We
make A into an algebra by defining the linear operations and multi-
plication pointwise. For I A and z D, we define l~(z)=I (:) and
sup{l/(z)l "zn}. Then A is a Banach u-algebra without unit. Given
where 0<< I, we will construct a positive functional p satisfying
(21.18.i), (21.18.ii), IlPll__<], and M(p)>=l. VWe begin by choosing a
1
nonnegative continuous function w on 0, 1 such that f w (t) d t
1 o
and w(], I)=0. Then we have f tw(t)dt<=]. For IA, we define
o
1
p (/) = f /(t) w (t) dr.
o
Clearly p is a linear functional and p is positive since
1 1
p(f'/) - f/(t)/(t) w(t) dt= f [/(t)l.w(t) dt>_ o.
o o
Condition (21.t8.i) is obviously satisfied, and (21.t8.ii) follows from the
inequalities
Ip (/)1 - (t)w(t)dt <fll(t)w(t)[.dt<flU(t)l(t)w(t)dt=p(/Ul),
o o
where z denotes the number max {w (t) " O_<_ t_<_ l }. If /A and
then SCHWARZ'S lemma for analytic functions shows that I/(t)] =<t for
0_--< t _< t and therefore
1 1
[P(I)I < f[l(t)[ w(t) dt < f t w(t)tit<=
o o
That is, llb I1_ -< .
{; if t>}
Let d denote the function on 0, I defined by d(t)= if t= "
For n-- 1, 2, ..., define/ (z) = 1 -- (I -- z) for z D. For t 0, 1 ], we have
,lirnoo/ (t) = d (t). Thus by LEBESGUE'S theorem on dominated convergence,
we have
1 1
- f w (t) d t- lim f/, (t) w (t) d t = lim p (/,),
0 n---> oo 0 n-+oo
and
1 1
-- f w (t) d t = lim f/ (t)/, (t) w (t) dt -- lim p (/'/,,).
0 n-+oo 0
Therefore lim P (//) = and M(p) = sup I P (/)I
.-oo (1/) p (I~1) o _>_ .j
(21.40) (a) Every finite-dimensional ~-representation T of a ~-algebra
A [unitary representation V of a group G is the direct sum of a finite
number of irreducible ~-representations unitary representations. If
the original representation is a continuous mapping of A [or G into the
algebra of all operators on the representation space, so are the direct
summands. We sketch the proof for a representation T of A; the proof
for V and G is similar. Let H be the representation space, of dimension n.
The proposition is true and trivial for n=l. Suppose that the prop-
osition is true for all T and H such that dim (H)< n. If T is reducible,
let HI, 0<dim(H1)<n, be an invariant subspace. Then H is also
invariant (2.9) and 0<dim(H)<n. By our inductive hypothesis,
T is the direct sum of irreducible components on H I and H, and there-
fore also on H. Continuity of the irreducible components of T is
obvious if T is continuous.
(b) A finite-dimensional representation, even one that is continuous
in the sense of (a), need not be the direct sum of irreducible representa-
334 Chapter V. Convolutions and group representations §22. Unitary representations of locally compact groups 335
tions. A simple example is the representation V,
of the additive group K by operators on K . The subspace Hi=
{(, 0):K} of K is invariant, so that V is reducible. The orthogonal
subspace H= {(0, ')"" K} is not invariant, and H I is easily shown to
be the only l-dimensional subspace of K invariant under V. Thus V
is not a direct sum of irreducible representations.
Consider the algebra of all matrices ( , where , K, and
/
(c)
let ( ): ( ). Then is a -algebra, and /the identity mapping
of onto itself is a representation that is reducible but is not the direct
sum of irreducible representations. [Argue as in (b).
(d) Uniqueness of the irreducible components of the representation V
in (a) will be discussed in Vol. II, (27.30); see also (27.44) and (27.56).
Notes
The theory of representations of groups and algebras has a long
history; see for example the notes to WEYL 3. The definition of
n-algebras is due to GEL'FAND and NAiMARK l, and much of the
present section is taken from GEL'FAND and NAMARK [2]. Positive
functionals on Ma(G ) make their appearance in GEL'rAND and RAI-
KOV 2], on n-algebras in GEL'rAND and NAIMARK l]. Cyclic -rep-
resentations of N-algebras with unit were defined by GEL'FAND and
NAiMARK l], and of groups by GODEMENT t]. Theorem (2t.lt) is due
to GODEMENT l]. Theorem (2t.18) is due to GODEMENT 2]; (2t.19),
(2.20) and (2.22) [for the case in which A has a unit to GEL'rAND
and NAiMARK 2. So far as we know, the function M(p) defined in
(2.20) is new, as are its applications in (2t.34) and (2t.39). We find
throughout the present section that some care is needed in going from
the case in which A has a unit to the general case which we treat.
Theorem (21.23) is due to GEffFAND and NAiMARK [2]; its analogue for
cyclic unitary representations of groups to GODEMENT Ill. I.E. SE-
GAL [2] has obtained the main results of the present section, for norm-
closed n-algebras of operators on Hilbert spaces [C*-algebras]. See
also RAIKOV [4]. Part of the proof of (21.32) is taken from RICKART Ill.
The main theorem of the present section, (2t.3 7), is due to GEL'FAND
and NAMARK [2] for Banach N-algebras with unit, and to I.E. SE-
GAL I2] for C*-algebras. Both proofs, and of course ours as well, depend
upon the KREN-MIL'MAN theorem.
§ 22. Unitary representations of locally compact groups
In the present section we study the connection between representa-
tions of a locally compact group G and representations of various
subalgebras of M(G). This culminates in the famous GEL'FAND-RAIKOV
theorem (22.2), which establishes the existence of a large number of
continuous irreducible unitary representations of G.
(22.1) Notation. Throughout (22.)-(22.2), G will denote an
arbitrary locally compact group. The symbol E will denote an arbitrary
reflexive complex Banach space, and E the linear space of all bounded
coў1"ugate-linear functionals on E. For oE and E, we write the
value of o at as (o, ). As pointed out in (B.2), every functional in
E has the form o-(o, ) for some fixed E. We therefore write
(, o)----(o, ) for E and oE, and identify E with E. Except
in (22.8), the symbol H will denote a Hilbert space, which may be
arbitrary or may be specified in the context. We identify H with H
in the usual way, writing as always (, ) for the inner product of
and in H.
For notational reasons it is convenient to consider representations of
G by operators on E ~ instead of on E. Since we are concerned only
with reflexive Banach spaces, this is a matter of convenience, nothing
more. If E is a Hilbert space H, the distinction vanishes.
(22.2) Definition. Let V be a representation of G by bounded
operators on E ~. If the function x-(Vo,) is Borel measurable
I),-measurable, ... for all oE N and E, then V is said to be weakly
Borel measurable weakly ?-measurable,.... If the function x- (Vo, )
is continuous for all oE and E, then V is said to be weakly contin-
uous. If the mapping x-Vo is a continuous mapping of G into the
Banach space E N for each o E, then V is said to be strongly continuous.
We first show how to obtain representations of M(G) and its sub-
algebras from certain representations of G.
(22.3) Theorem. Let A be a subalgebra o/ M(G), and let V be a
representation o/G by bounded operators on E such that:
(i) V is weakly ]#l-measurable ad weakly ]/1 *Iv] -measurable/or all
,vA;
(ii) a = sup {It,ll=
Then/or every/ A, there is a unique operator T, o E N such that
(iii) (T,oo,)--f (Vo,)d/(x) /or ooE and E.
The mapping #--> T, is a representation o[ A by bounded operators on E~,
/or which
(iv) I] T t] --<-- a ]1/ ]] /or / A.
336 Chapter V. Convolutions and group representations § 22. Unitary representations of locally compact groups 337
Proof. The integral in (iii) plainly exists and is a complex number for
every o)E and E. For a fixed o)E, consider the mapping
-+ f <V. o, > d# (,) ()
of E into K. This mapping is obviously conjugate-linear, and as
[J(Ve°,e)d/(x)l <c sup{llK, ll" xG} n e° .[ll. [l[, (2)
it is bounded. Hence there is an element of E, which we write as Tv m,
such that
for all E. The linearity of (m, ) in m and the linearity of f ... d#
show at once that the mapping m T,m is a linear mapping of E into
itself for each fixed . The inequality (iv) follows readily from (2). It
is also clear that T.+,= T. + g and T=z=T. for , v A and K.
We now show that T. = T T. Using (I 9.10), we have
= f f <g g , > dv (y) d (x)
-- f f <, V>dv(y)d(x)= f <To), VT> d(x)
-- f <T,>d(x)= <TT,>. 1
(22.4) Note. If eA for some aG, then it is clear that T,,=.
In particular, T,,==I [see (21.3), so that T obeys the convention
of (21.5) for representations of algebras having a unit.
We can also construct the adjoint of T,, which naturally is an operator
onE.
(22.5) Theorem. Let A and V be as in (22.3). For all xG, let
be the operator V 7, acting on E. The mapping xU, is a representation
o/G satis/ying (22.3.ii) and (22.3.i) /or the algebra A---{-" A}. Let
S be the representation o/the algebra A constructed/rom U by the process
o/(22.3). That is, we have
G
The relation
(ii) (r.)- =
1 Our hypothesis that E be reflexive is needed in this computation. Otherwise,
the adjoint V could not be regarded as an operator on E" see (B. 52).
holds/or all F A. I/E is a Hilbert space, so that E= E"--H, i/ "< A
whenever t A, and i/ all o/ the operators V are unitary 1, then S,= T,
and accordingly T is a -representation o/ the -algebra A.
Proof. It is obvious that U is a representation of G. Since
<U,,m>=(-m,>, (20.23.i) shows that (22.3.i) holds for all
/,, v a. Since ]]U,H = 1[-[, the relation (22.3.ii) holds with the same
value of . Noting again that (, m)= (m, ), we manipulate (i) as
follows for arbitrary A :
This proves (ii).
If E=E=H and g is unitary, then V=g, and thus T,= S,.
Equality (ii) then implies that Tf = T,., i.e., T is a -representation. n
(22.6) Theorem. Let A be a -subalgebra o/ M(G), let V be a represen-
ta,ion o/ G by uni,ary operators on a Hilber, space H sa,is/ying (22.3.i),
and le* T be the -representation o/ A construc,ed as in (22.3). Taeu every
closed subspace o/ H invariant under V is invariant under T. A partial
converse o/ this holds, viz. : let V be a weakly continuous unitary representa-
tion o/ G and let A have the property that/or every nonvoid open subset U
o/ G, there is a nonnegative A such that (U)= 1 and ( U')= O. Then
every closed sub@ace o/ H invariant under T is also invariant under V.
Proof. Let H 1 be a closed subspace of H invariant for all , and
let P be the projection operator onto H: thus P=P for all xG,
by (2.28). Then for all # A and , H, we have
(T, P, ) = f <p, ) d (x)= f (P , ) d (x)
= f <E,P>d(x)= <T.,P>=<PT.,>.
Thus T. P= P T., and by (2t.27) Hx is invariant for all G"
To prove the second statement of the theorem, we consider a subspace
H, invariant for T and the projection operator P onto Hx. Then for all
, H and # A, we have
()
<TaP, >-- f <P, > d,, (x).
G
Consider the bounded continuous function/(x)= (P, )-- (P, )
If /(a)O and [say Re/(a)>0, then Re /(x) fl>O for all x in a
In this case, (22.3.ii)is automatic: ]]Vx]] = I for all x.
Hewitt and Ross, Abstract haoc analysis, vol. I 22
338 Chapter V. Convolutions and group representations §22. Unitary representations of locally compact groups 339
neighborhood U of a. Choosing ,ac:A such that #>= 0, # (U)-- t, and
t, ( U') = O, we have
f ! (x) d/, (x) = f Re ! (x) d/, (x)+ i f Im! (x) d/, (x)= 0,
G U U
which contradicts (I). Therefore P V,=V,P for all xG; i.e., H is
invariant under V.
Theorem (22.3) does not tell the whole story on representations of
M(G) and its subalgebras. In fact, if G is nondiscrete, M(G) has a
-representation not of the form (22.3.iii) see (23.28.a). However, for
certain very special subalgebras A of M(G), all -representations of A
are obtained from representations of G as in (22.3).
(22.7) Theorem. Let A be a Banach -subalgebra o/M(G) with the
[ollowing properties:
(i) A and aG imply e,A;
(ii) every element o/the confugate space A* o/A has the/Grin f h d,
G
where h is a bounded/unction on G such that h,is Borel measurable
every a-compact set B C G.
Let T,be a -representation o/A by bounded operators on a Hilbert
space H such that/or every 0 in H, there is a ,u A/or which T, O.
Then there is a representation V o/G by u,itary operators on H such that
(22.3.i) holds and such that
(iii) (T,) = f (,) d(x) /or A and , H.
G
Proof. (I) Suppose first that T is a cyclic representation with cyclic
vectorH: thus the linear subspaceH' = {T,: A) of H is dense in H.
Consider the mapping
carrying H' into H', for a fixed xG. Let us show that (t) is well defined;
that is, if T,= T, then T,,,= T,,. Using the fact that (e,)N = , -,
(2o.23), and evident properties of the representation T, we can write
(T...(,_ . T...(,_ ) -- (T(,_-**_.**.(,_. ) / (2)
There*ore (t) defines a single-valued mapping of H' into itself. We
define the operator ' on H' by
As in (2), we see that
Given Tў in H', T,,,ў also belongs to H' and
Since ' is plainly linear, ' is a linear isometry of H' onto itself [see
(B.42)]. The equalities V,'=I and V.'V'=V., [for all x, y in G] are
evident. As pointed out in (B.59), each V.' admits a unique extension
V. over H that is a unitary operator on H. It is easy to verify that
V,= I, and that V.V= V.y: i.e., V is a unitary representation of G.
Like all -representations of Banach -algebras, T is bounded (21.22),
and IIT I[--< II# II" Hence for every 7 in H, the mapping
is a bounded linear functional on A. Let h be a function as in (ii) such
that
(T,,) = f h(y) d#(y) for all #a. (4)
By (9.0), we have
(Taў,)=(L,,ў,)=f h(y)d(e,,)(y)= f h(xy)d#(y). (5)
Let and v be any measures in M(G). Let B and C be g-compact
subsets of G such that Then it is clear that
Since BxCc{(x, y)GxG:xyBC}, we see that
for ]** ] x [v ]-almost all (x, y)GxG. Since hc is Borel measurable by
hypothesis (ii), we infer that the function (x, y)h(xy) is in
By (13.8), we know that the function
is ]#l-measurable for all,um(G). Since vectors g g (v M) are dense in H,
it follows that the function x(g, ) is [ l-measurable for all #M(G) :
thus (22.3.i) holds.
Let us now establish (iii). Fix v A. Then for every # A, h is plainly
in 1(ў* *]v[). Thus we can apply (19.10) and (5) to write
= f a(z)g. (z) = <T...ў, >= <T.T.ў, >.
This proves () for the case = T,ў for some vA. Such vectors being
dense in H, we infer that (iii) holds for all , in H.
(II) Suppose now that T is not a cyclic representation. We apply
Theorem (21.3) to T. The subspace N of (21.13) is {0}, and so H is the
direct sum of closed invariant cyclic subspaces Hr. Let T y denote the
operator T, with its domain restricted to Hr. By (I), there is a unitary
representation V {y} of G by operators on Hy such that
22*
340 Chapter V. Convolutions and group representations § 9.2. Unitary representations of locally compact groups 341
for ,tt A and all %, ]v Hr. Every vector e H has a unique representa-
tion e= ev, where e H,, only a countable number of e v are nonzero,
the series is convergent in the norm of H, and Ill= Y II ,ll", Let V be
the operator on H such that Ve= V,(/%. It is easy to see that x-V
is a unitary representation of G that is weakly I/,l-measurable for all
/,M(G) and for which (iii) holds. 1
We next discuss continuity of representations V of G.
(22.8) Theorem. Let V be a weakly -measurable representation o/G
by bounded operators on E~/or which (22.3.ii) holds, and suppose that the
Junction x-- (V, eo, ) is continuous at e/or all eo E N and E. Then the
representation V is strongly continuous. In /act, /or every E~ and
e > O, there is a neighborhood U o/e such that
(i) ]IV, -- Vy II < e/or x, y G such that y-1 x U.
That is, the mapping x-->V, is a le/t uni/ormly continuous mapping o[ G
into the metric space E ~. In particular, every weakly contimous V salis-
]ying (22.3.ii) is strongly continuous.
Proof. Let T be the representation of M(G) constructed from the
representation V as in (22.3). We first show that if E ~, then
@{T,@'#M(G)}-. Assume that @E ~ and that @ў(T,@'#M(G)}-.
As in (B.I 5), there is an element e of E such that
(Te, )=0 for all /,M(G), (I)
and
(, e)__ 1 (2)
[recall that E is reflexive]. From (2) and the continuity of (V,o, ) at e,
we infer that there is a neighborhood W of e such that 2 (W)< oo and
Re(V,o, )>Ѕ for all x W. Let v be the measure in 3I,(G) such that
dv = ewd2 e w is the characteristic function of W]. Then applying
(22.3.iii), we have
Re (T , ) = Re f V,,) dx = fRe (V,, e) dx > Ѕ (W).
W W
This contradicts (I).
Now consider any xG, eE, E and,uM(G). By (22.3.iii) and
(I 9.10), we have
G G
Thus we have
where the number is as in (22.3.ii). Now if d# = /d2, with/ 1 (G),
then d(e,./)=,_/d2, as (20.9.i) shows. From (20.4.i) we infer that
there is a neighborhood U of e such that
for all x U. Combining (3) and (4), we infer that
lie L e - L e II <
= (5)
for all x U.
As shown already, there is a 3& (G) such that
ч )lie - T. e II < (6)
If x U, where U is chosen for the/ just selected, (5) and (6) give
Finally, if x, y G and y-1 x U, we have
(22.9) Note. In view of (22.8) and an obvious calculation, a represen-
tation V oI G satisfying (22.3.ii) is strongly continuous iI and only iI it
is weakly continuous. We will henceforward write "continuous" instead
oI "strongly continuous" and "weakly continuous" in referring to such V's.
(22.10) Theorem. Ld T be a -represenaўion o/ўhe algebra M(G)
by operaors on a Hilber space H sўtch hat /or every + 0 iT H, there is a
/t M(G) /or hich Tў +0. There is a continuous unitary representation
V o/G by operaors on H such that (22.7.iii) holds. The representation V
is deermied by (22.7.iii) among all conўinuous unitary representations.
Proof. (I) For typographical reasons, this step of the proof is given
on page 508. (II) By (20.23.iv), M(G) is a -subalgebra oI M(G). By
(19.t8), it is a closed two-sided ideal in M(G). By (2.8), every bounded
linear functional on M (G) has the form
f (s) = f
where h(G) and / is the function in I(G) such that d = / dl. By
(I), we may suppose that he is Borel measurable for every C-compact
set B c G. Thus t&(G) satisfies the hypotheses oI (22.7). Let V be the
unitary representation oI G constructed in (22.7) for which (22.7.iii) holds.
We will show that V is continuous. Consider first the case in which
T is cyclic with cyclic vector ў. For jl (G), with/ (G) as above,
342 Chapter V. Convolutions and group representations §22. Unitary representations of locally compact groups 343
and xG, we have from (2.22) and (20.9.i) that
I1 , - T,, II- liT.:,,, - L il--< I1,,* - ' It" II It-- II,,-'l - III," It II.
By (20.4.i), the last expression is arbitrarily small for x in a suitable
neighborhood of e. Since vectors T, " are dense in H, we find that
for all x U, where U is a neighborhood of e that depends upon and e.
For arbitrary T, we use the construction and notation of part (II)
o (22.7). For n, we have I]V--II =- IVl--vll . This obviously
implies (I) for arbitrary T. For arbitrary yG, we have
so that the mapping z-- V, is a left uniformly continuous mapping of G
into the metric space H.
Suppose that V (1) and V () are continuous unitary representations
of G such that
f ::, ) I() = f v, ) I()
for all [ : (G). The argument used in (22.6) shows that (E(:), )=
((, ) for all , H and xG. Thus (=( for all xG.
We need one more preliminary to our main theorem.
(22.11) Theorem. For ,tM(G) and qg62(G ), let T,:#,. Each
T, is a bounded operator on the Hilbert space 2.(G), and the mapping
#--T, is a/aith/ul 1 ...-representation o/M(G).
Proof. The linearity of T, on .(G) is obvious from (20.a2.i), and
the boundedness of T,, with IIT II__< II ll, follows from (20.a2.ii). For
1 (G) ffl . (G), we have
( • ) • = • ( • )
from (19.2.iv). Thus T,,,(q))=T,(T,q)) for all q)g(G)fg.(G). Since
0o (G) c (G) ffl . (G), the subspace 1 (G) CI . (G) is dense in 2 (G)
[see (12.10)]. It follows that T,,,=T,T,. To show that T@0 if #=0,
consider an/oo(G) such that f/* d#0. The inequality
I,/(y)-,/(x)[ < [l,-,(s'*)-x-,(s'*)ll,, II,l
1 That is, T4= 0 if # 4: 0.
2 We call this the regular representation o[ 214(G). The mapping a-- Te a oI G
into operators on 2 (G) is called the regular representation o] G. For [ (G), we
have Teai--a-[" see (2o.12).
and the right uniform continuity of/* (4.t 5) show that #,/is continuous;
also we have ,/(e)= f/* d. Thus I[#*/ll = f [*/I d2 is positive"
i.e., T,/is not the zero element of (G).
It remains to show that T is a -representation. For , 0o(G),
we have by (20.23.iii) that
(, T,.) = f (x) f (yx) d(y) dx
= f f (x) (yx)dx d (y)
-- f f (y-x)(x)dxd(y) ' ()
= f f (y-1 x) d (y) (x) dx= (T u , ).
The two applications of FOBINI'S theorem are justified easily because
and are in 00 (G). Since o0 (G) is dense in (G), (I) holds for all
, in 2 (G), and this shows that T = T,..
(22.12) Theorem [GEL'FAND-RAIKOV]. Let G be a locally compact
group. For every a in G di//erent /rom e, there is a continuous, irreducible,
unitary representation V o/ G such that @ I. 1
Proof. Let U be a symmetric neighborhood of e such that 2(U)<
and a U . Let be the measure in M (G) such that d=v d2 recall
that u is the characteristic function of U]. Formula (20.9.i) shows at
once that d(e,)=vd2. Since aU , we see that II**-,-II=
liu--Uiil--22(U), so that e,@t/. By (22.), we thus have
T,:,, @T v, where T is the regular representation of M(G), restricted to
the -algebra M(G). By Theorem (2.37) applied to M(G) and T,
there is an irreducible -representation, say S, of M (G) such that
S,:,, # S,. ()
Theorem (22.t0) shows that there is a continuous unitary representation
V of G such that
<s,, > = f<<, > (x) (2)
for all vM(G) and all , in the representation space H of S and V.
From (t) and (2)we have
f <, > , (x)m f <, > ,,,(x) ()
for some vectors , in H. Theorem (9.t0) shows that
I <<,> ,ff(x):I <<,> ff(x) = f <,> f(x).
We abbreviate this assertion by the statement that G has su/jiciently many
representations of the sort described, similarly with other homomorphisms of G.
344 Chapter V. Convolutions and group representations §22. Unitary representations of locally compact groups 345
If E were the operator I, (3) would fail. By (22.6), the representation V
is irreducible" any closed subspace of H invariant under V is also in-
variant under the irreducible representation S, and hence is {0} or H.
For compact groups, Theorem (22.2) becomes particularly simple,
as the following theorem shows.
(22.13) Theorem. Every irreducible continuous representation V o/a
compact group G by unitary operators on a Hilbert space H is finite-
dimensional.
Proof. Let , , and be in H and consider the integral
f (V,, ) (V,,) dx. If we regard as fixed, this is a function that is
linear in and conjugate-linear in . Moreover, we have
Therefore by (B.60), there is a bounded operator Be on H such that
G
We next show that B commutes with every operator , y G. The left
invariance of the Haar integral shows that
G
and hence Bў=Bў. By (2.3o), Be is a multiple of the identity
operator I" Bў= (ў) I. Thus we have
G
and in particular
f [<K,>[=dx= (ў> i1[i
G
for all ў, H. Interchanging ў and in (t) and using (20.2.i) [being
compact, G is unimodular], we obtain
(> I[ I : f I <,>1 dx : f[<, >1 dx
G G
: fl <-, >l dx : f I<, >l dx : ()It,ll .
G
Hence there is a constant c such that ()=c[[ 2 for all H. Setting
--g and [ [= in (), we find that
fl <, >l d, : = (#)I[ll : c IItl - c.
G
Hence c is positive, since the continuous function x[ (, >1 has the
value t at e.
Now let 1,..., v,be orthonormal vectors in H, and apply (t) with
ў--v, and =v 1 (k= t, ..., n). We obtain
.V I <, 1>1 = dx= ()IIlll= = .
Summing (2) from to n, using the fact that {g}= is orthonormal,
and applying BESSEL'S inequality, we find that
=1 G G k=l G
Consequently, the dimension of H does not exceed /c.
(22.14) Corollary. Let G be a compact group. Theў G has sufficiently
many irreducible, continuous representations by unitary matrices.
Proof. Combine (22.2) and (22.3), and identify the group of
unitary transformations of K with its concrete representative
choose any orthonormal basis in K.
(22.15) Definition. Let S be a semigroup. Let Z be a complex-
valued function on S such that
Z(xY):Z(x)Z(y) for all x,yS.
Then Z is called a multiplicative/unction on S. If Z is also bounded and
not identically zero, Z is called a semicharacter o/ S. A semicharacter
of a group is called a character.
(22.16) Remarks. (a) A multiplicative function is a homomorphism
of a semigroup S into the multiplicative semigroup K. A semicharacter
is a homomorphism of S into the multiplicative semigroup
{zK:[z[ t} that is not identically 0. A character of a group G is a
homomorphism of G into the group T.
(b) Let V be a -dimensional unitary representation of a group G.
Then the representation space H of V has the form {:K) where
has norm . Defining Z(x) by the relation :Z(x), we plainly obtain
a character of G. Conversely, every character Z of G defines a -dimen-
sional unitary representation of G, so that there is a one-to-one corre-
spondence between all -dimensional unitary representations and all
characters. It is clear that V is continuous if and only if the corre-
sponding character is continuous. Henceforth we will identify -dimen-
sional unitary representations of a group with characters of the group.
(22.17) Theorem. Let G be a locally compact A belian group. Then G
admits su//iciently may continuous characters.
Proof. In view of (22.2) and (22.6.b), we need only to prove that
every irreducible unitary representation V of G is -dimensional. Let a
be any element of G. Then the unitary operator commutes with all
346 Chapter V. Convolutions and group representations § 22. Unitary representations of locally compact groups 347
of the operators V. Theorem (21.30) shows that V is a multiple of I"
V=z(a)I for all aG. Thus every subspace of the representation space
H is invariant under V, so that H has to be i-dimensional. l
Continuity of characters follows from -measurability. In fact, much
more can be proved.
(22.18) Theorem. Let G be a locally compact group with le/t Haar
measure . Let H be a topological group that is either -compact or contains
a countable dense subset H need not be locally compact. Let be a homo-
morphism o/G into H such that/or some -measurable set A c G/or which
0 < (A) < oo, z -1 (U A z (G)) A is -measurable /or all open subsets U
o/H. Then z is continuous on G.
Proof. Let W 0 and W be symmetric neighborhoods of the identity
in H such that Wc W 0. The hypotheses on H ensure that there is a
countable subset {Yl .... y,,...} of H such that U Wy,,=H. We have
' n=l
U (z - (Wy,)A)= A, and so there is at least one value of n for which
n=l
0< 2 (z -1 (Wy,) A) < oo. By (20.I 7), there is a neighborhood V of the
identity in G such that Vc(z-l(Wy,)A). (z-l(Wy,)A) -1. For x V,
we have x-- ab -1, where z (a) --wl y, and z(b)--w.y,,andwherewl, w. W.
Hence z (x) = z (a) z (b)-I __ WlW 1 W W0. Thus z is continuous at the
identity and therefore is continuous throughout G (5.40.a). l
(22.19) Corollary. Let G be a locally compact group azd z a homo-
morphism o/ G into ®(n,K); we write z(x)=(a(x))7,=l. Suppose
that there is a -measurable set A G such that O< (A) < c and each
/unction x-->ay(x) is -measurable on A. Then z is continuous o G.
In particular, every multiplicative complex/unction on G that is -meas-
urable on such a set A is continuous.
Proof. It is easy to show that {xA'(a,(x))7,= 1 1I} is t-measurable
for all open subsets 1I of ®(n, K). Then apply (22.18).
Miscellaneous theorems and examples
(22.20) Continuous unitary representations. (a) Let G be a locally
compact group and V a representation of G by unitary operators on a
Hilbert space H. The following properties of V are equivalent:
(i) V is strongly continuous;
(ii) V is weakly continuous;
(iii) the mapping x->V of G into H is left uniformly continuous
for all H;
(iv) the mapping x->V of G into H is continuous at e for all H;
(v) the function x-->(Vl, ) is left uniformly continuous for all
$, H;
(vi) the function x- (V $, $) is continuous at e for all $ H.
[The equivalence of (i) and (ii) was noted in (22.9). Taking into
account trivial implications, we see that it suffices to prove that (iv)
implies (iii) and (vi) implies (v).
The equality IlK, e_Ky ell = IiKy_,,e_ e [
shows that {iv) implies {iii). Suppose that {vi) holds. Then for e, / H
and x, yG, we have
and therefore
This last inequality and (vi) imply (v).
{b) Let G be a locally compact group and K a representation of G
by unitary operators on a separable Hilbert space H. Then the following
are equivalent:
(i) K is continuous;
(ii) the function x-->(V/, ) is -measurable for all /, H;
(iii) the mapping x-->V of G into H is -measurable for each fixed
H; i.e., the inverse image of any open set in H is -measurable.
Obviously (i) implies (ii). For , H, we have
this shows that (ii) implies (iii). Suppose that (iii) holds. To prove (i), it
suffices by (a) to show that ---V is continuous at e, where : H and
=t=0. For e>0, let A= {xG-IIV--II< -}. Clearly A is ;t-measurable
and symmetric. If x and y are in A, then IlV,,-l[_<_llV,,-Kl[+
so that It
remains to prove that A contains a neighborhood of e. Since H is sepa-
rable and metric, the subset {V'G} of H is also separable. Thus
there exists a sequence {,},--1 of elements in G such that {V,}__I is
dense in {V" G}. If yG, then for some positive integer k, we have
]IE$-$II=II$--E,,I[<-, and therefore yxA. In other words,
G =,;1U x,A. It follows that A contains a compact subset F having posi-
tive measure. By (20.17), F(F -1) contains a neighborhood of e.
348 Chapter V. Convolutions and group representations § 22. Unitary representations of locally compact groups 349
(C) (E. THOMA, oral communication.) Let G be a nondiscrete locally
compact group. Then G admits a cyclic, weakly -measurable, unitary
representation that is not continuous. Let l,. (G) be the Hilbert space
of all functions on G such that l(x)['<oo, with pointwise linear
operations and (, )---- (x) (x). This is of course . (G), where G d
denotes G with the discrete topology. Let be the function in
such that (e)----I and (x)=0 for x=e. For xG, let V be the operator
on 12(G ) such that V(y)--(x-ly). It is obvious that the mapping
x-->V is a cyclic unitary representation of G, with cyclic vector
is the function on G whose value at x is I and elsewhere is 0. Linear
combinations of such functions are dense in l,.(G). For , l,.(G), let
A and B be countable sets outside of which and , respectively, vanish.
If (, V) 4= 0, then x must be in the countable set B (A-l). Hence the
representation V is weakly -measurable. Since G is nondiscrete, (4.26)
shows that the set {xG: (V, 1)4 0} is open if and only if it is void.
Thus the function x-->(V, /) is continuous if and only if it vanishes
identically. Therefore V is not a weakly continuous representation.
(d) The representation /,-->T constructed in (22.3) may well be
identically 0: T may be the zero operator for all ,u A. For example,
consider the algebra M c (G) for a nondiscrete G and apply (22.3) to m (G)
and the representation V constructed in part (c).
(22.21) The definition of the regular representation T of M(G) given
in (22.11) is by no means arbitrary. Alternative candidates exhibit
serious drawbacks. A first naive candidate might be the following. For
#M(G) and ,.(G), define S=,/,. With such a definition, S will
not even satisfy the relation S.,= SS,. To correct this, we redefine
S =.#'. This is still unsatisfactory since -->S =.#' might not
map ,.(G) into ,.(G) (20.34). In any case, the relation S,=oS, will
not hold for all # M (G) and K.
There is, to be sure, another natural N-representation S of M(G) : use
right Haar measure instead of , and use the representation space
,. (G, e). In the notation of (20.32), this representation turns out to be
S=Ky.,# for ,.(G, ). The representation S is equivalent to the
representation T given in (22. ). The connecting isometry W between
,.(G, )and ,.(G, )is W()----* (20.32.b).
(22.22) Groups with no finite-dimensional unitary representations.
Let G be any topological group. We are interested here in those elements
of G that can be separated from e by a continuous, finite-dimensional,
irreducible, unitary representation of G. We denote by G 0 the set of
elements in G that cannot be separated from e by a continuous, finite-
dimensional, irreducible, unitary representation of G; i.e., x belongs to G o
if P--I for all such representations V.
(a) If K 4 I for some continuous, finite-dimensional, unitary represen-
tation V, then x is not in G 0. IBy (2.40.a), V is the direct sum of ir-
reducible representations V ) ..... V ("), which are clearly also continuous.
Then for some/'-----, ..., m, it must be the case that V, lil= I.
(b) The subset G o of G is a closed normal subgroup of G. This follows
from the identity Go={xG:V=I and V is a continuous, finite-
dimensional, unitary representation of G}.
(c) If H is a closed normal subgroup of G such that G/H is compact,
then GoCH. Let y be in GH'. By (22.14), there is a continuous,
finite-dimensional, irreducible, unitary representation V' of G/H such
! !
that Vn4= V--I. For xG, we define V= Vn, and obtain a continuous,
finite-dimensional, irreducible, unitary representation V of G for which
VI. ThusyGGandGocH.
(d) For any discrete free group G, we have G0--{e}. By (4.21.e),
the intersection of all normal subgroups having finite index is exactly {e}.
Hence Go-----{e} by (c).
(e) For our remaining examples, we need the following lemma
(v. NEUIANN and WIGNER i). Let A belong to lI(n), the group of
unitary n xn matrices. Suppose that for every m--i, 2, ..., there is an
integral multiple k m of m and a nonsingular matrix B m such that
A*'=BAB, 1. Then A--I. Let Xl .... , x, be the eigenvalues of A.
For any m, A " has Xl ", ..., x, as its eigenvalues and B,,,AB, has
x I .... , x, as its eigenvalues. Thus x, ..., x, must be a permutation
of x I .... , x,. Consider now a fixed x.,/'--i, ..., n. Since {kin}m= 1 is an
m = x for
infinite sequence and {@, ", ...}C{Xl, ..., x,,}, we have x i
some k m and kz with kz < k,,. That is, x --- and x. is a root of unity.
Now choose m 0 so that xF°=t for /'--, ..., n. Then xo=1 for
-"o is a permutation of Xl, . x,,, we see
-= l .... , z, and as x o .... , x, .. ,
that all of the eigenvalues of A are equal to . Consequently, we have
=.
(f) (v. NnUA and Wc,nn t.) Let G be a topological group.
Consider a fixed in G. If for every m= I, 2, ..., there is an integral
multiple k of m and an element in G such that = , then
belongs to G 0. [Let V be a continuous, finite-dimensional, irreducible,
unitary representation of G; we may suppose that the representation
space is K " Let A = V and B= Vy and apply (e) to infer that V=I.J
(g) Let G be the group of matrices 0 where x and are rational
numbers and =t=0; give G the discrete topology. Then G o consists of
350
Chapter V. Convolutions and group representations
the matrices having the form (;/. ,\ IY). EFor typographical convenience,
write (x, y)for ( /" For a real number 0, let Z((x, y))
we
-/
exp(2i.log[xl) and ((x, y))--I1--sgnx. Then Z and are con-
tinuous characters of G, that is, t-dimensional unitary representations
of G. For any (x, y)G for which Ix I 4=t, there exists an R such that
Z((x, y))=exp(2i.loglxl)4=. Also for any (--t, y)G, we have
((--t, y))=--t. Therefore we have GoC{(t, y)G'yQ}. For every
matrix (t, y) G and positive integer m, the equalities (m, O) (t, y) (m, O)-1 =
(1, my)=(t, y) hold. Setting k, equal to m, we see from (f) that
(t,y)G o. That is, {(t, y)G'yQ}cG0.
If G is the above group with its relative topology as a subgroup of
®(2, R), the same computations show that G0={(t, y)'yQ}. Also,
ifGisthegr°up°fmatrices (Xo Yt) wherexandyarerealandx=4=O'r/
then G0--/(lo t)'yRt when G is given its usual topology or the dis-
%\
crete topology.
(h) Let G be the special linear group ®g(2, K) with its usual topology
or with the discrete topology. Then Go= G" i.e., G admits no nontrivial,
finite-dimensional, unitary representations. EObviously we need only
/J
Gis discrete. Consider the element (a, /in G"
consider the case where
a is any complex number. Then for any positive integer m,
Letting k=m and applying (f), we find that t; i') G°" Since G O
is a
normal subgroup of G and '
(t )__( -;t)(t 0 -)( --0t) -1,
(t )isin G0 for all a/Ј" Now consider an arbitrary matrix (c a )inG.
If c =t= 0, then
and if c =0, then d 4= 0 and
Consequently G o-- G.
§ 22. Unitary representations of locally compact groups 351
(i) Groups with no nontrivial, finite-dimensional, unitary representa-
tions can also be found by the following argument, which is strongly
analogous to (f) supra.
Let G be a group [no topology and let H be an infinite subgroup of
G with the following properties:
(i) there are only finitely many conjugacy classes {e}, H1,..., m m
of H with respect toinner automorphisms of G;
(ii) for every H i, the smallest normal subgroup of H containing H i
is H;
(iii) the smallest normal subgroup of G containing H is G.
Then G admits no nontrivial homomorphism into a compact group.
[Let be a compact group and q a homomorphism of G into .
We prove first" (I) If q (Hi) I4= for all neighborhoods I of , then
(Hi)--{}. For every neighborhood of , there is a neighborhood
I of such that -lI C for all (4.9). If h, h lie in /-/., then
-1 hi t -- h2 for some t G. Thus if q (hi) I, then q (he) = q(t)-1 q(h) q(t) .
Therefore c(Hi) U and, as U is arbitrary, (I) follows.
(II) If 9 (H;)= [} o some i, then (ii) and (iii) imply that q (G)= {}.
(III) If 9 (H;) {} for i-- , 2, ..., m, (I) implies that q (H fl {e}') fll
-- for some neighborhood I of . This implies that q is one-to-one
on H and also that q (H) is a discrete subgroup of . Since H is infinite
and G is compact, this is impossible. Hence this case cannot occur.
(22.23) Some special representations. (a) Let G be a compact
group, and let x-+V, be a continuous representation of G by bounded
operators on a Hilbert space H. Then there is an inner product << ', />>
on H with respect to which: the operators V, are unitary; the mapping
x- V, is a continuous representation; and H is a Hilbert space. [By virtue
of (22.8) and the BANACH-STEINHAUS theorem (B.20), we have sup{llV, I] :
x G} < . Let (', ) denote the original inner product in H. For ', H
and a,x G, we have
Applying (22.8.i) to the foregoing, we see that the mapping x-->
is continuous on G" note that the mapping x- llv wll i a continuous
function on G.
Therefore we can define
where f ... d s denotes normalized Haar integration on G. We obtain in
this fashion a complex number for each pair (, ])HxH. It is obvious
352
Chapter V. Convolutions and group representations
that (B.39.i)--(B.39.iii) hold for the function <<, 9>>. Since (Vs, Vsў)
is positive for all sG and nonzero H, (B.39.iv) also holds. For every
x{ G, we have
since the Haar integral on G is right invariant. We also have
f IlK,g- gg[I. Ilgllds;
Theorem (22.8) now shows that the representation x is continuous
with respect to the inner product <<, >>.
It remains to show that H is complete in the metric induced by the
inner product <<ў,>. Write ў for <<ў,>. Let {ў,,, be a
Cauchy sequence. Then there is a subsequence {g}%1 for which, upon
applying (t2.4), we have
_ ў-ў+ < 2 - ( = , 2,...).
Summing this over h= , 2,..., and applying the monotone convergence
theorem, we have
Theorem (tt.27) implies that the sum is finite for
, E is a
k-almost all s<G. Thus for some a<G the sequence {
Cauchy sequence of vectors in the original norm for H, and hence has a
limit, say g. Since -: is a bounded operator, we infer that g has limit
_,g. Write -,ў=ў'. We then have
lim ў,--ў' = lim [f ((ў--ў'),(ў--ў'))ds]
It follows as usual that lira , --' 0.
(b) Let be a positive integer and let be a ubgroup of the linear
group (, ) that is compact in its relative topology as a subspace
of . Then there is a matrix T (, ) such that all of the matrices
T-MT for M are unitary. Choose any basi .... , % in and
for M()i, let M' be the linear operator on such tha
§ 22. Unitary representations of locally compact groups 3 53
M'(ek)= rn.k e i (h--I .... , n). The mapping M-->M' is obviously a
/'=1
representation of 9A by operators on K"satisfying the hypotheses of
part (a). Thus there is an inner product <<a, ?/>> for K n such that
<<M' de, M' j>> = <<a, ?/>> for all M 9A and all a, j K n. Let 1'1, -..,
be an orthonormal basis for K" with respect to the inner product
<<a, y>>. Let 2"= (t/.k)in, k=l be the matrix such that f--
/.=1
(k= 1 .... , n). Then all of the matrices T-1MT are unitary.]
(c) (WEII 4, p. 70.) Let G be any group not necessarily topo-
logical], and let z be a homomorphism of G into ®g(n, K); we write
z(x)--(a/.k(x))i",= 1. Suppose that all of the functions ai are bounded
on G. Then there is a matrix T®g(n, K) such that all of the matrices
T -1 z(x)T are unitary, lit is easy to see that the closure of z(G) in
((,, K) is a bounded, closed subset of K s' and hence is a compact
subgroup of ®g(n, K). Now apply part (b).
(d) Let G be any group and let x--> V, be a finite-dimensional represen-
tation of G by operators on K such that the functions x-->(V,y,
are bounded on G for all ?/K . Then there is an inner product
on K such that all of the operators V, are unitary with respect to this
inner product. This is merely a restatement of part (c).]
Notes
The first paper we know of dealing explicitly with in/inite-dirneusional
unitary representations of a group is WICKER [t], in which irreducible,
infinite-dimensional, unitary representations of the inhomogeneous
Lorentz group are computed. The regular representation V [V/=e./
of G by operators in g(G) appears implicitly in WEYL and PETER t
for compact Lie groups G, and by operators in g(G) in A.WEIL 4,
Ch. V. The first study of infinite-dimensional unitary representations
of general locally compact groups is in the fundamental paper of GEI'A
and RMtov [21; see also GEI'FA and RMtov [3. Representations
of G by operators on Banach spaces were introduced by I. E. SEGAL [31.
They have been further studied by LoosilS 2, §32, and by SHICA [1]
for compact groups.
Theorem (22.3) for unitary representations of M(G) is due to GOE-
IET [tl and for Banach space representations of M (G) to LooMIs 2,
Theorem 32B. For A=M(G), (22.7) appears in LoosIS 21, Theo-
rem 32C, and in Na1siat [t, §29, Theorem t. The fact that weak
continuity implies strong continuity for unitary representations seems
first to have been remarked by GO>EMEXT [t, p. 14, footnote t2. A
result similar to (22.8) for compact groups appears in SHICA [I. So
Hewitt and Ross, Abstract harmonic analysis, vol. I 23
354 Chapter V. Convolutions and group representations §23. The character group of a locally compact Abelian group 355
far as we know, (22.8) is new for the groups and representations con-
sidered. Theorem (22.t0) is based on NMMARK t 1, §29, Theorem t.
The basis of (22.t t) appears in A. WEIL [41, p. 48; the rest is closely
related to observations of GODEMENT [t.
Theorem (22.t2) is due, as noted, to GEL'FAND and RAiKov. The
proof given here is like the proof given by I. E. SEGAL [21. GOE-
NENT [tl has published a proof of (22.t2) very like the original one, as
have several other authors. In the general arrangement of our proof
[although not in its details I we have been influenced by NMSIARK t],
§29. It is worth noting once again that all proofs of (22.t2) depend
upon the KREiN-MIL'AN theorem (B.30) and hence in the end upon
compactness. In this (22.t2) resembles a remarkable number of the most
important facts in analysis.
The GEL'FANI)-RAiKOV theorem has provided the basis for the formi-
dable theory of unitary representations of locally compact, noncompact,
non-Abelian groups, as it has developed since t947. This theory is
outside of the scope of our work.
Theorem (22.t3) was proved for compact groups G with countable
open bases by A. GUREVI [t. In fact, she proved more: every con-
tinuous unitary representation of such a group is the direct sum of
irreducible, finite-dimensional, unitary representations. We will prove
this for all compact G in Vol. II, (27.44).1 KoosIs t and NACHBIN
have published proofs of (22.t3); the proof in the text is NACHBIN'S.
Corollary (22.t4) is due to WEYL and PETER Ill for compact Lie
groups and to VAN KAMPEN [t I for arbitrary compact groups. VAN
KAMPEN pointed out that every continuous function on a compact
group is almost periodic [see (t8.2) and then applied VON NEUMANN'S
theorem [4] on almost periodic functions and irreducible, finite-dimen-
sional, unitary representations. PONTRYAGIN [51 also published a proof
of (22.14) for compact groups with countable open bases.
Characters of groups have been known and exploited for many years.
For some of the history, see the notes to §§23--25. Semicharacters of
semigroups were introduced only recently: in independent papers of
. GCHWARZ [t] and HEWITT and ZUCKERMAN 31" Theorem (22.t7) was
proved by voN NEUMANN 4 for locally compact Abelian groups with
countable open bases; by J.W. ALEXANDER t for discrete Abelian
groups; by VAN KAMPEN [1 ] for arbitrary locally compact Abelian groups.
A proof of (22.a 7) based on the theory of Banach algebras Ibut postulating
the existence of Haar measure was given by GEL'FAND and RA]KOV
Theorem (22.a 8) generalizes (22.a9), which is a theorem of A. WEIL
pp. 66 --67.
, propos of (22.22), it is interesting to remark that locally compact
groups G with sufficiently many finite-dimensional, continuous, unitary
representations are fairly special. If such a group is connected, then it
has the form R'xGo, where n is a nonnegative integer and G o is compact.
This startling result was proved by FREUDENTHAL [21 for groups with
countable open bases. The general case is due to A. WEIL [4], pp. 26--
t30. There are many disconnected noncompact groups with sufficiently
many finite-dimensional, continuous, unitary representations: see
(22.22.d) for example. It would be interesting to have extensions of
FREUDENTHAL'S theorem to at least some classes of disconnected groups.
Chapter Six
Characters and duality of locally compact Abelian groups
This chapter is devoted to what might be called the "fine structure"
of locally compact Abelian groups. We will obtain extremely detailed
information about the structure of these groups. The basic tool in this
program is the character group, which is defined and studied in §23.
The character group is important in large measure because of the
PONTRYAGIN-VAN t{AMPEN duality theorem, which is stated, proved, and
utilized in § 24. In § 25, we apply the duality theorem to study a number
of special locally compact Abelian groups, and in §26 we apply it to
certain problems in structure theory and analysis.
§ 23. The character group of a locally compact Abelian group
(23.1) Definition. Let G be a group, not necessarily Abelian or
topological. Let Z1 and Z2 be characters of G. The product Z1Z2 is defined
as the ordinary pointwise product: I(X)=1 (X)
(23.2) Theorem. The set o/all characters o/G is an A belian group,
with identity element the/unction and Z-I= ,.
This theorem is obvious.
(23.3) Definition. Let G be a topological group. The group of all
continuous characters of G is called the character group o/G. Character
groups will be denoted by letters N, Y, 03 .....
Plainly every topological group has at least one continuous character,
namely, t. Some noncommutative locally compact groups have no
others (22.22.h). We shall see in the present section the usefulness of the
character group of a locally compact A belian group, and begin with a
general theorem.
(23.4) Theorem. Let G be a locally compact I group, and let A be a
closed subalgebra o/2VI(G) with the ]ollowing properties:
1 Not necessarily Abelian!
23*
:356 Chapter VI. Characters and duality of locally compact Abelian groups § 23. The character group of a locally compact Abelian group 357
(i) every bounded linear [unctional on A has the [orm #--> f h d/z,
where h is a bounded [unction on G such that h E is Borel measurable [or
all a-compact subsets E o[ G;
(ii) aG and #A imply ,,#A and #,eaA.
Let be a multiplicative linear [unctional on A. 1 Then there is a contin-
uous character Z o/G such that
(iii) (/Z)= fy.d# /or all /ZA.
I[ Z is any continuous character o[ G, the mapping
(iv) - f '.
G
is a multiplicative linear [unctional on M(G).
Proof. Let/Z, v be measures in A for which z(/z)4: 0, z (v)4:0, and
let a be any element of G. Then we have
Thus we have
putting Az--v, we have z(e,v)--z(v,e), and hence
z(,,) ,(,,v)
Let Z be the function such that z(a)--z(e-,,) for any such that
z) 0, and all aG. If (v) =0, then z(,,v) =0 for all bG. To see this,
choose A such that z()= l, and write
(,, ) = () (,, ) =, (, ,, ) = ( • ,) ff) = 0.
Hence we have z(a)z(v)=z(e-,v) for all vA and aG. Then for a, b
in G, we have
Since z is a bounded linear functional (C.2]), we have
so that Z is bounded. Since Z (e)= , it follows that Z is a character of G.
We now prove that Z is continuous. By hypothesis (i), we have
-c(/z)-- f h d/z for all/Z A, where h is a function as in (i). Choose # A
Recall that a multiplicative linear functional is not identically zero (C.2).
Our choice of instead of Z in this integral and in (iii) is arbitrary. Inversion
formulas to be presented in Vol. II use X instead of " we elect to use , first.
so that z(F)--], and let E be a a-compact subset of G such that
[[ (E')--O. For all Borel sets A cG and xG, we have
[/Zl (xA), so that e-, [/Zl ((x-E) ') = -'* l/z[ (x-(E')) =l/z[ (E')--O. Thus
for each x G, we have
Z (x) -- f h (y) d e-,,/ (y) -- f x-, (Y) h (y) d e-, ,,u (y)
G G
-- f ў,-e(x-ly) h(x-ly) d (y) = f e(Y) h(x-ly) d(y).
G G
Now let F be a compact subset of G such that 2(F)>0. The function
F-,eh is Borel measurable on G by hypothesis (i). The function
(X, y) % F(X) E(Y) iE(x-ly) h(x-lY) ()
is accordingly Borel measurable on GxG and vanishes outside of the
-compact set FE. The estimate
is immediate. Theorem (3.0) implies that the function () belongs to
(GxG, x[), and (t 3 .8) implies that the function
x f ()(y)_ (x- y) (x- y)
G
--ў(x) f (y) h(x-y) d(y) = ў(x)Z(x)
G
is -measurable on G note that ў (x) ў (y) ў-, (x - y) = ў (x) ў (y) for
all (x, y)GxG. That is, the function Z is -measurable on F, and so
by (22.9), is continuous on G.
To establish (iii), let be a measure in A such that z ()
be any measure in A, and let E and E2 be -compact subsets of G such
that ] (E) -- 0 (j-- t, 2). Then [[ • I1 ((EE>') 0, EE is -compact,
and so we can write
G
-f ўc(u) h(u) d,2(u) = f f ,(xy) h(xy) d2(y) d (x)
G G G
G G
E G
For xE, we have 2-EE2 x - xE= E2. Hence the last integral is
equal to
E G E G
Since is arbitrary, we have established (iii).
Finally, we establish (iv). Since is continuous and 2] =1, Z is in
(G, ]/z] ,]v) for all , vM(G). Theorem (19.10) then gives
358 Chapter VI. Characters and duality of locally compact Abelian groups §23. The character group of a locally compact Abelian group 359
Since f Z (x)de, (x)-- (e)= t, the mapping (iv) is not identically zero on
M(G)" thus it is a multiplicative [obviously linear I functional.
(23.5) Note. If G is nondiscrete, then M(G) admits at least one
multiplicative linear functional not of the form (23.4.iv). See (23.28.a).
(23.6) Theorem. Let G be a locally compact group. Let A be a sub-
algebra o/M(G) such that/or every nonvoid open subset U o/ G, there is a
in A + such that # (U) = t and # (U') = O. Then i/ 25 and are distinct
continuous characters o/ G, the multiplicative linear/unctionals # -> f d#
and #-> f d# are distinct on the algebra A.
G
Proof. Let aG be such that 25(a):4:v(a). Suppose also that
Re 25 (a) > Re (a). Then there are a neighborhood U of a and a posi-
tive number such that Re 25 (x) -- Re (x) >__ for all x U. Let # be
a measure in A + such that # ([7) -- t and # (U') -- 0. Then it is clear
that Re f , d#- Re fcp d# _>_ . Similar arguments apply to the cases
G G
Re 25 (a) < Re o (a) and Im 25 (a) 4: Im (a). [3
(23.7) Corollary. Let G be a locally compact group. Every multi-
plicative linear/unctional on 3I s (G) has the/orm
(i) -fd,
G
where 25 is a continuous character o/ G. Conversely, every such mapping
is a multiplicative linear /unctional on Ms(G ), and distinct continuous
characters o/ G produce distinct multiplicative linear /unctionals by the
mapping (i).
Proof. This follows at once from (t2.t8) and part (I) of the proof of
Theorem (22.0) page 5081, which show that Ms(G ) satisfies the hy-
potheses of (23.4), together with (23.4) and (23.6).
The character group of a locally compact group G is useful in studying
the structure of G only insofar as the elements of G commute with each
other. We state this in precise terms as follows.
(23.8) Theorem. Let G be a locally compact group. Let C o be the
group generated by all elements o/ the/orm a ba-lb -1, where a, bG" 1 Co
is a normal subgroup o/G. Let
A = {xG'25 (x)= t /or all continuous characters o/G}.
1 Co is called the commutator subgroup o[ G. See, however, the footnote on
page 142 of Volume II.
Then C=A. The character groups o/ G and G/A are isomorphic under
the mapping 25->251, where Z is a continuous character o/G and T.1 (x A)=25 (x)
/or all xG. I/H is a normal subgroup o/G, then G/H is A belian i/and
only i/HD C o. I/H is closed and G/H is A belian, then H D A and G/H
is topologically isomorphic with (G/A)/(H/A).
Proof. If xG and cCo, then XCX-I=(xcx-lc -1) cCoCo=Co, so
that C o is a normal subgroup of G. By (5-3), C- is a closed normal
subgroup of G.
Suppose that H is a normal subgroup of G. If G/H is Abelian, then
for a, bG, we have aHbH=bHaH and hence a-lb-labH. It follows
that C o c H. Conversely, if C o c H, then for a, b G, we have a -lb -lab
C o c H and so a H b H = b H a H. Thus G/H is Abelian.
If Z is a continuous character of G, then -1(t) is a closed normal
subgroup of G. As the intersection of closed normal subgroups, A is a
closed normal subgroup. Clearly all elements of the form aba-lb -1
belong to A. Therefore we have C-c A, since C- is the smallest closed
subgroup containing these elements. Assume now that C- . A and that
xA f (C-)'. Then G/C- is Abelian; it is locally compact and Haus-
dorff by (5.22) and (5.2t). By (22.a7), there is a continuous character
251 of G/C-ff such that Z (xC)4: t. If 9 denotes the natural mapping of
G onto G/C-, then Zl° q0 is a continuous character of G and Zl° q0 (x)@ 1.
That is, xў A. This contradiction shows that C-=A.
The last statement of the theorem now follows from (5-35).
Finally, for every continuous character Z of G, let Z1 be the function
on G/A such that ZI(XA)=z(x ). If xA=yA, then y-lxA, so that
25(y-1x)=25(y)-1Z(x)=l. Thus Z1 is well defined on G/A. It is plain
that Z1 is a character of G/A. If W is an open subset of T, then Z 1 (W)=
{xA G/A "251(xA)W}--{xA G/A "25(x)W}, which is the image of the
open set Z-I(W) under the natural mapping q0 of G onto G/A. Since q0
is open (5.t 7), Z1 is a continuous character of G/A. If 0 is a continuous
character of G/A, then o q0 is a continuous character of G and = (oq0)l.
Hence the mapping Z->Z1 is an isomorphism of the character group of G
onto the character group of G/A.
Theorem (23.8) shows that in studying character groups we lose
nothing by considering only A belian groups. The ensuing analysis will
therefore be couched in terms of Abelian groups. Throughout (23.9)
to (23.t6), G will denote an arbitrary locally compact Abelian group
and X will denote the character group of G.
(23.9) Definition. For every M(G), let/2 be the complex-valued
function on X such that
(i) fi(25) =
G
360 Chapter VI. Characters and duality of locally compact Abelian groups § 23. The character group of a locally compact Abelian group 361
This function is called the Fourier-Stiel@s trans/orm o/#. If M (G),
so that d#- [ d with [il(G), we write ] for// and call ] the Fourier
trans/orm o/the/unction [. For a subset A of M(G), . will denote the
set {fi'#A} of functions on X" similarly, will denote {]'/} for
c ().
(23.10) Theorem. The [ollowing relations hold:
(i) (# + v) '- fi + /or #, v M(G);
(ii) (#)'--f, [or o<K and #<M(G);
(iii) (#. v) - =/; [or #, v M(G) ;
(iv) (#-)'-- /or #M(G) "
(v) is(Z)--z(a) /or a(G and
(vi) sup (z) l: x} _<___ I1# II/or # M(G).
All of these relations are simple consequences of the definition. We
omit the verification.
(23.11) Theorem. Let # be a nonzero measure in M s(G). Then there
is a y. X such that fi (Z) 4 = O.
Proof. By (22.tt) and (21.37), there is an irreducible --representa-
tion S of Ms(G ) for which S, is not the zero operator. The algebra
M s (G) is commutative (t9.6), and see also (2o.24). Hence every operator
S, (vms(G)) commutes with all operators $t, (F3/Is(G)) and so by
(2.30), S, is a multiple of the identity operator: S,--z(v)I. Clearly
defined in this way, is a multiplicative linear functional on M s (G) and
z(#)4:0. Referring to (23.7), we see that z(#)--f.d# for some
zX. 0
(23.12) Note. The analogue of (23.t t) holds also for M(G): if #M(G)
and ,u 4: 0, then fi 4= 0. The proof is somewhat more complicated, how-
ever, and is postponed to Vol. II, (3 a. 5).
We now define a topology in X.
(23.13) Theorem. For /,...,/m(G), ,>o, and z0X, let
a (/,, ...,/m 'Z0) {Z X" IL (Z0) I < /or i- .... , m}. Let
the sets A (ll, ..., /m; e;Z0) be taken as a basis/or open sets in X. Under
the topology so delined 1 /or X, X is a locally compact Hausdor/] space.
Proof. By (23.7), X can be identified with the structure space of the
commutative Banach algebra M s(G) (C.23). The A-topology is just the
Gel'land topology for this structure space and is therefore locally com-
pact and Hausdorff (C.26).
(23.14) Theorem. Let I be a/unction in (G), and let be a positive
number. There is a neighborhood U o/ e depending on / and with the
1 We vill call it provisionally the A-topology for X.
lollowing property. I/ x0, x in G and ;go, Z in X are such that ](Y.o)--t,
x Uxo, and y. A (/, o/; /3 ; Z0), then
(i) Ix(x)-xo( o)l <
In particular, the mapping (x, l,)Z(x) o/ GxX into T is continuous in.
the given topology/or G and the A-topology/or X.
8,
Proof. Let U be a neighborhood of e such that Ils/-,! 11< if
st-lU (20.4.i). For arbitrary ;go, ;gX and x o, xG such that (Zo)--t,
we have
Iz(x)-z0(x0)l-lz(x)-zo(Xo)Z(Zo)llz(x)-z(x)(z)l
+ Iz()](z)- Z(Xo)(z)l + Iz (x0)(z)- Zo(Xo) ](z0)l
- It - ](z) l + I(J) (x) - (.ol) (z) l ч I(#)" (x) - (.ol)" (xo) l •
()
The last equality is verified by (23.t0.iii) and (23.t0.v) when we note that
(J) d-- (-,./) d3,. We also have from (23.t0.vi) that
I(J) (x)- (x) l IIJ- II1.
(2)
Now if (x, Z)(Uxo)x(A(/,,,o/; /3; ;go)), we infer from (t) and (2) that
Iz(x)-zo(Xo)l< .
(23.15) Theorem. For every compact set F cG and every e>O, let
P(F, e) be the set {zX:Iz(x )- t1< e/or all xF}. With all sets P(F, e)
taken as an open basis at the identity t in X, X is a topological group,
under the construction o/ (4.5). The P-topology is equal to the A-topology
o/(23.3), and so X under this topology is a locally compact Abelian group.
Proof. We verify (4.5.i) -- (4.5 .v), noting first that (4.5.ii) and (4.5 .iv)
are trivial for the sets P(F, e). I Iz(x)-[<T and I <x)-tI< 2,
then we have IZ(x)o(x)--tl<<_lZ(x ) o(x)--y(x)l+lo(x)_t]<e ' so that
P (F, e/2)- P (F, e/2) C P (F, e): this verifies (4.5.i). I x P (F, e), then
max {Iz (x) -- t I :xF}---- is less than e, since ;g is continuous and F is
compact. It is easy to see that z.P(F, e--)cP(F, e). This is (4.5.iii),
and thus we have shown that X is a topological group under the P-topol-
ogy. Given P (F, e) and P (F0, e0), we have
P(FUF o, min (e, o))c P (F, )n P (Fo, ,0),
so that (4.5.v) holds and the sets P (F, ) form a basis at t X.
We now prove that the A- and P-topologies are equal. Since
Z0" A (I1 .... ,1," c" 1)-- A (Z0ll, -.., Z0/m" " Z0), this will be accomplished
when we have shown that every P(F, ) contains a A(ll, ..., 1,; 6; t)
and vice versa. Suppose that F is a compact subset of G and that e is
positive. Let ! be a function in I(G) such that f/d--]()--t, and
let U be a neighborhood of e as in (23.t4). Since F is compact, a finite
number of neighborhoods Ux cover F- suppose that x 1 ..... xeF and
362 Chapter VI. Characters and duality of locally compact Abelian groups
UxDF. Now suppose that ;gCA (], 1/, x,/, .--, ,,/; e/3 ; t). For all
xcF, we have xUx for some k=t .... ,n and ZA(/,,k/;e/3;t).
Appealing to (23.14) and taking ;g0=l, we infer that
that is, Z P (F,
Finally, let A(/1, ...,/,; d; t) be arbitrary; we may suppose that
no [i is zero. Let F be a compact subset of G such that
.fl/i]d;<- for i= , .. ., rn.
F'
Let e----- min 11/11' "'" Illmlll " Then if Z P(F, e), we can write
Thus P (f, ) is contained in (11, ..., 1; ; ).
From this point on, we will always consider X as topologized with
the &- [or P-] topology.
(23.16) Corollary. Let F be a compact subset o/G such that (F)>0,
and let be any positive number less than . The neighborhood P (F, )
has compact closure in X.
Proof. Consider the function g=e(G). If Z P(F ), then we
have f
and thus 19(z)l
Since is in 0(X)(C.26), the set is compact.
Thus the closure of P (F, ) is also compact.
(23.17) Theorem. I/G is con, pact, then X is discrete; and i! G is
discrete, then X is compact.
Proof. Let A be a subgroup of the group T, and suppose that
Iz--t I<]/ for all zcA. Then it is easy to see that A--(}. Thus if G
is compact, the set P (G, ]/))- ( } is a neighborhood of the character
in X, so that X is discrete. If G is discrete, then M s (G) has a unit, so
that (C.25) implies that X with the A-topology is compact.
We wish to compute the character groups of a number of locally
compact Abelian groups, and we wish also to prove an important duality
theorem (24.8). We first establish some simple preliminaries.
(23.18) Theorem. Let G1, ..., G m be locally compact Abelian groups,
with character groups Xl , ..., Xm respectively. For every (Zl , "'', Zm) .=P1X/'
§ 23. The character group of a locally compact Abelian group 363
let )1 ..... ,] denote the/unction
(Xl .... , Zl (x l) (h) ...
de/ined on i Pt G i" Then the mapping (Z .... , Z) [Zl .... , Z, which we
denote by O, is topological isomorphism o/ P X i onto the character group
o/i PiGs. 1
Proof. It is obvious that every function IX1 .... , Xm is a continuous
character of i=P Gi, and that the mapping 0 is an isomorphism of P X
into the character group of iPGi. Let be any continuous character
of i Gi. For every (Xl, ..., x), we have
(xl ..... Xm) :(Xl' .... ,m)(, X,8,...,m)...(l,,...,m_l,Xm)
where e is the identity element of Gi. If we define Xi on Gi by
Z(xi)=(el .... , ei-1, x i, ei+ 1, ..., era), then it is plain that X X i and
• • P Xi onto the
that --IX1, .., Xm Thus the isomorphism 0 carries i=t
character group of i PtGi.
If P (F, e) is a neighborhood of the identity of the character group of
m
iPlGi, and if denotes the projection of F onto Gi, then
If P(, ei) is a neighborhood of the identity in Ni for each ], if F-=
P (U{ei}) and if e=min(el era) then
Relations (t) and (2) show that 0 is a homeomorphism.
We next point out a trivial but useful fact.
(23.19) Lemma. Let G be a compact group, not necessarily A belMs,
azd let Z be a continuous character o/G. The
(i) f Z (x) dx = 0
utsless Z = t, in which case the integral is t.
Proof. For all aG, we have
f Z(x) dx= f Z(ax) dx= Z(a) fz(x) dx. ()
If Z (a) + t, (t) can hold only if f
1 We abbreviate this assertion by the statement that the character group of
m
i 1G is the product i 1 Xi of the character groups.
364 Chapter VI. Characters and duality of locally compact Abelian groups § 23. The character group of a locally compact Abelian group 365
(23.20) Theorem. Let G be a compact Abelian group, and let Y be
a subgroup o] X such that ]or every a=e in G, there is a .Y such that
, (a) . Then Y is equal to X.
Proof. Let denote the linear space of all complex-valued functions
on G of the form yX, where ,..., are in and X,..., , are
in Y. Since Y is a subgroup of X, is a subalgebra of G(G) closed under
the formation of complex conjugates. If a, b are distinct elements of G,
then there is a xY such that x(ab-)l, and X(a)z(b). Thus
separates points of G. Bythe STONE-WEIERSTRASS theorem, is uniformly
dense in G(G). Now assume that there is a continuous character p of G
such that Y. Then there is certainly a function yXy such that
j=l
/=1
the Xy's are taken to be distinct. Then, taking note of (23.19), we write
m m m
'
/=1 =1
This contradiction proves the theorem.
Theorem (23.20) is useful in obtaining character groups of compact
Abelian groups.
(23.2) Theorem. Let {G,:,I} be a nonvoid /amily o/ compact
Abelian groups, and let X, be the character group o[ G,/or each I. For
an element (X,) i P'X, let , be the ]umtio on P G, such that
X,((x,))=X,(x,) oly a [inite ,umber o/ terms in the product are
different ]rom . All continuous characters o/ ,G, have the ]orm ,,
and the character group o] , G, is topologically isomorphic with the group
P* X, under the discrete topology.
Proof. The set of all characters X, is plainly a subgroup of the
character group of ,G, that separates points of ,G,. Thus we can
apply (23.20) to see that the characters X, are all of the continuous
characters of ,G,, and (23.7) to see that the character group of ,G,
is discrete. It is obvious that the multiplication of characters corresponds
to multiplication in the group P* X
Theorem (23.21) has an obvious complement, as follows.
(23.22) Theorem. Let {G,: I} be
discrete A belian groups, and let X, be the compact character group
that Z,((x,))--llx,(x,) only a /iў,ite number o/ terms in this product
are different/rom 1]. Every/mctioў X,] is a character o/ P*G and all
characters o/ P* G are/u,ctions [X,]. I/ P* G, is given the discrete topology,
tel t
the mapping (,)[, is a topological isomorphism o] ,X, onto he
[compact character group o/ P* G,.
Proof. It is trivial that all of the functions [X,] are characters of
P* G,. For each fixed z I, let G(0)be the subgroup of P* G, consisting
of all (x,) such that x,--e, for ,z. If is a character of P*G then
EI '
is a character of the subgroup Gd " write for with its domain re-
stricted to Gd . Then it is clear that = Z,. The mapping (,),
is obviously an isomorphism of ,P X, onto the character group of ,P* G,.
It is a routine matter to show that this mapping is continuous. Since
the groups in question are compact, the mapping is a homeomorphism.
We next identify the character group of a quotient group.
(23.23) Definition. Let G be a locally compact Abelian group with
character group X. For an arbitrary nonvoid subset H of G, let A (X, H)
be the subset of X consisting of all Z such that z(H)=(I}. The set
A (X, H) is called the annihilator o[ H in X.
(23.24) Remarks. (a) If H is as in (23.23) and H is the smallest
closed subgroup of G containing H, then it is clear that A(X, H) =A(X, H).
(b) It is trivial that A(X, (e})=X. It follows from (22.a) that
A (X, H) X if H (e}.
(c) It is obvious that A(X, H) is a subgroup of X. To see that
A(X, H) is closed in X, consider any X0A(X,H), and an element
a H such that 0 (a) 1. The P-neighborhood 0" P ({a}, 0 (a)-- ]) of
0 is contained in A (X, H)', so that A (X, H) is P-closed.
(d) Let H be a compact subgroup of G. Then it is easy to see that
P (H, /3) = (X, H), so that (X, H) is open.
(e) If H is a closed subgroup of G that is not locally null, then A (X, H)
is compact. This follows from (23.16) and (c).
(23.25) Theorem. Let G be a locally compact Abelia group with
character group X, ad let H be a closed subgroup o] G. Let Y be the
chrader group o/ the locally compact A belia, group G/H. The group Y
is topologically isomorphic with the group A(X, H). In [act, i/ denotes
the atural mapping x-xH o/ G onto G/H, the mapping o= ()
is a topological isomorphism o/ Y onto A (X, H).
Proof. Since is a continuous homomorphism and -(H})=H,
it is clear that o 9 A (X, H) for all y. Since (2) o 9-- (h.og) (p2o),
366 Chapter VI. Characters and duality of locally compact &belian groups § 23. The character group of a locally compact &belian group 367
is a homomorphism. A character 21C& (X, H) is constant on all cosets
xH, and so the function 1 on G/H defined by pl(xH)--,l(x) is well
defined; it is obviously a character of G/H. It is also simple to see that 1
is continuous on G/H. For e>0, there is a neighborhood U of e in G
such that [2;1 (x)-- t I < e for xc U. The set {xH" x U) is a neighborhood
of the identity H in G/H (5.t5) in which [l(xH)--t1< e. Thus 1 is
continuous at H and therefore continuous throughout G/H (5.40.a).
Since 21--1o F, we have proved that maps Y onto A(X, H). It is
obvious that o F--t only if --t, so that is an isomorphism.
Finally, we show that is a homeomorphism. If F is a compact
subset of G, then (wHiG/H: xF} is compact in G/H. Moreover, every
compact set in G/H has this form by (5.24.b). For s>0, we have
O({YY']y(xH) - for xcF))
- A (x, H): iX (x)-- < for xeF).
Thus maps an open basis at t in Y onto an open basis at t in A (X, H),
and is a homeomorphism. [
(23.26) Corollary. Let G be a locally compact A belian group with
character group X, and let H be a closed subgroup o/G. I/a is in G an
not in H, here is a zA (X, H)/or which z(a) t.
Proof. Let Y and O be as in (23.25). By (22.7), there is a yY such
that y(aH)=l. Then O(y)A(X,H) and O(p)(a)--p(aH):t.
(23.27) Some examples of character groups. We now list six ele-
mentary and important examples of character groups. Section 25 con-
tains an extended list of character groups.
(a) Consider the group T. The identity mapping exp(it)--exp(it)
(0_<_t<2) is obviously a continuous character of T, and by itself it
separates points of T. Its integral powers, that is, the functions
t, exp(it), exp(--it), exp(2it), exp(--2it), ..., exp(nit), exp (-- nit), ...
(n-- 0, t, 2, 3, ...), therefore exhaust all continuous characters of T (23.20).
As a group under multiplication, these functions are isomorphic with
the additive group Z. Thus we may say that the character group of T
is the discrete group Z.
(b) Consider the group Z. A character 2; of Z is plainly determined
by the number Z(t), since Z(n)-Z(t) (ncZ), and Z(t) can be any
number in T. Let Z denote the character of Z such that Z(I)-- ( T).
Then the mapping -+Z is obviously an isomorphism of T onto the
character group of Z, and it is also obvious that this mapping is contin-
uous" if [--t I is sufficiently small, then I '- t I is arbitrarily small for a
finite number of pre-assigned integers f. Hence the mapping is a homeo-
morphism, and we say that the character group of Z is T.
(c) Let m be an integer greater than t, and consider the finite cyclic
group Z(m), which we represent as (0, t, ..., m--t) with addition
modulo m. The mapping k-+exp (2) is a character of Z (m) that
separates points. As in (a) we see that every character of Z(m) has the
form k-+exp (2--A-), where/is a fixed integer such that O<_l<_m--1.
Thus the character group of Z (m) is isomorphic with Z (m).
(d) Let G be a finite [discrete Abelian group; by (A.27), G is iso-
morphic with Z(rr) xZ(m2) x...xZ (m,) for integers m 1, m, ..., ms
greater than t, each of which is a power of a prime. From (c) and (23.t8)
we see that the character group of G is isomorphic with G itself. Every
character of G has the form
(kl .... ks)-+exp 2i(kl +"'+ rns/j
, \ m 1 '
where 0 li< m i (f--t, 2,..., s).
(e) Consider the additive group R. For every fixed yR, the function
x--exp(ixy)--Z(x ) is a continuous character of R. We will show that
all continuous characters Z of R have this form. Let A--{x R: Z(x)--t).
Then A is a closed subgroup of R, and, as is easy to see, we have A--R
or A--{0) or A--{kb:kZ) for some positive real number b. If A--R,
then Z--t, so that Z(x)--exp(ixO)--Zo(X).
Assume that A--(0). Then 2;([0, tl)is a connected compact subset
of T containing t: that is, an arc or (t). By assumption, 2; ([0, t) is not
and so there is a number aCl0, t I such that z(a)--exp(2-*-),'--"
where m is a nonzero integer. Thus z(ma)--t, a contradiction to our
assumption 1.
Therefore we may suppose that A--{kb:k(Z}, where b is a positive
number. Since
and 0< - < b, we have Z ---- t. Since 2; -- Z ---- t, we must
have Z(-) = exp(-i) or Z(-)= exp(-- -i). If the second equality
holds, we look at , and so may suppose with no loss of generality that
Z ()--exp "(-- i)." This is the last choice available for a continuous Z.
Suppose that we have already proved that
We can also reason as follows. The set z(R) is a connected subgroup of T,
and hence is T or {1}. If A -- {0}, then Z is one-to-one, and so z(R) = T. By (S.29),
Z is an open mapping and hence a homeomorphism, which is plainly impossible.
368 Chapter VI. Characters and duality of locally compact Abelian groups § 23. The character group of a locally compact Abelian group 369
for some positive integer k. Then we must have
or
;g -- -- exp 2 + 1
( 2 i).
Assume that the second equality holds. Then ;g 2+1 ' 2 is a compact
connected set in T and must contain +t or --t. Thus for some
a 2+1 ' 2 ' we have x(a)--t or x(a)=--t. In either case,
b
and 2a < 2_ 1 __<b. This contradicts the fact that b is the smallest
positive number x such that ;g(x)=t, and so 2; - =exP\2--/.
It is now obvious that X (r b) = exp (2 a i r) for all dyadic rational numbers r.
By continuity we have x(xb)=exp(2rix) for all real numbers x:
equivalently,
Z(x)--exp(ix())--Z2,(x). 1
b
It is easy to see that the mapping Y-Zy is an isomorphism, so that
the group R is isomorphic with its own character group. It is also easy
to see that the ordinary topology of R agrees with the P-topology on the
group of characters ;gy. That is, the mapping Y-+Z is a topological
isomorphism of R onto its own character group.
(f) The character group of R"(n= 2, 3, ..-) is topologically isomorphic
with R " Every continuous character of R n has the form
(Xl, ..., xn)-->-exp(i(XlYl@...@
for some (Yl,-.., Yn)R. This follows at once from (e) and (23.t8).
Miscellaneous theorems and examples
(23.28) Comments on Theorem (23.4). (a) Let G be a nondiscrete,
locally compact group; it need not be Abelian. For every #M(G),
let (#)= #({x}). It is easy to see that is a multiplicative linear func-
tional on M(G) and that z (#) = z (#) for all # M(G). That is, the
mapping #-+(#) is a t-dimensional --representation of M(G). If ()
were representable in the form (23.4.iv), or even in the form (22.3.iii)
for some Borel measurable representation of G, then there would be a
continuous character ;g of G such that
G xEG
1 The group R has many 2 c to be exact] discontinuous characters. We discuss
these in (25.6).
for all #M(G)" the continuity of Z follows from (22.19). Thus Z would
have to be different from 0 while at the same time f, d#----0 for all
continuous measures #. This is clearly impossible, and so z has no
representation as f Ј, d#.
G
(b) Let G and 3(#) be as in (a) and let (#)--#(G), for all #M(G).
Then is also a multiplicative linear functional on M(G) use (t9.tt)
to show that is multiplicative 1. The functionals z and are equal on
M (G) but are different on M (G).
(c) Let Z be any character of G, continuous or discontinuous. Let
zz(#)-- #({x})" Then zz is a multiplicative linear functional on
M(G) that vanishes on M (G). The z of part (a) is Z" 1 of the present
construction.
(d) Let Z be a continuous character of G. Let z(#)- f d/. Then
z is a multiplicative linear functional on M(G). The functional of
part (b) is Ol.
(23.29) The annihilator A (X,/-/). (a) Let G be a locally compact
Abelian group with character group X, and let H be a closed subgroup
of G. If H is also open, then A (X, H) is compact. The converse is true
but requires the duality theorem (24.8). The group G/H is discrete by
(5.21), and therefore its character group is compact (23.t 7). Thus (23.25)
shows that A (X, H) is compact. 1
(b) Let G and X be as in (a). If H I and H2 are closed subgroups of
G and H a is the smallest closed subgroup containing H I and H2, then
A (X, HI) 0 A (X, H2) -- A (X, Ha). Note that H a is not necessarily H I H2"
see the last paragraph of the proof of (4.4).
(23.30) Extending characters of dense subgroups. Let G be a
topological group not necessarily Abelianl, and let H be a dense sub-
group of G. Every continuous character of H can be extended in one
and only one way as a continuous function on G, and this extension is a
Icontinuous character of G. To see this, note first that the character
of H is uniformly continuous in both the left and right uniform structures
of H. Like all uniformly continuous functions, it can be continuously
extended over G. It is easy to verify that the extended continuous
function is also a character.
(23.31) Some homomorphisms and automorphisms. (a) The only
continuous automorphisms of T onto itself are the identity mapping
exp (it) --exp (it) and the conjugate mapping exp (i) --exp (-- it). Clearly
a continuous automorphism of T must be a one-to-one continuous charac-
ter. Now apply (23.27.a).
Hewitt and Ross, Abstract harmonic analysis, vol. I 24
370 Chapter VI. Characters and duality of locally compact Abelian groups §23. The character group of a locally compact Abelian group 371
(b) Let be a continuous homomorphism of T into itself. If (T)=k { },
then (T)= T, and T/-I ( {t }) is topologically isomorphic with T. [This
follows at once from (23.27.a) and (5.27).
(c) The continuous homomorphisms of R into itself have the form
(x)=ex for some eR. [Let e=(t); then compute (r) for rational
r and use continuity.
(23.32) Groups with no continuous characters except 1. (a) (HEw-
ITT and ZUCKERMAN Eta, SMITH t.) Let E be any topological linear
space over R (B.5). Let Z be a continuous character of E, where E is
regarded as a topological Abelian group under +. Then there is a
continuous linear functional / on E (B.2) such that
(i) Z(x)=exp[2i/(x)
for all xE. Conversely, every function (i) is a continuous character
ofE. 1
Choose a balanced neighborhood U of 0 in E (B.5) such that
]Z(x)--ll<2 for all xU, i.e., Z(x)=--t if xU. For each xU, there
is exactly one real number /o(X) such that --Ѕ</0(x)<Ѕ and Z(x)=
exp2x i/o(X)). We now extend/0 to all of E: if xE, exU, and e0,
let
lo(X) ()
l (x) = - .
When we have shown that / is well defined, it will follow immediately
that / agrees with/0 on U. Let x be any nonzero element of E, and sup-
pose that e0 x and el X are in U. The subgroup {ex:eR} is a closed topo-
logical isomorph of R in E. The character Z is continuous on this sub-
group, and by (23.27.e), there is a unique flR such that Z(zx)=
exp2xi/51 for all eR. Therefore eo=/o(eoX)+no for some noZ.
1
If n04=0, then le0/5[>Ѕ. Therefore 2o/ .e0x=-fi belongs to V and
Z =--1. This contradiction shows that n0=0. Similarly
1
/0 (el x) and hence/5 -- --/0 (e0 x) = --1/0 (el x). Therefore / is well defined
and
Z(x) = exp [2xi/53 = exp[2xi/(x). (2)
To see that / is homogeneous, let fl be a nonzero real number and x
any nonzero element of E [the cases fl=0 and x=0 are trivial3. There
is a nonzero eR such that ex U and eflx U. By (t), we have
/0 (flx) = / (fix)
/(x) = V/0 (x) = .
It is also simple to show that / is continuous at 0. Let e be a number
such that 0< e< Ѕ, and let V be a neighborhood of 0 in E such that
1 For a theorem dealing with a similar situation on locally compact Abelian
groups, see (24.43) in/ra.
VcUand[z(x)--tl<2sin(xe ) forxV. ForxV, wehave
IZ(x)- t[ = 2sin (x I/(x)l)< 2 sin(xe),
which implies that I/(x)] < e.
Finally we show that / is additive. For x, y E, relation (2) shows at
once that/(x+y)--/(x)--/(y) is an integer. If W is a neighborhood of 0
such that ]/(x)l , I/(Y)I, and ]/(x+y)l are less than Ѕ for all x, yW,
then it is clear that/(x+y)=/(x)+/(y) for all x, y W. For arbitrary
x, yE, let e be a nonzero real number such that ex, e y, and e (x+y)
1
are in W. Then we have/(x+y)=-g/(z(x+y))=-g/(zx)+/(zy)=
/(x)+l(y).
The converse is trivial" every continuous linear functional / auto-
matically defines a continuous character of E by formula (i).3
(b) Let X be a locally compact Hausdorff space, and let I and t be
as defined in §tl. Suppose that 0<,(X)< oo and that ,((x})=0 for
every xX. Let 9X(X, ,) Fwe write this as 9X throughout the present
discussion be the set of all ,-measurable real-valued functions on X,
two functions in 9X being identified if they are equal ,-almost everywhere.
Define the metric (/, g) by
f !1-1 d,
(ii) p(/,g) = 1 + I/-1 "
x
With the topology induced by the metric p, 9X is a topological linear
space admitting no nonzero continuous linear functionals 1. Hence as a
topological Abelian group, 9X has no continuous characters save t. The
t is
space 9 is complete under the metric p. The function (t)= t +t
I+fll < Il+lfll
increasing in 0, oo, so that for e,/5R, we have 1 + +/1 = t + I1 ч I1
=< a+llll ч a + Iflw'lfll From this, the triangle inequality (/, h) < p (/, g) +=
(g, h) follows immediately. The other axioms for a metric are obvious.
For/, g, hgX, it is clear that (/+h, g+h)--(/, g). For eR and
the inequality (e/, 0) <= max (le ], t) (/, 0) is easy to check. Thus for
e, e0 R and/,/o 9X, we have
( 1, 0 I0) --<-- ( I, 10) + ( 10, -0 10)
<= max ([ 1, 1) (/-/0, 0)+ ((-- 0)/0,0).
Dominated convergence shows that lim ((-- 0)/0, 0) -- 0; from this
we infer that (,/)-+/is a continuous mapping of RxgX onto 9X. It
is also clear that (/, g)-+/+ g is a continuous mapping of 9Xx 9X onto
Therefore 92 is a topological linear space.
1 This property of 992 was pointed out to us by VICTOR L. KLEE.
24*
372 Chapter ҐI. Characters and duality of locally compact Abelian groups §23. The character group of a locally compact Abelian group 373
Since, vanishes for points, we can partition X into t-measurable sets
A1, ..., A, of arbitrarily small positive measure, say ,(A j)< e (1t.44).
Let ] be any function in gR, and let g.--m ] Aj ('---1, 2, ..., m). Then
m
g.. If q is a continuous
it is clear that (g., 0)< e and that ]=-.=
linear functional on r, then q9 (h) is arbitrarily small for (h, 0) suf-
ficiently small. Since Iqg([)l < max (I q(gl) I, [qg(g.> I, ..., Iqg(gm)l), it fol-
lows that qg([)--0.
The completeness of g) is proved by an argument similar to that of
(2.8), which we omit.
(c) Let p be a number such that 0< p < 1. Let X be a locally compact
Hausdorff space and let I and, be a functional and a measure for X as
constructed in §tt. Define (X, ,) as in (12.t); we abbreviate this as
p in the present discussion. For every e>0, let u={/p.lI/ll< }.
With the U,'s taken as an open basis at 0, g is a complete metrizable
topological linear space over R. [We first prove that
1
I1! ч gll < 2 --1 (11/11 ч Ilgll ) for /, g . (3)
If t is a nonnegative number, then
+ _>_ ( + ).
This is easy to see by considering the function (t)--t-+-t - (1 + t) ,
f0r which (0)=0 and ' (t)>0 for t>0. It follows that
I! (x)ч g Ig(x)l
wherever / and g are finite and so
Also the function o(t)=(t+t--)(t+t)- has exactly one minimum
value in [0, col, namely at t= 1, and (1)= 2 ---. This implies that
1 _1_ 1
Elllll; ч Ilgll;-I 2, l-II/ll, ч I1 11/I,
which together with (4) yields (3)-
It is easy to see from (3) that is a topological linear space. Since
has a countable open basis at 0, Theorem (8.3) implies that it admits
an invariant metric yielding the topology described above" in fact
ll/--g[l is such a metric. The proof that p is complete in the metric
is very like the proof of (t2.8), and is omitted.
(d) (DAY [t, proof adapted from ROERTSOX [t.) Let p, X, , and
i be as in (c), and suppose further that t({x}) =0 for all xX. Then g
admits no nonzero continuous linear functionals, and so ! as an additive
group admits no continuous characters save t. ILet q9 be a nonzero
linear functional on " we will show that it is discontinuous. There is
an/ such that q9(/)=t. Let I I be the functional --- f 1/I dt on
x
00 (X), and let ,! be the measure corresponding to I! as in § 11. Theorem
(t4.t7) shows that and it is clear that ,({x})--0 for
all xX. By (tt.44), there is a -compact subset E of X such that
f [11 d,= Ѕ f 1/I d,. Let g:=/e and h:--/,. Then f Ig:[ d, =
E X X
f lh[ d,= Ѕ f 11[ d,, and qg(g+h)--qg(gl) + qg(/h)--t. If qg(g)>Ѕ,
x x
let/----2g" otherwise, let /--2h 1. Then we have q9(/1)>--t and
1
2 [lg II -21-- ll/ll# • Continuing this process, we find a sequence/,/,...,
,... _
Inof functions in p such that 09(I=)_>__t and II/oli, 2('
Since 0< p< 1, we have lim II/ lip--0, and q9 is necessarily discontinuous.]
(23.33) Character groups of local direct products (BRACONNIER
[t]). Consider a family {G,},6 of locally compact Abelian groups, and
for each tI, suppose that H, is a compact open subgroup of G,. Let G
denote the local direct product of the groups G, relative to the open
subgroups H,. Write H for the compact open subgroup ,PH, of G.
Let X, and X be the character groups of G, and G, respectively. Identify
G, with the subgroup of G consisting of all (x,) such that x--e for
For Z X, the restriction zIG, is a continuous character of G,, which we
write as 2:,. Write z (2:)= (2:,) ,P X,. Then z is plainly a homomorphism
of X into P X,. Now look at the restriction 21/4. This function is a
continuous character of H, and by (23.21) we infer that
for all but a finite number of the indices t. That is, 2:, is in A(X,, H,)
for all but a finite number of indices t, so that z(2:) lies in the local direct
product of the groups X, relative to the compact open subgroupsA(X,, H,).
Denote this last group by the symbol Y.
Now let (2:,) be any element of Y. We will define a 2: X such that
z(Z)=(Z,), as follows. For (x,)G, let Z((x,))=[-[Z,(x,). Only a finite
number of factors in this product are different from t, since ;g, (H,)=t= {t }
for only finitely many , and x,ў H, for only finitely many . Plainly
is a character of G, and it is easy to see that ;g is continuous. Since
z(;g) = (;g,), we have shown that z maps X onto Y.
We now show that z is a topological isomorphism of X onto Y. It is
easy to show that z is an algebraic isomorphism. Hence it suffices to
show that z is a topological isomorphism carrying the compact open
subgroup A (X, H) of X onto a compact open subgroup of Y. It is clear
that z(A(X, H)) =,PA(X,, H,), and it is a routine matter to show that
z is continuous on A (X, H).
374 Chapter VI. Characters and duality of locally compact Abelian groups §23. The character group of a locally compact Abelian group 375
The foregoing may be summarized as follows. The character group
of G is topologically isomorphic with the local direct product of the
character groups of the G, relative to the annihilators of the H,.
(23.34) Other homomorphism groups. (a) Let G and H be topo-
logical groups with identity elements e and / respectively, and suppose
that H is Abelian. Let Horn(G, H) denote the set of all continuous
homomorphisms carrying G into H. For r, z Hom (G, H), let rz be the
mapping x-->r(x)z(x). Under this multiplication, Hom(G, H) is an
Abelian group with identity element the map x-->/and z-1 the mapping
xz(x)-=z(x-). It is trivial that rz is a homomorphism of G into H,
and obvious that rz is continuous. The group axioms are also easy to
verify.
(b) For a compact subset Y of G and a neighborhood U of / in H,
let P(F,U)--{rHom(G,H):r(F)cU}. The family of all such sets
satisfies (4.5.i) -- (4.5 .v) and accordingly defines a topology under which
Horn(G, H) is a topological Abelian group. Each of the properties
(4.5.i)--(4.5.v) is easy to verify. For example, to establish (4.5.iii), let
z be any element of P(F, U). Then z(F) is a compact subset of U. By
(4.t0), there is a neighborhood V of / in H such that z (F). V c U. Then
if rP(F,V) and xF, we have (zr)(x)--z(x)r(x)z(F).VcU, and
hence z(P(F,V))cP(F,U). Since H is a T1 space and points of G are
compact sets, it is easy to see that Hom (G, H) is a T space. 1
(c) Let G, G 2, ..., G,, be topological groups and H a topological
Abelian group. There is a topological isomorphism carrying Hom (Gx...
xG m, H) onto Hom(G, H)x...xHom(G,,, H). IThe proof is easily
adapted mutatis mutandis from the proof of (23.a8).1
(d) Let G be a topological group and H1, H2,..., Hm topological Abelian
groups. There is a topological isomorphism carrying Hom (G, H 1X... xH,)
onto Horn(G, H)x...xHom(G, H,). For zHom(G, Hx...xH,,)
and xG, we have z(x)= (Zl(X), ..., zm(x)), where zj(x)H i. It is easy
to see that the mapping x-zi(x) is an element of Horn (G, Hi). Denote
by 0 the mapping z-->(z, ..., Zm) just defined. It is simple to verify
that 0 is an isomorphism of Hom (G, H x... xHr ) onto Hom (G, H1)x..-
xHom(G, H,). Let F be a compact subset of G and neighborhoods
of the identity elements in H i (/'--1,2 ..... m). Then we have
O(P(F, Vx...xV,))--P(F, V1)x...xP(F, V,n), so that 0 is also a homeo-
morphism. 1
Notes
The word "character" is used by many \vriters in a sense different
from that established in (22.15) \ve \vill take up these other characters
in Vol. II, § 27. For Abelian groups the distinction disappears; and in this
sense one can say that characters for finite Abelian groups were invented
by FROENIUS 1. The idea of making the characters of a group into
a group as in (23.2) is due to PONTRYAGIN t; the theory was developed
much further in PONTRYAGIN 4. Several other writers in the early
t930's also computed character groups. Explicit computations of
characters and tantalizing hints of the modern theory of harmonic anal-
ysis on compact Abelian groups appear in WIENER and PALEY
A. HAAR 1 found very sophisticated results about characters of count-
ably infinite discrete Abelian groups G. He essentially defined the
character group X of G, identifying it with [0, t under a new multi-
plication, but not describing topology or Haar [! 1 measure in X. He
found the orthogonality relation (23.t9.i) and PARSEVAL'S equality see
Vol. II, (27.41) and (28.43). HAAR 21 also found interesting generaliza-
tions of his results to certain non-Abelian countable groups. HAAR'S
work in [t was simplified and extended by Sz.-NAGY [2, although still
without recognition of the r61e of the character group. FREUOENTHAL
pointed out that HAAR'S work can be interpreted in terms of PONTRYA-
OIN'S theory.
Theorem (23.4) for M,(G), where G is Abelian, is due to GEL'FAN
and RAIKOV [t [although with unneeded hypotheses 1. Our proof is
very like theirs. Theorem (23.6) for M,(G), G Abelian, is also due to
GL'FAD and RMKOV I1.
All of (23.9)--(23.13) is an application of GEL'FAND'S theory of Banach
algebras AppendixC to M(G) and (G). Theorem (23.t3) appears
explicitly in GEL'FAND and RAKOV 1. The P-topology of (23.15)was
invented by PONTRYAGIN: in 4 for countable discrete and compact
metric Abelian groups, and in 61, §30, for all locally compact Abelian
groups with countable open bases. VA t{AMPEN 11 used structure
theory to topologize the character group; his topology is the same as
PONTRYAGIN'S. The definition for general locally compact Abelian groups
is due to A. WEIL 4, §27, p. 100. A special case of Theorem (23.15)
was known to GEL'FAND and RAIKOV I11. The first proof of (23.15)
was given by YOSIDA and IWAMURA 1. Theorems(23.17), (23.18),
(23.21), (23.22), and (23.25) are due in one form or another to PONTRYA-
IN 41 and 6, VAST t{AMPEN 1, and A.WEIL 4. Lemma (23.19) goes
back to WEYL and PETER t at least; (23.20) is due to E. HEWITT 2 I.
The history of (23.27) is impossible to trace. Since 1920 everyone seems
to have known these facts, and nearly everyone has proved them.
The result in (23.32.a) is announced without proof by VILENKIN I13.
Necessary and sufficient conditions for the existence of sufficiently many,
or no, continuous characters on a topological Abelian group have been
given by FOLNER 1 , 2. Analogous conditions appear in COTLAR and
IICABARRA 1 . Comparatively little attention has been paid to comput-
ing character groups of topological Abelian groups that are not locally
376 Chapter VI. Characters and duality of locally compact Abelian groups §24. The duality theorem 377
compact. Of course the dual spaces of a host of topological linear spaces
are known, and so character groups of their additive groups are known
(23.32.a). For other computations, see VILENKIN 9. More comments
on this matter appear in the notes to §§ 24 and 25.
§ 24. The duality theorem
Making use of the structure theory of §9, of Theorem (22.17), and
of facts about character groups set forth in § 23, we prove here the famous
theorem of PONTRYAGIN and VAN t{AMPEN asserting that the character
group of a character group is the original group. Of course this assertion
is vague. Let us make it precise.
Throughout this section, G will denote a locally compact Abelian
group and X will denote the character group of G.
(24.1) Definition. For fixed x in G, let x' be the function on X
such that x' (;g)=Z (x) for all ;g X, and let z be the mapping z (x)= x'.
Let 03 be the character group of X. We call 03 the second character group
o/G. These notations will be fixed throughout (24.I)--(24.8).
The duality theorem asserts that z is a topological isomorphism of
G onto 03. We will prove this result in a number of steps.
(24.2) Theorem. The mapping is a continuous isomorphism o/G
into 03.
Proof. For Z, oX and xG, we have x'(z)=Zo(x)=z(x ) (x)--
x' (Z)x' (), by the definition of multiplication in X. Thus each x' is a
character of X. The character x' on X is continuous because
I x'(Z) -x'(9)1< e whenever ZV,-I P({x}, e). If x and y are distinct
elements of G, (22.17) shows that there is a 2;0 X for vhich )Co(X) =Z0(Y) ;
hence x'=y'. For x, yG and 2;X we have (xy)' (Z) =Z (x) z (y) -
x' (;g)y' (;g). We have therefore shown that z is an isomorphism carrying
G into 63.
We now show that z is continuous; by (5.40.a), it suffices to prove
that z is continuous at e. Let F be a compact subset of X and let e be
a positive number. Select any neighborhood U of e in G such that U-
is compact, and consider the neighborhood P(U-, e/2) of t in X. As F
is compact, a finite number of sets zIP(U-, e/2), ..., ZmP(U -,
cover F. Let V be a neighborhood of e in G such that VU and
12: (x)- ll< 2- for all x V and '--t, 2 .... , m. Now, in the P-topology
for 03, the set {I03:II(z )- t1< e for ZV} is a generic neighborhood of
the identity. If xVG and 2:V, then 12:.(x)--tl<- ff for all 1" and
e for some i. Hence IZ(x)--t I<
for all Z-F, and the mapping is continuous.
We now prove the duality theorem in an important special case.
(24.3) Theorem. Let G be a topological A belian group that is either
compact or discrete. Then the mapping "c o/ (24.) is a topological iso-
morphism o/G onto 03.
Proof. Suppose first that G is discrete. Then X is compact (23.17),
and the group z(G) of continuous characters of X separates points of X.
Theorem (23.20) shows that z(G)=(a). Since G and 03 are discrete, z is
trivially a homeomorphism.
Suppose next that G is compact. Then X is discrete and 03 is compact
(23.t7). Since z is continuous and one-to-one, z(G) is homeomorphic
with G and as G is compact, z (G) is a closed subgroup of 03. Assume that
z (G) 03" then (23.26) implies that there is a continuous character, say
of 03 such that (z (G)) = {1 } and 4= t. Since X is discrete, the preceding
paragraph shows that every continuous character of 03 arises from an
element of X. In particular, there is a 2;0 X such that (o)) -- o) (Z0) for
all c03. We find that o(x')=x' (Z0) =2:0(x) =t for all xG, while
This contradiction proves that z(G)=03.
In order to prove the duality theorem for all locally compact Abelian
groups, we must first establish (9.5), in order to be able to appeal to (9.8).
Three technical lemmas are required for this purpose.
(24.4) Lemma. Let G be a locally compact Abelian group, let H be a
closed subgroup o/ G, and let v be a continuous character o/H. Then i[
H is compact or open, there is a continuous character 7, o/ G such that
7,(x)=v(x) /or all xH. 1
Proof. Suppose first that H is compact. Then the extensible
continuous characters of H are a subgroup of the character group of
H that separates points of H (22.7). Hence all continuous characters
of H are extensible (23.20).
Suppose next that H is open. The character o can be extended to a
character of G by (A.7), and since o is continuous on H, the extension is
necessarily continuous on G (5.40.a).
(24.5) Lemma. Let G be a locally compact A belian group and let H
be an open subgroup o/ G. Let X be the character group o/ G and Y the
character group o/ H. The mapping Z-Zl H=a(Z ) is an open continuous
homomorphism o/ X onto Y with kernel A (X, H). Thus Y is topologically
isomorphic with X/A (X, H).
1 We abbreviate this assertion by stating that all continuous characters o] H
are extensible over G. The lemma is true for all H, as we will prove below (24.12).
We need the present lemma only for open subgroups H, but the proof for
compact subgroups H is so amusing that we include it.
378 Chapter VI. Characters and duality of locally compact Abelian groups §24. The duality theorem 379
Proof. The mapping a merely restricts the domain of definition
of Z. The definition of multiplication in X makes it obvious that a is a
homomorphism of X into Y, and our Lemma (24.4) implies that a (X) =Y.
If E is a compact subset of H, then a-l{zcY:]z(x)--l]<e for xE}=
P (E, e). Since E is compact in G, a is continuous.
Choose a compact subset E of H such that . (E)>0, write PЧ (E, Ѕ)
for the set {zcX']z(x)--t]< Ѕ for xE}, and write PҐ(E, Ѕ) for the set
{zY'Iz(x)--tI< Ѕ for xE}. We know from (23.t6) that Px(E, Ѕ)has
compact closure in X. Let X1 and Y1 be the subgroups of X and Y,
respectively, generated by Px (E, Ѕ) and PҐ(E, Ѕ). The obvious equality
a(Px(E, Ѕ)) =PҐ(E, Ѕ)implies that a(X)=Y1. As X is a-compact,
Theorem (5.29) shows that a is an open mapping of X1 onto Y. Since
X and Y1 are open subgroups of X and Ґ, respectively, we infer that (,
is an open homomorphism of X onto Y. ]
(24.6) Lemma. Let G be a discrete A belian group, and let P (F, ) be an
arbitrary neighborhood o/the identity in the character group X o/G. Then
there is a subgroup H o[ G having a/inite number o/generators such that
A (X, H) c P (F, ,).
Proof. This lemma is trivial. Since G is discrete, F is finite. Let H
be the smallest subgroup of G that contains F. ]
(24.7) Theorem. Let G be a compact Abelian group and let U be a
neighborhood o/e in G. There is a closed subgroup H o/G such that H C U
and G/H is topologically isomorphic with TxF, where a is a nonnegative
integer and F is a/inite group.
Proof. Consider the discrete character group X of G, and let
be any subgroup of X having a finite number of generators. By (A.27),
_ is isomorphic with ZxF, where a is a nonnegative integer and F is
a finite group. By (23.8), (23.27.b), and (23.27.d), the character group
of - is topologically isomorphic with TxF. By (24.3), the mapping z
defined in (24.) is a topological isomorphism of G onto the character
group of X. By (24.), the character group of :, that is TxF, is topo-
logically isomorphic with z(G)/A(z(G),-). By (24.6), the neighborhood
z(V) of z(e) in z(G) contains a subgroup of the form A(z(G),:). f]
Theorem (24.7), which we have just proved, is exactly Theorem (9.5).
The reader should note that our argument is not circular; we use neither
(9.5) nor any of its consequences to prove (24.7). Since the proof of (24.7)
does require some knowledge of characters, we have merely postponed
its proof from §9 to this more convenient place.
We can now prove the duality theorem in full generality.
(24.8) Theorem. Let G be a locally compact Abelia group, let X be the
character group o/G, and let 03 be the character group o/X. The mapping
z o/(24.t) is a topological isomorphism carrying G onto 03.
Proof. (I) Suppose first that G is compactly generated. We appeal
to the structure theorem (9.8), which tells us that G is topologically
isomorphic with RxZxF for nonnegative integers a and b and a
compact Abelian group F. The second character group of R can be
identified with R. By (23.27.e), every continuous character of R has the
form x-+exp(ixy)--Z(x ) for some yR, and the mapping Z-->y is a
topological isomorphism of the character group of R onto R. Hence every
function in the second character group of R has the form Z,-->exp(iay)
for some real number a. This is just the function a' operating on the
character group of R. Thus for the group R, the function z of (24.t)
maps R onto the second character group of R. By (24.3), the groups
Z and F also have this property. Theorem (23.t8) shows that the
function z defined for the group RxZxF maps this group onto its
second character group. Since z is continuous and G is a-compact,
z is a topological isomorphism of G onto its second character group
(5.29).
(II) Suppose now that G is any locally compact Abelian group, and
that is defined on G. Let H be a compactly generated open subgroup
of G: we may suppose that H contains any fixed compact subset of G
(5.t4). Let Y denote the character group of /-/. For x cH, it is plain
that z(x)<A(03, A(X, H)). Suppose that /<A(O3, A(X, H)). If Z and
0 are in X and I, IH---I, olH, then/(Z) =/(0). By (24.5), every character
,? in Y has the form zIH for some 2;(X. Hence if we assign to each
character ,? in Y the number /() where z]H---,?, we obtain a well-
defined character/0 of Y. The continuity of/0 follows from the fact that
the mapping Z-+zIH is open (24.5). By (I), we infer that /(Z)--Z(x)
for some xH and all zX. Thus z maps H onto the subgroup
A(C0, A(X, H)) of 03. Since H is open, G/H is discrete (5.2t). As the
character group of G/H (23.25), A(X, H) is thus compact (23.t7). The
annihilator of a compact subgroup is open (23.24.d), and so z(H)-
A(C0, A(X, H)) is an open subgroup of 03. By (5.29), z is a topological
isomorphism on H and by (5.40.a), it follows that z is a topological
isomorphism of G onto z(G).
(III) To complete the proof, we need only to prove that z maps G
onto 03. We will prove that for every/c03, there is an H as in (II) for
which /z(H). Let P(F, e) be a neighborhood of I in X such that
I/()--I]<V for all P(F, e). Let V be any symmetric neighborhood
of e in G with compact closure. Let H be the subgroup of G generated
by U. Then H is open and compactly generated (5.t3). Let Y1 be the
character group of G/H, and let be the topological isomorphism of
Y onto A(X, HI) given in (23.25). Since G/H is discrete, it contains a
subgroup C with a finite number of generators xH, xH1, ..., xH
such that A(YI, C)CЈ-(P(F,e)fqA(X,H)) (24.6). Now let H be an
380 Chapter VI. Characters and duality of locally compact Abelian groups §24. The duality theorem 381
open, compactly generated subgroup of G that contains U- and Xl, ..., X m
(s.4). I, (Ч, H), then y-(Hi)= {1} and y-(xi) = 1. Therefore we have
ZA(X, H1) and O-l(y-)A(Y 1, C). It follows that zP(F,e), and so
I/(Y-)-- t] <]/. Thus I/(Y-)-- 1[ <]/ for all Z in the subgroup A (X, H)
of X, which implies that /(y-)=t for all ZA(X, H). That is, / is in
A (C0, A (X,/-/)) = (H). FI
(24.9) Scholium. Theorem (24.8) permits us to identify G with its
own second character group, although conceptually the two objects are
of course quite distinct. Given a locally compact Abelian group G with
character group X, we consider the mapping of G x X into T defined by
(i) (x, y-) y- (x).
Then every continuous character of G and of X is obtained by fixing Z
or x. For a closed subgroup Y of X, we write A(G,Y) for the set
{x G: y- (x) = t for all y- e Y}.
We now draw some inferences from (24.8).
(24.10) Theorem. Let H be a closed subgroup o/ G. Then H--
A(G, A (X, H)).
Proof. It is obvious that HcA(G, A(X, H)). IfxcH, then by (23.26)
there is a y-eA(X,H) such that Z(x)q=t. Thus xcA(G, A(X,H)). [l
(24.11) Theorem. Let H be a closed subgroup o/ G. The character
group o/ H is topologically isomorphic with X/A (X, H), every co#inuous
character o/ H having the /orm x->y,(x) /or some y-6 X. Two characters
y- and Z in X de/ine the same character o/ H i/ and only i/ ZZge A (X, H).
Proof. Since G is the character group of X and H= A (G, A (X, H)),
Theorem (23.25) implies that H is the character group of X/A(X, H).
Therefore every continuous character of X/A(X,H) has the form
y-.A(X, H)-->y-(x) for some xeH. Hence by (24.8) every continuous
character of H has the form x-->Z(x ) for some y-X. The last statement
is obvious. I1
(24.12) Corollary. Let H be a closed subgroup o/ G. Every continuous
character V; o/ H admits an extension y- over G that is a continuous character
o/ G. I/aEGflH', then y- can be determied so that y-(a)
Proof. The existence of some extension y- follows at once from
(24. ). If Z (a)=t= the proof is complete. If Z(a)= , let /0 be a contin-
uous character of G/H such that va(aH)=i=1 (22.7), let be the natural
mapping of G onto G/H, and consider the character
(24.13) Theorem (24.8) tells us that every topological and algebraic
property of a locally compact Abelian group G must be describable in
terms of topological or algebraic properties of its character group X,
since X determines G both as a group and a topological space. It also
reduces the study of compact Abelian groups to the study of the algebraic
structure of Abelian groups with no topology" a field that is far from
exhausted at the present time. [See for example FUCHS [t].] Let us
now examine some of the most important of these dual properties of G
and X.
(24.14) Theorem. For arbitrary G and X, the equality ro(G)--ro(X)
obtains .
Proof. If O is finite, the theorem is trivial. We may thus suppose
that G and X are infinite, and accordingly that ro (G) is infinite. Let
be an open basis for G such that = ro (G). Since {U< " U- is compact}
is also an open basis for G, we may suppose that U- is compact for all
U. Let be the set of all functions on G of the form max{v,,
• .., v=} for {U,, U,., ..., U,,}c; plainly --W(G). Now let F be any
compact subset of G and W any open set in G such that W DF. Since
W is the union of sets in , there are a finite number of sets U, U2 .... , U=
in such that F c UU... U U c W. Thus we have u__< max {v,,
• .., v=}----<w. Since we can find a W for which (W F1F') is as small as
we like, it follows that I[u--g[ll is arbitrarily small for some ge. An
argument like that used in proving (t2.10) shows that the set of functions
of the form
(r+ is) ,+ ...+ (r, is, , (t)
[where F, ..., F,are compact and rl .... , r, s, ..., s,are rational num-
bers] is dense in (G). It follows that the set of all functions of the
form
(rl + is) g +...+ (r,,+ is, g, (2)
where g, ..., g, and r, ..., r, sl, ..., s,are rational, is also dense
in 1 (G). That is, (G) contains a dense subset of cardinal number ro (G).
Now consider the family a/ of all subsets of X that are finite inter-
sections of sets of the form
{z< sol <
where r and s are rational numbers, t is a positive rational number, and
/e . Note that all sets in ag are open in the A-topology. The family
ag has cardinal number not exceeding = ro (G) and is an open basis for
a topology in X in which all of the functions ] (/) are continuous.
The set of functions is uniformly dense in (G), since is dense in
(G) and ][1[_<_][[1 [see (23.t0.vi)]. Therefore all (gЈ1(G)) are
continuous in the topology defined by ag. As the A-topology is the
x For the definition of r0, see the introduction to § 3, P. 9.
382 Chapter VI. Characters and duality of locally compact Abelian groups {}24. 'he duality theorem 383
weakest topology for X under which all are continuous (23.t3), the
ag- and A-topologies are equal and
(x)< W-< 5 = m ().
Theorem (24.8) now implies that lv (G) <__ Iv (X).
(24.15) Theorem. Let G be a compact Abelian group. Then we have
(i) Iv (G) = dim (Ј (G))
Iu particular, G is metrizable i/ and only i/ X is a countable discrete group.
Proof. For the discrete space X, we have Iv(X)=X. Thus (24.t4)
implies that Iv(G)--X. Let 5g denote the set of functions , i ;gi for
i--1
i K and ;gi X. Plainly 5g is a subalgebra of g(G) closed under complex
conjugation and separating points of G [see (22.t7)]. Thus 5g is uni-
formly dense in g(G) and hence dense in g(G) (t2.10). Since X is an
orthonormal set in Ј(G) (23.t9), it follows that X is an orthonormal
basis in
The last statement follows from the elementary fact that a compact
Hausdorff space is metrizable if and only if it has a countable open
basis.
(24.16) Note. Since discrete Abelian groups exist having arbitrary
cardinal number, (24.15) and (24.8) show that dim (Ј(G)) can be an
arbitrary nonzero cardinal number for a compact Abelian group.
We next discuss connectivity properties of G and X.
(24.17) Theorem. Let B be the closed subgroup o/ X consisting o/ all
compact elements see (9.t0), and let C be the component o/ the identity
in G Isee (7.t). Then B--A(X, C) and C=A(G, B).
Proof. (I) Suppose that there is a compact subgroup Y of X such
that {t}.cY. From (23.24.b), (23.24.d), and (24.8), we infer that
A (G, Y) c, G and that A (G, Y) is an open subgroup of G. Consequently G
is not connected.
(II) Suppose that G is totally disconnected, and let ;g be any contin-
uous character of G. Let U be a neighborhood of e in G such that
I;g(x)--ll<V for all x<U. By (7.7), U contains a compact open sub-
group H. It is clear that ;g (H) -- {t }, i.e., ;gA (X, H). Since H is open,
(23.24.e) implies that A(X, H) is compact. Therefore Z is in B, i.e.,
B---=Xo
(III) We now turn to the general case. The locally compact Abelian
group G/C is totally disconnected (7.3). By part (II), the character group
of G/C contains only compact elements. Since the character group of
G/C is topologically isomorphic with A(X, C) (23.2S), we infer that
A(X, C)cB.
To prove the reversed inclusion, we argue as follows. The locally
compact Abelian group C is connected, and by part (I) its character
group contains no compact subgroup different from the identity. The
character group of C is topologically isomorphic with X/A(X, C) (24.tt).
Now if ZB and ZўA(X, C), there is a compact subgroup Y of X whose
image under the natural homomorphism of X onto X/A(X, C) is not
the identity. This image is a compact subgroup of X/A (X, C); the result-
ing contradiction shows that B cA(X, C), and thus B=A(X, C).
Applying (24.10), we have CA (G, A(X, C))--A(G, B).
(24.18) Corollary. The/ollowing properties o/ G are equivalent:
(i) G is O-dimensional;
(ii) every element o/ X lies in a compact subgroup o/ X ;
(iii) G contains a compact open o-dimensional subgroup H.
Proof. The equivalence of (i) and (ii) follows immediately from
(24.t7) and (3.5). The equivalence of (i) and (iii) follows from (7.7).
(24.19) Corollary. The group G is connected i/ and only i/ X contains
no compact subgroup different/rom {t }.
Proof. By (24.t7), we have C--A(G, B); and A(G, B)=G if and
only if B={I}.
(24.20) Corollary. Let G be a compact A belian group, with component
o/ the identity C. Let be the torsion subgroup 1 o/ X. Then q)=A (X, C)
and C = A (G, @). Also eO is isomorphic with the [discrete character group
ol /c.
Proof. The two equalities are (24.17) for compact G, since an element
of a discrete group lies in a compact subgroup if and only if it has finite
order. The last assertion follows from (23.25).
A curious consequence of (24.t8) is the following.
(24.21) Theorem. Let G be a torsion group. Then G and X are
O-dimensional .
Proof. Let H be an open compactly generated subgroup of G. It
follows from (9.8) that H is compact. Let Ґ be the character group
of H. If W<Y, then o(H) is a compact torsion subgroup of T and is
hence a finite cyclic group. Thus V: belongs to a compact subgroup of
Y and H is 0-dimensional by (24.t8). Consequently, G is 0-dimensional,
as H is open in G. Since every element of G obviously lies in a compact
subgroup of G, (24.t8) implies that X is 0-dimensional.
1 See (A.4).
See also (25.9) in/ra.
384 Chapter ҐI. Characters and duality of locally compact Abelian groups § 24. The duality theorem 385
We obtain more relations between G and X by considering the homo-
morphisms L, L (x)= x n, and the subgroups G()= L (G) and G()=/ 1 (e)
[see (A.2)I.
(24.22) Theorem. Let G be a locally compact Abelian group with
character group X, and let n be a positive i,teger. Then we have
(i) A (X, G
and
(ii) A (X, G(,)) = X(")-.
Proof. Let
and so Z (x)
for all x G, so that ;g A (X, G()). This proves (i).
To prove (ii), regard G as the character group of X. Then (i) implies
that A(G, X())=G(n), and (24.10) and (23.24.a) imply that
X(")- = A (X, A (G, X(")-)) -- A (X, A (G, X("))) -- A (X,
This proves (ii).
(24.23) Theorem. I/ G is divisible, then X is torsior-/ree. I/ X is
torsion-/ree, then G ( is dense in G/or n= 1, 2, 3,.... I/G is discrete or
compact, then G is divisible i/and only i/X is torsion-/ree .
Proof. If G is divisible, then G(")=G for all n, and (24.22.i) implies
that X(,)= {t}. Thus X is torsion-free.
Suppose that X is torsion-free. Then (24.22.ii) shows that A (G, X(,)) -
G=G (-. If G is discrete, then G (n is trivially closed. If G is compact,
G (" is a continuous image of the compact space G and is hence compact
and closed. Thus in these cases G--G ( for all n and G is divisible. F]
Another consequence of (24.22) follows.
(24.24) Theorem. Let D be the largest divisible subgroup o/G, let C
be the component o/e in G, and let q) be the torsion subgroup o/X. Then
we have
(i) CDD- r G(")-=A(G, (I)).
=1
I/G is compact, G()= G ()- and all irclusiom in (i) are equalities
Proof. We first prove that
° d ,).
1 There are nondivisible G's with torsion-free X's: see (24.44).
See (A.6).
a If G is noncompact, examples can be found to show that each inclusion in (i)
may be proper. See (24.44).
By (24.22.ii), we have A(G, X(,,)) = G(")-, and so
A (G, (I)) = A (G, -U-lX(n))=,,r"llA= (G, X(,,)) =,,=.r? G(")-.
The inclusion D C r? G () holds because D is divisible. Taking clo-
sures, we have
D C D-- C (°__1G(n, ) C nI(G''--) .=
This proves all of (i) except for the inclusion C c D.
To prove that CcD, it is convenient first to deal with the case in
which G, and hence C, are compact. In this case, the character group
of C is discrete and torsion-free (24.19), and so C is a divisible group
(24.23). Thus we have CcD. By (24.20), we have C= A(G, (I)), and so
by the foregoing we have
CDA (G, q))=C.
Thus (i) holds, and all inclusions become equalities, if G is compact.
It is now easy to show that C c D for all G. By (9.t4), C is topologi-
cally isomorphic with a group RnxF, where F is a compact connected
Abelian group and n is a nonnegative integer. The group F is divisible,
as proved already, and so C is a divisible group. This implies that
CD.
For convenience, we collect some of the foregoing results in one
statement.
(24.25) Theorem. Let G be a compact Abelian group. The/ollowing
properties are equivalent:
(i) G is connected;
(ii) X is torsion-/ree ;
(iii) G is a divisible group.
Dimension 0 can also be described, as follows.
(24.26) Theorem. Let G be a compact A belian group. The lollowing
assertions are equivalent:
(i) G has dimemion 0;
(ii) X is a torsion group;
(iii) ;g (G) is a/inite subgroup o/T/or every 7, X.
Proof. Corollary (24.t8) shows that (i) and (ii) are equivalent. The
equivalence of (ii) and (iii) is obvious.
Theorem (24.26) admits an interesting generalization: the dimension
of a compact Abelian group is equal to the torsion-free rank of its
character group. We first state and prove a preliminary result.
Hewitt and Ross, Abstract harmonic analysis, vol. I 25
386 Chapter VI. Characters and duality of locally compact Abelian groups
(24.27) Theorem. Let G be a discrete torsion-[ree A belia group o]
rank a. Then the character group X o] G contains a subset homeomorphic
with 0, I a considered as a topological space.
Proof. Let x,:,I) be a maximal independent subset of G of
cardinal number a (A.2). Let H be the free subgroup of G generated
by the x,'s. Then plainly a character of H can assume arbitrary values
at the x,'s. In particular, for a fixed element (t,),z in 0, t a, there is
exactly one character Z0 on H such that Zo(X,)=exp(2:it,) for
Let y be any element of G. Then we have y"= x, x,: ... x, for nonzero
integers n and a i and '1, ..., *kI. Since G is torsion-free, the element y
completely determines the elements x,l x,k and the ratios al_
Now write Z(Y)=exp 2i ,+... + -- ,k . The function Z is well
defined on G and is an extension of Z. To see that Z is multiplicative,
and x [some a's and b's may be 0 Then
let y"=x,o.-X, = ,1."x, •
(yz)"" x ''a+', • • • x,a + , and so Z(yz)--exp[2i (malta +nnb t,l+ • • •
-- rnamn + nb t,k)l, =Z (Y)Z (z). Each Z defined in this way is therefore a
character of G. We thus have a mapping ] of 0, into X: the element
(t,) in 0, t goes into the character Z of G constructed above. It is
obvious that ] is one-to-one. A routine argument establishes the con-
tinuity of ] and ]-.
(24.28) Theorem. Let G be a compact A belian group with character
group X. The dimension o] G (3. t) is equal to the torsion-]ree rank r o (X). x
Proof. Let n be a nonnegative integer, and suppose that q/is a
finite open covering of G for which every finite closed refinement of
has multiplicity at least n+ . That is, suppose that dim (G)_> n. For
every x6G, there is a neighborhood V of e such that x. V is contained
in some Uqz'. A finite number of the sets xV cover G: say
xIVU...Ux,V,--G. Let V--V-o.fV. By (24.7), there is a closed
subgroup H of G such that H V and G/H is topologically isomorphic
with TxF, where a is a nonnegative integer and F is a finite Abelian
group. Plainly the character group of G/H is topologically isomorphic
with ZxF; it is also topologically isomorphic with A (X, H) by (23.25).
Therefore we have r0 (X) _>-- r0 (A (X, n))>_a. We shall take it as known
that dim (TxF) -- a ; thus we have dim (G/H) ----- a. The construction
of V shows that if yG, the coset yH is contained in some Uq/. Since
y H is compact, there is a neighborhood W of e in G such that y HW
is contained in some Uqz' (4.t0). Choose y, ..., y in G so that
Here we make the convention that r 0 (X)=oo whenever r 0 (X) is infinite.
This is a well-known theorem in dimension theory. It is also well known that
its proof is not easy. See for example 1 . S. ALEKSANDROV , Ch. V.
§ 24. The duality theorem 387
3 yWy--G. Let / be the family of subsets of G/H of the form
=1
zH" z yk W}; plainly /: is a finite open covering of G/H. Now assume
that dim (G/H)< n. Then there is a covering {xH: x
of G/H, where the 's are compact subsets of G, that refines and has
multiplicity less than n+ . It is easy to see that {H .... , H} is a
closed covering of G that refines and has multiplicity less than n+
This contradiction shows that dim(G/H)n. Since we have already
proved that r 0 (X) a=dim (G/H), we infer that r 0 (X) dim (G).
In proving the reversed inequality r 0 (X) dim (G), we may suppose
that r0(X)>0. Then consider any positive integer kr0(X), and let
, ..., Z be k independent elements of X having infinite order"
1 g . .
...= andaZimply=a= • a 0. Lt Ybe the sub-
group of X generated by Z .... , Z and let be a subgroup of X such
that Y= t) and is maximal with respect to the property
Let X and suppose that some power of is in . If , there are
a power and a such that Z
.... Z, where not a of the
integers a are zero. Then "--- -
-- ... , and is in . Hence
n ..... na:O, a contradiction. Therefore
i.e., X/ is torsion-free. The cosets Z .... , Z are plainly a maximal
independent set in the group X/. From (24.27), we infer that the
character group of X/ contains a homeomorph of 0,
contains a homeomorph of [0, t . Since the character group of X/
is topologically isomorphic with A (G, ), we have proved that G contains
a closed subspace homeomorphic with 0, . Since dim (0, )= k,
we have proved that r 0 (X) dim (G).
The assertions (24.t8)--(24.26) depend upon the basic result (24.7)
[along with (24.8), of course. We next give a quite different application
of (24. 7), to the effect that every locally compact Abelian group admits
a group of the fo R (a:0, , 2, ...) as a direct factor, and that there
is a largest value of a, described simply in tes of the structure of G.
(24.29) Theorem. Let G be an arbitrary locally compact Abelian
group, with character group X, and let C and K, respectively, be the com-
ponents o/the identity in G and X. Let n be the nonnegative integer and E
a co.act connected A belian group such that there is topological iso-
morphism carrying R"xE onto C, as in (9. 4). Let m be the nonnegative
integer and F co.act connected A belian group such that there is
topological isomorphism z carrying RxF onto K. Then we have"
(i) G is topologically isomorphic with R"xA (G, z(R));
(ii) X is topologically isomorphic with RxA(X, (R")) ;
(i) re:n;
25*
388 Chapter VI. Characters and duality of locally compact Abelian groups §24. The duality theorem 389
(iv) n is the largest integer a such that, G or X contains a topological
i.'omorph oWthe group R .
Proof. Let B and B be the closed subgroups of G and X, respectively,
consisting of compact elements (9.t0). By (24.7) and (24.8), we have
B--A(X, C), K----A(X, B), B--A(G, K), C----A(G, B).
Since z is a topological isomorphism, z(R ") is locally compact in its
relative topology as a subgroup of X and is therefore closed (5.tt). It
follows that B F1 z(R ") --{t }. By (9.26.a), the subgroup BC of G is open
and closed 1. By (t), we have
Hence A(G, z(R")).C is an open and closed subgroup of G. If
A(G, z(R")).C were not equal to G, there would be a character
such that Z----t on A(G, z(R")) and on C vhile Z is not identically
(23.26). By (24.0), we would have
zeA(X, A((, (R')))ha(X, C)--(R") n B-{},
which is a contradiction. We infer that
A(G, (R')).C--G. (2)
We can write C--(E) (R"); since E is a compact group, (E) is a
compact subgroup of G, and hence a (E) c B. Thus a (E) c A (G, K)
A(G, z(R")), and so (2) implies that
The same argument shows that
A (X, (R")) • (R") -- X.
Thus if xA(G, A(X, a(R"))) =ў(R") and xA(G, z(R")), we have x=e.
That is,
A(O,, (R")) Fl (R")---- {e}. (4)
Since (R") is compactly generated, (6.21) shows that G is topologically
isomorphic with a (R")xA (G, z (Rm)). We have thus proved (i); (i) and
(24.8) yield (ii).
By (24. t), the character group of (R") is topologically isomorphic
with X/A (X, (R")), and by (ii) this group is topologically isomorphic
with R m. This implies at once that m----n, since the character group of
R" is topologically isomorphic with R"(23.27.f).
To prove (iv), suppose that G contains a topological isomorph of R a.
Theorem (9.14) shows that a<__ n. I1
(24.30) Theorem. A locally compact Abelian group G is topologically
isomorphic with R"Go, where G o is a locally compact Abelian group
containing a compact ope, subgroup. The group A(G, z(R")) o/ (24.29)
may be taken as G o . I/ G is also topologically isomorphic with R'xG1
and GI contains a compact open subgroup, then m--n.
Proof. Let G o be the group A (G, z(R")) of (24.29). Let H be a
compactly generated open subgroup of G 0. By (9.8), H is topologically
isomorphic with RaxZxF, where F is a compact group. If a>__ t, then
R"xG o would contain a topological isomorph of R ", contrary to
(24.29.iv). Thus we have a--0. Since Z is discrete, F is open in ZxF,
and its topological isomorph is open in H and hence in G 0.
We now prove the second statement. Let C o and C be the compo-
nents of the identities of G o and G1, respectively. Then plainly C o and
C 1 are compact and the component of e in G is topologically isomorphic
with both R"xC o and R"xC 1. Corollary (9.13) now implies that
Another application of (24.8) follows. We will use it in classifying
compact monothetic groups (25.t t) --(25.t 7).
(24.31) Theorem. Let G be a compact A belian group with character
group X. Let m be nonzero cardinal number. The ]ollowing conditions
are equivalent:
(i) G Fl{e}' contains a subset A o/ cardinal number m such that the
smallest subgroup H containing A is dense in G;
(ii) there is an index class I o! cardinal number m and an isomorphism
o[ X into ,PzT(o such that ,(z(X)) {t}/or each pro]ection ,,, and the
[unctions , o are distinct/or distinct I.
Proof. Suppose that (i) holds. A continuous character on G is
determined by its values on H, and these in turn are determined by its
values on A. For xX, let z() be the element ((a)) a of PaT(.
Plainly z is an isomorphism of X onto z(X). Since ae for aEA, there
is for each aA a character ZE X such that Z(a)t (22.7). Thus the
projection of z(X) onto the a-th axis contains a number different from t
for each a A. If a 4= b, there is a Z e X such that Z (a) (b). Thus (ii) holds.
Suppose conversely that (ii) holds. For every ,I, the mapping
Z-+,o z(Z) is a character of X different from the identity, and so by
(24.8) there is a point a,EG such that X(a,)--,oz(Z) for all zeX. It
is clear from (ii) that a, a,, if, =t= ,' and that no a, is e. If the subgroup
generated by {a,}, were not dense in G, there would be a X such
that t and (a,)--t for all ,. That is, ,oz()_--t for all ,, and
() = (t). This contradicts the hypothesis that z is an isomorphism. I1
390 Chapter VI. Characters and duality of locally compact Abelian groups §24. The duality theorem 391
(24.32) Corollary. A compact A belian group G is monothetic i/and
only i/ its character group is isomorphic with a subgroup o/ Td. 1
Proof. Barring the trivial case G={e}, this assertion is (24.3t) with
m=t. V
In various parts of harmonic analysis, continuous homomorphisms
into the additive group R are important. These can be analyzed on the
basis of Theorem (24. 7).
(24.33) Definition. Let G be a group. A homomorphism of G into
the additive group R is called a real character o/G. For real characters /
and g, the sum/ч g is defined pointwise. The set of all continuous real
characters of a topological group G is called the real-character group o/G.
(24.34) Theorem. Let G be a locally compact Abelian group, and let
B be the closed subgroup o/compact elements o/G. For x, y C G, there is a
continuous real character / o/ G such that/(x) q=/(y) i/and only i/ xy-lc B.
Thus G admits su//iciently many continuous real characters i/ and only i/
Proof. Since R has only one compact subgroup, namely {0}, it is
clear that/(B) = {0} if / is a continuous real character. Thus /(x) /(y)
implies that xy-lў B.
Next we examine the quotient group G/B. By (24.t7), we have
B--A(G, K), where K is the component of t in the character group X
of G, and so G/B is topologically isomorphic with the character group of
the connected group K. By (24.t9), G/B contains no compact elements
except the identity.
Our proof will thus be complete if we show that there are sufficiently
many continuous real characters on every locally compact Abelian group
G such that B = {e}. Let x be an element of G different from e, and let H
be a compactly generated open subgroup of G that contains x. By (9.8),
H is topologically isomorphic with RxZ b, where a and b are nonnegative
integers and a or b is positive. It follows that there is a continuous real
character/0 of H such that/o(X) 0. If we extend/0 in any manner
to a real character / of G Isee (A.7)I, the real character / will be contin-
uous on G because it is continuous on the open subgroup H. V
(24.35) Corollary. The /ollowing conditions o a locally compact
A belian group G are equivalent:
(i) G has su//iciently many continuous real characters;
(ii) the character group o/ G is connected;
(iii) G is topologically isomorphic with R"xF, where n is a nonnegative
integer and F is a discrete torsion-/ree A belian group.
1 Recall that T d denotes the group T with the discrete topology.
Proof. The equivalence of (i) and (ii) follows from (24.19) and (24.34).
The equivalence of (i) and (iii) follows from (24.30) and (24.34).
(24.36) Theorem. Let G be a locally compact Abelian group, H 1 a
closed subgroup o/ G, and/1 a continuous real character on H 1. Then there
is a continuous real character / o/ G such that/] Hi=/1.
Proof. (I) Suppose first that G=R c, where c is a positive integer.
By (9.1 t), H1 is topologically isomorphic with RxZ for nonnegative
integers a and b. Let xl, ..., x, Yl,-.., Y in R c be generators of H1 in
the sense that every vector in H 1 can be written uniquely as
(XlXl--"" " "-- (XaXa'- - mlYl + . . . + m,y, ,
where 1,-.., g=cR and m I .... , m,Z. A continuous real character /1
on H i assumes arbitrary real values at Xl,..., x,, Yl,..., Yb, and then
is completely determined"
/1 (XI Xl - "'" + (Xa Xa + ml YI + " " - mb Yb)
-- (XI/1 (1) -- " " " -- Xa/1 (X a) -- mi/1 (YI) +"" -- mb/1 (Yb)"
By (9.t t), the vectors Xl, ... , x, Yl,..., Y are linearly independent in
R , and we can extend them to a basis for R . It follows that/1 can be
extended to a real linear functional / on R , and that / is then also a
continuous real character on R .
(II) Suppose next that G=RCxF where c is a nonnegative integer
and F is a compact Abelian group. Let 9 denote the projection of RxF
onto R . By (5.t8), the group 9(H1) is closed in R . We define/, on
9 (H1) by the rule
/. (x) =/1 ((x, z)) where (x, z)
The function /. is well defined because /1 (({0}xF)H1)= {0}, and
is obviously a real character. Assume that/, is not continuous. Then/.
is not continuous at 0R; hence there exists an e>0 and a sequence
{(x, z)}=l of points in H1 such that lira x=0 and I/1 ((x,, z))l _>_ e for
n=t, 2 ..... There exists a subnet z# of the sequence {z,}= that
converges to an element z in F. We then have lim (x:, z:)=(0, z)
/ '
/1((0, z))=0, and ]/l((x#, z#))]_>e for all /5. This contradicts the con-
tinuity of/1, and so/. is continuous. By (I), /. can be extended to a
continuous real character/a on R . Defining / to be/ao 9, we obtain a
continuous real character on G that extends/1.
(III) Now suppose that G and H are arbitrary. Let H. be a com-
pactly generated open subgroup of G. Then H. is topologically iso-
morphic with RxZbxF (9.8), and since Z is discrete, G contains an
open subgroup H a topologically isomorphic with RxF. Part (II)
392 Chapter VI. Characters and duality of locally compact Abelian groups §24. The duality theorem 393
implies that/II(H1 f'lH3) can be extended to a continuous real character
]3 on H. On the open subgroup HaH 1 of G, let/4 (hahl)=Ja(ha)+ ]1 (ha).
It is easy to see that/4 is well defined, is a continuous real character on
HH 1, and extends/1. Since R is divisible,/4 can be extended to a real
character / on G (A.7). Since ]1H3H1 is continuous, / is continuous. [
(24.37) Definition. Let G 1 and G. be locally compact Abelian groups
with character groups X 1 and X. respectively. Let 99 be a continuous
homomorphism of G 1 into G.. For each Z. X., let 99" (Z.) be the function
Z. o 99. Then 99 (Z.) is obviously in X1, and 99 is a mapping of X. into
It is called the adoint o! 99.
(24.38) Theorem. The adoint mapping 99"" is a continuous homo-
morphism o/X. into X, and its kernel is A (X2, 99 (G1)).
Proof. Forz., y. X. and xG, we have (Z.p.)o99(x) =Z.(99(x))o.(99(x))
--Z.o99(x).o.o99(x). Thus 99 is a homomorphism. For a compact
subset F of G1, 99 (F) is compact in G., and plainly 99(P.(99(), e))c
Pl (F, e). Thus 99 is continuous. It is trivial that 99 (Z.) is identically
t if and only if Z. (Y) = t for all y 99 (G1). [
(24.39) Theorem. Let H 1 be a closed subgroup o/C 1. Then
-- (A(X,/-/1) rl -(X.))-A (X., (/-/)).
Proof. If Z.(x.)=t for all x.99(H), then Z.o99(H)={t}, so
that 99(Z2)A(X1, H1). Also, if Z.99-1(A(X1, H)N99"(X.)), then
9(Z2) (H)= {t}, so that Z2 (9(H1)) ={t}.
(24.40) Theorem. Suppose that
G 1 into G and that is an open mapping o/G 1 onto (G). Then is an
open continuous homomorphism o/X2 onto A(Xl, -1 (g)).
Proof. By (5.40.d), 9 (G1) is a closed subgroup of G. The kernel of
9 is plainly A (X, 9 (G1)). Let 00 be the mapping of X2/A (X2, 9 (G))
into Xl defined by E00(Z A(X, 9(G)))(x)=Zog(Xl), for all XlG 1.
Since 9 is open, 9(G) is topologically isomorphic with the group
G/9 -1 (e), and so 00 is the topological isomorphism 0 described in (23.25).
That is, 00 maps the character group of 9(G) [and hence of G/9-(e)
onto A(Xl, 9-1(e2)). If denotes the natural mapping of X2 onto
X/A (X, (G1)), then we have =0°. Since is open and 00 is a
homeomorphism, is open.
(24.41) We now list without proofs some additional properties of
adjoint homomorphisms.
(a) The homomorphism 9 satisfies =. This follows from (24.8).
(b) The homomorphism is one-to-one if and only if (G) is dense
in G2, and (X2) is dense in X if and only if is one-to-one.
(c) The homomorphism 99 is a topological isomorphism of X. onto
X1 if and only if 99 is a topological isomorphism of G 1 onto G..
(d) Let/, be the mapping x-+ x" of G into G. Then the adjoint homo-
morphism [ is the mapping Z--Z of X into X.
Miscellaneous theorems and examples
(24.42) Let H and H 0 be closed subgroups of G. Then Hf'lHo={e }
if and only if (A(X, H). A(X, H0))-= X. [Using (23.29.5) and (24.t0),
we see that
A (G, (A(X, H). A (X, Ho))-)= A (G, A(X, H))FIA (G, A(X, Ho))
-- H f?H0;
the assertion follows.]
(24.43) Real characters (DIXMIER [2]). Let G be a locally compact
Abelian group with character group X. Let ] be a continuous real
character of G. Then the mapping
(i) x-->exp El/(x)
is plainly an element of X. Every element of X has the form (i) if and
only if X is the union of one-parameter subgroups. [Let 99 be a continuous
homomorphism of R into X, and let 99~ be the adjoint homomorphism of G
into the character group R of R. For xG, 99"(x) is the real number such
that 99(t)(x)=exp[i99 (x).t] for all real numbers t [note that 99(t) is an
element of X. Thus if Z X and Z lies in a one-parameter subgroup of X,
then Z has the form (i). On the other hand, if [ is a continuous real
character of G, the characters x-->exp [itJ(x)] (tR) form a one-parameter
subgroup of X.]
(24.44) Examples relating to (24.23) and (24.24). (a) (H. FREU-
DENTHAL" see HARTMAN and RYLL-NARDZEWSKI [t].) There are locally
compact Abelian groups G that are not divisible, whose character groups
are torsion-free. [Let m be any infinite cardinal number, and let G be
the subgroup of T m consisting of all (x,) such that x,=+ for all but a
finite number of indices . Let H be the subgroup of G consisting of all
(x,) such that x,--t for all but a finite number of indices . It is easy
to see that G ( = H if n is even and G (' = G if n is odd. For a finite set
A of indices t, let UA be the set of all (x,) such that x,=l for t6A and
x,=+ t for teA. Let all UA be taken as an open basis at the identity (1 ,).
The subgroup UA is clearly topologically isomorphic with {--t, t }m, so
that UA is compact. Thus G is locally compact, and H is a dense subgroup
of G. If is a continuous character on G of finite order k, then {t}--
Y.(G(I)=Z(H)=z(G ). Thus X is torsion-free, and G is plainly non-
divisible.
394 Chapter VI. Characters and duality of locally compact Abelian groups
(b) The group G described in part (a) is obviously 0-dimensional, so
that the component of (t,) is ((t,)). The largest divisible subgroup of
G is H. Thus we have C D- H H---G for this group.
(c) The proper inclusion D- N G occurs in a number of discrete
Abelian groups. Let G be the set of all sequences (n;m., m 3, m4,...,
rn, ...), where n Z, rn {0, t, 2 .... , k-- t), and only a finite number of
entries are different from 0. We define the product of two of these
sequences by the formula
oo rn + m;, ' (mod 2) m + m (mod k)
-- ,+ n' + k ;m2+ m. ,. .... ... •
It is easy to see that G is an Abelian group. It is also clear that if
(n" m 2, m3,.., m, .. ) is in G Il then m--0. Hence N G Il contains
' " ' ў=2
only elements of the form (n; 0, 0, 0,...). Since
(0; 0, ..., 0(_, t(), 0(+,),...) ::- (n;0, 0,...),
we see that G I) consists exactly of the subgroup {(n" 0, 0, ...)'nZ).
/=1 oo
This group is reduced (A.5), and so D:D-: {(0" 0, 0,...)} IGI).]=
(d) The component C of a locally compact Abelian group is always
divisible, as shown in (24.24). The component C may fail to be divisible
for some non-Abelian, noncompact, locally compact groups 1. [The group
(2, R) is connected. First, the subgroups 0 t :xR and
{(; )',R} are obviously connected. For any (: )in (2, R)such
that b @ 0, we have
Since the component of E is a closed normal subgroup, ®(2, R) is con-
nected. If is any negative number different from -- t, then there is no
0
real matrix (z x )such that (z x wY): ( _). 1
(e) Another example illustrating (d) is given by ®(2, R). [Thi sub-
groups {( 0z):X>0, z>0, yR} and {( Yz)'.:>0 z>0, yR are
The definition of a divisible non-Abelian group is exactly the same as the
definition of a divisible Abelian group (A.5).
§24. The duality theorem 395
(c a )®(2, R)and b:[:0, then
obviously
connected.
If
-- -I bc--ad+a .
b 0 b
(c a ). must belong to the component
Thus
if
ad--bc>O,
the
matrix
of E; that is, {A (2, R) : det A > 0}. This inclusion is actually
an equality, since otherwise E would equal (2, R), contrary to the
fact that the continuous function AdetA maps (2, R) onto
]--, 0 U ]0, . If and are distinct nonzero real numbers and at
least one is negative, then there is no real matrix ( such that
(24.45) The component of e as a direct factor (BRACONNIER
(a) Let G be a locally compact Abelian group. If the component C of
is open, then G is topologically isomorphic with C x(G/C). By (24.24),
C is divisible. Now apply (6.22.b).
(b) If G is locally connected, then G is topologically isomorphic with
Cx(G/C). For, C contains a neighborhood of e, and (5.5) implies that C
is open. Now apply (a).
(24.46) Pure subgroups. (a) (HARTMAN and HULANICKI t.) Let
G be a locally compact Abelian group, H a closed subgroup of G, and X
and Y the character groups of G and H respectively. If every character
zIY admits an extension Z X having the same order as Z, then H is
said to have property (H). The subgroup H has property (H) if and only
if the annihilator A(X, H) is a pure subgroup of X. Suppose H has
property (H), that ZI X, and that zA(X, H). Then Z1 as an element
o/Y has order a, where a divides n. Let Z be an extension of Z1 in X
having order a. Then (2Z1):Z and zIA(X,H). Thus A(X,H)
is pure in X.
If A (X, H) is pure, Z0 is in Y, and the order of Z0 is n, let Z1 be an
arbitrary extension of Z0 such that ZI X. The character Z is in A (X, H).
Since A(X, H) is pure, there is a zA(X, H) such that Z:Z. Then
Z2Z1 is an extension of Z0 such that (2Z):
(b) Let G be a discrete Abelian group, and let H be a pure subgroup
of G such that G/H is divisible. Then A (X, H) is a pure subgroup of X.
Let Z0 be a character of H, of order n< . We will show that Z0 can
be extended to a character of G also of order n. This will show that
A(X, H) is pure, by part (a). Since G/H is divisible, every xG can be
written in the form x h y for some h H and y G. Define Z (x) : Z0 (h).
To see that Z is well defined, suppose that hy:h2y . Then hhl:
396 Chapter VI. Characters and duality of locally compact Abelian groups § 24. The duality theorem 397
(y.yi), and since H is pure, there is an element ha6H such that
h-- hi h 1. Thus we have t -- Z0 (ha)-- Z0 (h) -- Z0 (hi) Z0 (h2)-l. That is,
Z0 (hi)--Z0 (h2), and Z is well defined. To see that Z is a character of G,
let xl and x2 be any elements of G, and let x--hy 7 ('--, 2). Then
xl x. -- hi h. (Yl Y.), and as above we have Z(Xl x2) --Zo(hl h2) --Zo(hl) Zo(h2) -
Z(Xl) Z(Xg) • ]
(c) (BRACONNIER t.) The closure of a pure subgroup need not be
pure. Let I be any infinite index set. For each tI, let G,--Z(4),
which we represent as {0, t, 2, 3} with addition modulo 4. For tI, let
H,={0 2} Topologize G-- P G, as in (6.t5.c) Thus a generic neigh-
) " EI "
borhood of the identity has the form
UA={XG:x,{0,2} for all t and x,=0 for tA},
where A is a finite subset of I. It is easy to see that G is a locally compact,
Abelian, 0-dimensional group. Let G O be the subgroup consisting of all
xG such that (tI:x,=0} is finite. It is evident that G O is pure in G:
if y G and y" G 0, then there is a z G O such that z= y". The closure
G- of G O consists of all xG such that {tI:x,--t or 3} is finite. This
subgroup is not pure: choose x such that x,_{0,2} for all tI and
(tI:x,--2) is infinite. If y,-- for all t such that x,-----2 and y,--0 for
all other t, then y--x but there is no zG- such that z 2-- x. 1
(24.47) (KAKUTANI 41.) Let G be an infinite discrete Abelian group
with compact I character group X. Theorem (24.t5) shows that
ro(X)--G. The cardinal number of X itself is 2 a. Since elements of X
are complex-valued functions on G, it follows that X g (K) a -- ca -- 2 a.
To verify the reversed inequality, we first suppose that G is countable.
Since X is compact, (4.26) implies that X>= c--2 . Finally, suppose that
G>Lў 0. Let G--{x 1, x2, ..., xa, ...} be a well ordering of G, where
runs through all ordinals less than the smallest ordinal having cardinal
number G and where xl--e. For each g>>t, let H denote the group
generated by (xaG:fl<o}. Let A consist of those elements x a of G for
which xaў H a. The set A generates the entire group G. For, if A 0 denotes
the group generated by A and A 0q=G, there is a least element xa0 in
, n where
Gf3Ao. It is clear that a0>t and XaoCA. Therefore Xao--X...xa,
/5.< 0. Since all of the xa are in A 0, xao is also in A 0, a contradiction.
Since > tў 0, it follows that A--.
We now show how to construct a character Z of G by transfinite
induction. Let Z(Xl)--t and suppose that Z (x) has been defined for all
fl< . If xўA, i.e., x aH, then Z(x) is determined by the values
{Z (x)'fl<}. Suppose that x aA. I no power of x a lies in H a, then
we may plainly define Z(x) to be any number in T. If some power of
x lies in H and m is the least positive integer such that x H a, then
we may define Z(xa) to be any m-th root of Z(x). This procedure always
defines a character of G and moreover, for each xaA, at least two dif-
ferent choices of Z (xa) are possible. It follows that G admits at least 2
distinct characters. That is, X_>_ 2 - 2a.]
(24.48) (HEWlTT and STROMBERC, 1.) Let G be a locally compact
Abelian group with character group X. Let a be the least cardinal
number of an open basis at e in G and b the least cardinal number of a
family of compact subsets of X whose union is X. Then a and b are
equal. Let be an open basis at e such that eactl V in has com-
pact closure. The sets P(V-, })-, where V, are compact subsets of
X (23.t6), and it is obvious that their union is X. Hence >_ b. If =,
then also a-- t (23.7). If 4= , then b is infinite. Thus suppose that
l_>__tў 0 and that a/----{A').A} is any family of compact subsets of X
such that U A-- X and a/-- b. Plainly there are open sets Y D A such
aA
that the sets Y7 are compact. Let be the family of all finite unions of sets
{ 1 forallzB} whereB,
YU...UYT. The sets xG'lz(x)--t I<- ,
form a basis of neighborhoods at e in G of cardinal number b. In
fact, for an arbitrary neighborhood V of e in G, (24.8) shows that
there are a compact subset of X and a positive integer n such that
for all } V. There is Bin.
some
containing
(D" this proves that a_<_ .
(24.49) Remark on Theorem (24.30). We can obtain part of (24.30)
directly from (9.8) without appealing to the duality theorem. Let H
be a compactly generated open subgroup of G. By (9.8), H can be
regarded as R"xZxF, where F is a compact Abelian group and m
and p are nonnegative integers. The subgroup R"xF of H is plainly
open in G. Since R is a divisible group, the projection mapping of
R"xF onto R can be extended to a homomorphism of G onto R (A.7).
This homomorphism is continuous on G since it is continuous on R"xF
(5.40.a). By (6.22.a), G is topologically isomorphic with R'x(G/R").
Since the natural mapping of G onto G/R" is an open mapping (5.t7),
the image of R"xF is open in G/R '. That is, (R"xF)/R ' is open in
G/R'. Since (RxF)/R " is topologically isomorphic with F, we have
proved that G/R" contains a compact open subgroup.
Notes
The duality theorem (24.8) is the creation of L. S. PONTRYAGIN and
E. R. VAN KAMPEN" other writers have made contributions, as noted
below. Theorem (24.7) is due to VAN KAMPE>ў [t , p. 458. Theorem (24.3)
398 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 399
for G having a countable open basis was hinted at in PONTRYAGIN [t],
announced in PONTRYAGIN [3], and proved in PONTRYAGIN [4]. ALEX-
ANDER [t] proved (24.3) for arbitrary discrete Abelian groups; ALEX-
ANDER and ZIPPIN [t] simplified the argument. Theorem (24.8) was
first proved by VAN KAMPEN [t]. VAN KAMPEN'S topology for X is
formally different from PONTRYAGIN'S, but is actually the same. Our
proof of (24.8) is basically the same as VAN KAMPEN'S; we have also
been guided by PONTRYAGIN [7], Ch. 6. A.WEIL [4], §28, pp. 02--09,
has given a different proof of (24.8). RMKOV announced in [t], full
details in [3]] and CARTAN and GODEMENT [t] have given a short and
totally different proof of (24.8), based on Fourier transform theory.
We have chosen to give the "classical" structure-theoretic proof of
(24.8) for several reasons. First, we needed Theorem (9.8) for other pur-
poses; (9.8) is not a consequence of (24.8), while (9.8) is the key to our
proof of (24.8). Second, (24.0) and (23.2), which are needed for various
purposes, are not consequences of (24.8). Third, we feel that the classical
proof of (24.8) is more illuminating than the proof based on Fourier
transforms.
Theorems (23.2S), (24.t0), and (24.tt) are due to PONTRYAGIN [4] for
countable discrete groups; to VAN KAMPEN [t] for general locally com-
pact Abelian groups.
Theorem (24.t4) is due to KAKUTANI [4 for compact G, and in the
general case to PONTRYAGIN [7, §40, Theorem 57. Our proof is new.
Theorem (24.t7) and the proof given here are also PONTRYAGIN'S 7,
§40, Example 73, P. 282. Parts of (24.8), (24.25), and (24.26) are due,
in various forms, to PONTRYAGIN 4] and VAN KAMPEN [t]. Theorem
(24.2t) [but not our proof I is due to BRACONNIER [t 1, p. 5t. Various
parts of (24.23) and (24.24) are due to KAPLANSKY [t, p. 55 [without
proofs 1, to HARTMAN and RYLL-NARDZEWSKI [t , and to BRACONNIER [t ].
In (24.23), we correct an error in HALMOS It ].
Theorem (24.28) is due to PONTRYAGIN [4] for G with a countable
open basis and to VAN KAMPEN [t for arbitrary G. Our proof of (24.28),
and (24.27), are from PONTRYAGIN [71, §38, Theorem 47, p. 263. It is
interesting to note that ARHANGEL'SKII [t] and PASYNKOV [t] have
proved that all standard definitions of dimension are the same for locally
compact groups.
Theorem (24.29) is due to PONTRYAGIN [71, §40, Example 74, p. 283.
Its immediate consequence (24.30) is due to VAN KAMPEN [t, p. 46t;
a proof also appears in A. WEIL 4], § 29, p. t t0.
Real characters as defined in (24.33) have been studied by a number
of writers. Corollary (24.35) is due to G.W. MACKEY I21, and (24.36)
to J. DIXMIER [2] [our proof of (24.36) is different from DIXMIER'S.
RlSS [t, [2 has two notes on real characters and has utilized real
characters heavily in his theory of distributions on locally compact
Abelian groups I31. Real characters of certain semigroups have been
considered by DEVlNATZ and NUSSBAUM [t. DIXMIER [2 has also
characterized in a manner different from (24.43) the locally compact
Abelian groups such that every continuous character has the form
exp (i I) where ! is a continuous real character.
Definition (24.37) is due to FREUDENTHAL [3. The adjoint mapping
has been used by many writers, for example" A.WEIL [4, §28, p. t03"
ANZAI and KAKUTANI [t" BRACONNIER [t. Theorem (24.40) is due to
BRACONNIER [t .
Thousands of pages in research journals have been devoted to studies
of the structure of locally compact Abelian groups, of characters of all
sorts of groups, and of extensions of the duality theorem. We now
mention a few of the most interesting results and papers not dealt with
in the main text.
PONTRYAGIN [7, § 38, Theorem 48, p. 264, has characterized compact,
locally connected, Abelian groups G in terms of algebraic properties of
the character group X. A special case of his theorem is the following.
A compact Abelian group G is connected and locally connected if and
only if every finite subset of X is contained in a pure subgroup of X
having a finite number of generators. If X is second countable and G is
connected and locally connected, then G is also arcwise connected and
has the form R"xT m, where n is a nonnegative integer and m
this is due also to PONTRYAGIN I41, p. 380. DIXMIER I21 has constructed
an uncountable group X for which G is connected and locally connected
but not arcwise connected.
N. YA.VILENKIN [4], [7, [8, [t3] has made elaborate computations
of character groups, for both locally compact and non locally compact
Abelian groups. G.C. PRESTON t] has shown that continuous homo-
morphisms of 0-dimensional locally compact Abelian groups into certain
groups other than T can serve to produce theorems like (24.8), although
the "character" groups may be quite different from the character groups
treated in the main text.
Some attention has been given to extensions of (24.8) to topological
Abelian groups that are not locally compact. See the following" T. ISI-
WATA [t" S. KAPLAN tl, [2]" H. L.PTIN t, [2], [3, [41; H. SCH6NE-
BORN [I ], [2], [3]; M. F. SMITH t].
§ 25. Special structure theorems
We first construct the character groups of a number of specific
groups I(25.t) to (25.7). In doing this, the duality theorem (24.8) and
also Theorems (24.t0) and (24.11) are very useful.
400 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems 401
(25.1) The a-adic numbers. Consider the additive group -Qa of all
r-adic numbers, which is discussed in (t 0.2)--(t 0.). We will show that
the character group of Ј2a is topologically isomorphic with Q.., where
a*--a_, for all nZ. In particular, for any integer r> t, the character
group of Q. is Q. itself.
Consider first a continuous character Z of Q. different from the
character t. Then for some integer l, we have ]Z(a)--t]< for a
aA. Since A is a subgroup of Q., it follows as in (23.17) that
z(A)={t}. Now consider all integers l, negative or nonnegative, for
which z(A) = {t}. Since Z t, there is a least integer with this property;
it may or may not be negative, but in any case we denote its negative
by k. In other words,
(A)--{} if and only if m--k. ()
For nZ, let u ") be the element of Q. such that u")= b. (pz). By (),
we have Z(u (") = for n--k. Note that Z on A__ is determined by
its value at u (--1) because the set of multiples {u (--1), 2u(--), ...}
is dense in A__. It follows that Z(u --)) @ . For each nZ, let
be the unique number in 0, [ for which Z(u(-")=exp[2ai2.. Thus
we have .=0 for nk. Since a u(- ") -- a_. u(- " -- u -+, we infer
that a .= .1 (rood t), that is,
for some integer y. Moreover, we have y,--a --_xag
and y = a . -1 -- -1 >-- , so that 0 y, ag-- . Thus we
have defined an element = (y) of .. such that y=0 for n k and
Y+I0" since 0<+1< and a/+12, (2) shows that 0a/+x +1=
+ Y+I--Y+I. An elementary computation using (2) shows that
j=}+l a * ' (3)
for n= k+ t, k+ 2, .... Given a in Q., let us compute Z (a)- If aA_,
then Z(a)=t. Otherwise there is an integer m<--k such that
.. (--1) ,
a= x. u(")+ x.+u("+)+ • + x__ 1 + z, where the x i s are inte-
gers and zA_. Therefore we have
Z (a) = exp2a i(x. _+ X.+l i-.-1+ ""+ x__l i+1) • (4)
Relations (3) and (4) show that Z is determined by the sequence (y.
If Z is the character t, let =0 and .--0 for nZ. Then l =t (u(-")) =
exp 2ai. and the relation (2) holds for all nZ.
We now show how to obtain a continuous character of Q. from an
element Q... First consider the zero =0 of Q... Set .=0 for all
n6Z, and notice that (2) holds. We then require that Z(u(-")) --
exp2ai.] for all nZ; in other words, Z--t. Now consider any
element y in Q.. different from 0. Let k be the least integer such that
y,+l=0. Define the sequence {2.}.°°__+ 1 by (3), and for nk, let 2.--0.
It is easy to see that the relation (2) holds for all integers n when the
t.'s are so defined. For aA_, let Z(a)--t and for
define Z(e) by relation (4). To show that this defines a continuous
character Z of Q., it suffices to prove that Z is multiplicative" the
continuity of Z will then follow from the fact that Z is identically t on
the neighborhood A_ of 0. Suppose that a and z belong to Qa. If
either a or z belongs to A_ h, it is easy to verify that
z(+ z)-z(x) z(z); (5)
we omit the details. Suppose then that x, -- z, ----- O for n<l, that
xz-+- zzO, and that l<-- k. Let aq- z-----w= (w,) and it_ 1---0. Obviously
w. -- 0 for n < l, and for n _> l, we have x + z. + t._l-- t. a
is either 0 or t. From this and (2) it follows that
(X l 2_ l _A V Xl+l 2_l_ 1 2f_ ... 2f_ X__k_ 1 /k+l)
_ (2 z 2_ z _2f_ 2Z+l 2_Z_ 1 -A V ...-A V 2__k_ 1 2k ,+1)
(Tf,)l 2_ l 2f_ WI +1 /--I--1 -- .... -+- W--k--1 /k+l)
+ trY-z+/I+lY-/-1 - ""+ t--lY+I,
and since tzy_z+ tt+ly_t_l+...+ t_,-lY+l is an integer, (4) implies that
(5) holds.
Thus we have associated with every /in Q.. a continuous character
Zu of Q.. We will show that the mapping Y--Zu is a topological iso-
morphism of Q.,, onto the character group of Q.. Given Z in the character
group of Q., we construct the Y in ,Q,. and the sequence {2}°°___oo of
numbers that satisfy (2). The character Zu constructed for this /agrees
with Z on all u ("), and hence Z--Zu" The mapping Y--Zu therefore
maps Ј2a. onto the character group of .Q.. Let y and z be nonzero
elements of Q.., and let w--y+ z. Let 1 be the least integer such that
Y, +1-- Zl+l: = 0. Let (2.).%_oo, {#.}.°°=-oo, and {v.}.°°___oo be the sequences
of numbers defined as above that correspond to y, z, and w, respectively.
To show that ZuZ.--Zw, we need only show that Zu(u (-")
Z,.(u (-") for all nZ. This is obvious for nl. For n>l, we have
y.ч z.+ t._l--t.a*ч w." tz is 0 and each t. is either 0 or t. We have
v.--2.+,.--t, for n--l, 1+1,.... (6)
The relation (6) is trivial for n--l" and in general we have
Vn+wn+ (2n+pn--tn) + (Yn+l+Zn+l+tn--tn+la+l)
V+I -- a+ -- a+
2n + Yn+ __ ln + Zn+l __ tn -- 2n+l -{-/n+l -- tn+l
ag+l an+l +1 "
Hewitt and Ross, Abstract harmonic analysis, vol. I 26
402 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems 403
For n >= l, (6) shows that
Zw(u (-)) -----exp [2 i(2 + #--t)]
--exp 27 i ).. exp [27 i/tn- I --Zu(U (-'0) Z,(U(-')).
It is easy to see that the mapping BZu is one-to-one.
To complete the argument, we must show that the mapping yZu
is a homeomorphism. To avoid confusion, the neighborhoods of 0 in ,
that have the form A (0.4) will be written as A . We first observe that
for m Z. Let P (F, ) be a neighborhood of t in the chaacte group of .
Since F is a compact subset of a, F is a subset of some set A. Hence
by (7) we have
{Zy" Y .... A(,)+}c {Zy • ]Zy(X)-- ] < e for all XAm}
-- P(A, )cP(F, ).
Thus the mapping BZu is continuous; (5.29) now implies that BZu
is a homeomorphism.
It is useful to have an explicit formula for characters of Q. If
B Q* and Q, then
. ()
Zu(x) = exp 2 i xnan an+ " " as
This is verified by combining formulas (4) and (3). For each given x
and y, the sums in (8) are actually finite. In fact, if xi--O for j< m and
y = 0 for 1 k, then
(
where void sums are taken to be 0.
(25.2) The -adic integers. Consider the additive group A, of a
-adic integers, where = (a, ,...). We will find the character group
of A, with the aid of Theorem (24.t t). We extend to an arbitra
doubly infinite sequence (..., a_l, a, ,...) such that a2 for all
Z [for example, we might let a=2 for n<0. By (24.tt) and (25.t),
every continuous character Z of A, has the form ZI A, for some D.,.
Define in A, by =t for =0 and =0 for Z, n0. Then a
continuous character Z on A, is deteined by the value Z(), since
generates a dense subgroup of A,. It foows that the [discrete]
character group of A. is isomorphic with the subgroup {Z()"D.,}
of T. For D,,, relation (8) of (25.t) shows that
Z() = exp 2i Y= . (t)
the sums in (t) are finite for each y. Thus {Zu(u) • yY2a,) consists of all
numbers of the form exp[2zi( /- /]. We call this
group
Z (a°°) •
ao al •. a r
it is an obvious generalization of the group Z (p) defined in t.
Consider a number texp [2i (. .] in Z(a). We
may
sup-
a 0 a I •. a r
/J
pose that 0/< ao...a ,. Then 1 can be written uniquely as
l -- Yo al a..., a, + Y-1 a..., a r + ... + Y-,+I a, + y_,,
where each y. is an integer satisfying 0--<_ Yi < a-i" If we take y. to be 0
for/'ў {-- r, -- r+ t, ..., 0), then (t) becomes
Zu(u) = exp 27 i v = exp 2 i y_aA;., a, = t.
= aoa l'''a n = aoa l'''a r
Using this and (25.t .9), we see that the character Zt of Aa that corresponds
to t in Z (a ) is defined by
Z (x) = exp 2 i x Y
= an an+l •,, a s
-- exp 2 i x Y-s as+l "" an-1 + Y-n+1
,=. (2)
= exp 2 i x a 0 ... an_ 1
= = ao a 1 •.. a s
2il
= exp (x 0+ a 0x 1+ ... + a 0 a... a,_lX,)
a 0 a I •.. a r
__ xo +
for
(25.3) The a-adic solenoids. Let a denote the a-adic solenoid
described in (0.2). By (t0.t) Za is a compact group. The character
group of RAa is RZ(a), where Z (a ) is as defined in (25.2). The
continuous characters of RA are the unctions
where
in (t0.t2) and Theorem (23.25), the continuous characters of , can be
identified with the functions (t) that are equal to I at (t, ). It is easy
to see that the admissible values of are those such that + is
an integer for some choice of n and Z. Thus has to be a rational number
26*
404 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 405
m m
of the form . Moreover, if . has the form , then for
ao al " " a n aO al " " a n
l
some integer l, Ol< aoam...a , + is an integer, and (t) can
a0 al •.. a n
be written as
(,)__exp[2i(_ (x0 + a0XlAf_ ... 2f_ a0a,1 ... an_lXn)) [
(2)
- exp2i( (Xo+aoXl+...+aoa...a,lX,).
Thus, given a rational number - , (2) defines a continuous
0 1 " " "
character of , and distinct values of yield distinct characters.
Therefore we have a topological isomorphism of the character group of
, onto the [discrete] additive group of all rational numbers of the
m
form .
0 1 " " " n
Theorem (24.8) implies that two groups , and are topologically
if and only if the additive groups m
isomorphic
.... n=0, t ...;
mZ and b. b::. b " - 0, t, ... ; mZ are isomorphic.
(25.4) The discrete additive group Q. The character group of the
discrete additive group of rationals is topologically isomorphic with
, for any such that is equal to [or isomorphic with the set
-- "--0, t ." mZ This follows from (24.8) and (25.3).
0 l " n " " "
In particular, if a=(2, 3, 4, 5, ...), then a is the character group
of , since = '=0, t .... • Z . In this case, (25.3.2) [with an
inconsequential change of notation becomes
[
(,)exp 2i(-- (x,+ 2Xl+.--+(-- t) x_)) . (t)
Consider a fixed element (, )+ B in N,. By (24.9), we see that the
character Z of corresponding to this element is defined by
Z -- exp 2 i (-- (x 0 + 2 1 +''" + (-- t) n--)) " (2)
Note, in particular, tat Z(t) = exp [2i.
(25.5) The following characterization of the character group of
may also be illuminating. Let Z be a character of . For = t, 2, ...,
let=Z • Wehave
for n= t, 2,.... The relation (t) is also sufficient to produce a character
of " if {}L is a sequence of numbers in T such that (t) holds, then
the function Z on Q such that
(m) m
z = (2)
is well defined and is a character of Q" we omit the details. For
n=2, 3 .... , let/n be the mapping of T onto itself such that/n () =.
Then a character of Q can be identified with a sequence (-1, e., ...,
, .... ) T ° such that/n (n)--n-1 for n--2, 3, "" • Thus the character
group of Q is isomorphic to a projective limit as follows [see (6.t3)].
Consider the directed set {t, 2,...} and for m< n, let grim be the contin-
uous open homomorphism of T onto itself defined by g,m()=.-'--
Ignm is open by (.29)?. All that we have done is require
gn was computed by iteration. The character group of Q is isomorphic
to the projective limit of the inverse mapping system {t, 2,...},
and the P-topology of the character group of Q obviously agrees with
the Cartesian product topology of the projective limit.
(25.6) The discrete group of real numbers. To obtain the character
group of R a, consider any Hamel basis H for R over the rational number
field Q, so that every xR has one and only one representation as a
[finite sum, x-- kh k (k Q, h H). Thus R is algebraically isomorphic
k----1
to the group Qc.. Regarding Q and Qc. as discrete, we compute the
character group of Qc. from Theorem (23.22). Transferring this com-
putation back to R a, we see that the character group of R a is topologically
isomorphic with (a) c, where a--(2, 3, 4 .... ).
We can describe a character Z of R a somewhat more explicitly as
follows. The values of Z on H are quite arbitrary numbers of absolute
value t. For hH, the numbersz h ,Z -h ,...,Z . h ,... are
determined in accordance with (25.4) or (25.5). There are c--2 dis-
continuous characters of R.
(25.7) The discrete group T a. To find the character group of Ta,
we regard T a as Ra/Z , and compute all characters Z of R a such that
Z(Z)--{} (23.2s). This can be done by choosing a Hamel basis H for
R over Q such that t H, and then constructing all characters Z of R a
for which Z(t)= t, by the process of (2.6). Thus z(h) is an arbitrary
number of absolute value t for all ht in H, and Z at r h (rQ) is
(1)(n_2 3, ..)aredetermined
described as in (25.6). The numbers Z . , • ,
by the conditions Z . =t andz (+ =Z (n=2, 3,...). For
---(2, 3, 4, 5, ...), therefore, we see that the character group of T is
topologically isomorphic with ('a)exG1, where G 1 is a compact group
that is the proiective limit of a sequence of copies of T, with "first"
406 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems 407
element equal to (t). Note that G1 is the character group of the discrete
group Q/Z.
The group G1 can be identified in various ways. The group Q/Z is
isomorphic with P* Z (po) where P denotes the set of all prime numbers
pp '
(A.t4). By (25.2) and (24.8), the character group of Z(p °°) is/1. By
(23.22), the character group of Q/Z is topologically isomorphic with
eP.A , so that G: eP.A .
Finally, the group Q/Z is isomorphic with Z (a°°), where a: (2, 3,
4, 5,...). Therefore G is topologically isomorphic with A.
We now classify all compact Abelian torsion-free and torsion groups,
as follows.
(25.8) Theorem. A compact Abelian group G is torsion-/ree i/and only
i] it is topologically isomorphic with (2,,)mxpPpd p, where P is the set o]
all primes, m and 1t2,113 .... ,11,... are arbitrary cardinal numbers, and
a=(2, 3, 4,...).
Proof. By (24.23), G is torsion-free if and only if its character group
X is divisible. A divisible group has the form P* H, where each H, is
I '
either Q or z (poo) for some prime p, and of course all such groups are
divisible. Now apply (25.4), (25.2), and (23.22).
(25.9) Theorem. A compact Abelian torsion group G is topologically
isomorphic with
(i)
where there are only finitely many distinct positive integers b, and I is an
arbitrary nonvoid index set.
Proof. Let G<,)--{xG: x:e) (n--2, 3, 4 .... ). Each G<)is a closed
subgroup of G, and G: LJ G<). As pointed out in (5.28) some G<) has
n=l '
nonvoid interior, and so is open. Hence G/G(,) is discrete and compact,
and therefore finite. Let a-- G/G(). Then x a-- e for all xE G. That is,
the orders of all elements of G are bounded. The same is true of the
discrete character group X of G. By (A.25) X is isomorphic with P*Z (b,)
' tel '
where the b,'s are bounded. An appeal to (23.27.c) and (23.22) completes
the proof.
(25.10) Corollary. Let G be a compact A belian group not o] the ]orrn
(25.9.i). Then every neighborhood o] e in G contains an element o] infinite
order.
Proof. Let V be a neighborhood of e. Then F- is compact, every
subgroup G() has void interior, and by (5.28), we see that
We next make a detailed analysis of the structure of monothetic
groups.
(25.11) Theorem. Let G be a locally compact Abelian group with
character group X. Let S be the set o] elements a EG such that M--
(a: n Z}--- G. Then
S--{xEG:z(x):4:t ]or all zEX, Zt}.
The group G is monothetic i/and only i] S :4:
Proof. Suppose that M, G. Then by (24.2), there is a Z0E X such
that Z0 (a)- t and Z0:+: t. Conversely, if Ma--G, then plainly a character
2: such that Z (a)- t must be 1. The last statement is obvious.
(25.12) Theorem. There is a largest compact monothetic group Go,
in the sense that every compact monothetic group is a continuous hono-
morphic image o] G o. Furthermore, every continuous homomorphic image
o/ G o is a compact monothetic group.
Proof. Let G o be the character group of T described in (25.7). If
G is a compact monothetic group with character group X, then (24.32)
shows that X may be regarded as a subgroup of Td. By (24.tt), G is
topologically isomorphic with Go/A (G o, X). 0
(25.13) Theorem. A discrete Abelian group G is isomorphic with a
subgroup o/T i/ and only i/the p-ranks r (G) o/G are 0 or t /or all primes p
and the torsion-]ree rank ro(G ) o] G is less than or equal to c (A.t2).
Proof. If the ranks of G are as stated, imbed G in a divisible group E
such that rp (E) --r (G) for p----- 0, 2, 3, 5, 7, t t, ... (A.t6). By (A.t4),
E is isomorphic with
Q,01I, x P* (Z
pP
We also know that Td is isomorphic with
Qc. x P* z (poo).
pEP
Since r 0 (E) ____ c and rp (E) ____ t for all primes p, it follows that E and hence
G are isomorphic with subgroups of T d.
It is obvious that rp(G)--O or t and ro(G )_<c for every subgroup G
oT.
(25.14) Theorem. A compact connected A belian group G is mono-
thetic i] and only i] m (G) <___ c.
Proof. By (24.5), we have m(G)--X. Since G is connected, X is
torsion-free (24.25). Thus rp(X)--0 for all primes p. If X_<_c, then
obviously r0(X) < c. If r0(X)c, then (25.t3) shows that X is isomorphic
1 Ve call the elements o/ S generators o/G.
408 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 409
with a subgroup of T d, and hence X<__c. Now (25.t3) and (24.32) imply
that G is monothetic if and only if r 0 (X):< c. ]
(25.15) Corollary. The group T m is monothetic i/and only i/m<c.
Proof. This follows immediately from (25.4) and the fact that
r0 (T m) = max (m, 0)-
Alternatively, if m<__c, one can find directly an element of T m whose
powers are dense in Tm. Represent T m as the group of all functions (x,)
on a set I= {,} such that I= m, with values in T. Consider a Hamel
basis H for R over Q such that 1 H. Let p be a one-to-one mapping
of I into Hf {t}', and let a,=exp2:rip(,) for all ,I. Since every con-
tinuous character of T m has the form
.... x,.
for {'1 .... , ,}CI and nl, ..., nkZ, we see that only the character 1
maps (a,) onto t. Thus (25.tt) implies that (a,) is a generator of Tm. [-]
For a sequence a= (a 0, a 1, a., ...) of integers greater than t, Aa is an
infinite 0-dimensional compact monothetic group [see (t0.5) and (t0.6)].
We will now show that these are the only such groups.
(25.16) Theorem. Let G be an in[inite topological group. The ]ollowing
statements are equivalent:
(i) G is a O-dimensional compact monothetic group;
(ii) G is topologically isomorphic with some group
(iii) G is topologically isomorphic with a direct product pPp A p, where P
is the set o] all prime integers, and each A is {e}, Z (p'P) [r is a positive
integer, or A.
Proof. By (t0.5) and (t0.6), (ii) implies (i).
Suppose that G is a 0-dimensional compact monothetic group; we
will prove (iii). By (24.26) and (24.32) the character group X of G may
be regarded as a torsion subgroup of Ta. It follows that the p-primary
direct factor X of X (A.3) is a subgroup of z(p°°). Since all proper
subgroups of Z (poo) are finite cyclic groups with order a power of p,
we have X= P* Bp where B is {e}, z(p'p) or z(p°°). Theorem (23.22)
pp ' ,
now shows that G--PA, where each A is {e), Z (p'), or A p.
Finally, consider the direct product P,A as described in (iii). Let
P--{pP'A--A) and P:{pP'Ap:Z(p ") for some rt). It is
easy to see that there is a sequence a: (a o, a, a,...) of prime integers
consisting of infinitely many p's for each p P, exactly r p's for each
p P, and no other p's. In order to conclude that A is topologically
isomorphic with PA, it suffices to show that Z (a ) is isomorphic
with P*Bp where B is (e), z(p ") or Z(p °°) according as A is e),
pp '
z(p'), or Av. Since Z(a °°) is a torsion group, we can apply (A.3). If
p P, then the elements in Z (a °°) having order a power of p are exactly
the numbers of the form exp 12:ri-l, i.e., the group Z
If
Pc
then the elements having order a power of p are the numbers exp 2 i
where nr, i.e., the group z(p"). If pU, then no element of
Z () has order a power of p. HenceZ () is isomorphic with P* B
pp •
(25.17) Theorem. Let G be a compact A belian group with component
o/the identity C. The group G is monothetic i/and only i/ (G) c and
G/C is a [compact Abelian] group o/the/orm .
Proof. If G is monothetic, then clearly (G) : X c and the quotient
group G/C is monothetic. By (7.3) and (3.5), G/C is 0-dimensional. Thus
G/C has the form by (25.6).
Conversely, suppose that the conditions on G and G/C hold. Clearly
the torsion-free rank r0(X ) of X does not exceed X--(G)c. The
character group of G/C is the torsion subgroup of X [see (24.20),
and since G/C has the form . A (25.6), it follows that r() :0 or l
for all p. Since r():r(X) for every prime p, (25.3) and (24.32)
show that G is monothetic.
We now describe the compact solenoidal groups, defined in (9.2).
(25.18) Theorem. Let G be a compact A belian group with character
group X. The/ollowing conditions are equivalent"
(i) G is solenoidal;
(ii) G is connected and ro (G) c;
(iii) X is isomorphic with a subgroup o! Ra"
(iv) X is torsion-/ree and X < c;
(v) X is torsion-[ree and r0(X ) :< c. 1
Proof. Suppose that (i) holds, and let 9 denote a continuous homo-
morphism of R onto a dense subgroup of G. A character Z X is com-
pletely determined by its behavior on 9(R), and for zX, Z o9 is a
continuous character of R. If 2: and Z2 are distinct characters on 9 (R),
then obviously Z1 o and Z. o are distinct characters on R. The mapping
Z-->Z o 9 is therefore an isomorphism of X into the character group of R,
namely R (23.27.e), so that (iii) holds.
Suppose next that (iii) holds, and that z is an isomorphism of X
into Rd. For every R, the mapping z-+exp iz(Z)] is a character
of X, and so (24.8) shows that there is a unique point 9 () in G for which
expiz(Z)]_-:Z(q()) for all Z X. It is easy to see that q is a contin-
uous homomorphism of R into G" q is just the adjoint homomorphism
As in (25.3), ro(X) denotes the torsion-free rank of X.
410 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 411
of the homomorphism z. If (R) were not dense in G, there would be a
character 0X such that 0((R))--(l) and Z0q=l. Since z is an
isomorphism, we have z (0) q= 0 and hence exp i 0 (20)1 q= 1 for some
0R. This contradiction proves (i).
Since R is isomorphic with Qc., the equivalence of (iii), (iv), and (v)
follows readily from (A.16) and (A.14). The equivalence of (ii) and (iv)
follows from (24.15) and (24.25).
Theorem (10.t 3) shows that the a-adic solenoids defined in (t 0.12)
are solenoidal groups.
The following result is analogous to (25.2).
(25.19) Theorem. The group (Z,,) , where a=(2, 3, 4 .... ), is the
largest compact solenoidal group: a topological group G is a compact
solenoid i/ and only i/ G is a continuous homomorphic image o/
Proof. This follows at once from (25.t8) and (24.11), since the
character group of Ra is topologically isomorphic with
(25.20) Theorem. A locally compact Abelian group G is connected
i/ and only i/ the smallest closed subgroup o/ G that contains all one-param-
eter subgroups, say H, is G itsel/.
Proof. Let C be the component of e in G. Then C contains all one-
parameter subgroups of G and hence C D H. Thus C =G if H=G.
Conversely, suppose that G is connected. By (9.t 4) G is topologically
isomorphic with R'*xF, where F is compact, connected, and Abelian.
If the subgroup generated by one-parameter subgroups of F is dense
in F, then the same is obviously true of R"xF. Thus we may suppose
that G is compact as well as connected. Let X be the character group of G.
Assume that Hq= G. Then there is a character Z0 X such that Zo (H) -- {1 }
and Zoq= t. Since G is connected, Zo has infinite order (24.25), and there
is a real character z of X such that z(Z0)q=0 (24.34). Now for every
R, the mapping z-+exp[iz()] is a character of X and so
expiz(Z)]=Z(()), where is a continuous homomorphism of R
into G. For Z=Zo, we get Zo(())=exp[iz(Zo) , and since z(Z0)q=0,
there is an o R such that Zo ( (o)) q= 1. Since (R) is a one-parameter
subgroup of G, we have a contradiction.
We will now determine completely the algebraic structure of compact
Abelian groups. We require a lemma, which is of interest on its own
account.
(25.21) Theorem. Let H be a compact A belian group that is a pure
subgroup o/ some A belian group G. Then H is algebraically a direct
/actor o/G: G is isomorphic with Hx(G/H).
Proof. We emphasize that the direct product decomposition Hx(G/H)
is not topological" G need not have any topology at all. Let A be a
subset of G containing just one element in each coset xII. It is easy to
see that G admits H as a direct factor if and only if A can be chosen
to be a subgroup of H. For, if A is a subgroup of G, then HA--G and
H A = {e}, so that (2.4) applies. The converse is obvious.
For an A as just described, let a--, and consider the compact
group H a, which we will write as P H(,), where each H{,> is equal to H.
aA
For (x,)n a, let ў((x,)) be the set {axgX:aA}. It is plain that ў((x,))
contains exactly one element of each coset xH and that each set selecting
just one element from each coset xH can be obtained from a fixed A
as some #((x,)). Now, for a subgroup G o of G such that H C G o c G, let
g(G0) be the set of all (x)--H a such that ((x))G0 is a subgroup
of G o. As pointed out above, T(G0) is nonvoid if and only if H is a
direct factor of G o .
Given a, b A and (x) H a, consider (ax ) (bx). This element lies
in a coset cH, where c A is determined by a and b alone. Thus (x,) is
in g(Go) if and only if (ax ) (bx)--cx for all a, bA fG o. This
condition is equivalent to x, x x-X= a b c -. Hence we infer at once that
T(Go) is a closed subset of H
Let {G,},e be the family of all subgroups of G such that H G, G
and G,/H is finitely generated. It is obvious that G--tO G, and that
I '
for every {, ..., ,} C I, there is an 0 I such that G,0 D G,,U... O G,,,.
It is also apparent that g(G)=,9 (G,). By (A.27) and (A.22), H is a
direct factor of each G,; that is, g(G,) q=D for each I. We infer at
once that g(G,) q= D, since H a is compact. Therefore (G) is nonvoid,
and H is a direct factor of G.
(25.22) Theorem. Let G be a compact O-dimensional Abelian group.
Algebraically, G is isomorphic with a group
(i) Pp VA"x P Z (p',)
where P is the set o/all primes, a is an arbitrary cardinal number [pos-
sibly 0, I is an arbitrary index class [possibly void], and each r, is a
positive integer. Also every group o/ the ]orm (i) is [topologically] iso-
morphic to a compact A belian group.
Proof. Let X be as usual the character group of G. Then X is a
torsion group, and hence is the weak direct product of its p-primary
components X (A.3). By (23.21) and (24.8), G is topologically iso-
morphic with the complete direct product PpH, where H is the
character group of Xp. We may therefore restrict ourselves to the case
412 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 413
in which X itself is p-primary for some prime p. By (A.24), X contains
a subgroup B such that:
B is isomorphic with a weak direct product P*z(p',)" (1)
B is a pure subgroup of X; (2)
X/B is divisible. (3)
Now consider the subgroup A(G, B) of G. Since (2) and (3) hold,
(24.46.b) implies that A (G, B) is a pure subgroup of G. Since A (G, B) is
compact, (25.2) shows that A(G, B) is a direct factor of G. Thus G is
algebraically, although not in general topologically, isomorphic with
A(G, B)x(G/A(G, B)). The group A(G, B) is topologically isomorphic
with the character group of X/B. In view of (3) and the hypothesis that X
is a p-primary group, X/B is a weak direct product z(p°°)ap* (A.14).
Its character group is thus the full direct product Ap. The group
G/A(G, B) is topologically isomorphic with the character group of
A(Ч, A(G, B))=B. In view o* (), G/A(G, B)is topologically isomorphic
with P Z (p',). Hence G has algebraically the form (i)
iv •
The last statement is trivial: giving A p its usual topology and Z (p,)
the discrete topology, we see that each group (i) is a compact group.
(25.23) Theorem. Let G be a connected compact Abelian group, with
ro (G)=In. Then G is algebraically isomorphic with
(i) 2*Ч *z
pp '
where P is the set o/all primes, and each cardinal number bp is/inite or is
2 ep/or an i/inite cardinal ep<_In.
Proof. Since G is connected and compact, its character group X is
torsion-free and G itself is divisible (24.25). Theorem (A.14) implies that
G is isomorphic with a weak direct product
2* Ч * z (p)*. ()
PEP
We must show that n and bp are as described.
We begin by identifying bp for a fixed prime number p. Consider the
group X/X Ip/, where XIP/= {ZP:X X} (A.2). All elements of this group
save the identity have order exactly p note that p is a prime. Hence
X/X Ip/is a weak direct product Z (p)ep., where ep is a cardinal number,
possibly 0 (A.25) Iwe set ep=0 if and only if X= XIP/1. Since z(p) ev*
has cardinal number pep if ep < g0 and ep if ep >= g0, we see that ep--< X-- m.
By (24.22), we have G{p}=A(G, X/P/), and so G{p is topologically iso-
morphic with the character group of X/X Ip/. Since p is a prime, G{;
is exactly the subgroup of G of elements of order I or p. The character
group of Z (p)ep. is Z(p)ep, and so there are in G exactly pep elements of
order I or p. In the group (1), it is clear that there are p elements of
order I or p if bp is finite and bp such elements if bp is infinite. It follows
that bp = ep if lop is finite and b; = pe= 2ep if bp is infinite.
We next show that the cardinal number n in (1) must be 2 m. Let B o
be a maximal independent subset of X (A.10); thus B0=ro(X). Let
be the smallest subgroup of X containing B 0. Then Bt is a free Abelian
group, isomorphic with Z '°/x/*. For every Z6 X, Z q= 1, there are nonzero
integers n, m, m,., ..., m and elements , , ..., B o such that
, ,2 *. If we assign to Z the s-tuple ( , ) and the
(.m rn) of nonzero rational numbers, then Z is uniquely
s-tuple
\
/
determined by them; this is true because Ч is torsion-free. It follows
that ro(G)--In=X=max(g0, r0(X)). The character group of B is
T '0Ix/. A routine calculation shows that the torsion-free rank of T
is equal to (c)'°lx) = (2°) '°Ix) = 2 °" r0(X)__ 2max (0,,0(X)) __ 2 m. Since B is a
subgroup of X, the character group T '°lxl of B is topologically isomorphic
with G/A(G, B,). It follows that n=ro(G)>=ro(G/A(G, B))=2 m. Since
G= 2 m (24.47), we must have n-----2 m.
(25.24) Theorem. Let In be an in/inite cardinal number, and let
., , 3s, , ,... be cardinal numbers satis/ying the restrictions o/
(25.23). Then there is a compact connected Abelian group G with weight In
that is algebraically isomorphic with the group (25.23.i).
Proof. Let p be an arbitrary prime number, and let Lp be the addi-
tive group of all rational numbers m/n, where m is an arbitrary integer
and n is a nonzero integer prime to p. Let cp= 3p if 3p is finite and
cp= ep if bp is infinite and so equal to 2 e. Let X be the discrete group
(pP L p*) x Qm*. Let bp be the increasing] sequence of all integers greater
than I and prime to p, and let a=(2, 3, 4, ...). The character group,
say G, of X is topologically isomorphic with (P,L',) x' m. Since X--In,
P
we have r0 (G)= X = In, and Theorem (24.25) implies that G is connected.
Obviously we have r 0 (G) _<_ G = 2 m. Since Q is divisible, its character
group ' is torsion-free (24.23) and hence 2m= cm= (L' m) = max(g 0, r0('m))
=ro(X*)<=ro(G). Since G is divisible (24.25), it is the weak direct
product of copies of Q aild Z (poo) for primes p, aild tile number of cop-
ies of Q is exactly r0(G)=2 m. As in the proof of (25.23), in order to
infer that the decomposition of G contains 3p copies of z(p°°), we need
only to show that X/X Ipl is isomorphic with z(p) *. If q is a prime
distinct from p, then LP/=Lq, as is easy to see, and of course Q{pI= Q.
It follows that X/X (p) is isomorphic with (Lp/LP)) p*. For n prime to
The construction used here is due to R. S. PIERCE [oral communication].
ў14 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems
p, and m any integer, there is exactly one k6{O, 1, ..., p--l} such
that m--nk is a multiple of p. Thus each coset of L p) in Lp contains
exactly one k. It follows that L/L ) is isomorphic with Z (p), so that
X/X () is isomorphic with Z (p)c..
Theorems (2.22)-(25.24) can be combined to give a complete
characterization of the algebraic structure of compact Abelian groups.
(25.25) Theorem. Let P be the set o/all primes. For all p P, let ap
be an arbitrary cardinal number, possibly O, let I be an arbitrary index
class, possibly void, and let r, be an arbitrary positive integer/or each t I.
Let It be a cardinal number that is 0 or 2 /or an infinite cardinal m. For
all p P, let bp be a cardinal not exceeding It such that is finite or has
the/orm 2 e /or an infinite cardinal e m. Every compact A belian group
is algebraically isomorphic with a group
(i) ўPpAx P z(p',)x P*Z(p°°)*xQ *.
ўI pP
Also every group (i) is algebraically isomorphic with a compact Abelian
group.
Proof. Let G be a compact Abelian group. Then the component C
of e in G is the largest divisible subgroup of G (24.24), and so is an
algebraic direct factor of G (A.8). Thus G is isomorphic with (G/C)xC.
Since C is a connected compact group, it is isomorphic with a group
(25.23.i). Since G/C is 0-dimensional [(7.3), (5.22), and (3.), it is
isomorphic with a group (2.22.i). Hence G is isomorphic with a group (i).
Given a group of the form (i), it is isomorphic with the direct product of
two compact groups, as (25.22) and (25.24) show.
Miscellaneous theorems and examples
(25.26) Characters of . (a) If Q has its relative topology as a
subspace of R, then every continuous character of Q has the form
r--+exp(iry) for some yR. [By (23.30), every continuous character of
Q has a continuous extension over R.
(b) Let Z' denote the character group of Q as in (25.4), a-- (2, 3, 4,...).
The group Z is topologically isomorphic with [0, t[xfl when it is
made into a topological group as in (10.t 5). Let H denote the subgroup
of [0, l[xfl of all elements (, e) such that either x--0 for all but
finitely many n or else x,--n+ 1 for all but finitely many n. [This
subgroup is the dense subgroup (Rx{0)) described in the proof of
(10.13). Then H consists of exactly those elements of whose corre-
sponding characters are continuous when Q is regarded as a subspace of R.
[The proof of the bracketed remark is elementary and is omitted.
Suppose that (e, e)[0, t[xfl and that x--0 for nn o. If 2: is the
character of Q corresponding to (,e) and t is the number
-- (x0+2!x+... +n0!x0_), then using (25.4.2) we find that
z(r)--expI2ritr for all r6Q, and hence 2 is continuous. Since the
continuous characters form a group, every element in H corresponds to
a continuous character.
Now consider a continuous character 2 of Q such that 2(r)--
exp 2ritr for all r6 Q, where t is a negative real number. Then for
some value of in I0, 1 and some sequence {x0, x x .... , x0_} of inte-
gers where 0 x._<__ '+ 1, we have t---- -- (x 0+ 2 ! xx +..- + no! x0_). If
we set x.--0 for ]'_>__n o, then the character 2 corresponding to (,
satisfies the identity 2 (r) -- exp 2 r i t r for re Q. Since H is a group,
every continuous character of Q corresponds to an element in H.J
(c) (HALMOS I1.) The character group Z of Q is algebraically iso-
morphic with the additive group R a--(2, 3, 4 .... )1" [By (24.23), 2: is
a divisible torsion-free group. By Theorem (A.14), Z' is algebraically
the weak direct product of a certain number of copies of Q. Since the
cardinal number of Z is c, exactly c copies of Q are needed. The group
R is also of this form; thus the two groups are isomorphic. Hence we
can impose a topology on R making it into a compact Abelian group.J
(d) The character group Z of Q is connected but is not the union
of its one-parameter subgroups. [Every real character of Q has the form
r-+tr for some tўR. If L' were the union of one-parameter subgroups,
then (24.43) would imply that every character of Q has the form
r--explitr , tўR. This is plainly false" see for example (b).
(e) There is a character 2 of R such that 2(x) has infinite order in T
for all x@0. Such a 2; is necessarily discontinuous" see (23.27.e). Let
H be a Hamel basis for R over Q such that rCH. Let be a one-to-one
mapping of H onto H Vl{r)'. Let 2 be the character of R such that
(rh)--exp(ir(h)) if hH and rQ. A number in T has finite order
if and only if --exp(2ris) for some sўQ. Suppose that xўR and
.(x)--exp(2ris). Writing x----rxhx+...+r,,h,, where high and riўQ,
we have exp [i (rx z (hx) +... + r z (h,) -- 2 z s)l -- 1. Thus for some integer
k, we have rxz(hx)+...+rz(h)--z(2s+2k)=O. It follows that
rx--r. ..... rm--O , and hence x--0.
(25.27) Generators of monothetic groups (HALMOS and SAMEL-
SON ). (a) Let G be a compact, connected Abelian group such that
ro(G)-- 0, and let S be the set of all generators of G (25.11). Then S
is it-measurable and it (S)--1. Since ro (G)--0, X is countably infinite"
X----{I, Z, Z., Z, .-., Z,-..}. Let H,--{xG'z(x)--I }. Then H is a
closed subgroup of G and G/H,, is infinite because G is connected. There-
fore (H) ----- 0, and so (H,) -- 1. It follows from (25.11) that S-- V) H.
n=l
(b) The set S fails to be ).-measurable for some compact connected
monothetic groups. [Consider any group T m, where o'< mc, repre-
416 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems 417
sented as in (25.t5). It follows easily from (25.11) that (a,) T m belongs
to S if and only if a,= exp [2ait, where the numbers t, are distinct
and the set {t," tI}U {l} is rationally independent. As in (16.13.f) we
see that 2 (F)--0 if F is any compact subset of S, since S C {(x,) • x,:--
for all tI}. Now assume that F is a compact subset of S' such that
2(F)>0. Then by (19.30.b) and the argument used in (16.13.f), there
is a subset D of F such that D is 2-measurable, 2 (D)> 0, and D has the
following property" there is a countable subset X of I such that (x,) D,
(y,) T 1", and x,=y, for tE imply (y,)D. Thus the set D is "restricted"
only on the countable set of coordinates X. The set az(D) [az is the
projection map defined as in (6.8) has measure in P T(,) equal to (D).
Hence ax(D) contains a generator (a,),x of ,PxT(,) Isee (a). It is easy
to extend (a,),x to a generator (a,),x of Tm. Hence we have D -S', a
contradiction.3
(c) The group z(p') has p,_p,-a generators, since the congruence
ax=b (rood p') has a solution x for all b if and only if a and p', hence a
and p, are relatively prime. Thus the measure of the set of generators
1
of Z (p') is 1 -- -. Consider next the group A. It follows from (10.16.a)
that the set S of generators of A is A[. The measure of A 1 in A is
t
Isee (t 5.17.k), so that 2 (S) = 1 -- -. Now consider an arbitrary 0-
dimensional compact monothetic group, written as PpA, where each
A is as in (25 t6) By (23 2t) every continuous character of P A
• . . , pp P
different from I has the form (x)-+Z (x).. "Z (x), where (pa, p,...,
is a nonvoid subset of the set of primes for which A: (e), and each
is a continuous character of A different from 1. Let S denote the set
of generators of A, and S o the set of generators of P A. It is clear
PP
from (25.tt) and the preceding two sentences that So CPpS . On the
other hand, if (x)P S, and Z is a continuous character of P A
different from t, then )((x)) = exp 2 i a ч P- ч"" ч , where
a. is prime to Pi [see (23.27.c) and (25.2.2). Since the primes p. are
distinct, it is easy to see that )((x))=t=l. Thus S=PS, and so
2 (S,)=M (1 -- r--Z)' the product being taken over all the primes p for
pP
1
which A {e}. Since -=, we can select the A's so that 2
pP
is any number in the interval [0, 1 [.
(25.28) More about A a and Z (a). (a) Let a= (a0, a, a,...) be
an arbitrary sequence of integers greater than 1. Let P be the [perhaps
void set of all prime numbers p such that for every power p"there is
a k such that P" aoCfi"'a," Let P be the Iperhaps void3 set of all primes
not in P that divide some number aoaa...a . For pP, let r be the
largest power of p such that p'[ aOal...a for some k. Then Aa is topo-
logically isomorphic with pP,P A><Pp, Z (p'). Also Z (aoo) is isomorphic
with P*Z(poo)>< P*z(p') [Argue as in the proof of
P Pt P P, •
(b) Let a--(a o, a, a. .... ) and b--(b o, b, b. .... ) be arbitrary se-
quences of integers greater than t. The following statements are equiv-
Merit :
(i) A and A are topologically isomorphic;
(ii) A and A are algebraically isomorphic;
(iii) Z (a ) and Z (b ) are isomorphic;
(iv) Z (aoo)--Z (b°°);
(v) for every prime power p', p' divides some aoa...a if and only
if it divides some bobs.., b.
[By the duality theorem (24.8), (i) and (iii) are equivalent. It is
obvious that (i) implies (ii) and that (iv) implies (iii). It is easy to verify
that (iii) implies (v) and that (iv) and (v) are equivalent. It suffices then
to show that (ii) fails whenever (iii) fails. If Z (aoo) and Z (boo) are not
isomorphic, then for some prime p, the p-primary factors of Z (aoo) and
Z (boo) are different. Hence the corresponding factors of A and A b are
different. If Z(p') is a factor of A, and A, Z(pS), or {e} is the corre-
sponding factor of A b where s< r, then A has an element of order
and A b does not. If A is a factor of A and {e} is the corresponding
factor of A, then AP) A, and A p)-- A.
(25.29) (BRACOIER I1, p. 41.) Let G be a locally compact Abelian
group such that every element of G different from e has order p, where p
is a fixed prime. Then G is topologically isomorphic with
(i) Z (p)mxZ (p)*,
where m and rt are arbitrary cardinal numbers, Z(p) m has its usual
compact topology, and Z (p)u. is taken discrete. Conversely, every group
of the form (i) is locally compact and Abelian, and every element except
e has order p. [By (24.30), G contains a compact open subgroup H.
By (2.9), H is topologically isomorphic with z(p) m for some cardinal
number m. It is clear that if xH and k is an integer prime to p, then
there is an element yH such that yk-=-x. Consequently H is a pure
subgroup of G and is algebraically a direct factor of G (25.2t). It is
easy to see that H, being open, is also a topological direct factor of G
(6.22.a). Since G/H is a discrete Abelian group all of whose non-identity
elements have order p, G/H is isomorphic with Z (p)u. for some cardinal
number n (A.25).]
Hewitt and Ross, Abstract harmonic analysis, vol. I 27
418 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 419
(25.30) Remarks on direct factorization. (a) Let X be a discrete
Abelian group and let Y be a subgroup of X such that X/Y is isomorphic
with Z"* for some nonzero cardinal number a. Then Y is a direct factor
of X" X is isomorphic with Yx(X/Y). ISince X/Y is torsion-free, Y is
evidently a pure subgroup of X. Now apply (A.22).
(b) Let G be a compact Abelian group and H a closed subgroup of G
such that G/H is torsion-free. Then H is a topological direct factor of G"
G is topologically isomorphic with Hx (G/H). ILet X denote the character
group of G. Then A(X, H), being isomorphic with the character group
of G/H, is divisible (24.23) and hence is a direct factor of X" X is isomorphic
with A(X, H)xB, where B is a subgroup of X (A.8). It follows that B
is isomorphic with X/A (X, H), so that B is isomorphic with the character
group of H. Hence G is topologically isomorphic with (G/H)xH.
(c) Let G be a locally compact torsion-free Abelian group, and let C
be the component of e in G. Then C is a topological direct factor of G.
Suppose first that G contains a compact open subgroup H. By (7.8),
C is a subgroup of H, and so is compact. By (25.8), H is topologically
isomorphic with a group Z'xpPpA
p, and C is topologically isomorphic
with Z' m. Thus C is a topological direct factor of H, and there is a con-
tinuous homomorphism zr of H onto C whose restriction to C is the
identity map. As C is divisible (24.25), zr admits an extension zr' over
G that is a homomorphism of G onto C and is of course the identity on C
(A.7). Since :r is continuous on the open subgroup H, 7r' is continuous
on G. By (6.22.a), C is a topological direct factor of G.
For an arbitrary G, use (24.30) to write G as R"ЧG1, where G1
contains a compact open subgroup and is plainly torsion-free. Let C 1
be the component of the identity in G t. Then G is topologically iso-
morphic with R'*xC1x(G1/C1), and it is easy to see that R"xC1 is the
component of the identity in
(25.31) Subgroups /' as direct factors. An idea used in proving
(25.2t) can be used to show that subgroups T" are topological direct
factors of locally compact Abelian groups.
(a) Let G be a compact Abelian group, a a nonzero cardinal, and sup-
pose that G contains a subgroup H topologically isomorphic with T a.
Then H is closed, and G is topologically isomorphic with H x(G/H).
Informally, we write" subgroups T a are direct factors. Since compact
subsets are closed in Hausdorff spaces, H is a closed subgroup of G.
Let X and Ґ be the character groups of G and H respectively. Then Ґ
is isomorphic with X/A(X, H) and also with Z a*. By (25.30.a), A (X, H)
is a direct factor of X" X is isomorphic with A (X, H)xX/A (X, H) and
hence with A(X, H)xY. If we regard G as the character group of X
and use (23.a8) and (23.25), we see that G is topologically isomorphic
with (G/H)xH.
(b) Let G be a locally compact Abelian group containing a subgroup
H topologically isomorphic with T a for a nonzero cardinal number a.
Then H is a topological direct factor of G. By (24.30), G is topologically
isomorphic with R"xGo, where G o is a locally compact Abelian group
containing a compact open subgroup F. By part (a), H is a direct
factor of F; thus there is a continuous homomorphism zr of F onto H
whose restriction to H is the identity map. Since H is divisible, :r admits
an extension r' of G o onto H that is a homomorphism and is the identity
map on H. Since zr' is continuous on the open subgroup F, it is continuous
on G 0. By (6.22.a), Go is topologically isomorphic with Hx (Go/H), and
hence G is topologically isomorphic with Hx (Go/H) xR'.
(c) Theorem (25.2a) gives rise to the conjecture that a locally compact
Abelian group with a compact open subgroup might be the direct product
of a compact group and a discrete group. This is disproved by the
group Ј2 for an arbitrary prime p. Every compact subgroup of ,Q is a
group A k for some integer k (10.16.a), and no A is pure in f2, let alone
a direct factor.
(25.32) Minimal divisible extensions. It is proved in (A.15) and
(A.a6) that every Abelian group G can be imbedded in a minimal
divisible group E, and that E is unique up to an isomorphism leaving
the elements of G fixed. That is, E is as unique as it can possibly be.
We examine here a few of these minimal divisible extensions and a
natural topology for them.
(a) Let G be a topological Abelian group and E the minimal divisible
extension of G. Let be an open basis at e in the group G. Let be
taken as an open basis at e for the larger group E. As noted in (4.8.h),
E is a topological group containing G as an open subgroup.
(b) The minimal divisible extension of A is f2p, and for a finite
cardinal number n, the minimal divisible extension of A is f2. The
topology described in part (a) coincides with the usual topology of
when A and A are given their usual topologies. The set is a field
(a0.al) of characteristic 0 and hence is divisible. Therefore f2 is a
divisible group containing A. For every (x, x2,..., x,,)--xcO, there
is a nonnegative integer k such that pe is in A. It follows that
/3 is a torsion group. Since is torsion-free, (A.a7) shows that
f2 is the minimal divisible extension of 3. The remarks about the
topology of f2 are obvious, l
1 \Ve lapse into additive notation in dealing vith Y2, since the group operation
we are concerned vith in Y2p is addition.
27*
420 Chapter VI. Characters and duality of locally compact Abelian groups §25. Special structure theorems 421
(c) The strict analogue of (b) fails for in[inite cardinal numbers
The group in this case is of course a divisible supergroup of A, but
it is too large. Consider Q as the group of all p-valued functions (x,)
on a set I such that I--n. Let p be the function on defined in
(10.4). For (x,),Q, let v((x,))--sup((x,, 0)'tI}. Then the minimal
divisible extension of A is the subgroup of ,Q consisting of all (x,) for
which v ((x,)) is finite. We denote this group by the symbol Q'. Note
'-- U A where A is as defined in (10.4). It is easy to see
that 2
that ' is a divisible extension of A and also that p(x,)A if (x,) '
and k is a suitably selected nonnegative integer. Then argue as in part (b).
(d) Consider any group pPpA , where P is the set of all prime integers,
and n., n3, ns, nT, ... are arbitrary cardinal numbers. 1 The minimal
divisible extension E of P A u is a certain subgroup of P/Q' identified
pEP p P '
as follows. Write a generic element of this group as (x 2, x3, x s, x,...)
where x.Q '. Our group E is the set of all (x2, x, x s .... ) such that
all but a finite number of the x's lie in A . Suppose that/2 ' is topol-
ogized as in part (a), where A is furnished with its usual topology, and
that is topologized as in part (a). Then E is the local direct product
of the groups Y)' relative to the open subgroups A (6.t6). lit is easy
to see that E/pPpA" is a torsion group. In fact, if (x., x, xs .... )E,
xA" for P>P0, and v(x)--<_2 for p=2, 3, 5, ..., p0, then
(2.3" '" "Po) (x., x, xs,...) is in PA'. We must also show that E is
divisible. This fact rests upon a simple property of the fields Y2p. Let
/= (Y)F---oo be a nonzero element of Y2, let y,,,=t=O, and y=0 for l < m.
That is, a (/, 0) = 2 -". Then for every prime q 4= p, we have a - /, 0 =
(' ) 2 -'+ .
(/, 0), and % -/, 0 = Let (x., x, xs,...) be an element of E,
and suppose that xA ' for p>p0. For every prime q, the equation
(x., x3, xs, ...)= (qy., qy, qys .... ) has a solution for which v(y) =v(xp)
if p=q and ve(ye)=2ve(xe). Thus we have yA" for all p>max(Po,q).
Since q is an arbitrary prime, E is divisible.
(e) Consider two sequences of cardinal numbers n., n, n, n7 .... and
g, g, , g,.., as in part (d). Form the groups PvA and P A,
p PEP v
and the minimal divisible extensions E and E, respectively, of these
groups. Suppose that E and E" are topologically isomorphic. Then
n-- for all primes p. [The subgroup {(0, ..., 0, x, 0, ...)"
of E is exactly the set of all x in E such that lim p'x=0. Hence if E
and E" are topologically isomorphic, so are/2 ' and .Q'. Let be a
topological isomorphism of f2 ' onto f2'. Regard these groups as linear
t We make the convenient convention that A is the one-element group.
spaces over the field . Then it is easy to see that z is a linear mapping.
If n is finite, then n is the dimension of ' as a linear space over D,
and so n = gp if either n or g is finite. If n and g are infinite, consider
any compact open subgroup F of '. This subgroup is a finite union of
cosets of a group topologically isomorphic with A. The character group
of A is isomorphic with Z(p°°) n*, and (24.1 )implies that ro(A ) =
Z (po). = n. It follows that ro (F) = np, and we infer with no hesitation
that n = .
(25.33) The structure of torsion-free divisible groups (adapted
from MActўҐ [1 ). (a) Let G be a locally compact, divisible, torsion-free
Abelian group. Then G is topologically isomorphic with a group
(i) R'><>< Qr*><E,
where" 1 is a finite cardinal; q and r are arbitrary cardinals ; =
(2, 3, 4, 5,...); Qr. is topologized discretely; andE is the minimal divisible
extension of a certain group P,A', constructed in (25.32.d). Conversely,
every group of the form (i) is locally compact, divisible, torsion-free,
and Abelian. [By (25.30.c) G is topologically isomorphic with a group
RxCxG, (1)
where Ca is compact, connected and torsion-free and G is the quotient
group of G by its component of e. By (25.8), C1 is topologically iso-
morphic with for some cardinal q. We are left with the 0-dimensional
see (7.3) and (3.5)] group G. It is plain that G2 is also divisible and
torsion-free. Let H be a compact open subgroup of G (7.7). If H is the
identity, then G is discrete and is topologically isomorphic with Qr*
for some cardinal r (A.t4). If H is not the identity, then it is an infinite,
0-dimensional, compact, torsion-free group and necessarily has the form
PA (25.8). The intersection E o of all of the divisible subgroups of
G. that contain H is again a divisible subgroup of G. This follows from
the fact that G2 is torsion-free. It is trivial that there is no divisible
group D for which H D E 0 . Thus E o is a minimal divisible extension
of H and therefore is isomorphic as a group with the group E constructed
in (25.32.d). Since H is open in G., it is open in E 0 and so E 0 is topologi-
cally isomorphic with the group E of (25.32.d). As E 0 is operf in G.
and divisible, it is a topological direct factor of G (6.22.b): G is topo-
logically isomorphic with
(G2/Eo) xEo; (2)
and G/E o is discrete, divisible, and torsion-free. By (A.14), G/E o is
isomorphic with Qr. for some cardinal r. Combining (1) and (2), we
obtain (i). The converse is trivial. 1
We again agree that a group to the zero power is the one-element group.
422 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems
(b) Two groups G--RxxQr*xE and U-- Rx2:xQZ*x. of the
form (i) are topologically isomorphic if and only if p--,
and np--p 1 for p--2, 3, 5, 7, ... • lit is convenient to examine only one
group and to show that the numbers p, q, r, and np are defined in terms
of the structure of G; thus they must be preserved under a topological
isomorphism. The number p is the largest integer n such that G contains
a topological isomorph of R (9.t4). The subgroup Rx is the com-
ponent C of e in G, and ' is the subgroup B 0 of compact elements of C.
Since q is the torsion-free rank of the character group Qq* of ', q is
well defined.
It is evident that every element of E lies in a compact subgroup of E.
Hence the character group Ґ of E is 0-dimensional (24.t 8). The character
group X of G is topologically isomorphic with RxQ * x.F.r, xY.
Arguing as in the case of the cardinal q above, we see that r is well
defined. Obviously the group B of compact elements in G is 'xE.
Thus E is topologically isomorphic with BIB o, and hence is well defined
up to topological isomorphism. The uniqueness of the cardinal numbers
n2, n3, n5 .... that define E was proved in (25.32.e).
(25.34) Groups topologically isomorphic with their character
groups. If a locally compact Abelian group G is topologically isomorphic
with its character group X, we will say that G is sell-dual. The group
GxX, for arbitrary G, is a trivial example. If G1 and G. are self-dual,
then so is GIXG..
(a) Examples of self-dual groups already met with in the text are
the following"
(i) a finite Abelian group G (23.27.d);
(ii) R (23.27.e);
(iii) Y2 a where a={a}°°___oo is such that a=a_ for all kZ (25.1).
(b) More examples of self-dual groups are found from (23.33). Let I
be an index class. For each tI, let G, be a locally compact self-dual
Abelian group with character group X,. Suppose also that there is a
topological isomorphism z, of G, onto X, and a compact open subgroup
H, of G, such that z, (H,) = A (X,, H,). Then H, is topologically isomorphic
with the character group of G,/H, (23.25). Now form the local direct
product G of the G, relative to the compact open subgroups H,. It
follows immediately from (23.33) that G is self-dual.
(c) For a compact Abelian group H with character group Ґ, the
group HxY is a trivial example of a group satisfying the requirements
of part (b). Another example is the group Z (p"), written as {0, 1, 2, ...,
p"--1}, with the subgroup {0, p', 2p', ..., (p'--1)p'} [p a prime, r a
We write for the minimal divisible extension of pPepzJ,.
423
positive integer. Still another example is provided by any group Y2,
and its compact open subgroup A,, where r is an integer greater than 1.
(d) The foregoing examples illustrate the wealth of self-dual groups.
No complete classification of these groups is known to the writers.
(25.35) Products of {0, 1} again (adapted from VILENKIN [7]).
The following result is related to Theorem (9.15). Let G be a compact
Abelian group. For some cardinal number m, there is a continuous
mapping of {0, }m onto G; In can be taken to be max JR0, r(X)], where
X is the character group of G. [We break the proof up into two steps.
(I) If Y is a compact metric space, then there is a continuous mapping of
{0, }0 onto Y. This result and its proof are similar to the contents of the
footnote on page 98. The space Y is the union of closed sets B 1 .... , B=,
with nonvoid interior such that the diameter of each Be is less than 1.
Each Be is the union of a finite number of nonv0id closed sets BCx, Bee,
..., B [a 2 is the same positive integer for each i], where each Bii has
diameter less than . Continuing this construction in the indicated
manner, we find a sequence {a,} of positive integers and nonvoid
closed subsets Bi,ў,...ў of Y such that Bqi,...ў has diameter less than
2 -''+ B,...+ is a subset of Bi,... and U{Bi,,...'l<i<a
k= 1,..., m}= Y for each m. For each sequence i=(ix, i2,...), where
lia for k=l, 2, ..., define (i) to be the unique point in
Bў,,...ў. Then is a continuous mapping of .1{1 am} onto Y.
m=l " * "
The last two paragraphs of the proof of (9. 5) prove that , ..., am}
is homeomorphic with 0, }°.
(II) Let G be a compact Abelian group. The discrete group X can be
imbedded in a divisible group Y having the same rank as X see (A. )
and (A.6). By (A.4), Y is isomorphic with
e* (z
pp '
where P denotes the set of all primes. The character group H of Ґ is
topologically isomorphic with (Za)'°(Xlx P (/l),lx/, as (23 22) (2.)
pp • , ,
and (25.2) show. By (24. ), the character group G of X is topologically
isomorphic with H/A (H, X); that is, G is a continuous image of H.
It suffices then to prove that H is a continuous image of {0, 1) m where
m = max [o, r (X)I.
The groups Z7 a and A; are compact and metrizable by (24.48). Thus
by (I), ZT,, and A; are continuous images of {0, }°. It follows that
H is a continuous image of {0, t}m where m--max[Ro, ro(X ) +r2(X)
+r3(X) ч .... --max[Ro, r(X)l" see (A.2) and (A.,I3).
424 Chapter VI. Characters and duality of locally compact Abelian groups § 25. Special structure theorems 425
(25.36) The uniqueness of /' in the duality theorem (adapted
from PONTRYAGIN I7, Example 72). The choice of T as the value group
for characters of locally compact Abelian groups is natural enough on
various grounds: characters are l-dimensional unitary representations;
characters are generalizations of the famous functions exp (i x)-+exp (inx)
on T and x-+exp (i x y) on R; complex-valued functions are well-known
objects and hence presumably amenable to the techniques and concepts
of ordinary analysis. But even more, no other locally compact A belian group
would have given us Theorem (24.8). In fact, let H be a locally compact
Abelian group, and suppose that Horn (Horn (T, H), H) is topologically
isomorphic with T. Notation is as in (23.34). Thus we suppose much
less than the full duality theorem. It then follows that H-- T.
Let z be any element of Hom (T, H) not carrying T onto the identity
[ of H; clearly such a z exists. The kernel of z is a proper closed subgroup
of T and is hence finite. It follows that z (T) is topologically isomorphic
with T (5.39.j). Let H1 be the connected component of ! in H, and let
E be the largest compact subgroup of H1 see (9.14). Since z(T) is
compact and connected, it is clear that z (T) C E. By (25.31.a), z (T) is
a direct factor of E: E is topologically isomorphic with z(T)xE1, where
E1 is a closed subgroup of E. If aHom(T, H), it is clear that a(T)cE.
Thus replacing topological isomorphisms by equalities and using (23.34.d),
we have
Hom(T,H) -- Horn (T, (T)xE) -- Horn(T, (T))xHom(T,E). (t)
Applying (23.34.c) to (t), we obtain
Hom (Horn (T, H), H) -- Hom (Horn (T, (T)) x Horn (T, E), H) }
-- Horn (Horn (T, (T)), H) x Horn (Horn (T, E), H).
(2)
Since z (T) is topologically isomorphic with T, Hom (T, z (T)) is topo-
logically isomorphic with the [discrete character group Z of T. It is
an elementary matter to verify that Horn(Z, H) is topologically iso-
morphic with H. From (2) and our hypothesis, therefore, we have
T= HxHom (Hom (T, E), H).
()
Obviously the factor H in (3) is a connected subgroup of T. It follows
either that H--{[}, a palpable absurdity, or that H--T, as we wished
to show.
Notes
So far as we know, (25.t) is new. The result of (25.2) is stated by
ABE [1; it is of course only a slight generalization of the fact that the
character group of Ap isZ (p). This fact was published by W. KRULL
and has been published many times since.
As remarked in the Notes to §t0, the group a for a general a--
(a 0, a, a., ...) was constructed by VAN DANTZlG [6]. The fact that
is the character group of Qa if a=(2, 3, 4, ...) was pointed out by
ABE [1] and by ANZAI and KAKUTANI [1. Our computation in (25.4)
seems more explicit than theirs. BRACONNIER 1], p. 19, also mentions
the character group of Qa, but in describing it as a solenoid he refers to
VAN DANTZIG [1], where only the group X{,,,,, .... ) [which is not the
character group of Qa] is constructed. Other writers who at this period
studied the character group of Qa made no mention that it is
Isee for example HALMOS [1 and 1ViACKEY 11. The construction in
(25.5) appears in part in yON NEUMANN I4, p. 477.
The construction of a Hamel basis for R over Q is due, unsurprisingly,
tO G. HAMEL It. The computation of all characters of Ra given in
(25.6) appears in MAAK [1, §2 3, Satz4, pp. 89--90. We have no
reference for (25.7), although its content must be widely known.
Theorem (25.8) is a strengthened form of BRACONNIER It], p. 17,
Corollaire; (25.9) and (25.10) were suggested by a remark of W. RO-
DIN [t , Lemma 3.
Monothetic groups were introduced by VAN DANTZlG [1, and have
been studied by HALMOS and SAMELSON I1], ECKMANN [1, and ANZAI
and KAKUTANI I1. Much of (25.11)--(25.17) is drawn from these
sources. Theorem (25.3) was kindly communicated to us orally by
R. A. BEAUMONT and R. S. PIERCE. The implication (i) implies (ii) in
(25.6) is due to VAN DANTZlG [61; the rest of (25.6) appears to be new.
Essentially all of (25.t8) and (25.9) are due to ANZAI and KAKUTANI
Theorem (25.20) is stated in MACKEY
A large number of writers have contributed to (25.21)--(25.25), and
as frequently happens some duplication has occurred. The problem of
classifying the algebraic structure of compact Abelian groups was raised
by HALMOS 1. Incomplete results were obtained by KAPLANSKY
PP. 55--56; see also BALCERZYK I1. Theorem (25.21) [which is the key
to (25.25) is due to Log 1] and 2. Theorems (25.22), (25.23), and all
but the last sentence of (25.25) are due to HARRISON [1 and HULANICKI
I2 and I31- FUCHS I21 has given a simplified version of HULANICKI'S
construction. Finally, (25.23) was communicated to us orally in May 1959
by S. KAKUTANI as an unpublished result of S. KAKUTANI and T. NA-
KAYAMA. So far as we know, (25.24) is new.
426 Chapter VI. Characters and duality of locally compact Abelian groups § 26. Miscellaneous consequences of the duality theorem 427
The material presented in the text is only a part although we trust
an important part of what is known of the structure of various classes
of locally compact Abelian groups. The reader interested in pursuing
the subject of the present section further should first consult ]3RACON-
NIER [1], Ch. II. There is also a formidable sequence of papers by
N. YA. VILENKIN, of somewhat uneven merit, which should be examined.
These are: VILENKIN
§ 26. Miscellaneous consequences of the duality theorem
In this section we present a number of constructions and theorems
which depend in one way or another upon (24.8). We begin with a
study of automorphism groups, which do not ab initio depend on (24.8)
but whose most interesting application Eamong those given here does:
see (26.10).
(26.1) Definition. Let G be a topological group, and let ®(G) be
the set of all topological automorphisms of G onto itself, that is, the set
of all mappings of G onto G that are simultaneously automorphisms and
homeomorphisms. For , z E ®(G), let o z be as usual the composition
of and z: o z(x)= (z(x)) for all xEG. Let t be the identity mapping
on G: t(x)=x for all xEG. Let z-1 be the usual inverse of z; z-l(x)
is the unique! element yEG such that z(y)=x.
(26.2)It is trivial that ®(G) is a group under the operations described
in (26.1). In fact, ®(G) is a subgroup of the group of all one-to-one
mappings of G onto itself, referred to in (2.8.c).
(26.3) Definition. Let G and ®(G) be as in (26.). For a compact
subset F of G and a neighborhood U of e in G, let (F, U) be the set of
all zE ®(G) such that z(x)EUx and z-l(x)EUx for all xEF.
We will show that for a locally compact group G, the sets (F, U)
satisfy conditions (4. 5 .i) -- (4. 5 .v) and hence define a topology for ®(G)
under which ®(G) is a topological group. A lemma is needed.
(26.4) Lemma. Let F be a compact subset o[ G and U a neighborhood
o] e in G. For each z E 8(F, U), there is a neighborhood V o] e depending
upon F, U, and r. 1 such that Vz(x) c Ux ]or all xEF.
Proof. For xEG, let v/(x)--z(x ) x -1. Then ,p is a continuous mapping
of G into G. Since z(x) x-lE U for all xEF, it follows that ,p(F) is a com-
pact subset of U. By (4.10), there is a neighborhood V of e in G such that
Vo(F)cU. Now if xEF, then Vo(x)cU so that Vz(x) x-U and
V(x) c Ux.
1 The reader should note that z-l(x) is not necessarily the same as z(x-x).
The symbols x -1 and z-1 have different, and unrelated, meanings.
(26.5) Theorem. Let G be a locally compact group. As F runs through
all compact subsets o/ G and U through all neighborhoods o/ e in G, the
sets 8(F, U) o/ (26.3) describe a/amily o/ sets satis/ying axioms (4.5.i)--
(4.5.v) and so in accordance with (4.5) de/ine a topology on ®(G) under
which it is a topological group.
Proof. To verify (4.5.i), let (F, U) be an arbitrary neighborhood
of t; let V be a neighborhood of e in G such that V"C U and V- is
compact. Suppose that r, :E (V-. F, V). Then for all xEF, we have
v
and z(x)E V.Fc V-.F. Hence r (z(x)) z(x)-E V, and so
v .c v.
Similarly we see that
.
is in U for all x EF. Since V-.F is compact (4.4), axiom (4.5.i) is estab-
lished.
Since (F, U)= (93 (F, U))-I, axiom (4.5.ii) is trivially satisfied.
We now verify (4.5.iii). Given a compact set FcG, a neighborhood
U of e in G, and a fixed zE (F, U), let V be a neighborhood of e in G
such that Vz(x)c Ux and V:-(x)c Ux for all xEF see (26.4) above.
Let W be a neighborhood of e such that :(W)C V. Then the inclusion
wn v)c u)
obtains. In fact, let a be any automorphism in 3(FU-I(F), WN Y),
and let x be any element ofF. Then a(x)--wx where wE W, and so we
have
'o o'(x)-- '(w) .'(x) E Vr.(x)C Ux. (2)
Since z -* (x) E z - (F), we also have
r-o -(x) V-(x) c Ux. (3)
The relations (2) and (3) imply (1).
Axiom (4.5.iv) is simple: the inclusion
TO } (T -1 (_'), T-I(U)) o T -1 C } (F, U)
is immediate. Axiom (4.5.v) is also obvious.
It remains to show that ®(G) is a T o group. If in ®(G) is not the
identity automorphism t, then there is an aG such that (a)@a. Since
G is a T group (4.8), there is a neighborhood U of e in G such that
z(a)ўYa. Then 3({a}, U)is a neighborhood oft not containing z. It
follows that ®(G) is a T O group. [
428 Chapter VI. Characters and duality of locally compact Abelian groups § 26. Miscellaneous consequences of the duality theorem 429
(26.6) The group ®(G) need not be locally compact for locally com-
pact G. Even if G is compact and Abelian, ®(G) can fail to be locally
compact" see (26.t8.k).
The two following theorems are of some independent interest but for
our purposes are mainly lemmas leading up to (26.0).
(26.7) Theorem. Let G be a topological group and let H be a locally
compact normal subgroup o/ G. For sG, let s denote the inner auto-
morphism x--,sxs -1 o/G onto G. The mapping
(i) s-->s] H
is a continuous homomorphism o/ G into ®(H).
Proof. It is obvious that (i) is a homomorphism of G into ®(H).
By (5.40.a), it suffices to prove that (i) is continuous at e. Thus let F
be any compact subset of H and U any neighborhood of e in G. Let V
be a neighborhood of e in G such that VC U. Since F is compact in H,
it is obviously compact in G. Use (4.9) to choose a symmetric neighbor-
hood W of e in G such that W C V and xWx-1C V for all xF. We
claim that Os[H is in (F, U ClH) for all sW. In fact, if sW and
x F, we have
qs (X)" X -1-- (SXS -1) X-1--S (ms -1X -1) [{rv C V°C U, (t)
and plainly (sxs -) x - is in H. Since s-l W, we also have
)'1 (X). X,-I=s-(X, ) • X,-I V f3m. (2)
The computations (1) and (2) show that OslH is in
(26.8) Theorem. Let G be a O-dimensional locally compact group.
Then ®(G) is also o-dimensional; in/act, open subgroups [orm an open
basis at t. 1
Proof. Let H be any compact open subgroup of G, and let F be a
compact subset of G such that FDH. If , z (F, H) and xF, then
for some hH we have 3-1(x)=hx and hence o3-(x)=o(hx) =
o(h) o(x)Hh Hx=Hx. Similarly we have
Therefore oo 3-1 3(F, H) and (F, H) is an open subgroup of ®(G).
Now if G is 0-dimensional, F is compact in G, and U is a neighborhood
of e in G, there is a compact open subgroup H of G such that HC U
(7.7). Thus we have (F U H, H) c (F, U), so that open subgroups are
a basis at t.
The next result, also useful for (26.0), is of considerable importance
on its own account.
1 Since (G) need not be locally compact, we cannot use Theorem (7.7)" our
assertion about open subgroups is stronger than the assertion that @(G) is 0-
dimensional.
(26.9) Theorem. Lel G be a locally compact Abelian group wilh
character group X. For every 3 ®(G), lel 3" be the ad]oint homomorphism
de/ined in (24.37), i.e., let 3"(;)(x)=z(3(x)) /or all Z6X and xG. The
mapping
(i) 3-+ 3
is a topological anti-isomorphism o/®(G) onto ®(X).
Proof. As noted in (24.41.c), every 3~ is an element of ®(X). Since
continuous characters of G separate points of G, it is clear that 31
implies 31"4=3. If o is an element of ®(X), then, regarding G as the
character group of X, we see that o- is an element of ®(G), and by
(24.4t.a), (o)-=o. Thus (i) maps ®(G) onto ®(X) and is one-to-one.
For o, 3 ®(G), for Z X, and xG, we have
(z)(x)=z (oo (x)) = z (o = o (z) (x)) = o o~)(z)(x).
Thus (i) is an anti-isomorphism.
It remains to show that (i) is a homeomorphism. According to (24.8),
the sets Kx =lx(x)-al for all X6q)} form an open basis at e in G
as runs through all compact subsets of X and e runs through all posi-
tive real numbers. Thus a generic neighborhood of t in ®(G) can be
described as
gt(F, *, e) = {3 ®(G)" Ix (3(x) x
]2(3-1(X) x-l) - t I <? for all xF and ZO}.. (t)
Here F is compact in G, • is compact in X, and e is a positive real
number. We have also
Z (3(x) x -1) =Z (3(x)) Z (x-l)= 3"(Z)
and similarly
Z ( 3-1 (X) X -1) = ((3") -1 (Z) Z -1) (X).
It follows that gf(F, (i), e) is a set of exactly the same sort as
but in the group (X). Thus (i) is a homeomorphism.
With the foregoing results we can prove an interesting fact about
the centers of connected groups.
(26.10) Theorem. Let G be a connected group and H a locally compact
normal subgroup o/G. Suppose either that H is O-dimensional or that H
is A belian and has the property that every element o/H lies in a compact
subgroup o/H. Then H is contained in the center o/G.
1 For analogous but much simpler facts about the center of a connected group,
see (7.17), P. 65.
430 Chapter VI. Characters and duality of locally compact Abelian groups §26. Miscellaneous consequences of the duality theorem 431
Proof. If H is 0-dimensional, then ®(H) is 0-dimensional (26.8).
If H is Abelian and every element of H lies in a compact subgroup of H,
then the character group Y of H is 0-dimensional (24.t8), so that ®(Y)
is 0-dimensional. By (26.9), ®(H) is also 0-dimensional. By (26.7), the
mapping s-+sIH of G into ®(H) is continuous. Since G is connected,
each s must be the identity on H. That is, sxs -1-- x for all xH and
s G, and H is contained in the center of G.
We now turn to another construction, using subsets of the character
group of a locally compact Abelian group G to imbed homomorphic
images of G in compact Abelian groups.
(26.11) Definition. Let G be a locally compact Abelian group with
character group X, and let i- be any nonvoid subset of X. We define a
mapping b r of G into P T(z ) each T(z ) is a replica of T as follows" for
xEV
each xG, qr(x)--(;(x)) P T(z ) It is very easy to see that b r is a
zEF "
continuous homomorphism, and that qr is one-to-one if and only if V
separates points of G. Now let brG----- (qr(G))- rclosure in P
xEI" "
Such groups bG are called Bohr compactiiications o G. The group
bxG will be written bG.
(26.12) Theorem. The character group o/ brG is topologically iso-
morphic with the subgroup Y. o/ X generated by ; in particular the
character group o/ b G is topologically isomorphic with X. Thus brG is
topologically isomorphic with the compact character group o/Y-d, and bG
is topologically isomorphic with the character group o[ X.
Proof. For Zi- let :z z denote the projection of P T(z ) onto T(z ).
' xEF
Fo x C, w hav (@ (x)) -- 2 (x), an qalit whioh is tautological
but nonetheless useful. The projections are continuous characters
of the entire group P T(z ) and when restricted to brG are of course
zEr
continuous characters of brG. They also separate points of brG and so
by (23.20), the character group of brG consists of all functions
,,_... am where a Z m are integers. It is clear that a',a """
on brG if and only if 2'2 .... 2m-- t on G. Thus the character group of
brG is topologically isomorphic with X. The remaining statements about
brG and bG follow at once from the foregoing and (24.8).
Another characterization of Bohr compactifications follows.
(26.13) Theorem. Let G and X be as in (26.tt). Let H be a compacl
A belian group with discrete character group Y. The [ollowing assertions
are equivalent:
(i) there is a continuous homomorphism 9 o[ G onto a dense subgroup
o! H;
(ii) H is topologically isomorphic with a Bohr compacti/ication o/G;
(iii) Y is topologically isomorphic with a subgroup o/X.
Proof. If (i) holds, then the adjoint homomorphism 9 " of (24.37)
is an isomorphism carrying Y into X (24.4t.b) ; that is, (iii) holds. If (iii)
holds, and z is an isomorphism of Ґ into Xd, then z is also a continuous
isomorphism into X. Hence z~ is a continuous homomorphism of the
character group of X onto a dense subgroup of the character group of Y.
By (24.8), (i) thus holds.
It is trivial that (ii) implies (i). If (i) holds, take i- to be the group
of all characters Z o 9 of G, with ;E Ґ. Since H and bcG have isomorphic
character groups, they are topologically isomorphic. V]
With th construction of b G we can prove the following three
approximation theorems.
(26.14) Theorem. Let G be an infinite locally compact Abelian group;
let Z, Z, ..., Zm be elements o/the character group X o/G; and let e be a
positive real number. Then there is an x G such that x = e and ]Zi(x) -- ] < e
/or -----1, 2 .... , m. Dually, let x, x,,..., x m be elements o/ G and e a
positive number. There is a z,X such that Z= and ]Z(xi)--]<e [or
i=,2,...,.
Proof. The first statement is obvious if G is nondiscrete: there is a
neighborhood U of e such that 12; (x)-- l] < e for 1"--l, ..., m and x E U;
and clearly U is not equal to {e}.
Assume now that G is discrete and that the first assertion fails.
Write b for the isomorphism b x of (26.1t). Then b(G) is a discrete
subgroup of b G, since the topology of b(G) is simply that in which
neighborhoods of q(e) are sets {qb(x):xG and IZ(x)--[<e for
1-----t, 2, ..., m}. Since discrete subgroups of bG are closed (5.t0), it
would follow that q(G)--b G, and that b G is an infinite discrete compact
group. This is an obvious impossibility. The second statement follows
from the first and (24.8). V]
(26.15) Theorem. Let G and X be as in (26.ti), and let V-----. be any
subgroup o/ X. Let / be any character o/the group .d, let Za, Z,. ..... m
be arbitrary elements o/., and let e be a positive number. Then there is a
point x E G such that
(i) I/(2)-- Z (x) l < e (i-- , 2, ..., m).
Proof. Consider the compact group bxG. As shown in the proof of
(26.2), every continuous character of bxG has the form (tz)zEX-+tz0 for
some fixed 20EY. Theorem (24.8) shows that every character ] of Yd
This assertion is an abstract form of KRONECKER'S approximation theorem.
For various applications, see (26. 9).
432 Chapter VI. Characters and duality of locally compact Abelian groups §26. Miscellaneous consequences of the duality theorem 433
has the form/(2:)=t z for some fixed element (tz) of bxG. Since bxG is
the closure in the Cartesian product topology of bx(G ), (i) follows.
(26.16) Corollary. Let G be a locally compact Abelian group. Then
every character p o/G is the pointwise limit o/continuous characters, in the
sense that [or every e>0 and every finite subset {xl, x2, ..., Xm} o[ G,
there is a continuous character 2: o] G such that
Iz (x)- v (x)l < (i= , 2, ..., ).
Proof. Consider X as the group G of (26.t 5), consider G as the group
and apply (26. ).
Many classical theorems in the theory of equidistributed sequences
can be subsumed under a general theorem concerning Haar measure on
certain compact groups. It seems worthwhile to present this general
theorem; some of its applications appear in (26.20).
(26.17) Theorem. Let G be a locally compact, a-compact, A belian
group, with Haar measure 2, and let (H}°° 1 be a sequence o/sets as in
(t 8.t 4) such that,,oolim --) ] d is the mean value o/ / /or every continuous
almost periodic/unction ] on G. Let H be any compact A belian group, and
let q be a continuous homomorphism o/ G onto a dense subgroup o/ H.
Let # denote normalized Haar measure on H. Then /or every gўў(H),
g o q is almost periodic and
f
(i) g d # -- ,oolim (H,) (g o 9) d .
H Hn
Furthermore, iA U is an open subset o] H such that # (U)--# (U-), then
(ii) #(U) = lirnoo --(H-) (}v o 9) d t.
Hn
Proof. For a real-valued, bounded, t-measurable function h on G,
f
let us write (h) =,-oolirn -(-) h d t and N (h) = noolim -H h d t", for a
Hn Hn
complex-valued, bounded, .-measurable h, write o (h) =,lirn - h d
if the limit exists. Let 2: be a continuous character of H different from t.
Then 2: * 0 is a continuous character of G and, since (G) is dense in H,
2: 0 is different from 1. As (t 8.2) immediately shows, 2: o q) is almost
periodic. Let M be the mean value for g[(G) constructed in (18.8). It
is simple to prove that M(2:.)=0. From this and (t8.t4) we see that
o(2:.9)=0. By Lemma (23.t9), we have f2: d#=0. In other words,
f Z d#-o(z°9 )
H
for continuous characters 2: on H different from 1; this equality also
holds for 2:-- 1 since f 1 dt-- 1 =o (1 o 9). Since f and o are plainly
H H
linear and can be interchanged with uniform limits of sequences, we
infer that
f g d# -- o (g o )
H
for every complex-valued function g on H that is the uniform limit of
a sequence of linear combinations of continuous characters of H. The
STONE-WEIERSTRASS theorem shows that these functions g are all of the
continuous functions on H. This proves relation (i). Since every g in
@(H) is a uniform limit of a sequence of linear combinations of continuous
characters, the same is true for g o 9, so that g o 9 is almost periodic (t8.3).
To prove (ii), let {/}=1 and {g}=l be sequences of functions in
+(H) such that /=</+, g,,+<=g, O<=/u<=u_<=g,, , and
lim f ] d# -- lim f g, d,** ----- # (U) = # (U-). The regularity of Haar
n--,oo H n--- oo H
measure ensures the existence of such sequences of functions. It is
obvious that
(/no ) =<_ (o ) -<_ (o ) __< (-o ) __< (gno ).
Taking limits as n-+ in these inequalities, and using (i), we find
, (u)=< _(o )__< ( o )<= (- o )_<_# (u-).
Since #(U)=,(U-) by hypothesis, (ii) follows. ]
Miscellaneous theorems and examples
(26.18) Examples of automorphism groups. (a) The group Z ad-
mits just two automorphisms: and the mapping x----x. The group
®(Z) is obviously the discrete two-element group.
(b) The group ®(T) contains just two elements: , and z-+ (23.31.a).
The group ®(T) is the discrete two-element group. [This also follows
from (a) and (26.9).
(c) It is elementary to show that every automorphism z ®(R) has
the form z(x)--ox for a nonzero real number . It is also easy to see
that the topology of ®(R) is the topology of --oo, 0I t3 0, oo[ as a sub-
space of R. Thus ®(R) is topologically isomorphic with the multiplica-
tive group of all nonzero real numbers.
(d) Let r> t be an integer that is a prime power. The automorphism
group ®(.(2,) is topologically isomorphic with the multiplicative group of
all nonzero r-adic numbers, with its relative topology as a subspace
of .Q,. This is proved by an argument like that used to prove (c).
Hewitt and Ross, Abstract harmonic analysis, vol. I 28
434 Chapter VI. Characters and duality of locally compact Abelian groups § 26. Miscellaneous consequences of the duality theorem 435
(e) The automorphism group ®(Ap), where p is an arbitrary prime
number, is topologically isomorphic with the multiplicative subgroup of
the ring Ap consisting of all x----(x 0, xl, x,o, ...)cA for which x0=0;
this subgroup has its relative topology as a subspace of A. Consider
any xcA, such that x00, and define z by z(v)xv for all vA.
It is easy to see that z (Ap). Now let z be any element of (Ap).
Write u for the element (, 0, 0,...) Ap. Since multiples of u are dense
in A p, the same is true of z (u). Thus z (u)x where x o + 0, from which
it follows that zz. Hence the mapping xz is an isomorphism
onto (A); it is also easy to see that (A,A)=z'xu+A) for
k.
(f) If G is a finitely generated discrete group, then (G) is discrete.
gene at th n
(g) Let m be a positive integer. Then the automorphism group (Z )
is isomorphic with the discrete group of all mm matrices A having
integer entries and for which detA-- or --l. For such a matrix
A = (ai)i,=t, the corresponding automorphism z of Z is given by
(, • .. m) - Z m ()
' ]=1 i,''"i=l i •
The only nonobvious point is that detA = I. The matrix A - is the
matrix corresponding to z -. Both det A and det A - are nonzero integers,
and (detA)(detA-)--detAA---detI--t. The group (Z ) is discrete
by part (f).J
(h) Theorem (26.9) shows that the automorphism group of T is
topologically isomorphic with (Z), which is described in part (g).
Specifically, if z (Z ) and A- (ai)i,= is the corresponding matrix,
then
(z, ..., z) = (, ..... z2 , ..., ... 2 ), (2)
for (zl,..., Zm) T . By Theorem (26.9), all elements of ( Fro) have the
form (2). Write z'(zl,..., Zm)--(w,..., Win). Then w i is the value at
((lj, (21", "''' mi) CZm of the character of Z defined by z(z 1, ..., Zm);
equivalently, w i is the value at (li, b2i, ..., rni)---(ali, a2], ''', ami)
of the character of Z m defined by (zl, ..., Zm). It follows that w i =
o. amJ
(i) The automorphism group of R is topologically isomorphic with
®(R, m) for every positive integer m. This is shown as in (g); the
topology of ®(R ) is easily shown to be that of ((R, rn).
(j) The situation alters radically when we consider the discrete group
Z m* for an infinite cardinal number m. It is trivial that Z m* contains a
maximal independent set B of cardinal number m, and that every element
of Z m* can be written in exactly one way as klb+ ... + k b for k. Z and
bi6B for an obvious reason we write Z m* additively. An element
z (Z m*) is thus completely determined by its values on B, and of these
all that can be said is that z(B) is another maximal independent set in
Z m*, in terms of which every element of Z m* can be uniquely expressed.
Clearly neighborhoods (bl,..., bin), 0) are an open basis at, in ® (Zm*),
and each ((b, ..., bm), 0) is an open and hence closedJ subgroup of
®(Z*). I ®(Z *) were locally compact, some ((b, ..., bin), 0) would
be compact. Let bin+ 1 be any element of B different from b, ..., bm.
If ({b,..., bm}, O) were compact, we could choose ў),..., ў') in
({b, ... bin}, O) such that O z() ({bm+}, 0) ({b,... bin}, 0). For
' k=l '
an automorphism (r in [ Z () ({bm+l}, 0), the element c(bm+)is one
k=l
of the elements T(1)(b/+l) .... , T(n)(b/+I). There is evidently an element
cc({b, ..., bin}, 0) such that a(bm+)lies outside of any preassigned
set of cardinal number less than In. Hence ({b i, ..., bin}, 0) is not
compact, and ® (Z m*) is not locally compact.
(k) For an infinite cardinal In, the compact Abelian group T m has an
automorphism group that is not locally compact. [This follows from
part (j) and (26.9).]
(26.19) Approximation theorems (HEWITT and ZUCKERMAN
By using (26.t 5) and (26.t6) together with the computations of characters
given in § 25, we obtain sharpened forms of KRONECKER'S approximation
theorems as well as new approximation theorems. The following ex-
amples, while obviously not exhaustive, illustrate the method adequately.
(a) Let a and b be positive integers; let h, h., ..., h a be real numbers
b
1 li' h (f -- 2 a)
that are linearly independent over Q; let i-- = mi,! ' ' "'" '
where the l's are integers and the m's are positive integers. Let , ...,
be numbers in 0, l and x(1),..., x () be elements of A a--(2, 3,4, 5, ...).
Let e be a positive number. There is a real number t such that
b
(i) r/i__{__ mi,!l" (__ x(0)__ 2,x( ). ....
(mi, l)t x () '} < e (mod l)
'x
-- . mi, k--2]
for f=l, 2 .... , a. Here we use (26.6). For the roup G of (26.16),
take R. Computing an arbitrary character of R by (25.6) and (25.4),
we obtain (i).
(b) One form of the classical approximation theorem of I(RONECKER is
obtained by putting i = hi -- l, ..., a and a= b and x ()-..- --x ()= 0
in (a). Thus we have
1 For a real number , the expression I1 < (mod 1) means that
for some integer h.
28*
436 Chapter VI. Characters and duality of locally compact Abelian groups §26. Miscellaneous consequences of the duality theorem 437
(ii) Ihit-jl<e (modl) (]'=t,2 ..... a),
where 1 ..... a are arbitrary real numbers and hi, ..., ha are rationally
independent real numbers.
(c) Let h 1, h2,..., h b be real numbers such that t, h 1, h, ..., h are
b
li' h
linearly independent over Q, and let h0=t. Let i-- mi' !
=0
(= , 2, ..., a), where the l's are integers and the m's are positive integers.
Let =(2, 3, 4, 5, ...). Let (0,o, ...,( be elements of A," let
, e, ..., be numbers in [0, t [; let 0=0; and let e be a positive
real number. There is then an integer n such that
(iii) in- {o mi,li' (_ x, - 2'. @' ....
-- (mi, ) t x( < e (mod )
• m,-2J
for '=1, 2, ..., a. [Here we use (26.6) and (2S.2). The expression {...}
in (iii) is obtained from the value of an arbitrary character of the group T
computed at exp[2ii, and in from the value of a continuous
character of T computed at exp [2ii.
(d) We obtain another version of NRONECKER'S approximation theo-
rem by putting i=h i [j=l, ..., a and a=b and (0 ..... (v=0
in (c). Thus we have
(iv) (modt) (/=,2,...,a),
for some integer n, where 1, ..., a are arbitrary real numbers and
/h, ..., ha are real numbers such that t, hi, ..., ha are linearly independent
over Q.
(e) Let btil= ([3(,il,..., fll) (j= 1, 2, ..., m) be elements of R"such that
bin,..., b ('1, (t, 0,..., 0(), (0, t, 0,..., 0() .... , (0, 0,..., O, t(.) are
linearly independent over Q in the vector space R n. Let el, ..., em be
any real numbers and e a positive number. There is an element
(kl, k,., ..., k,) Z"such that
(v) loci-- ,kflli'l <e (modl) ('= t, 2,..., m).
/=1
[Apply (26.t6) to 7* and (T)'. To obtain the character on (T) ', regard
(T)"as (Rd)"/Z" and find a character of (Rd)" such that (b (il) --
exp [2 i%. for = t, ..., m, and (Z') = t. To obtain characters on
use (23.t 8).]
(f) Let E be a topological linear space over R, let /1,/2,...,/m be
continuous linear functionals on E that are linearly independent over Q,
let el, ..., em be any real numbers, and let e be a positive number. There
is an xE such that
(vi) I%--/i (x)! < e (mod 1) (i-- t, 2,..., m).
[Use (26.15) with E as the Abelian group G. As the group Y, take the
group of all characters x-+exp[2i[(x) for ]E*. If/1,/., ...,/m are
elements of E* independent over Q, then some linear functional F
E* has the property that F(].)=ei. Thus exp[2iF(]j) =exp 2i%.'
(vi) follows at once from (26.15).
(g) Let aў, aў., ..., alal be elements of the group A [p is any prime 1
that are independent:
rn]x(i)=O and m]Z imply rn 1 .....
i=1
Let a, e. .... , ea be any real numbers, and let e be any positive real
number. There are positive integers l and n such that
(vii) i-- l -p,_k < s (mod t) (i = t, 2,..., a).
k----0
[The p-adic integers a (a), aў2), ..., a lal generate a free subgroup of A p,
and hence a character of Ap can assume arbitrary values of absolute
value 1 at ae (a) .... , a (). Continuous characters of Ap are given by (25.2.2).
Combining these observations with (26.t6), we obtain (vii).]
(h) If is any character of Q and rl, r. .... , r m are any elements in Q,
there is a continuous character ;g of Q such that l.(ri)--va(r) for
/'=1, ..., m. [Although this is a strong approximation theorem, it is
really quite elementary and does not depend upon (26.t5) or (26.16).
Let a=(2, 3, 4 .... ) and let (, a) be the element of
that corresponds to Va as in (25.4). Let n be a positive integer such that
n!rjisanintegerfor'=t, ...,m. Let t=-- (x0+2! x+ ...+ (n--t)!x,_.)"
by (25,4.2) we have (ri)--expI2nirit 1. Define Z by z(r)=exp2nirt l
for all r Q.I
(26.20) Examples on equidistributions. Theorem (26.7) has many
applications. The following are a small sample.
(a) Let H be a compact monothetic group, with {x'}=_oo dense in H.
Let Ube any open subset of H such that # (U) =#(U-) # is normalized
Haar measure on Hi. Then
m
(i) / (U)= lim ,
,,,oo 2rn + . v (xi) -- lim v (xi)
m--oo m -- a + I .
] = --m
-- lim
-oo + + . v(x).
The first equality follows from (26.17) by taking G=Z, 0(n)= x", and
H,,= {-m, --m+ t .... ,0 ..... m-- t, m}. The proofs of the remaining
equalities are similar; see (t8.15.b). 1
(b) Let H be a compact solenoidal group, and let 9 be a continuous
homornorphism of R onto a dense subgroup of H. Let,, and U be as in (a).
438 Chapter VI. Characters and duality of locally compact Abelian groups Appendix A. Abelian groups 439
Then T
f
(ii) #(U)=rlrn- u(9(t))dt=limT--o T--a- u(q)(t)) dt
lim f
r + .-
--T
[The proof is similar to the proof of (a); use (t8.15.a).]
(c) Let al, ..., a m be irrational real numbers. Let U be an open
subset of T" such that #(U)=#(U-) [# denotes normalized Haar
measure on T"]. For every positive integer n, let A (n) be the number of
distinct m-tuples (kl, ..., k) of integers such that Ik]] =< n for i= t, ..., m
and (exp2ziak] .... , exp27iamkm] ) lies in U. Then
(iii) ,-oolim (2n + 1)" = # (U).
[The mapping (k,...,km)-+(exp[2ziakl],...,exp[2ziamk,,])is
a [continuous] homomorphism of Z" onto a dense subgroup of T". Let
H, be the set of all m-tuples (k,..., kin) of integers such that ]ki] < n for
i= 1, ..., m. Since {n,}°= satisfies (t8.t0.i), (26.t 7) implies (iii).]
(d) Let a ..... am, U, and # be as in (c). Suppose further that the
set {al, ..., am, t } is linearly independent over Q. For positive integers n,
let B(n) denote the number of integers k such that [k] =< n and
(exp[2:ia k], ..., exp[27iamk]) lies in U. Then
_ (u).
(iv) lim
n-oo 2n
[The set {(exp[2zialk], ..., exp[2ziamk)'kZ} is a dense subgroup
of T by virtue of (26.19.d). Now apply part (a).]
(26.21) The modular function for
and 76). Let G be any locally compact group with Haar measure 2. The
modular function z-+A(z) defined in (t5.26) is a continuous homo-
morphism of ®(G) into the multiplicative group 0, o[. [In view of
(15.26), we need to prove only continuity. Let ]o(G) be such that
f / d2= t, and let F be a compact subset of G such that ] (F')=0. Let
V be a fixed neighborhood of e with compact closure. For e> 0, let U
be a neighborhood of e such that UcV and ]](y)--/(z)l<e if yz-U
(t5.4). Now let z be any automorphism in !3(U-F, U). Then
z-(F)cU-F, and ][(z(x))--[(x)l<e if xU-F. Therefore we have
]A(-) 1] --]A(-c)-- ] f /d2=[ f /(-c(x))dx-- f /(x)dx]
<_ f]/(-c(x))--l(x)]dxg f edx=e;(U--F) <__e;(V-F).]
U -F U -F
Notes
Definition (26.3), (26.5) [barring changes needed to show that (F, U)
is open], (26.8), and (26.9) are taken from BRACONNIER [1], Ch. IV.
Theorem (26.t0) generalizes IWASAWA [2], Theorem 4, p. 515. Auto-
morphism groups were studied earlier by ABE [t 1, but with a topology
different from that defined in (26.3). A detailed investigation of homeo-
morphism groups of locally compact Hausdorff spaces has been made
by ARENS [1]. For results related to (26.10), see HOFMANN [t].
We note also the following interesting and apparently difficult theorem
of IWASAWA [21. Let G be a compact group and,(G) the group of all inner
automorphisms of G. Then ,(G) is a compact normal subgroup of ®(G),
and ®(G)/,(G) is a 0-dimensional group.
The constructions and results of (26.)--(26.13) are due to ANZAI
and KAKUTANI tl. Theorem (26.4) seems to be new; (26.5) and
(26.t6) are from HEWlTT and ZUCKERMAN [t]. Theorem (26.t7) appears
to be new.
Appendix A
Abelian groups
In this appendix, we set forth part of the algebraic theory of Abelian
groups. We limit ourselves to those parts of the theory that are needed
for our study of the structure of locally compact Abelian groups
and the duality theory of locally compact Abelian groups [§§23--25].
A more complete treatment appears in KAPLANSKY t I and a far more
complete treatment in FUCHS t .
We will use multiplicative notation for the group operation except
in dealing with Z, Q, R, Z (n), and a few linear spaces. The exceptions
will be specifically pointed out as they occur. Throughout this appendix,
"group" means "Abelian group" and the symbol "G" will be used to
denote an arbitrary Abelian group.
(A.1) Definition. If every element of a group G has finite order,
then G is called a torsion group. If every element of G except e has
infinite order, then G is called a torsion-/ree group. A torsion group G
is said to be p-primary Ifor a prime Pl if the order of every element is a
power of p. If G is a torsion group and if the order of every element is
less than some fixed positive integer, then G is said to be of bounded
order.
(A.2) Definition. Let G be a group. For n=2, 3,..., let/ be the
mapping of G into G defined by/,x)= x". We define G(")=/(G) and
G(.) =/-;: (e).
We first prove an elementary but very useful theorem on the decom-
position of torsion groups.
(A.3) Theorem. Let G be a torsion group. For each prime p, let Gp
be the subset o/ G consisting o/ elements whose order is a power o/ p. Then
440 Appendix A. Abelian groups Appendix A. Abelian groups 441
each Gp is a p-primary group and G is isomorphic with P*G, [P is the set
pp
o/all primes].
Proof. Obviously each Gp is a p-primary group. To show that G is
isomorphic with P*G, we apply (2.5). Consider any element xG having
pEP
order n. Then n--l . "'P, where Pl, P2, .-., P are distinct primes.
If n--n/p , then x has order p and belongs to G. Since nl, m. .... , nk
are relatively prime, there are integers aa, a. .... , a such that aln+
a2 n + ... + a n = 1. Then
x-- (x') al. (x"") .... (x"*)*G,.G.. . . .Gp,,
and (2.5.i) is proved.
Suppose now that xG and that x--xx....x,, where xiG and
the pi are all different from p. Then x has order a power of p and
xl x. • • .x has order r""r"" "P, for some integers r , r ,..., r . It follows
that x must be e" so that (2.5.ii) is established.
(A.4) Theorem. Every group G contains a largest torsion subgroup F;
the equality F:,,xG<, 0 obtains; and G/F is torsion-]ree.
Proof. Let F be the set of elements in G having finite order; then F
is plainly the largest torsion subgroup of G and F--,,=G<, O. Suppose
that (xF)":x'*F:F for some n. Then x"F and x"':e for some m;
i.e., xF and xF--F.
We next introduc a very important if wry special class of groups.
(A.5) Definition. A group G is said to be divisible if G<'O--G for
n--2, 3, -.. • In other words, given xG and n, there is a yG such that
x--y. If {e} is the only divisible subgroup of a group G, then G is said
to be reduced.
A direct product or weak direct product of groups is divisible if and
only if each factor is divisible. A homomorphic image of a divisibl
group is again divisible. The group Q and the groups Z (poo) are divisible.
We will prove in (A.t4) that every divisible group is a weak direct
product of these groups.
(A.6) Theorem. Every group G contains a largest divisible subgroup D.
Proof. Let D be the smallest group containing all divisible subgroups
of G. If x is in D, then x-- x...x, where x. belongs to a divisibl group
D i for 1":1, ..., m. For n> t, we have x-y 7 for some yiDi. Then
Xl"" Xm-- (Yl" " " Ym) and y... Ym D; that is, D is divisible.
By divisible subgroup we of course mean a subgroup that is divisible when
looked at by itself.
The next theorem has several useful applications in the theory of
topological groups.
(A.7) Theorem. Let H be a subgroup o/G. Every homomorphism q
o/ H i#o a divisible group D can be extended to a homomorphism o/G
into D.
Proof. By ZORN'S lemma Ior by a well-ordering argument/, it
suffices to choose any x CH and extend 9 to the group
Ho--{x"h" hH and nZ}.
If x"ўH for all n_>_2, let v(x"h):q(h) for x"hH o. Then v is a well-
defined homomorphism that extends 9. If x" H for some n_>_ 2, we take
k to be the least such integer. There is an element zD such that
zn--q(x). Defining v(x"h) to be z"q(h) for x"hHo, we again obtain
a well-defined homomorphism that extends . [
(A.8) Theorem. Let H be a divisible subgroup o/ G and let A be a
subgroup o/ G such that n (A---=-{e}. Then there is a subgroup A o o/ G
such that H (Ao--{e}, A ca o, and HAo--G; i.e., G is isomorphic with
HЧA o. In particular, every divisible subgroup of a group is a direct
/actor o/the group.
Proof. Since H fA--{e}, HA is isomorphic with HxA. Let
denote the projection of HA onto H" then (x):x for xH and
A---(e). By (A.7), can be extended to a homomorphism of G
onto H; note that ov--v. Let Ao--v-(e). Then AofH--{e } and
A ca o. Moreover, if xG, then x--[xv(x-)]v(x); and we have
x(x-)Ao and p(x)H.
Before completing our study of divisible groups, we will define and
study the notion of rank. The following simple fact is useful for this
purpose.
(A.9) Theorem. Let p be a prime integer and let {0, t, ..., p--1} be
a realization o/ the c,clic group z(p). Define multiplication i, Z (p) as
ordinary multiplicatio, modulo p. Then Z (p) is a/ield. Let G be a group
such that every element except e has order p, and regard G as an additive
group. For a {0, 1, ..., p-- 1} a,d x G, we de/ine a x to be, as usual, the
sum x+ x+...+ x [a times. Then G is a linear space over the/ield Z (p).
The proof of this theorem is elementary, and we omit it. The order
restriction on G is needed in order to prove that
and [(a+ b)modp].x-= ax+ bx
I() mode/, x: ()
for a, b{0, ,..., p---1) and xCG.
(A.10) Definition. A finite subset {Xl, ..., x) of a group G is said
to be independent if it does not contain e and if x .... x*= e implies that
442 Appendix A. Abelian groups Appendix A. Abelian groups 443
x ...... x--e [each n. is an integer. An infinite subset L of G is
said to be independent if every finite subset of L is independent 1. If an
independent set L generates the group G, then L is called a basis o] G.
The following rather cumbersome theorem will allow us to define
various notions of rank.
(A.11) Theorem. Let G be any group. Let be the ]amily o/ all
independent sets L in G consisting only o/elements whose order is infinite
or a power o/ a prime and such that L is maximal with respect to these
properlies. Similarly, let o 1 be the [amily o/independent sets L o
in G consisting o/elements whose order is infinite [a power o/p and such
that L o ILpl is maximal with respect to these properties" p denotes any
prime integer. Then ,, o, and are nonvoid. Every set L in is
o/
o/all primes 1. All the sets in s' have the same cardinal number; all the sets
in o [ have the same cardinal number.
Proof. (I) An elementary ZORN'S lemma argument shows that
0, and @ are all nonvoid, since the void set is independent.
(II) Consider a set L in , and let L 0 [Lp consist of the elements x
in L having infinite order [order a power of p. If L0 does not belong to
0, there must exist an element a G V)L6 having infinite order and for
which L 0 U {a} is independent. Then L U {a} is easily seen to be independ-
ent, contradicting the hypothesis that L . It is a little more delicate
to show that L; belongs to . Ii not, there is an aG having order p'
such that L;U{a} is independent and L U{a} is dependent. Then
a"SoS;t - e where 0< n< p', s o is a product of powers of elements from L 0,
s; is a product of powers of elements from L;, and tp is a product of
powers of elements from U Lq. There is a positive integer m relatively
q4:P
prime to p such that t = e. Thus a n -- a n
sg's e and 4= e. Since L U {a}
is independent, s o cannot be equal to e. Now for some power p*, we have
s (a '-''
-- -0 sp) =e so that L 0 is not independent; this is a contra-
diction.
(III) In view of part (II), sets in have the same cardinal number,
if the corresponding assertions for 0 and are true. Let us prove these
facts" we deal first with . For this purpose, we may suppose with no
loss of generality that G itself is p-primary" then we want to show that
all maximal independent sets in G have the same cardinal number. We
will also write G as an additive group. By (A.9), the subgroup G(=
{xG'px=o} is a linear space over z(p). Since all bases of G() have
the same cardinal number this is a well-known fact in the theory of
linear spaces, it suffices to prove the following.
It is also convenient to consider the void set as an independent subset of G.
Let L be a maximal independent set in G. For xL, let pt* be the
order of x. Then .={p-x'xL} is a basis for G(). It is obvious that
L is linearly independent in G(p). If L is not maximal, there is an element
a G(p V) (L)' such that L U {a} is linearly independent. Since L U {a} is
dependent in G, we have
na--na x+ . " .+ n x (t)
for some integers ni and n where 0< n< p. Then
o--pna=pn xl + . . .+ pnx;
hence pnixi--O for each " and pni--mipn for certain integers m.. In
view of this, relation () becomes
n a = nhp - x + . . . + rn p - x ;
since n a4=O, this violates the linear independence of L U {a} in G(.
(IV) We now show that all sets in o have the same cardinal number
in the case that G is torsion-free. We return to multiplicative notation.
Suppose first that G contains a finite maximal independent set L-
{21, x. .... , x}. Let {Yl, Y., ..., Y,} be any finite independent subset
of G. We will prove that r<=k. The maximality of L implies that
y--x .... x *, where n 4=0. Since G is torsion-free, not all m. are 0;
by relabelling if necessary, we may suppose that rn 1 0. It is then easy
to see that {Yl, x. .... , x} is a maximal independent set. Thus if r_> 2,
we have y= y[' x'-.., x *, where at least one of s. .... , s is not zero. Let
us say that s.=t=0; then {y, y., x, ..., x} is a maximal independent
set. The process continues until we run out of y's or x's. If r> k, we
find that {Yl, ..., Y} is a maximal independent set properly contained
in the independent set {Yl,..., Y,}. This contradiction proves (IV) if
G contains a finite maximal independent set.
We complete the proof of (IV) by showing that any infinite set L in
has cardinal number G. It is trivial that L_<_ G. If z G, then z= x.-. x
for some integers n, n 1, ..., n where n=t=0 and x 1 .... , xL. We asso-
- ni for
ciate with z the element q) (z) -- (s,),e L in QL. such that s,=--
'=l,...,k and s,--0 for xў{xl,...,x}. If zw, then we have
(z)0(w) For if z x x and w
..... = • • • x , where rn. n = m n
for '=1 ..... k, then zm'=w '' and so z=w, since G is torsion-free.
Similarly one sees that 0 (z) is uniquely determined by z although this
fact is not essential for the proofS. Consequently, 9 is a one-to-one
mapping of G into QL.; and so we find that U_<
(V) Now let G be any group. Let F be the torsion subgroup of G
(A.4) and let L be in '0- We will show that {xF" xL} is a maximal
444 Appendix A. Abelian groups Appendix A. Abelian groups 445
independent set in the torsion-free group G/F; part (IV) then implies
that all elements in 0 have the same cardinal number. Suppose that
( xl F)' . . . ( xaF) "*= F
for xl, ... , xL. Then x... x, belongs to F and hence for some integer
m=>l, we have (x .... x*)'=e. This implies that x'=e and thus
hi=0 for 1"=1, ..., k. Hence {xF" xL} is independent in G/F. If it is
not maximal, then there is an a in G (L' having infinite order such that
{xF'xL}U{aF} is independent. However, a"=x...x'" for some
xl, ..., xsL and n0, so that (aF)"= (xiF) ' .... (xsF)'% contrary to the
choice of a.
(A.12) Definition. Let G, , 0, and be as in (A.tl). The
cardinal number of any set in is called the rank o! G and is denoted
by r(G). Similarly, the cardinal number of any set in o [ is called
the torsion-]ree rank [the p-rank] o/G and is denoted by ro(G ) [rp(G).
(A.13) Theorem. Let F be the torsion subgroup o] G (A.4). Then
ro(G)--ro(G/F)--r(G/F) and rp (G)=r(F) /or all primes p.
Proof. The equalities r (G) = r (F) and r o (G/F) = r (G/F) follow imme-
diately from the definitions. The equality ro(G)--r(G/F ) is proved in
part (V) of the proof of (A.t t).
We now identify all divisible groups.
(A.14) Theorem. Let G be a divisible group. Then G is isomorphic
with
x (z
pEP '
where P denotes the set o] all primes.
Proof. Let F be the torsion subgroup of G. It is obvious that F is
divisible. Theorem (A.8) shows that G is isomorphic with FxH, where
H is isomorphic with G/F.
In order to apply linear space theory, we write H as an additive
group. Theorem (A.4) states that G/F, and hence H, is torsion-free and,
being a homomorphic image of G, it is also divisible. Let x be in H and
n a positive integer. Then x--ny for some y H. The element y is unique
since ny--ny 0 implies that n(y--y0)--0 and hence Y--Y0. In other
words,--x is uniquely defined, and it follows that rx is uniquely defined
for all r Q and x H. It is now easy to show that H is a linear space over
the field Q. As such, H contains a basis having cardinal number
say. Thus H is isomorphic with Qn0..
The divisible torsion group F is isomorphic with P*G. where each G
pEP v '
is a divisible p-primary group (A.3). Fix p and consider a family of
subgroups {A t},EI of Gp such that each A, is isomorphic with Z (po) and
if xtx2...x--e where x.A, and the t i are distinct, [
then x]-- e for /'---- t, ..., k.
By virtue of (2.5), the smallest subgroup of Gp that contains all of the A,
is isomorphic with a weak direct product of groups Z (poo). Let ' denote
the family of all such families of subgroups of G. It is easy to see that
ZORN'S lemma applies" that is, there is a maximal family {B,},Ex in '.
Let B be the smallest subgroup of G that contains all of the B,'s.
Then B is the weak direct product of Z (p°°)'s and is divisible. If B G,
(A.8) shows that Gf contains a nontrivial divisible subgroup C such that
C f')B--{e). Using the divisibility of C, we find a sequence x, x2, ... of
elements of C such that x{--e, x{--x, and so on. Consider the group B 0
generated by {x)°°__ and map each xi onto the complex number
exp 2 i - . This mapping can obviously be extended to an isomorphism
of B 0 onto z(p°°). Thus {B,},ExU {B0} belongs to . This contradiction
proves that B----- Gp.
Summarizing, we see that G is isomorphic with
Q 0* x e* (z
pEP '
for some cardinals n o and n. The torsion-free and p-ranks of this
product are no and n, respectively. Hence n0=r0(G) and n--r(G)
for p P.
For some of the refined analysis of locally compact Abelian groups,
we need to imbed an arbitrary group in a divisible group, and in fact in
a particular way.
(A.15) Theorem. A group G can be imbedded in a divisible group E
that is minimal in the sense that i] G D E and D is a divisible group,
then D -- E.
Proof. First we show that G can be imbedded in some divisible
group E 0. Let m--G and for each x G, let Z, be the group Z. Consider
Z m* ----- P*Z,, and for (k,) Z m* let
xEG '
for each (k,), this is a finite product. Then 9 is a homomorphism of Z m*
onto G, and G is isomorphic with Z"*/N where N---(e). Now imbed
Z "t*/N in Q,t./N. This group is divisible, since it is a homomorphic image
of Qrn.. Let Eo--Q"*/N, and identify G with its isomorphic image
contained in E 0 .
446 Appendix A. Abelian groups Appendix A. Abelian groups 447
Let H be a divisible subgroup of E o maximal with respect to the
relation G(H--{e}. By (A.8), there is a subgroup E of E o such that
G cE, E VH--{e}, and EH--E o. Thus E is a direct factor of E o and
hence is divisible. Suppose that GcD E and that D is divisible. By
(A.8), there is a divisible subgroup H o of E such that H o : {e}, DHo--E,
and D VHo--{e}. Now HH o is a proper supergroup of H, is isomorphic
with HxHo, and satisfies the relation (HHo)(G= {e}. Thus HH o is a
divisible supergroup of H that is disjoint from G. This contradicts the
maximality of H.
(A.16) Theorem. Let G be a group and E a divisible extension o] G
that is minimal in the sense o] (A.I 5). Then ro(G)--ro(E) and rp(G)--rp(E)
[or all primes p. I! E 1 is another minimal divisible extension o/G, there
is an isomorphism carrying E onto El that is the identity mapping on G.
Proof. It is obvious that r o (G) =< r o (E) and rp (G) __< r (E) for all
primes p. Consider now any L o in o for the group G. Then L o is also
in o for the group E. To see this, let H be the group generated by Lo,
and assume that a in E has infinite order and aўH for n--t, 2 .....
There is a subgroup Qo of E containing a and isomorphic with the rational
number field Q: to see this, use the decomposition of E given in (A.t4).
Now QofG -- {e}, since otherwise some integral power a' of a would lie
in G and LoU{a" would be an independent set in G containing only
elements of infinite order. By (A.8), there is a divisible subgroup D of
E such that GD, D( Qo--{e}, and DQo--E. This contradicts the
minimality of E. Hence L o is in o for the group E; this proves that
r o (E) =< r o (G). Similarly, we have r (E) =< r (G) for all primes p.
Let E be any minimal divisible group containing G. By (A.7), the
identity mapping of G onto itself can be extended to a homomorphism
9 of E into E. Since 9(E) is a divisible group containing G, we have
(E)--E. Assume that is not one-to-one, i.e., -l(e)={e}. Since
9-1 (e)VIG--{e}, there is an element a in 9- (e) such that a = e and such
that a G implies a "- e. As in the previous paragraph, we see that there
is a divisible group D such that G cD and a cD. This contradicts the
minimality of E.
(A.17) Theorem. Let G be a group and E a minimal divisible ex-
tension o/G. Then E/G is a torsion group. I/G is imbedded in a divisible
torsion-/ree group D and DIG is a torsion group, then D is a minimal
divisible extension o/G.
Proof. Let
E o-- {x E: x G for some positive integer n).
Given xE o and an integer k>__l, we have x'*G for some n and z*--x
for some z E. Since z *n is in G, z belongs to E o. Therefore E o is divisible
and Eo--E. Hence E/G is a torsion group.
Suppose that D is as in the second statement of the theorem and
that G D o D, where D o is divisible. If x D, then x G for some
integer n. Since D o is divisible, x--- - y" where y D o. Since D is torsion-
free and (xy-)"-- e, it follows that x-- y Do . That is, Do is equal to D.
(A.18) Note. A subgroup H of a group G that is generated by an
independent set is the weak direct product of cyclic groups, the generators
of these cyclic groups being the elements of the independent set. We
omit the proof of this elementary fact.
For our study of the structure Of compact Abelian groups, we require
another group-theoretic concept, which we will now introduce and discuss.
(A.19) Definition. Let G be a group. A subgroup H of G is said
to be pure [in G if whenever x G and x" H, there is an element y in H
such that y-- x .
(A.20) Theorem. Let G be a group and H a subgroup o] G. The
]ollowing statements are equivalent:
(i) H is pure in G;
(ii) H"-- H ( G or n-- 2, 3, ... ;
(iii) [ (H)--= H. G ]or n--2, 3,..., where ] is the mapping defined
in (A.2).
Proof. The inclusion H"cH(G" always holds. It is easy to see
that the purity of H in G is equivalent to the inclusion H H V G ").
The inclusion ] (H) H.Goo always holds. Suppose that H is pure
in G and that x[7, (H). Then x"H and x"--y" for some yH. It
follows that y- x G) so that x -- y (y- x) H. G. Thus ] (H) H. G).
Finally, suppose that ](H)H.G and that x"H. Then x--hy
where hH and y"--e. Thus h -- x" and H is pure in G.
(A.21) Theorem. Let H be a pure subgroup o] G. Then every coset
in G/H contains an element y such that the order o[ y in G is equal to the
order o/y H in G/H.
Proof. If a coset has infinite order in G/H, then all of its elements
have infinite order. Suppose then that the coset, say x H, has order n.
Then x H and H contains an element z such that z -- x . Let y--xz-;
obviously y x H and y"--e. If the order of y were less than n, then the
order of y H would also be less than n. Hence the order of y is exactly n.
(A.22) Theorem. I! H is a pure subgroup o] G, and G/H is iso-
morphic with a weak direct product o[ cyclic groups, then G is isomorphic
with Hx (G/H).
Proof. For each cyclic factor of G/H, choose a generator xH. By
virtue of (A.2), we may suppose that the order of each x in G is equal
to the order of x,H in G/H. Let N be the subgroup of G generated by
448 Appendix A. Abelian groups Appendix A. Abelian groups 449
all the x,. Let us show that HN--G and H f N--(e}. This will imply
that G is isomorphic with HxN and that N is isomorphic with G/H (2.4).
Consider any y in G. Then yH--(x,H)" .... (x,H) " for some integers
nl, ..., n and indices ,1 .... , ,. Consequently, y-- xl.., x,h for some
hH; i.e., y belongs to NH and hence G--NH. Suppose that h belongs
to both H and N" then h--x.-, x: and (x,IH) ' .... (x,,H) ''-- H. Thus
each m i is a multiple of the order of x,H and hence of the order of x.
This implies that ' •
x,--e for all ', so that h-----e This proves the relation
nN--{e}. ]
Theorem (A.24) in]ra is another group-theoretic fact needed in the
detailed study of compact Abelian groups. We first state and prove a
lemma.
(A.23) Lemma. A nondivisible p-primary group G contains a non-
trivial cyclic subgroup that is pure in G.
Proof. (I) We first assume that for every x in G having order p and
for every positive integer k, there is a yG such that yP-----x. We will
prove that in this case G must be divisible, which is contrary to our
hypothesis. Suppose that x is in G, q is a prime different from p, and
that pt is the order of x. Then 1--apt+ bq for some integers a and b,
and hence x--(xP')a(x)a--(x) . To show that every element x in G is
divisible by p, we apply finite induction to k, where p is the order of x.
The assumption yields this for k--t. Suppose that all elements of order
less than or equal to p- are divisible by p. Let x have order p; by the
assumption x#-'--y for some yG. Since (yx-)-X--e, yx - is
divisible by p" yPx---z for some zG. Thus x--(yz-) , and G is
divisible.
(II) Choose an element x in G and a positive integer k such that x
has order p and x--y - for some y in G, but for which the equation
x--z has no solution z. Let H be the subgroup of G generated by y.
Suppose that w'--y for integers t and m, where 0< t< p. Then t-----up
where l <= k-- 1 and u is prime to p. We show that p+ does not divide m.
If pt+ divides m, then we have y"P'--w + for some integer r. Now
t--au+ bp for some integers a and b, so that
which is a contradiction. Since the greatest common divisor of m and
p is pV where l' l, there are integers c and d for which t:upt--cm+ dp .
Hence w:
To prove Theorem (A.24), we need another concept. A subset L of
a group G is called pure independent if it is independent and the smallest
subgroup of G containing L is pure in G. The usual ZORN'S lemma
argument shows that every pure independent subset of a group is
contained in a maximal pure independent subset.
(A.24) Theorem. Let G be a p-primary group. Then G contains a
subgroup B such that:
(i) B is isomorphic with a weak direct product o[ cyclic groups;
(ii) B is a pure subgroup o/ G ;
(iii) the quotient group G/B is divisible.
Proof. Let L be a maximal pure independent subset of G. If L is
void, (A.23) shows that G is divisible, and in this case we take B--
If L is nonvoid, let B be the subgroup of G generated by L. Then asser-
tions (i) and (ii) are immediate. If G/B is not divisible, then again by
(A.23), G/B contains a nontrivial pure cyclic subgroup; this subgroup
is generated by xoB, say. Since B is pure in G, we may suppose that x 0
has the same order in G as xoB has in G/B (A.2t). It follows that Lt3 (x0}
is an independent subset of G. To arrive at a contradiction, we will show
that the group B 0 generated by L t3{x0} is pure in G. Suppose that
bx--y" for some yG and b B. We then have xB--y'B. Since the
group generated by xoB is pure in G/B, there is an integer k such that
x B-- (x0 B)m. Since also x B = y B, we find that (Xo y-)-- xo y-
belongs to B. The purity of B yields a bomB for which (xoy-)'--b'.
Thus y--(xob) '', and the group B 0 is pure in G.
We now classify groups of bounded order.
(A.25) Theorem. Every group G o] bounded order is isomorphic with a
weak direct product o! cyclic groups ,PZ (if,0, where only ]initely many
distinct primes p, and positive integers r, occur. Conversely, every such
group has bounded order.
Proof. By (A.3), we may suppose that G is p-primary. Let B be a
subgroup of G as in (A.24). Since G/B is divisible and of bounded order,
it follows that G/B--{B). Therefore G:B and G is isomorphic with
a weak direct product of cyclic groups. Since each factor is p-primary
and of bounded order, it has the form z(p'). The remainder of the
theorem is obvious .
If a group G is generated by some finite subset, then G is said to be
finitely generated. Using the next theorem, we will prove the fundamental
theorem (A.27) which characterizes all finitely generated [Abelian]
groups.
This deduction of (A.25) from (A.24) was pointed out to us by R. S. PIERCE.
Hwitt arid Ross, Abstract harmonic analysis, vol. I 29
450 Appendix A. Abelian groups Appendix B. Topological linear spaces 451
(A.26) Theorem. Let n be a positive integer, and let H be a subgroup
o[ Z" diJJerent [rom {0}. Then there are a basis {x 1 .... , a%} [or Z [see
(A.t0)] and a sequence {kl, ..., k,} o[ positive integers such that r<= n,
(i) {klOgl,... , kt, ogt, } is a basis/or H,
and
(ii) kj divides kj+l [or = 1,..., r-- t.
Proof. The number n is the cardinal number of all bases of Z .
If n = t, then Z is cyclic and the theorem is obvious. Suppose that the
theorem is true for n--t. For every hH and every basis {Yl,...,
of Z", we can write h=lyч...ч l,,y,,. There is a least possible positive
value k 1 for the integers l. as h runs through all nonzero elements of H
and {y, ..., y,,} runs through all bases for Z'. Let/t o be an element of H
such that
ho : kl Yl @ m2 Y2 @"" @ mn Yn , (t)
where {Yl,'", Y,,} is a basis for Z and m2, ... , m n are integers.
Let us show that k divides m i for '=2, ..., n. If mi=aikl+b
where 0--< bi < kl, then we have
ho=kx + b2Y2@ " .@ b,,y,,
where xl=yl+ a2Y2+'"+ a,,y,,. It is easy to see that {Xl, Y2, ..., Y,,}
is a basis for Z " The minimality of k 1 implies that each b i is zero. Note
that h 0 = kx.
Now let G be the subgroup of Z"generated by {Y2, ..., Y,}. Then G
is isomorphic with Z "1. If GClH={O}, then we have g={lkl3P, l'lZ},
and the theorem is evidently true for H. Otherwise, our inductive
hypothesis implies that there are a basis {x 2, ..., x. for G and a sequence
{k2,..., k,} of positive integers such that
{k 2 1 2 .... , k, x,} is a basis for G Cl H
and
k. divides ki+ for '=2, ..., r--.
It is easy to see that {Xl, ..., x. is a basis for Z "
Suppose now that h is any element in H; we write h=llXl+ .. .+ l..
Write ll = Sl kl+ c, where 0 < c < kl. Then we have
h-- s 1 ho--coe 12r- 12x2- .. .2r- lnx n ,
and the minimality of kl implies that c=0. Thus h--Slh o belongs to
G (3H, so that h-- Sl llO---s2k2og2+- " "+ - srkrxr and h--SlklOgl-+ - s2k2x2-t-
• ..+s,k,x,. Since {klal, ...,kroer} is an independent set, we have
proved (i).
It remains only to verify that k 1 divides k 2. Let k.--akla t- b where
0--<_ b < kl. Then {x 1 + aa%., a%. .... , a%} is a basis for Z"and
klX 1-- k2x2: k (x 1-- ax2) - bx 2 .
The minimality of k 1 shows that b=0.
(A.27) Theorem. A finitely generated group G is isomorphic with the
direct product o[ a [inite number o[ cyclic groups, where each [actor is
inJinite or has order a power o[ a prime.
Proof. Let {Yl .... , y,} b a set that generates G. For (ml, ... , m.
in Z , let
j=l
Then 0 is a homomorphism of Z"onto G, and G is isomorphic with Z'/H,
where H=0 -1 (e). If H= {0}, then G is isomorphic with Z'. Otherwise,
(A.26) shows that there are a basis {x I .... , x, for Z" and a sequence
{kl,..., k,} of positive integers such that {klXl,..., k,x,} is a basis for H.
The mapping W of Z"onto Z (kl) x... xZ (k,) xZ"' defined by
) (mlXl-'' "- mn,) = (m 1 (mod kl) , ..., m, (mod k,), mr+i, ..., m,),
is a homomorphism of Z"onto Z (kl) x... xZ (k,) xZ'-' whose kernel is H.
It follows that G is isomorphic with Z(kl)X...xZ(k,)xZ '. Finally,
by (A.25) each Z (k,) is a product of cyclic groups, each having order a
power of a prime.
Appendix B
Topological linear spaces
In his appendix, we sketch the theory of topological linear spaces.
Because of the plethora of lucid textbooks on various parts of the
subject, we have merely given references for many classical theorems.
We thought it worthwhile to prove the KI-IN-ML'tAN theorem, how-
ever, because of ffs vital r61e in he theory of representations [§ 22] and
the fac that no simple proof seemed readily citable.
(B.l Definition. Let F be a field and E an Abelian group, wrfften
addffiwly. Suppose that for each 0F and E, an element 0 in E,
called the #rodut o/0 and , is defined, for which the following axioms
hold:
(i) o(x+ y)=ex+oy;
(ii) (+/5)x=x+x;
(iii) (eft) x=e (fix);
(iv) tx= x;
29*
452 Appendix B. Topological linear spaces Appendix t3. Topological linear spaces 453
where x, y E and v.,/5 F. Then E is said to be a linear space [or vector
space over F, and F is called the scalar [ield [or E. The symbol 0 denotes
the identity element of the additive group E and also zero in F. The
linear space E is called a real or complex linear space if F is R or K,
respectively. For a subset A of E and mF, mA denotes the set {rex: xA}.
Whenever ,an ex,,pression such as "linear space", "Banach space",
"Hilbert space ', or linear functional" appears in this monograph and
the scalar field is not specified, the scalar field is the complex number
field K.
(B.2) Definitions. Let E be a linear space over a field F. A subset
E 0 of E that is itself a linear space over F is called a linear subspace o] E.
Let E 0 be a linear subspace of E and consider the quotient group E[E o.
If x and x' belong to the same coset of E 0 and m belongs to F, then ex
and ex' belong to the same coset. Therefore the group E[E o admits a
well-defined scalar multiplication, namely o(xEo)--(ox)E o. Then E[E o
is a linear space over F; it is called the quotient space or diJJerence space
o[ E by E o.
Let E be a real linear space. For x 1, ..., XrE, an element
elxl+'"чex, where el, ..., eR, each ej is nonnegative, and
e+...+e= t, is called a convex combination o[ Xl,..., Xm. A subset
A of E is said to be convex if each convex combination of elements in A
again belongs to A.
The direct sum of linear spaces E,, tI, is the usual direct product
P E, of the additive groups E,. Scalar multiplication is defined
coordinatewise. We frequently write the direct sum of a finite family
of linear spaces as E 1 ®... ®E.
Let E and E' be linear spaces over the same field F. A mapping T
of E into E' is said to be additive if T(x+ y)= T(x)+ T(y) for all x, yE,
and T is said to be homogeneous if T(ox)=oT(x) for all xE and F.
A function that is both additive and homogeneous is said to be a linear
Junction. Thus a function T is linear if and only if T(ox+ y) =eT(x) + T(y)
for x, yE and eF. Suppose that E and E' are linear spaces over K.
An additive function T such that T(ox)=T(x) for xE and eK is
said to be conjugate-linear. The term "linear transformation" is
synonymous with "linear function". An operator is a linear transforma-
tion of a linear space into itself.
A linear or conjugate-linear function on a linear space E into its
scalar field is called a linear [or con#gate-linear Junctional. We empha-
size that linear functionals always assume values in the scalar field of E.
(B.3) Theorem. Let E be a complex linear space, and let L be a
linear/unctional on E. For x E, write L(x) = L (x) + iL 2 (x), where L1
and L. are real-valued. Then we have:
(i) Li(x + y)=Li(x)+Li(y ) [or x, yE and i=l, 2;
(ii) Li(ox)=oLi(x ) ]or xE, oR, and i=l, 2;
(iii) L2(x)=--La(ix) ]or xE.
Proof. Properties (i) and (ii) are obvious from the additivity and
homogeneity of L. Property (iii) is proved from the identities La(ix)+
iL(ix)=iL(x)=iL(x)--L(x).
(13.4) Definition. Consider the Cartesian product ExE' of complex
linear spaces E and E'. A bilinear Junctional L is a functional on ExE'
that is linear in the first variable and conjugate-linear in the second
variable. In other words,
(,,+ y, ,,')=(,,, ,,')+ (y, x')
and
L(x, .x'+ y') = L(x, x')+ L(x, y')
for x, yE, x', y'E', and ,K.
(13.5) Definition. Let E be a real or complex linear space that is
also a topological T O space. Suppose that the mapping (x, y)--x + y of
ExE onto E is continuous and that the mapping (, x)--ox of FxE
onto E is continuous. Note that F is either R or K. Then E is said to
be a topological linear space. A topological linear space over R is said
to be locally convex if there is an open basis at 0 consisting of convex sets.
A neighborhood U of 0 in a topological linear space E is said to be
balanced if x U, ,F, and ]e I =< t imply ,x U.
Theorem (8.4) applies to every topological linear space E, since E
is a topological group under addition. Hence a topological linear space
is completely regular.
(B.6) Theorem. Let E be a topological linear space. The/amily o[
all balanced neighborhoods o/0 is an open basis at O.
Proof. Let U be any neighborhood of 0. By the continuity of scalar
multiplication at 0, there is a positive number e and a neighborhood W
of 0such that I[<e and xWimplyoxU. Let V=U{W-II<e }.
Then V is a balanced neighborhood of 0 and V c U.
(13.7) Definition. Let E be a linear space over F, where F is R
or K. A norm on E is a real-valued function [whose value at xE is
denoted by IIll having the following properties"
(i) [1 x 1[ > 0 whenever x =4= 0;
(ii) Jinx 11 = [ 1.11 x 1[ for all x E and F;
(iii) ]1 x + y [1 =< 1[ x [1 + 1[ y 1] for all x, y E.
Property (iii) is known as the triangle inequality. We call E a normed
linear space. The set {xE:llxl]<= } is called the unit ball o[ E.
454 Appendix B. Topological linear spaces Appendix B. Topological linear spaces 455
Let E be a normed linear space, and for x, y E, let d (x, y)= II x-y.
Then d is a metric on E. [The equality Iio1[-o onows from (B.7.ii).]
With the topology induced by d, E is a locally convex topological linear
space; this topology is called the norm topology o! E.
The fields K and R are always normed by I[1[--Il; this amounts
to defining lit I[= .
(B.8) Definition. Let T be a linear transformation of a normed
linear space E into a normed linear space E', where E and E' are both
real or both complex. If there is a positive number A such that
[[T(x)I[<A IIll for all xE, then T is said to be bounded. The infimum of
all such values of A is defined to be the norm o/T and is denoted by IITII.
If E' is the scalar field [i.e., R or K], then T is called a bounded linear
/unctional.
(B.9) Theorem. I/ T is a bounded linear trans/ormation o/a normed
linear space E into another normed linear space E', then
(B.10) Theorem. A linear trans/ormation T mapping a real or com-
plex normed linear space into another such space is bounded i/and only i/
it is continuous.
Theorems (B.9) and (B.t0) are very simple; we omit their proofs.
(B.11) Theorem. Let E be a real or complex normed linear space
and D a dense linear sub@ace o/E. A bounded linear/unctional L defined
on D can be extended to a bounded linear/unctional L o on E. Moreover,
IILolt = IlL It, d h x,so is qe.
Proof. If xE and lim IIx--,1[--0, where each x, is in D, then
{L (x,) },°°= 1 is a Cauchy sequence in K or R and so has a limit. Define
L o (x) to be this limit. This defines L o uniquely on E, L o is a linear func-
tional on E, and IILoll--IILII. a
We next state the essential part of the HAHN-BANACH theorem for
real linear spaces and some of its consequences.
(B.12) Theorem. Let H be a linear subspace o! the real linear space E.
Let p be a real-valued/unction on E such that
(i) p (ox)=op (x) i/o>= 0 and x E;
(ii) p(x+ y)_p(x)+p(y) /or all x, yE.
Let L be a linear/unctional on H such that L(x) <= p (x) /or x H. Let x o
belong to E f?H', and let H o be the linear subspace o/E generated by Xo
and H. Then we have
sup{--p (-- h-- Xo)--L(h)'hH}<=inf{p(h+ Xo)--L(h)'hH}.
Let o be any real number such that sup{--p(--h--xo)--L(h).hH}<:
%<=inf{p(h+xo)--L(h)'hH}. Then L can be extended to a linear
Junctional L o on H o such that L0(x0)=e 0 and Lo(x)<=p(x ) /or all xH o.
(B.13) Theorem. Let E, H, p, and L be as in (B.2). Then L can be
extended to a linear/unctional on E such that (x)<=p (x) /or all xE.
(B.14) Theorem. Let E be a real or complex normed linear space and H
any linear sub@ace o/E. Then any bounded linear/unctional L defined
on H can be extended to a linear [unctional on E such that the norm o/
on E is equal to the norm o/L on H.
(B.15) Corollary. Let H be a linear sub@ace o/ a real or complex
normed linear space E. For every x E f? (H-)', there is a linear/unctional
L o E s L (H) ---- {O}, L(x)--, IILI[-/, where d=
All of Theorems (B.t2)--(B.t 5) have been referred to as "the HAHN-
BANACH theorem".
Proofs of Theorem (B.t}) appear in : DAY [6], p. 9; DUNORD and
SCHWARTZ [t], p. 62; HILLE and PHILLIPS [t, p. 29; K6THE [t, p. t93;
MONROE [t, p. 56; and NAIMARK [t, §(I.t.9). Theorem (B.t4) and
Corollary (B.t 5) are stated and proved in DUNORD and SCHWARTZ
and HILLE and PHILLIPS [t.
(B.16) Definition. A [real or complex normed linear space E which
is complete in the metric induced by its norm is said to be a [real or
complex Banach space.
(B.17) Theorem. Suppose that E is a real or complex Banach space
and that H is a closed linear subspace o/E. For x+ H E/H, let
IIx+ HII = inf (lly II" y +
With this norm and with the linear operations o/ (B.2), the di//erence space
E/H is a Banach space.
Proof. If II + HI[--o, there is a sequence {h,,°__ 1 in H such that
2imoo II + , II = o. Since H is closed and lim II - (- h)II- o, belongs
to H and xчH=H. Thus (B.7.i) holds. It is trivial to verify that
IIчgII-Il.[lчgll for in the scalar field and xE, and that
[Ix+ y+H][ []x+gl[+ [ly+g[] for x, yE.
To show that E/H is complete, let {x,+H}, be a sequence of
cosets in E/H such that lira II- + HII: o. We cn now choose
subsequence [which for convenience we again write as {x. H}.] such
that I1-+ 11<2 - for . In particular, for each n, there is
an h,H such that ][x,+--x,+h,[[<2-". Let y,=x,+h,_l+...+h 1
for n=t, 2 ..... Then {y,}= is a Cauchy sequence in E, since for
456 Appendix B. Topological linear spaces Appendix t3. Topological linear spaces 457
m>_ n, we have
[[Y,-- Yn][= , [x+l-- x + hl < IlXk+l-- X k- hl[< 2 -'+1.
k=n k
Since E is complete, there is a y E such that illy-yfl =0. Therefore
2i ]I(Y
because x,+ H = y,, + H for all n. That is, y+ H = lim (x, + H), and so
E/H is complete.
We next take up the BANACH-STEINHAUS theorem. Some preliminary
results on category are needed.
(B.18) Definition. Let X be a topological space. A subset A of X
is nowhere dense in X if the interior of A- is void. A subset A of X is
said to be of the [irst category i X if it is the union of a countable family
of nowhere dense sets. If a subset A of X is not of the first category,
then it is said to be of the second category in X.
(B.19) Theorem. Let X be a complete metric space, a locally countably
compact regular space, or a locally compact Hausdor[[ space. Then X is
o[ the second category in itsel[.
The first assertion is proved in KELLEY [2], p. 200. For the others,
see (5.28)supra.
(B.20) Theorem [BANACH-STEINHAOS]. Let E and E' be Banach
spaces over the same scalar/ield. Let be a set o/bounded linear trans/orma-
tions o/E into E'. I/the set A o/points x/or which sup{liT(x)
is o/ seco ceeo , e s {llrl[ : r } o/.,Vs is Vo.e.
I oer os, ee is
all T6.
Proof. For m = t, 2 ..... let A denote the set of elements x in A
such that sup{]T (x) [1 : TN} m. For each T the set {xE: ]T(x)] m}
is closed because the mapping xllT(x)]] is continuous. Therefore
A = r{x E" ]T (x)1 m} is closed. Since A -- =U A is of the second
category in E, some A, say Amo, must have nonvoid interior. That is,
there is an x 0 in E and an e > 0 such that
{e: lls-x0ll<,)c A..
For any nonzero element x in E, the element x + x o belongs to the
set {yE" Ily--Xol < e} and therefore we have T 211II x + Xo m0
for all T9/. Consequently, we have
__-.- 9.11-.i I x ч x0 ч liT (-
and hence Ilzll < gin° for all T6N
xo)ll 1 < 21111
-- Fg.mo ,
(13.21) Definition. Let E be a real or complex topological linear
space. The set of all continuous linear functionals on E is denoted by E*;
E* is called the con1"ugate space o[ E. For L, M E* and x E, we define
(LчM)(x)=L(x)чM(x). Also for LE* and a scalar , we define
(L) (x)=. L(x).
(B.22) Theorem. Let E be a real or complex normed linear space.
With the linear operations de/ined in (B.2) and the norm de/ined in (B.8),
E* is a Banach space.
Proof. We omit the proof that E* is a normed linear space. Let
{L.°°__ be a Cauchy sequence in E*. For xE, we have [L,x)--L,,,(x)[<__
][x[['[[L,Lml I so that {L.x)}.°= 1 is a Cauchy sequence of numbers.
We define L(x) to be lim L.x). Since
IczL(x) q- L(y)- L(ox q-
< I cz L(x) -- o L,(x)] + [L(y) -- L, (y) l + IL, (ox + y) - L(ox +
it follows that L(ex+ y)=eL(x)+ L(y) for x, yE and m in R or K.
Suppose that I[L--LmI[ for , m. Then for xE, we have
I()- (x) I(x)- (x)l + [ (x)-
Taking the limit as m, we find that ]L(x)--L,(x)]e[lx]] for xE.
This shows that L is bounded and that 2i ILL--L,,][=0.
(B.23) Let E be a real or complex normed linear space. For each x
in E, let x' denote the bounded linear functional on E* defined by
(i) x'(L)=L(x) for LE*.
Plainly x' belongs to the conjugate space E** of E*, and
The mapping xx' is thus a linear norm-preserving mapping of E
into E**. The space E is said to be reflexive if this mapping carries E
onto E** E must be complete for this to occurS.
In addition to the norm topology, there are other important topol-
ogies that may be defined on normed linear spaces. We define two
such topologies.
458 Appendix B. Topological linear spaces Appendix B. Topological linear spaces 459
(B.24) Definitions. Let E be a real or complex normed linear space.
For each xE, each finite subset {L 1, ..., L,,} of E*, and each e>0,
define
(i) U(x;L,...,L,;e)--{yE:IL(y)--L(x) I <e for k -- , ..., n}.
The sets (i) are an open basis for a topology for E; this topology is called
the weak topology [or E. It is the weakest topology on E for which each
L in E* is continuous.
Consider the conjugate space E*. For each LE*, each finite subset
{xl,..., x,} of E, and each e> 0, define
(ii) U(L;x 1 .... ,x,,;e) = {ME*:IM(xk)--L(xk) I < e for k=t,...,n}.
The sets (ii) are an open basis for a topology for E*; this topology is
called the weak-, topology [or E*. It is the weakest topology on E* for
which each x' in E** is continuous on E*.
The weak and weak-, topologies on E* coincide if E is reflexive.
(B.25) Theorem [ALAOGLLI]. Let E be a real or complex normed
linear space. Then the unit ball
in the weak-, topology o/ E*.
Proof. For each xE, let I, denote the set of scalars e such that
]0]<___llxl]. 1 For LS*, let z(L) be the element in ,PEI, such that
(z(L)),=L(x). It is elementary to verify that is a homeomorphism
of S* into ,PEI,. By the TIHONOV product theorem, ,PI, is compact;
hence it suffices to show that z(S*) is closed in P
Let (t,) be in z(S*)-c,PI,. Let x and y be fixed in E. For
there is an L in S* such that It,--L(x)l< e, Ity--L(y)l<e, and
1,+-- L(x+ Y)I < e" It follows that It,+ ty-- t,+:l < 3 e. Thus x-+t, is
additive; homogeneity is proved in like manner. Since I t, l<_--_IIxl], the
mapping x-t, is a bounded linear functional on E whose norm does not
exceed t; hence (t,) belongs to z(S*).
We now take up the I{REIN-MIL'MAN theorem. Some preliminaries
are needed.
(B.26) Lemma. Let E be a real, locally convex, topological linear space,
C a closed convex subset o/E, aўd z a point in E Cl C'. Then there is a
continuous [real linear [unctioal L o on E such that
(i) inf {L o (x): x C} > L o (z).
Proof. Since (i) is equivalent to the inequality
inf{L o (x)'x C-- z} > Lo (0)--0,
1 For a real space E, I x is a closed interval in R. For complex space E, I is a
closed disk in K. In both cases I, is compact.
we may. suppose that z--0. Let U be a convex neighborhood of 0 such
that U- is disjoint from C, and let V--U Cl (--U). Let c be any point
in C. Then Co--V-+ (--C)+ c is easily seen to be a convex set such
that OVC o and cўC o If xE, then
• --x belongs to C o if e is a
sufficiently large real number; we define
p(x) inf {0R • 0 > 0, }
= -x Co •
It is trivial to verify that p satisfies (B.2.i). Suppose that --t x and t
belong to C o. Then e - y
o + (x + y ) -- o + ;
as this is a convex combination of 1
-- x and y, (x + y) belongs
?-
to C o. This shows that p(xч y)<__p(x)+p(y); i.e., (B.t2.ii) holds.
Note that p(c)_>_ t. For eR, let L(ec)--o. Then L is a linear func-
tional on {oc'eR}. If e_>--0, then L(oc)=e<=op(c)=p(oc); if e<0,
then L(oc)=e<o<=p(ec). Thus the HAHN-BANACH theorem (B.13)
shows that L can be extended to a linear functional L o on E such that
L o (x) ____ p (x) for all x E.
For every x V- and yC, the element x--y+ c is in Co, so that
p(x--y+c)<_l. Therefore Lo(x--y+c)<=t. Since Lo(c)=t and L o is
linear, we infer that Lo(x ) <=Lo(y ). Therefore we have
sup{Lo(x)'x V-}<inf{Lo(y)'yC}<=Lo(c)=t . (2)
There is a positive real number d such that dcV. Thus Lo(dC)=d ,
and so sup{Lo(x)'xV- } is positive. In other words, we have
inf {L o (y)" y C} > 0 = L o (0).
To see that L 0 is continuous, consider any e> 0. If x belongs to e V,
then t (-)
V x belongs to V and (2) implies that L o (x)=eL o x <_ e. Since
V= V, we also have --L 0(x)<_e so that IL0(x) l<_e. Thus L 0 is
continuous at 0, from which it follows that L 0 is continuous everywhere
on E (5.40.a).
(B.27) Definition. Let C be a convex subset of a real linear space E.
A convex subset B of C is said to be an extreme subset o/C if the relations
x, yC, ex+(t--e)yB, and 0<0<t imply that x and y are in B.
If {x} is an extreme subset of C, then x is said to be an extreme point o! C.
(B.28) Remarks. Let C be a convex subset of a real linear space.
The intersection of a family of extreme subsets of C is an extreme subset
1 Note that the closure in E of a convex set is convex.
460 Appendix B. Topological linear spaces Appendix B. Topological linear spaces 461
of C. If B is an extreme subset of C, and A is an extreme subset of B,
then A is an extreme subset of C. These statements are easy to verify;
we omit the details.
(B.29) Lemma. Let C be a compact convex subset o/a real topological
linear space E, let L be a continuous linear /unctional on E, and let
=min{L(x):xC}. Then the set S--{xC:L(x)=fl} is a nonvoid com-
pact extreme subset o/ C.
Proof. Plainly S is compact, convex, and nonvoid. For xC, yC,
and ax+ (t--a)y S, assume that x, say, belongs to C N S'. Then L(x)
and hence L(x+(t--)y)=L(X)+o(t--)L(y)>fl+(t--)L(y)>=
+ (t--) fl--fl, which is a palpable contradiction.
(B.30) Theorem [ KREiN-MIL'MAN]. Let C be a compact convex subset
o/ a real, locally convex, topological linear space E. Let B denote the set
o/all convex combinations o/extreme points o/C. Then
Proof 1. The fact that C has any extreme points at all is itself non-
trivial. This will follow from step (I) of the proof, since C is a compact
extreme subset of itself.
(I) Every nonvoid, closed, extreme subset A of C contains an extreme
point of C. Let be the family of all nonvoid closed subsets of A that
are extreme subsets of C. If is a subfamily of 5' linearly ordered by
inclusion, then N{B:BoW} is nonvoid since C is compact. Moreover,
by (B.28), {B : B } is an extreme subset of C and hence belongs to
ZORN'S lemma implies the existence of a minimal set B 0 in 5'. Assume
that B 0 contains two distinct points y and z. A special case of (B.26)
shows that there is a continuous linear functional L on E such that
L(z)<L(y). I ----min{L(x):xBo}, then {xBo:L(x)----fl} is a nonvoid
extreme subset of B 0 (B.29), and it does not contain y. But by (B.28),
{x Bo:L(x ) =/5} must also be an extreme subset of C, and this contradicts
the minimality of B 0.
(II) Let B denote the set of all convex combinations of extreme
points of C. Then B- is a compact convex subset of C. Assume that
C (B-)' is nonvoid and contains, say, x 0. By (B.26), there is a con-
tinuous linear functional L on E such that min{L(x):xB-}>L(x0).
Let fl----min{L(x):xC}; then {xC:L(x)=fl} is a nonvoid closed
extreme subset of C disjoint from B-. By step (I), the set {xC:L(x)--fl}
contains an extreme point of C. This is plainly impossible.
We now take up the problem of representing linear functionals on
certain spaces of functions as differences of nonnegative linear func-
tionals. The matter can be treated in more generality than is done here,
This proof is taken from KELLEY
but at the cost of technical complications. See for example BOURBAKI [2,
pp. t 7--40, or BIRKHO [2, pp. 238--248.
(B.31) Definitions. Let Y be a nonvoid set, and let be a real
linear space of real-valued functions on Y [linear operations are point-
wise] such that/ implies [/]. Then if/,g3, we have max(/, g)--
Ѕ(/+g+[/--g[) and min(/,g)--Ѕ(/+g--[/--g]). The symbol ч
will denote the set of all nonnegative functions in . A linear functional
L on is said to be nonnegative if L(/) >_ 0 for all/ч. A linear functional
L on is said to be strictly positive if L(/)>O for all / in ч except
/--0. A linear functional L on will be called relatively bounded if
for every /ч, the set {L(g):g, [g]<___/} is a bounded set of real
numbers 1.
Throughout (B.32)--(B.37), Y and will be as in (B.3t).
(B.32) Lemma. Let L be a real-valued/unction on ч such that"
(i) L(/+ g)--L(/)+ L(g) /or/, gч [L is additive];
(ii) L(/)----mL(/) /or/ч and m>_O L is nonnegative homogeneous].
Then L can be extended in exactly one way so as to be a linear/unctional on .
Proof. Since /-- [/] + / ]/[- / every / in is the difference of
2 2 '
two functions in ч. If/=/1--/. where /1,/2 ч, then define L, (/) --
L(/1)--L(/2). If /---- /1-- /. -- /3-- /4 with all /i ч, then /1+/4--/3+/.,
and so by (i), L(/1)--L(/)----L(/)--L(/). That is, L, is defined uniquely
on and is an extension of L. It is easy to verify that L, is a linear
functional on .
(B.33) Lemma. Every nonnegative linear /unctional L on is
relatively bounded.
Proof. If/ч, g, and [g[<___/, then/--g>--O and/+g>=O. There-
fore L(/)--L(g)>=O and L(/)+L(g)>=O, so that [L(g)[L(/).
The linear space * of all relatively bounded [real linear functionals
on admits a partial ordering: for L, M*, we write L >_ M if L--M
is a nonnegative linear functional.
(B.34) Theorem. Under the partial ordering >=, * is a lattice. That
is, given L, M*, there is a least element max (L, M) o/ * greater than
or equal to L and M, and there is a greatest element rain (L, M) o/ * less
than or equal to L and M. For/ч, we have
(i) max(, M)(/) = su{L()+ M(/--) : ч, <__/}
and
(ii) min(L, M)(/) = inf{L(g)+M(/--g):g6 ч, g/}.
1 If is a normed linear space such that llgll_ll/ll whenever Igl-l/I and
/, g , then a bounded linear functional on is relatively bounded.
462 Appendix B. Topological linear spaces Appendix B. Topological linear spaces 463
Proof. If I*, I>=L, and I>=M, it is clear that I(/) must be greater
than or equal to the right side of (i) for [+. Let I1(/) denote the
right side of (i)" note that /1(/) is a real number for all [ч since L
and M are relatively bounded. We next show that I 1 is additive and
nonnegative homogeneous on +. Since L(og)--eL(g) and M(og)=eM(g)
for eR and g, it follows at once that Ii(o[)=oIi(/) for [+ and
.->- O.
To prove that I1 is additive on -+, take [1 and [,. in +, and 0 < gi </i
(}'-- 1, 2) such that L(gi) + M(/i- gi) + - > I1 (/i)" Then
I1 (/1 @/2) L (gl-Jr- g2) @ M (/1 _Ay/2-- gl-- g2) > I1 (/1) _Ay I1 (/2) -- 8.
Hence we have
I1 (/1 @ ]9.) >_--- I1 (/1)@ I1 (]9.).
To prove the reversed inequality, choose gч such that g<=]l+]9, and
L(g) A V M(/1-@/2-- g) Av 3 > I 1 (/1-2y/2). Write gl-- min (g,/1) and g.= g-- gl.
Then we have 0 =< gl </1, 0 = gg. <=/9., gl@ g2-- g, and so
I1 (/1 @/2) <Z L (gl @ g2) @ M (/1 _2y 12- gl- g2) @
-- L (gl) @ M (/1 -- gl) @ L (g2) -- X (/2-- g2) -- e < I 1 (/1) - I1 (]2) @ e.
Therefore I is additive on +. By (B.32), I1 can be extended uniquely
to a linear functional on " clearly this extension is the least element
max (L, M) greater than or equal to L and M. Obviously L is relatively
bounded, and by (B.33), max(L, M)--L is relatively bounded" hence
max (L, M) is relatively bounded.
Relation (ii) follows from the identity min(L, M) =--max(--L, --M).
(B.35) Theorem. The space * is a complete lattice. That is, i/
{L,}r is any subset o/ * that is bounded above [below], then this set has
a least upper [greatest lower] bound.
Proof. Suppose that Lv<_M for all y_P. For [+, let
I(/) = sup {max (L,,,..., Lrm ) (/)" {)"1, • • •, )"m} C/-}.
It is easy to show that I is additive and nonnegative homogeneous on ч,
so that I can be extended to a linear functional I o on (B.32). It is also
easy to see that I o is the least linear maiorant of all of the L s. For any
o1 , Io--Lvo is nonnegative and hence relatively bounded (B.33).
Therefore Io= (Io-- Lr0)+ Lo is relatively bounded.
Sets {L}r bounded below are treated by examining {--L}r.
(B.36) Theorem. A linear /unctioT, al o can be expressed as the
di//erence o/two nonnegative linear/unctionals i/and only i/it is relatively
bounded.
Proof. If a linear functional is the difference of nonnegative linear
functionals, then it is relatively bounded by (B.33). To prove the
converse, consider a relatively bounded linear functional L on . Define
I1 = max (L, 0) and I. = I-- L. Then I 1 and I. are nonnegative linear
functionals on and L-- I-- I..
(B.37) Theorem. Let L be a relatively bounded linear/unctional on .
Then there are unique nonnegative linear/unctionals I 1 and I such that
L = I-- I and min (I, I) = 0. Furthermore, we have I = max (L, 0) and
19.=-- min (L, 0).
Proof. Suppose that L=J1-J9 . where J1 and J are nonnegative
linear functionals on . Then for/ч, we have
0 G min (fx, Jg.) (/) = inf {Jx (/- g) + Jg. (g) g ч, g G/}
= inf {Jx (/) -fx (g) +fx (g) - L(g)'g +, gG/}
-- inf {J (/) - L (g) " g +, g G /}
= fl (/) -- sup {L(g) • g+, g =</} = Ix (/) -- max (L, 0)(/).
Therefore we have rain (fx, fg.) =fx-- max (L, O) -->_ 0. It follows that
min (fl, fg.)=0 if and only if fx--max (L, 0).
The foregoing together with (B.36) shows that L=Ii--Ig. , where
I1-- max (L, 0) and min (I1, 19.) = 0. Since -- L = I-- I1, we must also
have 19. = max (-- L, 0) =-- min (L, 0).
Our final result along these lines concerns complex functionals on
complex linear spaces.
(B.38) Theorem. Let Y be a nonvoid set and a complex linear
space o/complex-valued/unctions on Y. Let " and ч denote the sets o/
real-valued and nonnegative real-valued /unctions in , respectively.
Suppose that/ implies and that every/" can be written in some
/ashion as the di//erence o/gwo/unctions in ч. Let 9 be any complex-
valued/unction de/ined on ч such that"
(i) I(/1-@/) --- I(/1) _2y I19(/9.) /or /1,/ + ;
(ii) #(e/)=e#(/)/or/ч and >0 [a9 is nonnegative homogeneous].
Then 9 can be extended in exactly one way so as to be a complex linear
/unctional on .
Proof. Just as in Lemma (B.32), can be extended uniquely to a
function #, on ' that is additive and real homogeneous. If ]F, then
1
Re [= :- (/ + ) and Im /= :7 (/-- ) are also in . Hence [ can be
uniquely written as [=1+ i] where /el, [9.'. The only possible way
to extend #, linearly over is to write # ([) = #, ([1) + iag, (1.). An
obvious calculation shows that #, is additive and complex homogeneous
on.
464 Appendix B. Topological linear spaces Appendix B. Topological linear spaces 465
We now sketch the theory of inner product and Hilbert spaces.
We give definitions largely to establish our notation. Theorems proved
in HAIMOS [3 are stated without proof.
(B.39) Definition. Let E be a complex linear space. Suppose that
there is a complex-valued function on ExE, whose value at (x, y) is
written as (x, y), having the following properties:
(i) (x + y, z) = (x, z) + (y, z) for x, y, z E ;
(ii) (ax, y) = a (x, y) for x, y 6 E and K;
(iii) (x, y) = (y, x) for x, yE;
(iv) (x,x)>0 if x4=0.
Then the function (x, y)-- (x, y) is called an inner product and E is
said to be an inner product space.
(B.40) Note. If E is an inner product space, then there is a natural
norm for E; for xE, define I]x]l= V(x, x). This is a legitimate norm
for E. This is not trivial; in fact, the triangle inequality depends upon
the CAUCHY-BUNYAKOVSKI]-SCHWARZ inequality: [(x, y)] ___< ]lxl]. ]]y]]
Isee HALMOS [3, P- t6.
(B.41) Theorem. I[ E is an inner product space, then
t t i i
(i) (x, y)= tlx + yll tlx-- YlI + - I[x + i Yll-- -4-Ix--
/or x, yE.
This is proved by a direct computation.
(B.42) Definition. A linear isometry is a linear mapping T of a
normed linear space E onto another normed linear space E' that preserves
the norm; i.e. []T(x)[ I --[]xl[ for all xE.
(B.43) Theorem. A linear isometry T between two inner product
spaces E andE' also preserves the inner product; i.e., (T(x), T(y) ) = (x, y)
/or all x, y E.
This follows from (B.4t).
The most important class of inner product spaces for our purposes are
those for which the associated norm induces a complete metric.
(B.44) Definition. An inner product space that is a Banach space
under the norm ]lxl[ = (x, x) is called a nilbert space.
In accordance with a common convention, we will denote Hilbert
spaces by H, H', etc. rather than by E, E', etc. In consonance with the
notation employed in the main text Chapter Five, passim], we write
elements of an abstract Hilbert space as , , co .....
(B.45) Theorem [F. RIESZ]. Let H be a Hilbert space. For every
bounded linear [unctional L on H, there is a unique H such that
L()= , 1) /or all H. For every conjugate-linear bouded /uuctional
M o H, there is a unique o)H such that M()=((o, ) /or all H.
(B.46) Corollary. The mapping L-- described in (B.45) is a con-
]ugate-liuear isometry carrying the space H* onto H.
A proof of (B.45) is given in HALMOS 3, P. 31.
(B.47) Definition. Let H be a Hilbert space. Two elements and
in H are said to be orthogonal if (, ])=0. More generally, two subsets
A and B of H are orthogonal if (, )=0 for all A and ]B. If
and ] are orthogonal, we write _1_ ]; similarly A_[_ B means that A
and /3 are orthogonal. For a subset A of H, A ± will denote the set
{ H : _[_ r] for all ] A }.
A single subset A of H is said to be orthogo,al if (, r])=0 for all
, rigA such that @r]. If we also have (, )--t for all A, then A
is said to be orthonormal. An orthonormal basis o/ H is a maximal
orthonormal subset of H.
(B.48) Theorem. Let M be a closed subspace o! a Hilbert space H.
The M ± is a closed subspace o/H, M ± is orthogonal to M, M ± ± = M, and
H=M@M ±.
See HALSOS [31, § 12, for the proof.
The next theorem justifies the definition following it. It is proved in
HALMOS [3], P" 29.
(B.49) Theorem. Let H be a Hilbert space. Then H has at least
one orhozormal basis, and any two orthonormal bases o/ H have t.he same
cardinal number.
(B.50) Definition. The dimensio,n of a Hilbert space H is the cardinal
number of any of its orthonormal bases.
We now consider adjoint operators. For the generality required in
§ 22, we must use an arbitrary complex normed linear space E, and define
a space E of functionals on E that imitates so far as possible the
formation of inner products in an inner product space . Again in conso-
nance with the notation of Chapter Five, we write elements of linear
spaces as s e, ], co, .... In handling operators T on E, it is standard and
convenient to write T for T()" we shall frequently do this in what
follows.
(B.51) Definitions. Let E be a complex normed linear space. Let
E denote the set of all bounded conjugate-linear functionals on E.
These functionals are added and multiplied by complex numbers point-
wise on E, and thus form a normed linear space. For (oE and eE,
we write the value of co at as (o), e). .
1 The following treatment of adjoint operators was suggested to us in con-
versation by N. ARONSZAJN.
2 A handy mnemonic is the rule "functionals come first".
Hewitt and Ross, Abstract harmonic analysis, vol. I 30
466 Appendix B. Topological linear spaces Appendix B. Topological linear spaces 467
(B.52) Remarks. The mapping (o), )--(o), ) has some of the for-
mal properties of the inner product in an inner product space. Thus,
for o's in E-, 's in E, and in K, we have"
<, }> : <, }>;
Since E and E are in genera] different entities, we can no longer
write
There exists a one-to-one correspondence between E and E*, as
follows. If oE and L(}):o(}):<o,}), then L belongs to E*.
Conversely, if L E* and o) () : L(, then belongs to E. Using this
correspondence, we make E into a complex Banach space. If E is
reflexive, ever element in E is determined b an element in E.
That is, for each E, the functional <, ) on E belongs to E
and all elements of E have this form. Moreover, this correspondence
is a one-to-one, linear, norm-preserving map of E onto E.
Consider a Hi[bert space H. Then H and H* can be identified, as
(B.46) shows. It is plain that H and H can also be identified" ever
bounded conjugate-linear functional on H has the form }<, ) for
some H, as (B.45) implies. Here <, ) denotes the inner product
in H.
(B.53) Definition. Let E be a complex normed linear space and let
T be a bounded operator I on E. For each in E, the mapping
}<, T) belongs to E; we define this mapping to be To. Thus
for oE and E. The function T carrying E into E is called the
adoin/ o/ the operator T.
It is easy to show that T is a bounded linear transformation, i.e.,
T is a bounded operator on E.
(B.54) Discussion. For an operator T on E, there is of course an
adjoint operator T on E. If E is a reflexive Banach space, then
E can be identified with E, as noted in (B.52), and T may be
regarded as defined on E. In fact, if the element in E that corresponds
to in E is denoted by 0, then the equality (B.3.i) takes on the form
<}0,
1 Recall that "operator on E" means "linear transformation carrying E into
itself" (B.2).
for }0 E "' and o) E ~. Identifying E "' with E, we obtain
(i) <To), }> -- <o), T~}>
for }E and
For the case of Hilbert spaces, the above definition of adjoint operator
reduces to the ordinary one.
(B.55) Theorem. Let H be a Hilbert space. I[ T and U are bounded
operators on H and is in K, then:
(ii) T= T;
(iii)
(iv) (r+ g)-= r+ U;
(v) (ru)"= ur.
For the proofs, see HALMOS [3, P. 39.
(B.56) Definitions. Let H be a Hilbert space and T a bounded
operator on H. If T= T-, then T is said to be Hermitian. It is easy
to show that a bounded operator T is Hermitian if and only if <T},
is real for all } H. If TT"-- T" T, then T is said to be normal. If T is
a linear isometry of H onto H, then T is said to be unitary. A bounded
operator T on H is said to be positive-definite if <T}, } =>_0 for all
}H. In particular, every positive-definite operator is Hermitian.
Let M be a closed linear subspace of H. Every element } in H can
be written in exactly one way as }1+}2, where }IM and .M ± (B.48).
The mapping P defined by P}--} is called the proection o[ H onto M.
For a Hilbert space H, 3(H) will denote the linear space of all
bounded operators on H.
(B.57) Theorem. A [unction P mapping a Hilbert space H into
itsel[ is a pro]ection i[ and only i[ it is an Hermitian idempotent operator.
Whenever this is the case, P is the proiection o/ H onto M= { H: P=},
This theorem is treated in HALMOS [3, §26.
(B.58) Theorem. For a bounded operator T on a Hilbert space H,
the/ollowing statements are equivalent:
(i) r--0;
(ii) < r, ] } = 0 [or all , H;
(iii) <T}, }=0 ]or all }H.
For the proof, see HALMOS [3, § 22.
(B.59) Theorem. Let H' be a dense sub@ace o! a Hilbert space H.
I[ T' is a linear isometry o[ H' onto a dense subspace o] H, then T' can be
extended uniquely to a unitary operator T on H.
30*
468 Appendix B. Topological linear spaces Appendix C. Introduction to normed algebras 469
This theorem is very like (B.I 1). We omit the proof.
(B.60) Theorem. Let H be a Hilbert space. Suppose that L is a
bilinear [unctional on HxH that is bounded: i.e., there is a positive real
number A such that [L(, )[ <=A II l[. II ll/or all , H. The,, there exists
a unique bounded operator T on H such that
(i) L(, )= (T, )
[or all , H.
For a proof, see HA,.os [3, §22.
We will have occasion to consider direct sums of arbitrarily many
Hilbert spaces and also direct sums of operators.
(B.61) Definition. For an arbitrary family {H}r r of Hilbert
spaces, the direct sum H= @H r is defined as follows. Let H be the
additive subgroup of the additive group rPrHr consisting of all (r)
for which Y, I1 11"< oo. Define scalar multiplication in H coordinatewise,
and an inner product in H by
(i) ((), (r)} = Y (r,
With the above definitions, H--@H r is a Hilbert space. Elements
of H are often written as
(B.62) Definition. Let {Hr}r r be a family of Hilbert spaces and
for each 7-P, let T r be a bounded operator on H r. Suppose also that
oo. We define the sum T= Tv onHby the following
relation" rr
(i) T((r))=
The function T is a bounded operator on H and lIT H = sup{llr[I.,/}.
We omit the proofs of these statements.
(B.63) Theorem. Let H be a Hilbert space and suppose that
is a [amily o[ sub@aces o[ H having the property that HrCH whenever
= , and such that Cl H# = {0}. Then n is isometric with @ n.
Proof. Let H 0 consist of those elements (r) in @ H- such that
rHr for all yF and r=0 except for finitely many y. Then H 0 is a
dense linear subspace of @ H-. For (r) H0, define
rEr rEF
It is easy to verify that is a linear mapping of H 0 into H. If (r)EH0,
then ( ((r)), ((]r)) } = (rr r ,rrr} =rr (r, r} = ((r),
is a linear isometry of H 0 onto z(H0). To see that z(H0) is dense in H,
suppose that EH and that (,}=0 for all Ez(H0). Then EH# for
all y/', and hence =0. As is arbitrary, z(H0) is dense in H. A
routine argument shows that z can be extended in exactly one way to
be a linear isometry of @ H-onto H.
Two more results needed in the text rightfully belong in this appendix.
However, it is convenient to state and prove them in Appendix C after
the elementary theory of Banach algebras has been described. These are
the spectral theorem (C.40)--(C.42) and the fact that a positive-definite
operator on a Hilbert space has a unique positive-definite square root
Appendix C
Introduction to normed algebras
In this appendix we sketch the elementary theory of Banach algebras.
We give only that part of the theory needed for our study of M(G)
and its subalgebras. A discussion of comparable scope appears in
LOOMIS [2. Far more complete treatments appear in TAMARK Et and
RICKART ItS. We include a proof of the spectral theorem, since it will
follow simply from machinery needed for other purposes.
(C.1) Definitions. Let A be a linear space over a field F. Suppose
that for each pair x, y of elements from A a product xy is defined, that
under this operation A is a semigroup, and that this product is related
to the linear operations in A as follows:
(i) x (y + z) = xy + x z,
(ii) (x+ y)z= xz+ yz,
(iii) (ex) y= x(ey)=e(xy),
for all x, y, zA and all eF. Then A is said to be an algebra over F.
An algebra with unit is an algebra having a multiplicative identity u @ 0;
a commutative algebra A is one for which xy--yx for all x, yEA. Real
and complex algebras are algebras over R and K, respectively. We recall
that an algebra with an unspecified scalar field is a complex algebra (B. t).
A Banach algebra normed algebra A is a Banach space Inormed
linear space I that is also an algebra and satisfies the following condition:
(iv) Ilxyll < Ilxil.
for all x, yA. If A has a unit u, we will often require that
Iv/
A linear subspace I of A is a le/t ideal if xI, zA imply zxI. A
linear subspace I is a right ideal if xI, zA imply x zI. A two-sided
ideal is a left ideal that is also a right ideal. An ideal I of A such that
I4= A is a proper ideal. An algebra having no proper two-sided ideals
except {0} is said to be simple. In a commutative algebra, the distinctions
among left, right, and two-sided ideals vanish. In this case, we write
"ideal" for these entities.
470 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 471
An algebra homomorphism, as one might expect, is a mapping z of
an algebra A into another algebra B that is linear and also multiplicative"
z(xy) =z(x)z(y)for x, yA.
(C.2) Theorem. Let A be a normed algebra and I a closed two-sided
ideal in A. Let A/I denote the normed linear space A/I described in
Theorem (B.I 7). For x + I, y + I A/I, de/ine (x + I) (y + I) -- xy+ I.
With this multiplication, A/I is a normed algebra, and A/I is a Banach
algebra i/A is. I/A has a unit u and lib]I--1, then lib+ I][--.
Proof. It is easy to see that the product is well defined and that
A/I is an algebra. By (B.7), A/I is a normed linear space which is
complete if A is complete.
We next verify that (C..iv) holds for A/I. Consider x+ I, y+ I in
A/I and e>0. For some v and w in I, we have tim+vii. Ily+wll<
IIxч ll Ilyч llч . Since vy+ xw+ vw belongs to I, we have
Ilxyч ll--< Ilxyч vyч xч vll- II(xч v)(yч )ll
-<_ IIxч vii. ][y+ ll < Ilxч zl[ Vч zll ч .
Therefore IIxч 11--< IIxч 11 Ilyч zll that is, (C..iv) holds in A/I.
Suppose finally that A has a unit u and that II"ll-a. It is obvious
that II-ч ll <- a. Moreover, we have II-ч 11- II<-ч )ll =< tl-ч ll , so
that 1=< Iluчll.
We now show that any algebra can be imbedded in an algebra with
unit.
(C.3) Theorem. Let A be an algebra. Let A be the set o/ all pairs
(x, ) with xA and K. De/ine
(i) . (x, 2) -- (x, 2),
(ii) (x,)+(y,#)--(x+ y,+#),
(iii) (x, ) (y, #) -- (xy+ ly+#x,:t),
/or (x, ), (y, lt)A, and czK. Then A,, is an algebra with unit (0, ),
and its subset {(x, 0)'xcA} is isomorphic with A in every way. The
algebra A,, is commutative i/ and only i[ A is commutative. I/ A is a
normed [Banach algebra, then A, with the norm
(iv) II(x, 2)iI = Ilxll + It I,
is a normed [Banach algebra.
Proof. All necessary verifications are easy. For example, the in-
equalities
-< Ilxll-Ilyll ч Il-Ilyllч I.I-Ilxli ч I1-I.I
- (llxll ч I1)(llyll ч I I) --II(x, )ll. II(y, :,)11,
establish (C.l.iv) for the algebra A u. [
We next establish some technical properties of normed algebras.
(C.4) Theorem. Let A be a normed algebra. For every x in A the
limit lim t[x'[I 1/' exists,
(i) ,li II:[[ 1/-- inf {[[:l[ 1/- n-- 1, 2 .... },
and
(ii) lim Ix-IIl/lxll .
Proof. Let -- inf([:ll 1/. = a, 2 .... }. obviously e lim II:[ll/. Con-
sider e > 0 and choose a positive integer m for which II:lll/< + . or
every positive integer , we write n = a.+ b.where 0 b,.< m. Then
II:ll II:[I IIxlt , and therefore
II:ll 1/" I1:11 / IlxllO/ ( + )/-IlxllO/.
Since i
It follows that : lira [l:l[/. R]tion (ii)is obvious, since II:l[ Ix]l
for all n.
For algebras without unit, the notion of quasi-inverse is important.
(C.5) Definition. Let A be an algebra and x an element of A.
An element y in A is called a le/ quasi-inverse right quasi-inverse /or x
if x+ y--yx=O x+ y--xy=O. If y is both a left and a right quasi-
inverse for x, then y is called a quasi-inverse/or x.
Suppose that A has a unit u. Then x has a quasi-inverse y if and
only if x--u has a multiplicative inverse y--u. The element u--x
has an inverse z if and only if x has a quasi-inverse u--z. Thus quasi-
inverses are substitutes for inverses in algebras without unit.
Let A be an arbitrary algebra. II x in A has a left and a right quasi-
inverse, these elements are equal. Therefore quasi-inverses are unique.
(C.6) Theorem. Let A be a Banach algebra and let x be in A.
]i II:lIm/< a, h x hs qsi-iverse y ad
(i) y=-- E x.
=1
Proof. From the hypothesis lira I[x"[/"< 1, it follows that the series
It:ll converges. If y,=-- x for n=t, 2, ..., then {y,}, is a
Cauchy sequence in A, since ][y.- y.] = x IIxtl for > m.
k=m+l =m+l
Let y in A be the limit of the sequence {3.}.=t. Clearly, we have
xy.=y.x--x+ y,.,
At least three different tertns have been employed for what we call quasi-
inverses- we follow RICKART [, p. 6, but not LOOMIS [2, p. 64 or HILLE and
PHILLIPS [1], pp. 680--6S.
472 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 473
for each n. Taking limits in (t) as n-->oo, we obtain xy--yx--x+ y"
that is, x+ y--xy--x+ y-- yx--O.
(C.7) Corollary. Let A be a Banach algebra with unit u. I! x is in A
and lim ]l(u--x)"llt/< t, then x -x exists and
(i) x -x- u + (u-- x) .
(C.8) Corollary. Let A be a Banach algebra and x an element o] A
(i) ч IIll = = -IIll "
Proof. By (C.6) and (C.4.ii), x has a quasi-inverse y. Since
x+ y-- xy--O, we have Ily][-- ]]xy- x][ ___< ]]xl]. Ily]] + ]]x]] and ]]x]] =
Ily- xy II
(C.9) Theorem. Let A be a Banach algebra, l/ x and y are in A,
i! y hs qusi-ivrs y', d i! IIxll ч Ity' I' then
(i) x+ y has a quasi-inverse z,
and
(ii)
- -- (I '11+ )I111 "
The set E--{yeA'y has a quasi-inverse) is open in A. Let y' denote
the quasi-inverse o] y; then the mapping y--> y' is a homeomorphism o[ E
onto E.
Proof. Since Ilx-y'xll<__ Ilxll(ч Ily'll)<, x-y'x has a quasi-
inverse v (C.8). Then -- vy' + y'+ v is a le[t quasi-inverse for x + y since
(x+ y) + (-- y'+ y'+ )-- (-- y'+ y'+ ) (x+ )
=(y+ y'--y' y)-- (y+ y'--y' y) + [(x-- y' )+ -- (x-- y' )] = 0.
Similarly, x--xy' has a quasi-inverse w and --y' w+ y'+ w is a right
quasi-inverse of x+ y. Corsequertly z --- v y' + y' + v is tile quasi-
inverse of x + y. This proves (i).
Using (C.8.i), we obtain
< I- y'l I111 (lly'll +
I1 '-11 (IlY'IIч ) < •
- - - - -(lly'll+)llTll
This proves (ii).
Suppose now that y is in E, and consider any yea for which
II-yll< + Ily'll" Then by (i), the element v--yчy-v has a quasi-
inverse. That is, the set E is open in A. Plainly the mapping y-->y'
is one-to-one and onto and is its own inverse. If y eE and II--
then (ii) implies that
1+ y'l'
iIv,_,ll< - (' ч
-- -(y' ч) -yl" ()
I IIv-ll i uiўienty m, th right id o () i rbitrariy ma.
This implies that the mapping yy' is continuous.
(C.10) Corollary. Let A be a Banach algebra with unit u. The set
E o {y A" y- exists} is open and the mapping y y- is a homeomorphism
o/ E o onto Eo.
Proof. This follows from (C.9) and the identity y-=u--(u--y)',
which is valid for y e Eo, where ' is as in (C.9).
(C.11) Theorem [GEL'FAND-MAZOR]. Let A be a Banach algebra
with unit u in which every nonzero element has an inverse. Then
A = {2u" }, and hence A can be identi/ied with .
Proof. Let x be an element in A and assume that x--2u 0 for all
2e. Then (x--2u) -1 exists for all 2e. Let L be any bounded linear
functional on A u tat L(x-)=, ana let ()=L((x--u)-) for
2e. Then g(0)=" and we will show that g is an entire function. It
is easy to verify that (x-- 2u)-(x - 2oU)-= (x - 2oU)-(x - 2u) - for all
2, 20e. Using this, we find that
(x - u)-- (x- o)-= (x- )-x (x- 0,)-[(- 0)- (-
= (x- 2u) -x (x- 2oU) - (2- 20),
and therefore
g (2) - g (2o)
o - o L ((x- u)-,-(x- 0u)-)
_ _ [ ()
- L ((x- u)- (- 0u)-).
The mapping 2x--2u is plainly continuous. By (C.0), the mapping
2(x--u) - is also continuous, as is the mapping 2(x--2u)-(X--2oU) -.
Since L is bounded, it follows that
i ((x-u)-(X-oU)-) = L ((X-oU)-). (2)
o
Relations () and (2) show that g has a derivative at 2o. Thus g is an
entire function.
For 2@0, we have (x--2)---2 - -- . Using continuity of the
mappingyy-,weobtain (---=--- and hence
k /
lim (x--) -= lim --2 --=0-
474 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 475
these limits are taken in the norm topology of A. Since L is bounded,
we infer that
lim g(2) = lim L ((x-- ,U) -1) --- 0. (3)
Accordingly, g is bounded and by LIOUVlLLE'S theorem, g is a constant.
We infer also from (3) that g(2):0 for all 2cK. Since g (0): 1, we have
a contradiction.
(C.12) Definition. Let A be an algebra over t7. A multiplicative
linear /unctional on A is a nonzero linear functional on A that is
multiplicative : z (xy) : z (x) z (y) for all x, y C A. In other words, is an
algebra homomorphism of A onto K. 1
The following theorem applies in particular to multiplicative linear
functionals, which play a fundamental r61e in the theory of commutative
Banach algebras.
(C.13) Theorem. Let A be an algebra over a/ield F, and let z be an
algebra homomorphism o/A onto a simple algebra B over F that is di//erent
/rom {0}. Then z-(O) is a maximal proper two-sided ideal in A. I] M
is a maximal proper two-sided ideal in A, then AIM is simple.
The proof is easy and is omitted.
The study of multiplicative linear functionals on commutative Banach
algebras can be reduced to the study of such functionals on commutative
Banach algebras with unit because of the following theorem.
(C.14) Theorem. Let A be a Banach algebra and let A, be as in (C.3) ;
A is identi/ied with {(x, 0):xcA}. Every multiplicative linear ]unctional
z on A admits a unique extension z, on A, that is a multiplicative linear
/unctional on A,. Every multiplicative linear /unctional on A, is a
multiplicative linear/unctional on A when restricted to A, except/or the
/unctional Too de/ined by zoo((x, 2)) :2.
Proof. Let z be a multiplicative linear functional on A and suppose
that z, is a multiplicative linear functional on A, that extends z. Then
z,((x, 0))=z(x) is defined for all xcA and z,((0, 1))=1 since (0, 1) is
the unit of A,. It follows that z,((x, 2))=z(x)+2 for all (x, 2)cA,.
Hence there is at most one extension of z over A, that is a multiplicative
linear functional. Further, it is easy to see that z, as defined above is a
multiplicative linear functional on A,.
The remainder of the theorem is likewise easily proved" we omit the
details.
(C.15) Definition. Let A be any algebra. A left ideal I in A is said
to be a regular le/t ideal if there is an element v in A such that
(x+ I) (v+ I)=xv+ I= x+ I
1 It is arbitrary but convenient to require that multiplicative linear functionals
be nonzero; see for example (C.I 7).
for all xA. We say that v is a right unit relative to I. Regular right
ideals are defined analogously. A two-sided ideal I in A is regular if
A/I has a unit v+ I:
vx+I=xv+I=x+I
for all x A.
Obviously every left, right, and two-sided ideal in an algebra with
unit is regular.
(C.16) Theorem. Let A be an algebra and I a proper regular le/t
ideal in A. Then I is contained in a maximal proper le/t ideal which is
also regular. I/A is a Banach algebra and I is a maximal proper regular
le/t ideal in A, then I is closed. Similar assertions hold/or right and two-
sided ideals.
Proof. Let v be a right unit relative to I" then plainly v is not in I.
Let x ў denote the family of all left ideals J such that I:J and vcJ.
i is a subfamily of oў linearly ordered by inclusion, then U {J:J oW}
also belongs to x ў. Thus ZOR'S lemma applies, and x ў contains a
maximal element J0. The ideal J0 is a maximal proper left ideal con-
taining I, and it is regular with relative right unit v.
Now let A be a Banach algebra and let I be a maximal proper regular
left ideal in A with relative right unit v. It is simple to verify that I-
is a left ideal in A; we omit the argument. If v is a right unit relative
to I, then clearly v is a right unit relative to I-; that is, I- is a regular
left ideal. Assume that I c, I-; then I---A since I is maximal. Thus
there is a z in A such that []zl[< 1 and v--z belongs to I. By (C.8), z has
a quasi-inverse z': z+z'--z'z:O. Now we have
v = (v- z)- z'(v- z) + z' v- z'.
Obviously v--z and z'(v--z) belong to I; z' v--z' also belongs to I since
z'+I=z'v+I. Therefore vI, and it follows that I=A, contrary to
hypothesis.
The statements involving right and two-sided ideals are proved
similarly. [
Theorem (C.13) implies that the kernel of a multiplicative linear
functional on a commutative Banach algebra A is a maximal proper
regular ideal in A. We now prove the converse of this statement.
(C.17) Theorem. Let A be a commutative Banach algebra. Then
every maximal proper regular ideal M in A is the kernel o/ a [unique]
multiplicative linear/unctional z on A.
Proof. By (C.16), M is closed. By (C.2), AIM is a Banach algebra.
Since M is maximal, AIM is simple (C.t 3)- Since M is regular, AIM has
a unit v + M. Since A is commutative, AIM is also commutative. From all
476 Appendix C. Introduction to normed algebras Appendix C. Introduction to norrned algebras 477
this, it is easy to see that every nonzero element of AIM has an inverse.
Theorem (C.11) implies that A/M--v+M:cK). For each x in A,
there is a unique complex number z(x) such that xcz(x)v+M. It is
easy to check that z is a multiplicative linear functional on A, that
M--z-l(0), and that z is uniquely determined by these two require-
ments.
We next define the spectrum of an element in an algebra, and proceed
to relate this notion with multiplicative linear functionals (C.20).
(C.18) Definitions. Let A be an algebra with unit u and let x be an
element of A. The spectrum o] x is the set of all in K for which
(x--u) -1 does not exist. Plainly, a nonzero in K belongs to the spec-
trum of if and only if --- does not exist. This motivates our
next definition.
Let A be an algebra without unit. The sectmm of an element in
A is defined to be the set of all nonzero . in K for which /. does not
have a quasi-inverse, together with 0.
The virtue of having 0 in the spectrum of all x in A whenever A fails
to have a unit is seen in the following theorem.
(C.19) Theorem. Let A be an algebra without unit, and let A be as
in (C.3). I] xcA, then the spectrum o] x as an element in A coincides
exactly with the spectrum o] x as an element in A.
Proof. This is immediate from the definitions. Note that an element
(x, 0) in A has no inverse in A.
(C.20) Theorem. Let A be a commutative Banach algebra and let x
be an element o[ A. I] A has a unit, then belongs to the spectrum o] x
i] and only i] there is a multiplicative linear ]unctional z on A such that
z (x)-----. I] A has no unit and 4: O, then belongs to the spectrum o] x
i] and only i] there is a multiplicative linear ]unctional z such that z (x)--.
Proof. Suppose that A has a unit u, and consider x cA and c.
If ; belongs to the spectrum of x, then (x--u) -1 does not exist. Hence
{(x--u)z :zcA) is a proper ideal containing x--u. By (C.16), there is
a maximal proper ideal M in A such that ((x--u)z:zA)cM. By
(C. 7), M is the kernel of some multiplicative linear functional z. Since
z(x--u)-----O, we have z(x)-=z(u)-----. On the other hand, if (x--u) -
exists, then for every multiplicative linear functional z, we have
z(x-- .u)4=0, since otherwise z (u) --- z( (x-- u)(x-- u)-l)--0. That is,
we have z (x) .
The second statement of the theorem is proved from the first state-
ment together with (C.4) and (C.t9).
(C.21) Theorem. Let A be a Banach algebra. Every multiplicative
linear [unctional z on A is bounded, and ]lzll 1. I] A has a unit u and
Proof. Consider x in A and assume that z(x)= where ]el > ]lx]l.
Since ]]x/czl]< t, x/z has a quasi-inverse y (C.8). Therefore we have
-- +y-- --=0, +(y)-- (y)=0, and hence t + (y) -- (y)
=0. This obvious contradiction shows that ]z (x)! [Ixll and hence that
If A has a unit u and Ilull=a, then ]z[]t since z(u)=t.
(C.22) Theorem. Let A be a commutative Banach algebra.
spectrum o/every element x in A is compact and nonvoid.
The
Proof. In view of (C.19) and (C.3) , we may suppose that A has a
unit u. If ; belongs to the spectrum of x, then u-- has no inverse in A.
Hence by (C.7), we have
-2 .lirnoo u-- u-- >=1.
Thus ll-<-IIll and the spectrum of x is a subset of
Since the set {yc A'y - exists} is open (C.t 0) and the mapping 2- (x-- 2u)
is continuous, we see that the set {2cK" (x--2u) - exists} is open. Thus
the spectrum of x is closed and bounded, i.e., compact. The argument
used in proving (C.tl) shows that (x--2u) - fails to exist for some
complex number 2. That is, x has nonvoid spectrum.
Let X be a locally compact Hausdorff space. The algebra 0(X)
[all operations are pointwise, with the norm ][/[[.= rnax {[/ (x)[ :xX} is
obviously a commutative Banach algebra. Its structure is particularly
simple: and a great deal of effort is expended in harmonic analysis in
trying to do for M(G) and its subalgebras things that are obvious for
algebras 0 (X). The algebra 62o (X) has a unit if and only if X is compact;
the spectrum of/0 (X) is the set/(X) [plus {0} if X is noncompactl ;
every multiplicative linear functional on 0(X) has the form/--/(p) for
some pc X (C.29) ; all closed ideals of 62 o (X) are easily identified (C.30).
It turns out that a reasonably large class of commutative Banach algebras
are algebraically [but not necessarily normwise isomorphic with sub-
algebras of certain algebras 62o (X). To establish this fact, we first make
a definition.
(C.23) Definition. Let A be a commutative Banach algebra. Let X
denote the set of all multiplicative linear functionals on A" X is called
478
Appendix C. Introduction to normed algebras
the structure space/or A. a For every xA, the Fourier trans[orm is
defined by
(i) 2 (r) = z (x)
for all rX. Observe that is a function on X. The Gel'land topology
]or X is the weakest topology on X under which all of the functions 2
are continuous. Unless the contrary is specified, X has this topology.
The family of functions 2 on X is denoted by A.
For the remainder of Appendix C, if a normed algebra has a unit u,
it will be tacitly assumed that II"lt = t.
(C.24) Theorem [ GEL'FAND]. Let A be a commutative Banach algebra
and let x A . Then
Proof. The first equality in (i)is merely the definition of I111,. In
what follows, I111 means I111,. Let A, be as in (C.3) and let X, denote
its structure space. Since zoo((x, 0))=0 for all xA, we find that
sup{[z(x)l"rX}=sup{l((x, 0))1 "zX,} for xA [see (C.t4)]. Hence it
suffices to prove the present theorem for the case in which A has a
unit u. Plainly, we may also suppose that x 4=0.
Using (C.2t), we have
I(x)l - l(x)l <--I111. IIxll -
or all X and n = 1, 2 ..... Thus 1 (x)] <= lim II x-II " and
II # II <-- .li II x-II TM. ()
1
Consider first a complex number 2 such that 121 < I-" Then 112x]l< 1;
hence by (C.7), (u--2x) -a exists and
(u- 2x) -1-- u + Z (2x) k', (2)
this series converges in the norm topology of A. If 2 is in K and
0< I1< l not i-!, then - > I(x)l for an x, so that by
(C.20), x- -u and (u--2x) -1 exist. Let L be an element in A*, the
1
conjugate space of the normed linear space A. For 2 in K and Ixl < I' let
g(2)=L((u--2x)-). By the argument used in the proof of (C.It), it
can be shown that g is analytic in the open disk K'12] < I- "we
omit the details. The continuity of L and the identity (2) imply that
g() = ((u- x)-) =(.) + Z (x )
1 Many writers call X the maximal ideal space/or A • this is reasonable in view
of Theorem (C.I 7) and the remark preceding it.
Appendix C. Introduction to normed algebras 479
for all 2 such that [21 < _<_ I-" Thus the series L(u) + .Y, 2L(x ) is
the Taylor series for g. Since g is analytic in 2K 11 < , it follows
from the elementary theory of analytic functions that the series L(x )
k=l
1
converges absolutely not only for I1 <- but also for [1 < . The
point of all this is that
lim IL(x)l = 0
1
for all 2 such that [21 < and for all LA*.
We now use the BANACH-STEINHAUS theorem (B.20). Choose any 2 K
1
such that[2[<. For eachn=l,2,...,andLA*,let
T,(L)=L(2"x").
Clearly T,is the functional (2"x"' on A* where y' is as defined in (B.23).
As a result, we have
liT. II- II x" x-II (4)
for all . By (), we have sup{llE()ll=, 2,
, 2, ...}< for each LA*. Hence, by (B.20) there is a>0 such that
IIEIl for all . Using (4), we see that II"x"ll, and hence that
II"lla/" , or a . Thereore.lim I1[11/ whenever I1< I111 "
It follows that lira tl-tlX/ I111; this together with (1) proves (i).
(C.25) Theorem. Let A be a commutative Banach algebra with unit
Under the Gel'land topology, X is a compact Hausdor// space. The mapping
x is an algebraic homomorphism o/ A onto a subalgebra A o/ (X),
" I1 II. I1 I1/or x A . The algebra Y separates points o/X and contains
all constant/unctions. The mapping x is an isomorphism i/ and only
Proof. It is easy to see that the following sets form an open basis
for the Gel'fand topology of X"
-,. (0 = { x 1 ()- (0 I < or },
where F is a finite subset of A, z0X, and e is a positive real number.
It then follows that X can be regarded as a subspace of the product
pe z:& tl I111 im ct, the element z in X is identified
with the element (z(x))= ((z))in Y. To see that X is closed in Y,
consider (t) in X-. It is easy to verify that xt is linear and multi-
plicative Ecompare with the proof of (B.25). Also, we have
since z(u):t for all zX. Thus xt is not the zero functional and
480 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 481
hence (tx) belongs to X. We have shown that X is homeomorphic with
a closed subspace of the compact space Y, and is therefore itself compact.
It is routine to verify that the mapping x-->? is an algebraic homo-
morphism, that I11],_<_ Ilxll for xcA, that A separates points of X, and
that A contains the constant functions. The mapping x-->? is an iso-
morphism of A into (X) if and only if @ 0 whenever x 4= 0; i.e., if and
only if I]? 11,> 0 whenever x 4= 0. By (C.24), the last condition is equivalent
to the relation lim []xl[/>0whenever x4=0.
(C.26) Theorem. Let A be a commutative Banach algebra and let
be as in (C.3). Let X and X, be the structure spaces/or A and A,,, respec-
tively. Then X,,=XU {too} (C.14), X is a locally compact Hausdor// space,
X,, is the one-point compacti/ication o[ X, and A co (X).
Proof. As in (C.25), a generic neighborhood of r in X is given by
VF,,(z)--{z'X'lz'(x)--z(x)l< e for
where F cA is finite and e> 0. For r in X, let r, denote its extension
to A,. Suppose that F,,= {(x, O) C A,," x6F}. If I(x)l< or a x,
then
v, ()= {x. ' u,()} {}-
otherwise, we have
Thus neighborhoods in X are relatively open as subsets of X,. A like
argument shows that subsets of X that are relatively open in X, are
also open in the Gel'fand topology of X. In other words, the Gel'fand
topology of X is the relative topology of X as a subset of X,. Since
{zoo} is closed, X is open in the compact Hausdorff space X and hence
is locally compact.
Consider any element x in A and e >0. Let F,= {(x, 0)}; then we have
r&.., (,-oo)-- {,-oo} u {,-,,< x,,. I,-,,((x, 0))- ,-oo ((x, 0))1 < }
- (,-} u {,-,, x,, I,(x)l < }.
This means that 11 is arbitrarily small in neighborhoods of zoo. Equiv-
alently, I?[ is arbitrarily small outside of compact sets in X, and so
o(X). [3
(C.27) We now take up certain Banach M-algebras" for the definition
of a M-algebra, see (2.6). We recall that in a normed M-algebra A, the
identity
(i) II x tl --- II -ll
for all xA is postulated. We shall also need to consider an additional
property, which may or may not be satisfied"
(ii) I[xxl[-
The following theorem is of great importance.
(C.28) Theorem [GEL'FAND-NAiMARK]. Let A be a commutative Ba-
nach M-algebra satis/ying (C.27.ii), and let X be the structure space o/A.
Then the mapping x-- is a norm-preserving algebra isomorphism o/ A
onto o(X). Furthermore, we have 2 % = /or all x A.
Proof. This simplification of the proof given in the first edition of
this book was kindly shown to us by FRED THOELE.
(I) We first prove" if x is in A and x = x-, then z (x) is a real number
for all z X. Let t be an arbitrary real number and consider any element y
in A such that IlylI <. Then we have
I,- (y) I1,- (x) ч ўtl" = I(xy ч ity)I < II,-II, llxy ч ўtyll"
[note (C.21)i and
[Ixy +/tylI -II(xy +ity) (xy +ўty)'-II --II(xy +ity) (xy~--ity")] I
--l[xyy - +tyy-[] <= [[yy-][ (t[x[[ +t )
= IlY tl (ll x II + ,.) __< II II ч t2,
and therefore
I-(y)l, I -(,) -+-ў,1 < II-II,(llxll +,). ()
The supremum of the left side of (1) as y runs over all y in A such that
[[yl[ =< t is exactly IItll (x) ч ўt I and hence we have
Now write (x)--a +lb. We have
[(x)+itl= la+ib+itl=a+b+2bt+t ,
+ + 2 t + t. a tl x II . +
and
a + b + 2bt <__ Ilxl] . (2)
Since (2) holds for all real numbers t, the number b must be 0.
(II) For every x in A, we can write
Hewitt altd Ross, Abstract harmonic analysis, vol. I 31
482 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 483
x- 2 + i--2---= x + ix,.,
where xl= xl and x2= x. Then z(x) = Z(Xl)+ iz (x2) and z(x-) =
z((xl+ ix=)-)--z(x--ix=)=z(x)-- iz(x,)= z(x). In other words, x=
for all xA. Thus A is a subalgebra of g0 (X) that separates points of X
and is closed under complex conjugation. The STONE-WEIERSTRASS
theorem [see (d) of the footnote on p. 151 implies that A is dense in
the uniform topology of g0 (X).
Next we show that ]]?]1= IIxll for xA" it is evident from (C.2t)
that ]lll,<: I]xl]. Suppose first that x=x-. Then I]x=ll = ]lxxll= ]lxl]
and IIx'.ll -Ilxll. Similarly, we have Iix'lI -IIx=ll and hence
llx"lI - Ilxll. By finite induction, we have IIx"lll/- IIxll or all n. By
(C.24), we have
Consider an arbitrary in A" plainly (-)-= -. Therefore
II I1" - II 11 - II ' I1 - II # 11 - II # II, - II1#1 I1 - II # I1.
Since the mapping --># preserves norms and A is complete, is com-
plete. Consequently, A is closed in 0 (X) and so A--o (X).
Theorem (C.28) characterizes very simply in terms of the behavior
of norm and adjoint the Banach algebras g0(X) for locally compact
Hausdorff spaces X. Vv'e complete our analysis of these algebras by
identifying all of the closed ideals and multiplicative linear functionals
in them.
(C.29) Discussion. Let Y be a locally compact Hausdorff space.
For p U and lg0(U), let ,(1)=1(). It is obvious that r is a multi-
plicative linear functional on g0(Y). Theorems (C.t}) and (C.6) imply
that r'(0)={lg0()"I(P)=0} is a maximal proper regular closed
ideal in 0 (Ґ). We will show that these are the only maximal proper
closed ida.is i %() .d that the structure space of g0() is
(C.30) Theorem. Lst Y bs a locall2 compact Hassdorll spas
lt , a ror dossd dal n g0 (Ґ)- Thrs ssts a snqss dosed
E o/ Y such that
(i) -- {l g0 (Y)" l (E) -- 0}.
Also every set as in (i) is a closed ideal o/o (Y).
Proof. Let E={pY'/(p)=o for all/,} and E={/go(Y)'/(E)
=0}" plainly E is closed in Y and
Let F be a compact subset of YrE'. For each xF, there is a
function /x such that /x(x):4:0. Then hx=/x/, belongs to and
hx(x)>0. For each xF, there is a neighborhood U, of x such that
h,(y)>0 for all yU,. There are points xl, ..., x, in F such that
U U, D F. Let h= h," then h belongs to ,, and h (y)> 0 for all yF.
=I =I
Let consist of those functions in 00 (Y) such that {yY" v (y) 4:0}-
f-) E--. Consider any v in and let F-- {y Y'v (y) :4: 0}-. The
previous paragraph shows that there is a function h in such that
(y)
h (y) >> 0 for all y F. We define g (y) -- -h- for y F and g (y) -- 0 for
ycF. It is easy to see that g belongs to g00(Y) and hence that g.
belongs to g. We have shown that c g. A very routine argument shows
that is uniformly dense in e. 1 Since and e are closed, it follows
that
The uniqueness of E is trivial to prove, as is the fact that every
is a closed ideal in 0 (Y).
(C.31) Corollary. Let Y be a locally compact Hausdor// space. For
each point p in Y, let ={/go(Y):/(p)--O}. Then every is a
maximal proper closed ideal in the Banach algebra o ( Y), and there are no
other maximal proper closed ideals in (o(Y). Every proper closed ideal
is the intersection o/maximal proper ideals.
The last statement follows from the identity -- r .
(c.32) Corollary. Let Y be a locally compact Hausdor// space. Then
the structure space X o/o ( Y) is identical with {z:p Y} and the mapping
p-->r is a homeomorphism o/ Y onto X.
Proof. We have already seen that {z'p Y}cX. Since the only
maximal closed ideals in g0(Y) are the ideals J=z(0), it follows
that {z" p Y} = X (C. 17).
For a finite subset of rS 0 (Y), we have
u,,(,)-- {,<x. I,(1)-(1)[ < * for l<}
= {**x. I1(#)- 1()1 < or 1<}.
It follows that U,e(Tq) is the image of an open set under the correspond-
ence p--.r. If U is an arbitrary open neighborhood of q in Y and / in
ў0(Y) is chosen so that/(q) =t and / (U') = O, then {p Y" r U{/I, 1 (re)} c U.
Consequently, the mapping p-.z is a homeomorphism. [
If 9 and >0, then F0= {y Y" I(y)l } is compact and disjoint
from E. Let f be a function in o(Y) such that /(Fo):l , /(Y)c[0, 1], and
{yY'/(y).O}--E:. Then/.9 and
31"
484 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 485
Theorems (C.28), (C.34), and (11.37) can be applied in a striking way
to give a proof of the spectral theorem for [bounded I Hermitian opera-
tors on a Hilbert space (C.42). Throughout (C. 33)-(C.43), H will denote
an arbitrary Hilbert space. Ancillary notation and terminology are as
in (B.53) and (B.56); I will denote the identity operator on H.
(C.33) Lemma. Let H be a Hilbert space and A a normal operator
in 3(H). Then A -1 exists in 3(H) i/and oily i/there is a positive real
number cz such that ][A 1[ >= cz llll /or all H.
Proof. If A - exists, take e= IIA-ll -. Suppose conversely that
our condition holds. Since (A, A) = (AA, ) = (AA
(A,A), we have IIAll= IIAll I111 or all H. It onos
that A and A are one-to-one. Let in Hbe such that (A,)=0
for all H. Then (, A)= 0 for all , so that A = 0, and hence
=0. Therefore A(H) is a dense subspace of H. If A,, then
][--m] - ]A--A][, so that {},% is a Cauchy sequence in H.
Hence , for some H, and by continuity of A, A=. We have
thus proved that A (H)= H. Since A is one-to-one, its inverse A - exists.
It is obvious that A - is linear and bounded, with norm not exceeding
(C.34) Theorem. Let A be an Hermitian operator in the algebra (H).
Then the spectrum o/A is real. I/A is positive-de/inie, then the spectrum
o/ A is also nonnegaHve.
Proof. If 26K and t is not real, then we ave
I- l. IIll -I< A - , > - <, A --
for all $H. By (C.33), (A--2I) - exists and 2 does not belong to the
spectrum of A.
If A is positive-definite and 2 is real and negative, then we have
Once again (C.33) shows that (A--2I) - exists.
(c.3s) Theorem. Let A be a positive-de/inite operator on a Hilbert
space H. Then there is a unique positive-de/inite operator B on H such
that B=A. An operator T in N(H) commutes with A i/ aud only
commutes with B.
Proof. (I) Let N be the smallest norm closed subalgebra of
N(H) that contains A, I, and all operators (A--2I) -1 that are in
for complex numbers ;. It is easy to show that 9 is a commutative
Banach -algebra with unit satisfying the hypotheses of (C.28). Every
in the structure space X of 9 is completely determined by the number
z (A). By (C.20) and (C.34), every number z (A) is real and nonnegative.
Hence we can and do identify X with a certain compact subset of
[0, [ (C.22). We write A(x)--x for all xX. By (C.28), 9 can be
identified wffh (X). 1
Let (x)= xl for all x X, and let {%},o__1 be a sequence of polynomials
in x with real coefficients such that lirnoo max {l, (x)--v/(x)l'x X}=0.
By (C.28), there are operators C, CI, C2, ... in 91 such that v=C and
%---C,. Each C, is a polynomial in the operator A with real coefficients
and hence is Hermitian. Therefore C is Hermitian, since it is the limit
in the norm topology of the Hermitian operators C,. Let B = C 2" as the
square of an Hermitian operator, B is positive-definite. It is also
obvious that B=A.
The operator B is obviously the limit of polynomials in A. Hence
every operator in 3(H) that commutes with A also commutes with B.
(II) We now prove the uniqueness of B. Suppose that Bo=A and
that B 0 is a positive-definite operator. By (I), there are positive-definite
operators C and C O such that C"= B and Co = B 0 . Since Boa = B-- A Bo,
B 0 also commutes with B. Let be any element in H and = (B-- B0)
Then
= <(B+ B0) (B-- B0), > = <(B-- B0 ) , > = <0, > = 0.
Therefore C=Co=O , B--C--O, and B0?=0. Finally, we have
II(e- e0)l[ - <(B- Bo)=, > -- <(B-- B0) , > = <0, > -- 0,
so that B e- B0 e. F]
The spectral theorem, as we shall see, is really a theorem about
,-,-representations of algebras g0(X) by bounded operators on H. We
use some properties of --representations established in §21 to study
these representations. In the two following theorems, we find all
-representations of an algebra go (X), in terms of certain measures on X.
1 The STONE-WEIERSTRASS theorem [see footnote to p. 151 ] implies that if it/g
and it is not in the spectrum of A, then the function x -+ (x -- it)-1 on X is the uniform
limit on X of polynomials in x. Transferring this assertion back to the algebra ,
we see that every operator (A --iti)-I in 3 (H) is the norm limit of polynomials in
the operator A.
486 Appendix C. Introduction to norrned algebras Appendix C. Introduction to norrned algebras 487
(C.36) Theorem. Let X be a locally compact Hausdor// space, and
let T be a cyclic --representation o/o (X) by operators on a Hilbert space H.
Then there are a nonnegative measure on X as in § 1 t and a linear isometry
W o/H onto 2 (X, ) such that
(i) WTtW-(_)--/ /or all 2(X, t) and [o(X).
That is, T is equivalent to multiplication by / in (X, ).
Proof. Let be a cyclic vector in H of norm 1. For/0(X), let
p (/)= (Tt, ). The functional p is obviously linear and positive. Since p
arises from a --representation, (21.t8.i) and (21.18.ii) hold for p. Since
0(X) is a Banach --algebra, p is bounded (21.20)" i.e., [p(/)]
for all/Co(X). If/C (X), there is a g (X) such that
Hence we have p (/)=p (g-g)>= O. This means that p is nonnegative in
the sense of (11.4) and so there is a measure , as constructed in § t t for
which
p(/)= (Tt,)=f/dt for all /oo(X). (1)
x
For every compact subset F of X, it is easy to see that ,(F)=< ; hence
we have ,(X)___< oo. Therefore the mapping /-+ f / d, is continuous
x
in the norm topology of 0 (X)" as p is also continuous in this topology,
equality (1) must persist throughout 0(X). For /o(X), let W(T/)
be the function/(o(X) C (X, ). It is easy to see that W is a linear
isometry of {Ttў "/c 0 (X)} [which is by hypothesis a dense linear sub-
space of H l onto o(X), which we regard as a subspace of (X, ).
Since 0(X) is dense in .(X, ) (12.10), (B.59) implies that W can be
extended to a linear isometry carrying H onto (X, ,). For every
g o (X), we have
W-a g-- Tgў •
hence
finally
T/W-g=TtT=T/ for /0(X);
Since 0 (X) is dense in . (X, ), this proves (i).
(C.37) Theorem. Let X be a locally compact Hausdor// space and
let T be any --representation o/ o(X) by operators on a Hilbert space H
such that/or every nonzero in H, there is an/o(X) /or which TIO.
Then T is equivalent to a representation S o/ the /ollowing sort. There is a
locally compact Hausdor// space Y which is the union o/pairwise disjoint
homeomorphs (X) o/ X and in which a subset is open i/ and only i/its
intersection with each z,(X) is open. There is a nonnegative, possibly in-
/inite, measure t on Y as constructed in §tl. For/o(X), let / be the
/unction in (Y) such that /((x))=/(x). Then S t is the operator on
.(Y, t) such that St=(q/). /or all C.(Y, t).
Proof. Let N and {Hv}r be as in (21.13), for the representation
T of the algebra 0 (X). Our hypothesis implies that N= {0}. For each
F and/o(X), let , denote ]H, and let Tv be the representation
/,v of 0(X) by operators on Hr. Since T is cyclic, (C.36) shows
that there is a finite, nonnegative measure tv as in 11 on X such that
Tv is equivalent to multiplication in (X,t): let be the linear
isometry of Hv onto (X, iv) such that (C.36.i) holds.
Since N={0}, H is the Hilbert space direct sum @Hr. We form
a corresponding concrete realization of the direct sum @(X, t)as
follows. For each 7F we consider a homeomorphism of X onto a
space (X) such that the spaces v(X) are pairwise disjoint. Write Y
for verU v(X), and let A c Y be open if and only if A(X) is open for
all F. It is trivial that Y is a locally compact Hausdorff space in
which each v(X) is open and closed. For 00(Y), let be ] v(X),
and let I()= f (o vv) dry. It is easy to see that I is a nonnegative
EF x
linear functional on 00 (Y) in the sense of 11, and that the correspond-
ing measure on Y is in an obvious sense the sum of the measures v.
For in (Y, ), let be the function equal to on v(X) and 0
elsewhere on Y. The space of all 's is identifiable qua Hilbert space
with (X, v), and we can [again in an obvious senseJ look at
as the Hilbert space direct sum of the spaces
in the original Hilbert space H has a unique representation , where
each is in H v. Now define W() as (v), where we regard
(X, v) as imbedded as a direct summand in (Y, ). Plaly W is a
linear isometry of H onto (Y, ). For E (Y, ), we have
so that
= Z
Applying W to this identity, we obtain
W W-: () = Z Tt, -: ().
Since ,-1()= (/o ) • for VF, we see that
W W -1 () = 2 (/o T;1). ?
which in turn implies the present theorem.
488 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 489
(C.38) Note. We can deal with the case Nq={0}, but in a totally
uninteresting way. A measure # can plainly be imposed upon X or the
union of a number of replicas of X, say Y, so that N is identifiable with
2 (Y, #). Then T restricted to N is plainly equivalent with multiplication
in 5g2(Y, #) by the function 0.
(C.39) Discussion. The concrete realization of the representation
[-T described in Theorem (C.37) can be used to write the operators
T as "integrals" of certain projections. All notation in the present
discussion is as in (C.37). Since H and (Y, t) are connected by a
linear isometry under which T and multiplication by /are equivalent,
we lose no generality by considering only .(Y, t) and multiplication
by ]: an abstract formulation appears in (C.40) in/ra.
For asubset A of X, let A be the subset Urn(A). Let 9 be the
family of all subsets of Y having the form 0E for Borel subsets E of X.
For every 0E., the characteristic function 0E is a bounded Borel
measurable function on Y, so that 0E " is in (Y, ) for allff .(Y, ,).
The mapping
-0" =M () ()
is plainly an idempotent Hermitian operator on 2(Y, )" that is, M E is
a proiection, and it is projection onto the subspace of all C99,.(Y, )
that vanish off of )E. 1 The following identities are immediate con-
sequences of (1)"
M,o...o ,,o...-- ,, M,, (2)
if El,..., E,,... are pairwise disjoint Borel sets in X;
M 1 ME, • • " ME k-- MEan ... n Ek (3)
for any Borel sets El,..., E, in X;
Mx=I , (4)
where I is the identity operator;
M--O, (5)
where 0 is the zero operator.
Now consider any /c go (X) and any e > 0. There is plainly a partition
E,..., E,,, of X into Borel sets such that I/(x)--/(x')l < e for x, x' in E..
Let %. be any complex number such that I/ (x) -- %.[ < e for all
We can obviously define the mapping ( --> e A • ( for every t-measurable subset A
of Y. For our present purpose, we wish to consider only sets E.
Then for every = (Y, ,), we have
m 2
y --
< [sup {l/(y ) -- czio,(y) l'y v}]=f Iўl=a, •
]=1 y
equivalently,
The inequality (6) is an abstract formulation of the spectral theorem.
That is, there is a mapping EM of the Borel subsets of X into the
family of all projections on (Y, e) satisfying (2), (3), (4), and (5)
such that St is arbitrarily close to linear combinations of the projections
M, in the sense of (6).
The projections M have another important property. Let B be any
bounded operator on (Y, ) that commutes with all of the operators S 1.
Then B commutes with all Mz. To plove this, let $ and be any elements
of (Y, ). Since (S/B,)= (BS,)= (SI$, B), we have
f (/)(B;) d = f (/) (Bv) d.
Y Y
We wish to prove that
oE oE
for all Borel sets E C X. This will show that (M B , ) = (M$, B)
for all and , and hence that M B= BM. Let = (B$)--$ (B.
Then belongs to (Y, ),
f (/) d = 0
Y
for all ]o(X), and we wish to prove that
f d = 0
OE
for all Borel sets E in X. Assume that fde=0 for some Borel
oE
set EcX. The function is 0 outside of the union of a countable
subfamily of the sets (X): for convenience, we write {%(X)}= for
this family. Since f I] d<, there is a positive integer m for
n=l zn(X)
which Ј f 1 dr< j . For each integer n=l, ..., m, there are
n=m+l zn(X)
I1
a compact set C E and an open set U,D E such that f ] 1 d t < .
m m
Let U= U and F= U and choose a g in g00 (X) satisfying g (F) --t
n=l n=l
490 Appendix C. Introduction to normed algebras Appendix C. Introduction to normed algebras 491
g (U')=0, and g (X)c [0, 1 . Plainly, we have
c n=m+l vn(X)
,=u+ (x) (9)
n=m+l v(X)
and for n=,..., m, we have
[ f (-) f I1 f I1 < I1 (ao)
v() (u F') ( [ F) 2 m
Relations (9) and (0) impl that
Since f dt = , it follows that f (On) dt 40, contrary to (7').
oE Y
(c.40) Theorem. Let X be a locally compact Hausdor// space, and let
[ be any -representation o/o (X) by operators on a Hilbert space H.
Then there is a mapping E o/the Borel sets E c X onto pro/ections
in H with the/ollowing properties:
(i) Pe, o v... v v...-- i/ the sets E, E, ..., E,,, ... are pairwise
disjoint;
(iii) the pro/ections commute with all bounded operators B that
commute with all operators .
Furthermore, i/ {E, ..., Era} is any partition o/ X ito Borel sets such
that I/(x)-/(x')l< /o x,x'E and l/(x)--l< /o xE
2..., m), then
=1
[or all H. Finally, i/ , are any eZemenls o/ H, lhe mapping
E(. ) =ў.(E) is a compZe meas.re on X in he sense o/ 4.
(v) <E. > = f / .
x
/or all / 0 (X).
(C.41) Theorem. Let 91 be any commutative algebra o/ bounded
operators on a Hilbert space H such that AC 91 whenever A 91 and such
that 91 is closed in the norm topology/or the algebra o/all bounded operators
on H. Let X be the structure space o/ 91 and A the Fourier trans/orm o/
A 91; thus (A" A 91}-- o (X). Then all o[ the assertions o/ (C.40) hold
with T t replaced everywhere by A and the mapping ]--> T t replaced by the
identity mapping A---A. The relation (C.40.iv) becomes the ]ollowing.
Let E 1 .... , E m be any partition o/X into Borel sets such that ]A (x) --A (x')l
[or x, x'E i, and [A(x)--og[< s [or xE if=t, 2 .... , m). Then
(iv) A $- Y, ei P,ў =< e 1$ ]
[or all H. The relation (C.40.v) becomes
(v) <Aў, V> = f A
x
Having come this far, we may as well state the classical form of the
spectral theorem for bounded Hermitian operators.
(C.42) Theorem. Let B be an Hermitian operator on a Hilbert space H.
There is a one-parameter /amily o/ proiections {B:tR} on n with the
[ollowing properties:
(ii) every commutes with every operator that commutes with B;
(iii) /or every, H, lim IB- $-B $ I[ = 0-
(iv) there are real numbers a and b, a<b, such that B=0 /or ta
and =I /or tb;
(v) /or every $ H and every subdivision a = t o< t < . . . < t m = b such
i=l
b
(vi) /or , H, we have (B, ) = f t d (ў, ), the integral being an
ordinary Riemann-Stielties integral.
Proof. Let N be the smallest norm closed subalgebra of (H)
containing the operators B and I. Arguing as in the proof of (C.35),
we see that the structure space X of N can be identified with a compact
subset of R, where we have B(x)=x for all x(X. Choose real numbers a
and b such that aminX and b>maxX. Let =,ў[nx, where is
the projection operator defined in (C.39) and (C.40). All assertions of
the present theorem now follow from (C.41).
(C.43) Note. We could also give the classical spectral theorem for
a bounded normal operator on H; the assertion and proof are obvious
from what has already been done.
Bibliography
ABE, MAKOTO: [I ] ber Automorphismen der lokal-kompakten abelschen Gruppen.
Proc. Imp. Acad. Tokyo 16, 59--62 (1940).
ALEKSANDROV, A.D. [AaeICaHAp0B, A. .]: [I] On groups with an invariant
measure [O rpylIIIaX c HHBapHaHTH0it iep0it]. Dokl. Akad. Nauk SSSR, N. S.
34, 5--9 (1942).
ALEKSANDROV, P. S. [AaeIcaHpOB, II. C.]: [1] Combinatorial topology [HOM6H-
HaT0pHaI TOII0IOI'HI]. OGIZ, Gostehizdat, Moscow, 1947. English translation,
Graylock Press, Rochester, New York, 1956.
--, and H. HoPF: [I] Topologie I. Grundlehren der Mathematischen Wissenschaf-
ten, Bd. XLV. Berlin: Springer 1935.
ALEXANDER, JAMES W.: [I] On the characters of discrete abelian groups. Ann. of
Math. (2) 35, 389--395 (1934).
--, and LEON W. COHEN: [1] a classification of the homology groups of compact
spaces. Ann. of Math. (2) 33, 538-- 566 (1932).
--, and LEO ZIPPIN: [I] Discrete abelian groups and their character groups. Ann.
of Math. (2) 36, 71--85 (1935).
ANZAI, HIROTADA, and SHIZUO KAKUTANI: [1] Bohr compactifications of a locally
compact abelian group. I, II. Proc. Imp. Acad. Tokyo 19, 476--480, 533--539
(1943).
APOSTOL, TOM M. : [1] Mathematical analysis. Reading, Mass. : Addison-Wesley
Publ. Co. Inc. 1957.
ARENS, RICHARD F.: [1 Topologies for homeomorphism groups. Amer. J. Math.
68, 593--610 (1946).
ARI-IANGEL'SKI, A. [ApxaHPeaBcHi, A.: [I On the identity of the dimension
ind G and dim G for locally bicompact groups [O coagte, paepoc
ind G I dim G g 0aH0 6IIHOMrlaHTHBIX rpynnJ. Dokl. Akad. Nauk SSSR,
N.S. 132, 980--98t (1960).
AUBERT, K.E.: [t] Convex ideals in ordered group algebras and the uniqueness of
the Haar measure. Math. Scand. 6, 181--188 (1958).
BAER, REINHOLD: [1] Zur Topologie der Gruppen. J. Reine Angew. Math. 160,
208-- 226 (1929).
--, and FRIEDRICH LEVI: [I] Stetige Funktionen in topologischen Rgumen. Math.
Z. 34, 110--130 (1932).
BALCERZYK, S.: [1] On algebraically compact groups of I. Kaplansky. Fund.
Math. 44, 91--93 (1957).
BEURLING, ARNE: [I] Bur les int6grales de Fourier absolument convergentes et
leur application g une transformation fonctionnelle. Nionde Skandinaviska
Matematikerkongr., Helsingfors, 1938, 345--366. Helsingfors: Mercator 1939.
BIRKHOFF, GARRETT: [I] A note on topological groups. Compositio Math. 3,
427--430 (1936).
-- [2] Lattice theory. Amer. Math. Soc. Colloquium Publications, Vol. XXV,
2nd Edition. Amer. Math. Soc., New York, N.Y., 1948.
BOCHNER, SALOMON: [1] Vorlesungen fiber Fouriersche Integrale. Leipzig: Aka-
demische Verlagsgesellschaft 1932. Reprinted by Chelsea Publ. Co., NewYork,
1948.
BOURBAKI, NICOLAS: [I] t16ments de math6matique. III. Premi6re pattie. Les
structures fondamentales de l'analyse. Livre III. Topologie g6n6rale. Chap. III.
Bibliography 493
Groupes topologiques (Thdorie dldmentaire). Actualitds Sci. et Ind. 916. Paris:
Hermann & Cie. 1942.
BOURBAKI, NICOLAS" [2] tldments de mathdmatique. XIII. Premi6re partie. Les
structures fondamentales de l'analyse. Livre VI. Intggration. Chap. I--IV.
Actualitds Sci. et Ind. 1175. Paris: Hermann & Cie. 1952.
-- [3] 16ments de math6matique. XXI. I. Les structures fondamentales de
l'analyse. Livre VI. Int6gration. Chap. V. Int6gration des mesures. Actualit6s
Sci. et Ind. t244. Paris: Hermann & Cie. t956.
-- [4] 16ments de math6matique. Premi6re partie. Livre VI. Int6gration.
Chap. VI. Int6gration vectorielle. Actualitds Sci. et Ind. 128t. Paris: Her-
mann & Cie. t 959.
BRACONNIER, JEAN: [1] Sur les groupes topologiques localement compacts. J.
Math. Pures Appl., N.S. 27, I -- 85 (I 948).
BUCK, R. CREIGHTON: [t] Generalized group algebras. Proc. Nat. Acad. Sci. U.S.A.
36, 747--749 (1950).
-- [2] Operator algebras and dual spaces. Proc. Amer. Math. Soc. 3, 68t--687
(1952).
CALDER6N, ALBERTO P.: [1] A general ergodic theorem. Ann. of Math. (2) 58,
182--191 (1953).
CARTAN, gLIE: [I] La Th6orie des Groupes Finis et Continus et l'Analysis Situs.
Mdmorial des Sci. Math., Fasc. 42. Paris: Gauthier-Villars & Cie. t930.
CARTAN, HENRI: [1] Sur la mesure de Haar. C.R. Acad. Sci. Paris 211, 759--762
(1940).
-- [2] Sur les fondements de la thdorie du potentiel. Bull. Soc. Math. France
69, 71--96 (1941).
--, and J.A DI-UDONN-: [1] Notesde tdratopologie. -- III. Rev. Sci. 77, 4t3--414
(1939).
--, and ROGER GODEMENT: [1 Thdorie de la dualit6 et analyse harmonique dans
les groupes ab61iens localement compacts. Ann. Sci. cole Norm. Sup. (3) 64,
79--99 (1947).
CU-VA-L.y, O-AUD.: [t] Theory of Lie groups. I. Princeton, N.J.: Princeton
University Press 1946.
COHEN, LEON W. : See ALEXANDER, JAMES W.
COHEN, PAUL J.: [1] On a conjecture of Littlewood and idempotent measures.
Amer. J. Math. 82, 19t -- 212 (I 960).
COTLAR, MlSCHA, and R. RICABARRA: [1 ] On the existence of characters in topo-
logical groups. Amer. J. Math. 76, 375--388 (t954).
DANIELL, P.j.: [I] A general form of integral. Ann. of Math. (2) 19, 279--294
(1917--t918).
DANTZlG, D. VAN : [1] tdber topologisch homogene Kontinua. Fund. Math. 15,
102-- 125 (1930).
-- [2] Studien over topologische Algebra. Dissertation, Amsterdam 1931.
-- [3] Zur topologischen Algebra. I. Komplettierungstheorie. Math. Ann. 107,
587--626 (1932).
-- [4] Zur topologischen Algebra. II. Abstrakte bv-adische Ringe. Compositio
Math. 2, 201 -- 223 (1935).
-- [5] Nombres universels ou v!-adiques avec une introduction sur l'alg6bre topo-
logique. Ann. Sci. cole Norm. Sup. (3) 53, 275--307 (1936).
-- [6] Zur topologischen Algebra. III. Brouwersche und Cantorsche Gruppen.
Compositio Math. 3, 408--426 (1936).
DAUBLEBSKY YON STERUECK, R. : [I ] Ableitung zahlentheoretischer Relationen mit
Hilfe eines mehrdimensionalen Systemes yon Gitterpunkten. Monatsh. Math.
5, 255--266 (1894).
4 94 Bibliography Bibliography 4 95
DAVIS, HARRY F. : [1] A note on Haar measure. Proc. Amer. Math. Soc. 6, 318--321
(1955).
DAY, MAHLON M. : [I ] The spaces LP with 0 < p < I. Bull. Amer. Math. Soc. 46,
816--823 (1940).
-- [2] Ergodic theorems for Abelian semi-groups. Trans. Amer. Math. Soc. 51,
399--412 (t942).
-- [3] Amenable groups. Preliminary report. Bull. Amer. Math. Soc. 56, 46
(1950).
-- [4] Means for the bounded functions and ergodicity of the bounded representa-
tions of semi-groups. Trans. Amer. Math. Soc. 69, 276--291 (1950).
-- [5] Amenable semigroups. Illinois J. Math. 1, 509--544 (1957).
-- [6] Normed linear spaces. Ergebnisse Math., neue Folge, H. 21. Berlin-G6t-
tingen-Heidelberg: Springer 1958.
DEVINATZ, A., and A.E. NUSSBAUM : [1 ] Real characters of certain semi-groups with
applications. Duke Math. J. 28, 221-237 (196t).
DIEUDONNI, JEAN: [1] Sur les fonctions continues numdriques definies dans un
produit de deux espaces compacts. C.R. Acad. Sci. Paris 205, 593--595 (1937).
-- [2] Notes de t6ratopologie. -- I. Rev. Sci. 77, 39--40 (1939).
-- See CARTAN, HENRI.
DIRICHLET, P. G. L. : [1 ] Sur la convergence des sdries trigonom6triques qui servent
/ reprdsenter une fonction arbitraire entre des limites donndes. J. Reine Angew.
Math. 4, 157-- 169 (I 829).
DIXMIER, JACQUES: [1 ] Les moyennes invariantes dans les semi-groupes et leurs
applications. Acta Sci. Math. Szeged 12. Leopoldo Fejdr et Frederico Riesz
LXX annos natis dedicatus, Pars A, 213--227 (1950).
-- [2] Quelques propridtds des groupes abdliens localement compacts. Bull. Sci.
Math. $1, 38--48 (1957).
DUNFORD, NELSON, and JACOB T. SCHWARTZ: [I ] Linear operators. Part I: General
theory. NewYork, N.Y. : Interscience Publishers, Inc. 1958.
ECKMANN, BENO: [1] ber monothetische Gruppen. Comment. Math. Helv. 16,
249--263 (1943--1944).
EDWARDS, ROBERT E. : [1 ] A theory of Radon measures on locally compact spaces.
Acta Math. 89, 133--164 (1953).
ERDS, PAUL" [I] The dimension of the rational points in Hilbert space. Ann. of
Math. (2) 41, 734--736 (1940).
FOLNER, ERLING: [1] Generalization of a theorem of Bogolioflboff to topologica.
Abelian groups with an appendix on Banach mean values in non-Abelian groups.
Math. Scand. 2, 5-- 18 (1954).
[2] Note on a generalization of a theorem of Bogolioflboff. Math. Scand. 2,
-- 224--226 (1954).
-- [3] On groups with full Banach mean value. Math. Scand. 3, 243--254 (1955).
-- [4] Note on groups with and without full Banach mean value. Math. Scand.
5, 5--11 (1957).
FREUDENTHAL, HANS: [1] Einige S/tze fiber topologische Gruppen. Ann. of Math.
(2) 37, 46--56 (1936).
-- [2] Topologische Gruppen mit gentigend vielen fastperiodischen Funktionen.
Ann. of Math. (2) 37, 57--77 (1936).
-- [3] Entwicklungen yon R/umen und ihren Gruppen. Compositio Math. 4,
145--234 (1937).
-- [4] Die Haarschen Orthogonalsysteme von Gruppencharakteren im Lichte der
Pontrjaginschen Dualitttstheorie. Compositio Math. 5, 354-- 356 (I 938).
FROBENIUS, G.: [I] (lber Gruppencharaktere. Sitzungsber. Preuss. Akad. Wiss.,
phys.-math, t(1. 1896, 985--t02t.
FUCHS, L.: [I] Abelian groups. Budapest: Publishing House of the Hungarian
Academy of Sciences 1958. London: Pergamon Press t960.
-- [2] On character groups of discrete Abelian groups. Acta Math. Acad. Sci.
Hungar. 10, t33--140 (1959).
GELBAUM, B., G.t(. t(ALISCH and J.M.H. OLMSTED: [I] On the embedding of
topological semigroups and integral domains. Proc. Amer. Math. Soc. 2,
807--82t (t95t).
GEL'FANn, I.M. [I'eabaH, H.M.]: [I] t3ber absolut konvergente trigonometri-
sche Reihen und Integrale. Mat. Sb., N.S. 9 (51), 51--66 (1941).
--, and M.A. NAiMARK [M.A. Haiap]: [t] On the imbedding ofnormed rings
into the ring of operators in Hilbert space [O BameH H0pp0BaHH0r0
oaba B HOJIIO oHepaTOpOB B rna6epTOBOM HpocTpaHCTBe]. Mat. Sb., N. S.
12 (54), 197--2t7 (t943).
-- [2] Normed rings with involution and their representations [HopMpOBanHe
oa c nBOmIei X HpecwaBen]. Izv. Akad. Nauk SSSR, Set.
Mat. 12, 445--480 (1948).
--, and D. A. RAiOV [. A. PaiOB]: [I] On the theory of characters of com-
mutative topological groups [H eop xapaepoB 0yaBHX oHo-
aorecx rpylm]. Dokl. Akad. Nauk SSSR, N.S. 28, 195--t98 (t940).
-- [2J Irreducible unitary representations of locally bicompact groups [Henp-
B0IMNe yHiTapHIIe HpeCTaBaeH oa5tto 6IOMHaTttNX rpyHH]. Mat.
Sb., N.S. 13 (55), 30t--3t6 (t943).
-- [3] Irreducible unitary representations of locally bicompact groups [HelIp-
BOIIiIIe yHnTapHe HpecTaBeHna ouabHO 6IIHOMIIaHTHI)IX rpyHH]. Dokl.
Akad. Nauk SSSR, N.S. 42, t99--20t (t944).
GLICKSBERG, IRVING: [t] The representation of functionals by integrals. Duke
Math. J. 19, 253-26t (t952).
GODEENT, ROGER: [t] Les fonctions de type positif et la th6orie des groupes.
Trans. Amer. Math. Soc. 63, t -- 84 (1948).
-- [2] Sur la th6orie des repr6sentations unitaires. Ann. of Math. (2) 53, 68--t24
(1951).
-- See CARMAN, HENRI.
GRAEV, M.I. [I'paeB, M. If.I: [1] On the theory of complete direct products of
groups [H eopnn H0aHUX npux HponaBeeHni rpyHH]. Mat. Sb., N. S.
17 (59), 85--104 (1945).
-- [2] Free topological groups [CB060Hhte oHoaornqecne rpyHHU]. Izv. Akad.
Nauk SSSR, Ser. Mat. 12, 279--324 (1948).
-- [3] On free products of topological groups [O eB060HhtX Hp0naBeeHntX ono-
aornqecrnx rpyHH]. Izv. Akad. Nauk SSSR, Ser. Mat. 14, 343--354 (1950).
-- [4] Theory of topological groups. I. Norms and metrics on groups. Complete
groups. Free topological groups. [Teopn oHoaornqecnx rpyHH. I. Hopuu
n MeTpnn Ha rpyHHax. HOHIIe rpyHHI. CBO60HIIe TOHOornqecne
rpynnu.] Uspehi Mat. Nauk, N.S. , vyp. 2 (36), 3--56 (1950).
GUREVI6, ANNA [Fypeanq, AHHa]: [I] Unitary representation in Hilbert space of
a compact topological group [YHnTapHoe HpecTaaaeHne B rna6epTOaOM
HpocTpaHCTBe OMHaTHOt TOHOaornqeco rpynHu]. Mat. Sb., N. S. 13
79--86 (t943).
HAAR, ALFRED: [1] ber unendliche kommutative Gruppen. Math. Z. 33, 129-- 159
(1931). Also in Gesammelte Arbeiten, pp. 486--516. Budapest: Akad6miai
Kiad6 1959.
496 Bibliography Bibliography 497
HAAR, ALFRID" [2] {ber die Gruppencharaktere gewisser unendlichen Gruppen.
Acta Litt. Sci. Szeged 5, 172--186 (1932). Also in Gesammelte Arbeiten,
pp. 562--576. Budapest" Akaddmiai Kiad6 1959.
-- [3] Der Ma[3begriff in der Theorie der kontinuierlichen Gruppen. Ann. of Math.
(2) 34, 147--169 (1933). Also in Gesammelte Arbeiten, pp. 600--622. Budapest"
Akad6miai Kiad6 1959.
HALL, MARSHALL, JR." [1] A topology for free groups and related groups. Ann. of
Math. (2) 52, t27--139 (1950).
HALMOS, PAUL R." [1] Comment on the real line. Bull. Amer. Math. Soc. 50,
877--878 (1944).
-- [2] Measure theory. NewYork, N.Y." D. Van Nostrand Co. 1950.
-- [3] Introduction to Hilbert space and the theory of spectral multiplicity.
NewYork, N.Y." Chelsea Publ. Co. 1951.
--, and HANS SAMELSON" [1] On monothetic groups. Proc. Nat. Acad. Sci. U.S.A.
28, 254--258 (1942).
--, and HERBERT E. VAUGUAN" [1] The marriage problem. Amer. J. Math. 72,
214--215 (1950).
HaE-, GEORa" [1] Eine Basis aller Zahlen und die unstetigen L6sungen der
Funktionalgleichung" /(x-t-y)=f(x)-t-f(y). Math. Ann. 60, 459--462 (1905).
HAD, G.H., J.E. LTT-EWOOD and G. P6-A" [I] Inequalities. Cambridge"
Cambridge University Press 1934.
HARRISON, D.K." [1] Infinite abelian groups and homological methods. Ann. of
Math. (2) 69, 366--391 (1959).
HARTMAN, S., and A. HULANICKI" [1 ] Les sous-groupes purs et leurs duals. Fund.
Math. 45, 71--77 (1957).
--, and JAN MYCIELSKI" [1 ] On the imbedding of topological groups into connected
topological groups. Colloq. Math. 5, 167--169 (1958).
--, and C. RYLL-NARDZEWSKI" [I] Zur Theorie der lokal-kompakten abelschen
Gruppen. Colloq. Math. 4, 157--188 (1957).
HAUSDORFF, FELIX" [I] Mengenlehre, 3. Aufl. G6schens Lehrbticherei, Bd. 7.
Berlin u. Leipzig" W. de Gruyter 1935. Reprinted by Dover Publications,
NewYork, N.Y., 1944.
HELSON, HENRY" [1 ] On the ideal structure of group algebras. Ark. Mat. 2, 83--86
(1952).
HENSE-, KURT" [I] Zahlentheorie. Berlin u. Leipzig" G.J. G6schensche Verlags-
handlung 1913.
HEWITT, EDWIN" [1 ] Integral representation of certain linear functionals. Ark. Mat.
2, 269-- 282 (1952).
-- [2] Linear functionals on almost periodic functions. Trans. Amer. Math. Soc.
74, 303--322 (1953).
-- [3] Remarks on the inversion of Fourier-Stieltjes transforms. Ann. of Math.
(2) 57, 458--474 (1953).
-- [4] Compact monothetic semigroups. Duke Math. J. 23, 447-457 (1956).
- [5] A survey of abstract harmonic analysis. Some aspects of analysis and
probability, pp. 105--168. Surveys in Applied Mathematics, Vol. 4. NewYork,
N.Y." John Wiley and Sons, Inc. 1958.
--, and KARL R. STROMUERG" [I] A remark on Fourier-Stieltjes transforms. An.
Acad. Brasil. Ci. 34, 175-- t 80 (I 962).
- [2] Real and abstract analysis. Graduate Texts in Mathematics, Vol. 25, 2nd
printing, corrected. Berlin, Heidelberg, New York" Springer 1969.
HEI,VITT, EDWIN, and HERBERT S. ZUCKERMAN: [1] A group-theoretic method in
approximation theory. Ann. of Math. (2) 52, 557--567 (t950).
-- [2] On convolution algebras. Proc. Intern. Cong. Mathematicians Cambridge,
Mass., 1950. Amer. Math. Soc., Providence, R. I., 1952. Vol. I, p. 455.
-- [3] Finite dimensional convolution algebras. Acta Math. 93, 67--119 (t955).
-- [4] The /1-algebra of a commutative semigroup. Trans. Amer. Math. Soc. 83,
7O--97 (1956).
-- [5] The irreducible representations of a semigroup related to the symmetric
group. Illinois J. Math. 1, 188-- 213 (I 957).
HILLE, EINAR, and RALPH S. PHILLIPS: [1] Functional analysis and semi-groups.
Amer. Math. Soc. Colloquium Publications, Vol. XXXI. Revised Edition.
Amer. Math. Soc., Providence, R. I., 1957.
HOFMANN, I{ARL HEINRICH: [I] Eine Bemerkung fiber die zentralen Untergruppen
in zusammenhgngenden topologischen Gruppen. Arch. Math. 9, 33--38 (1958).
HOPF, H. : See ALEKSANDROV, P. S.
HULANICKI, A. : [1] On locally compact topological groups of power of continuum.
Fund. Math. 44, 156--t58 (t957).
-- [2] Algebraic characterization of abelian divisible groups which admit compact
topologies. Fund. Math. 44, 192--197 (1957).
-- [3] Algebraic structure of compact abelian groups. Bull. Acad. Polon. Sci., S6r.
Sci. Math. Astr. Phys. 6, 71--73 (1958).
-- [4] On the topological structure of 0-dimensional topological groups. Fund.
Math. 46, 317--320 (1959).
-- See HARTMAN, S.
HURWlTZ, ADOLF: [t] llber die Erzeugung der Invarianten durch Integration.
Nachr. k. Ges. GOttingen, math.-phys. K1. 1897, 71--90. Also in Mathematische
Werke, Bd. II, pp. 546--564. Basel: Birkhguser 1933.
ISlWATA, TAKESI: [I] Duality of topological groups. Sci. Rep. Tokyo Kyoiku
Daigaku, Sect. A. 5, 82--87 (1955).
IVANOVSKI1, L.N. [I4BaHOBCii, O-I. H.]: [I] On an hypothesis of P.S. Aleksan-
drov [06 0)HOR rnoTeae II. C. AaecaHalpoua]. Dokl. Akad. Nauk SSSR,
N.S. 123, 785--786 (1958).
IWAMURA, TURANE" See YOSIDA, K)SAKU.
IWASAWA, KENKICHI: [I ] Einige Sgtze fiber freie Gruppen. Proc. Imp. Acad. Tokyo
19, 272--274 (1943).
-- 2] On some types of topological groups. Ann. of Math. (2) 50, 507--558
(1949).
JoNEs, F. BURTON: [1] On the first countability axiom for locally compact Haus-
dorff spaces. Colloq. Math. 7, 33--34 (1959).
KAVrANI, SUIZUO: [I] llber die Metrisation der topologischen Gruppen. Proc.
Imp. Acad. Tokyo 12, 82-- 84 (1936).
-- [2] On the uniqueness of Haar's measure. Proc. Imp. Acad. Tokyo 14, 27--31
(1938).
-- 3] Concrete representation of abstract (M)-spaces (A characterization of the
space of continuous functions). Ann. of Math. (2) 42, 994-- 1024 (1941).
-- 4] On cardinal numbers related with a compact abelian group. Proc. Imp.
Acad. Tokyo 19, 366--372 (1943).
-- [5] Free topological groups and infinite direct product topological groups.
Proc. Imp. Acad. Tokyo 20, 595--598 (1944).
--, and KUNIUKO KODAIRA: 1 ] Uber das Haarsche Mal3 in der lokal bikompakten
Gruppe. Proc. Imp. Acad. Tokyo 20, 444--450 (1944).
Hewitt and Ross, Abstract harmonic analysis, vol. I 32
498 Bibliography Bibliography 499
t(AKUTANI, SHIZUO, and JOHN C. OXTOBY: [I] Construction of a non-separable in-
variant extension of the Lebesgue measure space. Ann. of Math. (2) 52, 580--
590 (95o).
-- See ANZAI, HIROTADA.
KALISCH, G. If. : See GELBAUM, lB.
KAMPEN, E.R. VAN: [t] Locally bicompact Abelian groups and their character
groups. Ann. of Math. (2) 36, 448--463 (1935).
IZAPLAN, SAMUEL: [1] Extensions of the Pontrjagin duality. I. Infinite products.
Duke Math. J. 15, 649-658 (1948).
- [2] Extensions of the Pontrjagin duality. II. Direct and inverse sequences.
Duke Math. J. 17, 4t9-435 (t950).
IfALANSKY, IRVING: [I ] Infinite abelian groups. Ann Arbor, Michigan: University
of Michigan Press 1954.
KASUGA, TAKASHI: [t] On the isomorphism of topological groups. Proc. Japan
Acad. 29, 435--438 (1953).
KAWADA, YUKYOSI: [t ] llber den Mittelwert der mel3baren fastperiodischen Funk-
tionen auf einer Gruppe. Proc. Imp. Acad. Tokyo 19, 264--266 (1943).
IfEINER, HORST: [t ] Verallgemeinerte fastperiodische Funktionen auf Halbgruppen.
Arch. Math. 8, t29--134 (1957).
IfESS.y, JoHN L.: [1] Note on a theorem of Krein and Mil'man. J. Osaka Inst.
of Sci. and Tech. 3, t--2 (t951).
-- [2] General topology. NewYork, N.Y.: D. Van Nostrand Co. 1955.
IfEMPERMAN, J.H.B.: [t] On small sumsets in an abelian group. Acta Math.
103, 63--88 (1960).
KERTSZ, A., and T. SzEL: [t] On the existence of non-discrete topologies in in-
finite abelian groups. Publ. Math. Debrecen 3, 187--t89 (1953).
IfST.N, HARRY: [t] Full Banach mean values on countable groups. Math. Scan&
7, t46--I 56 (t959).
KNESER, MARTIN: [t] Summenmengen in lokalkompakten abelschen Gruppen.
Math. Z. 66, 88--tt0 (t956).
KODAIRA, IfUNIHIKO: See IfAKUTANI, SHIZUO.
IfOOSlS, PAUL : [I ] An irreducible unitary representation of a compact group is finite
dimensional. Proc. Amer. Math. Soc. 8, 7t2--715 (1957).
KSTHE, GOTTFRIED: [t] Topologische lineare R/iume I. Grundlehren der Mathe-
matischen Wissenschaften, Bd. t07. Berlin-GSttingen-Heidelberg: Springer
t 960.
KR.iN, M. G. [HpeiH, M. I'.]: [t ] Integral equations on the half-line with kernel
depending on the difference of the arguments [HHwerpasHe ypaHeH Ha
II0IyHpaM0i C Ip0M, 8aBIICHILIHM 0T paSHOCTH apPyMeHTOB]. Uspehi Mat.
Nauk 13, vyp. 5 (83), 3--t20 (t958).
KRISTENSE, LEIF: [t] Invariant metrics in coset spaces. Math. Scan& 6, 33--36
(1958).
KRvr.r-, Wor-GaNG: [t ] Ober separable, insbesondere kompakte separable Gruppen.
J. Reine Angew. Math. 184, t9--48 (t942).
IfRo, A.G. [Hypom, A. P.] : [t] Theory of groups, 2rid Edition. [Teopnn rpyrm,
nataHne uwopoe]. Gostehizdat, Moscow, 1953.
Kz'mov, V. [HyabMnH0, 13.]: [I] On an hypothesis of P.S. Aleksandrov in
the theory of topological groups [O raHoweae 11. (i. AaeeaHtpoua u weopna
woHoaornneenx rpyrm]. Dokl. Akad. Nauk SSSR, N. S. 12, 727-- 729 (t959).
L..sa, I-I.NRI: [1] Legons sur l'int4gration et la recherche des fonctions
primitives. Paris: Gauthier-Villars & Cie. a 904.
LEHMER, D.H. : [I] Arithmetic of double series. Trans. Amer. Math. Soc. 33,
945--957 (1931).
LEJA, F. : [I] Sur la notion du groupe abstrait topologique. Fund. Math. 9, 37--44
(1927).
LEPTIN, HORST: [I] lber eine Iflasse linear kompakter abelscher Gruppen. I,
II. Abh. Math. Sem. Univ. Hamburg 19; 23--40 (1954); 221--243 (1955).
-- [2] Abelsche Gruppen mit kompakten Charaktergruppen und Dualittstheorie
gewisser linear topologischer abelscher Gruppen. Abh. Math. Sem. Univ.
Hamburg 19, 244--263 (1955).
-- [3] Zur Dualitttstheorie projektiver Limites abelscher Gruppen. Abh. Math.
Sem. Univ. Hamburg 19, 264--268 (1955).
-- [4] Bemerkung zu einem Satz von S. Kaplan. Arch. Math. 6, 139--144 (1955).
LEVI, FRIEDRICH: See BAER, REINHOLD.
LITTLEWOOD, J.E.: See HARDY, G.H.
LOOMIS, LYNN H.: [I] Haar measure in uniform structures. Duke Math. J. 16,
193 - 208 (I 949).
- [2 An introduction to abstract harmonic analysis. Princeton, N. J.: D. Van
Nostrand Co. 1953.
-- [3] Linear functionals and content. Amer. J. Math. 76, 168--182 (1954).
LORENTZ, G. G. : [1] A contribution to the theory of divergent sequences. Acta
Math. 80, 167 -- 190 (1948).
Log, J. : [I ] Abelian groups that are direct summands of every abelian group which
contains them as a pure subgroup. Bull. Acad. Polon. Sci., S6r. Sci. Math.
Astr. Phys. 4, 73 (1956).
-- [2] Abelian groups that are direct summands of every abelian group which
contains them as pure subgroups. Fund. Math. 44, 84--90 (1957).
LUTHAR, INDAR S. : [1 ] Uniqueness of the invariant mean on an Abelian semigroup.
Illinois J. Math. 3, 28--44 (1959).
LYUBARSKIi, G. YA. [JIm6apcgt, F. ]:[.]: [t] On the integration in the mean of
almost periodic functions on topological groups [06 lHTerpIp0BaHml B cpeAHei
II0qTH Hepno)nqecix yHIIIIlft Ha TOHOaornqecnx rpyHHax]. Uspehi Mat.
Nauk, N.S. 3, vyp. 3 (25), 195--201 (1948).
MAAK, WILHELM : [I ] Fastperiodische Funktionen. Grundlehren der Mathemati-
schen Wissenschaften, Bd. LXI. Berlin-G6ttingen-Heidelberg: Springer 1950.
-- [2] Integralmittelwerte von Funktionen auf Gruppen und Halbgruppen. J.
Reine Angew. Math. 190, 34--48 (1952).
MACBEATH, A.M.: [1] On the measure of product sets in a topological group. J.
London Math. Soc. 35, 403--407 (I 960).
MACKEY, GEORGEW.: [1] A remark on locally compact Abelian groups. Bull.
Amer. Math. Soc. 52, 940--944 (1946).
-- [2] The Laplace transform for locally compact Abelian groups. Proc. Nat.
Acad. Sci. U.S.A. 34, 156--t62 (t948).
-- [3] Borel structures in groups and their duals. Trans. Amer. Math. Soc. 85,
134--165 (1957).
MARKOV, A.A. [Mapo, A. A.]: [I ] ber endlich-dimensionale Vektorrtume. Ann.
of Math. (2) 36, 464-- 506 (1935).
-- 2] On free topological groups [O CB060/m, IX woHoJIornqecnx rpyrmax]. Dokl.
Akad. Nauk SSSR, N.S. 31, 299-- 301 (1941).
-- [3] On the existence of periodic connected topological groups [O CyIIIeCWBOBaHnn
Hepnonqecnx CBaaHIX TOII0a0rnqecHnx rpyHn]. Izv. Akad. Nauk SSSR,
Ser. Mat. 8, 225--232 (1944).
32*
500 Bibliography Bibliography
MARKOV, A.A. [MapoB, A.A.]: [4] On free topological groups [O CB060HBIX
wonoaorHqecHnx rpynnax], lzv. Akad. Nauk SSSR, Ser. Mat. 9, 3--64 (t945).
MONTGOMERY, DEANE, and LEO ZIPPIN: [I] Topological transformation groups.
Interscience Tracts in Pure and Applied Mathematics t. New York, N.Y.:
Interscience Publishers, Inc. t955.
MuNN,W. D.: [ t ] On semigroup algebras. Proc. Cambridge Philos. Soc. 51, t - t 5 (1955).
MUNROE, M. E. : [1] Introduction to measure and integration. Cambridge, Mass. :
Addison-Wesley Publ. Co. t953.
MYCIELSKI, JAN: See HARTMAN, S.
NACHBIN, LEOPOLDO: [t ] On the finite dimensionality of every irreducible represen-
tation of a compact group. Proc. Amer. Math. Soc. 12, t I- t2 (196t).
NAMARK, M.A. [Ha/IMap, M.A.]: [t] Normed rings [H0pMHp0BannNe oabtIa].
Gostehizdat, Moscow, t956. Revised German translation : Normierte Algebren.
Berlin: VEB Deutscher Verlag der Wiss. t959. English translation: Normed
rings. Groningen: P. Noordhoff N.V. t 959.
-- See GEL'FAND, I.M.
NAKAYAMA, TADASI: [1] Note on free topological groups. Proc. Imp. Acad. Tokyo
19, 471--475 (1943).
NEUMANN, JOHN VON: [I] Zur Prtiferschen Theorie der idealen Zahlen. Acta Litt.
Sci. Szeged 2, 193-- 227 (t 926).
-- [2] Zur allgemeinen Theorie des Mal3es. Fund. Math. 13, 73--1t6 (t929).
-- [3] Die Einffihrung analytischer Parameter in topologischen Gruppen. Ann.
of Math. (2) 34, 170--190 (1933).
-- [4] Almost periodic functions in a group. I. Trans. Amer. Math. Soc. 36,
445 --492 (1934).
-- [5] Zum Haarschen Ma[3 in topologischen Gruppen. Compositio Math. 1,
106--114 (1934).
-- [6] The uniqueness of Haar's measure. Mat. Sb., N.S. 1 (43), 721--734 (1936).
--, and EUGENE P. WIGNER: [t] Minimally almost periodic groups. Ann. of Math.
(2) 41, 746--750 (t940).
NIKODi'M, OrroN M.: [1] Sur une gdn6ralisation des intdgrales de M.J. Radon.
Fund. Math. 15, 13 t -- 179 (1930).
NUMAKURA, I/:ASUM: [I] On bicompact semigroups. Math. J. Okayama Univ. 1,
99--108 (1952).
NUSSBAUM, A.E. : See DEVNArZ, A.
OLMSrED, J.M.H. : See GELBAUM, 13.
OXrOV, JOHN C. : [1 ] Invariant measures in groups which are not locally compact.
Trans. Amer. Math. Soc. 60, 215--237 (t946).
-- See t{AKUTANI, SHIZUO.
PALEY, R. E. A. C. : See WIENER, NORBERT.
PASYNKOV, I3. [IIae/,IHI0B, ]3.]: [I] On the coincidence of different definitions of
dimension for locally bicompact groups [O e0BHaeHHH paaaUHmX onpe)e-
aeHiI paaMepH0CT aa aoaaHO 6oMnaTHmX rpylm]. Dokl. Akad. Nauk
SSSR, N.S. 132, 1035--1037 (1960).
PRIS, JOSEPH" See VOLTERRA, VITO.
PETER, F. : See WEYL, HERMANN.
PHILLIPS, RALPH S. : See HILLE, EINAR.
PITT, H. R. : See WIENER, NORBERT.
P6LYA, G.: See HARDY, G.H.
PONTRYAGIN, L. S. [HoHTparH, 2I. C.]: [I] Der allgemeine DualitAtssatz ffir ab-
geschlossene Mengen. Verhandlungen des Internat. Math.-Kongr. Ziirich, 1932.
Ziirich u. Leipzig: Orell Fiissli, Bd. II, pp. 195--197.
PONTRYAGIN, L. S. [HOHTparH, JI. C. : [2] Sur les groupes topologiques compacts
et le cinquime problme de M. Hilbert. C. R. Acad. Sci. Paris 198, 238--240
(1934).
-- [3] Sur les groupes abdliens continus. C.R. Acad. Sci. Paris 198, 328--330
(1934).
-- [4] The theory of topological commutative groups. Ann. of Math. (2) 35,
361 -- 38} ( 934).
-- [5] Linear representations of compact topological groups [aInee npeaI-
CTaBaeItHa HOMIaHTHhIX TOnOormecgHx rpynn]. Mat. Sb., N.S. 1 (43),
267--271 (1936).
-- [6] Continuous groups [HenpepBHe rpynn]. Moscow, 1938. English trans-
lation: Topological groups. Princeton, N. J. : Princeton University Press t939.
- [7] Continuous groups, 2nd Edition [HenpepBHe rpynnI, aaHe BT0p0e].
Gostehizdat, Moscow, t954. German translation: Topologische Gruppen, I, II.
Leipzig: 13. G. Teubner t957 and 1958.
PRESTON, GERALD C. : [1] On locally compact totally disconnected Abelian groups
and their character groups. Pacific J. Math. 6, 121--134 (1956).
PROER, HEINZ: [1] Neue 13egrfindung der algebraischen Zahlentheorie. Math.
Ann. 94, 198--243 (t925).
RADON, JOHAYY: [t] Theorie und Anwendungen der absolut additiven Mengen-
funktionen. Sitzungsber. math.-naturwiss. K1. Akad. WiNs. Wien 122, Abt. IIa
(t913), t295-- 1438.
RAiOV, D.A. [Pai0B, . A.]: [1 Generalized duality theorem for commutative
groups with an invariant measure [O606meHHbi aaoH B0iCTBeHH0eT
0MMyTaTBHX rpynn C HBapaHTH0i Mep0i]. Dokl. Akad. Nauk SSSR,
N.S. 30, 589--591 (1941).
-- [2] A new proof of the uniqueness of Haar's measure [HoBoe oaaaTeeTBO
eHCTBeHH0CT Mep Xaapa]. Dokl. Akad. Nauk SSSR, N. S. 34, 211--212
(1942).
-- [3] Harmonic analysis on commutative groups with Haar measure and the
theory of characters [FapMoHnqeeH/I aHaaa Ha HOMMyTaTtIBItIIX rpynnax
weopn xapawepoB]. Trudy Mat. Inst. Steklov 14 (1945). German trans-
lation in Sowjctische Arbeiten zur Funktionalanalysis, pp. 1 t --87. Berlin: Kul-
tur u. Fortschritt 1954.
-- [4] On the theory of normed rings with involution [I we0p n0pMp0Bannx
HoaetI c mlBOamtIei]. Dokl. Akad. Nauk SSSR, N.S. 54, 387--390 (1946).
-- See GELtFAND, I.M.
RAIMI, RALPH A.: [t] On a theorem of E. Folncr. Math. Scand. 6, 47--49 (1958).
RICABARRA, R. : See COLLAR, MISCHA.
RICKART, CHARLES E.: [t] General theory of Banach algebras. Princeton, N.J.:
D. Van Nostrand Co. 1960
RIESZ, FRIDRIC" [1] Sur les opSrations fonctionnelles linaires. C.R. Acad. Sci.
Paris 149, 974--977 (1909). Also in Gesammelte Arbeiten Bd. I, pp. 400--402.
Budapest : Akadmiai Kiad6 1960.
-- [2] Untersuchungen fiber Systeme integrierbarer Funktionen. Math. Ann. 69,
449--497 (1910). Also in Gesammelte Arbeiten, Bd. I, pp. 441--489. Buda-
pest: Akaddmiai Kiad6 1960.
-- [3] Les systmes d'quations lindaires h une infinit6 d'inconnues. Paris:
Gauthier-Villars & Cie. 1913. Also in Gesammelte Arbeiten, Bd. II, pp. 829-- I 016.
Budapest : Akaddmiai Kiadd 1960.
RISS, JEAN: [] Representations continues des groupes topologiques abliens dans
le groupc additif dcs Hombres rdels. C. R. Acad. Sci. Paris 224, 987--988 (1947).
502 Bibliography Bibliography 503
Rlss, JEAN: [2] Sur les representations r6elles des groupes topologiques abdliens.
C. R. Acad. Sci. Paris 224, 1095--1097 (1947).
-- [3] ]ldments de calcul diffdrentiel et th6orie des distributions sur les groupes
ab61iens localement compacts. Acta Math. 89, 45--105 (1953).
ROBERTSON,WENDY: [1 ] Thesis, Cambridge University 1954.
ROBISON, GERSON B. : [1] Invariant integrals over a class of Banach spaces. Pacific
J. Math. 4, 123-- 150 (1954).
ROSEN, WILLIAM a.: [1] On invariant means over compact semigroups. Proc.
Amer. Math. Soc. 7, 1076--1082 (1956).
RUDIN, WALTER: [1] Independent perfect sets in groups. Michigan Math. J. 5,
159-161 (1958).
- [2] Measure algebras on abelian groups. Bull. Amer. Math. Soc. 65, 227--247
(1959).
RYLL-NAZEWSXI, C. : See HATMAN, S.
SAILS, STANISLAW : [1 ] Theory of the integral, 2nd Edition. Monografie Matematyczne,
Tom VII. Warszaxva-Lw6xv 1937.
SAM.LSON, HANS: See HALMOS, PAUL R.
SCHON.ON, H.INZ: [I] lber Linearformenmoduln unendlichen Ranges. I. Pri-
mgre, kompakte Linearformenmoduln. I I. Nichtarchimedisch perfekt bewertete,
operatorreduzierte Linearformenmoduln. J. Reine Angew. Math. 189, t 68-- t 85,
193--203 (1952).
-- [2] lber gewisse Topologien in Abelschen Gruppen. I, II. Math. Z. 59,
455--473 (1954); 60, 17--30 (1954).
-- [3] lber den Zusammenhang zwischen Dualitgts- und Vollstgndigkeitseigen-
schaften bei gewissen topologischen Abelschen Gruppen. Math. Z. 65, 429--44t
(1956).
SCU.IE, OTTO: [1] Abstrakte kontinuierliche Gruppen. Abh. Math. Sem. Univ.
Hamburg 4, 15--32 (1925).
SCHUR, IssAI: [1] Neue Anwendungen der Integralrechnung auf Probleme der
Invariantentheorie. I, II, III. Sitzungsber. Preuss. Akad. WiNs., phys.-math.
K1. 1924, 189-- 2O8, 297-- 321, 346-- 355.
SCHWARTZ, JACOB T. : See DUNFORD, NELSON.
SCHWARZ, TEFAN: [1] Theory of characters of commutative semigroups [Teop
xapawepoB HOMMyTaTHBHBIX noayrpynn]. Czechoslovak Math. J. 4 (79), 219-
247 (1954).
- [2] On the existence of invariant measures on certain types of bicompact
semigroups [O CyIIIeCTBOBaHHH HHBapHaHTHbIX Mep Ha neE0w0pbIX THIIaX
6HI0MIIat;THtdX noayrpynn]. Czechoslovak Math. J. 7 (82), 165-182 (1957).
SEGAL, IRVING F.. : [I] The group ring of a locally compact group. I. Proc. Nat.
Acad. Sci. U.S.A. 27, 348--352 (1941).
-- [2] Irreducible representations of operator algebras. Bull. Amer. Math. Soc.
53, 73--88 (1947).
-- [3] The group algebra of a locally compact group. Trans. Amer. Math. Soc.
61, 69--105 (1947).
SHIGA, K6JI: [1 ] Representations of a compact group on a Banach space. J. Math.
Soc. Japan 7, 224-248 (1955).
SIr-V.RMAN, ROU.RT J.: [1] Invariant linear functions. Trans. Amer. Math. Soc.
81, 411--424 (1956).
-- [2] Means on semigroups and the Hahn-Banach extension property. Trans.
Amer. Math. Soc. 83, 222--237 (1956).
-- [3] Invariant means and cones xvith vector interiors. Trans. Amer. Math. Soc.
88, 75-- 79 (1958).
SMITH, MARIANNE FREUNDLICH: [1] The Pontrjagin duality theorem in linear
spaces. Ann. of Math. (2) S6, 248--253 (1952).
gREiDER, YU.A. [IIIpeiaep, IO. A.]: [I] Construction of maximal ideals in rings of
completely additive measures [CTp0eHHe MaHcHMaJIbHt,IX ReaalOB B oabtlax
BH0ane aHTHBHtdX Mep]. Dokl. Akad. Nauk SSSR, N.S. 63, 359--361 (1948).
-- [2] Construction of maximal ideals in rings of measures with convolution
[Cwp0ente MaHcHMaa,H,IX ualeaaou oabtIax Mep CO cBepwH0fl]. Mat. Sb.,
N.S. 27 (69), 297-- 318 (1950).
SWEINHAUS, HUOO: [I] Additive und stetige Funktionaloperationen. Math. Z.
5, 186--221 (1919).
STOE, A.H.: [1] Paracompactness and product spaces. Bull. Amer. Math. Soc.
54, 977--982 (1948).
STROMURa, KARr- R. : See HEWITT, EDWIN.
STUUr-E, RaIMON A. : [1] Almost periodic functions on locally compact groups.
Proc. Nat. Acad. Sci. U.S.A. 39, t22--t26 (1953).
gwI.czowsI, S.: [I] Measures equivalent to the Haar measure. Proc. Glas-
gow Math. Assoc. 4, 157-- 162 (1960).
-- [2 Jacobians for measures in coset spaces. Proc. Glasgoxv Math. Assoc. 4,
208--212 (1960).
Sz.r-E, T." See K.TkSZ, A.
Sz.-NAOV, Bk.A: [1] Sur la mesure invariante darts des groupes topologiques.
C. R. Acad. Sci. Paris 202, 1248--1250 (1936).
-- [2] Zur Theorie der Charaktere Abelscher Gruppen. Math. Ann. 114, 373--384
(1937).
TASI, AI.R.: [lJ Ideale in vollst/indigen Mengenk6rpern. I. Fund. Math. 32,
45--63 (1939).
T6AMA, HIAU: [I] On Haar measure of some groups. Proc. Japan Acad. 24,
13--16 (1948).
URUAI, K. : [I] On the isomorphism of Haar measures. Fund. Math. 46, 277--284
(1959).
VAUGHAN, HERBERT E. : See HALMOS, PAUL R.
VILENKIN, N.yA. [BIIJIenItln, H. J=[.] : [1] Direct decompositions of topological
groups. I, II, III, Corrections [IIpMNe pa3aomena wonoaioruqecx rpynn.
I, II, III, HcnpauaeHI]. Mat. Sb., N.S. 19 (61), 85--154, 311--340 (1946);
22 (64), 191--192 (1948); 29 (71), 519--528 (1951).
-- [2] On the theory of general topological groups I veopu o6mx vonoaoru-
nec,x rpynn3. Dokl. Akad. Nauk SSSR, N.S. 58, 1573--1575 (1947).
-- [3] On the theory of weakly separable groups [I veopuu ca6o cenapa6eanix
rpynn]. Mat. Sb., N.S. 22 (64), 135-- 177 (1948).
-- [4] Fibered abelian topological groups and their character theory [Boouucvie
a6eeuIe vonoaornec,e rpynni x eopn xapa,vepou]. Mat. Sb., N. S.
24 (66), 189--226 (1949).
-- [5] On the classification of separable and coseparable topological abelian groups
[O aaccatI cenapa6eaHX ocenapa6ea,HiX Tonoaorecx
a6eaeBx rpynn]. Mat. Sb., N.S. 27 (69), 85--102 (1950).
-- [6] On the theory of general noncommutative topological groups [I Teop
o6mux He0sMyTaTUUHblX TOnOaoruuecax rpynn]. Dokl. Akad. Nauk SSSR,
N.S. 71, 1013--1015 (1950).
-- [7] Theory of topological groups. II. Direct products. Direct sums of groups
of rank 1. Locally bicompact abelian groups. Fibered and weakly separable
groups [Teopa TonoaorqecHx rpynn. II. HpnMNe npOgaBeReHn. HpnMIe
504 Bibliography Bibliography 505
CyMMII rpyrm paHra I. 3IoIaJIt, HO 6tlI0MIIaITHIIe a6eaeBI rpynni. Bo3Iom
HnCTIe n c3ia6o cenapa6e3iHie rpynnI]. Uspehi Mat. Nauk, N.S. 5, vyp. 4
(38), t9--74 (t950).
VILENKIN, N.YA. [BtI3IeHItlH, H. ]:[.] : [8] Direct and inverse spectra of topological
groups and their character theories [HpMe 06paTHIe cneTpI TOnO3IOrm
qecnx rpynlI x Teopn xapaTepOB]. Izv. Akad. Nauk SSSR, Ser. Mat. 15,
503--532 (1951).
-- [93 Direct operations on topological groups [IIpMue onepaunn na onoao-
rnqecnMn rpynnaMn]. Mat. Sb., N.S. 29 (71), 371--402 (1951).
-- [10] On the classification of zero-dimensional locally compact periodic abelian
groups without elements of finite order [I aaccnnaunn nyaMepnux
3IOHa3IH0 HOMIIaHTHhlX neponecx a6e3ieBiX rpynn 6ea 03IeMeHTOB
0HeH0r0 ilopua]. Mat. Sb., N.S. 28 (70), 503--536 (t95).
-- [tt On the existence of locally compact groups with given Ulm factors [O
cymeCTBOBaHtltl JIoHaJIt, HO HOMHaHTHIaIX rpynn c 3a/aHHIIMII yJIt, MOBCHtlMtl
agwopaM]. Mat. Sb., N.S. 29 (71), 13--30
-- [t2] On the isomorphism of locally compact zero-dimensional abelian groups
with isomorphic factors [O5 30MOp(3Me JIoam, no HOMIIaHTHI)IX nyJIMep-
HIX aSeieBiX rpy c II30MOp(DHt,IMII (awopaM]. Mat. Sb., N.S. 29 (71),
3t--62 (t951).
-- [13] Theory of characters of topological abelian groups with a given bounded-
hess [Teop xapawepoB wonoaorecnx a6eaeBx rpynn e aao
orpaeocwm]. Izv. Akad. Nauk SSSR, Ser. Mat. 15, 439--462 (1951).
-- [143 Generalized normal divisors of topological groups and their applications
to combinatorial topology [O6o6uloIe opMaae eawea wonoaorm
qeenx rpynn x npaome oM6nawopot wonoaor]. Trudy Mos-
kov. Mat. Ob§6. 3, 15--88 (1954).
-- [t5] On the classification of zero-dimensional locally compact abelian groups
with an everywhere dense set of elements of finite order [I
yaMepx aoao oMnawx a6eaeBx rpynn co BCmy
MomecWBO aeMewoB oeoro nopa]. Mat. Sb., N.S. 34 (76), 55--80
(1954).
-- [t6] On a class of locally compact zero-dimensional topological groups [06 0HOM
aacce 310aJI,H0 0MtlaTHIX HyYIbMepHbIX TOlIO3IOrecx rpynn]. Mat.
Sb., N.S. 40 (82), 479--496 (1956).
-- [17] On the dyadicity of the group space of bicompact commutative groups
[O anH0CT rpynnoro np0cTpaHCTBa 60MnaTHIX rpynn]. Uspehi
Mat. Nauk, N.S. 13, vyp. 6 (84), 79--80 (1958).
VOLTERRA, VITO, and JOSEPH PRS: [1 ] Lemons sur la composition et les fonctions
permutables. Paris: Gauthier-Villars & Cie. 1924.
WAERDEN, ]ARTEL L. VAN DER: [1 ] Algebra. Teil I. 4. Aufl. Grundlehren der
Mathematischen Wissenschaften, ]Bd. XXXIII. ]Berlin-G6ttingen-Heidelberg:
Springer t 955.
WALLACE, A.D.: [t] The structure of topological semigroups. Bull. Amer. Math.
Soc. 61, 95--t12 (1955).
WEIERSTRASS, KARL: [t] t3ber die analytische Darstellbarkeit sogenannter will-
kiirlicher Funktionen reeller Argumente. Sitzungsber. Preuss. Akad. Wiss.
1885, 633--640, 789--906. Also in Math. Werke, Bd. III, pp. 1--37. Berlin:
Mayer u. Mfiller t903.
WEIL, ANDR#.: [1] La mesure invariante dans les espaces de groupes et les espaces
homognes. Enseignement Math. 35, 241 (1936).
WEIL, ANDRe: [2 3 Sur les groupes topologiques et les groupes mesur6s. C. R. Acad.
Sci. Paris 202, t 147-- 1149 (I 936).
-- [3] Sur les espaces structure uniforme et sur la topologie g6n6rale. Actualit6s
Sci. et Ind. 551. Paris : Hermann & Cie. 1938.
-- [4] L'int6gration dans les groupes topologiques et ses applications. Actualit6s
Sci. et Ind. 869, 1145. Paris: Hermann & Cie. 1941 and t95t.
WENDEL, JAMES G.: [13 Left centralizers and isomorphisms of group algebras.
Pacific J. Math. 2, 251--261 (1952).
WELL, HERMANN." [1 Theorie der Darstellung kontinuierlicher halb-einfacher
Gruppen durch lineare Transformationen. I, II, III, Nachtrag. Math. Z. 23,
271--309 (1925); 24, 328--376, 377--395, 789--791 (1926). Also in Selecta
Hermann Weyl, pp. 262-- 366. ]Basel : BirkhXuser 1956.
-- [23 Integralgleichungen und fastperiodische Funktionen. Math. Ann. 97,
338--356 (t927). Also in Selecta Hermann Weyl, pp. 367--386. Basel: ]Birk-
huser 1956.
-- [33 The classical groups, their invariants and representations. Princeton, N. J. :
Princeton University Press 1939.
-, and F. PETER: [13 Die VollstXndigkeit der primitiven Darstellungen einer ge-
schlossenen kontinuierlichen Gruppe. Math. Ann. 97, 737-- 755 (1927). Also
in Selecta Hermann Weyl, pp. 387--404. ]Basel: ]BirkhXuser t956.
WIENER, IORBERT, and R. E. A.C. PALEY: [I ] Analytic properties of the characters
of infinite Abelian groups. Verhandlungen des Internat. Math.-IZongr. Zfirich,
1932. Zfirich u. Leipzig: Orell Fiissli, Bd. II, p. 95.
-- [23 Characters of Abelian groups. Proc. Nat. Acad. Sci. U.S.A. 19, 253--257
(1933).
--, and H.R. Pxx: l] On absolutely convergent Fourier-$tieltjes transforms.
Duke Math. J. 4, 420-436 (1938).
WNER, EUgEnE P. : l ] On unitary representations of the inhomogeneous Lorentz
group. Ann. of Math. (2) 40, t49--204 (1939).
-- See NEVMA, JOU VOW.
YOSIDA, KSAKU, and TURANE IWAMURA" [1] Equivalence of two topologies of
abelian groups. Proc. Imp. Acad. Tokyo 20, 451--453 (1944).
YOUNG, W.H.: [I] On classes of summable functions and their Fourier series.
Proc. Roy. Soc. London, Set. A 87, 225--229
ZIPPIN, LEO: See ALEXANDER, JAMESW., and MONTGOMERY, DEANE.
ZUCKERMAN, HERBERT S. : See HEWITT, ED'vVIN.
ZYGMUND, ANTONI : [I ] Trigonometric series, 2nd Edition. 2 Vols. Cambridge : Cam-
bridge University Press 959.
Index of symbols 507
Index of symbols
92[c(G ) [continuous almost periodic
functions] 24 7
(G) [almost periodic functions] 247
A (G, Ґ) 380
A u 470
A(X, H) [annihilator of HI 365
b G [Bohr compactification] 430
brG 430
3(H) [bounded operators] 467
3(X) [bounded functions] 230
c 2
(g(X), (go(X), (goo(X) tt9
Da/ 246
det A [determinant] 7
dim (X) [dimension] 15
b(X) 9
A [modular function] 195-- 196
A (h ..... /m ; e ; Zo) 360
A a [a-adic integers] t09
A r [r-adic integers] 109
dzy [Kronecker delta] 3
e [identity of a group] 4
E,, [G(/) =/(a)] 265
exp 3
exp (A) 29
e a [point mass at a] 270
G a [discrete group G] 127
(3 (n, F) [nonsingular matrices
over F] 7
G(") [= {xn:xEG}] 439
G(n ) [={xEG:xn=e}] 439
Gp 439
Hom(G, H) 374
[identity operator] 312, 484
[nonnegative linear functional
on 100 ] 120
I 121
122
I3 159
I., i., . 150
e(A) [= i(A)] 123
K [complex numbers] 3
K n 3
p, p(x, ) 135
p(G) [=p(G, 2)] 264
h, (x) 3
oo, oo (X, ') 14t
2 [left Haar measure] t93
Ak [subgroup of Qa] 110
', [e-measurable sets] t25
+ [lower semicontinuous
functions] t 21
m () [multiplicity] 15
M(G) 269
M + (G), M (G), etc. 269
M a (G), Mr, (G), M d (G), M s (G) 269
max(L, M), min(L, M) 461
min (, 7) 169
J (n, F) [matrices over F] 7
M(p) 3t9
M(X) 17o
[locally t-null sets] 124
[upper semicontinuous
functions] 12t
O(G/H) [quotient topology] 36
(n) [orthogonal group] 7
Qa [a-adic numbers] 108
Qr [r-adic numbers] 109
P ] 136, 141
P'
P(F, e) 361
(X) [all subsets of X] 2
[rational numbers] 3
R [real line] 3
R n 3
r(G) [rank of G] 444
r 0(G) [torsion-free rank of G] 444
rp(G) [p-rank of G] 444
9a [inner automorphism] 4
@ 140
®(G) [topological automorphisms] 426
sgn [signum] 3
S() [support of ] 124
®(n, F) [special linear group] 7
(G), S(G) [uniform structures] 21
Sn_ [sphere in R n] 3
®n [symmetric group] 8
®(n) [special orthogonal group] 7
®lI(n) [special unitary group] 7
'a [a-adic solenoid] 114
T [---{exp(2:gix)'OGx< I}] 3
trA [trace] 7
u [unit of an algebra] 469
1I (n) [unitary group] 7
3(F, U) 426
lv (X) [weight of X] 9
[structure space] 477--478
[character group] 355
Z [integers] 3
Z(m) [cyclic group with m elements] 5
Z (aoo) 403
z (p ) 3
[characteristic function] 2
[aP (x H) --- a x HI 4 -- 5
[vdid set] 2
L*M [convolution] 262
/*g, /,#,t,/ 286
,u • v 266
L;>M 46t
=<r/ 169
l<g, 14=o 120
p>-q [positive functionals] 325
A B [symmetric difference] 2
/og [composition] 2
G@H [semidirectproduct] 7
EI(DE. [direct sum] 452
A _[_B, ў_l_r/ 465
A/I 470
G/H [left cosets of HI 4
/IA 2
II all [norm in K n] 3
Illlls> a35
II/11. [uniform norm] t 19, 230
IIIIG 14t
117"11 454
II*ll 453
Ilxll [=]/x, x)] 464
I1,1 7o
] t67
169
I@1 t68
(a, ) [inner product in K n] 3
(],g) [in] t39
(x, y) [inner product] 464
(, ) 465
[cardinal number of A] 2
[conjugate of matrix] 7
[={k'xA}] 478
[= {D:#6A}] 360
[= {:/E}] 360
[Fourier transform] 360
[Fourier transform] 478
359
A ± 465
A-- [closure] 9
A' [complement] 2
A o [interior] 9
E ~ 465
g~ [adjoint] 300
L ~ [adjoint] 299
T ~ [adjoint] 466
x ~ 313
/~ [adjoint] 300
508 Index of symbols
A*
E*
/*
L!
M/
[adjoint homomorphism] 392
7
[conjugate space] 457
461
l(x) =l(xa)] 184
[d(x) =l(ax)] 184
Li(x) = L fix)] 262
[l(x) = s ()] 262
[transpose of matrix] 7
2R°, o 2
G m, G m * 6
X m 3
P G, P* G, 6
PX l 2
(x,) [element of,g/X,] 3
AB, A -1, aB, Ba, A n 4
@ H, [direct sum] 468
/, T 468
r 468
wd 178
fl(*) d [=ld] 194
f / ,64
A
2
[a, hi, [a, [, etc. 3
3
Step (I) of the proof of Theorem (22.10), page 341. (I) In order to
apply (22.7) in this proof, we need the following fact. Let h be a real- or
complex-valued 2-measurable function on G. Then there is a 2-measurable
function h' on G such that h--h' locally 2-almost everywhere, and h' B
is Borel measurable for all a-compact sets B c G.
We may suppose that h is nonnegative. By (5.7), G contains an open
and closed a-compact subgroup H. Let {xH}eA denote the distinct
cosets of H, and let h a =h,,n. For each fixed e A we have xH
where {F}.°__ is an increasing sequence of compact sets. Each function
min(h, n) belongs to I(G) and hence by (11.41), min(h,n),=g,,
2-almost everywhere for some Borel measurable function g. It follows
that lim g,(x) exists and is equal to h=(x) 2-almost everywhere. Let
h; = (lim g)..H for A.
Now for xG, we define h' (x)=h'=(x), where x x=H. If B is a-com-
pact, there exist at most countably many distinct elements e, e2, ...,
in A such that B C xH. From the foregoing, h'-
' " " " n=l
is Borel measurable and therefore h' is Borel measurable.
Index of authors and terms
91(G) 247
--, existence of invariant means 250
--, uniqueness of invariant means 252
-adic integers 109 [see also
-adic numbers 109 [see also
-adic solenoid 114
AUE, M. 425, 439
Abelian group [see also group]
-- --, nondiscrete topologies 27
absolutely continuous measure 180, 269
additive function 452
adjoint in an algebra 313
-- in )*(G) 310
-- in '(G) 299
-- in*
(G) 3o
-- in 214r(G) 300
adjoint homomorphism 392
adjoint operator 466
ALAOaLU'S theorem 458
ALEKSANDROV, A.D. 215
ALEKSAnIgOV, P.S. 47, 386
ALEXAnDEg, J.W. 78, 354, 398
algebra 469
-- with adjoint operation 313
--, convolution 263
-- homomorphism 470
-- imbedded in algebra with unit 470
-- ofsets 118
--algebra 313
almost everywhere 124
almost periodic function 247 [see also
-mesh 13
annihilator 365
ANZA, H. 399, 425, 439
approximate unit 303
approximation theorems 431 -- 432, 435
arbitrarily small sets 62
arcwise connected space 11
AENS, R.F. 439
AHAEI/SKIi, A. 398
ARONSZAJN, N. 465
AUBERT, K.g. 214
automorphism, inner 4
--, topological 426, 208
automorphism group 426--429
-- --, examples 433
-- --, inner automorphism subgroup
439
-- --, modular function of 438
-- --, non locally compact 435
BAER, R. 3t, 51
:BAIRE category theorem 42, 456
Baire measurable function t 18
Baire sets I t8
and Haar measure 280
balanced neighborhood 453
BALCERZYK, S. 425
Banach algebra 469
Banach -algebra 313
Banach fields, characterized 473
Banach space 455
-- --, reflexive 457
-- --, weak topology 458
-- --, weak-, topology 458
BANACH-STEIHAUS theorem 456
basis of a group 442
-- of a measure space 215
--, orthonormal 465
BEAUMONT, R.A. 425
BEURLING, A. 261, 282
bilinear functional 453
-- --, bounded 468
BIRKHOFF, G. 83, 243, 46t
BOCHNER, S. I, 282
Bohr compactification 430
Borel measurable function I t8
Borel sets 118
bounded bilinear functional 468
bounded linear function 454
-- -- --, relatively 461
bounded order 439
-- --, characterized 449
BOURBAKI, N. 32, 46, 52, 134, 135, 150,
166, 184, 461
BRACONNIER, J. 56, 60, 208, 278, 373,
395, 396, 398, 399, 417, 425, 426, 438
BucK, R.C. 275, 283
BUNYAKOVSKII'S inequality 137
510 Index of authors and terms Index of authors and terms 511
o(X) 119
--, characterized as Banach algebra 481
--, closed ideals in 482
--, conjugate space of 170, 175
--, multiplicative linear functionals of
483
--, structure space is X 483
' (G), adjoint operation in 299
-- as convolution algebra 265
-- equivalent with hi(G) 269
00 (x) 119
--, nonnegative linear functionals on
120
--, unbounded linear functionals on 167
CALDER6N, A.P. 261
cancellation laws 98-- 99
cancellation semigroup 258
Carath6odory outer measure 123
cardinal number of character groups
382, 396
of nondiscrete locally compact
groups 31
-- --, notation for 2
CARTAN, ]. 32
CARTAN, I-I. 57, t35, 214, 283,398
Cartesian product of sets 2
category, Baire 456
-- theorem of BAIRE 42, 456
CAUCHY'S inequality t 37
CAU CHY-BUNYAKOVSKII- S CHWARZ
inequality 464
center of a topological group 46, 64,429
character(s) 345
--, extensibility of 380
--, A-measurable implies continuous 346
--, product of 355
--, real 390, 393
--, sufficiently many 345
character of a measure space 215
and dimension of . 225
character group 355
of closed subgroup 380
of Z] a 402
-- --, A-topology of 360
of finite group 367
of local direct product 373
of Qa 400
with only one element 350, 370
-- --, P-topology of 361
of products 362-- 365
of Qd 404, 4t4
of quotient group 365
of R 367
character group of R d 405
of X a 403
-- -- and structure space of 7tla(G )
358, 360, 361
of T 366
-- -- of T d 405
topologically isomorphic with the
group 422
-- -- of weak direct product 364
-- -- of Z 366
of Z (a °°) 402--403
characteristic function 2
CHEVALLEY, C. 67, 106, 214
cluster point [of a net] 14
cofinal set 2
COHEN, L.V. 78
COHEN, P.j. 84
commutative algebra 469
commutator subgroup 358
compact elements 92, 103
compact groups, algebraic structure
of Abelian 410--414
have discrete character groups
362
-- --, small subgroups 61--62
-- --, torsion characterized 406
-- --, torsion-free characterized 406
compact space 11
compactification, Bohr 430
compactly generated group 35
complete lattice 462
complete measure space 216
complete regularity of groups 70
completely regular space 9
completely simple semigroup t01
complex algebra 469
complex linear space 452
complex measure(s) 118
-- --, FUBINI'S theorem for 183
-- --, product of t82
complex n-dimensional space 3
component 11
-- of e as direct factor 395
-- of e in topological groups 60--63
composition of functions 2
concentrated [a measure] 180
conjugacy classes of a group 5
conjugate elements of a group 5
conjugate of a matrix 7
conjugate space 457
of(0 t70, 175
of p 148
conjugate-linear function 452
connected groups 62, 383, 385, 390, 410
-- --, center of 64, 429
connected space 11
continuous measure 269, 132
continuous representation 341
-- --, strongly 335
-- --, weakly 335
continuous unitary representations,
equivalences 346
convergent net 14
convex combination 452
convex set 452
in ordered group 25
convolution 262
--, associativity of 262
-- of functions and measures 286,
290--298
-- involving unbounded functionals 278
-- of measures 266--269
--, using right Haar measure 306
convolution algebra 263
-- --, 3"(S) 275
-- --, '[u(G) and *ru(G) 275
-- --, (G) 265
-- --, l 1 (S) 263
-- --, noncommutative 275--276
correspondence 2
CORSON, H.H. 81
COTLAR, M. 375
countable set 2
countably additive measure 118
countably compact space 11
covering [cover] 2
cyclic groups, finite 5
cyclic representation 315
cyclic vector 315
DANIELL, P.J. 134
DANTZIG, D. VAN 32, 51, 67, 78, 83, 117,
425
DAUBLEBSKY VON STERNECK, R. 282
Davis, H.F. 215
DAY, M.M. 238, 241, 245, 372
A-topology 360
Aa t09, 416--417
--, character group of 402
--, Haar measure on 202
/lp 109
--, automorphism group of 434
--, minimal divisible extension of 419
DEVINATZ, A. 399
DIEUDONNt, J. 9, 25, 57
difference space 452
dimension 15
-- of a Hilbert space 465
direct protuct, character group of
362--365
of groups 6
-- --,local 56--57
of topological groups 52
, weak 6
direct sum of Hilbert spaces 468
of linear spaces 452
directs [directed set] t4
DIRICHLET, P.G.L. 281
discontinuous measure, purely 269
discrete group 24
has compact character group 362
-- --, Haar measure on 198
divisible group 440
-- --, characterized 444
-- --, imbeddings in 445--447
-- --, torsion-free characterized 42t
DIXMIER, J. 245, 393, 398, 399
dominated convergence theorem t81-
182
dual group 355 [see character group]
duality theorem 378
-- --, uniqueness of T 424
DUNFORD, N. t49, t66
ECKMANN, B. 425
EDWARDS, R.E. 135
empty set 2
e-mesh 13
equidistributions 432, 437
equivalent representations 314
equivalent uniform structures 21
ERD6S, P. 64
essentially bounded functions 141
exponential of a matrix 29
extensibility of characters 380
extensibility of real characters 39t
extensible positive functional 317
extensions of Haar measure 216
extensions of measure spaces 216
extreme point 459
extreme subset 459
factor 6
field of scalars 452
finite cyclic groups 5
finite intersection property 2
finite-dimensional representation, direct
sum of irreducibles 333--334
-- --, no unitary 348--351
finitely additive measures 118
512 Index of authors and terms Index of authors and terms 513
finitely additive measures, invariant
242-- 245
finitely generated group 449
--, characterized 451
first category 456
first isomorphism theorem for groups 5
-- for topological groups 44
FOLNER, E. 245, 375
Fourier transform 360, 478
Fourier-Stieltjes transform 360
free Abelian group 8
free group 8
free topological group 72
FREUDENTHAL, H. 32, 49, 51, 52, 60,
355, 375, 393, 399
FROBENIUS, G. 213, 375
FUBINI'S theorem 153-- 156
for complex measures 183
FUCHS, L. 381, 425
functional, bilinear 453
--, bounded bilinear 468
--, conjugate-linear 452
--, linear 452
--, multiplicative linear 474
--, nonnegative 461
--, positive 316
--, relatively bounded 461
--, strictly positive 461
--, total variation of 168
fundamental theorem of Abelian groups
451
GELBAUM, B. 104
GEL'FAND, I.[. 282, 311, 334, 353,
354, 375
GEL'FAND'S theorem 478
Gel'fand topology 478
GEL'FAND-/IAZUR theorem 473
GEL'FAND-NAMARK theorem 481
GEL'FAND-RMKOV theorem 343
general linear group 7 [see also
generator of monothetic group 105, 407,
415
GLICKSBERG, I. 135
((n, F) 7
--, Haar measure on 201, 209
-- has inequivalent uniform structures
28--29
--, left invariant metric on 78
-- is locally Euclidean 29--31
GODEMENT, R. 309, 311, 332, 334, 353,
354, 398
GRABAR', L. 80
GRAEV, M.I. 47, 50, 60, 74, 79, 82, 83
group [see also topological group]
-- [= Abelian group in Appendix A] 439
-- of bounded order 439
--, bounded order characterized 449
--, divisible 440
--, finite cyclic 5
--, finitely generated 449
--, free 8
--, free Abelian 8
--, linear 7
--, p-primary 439
--, permutation 8
--, reduced 440
--, symmetric 8
--, topological [see topological group]
--, torsion 439
--, torsion-free 439
GUREVI, A. 354
HAAR, A. 1, 32, 214, 375
Haar integral [left] 184-- 194
[see also Haar measure]
-- --, examples of 198, 209
-- --, existence and uniqueness of
185--193
-- --, right 195
Haar measure [left] 194
and Baire sets 280
-- --, examples of 198, 209
-- --, extensions of 216
-- --, right 195
-- --, uniqueness of 193--194
HAHN-BANACH theorem 454--455
HALL, M. JR. 26
HALMOS, P.R. 229, 261, 280, 398, 415,
425
HAMEL, G. 425
HARDY, G.H. 149
HARRISON, D. I(. 425
HARTMAN, S. 68, 393, 395, 398
NELSON, H. 311
HENSEL, K. 117
Hermitian element 313
Hermitian matrix 7
Hermitian operator 467
-- --, spectral theorem for 491
-- --, spectrum is real 484
HEWITT, E. 105, 135, 184, 261, 282,
283, 312, 354, 370, 375, 397, 435, 439
Hilbert space 464
-- --, dimension of 465
Hilbert space, direct sum of 468
HOFMANN, K.H. 64, 439
H6LDER'S inequality 137
homogeneous function 452
homogeneous space 9
-- --, quotient spaces 37
homomorphism, adjoint 392
--, algebra 470
homomorphism group 374
HOPF, H. 47
HULANICKI, A. 79, 106, 395, 425
HURWITZ, A. 213
ideal of an algebra 469
--, regular 475
--, regular left 474
-- in a semigroup 100
idempotent 99
independent subset of a group 441 --442
-- -- --, pure 448
infinity, conventions regarding 119
inner automorphisms 4
-- --, topological group of 439
inner product 464
in K n 3
for z 139
inner product space 464
integral, Lebesgue 119
invariant finitely additive measures
242--245
invariant linear functional 184
invariant means 230
for .I(G) 250--252
for 3' (G) 239
for 3 r (R) 240
for3 r(S) 237
-- --, groups not admitting 236
-- --, uniqueness of 241
invariant pseudo-metric 67
invariant set function 185
invariant subspace [under a representa-
tion] 313
inverse mapping system 55
inverses, form open set 473
inversion invariant functional 184
inversion invariant set function 185
inversion mapping 4
t-almost everywhere 124
t-measurable function 125
t-measurable set 125
t-null function 124
t-null set 124
irreducible representation 323
Hewitt and Ross, Abstract harmonic analysis, vol, I
irreducible representation, equivalent
properties 324
irreducible set of operators 323
ISlWATA, T. 399
isometry, linear 464
isomorphism theorems 5, 44--45
IVANOVSKIi, L.N. 106
IWAMURA, T. 375
IWASAWA, K. 26, 439
JAMISON, R. E. 132
JONES, F. B. 79
KAKUTANI, S. 78, 81, 83, 134, 214, 215,
229, 280, 396, 398, 399, 425, 439
KAKUTANI-OXTOBY extension of
Haar measure 216
KALISCH, G.K. 104
KAMeEN, E.R. VAN 32, 51, 67, 104, 06,
354, 375, 397, 398
KAeLAN, S. 399
KALANSKV, I. 398, 425
KASUGA, T. 58
I{AWADA, Y. 261
KEINER, H. 259
KELLEY, J.L. 1, 460
KEMPERMAN, J. H.B. 229
kernel of a semigroup 101
KERTtSZ, A. 27
KESTEN, H. 245
KLEE, V. L. 371
KNESER, M. 229
KODAIRA, K. 83, 280
K6NIG'S theorem 21 7
KoosIs, P. 354
KREiN, M.G. 282
KREiN-MIL'MAN theorem 460
applied 330
I{RISTENSEN, L. 77
KRONECKER approximation theorems
431, 435, 436
Kronecker's delta function 3
KRULL, W. 425
KUROg, A.G. 26
KUZ'MINOV, V. 106
1 (G), isomorphic with M a 272
[see also M a (G) ]
lattice 461
LEBESaUE, H. 150, 283
Lebesgue integral 119
LEBESUE'S theorem oft dominated
convergence 181 -- 182
33
5t4 Index of authors and terms Index of authors and terms 5 ! 5
LEBESGUE-RADON-2qlKODrM theorem
144
left cancellation law 98--99
left Haar integral 184-- 194
[see also Haar integral]
left Haar measure 194 [see also Haar
measure]
left ideal of an algebra 469
-- --, regular 474
ill a semigroup 100
left invariant functional 184
left invariant mean 230
[see also invariant means]
left invariant pseudo-metric 67
left invariant set function 185
left quasi-inverse 471
left translate of a function 184
left translation 4
left uniform structure 21
left unit relative to I 474--475
LEHMER, D.H. 282
LEJA, F. 31, 67
length of reduced vord 8
LEPTIN, H. 399
LEVI, F. 51
Lindel6f property 11
linear function 452
-- --, bounded 454
-- --, norm of 454
linear functional(s) 452
-- --, bounded 454
-- --, infinite product of 159
-- --, inversion invariant 184
left invariant 184
-- --, multiplicative 474
-- --, nonnegative 461
-- --, nonnegative on G00 120
-- --,normof 454
-- --, positive 316
-- --, product of t52, 159
-- --, relatively bounded 461
-- --, strictly positive 461
-- --, two-sided invariant t84
unbounded on 00 167
linear groups 7, 29-- 31
topologies in 24
linear isometry 464
linear space(s) 451 --452
, Banach 455
-- --, conjugate of 457
-- --, direct sum of 452
Hilbert 464
,
-- --, inner product 464
linear space(s), locally convex 453
-- --, normed 453
-- --, topological 453
linear subspace 452
linear transformation 452
LITTLEWOOD, J.E. 149
local direct product 56--57
-- -- --, character group of 373
-- -- --, self-dual 422
locally almost everywhere 124
locally arcwise connected space 12
locally bounded group 215
locally compact groups, normality of 76
locally compact space 11
locally connected compact Abelian
groups 399
locally connected space 11
locally convex space 453
locally countably compact space 11
locally Euclidean space 13
-- -- --, linear groups are 293t
locally finite 13
locally ,-almost everywhere 124
locally ,-null function t24
locally ,-null set t24
locally null function 124
locally null nonnull sets 127, 228
locally null set 24
LooMIs, L.H. 32, t35, 215, 31t, 353
LORENTZ, G.G. 245
Log, J. 425
lower semicontinuous function t2t
LUTHAR, I.S. 245
LYUBARSKII, G. YA. 26t
2$1(G) 269
--, adjoint operation in 300
-- commutative if and only if G is 302
-- is conjugate space of G0 (G) 170
-- = Md ( M s ( M a 273
--, 2la is closed ideal 272
--,/II c is closed ideal 271
--, M d is closed subalgebra 270
--, M, is closed linear subspace 273
--, l-dimensional ideals of 309-- 310
M a (O) 269
-- has approximate unit 303
-- commutative if and only if G is 302
--, ideals characterized 303
--, isomorphic with 272
-- has no unit 303
MAAK, W. 257, 259, 26, 425
MACBEATH, A.M. 229
VIACKEY, G.W. 229, 398, 421, 425
mapping 2
VIARKOV, A. A. 32, 51, 83, 104, 106
marriage lemma 248
matrix [special types defined] 7
matrix group 7 [see also linear group]
maximal ideal space 478
mean 230 [see also invariant means]
measurable function t 18, 125
measurable representation, weakly 335
measurable set t25
measure(s) t 18
--, absolutely continuous 180, 269
--, Carath6odory outer 123
-- concentrated on a set 180
--, continuous 269
--, convolution of 266
--, Haar 194 [see also Haar measure]
--, invariant finitely additive 242--
245
--, inversion invariant 185
--, left invariant 185
--, product of t 52
--, product of complex 182
--, purely discontinuous 269
--, regular 127
--, singular 180, 269
--, support of 124
--, total variation of 169
measure space, basis of 215
-- --, character of 215
-- --, complete 216
-- --, extension of 216
mesh 13
metrizability of topological groups 70
MICHAEL, E.A. 83
MICHELOW, J. t 13
minimal divisible extension of a group
49, 445--447
)/[INKOWSKI'S inequality t38
modular function t96
on closed normal subgroups 206
-- -- for (G) 438
monothetic group 85, 390, 407--409
-- --, generators of 105, 407, 415
-- --, largest 407
monothetic semigroups 105
IONTGOMERY, D. 32, 5, 66, 76, 83,
106
multiplicative function 345
multiplicative linear functional 474
multiplicity [of family of sets] 5
MuNN, W. D. 282
MYClELSKI, J. 66
n-dimensional space, complex and real 3
-- --, compact Hausdorff 15
NACHBIN, L. 354
NTAMARK, M.A. 135, 166, 334, 353, 354
NTAKAYAMA, T. 83, 425
natural mapping 4
neighborhood [always open] 9
net t 4
NTEUMANN, J. VON 26, 32, 1 ! 7, 134, 214,
243, 245, 283, 349, 354, 425
NIKODM, O.M. t 50
nondiscrete topologies for Abelian
groups 27
-- -- for Z 27
nonmeasurable sets 226
nonnegative linear functional 461
one00 t20
nonnormal groups 74--76
norm 453
-- ink n 3
-- in l (X) 3
-- inP.p(X,t) 135
-- of linear function 454
--, uniform 119, 230
norm topology 454
normal operator 467
-- --, existence of inverse 484
normal subgroup 16
normal topological group 16
normality of locally compact groups 76
normed algebra 469
normed --algebra 313
normed linear space 453
-- -- --, reflexive 457
-- -- --, weak topology 458
-- -- --, weak-, topology 458
nowhere dense 456
null function 124
null set t24
2qUMAKURA, K. 106
NUSSBAUM, A.E. 399
(n) 7
-- is compact 29
-- is locally Euclidean 29--31
OLMSTED, J.M.H. 104
og-functions 259
/2 108
--, character group of 400
--, closed subgroups of 116
--, Haar measure on 202--203
r 109
33*
5 ! 6 Index of authors and terms Index of authors and terms 517
/2r, automorphism group of 433
--, character group is Qr 400
one-parameter subgroup 85
open and closed subgroups 33--34, 62
operators 452
--, adjoint 466
--, Hermitian 467
--, normal 467
--, positive-definite 467
--, projection 467
--, sum of 468
--, unitary 467
ordered groups 24
orthogonal elements 465
orthogonal group 7 [see also (n) J
special 7 [see also (n)]
orthogonal matrix 7
orthogonal set 465
orthogonal sets 465
orthonorlnal basis 465
orthonormal set 465
OXTOBY, J.C. 215, 229
p-adic integers 109 [see also
p-adic number field 112 [see also
p-adic numbers 109 [see also
p-primary group 439
p-rank of a group 444
P-topology 361
PALEҐ, R.E.A.C. 375
paracompact space 13
paracompactness of locally compact
groups 76
partition of a set 2
partitions of unity 9--10
PASYNKOV, B. 398
PRfS, J. 282
permutation group 8
permutations 8
PETER, F. 213, 283, 311, 353, 354, 375
PERCE, R.S. 413, 425, 449
PITT, H.R. 282
P6LYA, G. 149
PONTRYAON, L.S. 32, 51, 60, 80, 83,
103, 106, 354, 375, 397, 398, 399, 424
PONTRYAGIN-VAN IAMPEN duality
theorem 378
--, uniqueness of T 424
positive functional 316
-- --, extensible 317
-- --, nonextensible 331
positive linear functional, strictly 461
positive-definite operator 467
positive-definite operator, spectrum is
nonnegative 484
-- --, square root of 484
PRESTO., G.C. 399
primary group 439
product of characters 355
-- of complex measures 182
-- of functionals 152, 159
-- of groups 6
-- of measures 152
-- of sets 2--3
product of topological groups 52
-- -- --, character group of 362--365
projection 54
projection operator 467
projective limit 56
proper ideal, of an algebra 469
proper subgroup 4
PRt3FER, H. 117
pure independent set 448
pure subgroups 447, 395
purely discontinuous measure 269
Q [rational numbers 3
--, character group of 4 04, 414
quasi-inverse(s) 471
--, form open set 472
quaternions, Haar measure on 210
quotient group 4, 40
-- --, character group of 365
quotient space 4, 452
-- --, homogeneity of 37
-- --, topology of 36
R real line3 3
--, automorphism group of 433
--, character group is R 367
--, compact connected topology for 415
--, continuous homomorphisms of 370
--, Haar measure on 198
--, invariant mean for c(R) 256
--, invariant means for 3 r(R) 240
-- is open continuous homomorph of
totally disconnected group 50
--, topologies in 27
R n 3
--, automorphism group of 434
--, characterized 104
--, closed subgroups of 92
r-adic integers 109 [see also A p]
r-adic numbers 109 [see also f2]
RADON, j. 150
RADON-NIKODrM theorem 144
RAiKOV, D.A. 214, 311, 334, 353, 354,
375, 398
RAIMI, R.A. 245
rank of a group 444
real algebra 469
real characters 390, 393
-- --, extensibility of 391
real-character group 390
real linear space 452
real lnatrix 7
real n-dimensional space 3
reduce 323
reduced group 440
reduced word 8
reducible representation 323
reducible set of operators 323
refinement 3
reflexive space 457
regular ideal 475
regular left ideal 474
regular measure 127
regular representation 342
regular topological space 9
relatively bounded linear functional 461
relatively invariant functionals 203
--, examples 212
representation(s) of an algebra 312
-- of an algebra with unit 313
--, continuous 34
--, cyclic 315
--, equivalent 314
-- ofagroup 312
--, invariant subspace under 313
--, irreducible 323
--, reducible 323
--, regular 342
-- of a semigroup 312
--, strongly continuous 335
--, sufficiently many 343
--, weakly continuous 335
--, weakly measurable 335
representation space 313
-representation 314
RICABARRA, R. 375
RICHARDSON, R.W. 94
RICKART, C.E. 334
RIESZ, F. 134, 150
F. RIESZ'S representation theorem 129
for Hilbert spaces 464
right [see also left
right Haar integral 195
right Haar measure 195
right unit relative to I 474--475
ring of sets 118
RlSS, J. 398
ROBERTSON, \V. 372
ROUlSON, G.B. 237
ROSEN, W. G. 215
RUDIN, V. 184, 425
RYLL-NARDZEWSKI, C. 393, 398
SAKS, S. 166
SAMELSON, H. 415, 425
scalar field 452
SCH6NEBORN, H. 399
SCHREIER, O. 31, 64, 67
SCHUR, I. 213
SCHUR'S lemma 324
SCHWARTZ, J.T. 149, 166
SCHWARZ, . 215, 354
SCHWARZ'S inequality 464
second category 456
second character group 376
second isomorphism theorem for groups
5
for topological groups 45
sections 153
SEL, I.E. 311, 334, 353, 354
self-dual groups 422
semicharacter 345
semicontinuous functions !21
semidirect product of groups 6--7
-- --, Haar measure on 210
-- -- of topological groups 58-- 59
semigroup 4
--, cancellation 258
--, topological 98, 233
separate points 151
SHIGA, I. 353
-algebra of sets 118
-compact spaces 11
-- --, products of 13-- 14
-finite set function 118
-locally finite 13
-ring of sets 118
signun [sgn] 3
SILVERMAN, R.J. 245
simple algebra 469
simple semigroup 100-- 101
singular measure 180, 269
skew-Hermitian matrix 7
skexv-symmetric matrix 7
(n, F) 7
--, homomorphisms into 0, xa[ 22--
213
5t8 Index of authors and terms Index of authors and terms 519
a.(n, F) has inequivalent uniform
structures 28--29
-- is locally Euclidean 29--31
-- has no finite-dimensional unitary
representations 350
SMITH, M.F. 370, 399
®9 (n) 7
-- is compact 29
-- is locally Euclidean 29--31
solenoidal groups 85, 409--410
-- --, a-adic tt4
-- --, largest 4t0
special linear group 7 [see also a.(n,F)]
special orthogonal group 7 [see also
special unitary group 7 [see also all (n) ]
spectral theorem 488--49t
applied 325
for Hermitian operators 491
spectrum 476
-- is compact nonvoid 477
square root of positive-definite operator
484
REtDER, YU. A. 3t t
STEINHAUS, n. 150
STONE, A.H. 83
STONE-WEIERSTRAS$ theorem 151,
2Sl--282
strictly positive linear functional 461
STROMBERG, K.P,. 397
stronger topology 9
strongly continuous representation 335
STRUBLE, :R.A. 261
structure space 477 -- 478
structure theorem for locally compact,
compactly generated Abelian groups
9O
all (n) 7
-- is compact 29
-- is locally Euclidean 29--31
subgroup, commutator 358
--, normal t6
--, one-parameter 85
--, open and closed 33--34, 62
--, proper 4
--, pure 447
subnet 14
sufficiently many 343
characters 345
representations 343
sum of Hilbert spaces 468
sum of linear spaces 452
sum of operators 468
support of a measure t24
Sugkevi6 kernel 101
WIERCZKOWSKI, G. 229
symmetric difference 2
symmetric group an 8
symmetric matrix 7
symmetric set t9
SZELE, T. 27
,z.-NAGY, B. 214, 375
T3
--, automorphism group of 433
--, character group is Z 366
--, continuous automorphisms of 369
--, Haar measure on 199
TARSKI, A. 220
THOELE, F. 48t
THOMA, E. 103, 348
topological automorphism 426, 208
topological field 112
topological group 16
always T o after page 83
-- --, automorphism group of 426--429
-- --, compactly generated 35
-- --, discrete 24
-- --, free 72
-- --, homomorphism groups of 374
-- --, independence of axioms 25
-- --, monothetic 85
-- --, normal t6
-- --, ordered 24
-- --, quotient 40
-- --, real-character 390
-- --, self-dual 422
-- --, solenoidal 85
-- --, unimodular 196
topological linear space 453
-- -- --, locally convex 453
with only one linear
functional 371 -- 373
topological ring 112
topological semigroup 98, 233
topologically isomorphic 41
topology, A- 360
--, Gel'fand 478
--, norm 454
--,P- 361
--, stronger 9
--, weak 458
--, weaker 9
--, weak-. 458
--, Well 229
torsion group 439
-- --, compact characterized 406
torsion-free group 439
torsion-free group, compact character-
ized 406
-- --, divisible characterized 421
-- rank 444
total variation of a measure 169
totally bounded metric space 13
totally disconnected group, small
subgroups of 62
totally disconnected space 11
T6YAMA, H. 214
trace of a matrix 7
transform, Fourier 360, 478
.
--, Fourier-Stieltjes 360
transformation 2
transitive set of mappings S
translate of a function 184
translation 4
translation of functions, continuity of
285, 305
transpose of a matrix 7
triangle inequality 453
two-sided ideal of an algebra 469
in a semigroup 100
two-sided invariant functional 184
two-sided invariant mean 230
[see also invariant means]
two-sided invariant pseudo-metric 67
1I () 7
-- is arcwise connected 64
-- is compact 29
-- is locally Euclidean 29--31
uniform nearness 20
uniform norm t 19, 230
uniform structures 21
-- --, equivalent 21
-- --, examples 28
uniformly continuous mapping 21
unimodular group 196
-- --, compact 196
-- --, equivalent uniform structures
278
uniqueness of T in duality theorem 424
unit, approximate 303
-- relative toI 474--475
unit ball 453
unitary group 7 [see also
-- --, special 7 [see also 1I (n)]
unitary matrix 7
unitary operator 467
unitary representations, continuity
equivalences 346
-- --, no finite-dimensional 348
upper semicontinuous function 121
URBANIK, I. 229
variation of a measure 169
VAUGHAN, H.E. 261
vector, cyclic 315
vector space 451--452
VILENKIN, N. YA. 52, 60, 82, 106, 375,
376, 399, 423, 426
VITALI, G. 283
\rOLTERRA, V. 282
WAERDEN, B.L. VAN DER 1
WALLACE, A.D. 106
veak direct product of groups 6
, character group of
364--365
weak topology 458
weaker topology 9
weakly continuous representation 335
weakly measurable representation 335
weak-, topology 458
WEIERSTRASS, K. 281, 282
VEIERSTRASS'S approximation theorem
281 --282
weight function 144
\VEIL, A. 1, 32, 59, 60, 83, 106, 134,
135, 214, 229, 283, 311, 353, 354,
355, 375, 398, 399
Well topology 229
WENDEL, J.G. 311
\VEYL, H. 213, 214, 282, 283, 311, 334,
353, 354, 375
WIENER, N. 282, 375
WENER-Hovv equation 282
WIGNER, E. . 26, 349, 353
xvord 8
YOSIDA, I. 3 75
YOUNG, W. H. 150
Z [integers] 3
--, character group is T 366
--, invariant mean for 9 (Z) 256
--, nondiscrete topology for 27
zero of semigroup 257
0-dimensional group, small subgroups of
62
0-dimensional space 11
ZIPPIN, L. 32, 51, 66, 76, 83, 106, 398
ZUCKERMAN, H.S. 282, 283, 312, 354,
370, 435, 439
ZYGMUND, A. l, 282
Grundlehren der mathematischen Wissenschaften
Continued from page ii
81. Schneider: Einfiihrung in die transzendenten Zablen
82. Specht: Gruppentheorie
84. Conforto: Abelsche Funktionen und algebraische Geometrie
86. Richter: Wahrscheinlichkeitstheorie
88. Mfiller: Grundprobleme der mathematischen Theorie elektromagnetischer Schwingungen
89. Pfluger: Theorie der Riemannschen Flichen
90. Oberhettinger: Tabellen zur Fourier-Transformation
91. Prachar: Primzahlverteilung
93. Hadwiger: Vorlesungen fiber Inhalt, Oberfliche und Isoperimetrie
94. Funk: Variationsrechnung und ihre Anwendung in Physik und Technik
95. Maeda: Kontinuierliche Geometrien
97. Greub: Linear Algebra
98. Saxer: Versicherungsmathematik. 2. Teil
99. Cassels: An Introduction to the Geometry of Numbers
100. Koppenfels/Stallmann: Praxis der konformen Abbildung
101. Rund: The Differential Geometry of Finsler Spaces
103. Schfitte: Beweistheorie
104. Chung: Markov Chains with Stationary Transition Probabilities
105. Rinow: Die innere Geometrie der metrischen Riume
106. Scholz/Hasenjaeger: Grundzfige der mathematischen Logik
107. K6the: Topologische lineare Riume I
108. Dynkin: Die Grundlagen der Theorie der Markoffschen Prozesse
110. Dinghas: Vorlesungen fiber Funktionentheorie
111. Lions: Equations diff6rentielles op6rationnelles et problmes aux limites
112. Morgenstern/Szab6: Vorlesungen fiber theoretische Mechanik
113. Meschkowski: Hilbertsche Riume mit Kernfunktion
114. MacLane: Homology
115. Hewitt/Ross: Abstract Harmonic Analysis. Vol. 1: Structure of Topological Groups,
Integration Theory, Group Representations, Second Edition
116. H6rmander: Linear Partial Differential Operators
117. O'Meara: Introduction to Quadratic Forms
118. Schiike: Einffihrung in die Theorie der speziellen Funktionen der mathematischen Physik
119. Harris: The Theory of Branching Processes
120. Collatz: Funktionalanalysis und numerische Mathematik
121. Dynkin: Markov Processes
122. Dynkin: Markov Processes
123. Yosida: Functional Analysis
124. Morgenstern: Einffihrung in die Wahrscheinlichkeitsrechaung und mathematische Statistik
125. Ito/McKean: Diffusion Processes and Their Sample Paths
126. LehtoNirtanen: Quasikonforme Abbildungen
127. Hermes: Enumerability, Decidability, Computability
128. Braun/Koecher: Jordan-Algebren
129. Nikodym: The Mathematical Apparatus for Quantum Theories
130. Morrey: Multiple Integrals in the Calculus of Variations
131. Hirzebruch: Topological Methods in Algebraic Geometry
132. Kato: Perturbation Theory for Linear Operators
133. Haupt/Kfinneth: Geometrische Ordnungen
Grundlehren der mathematischen Wissenschaften Grundlehren der mathematischen Wissenschaften
134. Huppert: Endliche Gruppen I
135. Handbook for Automatic Computation. Vol. 1/Part a: Rutishauser: Description of ALGOL 60
136. Greub: Multilinear Algebra
137. Handbook for Automatic Computation. Vol. 1/Part b: Grau/Hill/Langmaack: Translation of
ALC, OL 60
138. Hahn: Stability of Motion
139. Mathematische Hilismittel des lngenieurs. Herausgeber: Sauer/Szab6. 1. Teil
140. Mathematische Hilismittel des Ingenieurs. Herausgeber: Sauer/Szab6. 2. Teil
141. Mathematische Hilismittel des Ingenieurs. Herausgeber: Sauer/Szab6. 3. Teil
142. Mathematische Hilismittel des Ingenieurs. Herausgeber: Sauer/Szab6.4. Teil
143. Schur/Grunsky: Vorlesungen fiber Invariantentheorie
144. Weft: Basic Number Theory
145. Butzer/Berens: Semi-Groups of Operators and Approximation
146. Treves: Locally Convex Spaces and Linear Partial Differential Equations
147. Lamotke: Semisimpliziale algebraische Topologie
148. Chandrasekharan: Introduction to Analytic Number Theory
149. Sario/Oikawa: Capacity Functions
150. Iosifescu/Theodorescu: Random Processes and Learning
151. Mahdi: Analytical Treatment of One-Dimensional Markov Processes
152. Hewitt/Ross: Abstract Harmonic Analysis. Vol. 2: Structure and Analysis for Compact
Groups. Analysis on Locally Compact Abellan Groups
153. Federer: Geometric Measure Theory
154. Singer: Bases in Banach Spaces I
155. Mfiller: Foundations of the Mathematical Theory of Electromagnetic Waves
156. van tier Waerden: Mathematical Statistics
157. Prohorov/Rozanov: Probability Theory
158. Constantinescu/Comea: Potential Theory on Harmonic Spaces
159. K6the: Topological Vector Spaces I
160. Agrest/Maksimov: Theory of Incomplete Cylindrical Functions and Their Applications. In
preparation
161. Bhatia/Szeg6: Stability Theory of Dynamical Systems
162. Nevanlinna: Analytic Functions
163. Stoer/Witzgall: Convexity and Optimization in Finite Dimensions I
164. Sario/Nakai: Classification Theory of Riemann Surfaces
165. Mitrinovic: Analytic Inequalities
166. Grothendieck/Dieudonn6: El6ments de G6ometrie Alg6brique 1. En pr6paration
167. Chandrasekharan: Arithmetical Functions
168. Palamodov: Linear Differential Operators with Constant Coefficients
169. Rademacher: Topics in Analytic Number Theory. In preparation
170. Lions: Optimal Control Systems Governed by Partial Differential Equations
171. Singer: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces
172. Bfihlmann: Mathematical Methods in Risk Theory. In preparation
173. Maeda/Maeda: Theory of Symmetric Lattices
174. Stiefel/Scheifele: Linear and Regular Celestial Mechanics. Perturbed Two-body Mo-
tion-Numerical Methods--Canonical Theory
175. Larsen: An Introduction to the Theory of Multipliers
176. Grauert/Remmert: Analytische Stellenalgebren
177. Flfigge: Practical Quantum Mechanics I
178. Flfigge: Practical Quantum Mechanics II
179. Giraud: Cohomologie non ab61ienne
180. Landkof: Foundations of Modern Potential Theory
181. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications I
182. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications II
183. Lions/Magenes: Non-Homogeneous Boundary Value Problems and Applications III
184. Rosenblatt: Markov Processes, Structure and Asymptotic Behavior
185. Rubinowicz: Sommerfeldsche Polynommethode
186. Handbook for Automatic Computation. Vol. 2. Wilkinson/Reinsch: Linear Algebra
187. Siegel/Moser: Lectures on Celestial Mechanics
188. Warner: Harmonic Analysis on Semi-Simple Lie Groups I
189. Warner: Harmonic Analysis on Semi-Simple Lie Groups II
190. Faith: Algebra: Rings, Modules, and Categories I
191. Faith: Algebra II: Ring Theory
192. Mal'cev: Algebraic Systems
193. P61ya/Szeg6: Problems and Theorems in Analysis I
194. Igusa: Theta Functions
195. Berberian: Baer *-Rings
196. Athreya/Ney: Branching Processes
197. Benz: Vorlesungen fiber Geometric der Algebren
198. Gaal: Linear Analysis and Representation Theory
199. Nitsche: Vorlesungen fiber Minimalflichen
200. Doid: Lectures on Algebraic Topology
201. Beck: Continuous Flows in the Plane
202. Schmetterer: Introduction to Mathematical Statistics
203. Schoeneberg: Elliptic Modular Functions
204. Popov: Hyperstability of Control Systems
205. Nikol'skii: Approximation of Functions of Several Variables and Imbedding Theorems
206. Andr6: Homologie des Algbres Commutatives
207. Donoghue: Monotone Matrix Functions and Analytic Continuation
208. Lacey: The Isometric Theory of Classical Banach Spaces
209. Ringel: Map Color Theorem
210. Gihman/Skorohod: The Theory of Stochastic Processes I
211. Comfort/Negrepontis: The Theory of Ultrafilters
212. Switzer: Algebraic Topology--Homotopy and Homology
213. Shafarevich: Basic Algebraic Geometry
214. van der Waerden: Group Theory and Quantum Mechanics
215. Schaefer: Banach Lattices and Positive Operators
216. P61ya/Szeg6: Problems and Theorems in Analysis II
217. Stenstr6m: Rings of Quotients
218. Gihman/Skorohod: The Theory of Stochastic Processes II
219. Duvant/Lions: Inequalities in Mechanics and Physics
220. Kirikov: Elements of the Theory of Representations
221. Mumford: Algebraic Geometry I: Complex Projective Varieties
222. Lang: Introduction to Modular Forms
223. Bergh/L6fstr6m: Interpolation Spaces. An Introduction
224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order
225. Schfitte: Proof Theory
Grundlehren der mathematischen Wissenschaften Grundlehren der mathematischen Wissenschaften
226. Karoubi: K-Theory. An Introduction
227. Grauert/Remmert: Theorie der Steinschen Riume
228. Segal/Kunze: Integrals and Operators
229. Hasse: Number Theory
230. Klingenberg: Lectures on Closed Geodesics
231. Lang: Elliptic Curves: Diophantine Analysis
232. Gihman/Skorohod: The Theory of Stochastic Processes III
233. Stroock/Varadhan: Multidimensional Diffusion Processes
234. Aigner: Combinatorial Theory
235. Dynkin/Yushkevich: Controlled Markov Processes
236. Grauert/Remmert: Theory of Stein Spaces
237. KSthe: Topological Vector Spaces II
238. Graham/McGehee: Essays in Commutative Harmonic Analysis
239. Elliott: Probabilistic Number Theory I
240. Elliott: Probabilistic Number Theory II
241. Rudin: Function Theory in the Unit Ball of C R
242. Huppert/Blackburn: Finite Groups II
243. Huppert/Blackburn: Finite Groups III
244. Kubert/Lang: Modular Units
245. Cornfeld/Fomin/Sinai: Ergodic Theory
246. Naimark/Stern: Theory of Group Representations
247. Suzuki: Group Theory I
248. Suzuki: Group Theory II
249. Chung: Lectures from Markov Processes to Brownian Motion
250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations
251. Chow/Hale: Methods of Bifurcation Theory
252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampre Equations
253. Dwork: Lectures on 13-adic Differential Equations.
254. Freitag: Siegelsche Modulfunktionen
255. Lang: Complex Multiplication
256. HSrmander: The Analysis of Linear Partial Differential Operators I
257. HSrmander: The Analysis of Linear Partial Differential Operators II
258. Smoller: Shock Waves and Reaction-Diffusion Equations
259. Duren: Univalent Functions
260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems
261. Bosch/G'tintzer/Remmert: Non Archimedian Analysis--A System Approach to Rigid Analytic
Geometry
262. Doob: Classical Potential Theory and Its Probabilistic Counterpart
263. Krasnosel'skiI/Zabreo: Geometrical Methods of Nonlinear Analysis
264. Aubin/Cellina: Differential Inclusions
265. Grauert/Remmert: Coherent Analytic Sheaves
266. de Rham: Differentiable Manifolds
267. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I
268. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. II
269. Schapira: Microdifferential Systems in the Complex Domain
270. Scharlau: Quadratic and Hermitian Forms
271. Ellis: Entropy, Large Deviations, and Statistical Mechanics
272. Elliott: Arithmetic Functions and Integer Products
273. Nikol'skii: Treatise on the Shift Operator
274. H6rmander: The Analysis of Linear Partial Differential Operators III
275. H6rmander: The Analysis of Linear Partial Differential Operators IV
276. Liggett: Interacting Particle Systems
277. Fulton/Lang: Riemann-Roch Algebra
278. Barr/Wells: Toposes, Triples and Theories
279. Bishop/Bridges: Constructive Analysis
280. Neukirch: Class Field Theory
281. Chandrasekharan: Elliptic Functions
282. Lelong/Gruman: Entire Functions of Several Complex Variables
283. Kodaira: Complex Manifolds and Deformation of Complex Structures
284. Finn: Equilibrium Capillary Surfaces
285. Burago/Zalgaller: Geometric Inequalities
286. Andrianov: Quadratic Forms and Hecke Operators
287. Maskit: Kleinian Groups
288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes
289. Martin: Gauge Field Theory and Complex Geometry
290. Conway/Sloane: Sphere Packings, Lattices and Groups
291. Hahn/O'Meara: The Classical Groups and K-Theory
292. Kashiwara/Schapira: Sheaves on Manifolds
293. Revuz/Yor: Continuous Martingales and Brownian Motion
294. Knus: Quadratic and Hermitian Forms over Rings
295. Dierkes/Hildebrandt/Kiister/Wohlrab: Minimal Surfaces I
296. Dierkes/Hildebrandt/Kiister/Wohlrab: Minimal Surfaces II
297. Pastur/Figotin: Spectra of Random and Almost-Periodic Operators
298. Berline/GetzlerNergne: Heat Kernels and Dirac Operators
299. Pommerenke: Boundary Behaviour of Conformal Maps
300. Orlik/Terao: Arrangements of Hyperplanes
301. Loday: Cyclic Homology
302. Lange/Birkenhake: Complex Abelian Varieties
303. DeVore/Lorentz: Constructive Approximation
304. Lorentz/v. Golitschek/Makovoz: Constructive Approximation. Advanced Problems
305. Hiriart-Urruty/Lemar6chal: Convex Analysis and Minimization Algorithms I. Fundamentals
306. Hiriart-Urruty/Lemar6chal: Convex Analysis and Minimization Algorithms II. Advanced
Theory and Bundle Methods
307. Schwarz: Quantum Field Theory and Topology