NORTH-HOLLAND
MATHEMATICS STUDIES
24
Notos de Motemotico
editor: Leopoldo Nochbin
Topological Algebras
EDWARD BECKENSTEIN
LAWRENCE NARICI
CHARLES SUFFEL
NORTH-HOLLAND
TOPOLOGICAL ALGEBRAS

NORTH-HOLLAND
MATHEMATICS STUDIES
24
Notas de Matematica (60)
Editor: Leopoldo Nachbin
Universidade Federal do Rio de Janeiro
and University of Rochester
Topological Algebras
EDWARD BECKENSTE1N
St John's University, Notre Dame College, Staten Island, New York
LAWRENCE NARICI
St John's University, Jamaica, New York
CHARLES SUFFEL
Stevens Institute of Technology, Hoboken, New Jersey
1977
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM • NEW YORK • OXFORD
© North-Holland Publishing Company - 1977
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Library of Congreas Cataloging In Publication Data
Beckenstein, Edward., 1940-
Topological algebras.
(Notas de matemfitica ; 60) (North-Holland, mathema-
tics studies ; 24)
Includes index.
1. Topological algebras. I. Narici, Lawrence,
joint author. II. Suffel, Charles, joint author.
III. Title. IV. Series.
QA1.B86 no. 60 [QA326] 510'.8s [512'.55] 77.1107
ISBN 0-7204-0724-9
PRINTED IN THE NETHERLANDS
Kilimanjaro is a snow-covered mountain 19,710 feet high, and is said
to be the highest mountain in Africa. Its western summit is called by the
Masai "Ngaje Ngai," the House of God. Close to the western summit there is
the dried and frozen carcass of a leopard. No one has explained what the
leopard was seeking at that altitude.
From "The Snows of Kilimanjaro," by
Ernest Hemingway
For Paul and Maria, Dori and Chuck,
and Marshall and Cheryl.

PREFACE
Let T be a completely regular Hausdorff space and let F stand for the
real numbers R or the complex numbers C without specifying either.
Three main subjects are dealt with in this book: (1) general
topological algebras; (2) the space C(T,F) of continuous functions mapping
T into F as an algebra only (with pointwise operations); and (3) C(T,F)
endowed with compact-open topology as a topological algebra C(T,F,c). We
wish to characterize the maximal ideals and homomorphisms of C(T,F) and the
closed maximal ideals and continuous homomorphisms of topological algebras
in general and C(T,F,c) in particular. In addition a considerable inroad
is made into the properties of C(T,F,c) as a topological vector space in
Chapter 2. Naturally enough, many of the results about C(T,F,c) serve to
illustrate and motivate results about general topological algebras.
Attention is restricted to the algebra C(T,R) of real-valued
continuous functions in Chapter 1 and to the pursuit of the maximal ideals
and real-valued homomorphisms of such algebras. The clue to their identity
and capture is found in the case when T is compact. The collection of
functions x 6 C(T,R) which vanish at the point t is a maximal ideal whether
T is compact or not, but when T is compact, every maximal ideal of C(T,R)
is of this type. For noncompact T, the maximal ideals of C(T,R) are tied
to the points of the Stone-Cech compactification 0T of T in a very similar
way. When T is compact, all homomorphisms of C(T,R) are evaluation maps,
maps t* taking functions x 6 C(T,R) into their values x(t) at t. (Note
that the kernel of t* is Mfc.) By a quirk of nature, this remains
essentially true even for noncompact T, but the whole story is a little
more complicated. Gnerally the homomorphisms of C(T,R) are evaluation
maps but associated with the points of the repletion uT of T, a certain
subspace of £5T^ rather than just T. The quirk mentioned above by which the
homomorphisms of C(T,R) are usually given by just the points of T is that
UT = T for most spaces.
These things and others are discussed in Chapter 1. Rather than deal
with them from the z-ultrafilter point of view however, as Gillman and
Jerison do for example, we have used uniform spaces as the habitat for the
development of the theory. The idea that such an environment provides a
felicitous setting for the development of the theory of rings of continuous
functions is due to Nachbin and Chapter 1 owes a great deal to the way in
which Warner carried such a development through in a set of lectures given
vii
viii
PREFACE
at Reed College. Some background results on uniform spaces are given
without proofs in Chapter 0 with references given for the details.
As mentioned above, the Stone-Cech compactification gT and repletion
UT of T plays an important role in the development of the algebraic
properties of C(T,R). Thus, significance attaches to obtaining them and
viewing T as a uniform space enables a simple and direct realization of
each. The theory of uniform spaces provides that every Hausdorff uniform
(= completely regular) space T has a unique completion. With no further
fuss this fact produces one form of the Stone-Cech compactification of T,
as the completion of T with respect to the weakest uniformity with respect
to which each bounded continuous function is uniformly continuous.
(Mercurial entity that it is, (JT emerges as a space of measures in
Section 1.7 and as a space of homomorphisms of a Banach algebra in
Section 4.10.) The repletion is obtained similarly — as a completion of T
with respect to a different uniformity.
An interest that is always present when studying C(T,F) is the
correlation of algebraic properties of C(T,F) with purely topological
properties of T. For examples; (1) If consideration is restricted to
compact spaces, S and T say, then C(S,R) and C(T,R) are isomorphic as
algebras if and only if S is homeomorphic to T; (2) T is connected if and
only if 0 and 1 are the only idempotents in C(T,F). When C(T,F) takes on
the compact-open topology to become C(T,F,c), the scope for possible inter-
actions broadens as one now seeks interplay between topological properties
of T and topologico-algebraic properties of C(T,F,c). In this spirit, in
Chapter 2, the famous theorems of Nachbin and Shirota are presented which
settled Dieudonne's question: Must a barreled topological vector space be
bornological? Nachbin and Shirota independently obtained negessary and
sufficient conditions on T for C(T,R,c) to be barreled and for it to be
bornological. The condition for bornologicity is especially simple:
C(T,R,c) is bornological if and only if T = uT. Referring to the necessary
and sufficient conditions on T which makes C(T,R, c) barreled as "condition
NS", one can investigate the question: Is there a T which satisfies
condition NS but for which T uT? I.e. is there a T for which C(T,R,c) is
barreled but for which T UT? There are such spaces (of ordinals,
predictably enough) and so bornologicity is not implied by barreledness.
All of Chapter 2 is devoted to correlating topological properties of T with
topological vector space properties of C(T,F,c). In particular, in
PREFACE
ix
addition to barreledness and bornologicity, conditions which guarantee or
characterize metrizability, completeness, and separability of C(T,F,c)
are obtained.
Another compactification, the Wallman compactification, plays an
important role in characterizing the maximal ideals of certain topological
algebras (Chapter 5). To develop the Wallman compactification however some
knowledge of lattice theory is required. What is needed, together with the
Wallman compactification itself, is presented in Chapter 3. In Chapter 4
the general subject of commutative topological algebras (with identity) is
introduced and developed. To be more accurate, it is the theory of com-
mutative locally m-convex algebras that is developed there. For just as
topological vector spaces display an almost disappointing similarity to
topological groups without the added assumption o* local convexity, general
topological algebras are similar to the point of disinterest to topological
rings without "local m-convexity". With this added property, scalar
multiplication plays an important role.
In Chapter 6 a special type of algebra is dealt with which we call an
LB-algebra • The reason for the "LB" is that they are essentially just
inductive Limits of Banach algebras.
In dealing with algebras of continuous functions as algebras we have
stuck to real-valued functions in the text. As pointed out in Section 1.4
tho, most of what appears in Chapter 1 remains true for algebras of
complex-valued functions. In dealing with topological algebras, where
whether the underlying field is R or C can make a significant difference,
we have tried to treat the real and complex cases on an equal footing
wherever possible, and pointed out where it is not possible. In excursions
in the exercises we consider algebras of К-valued functions where К is a
topological field or, more specially, a nonarchimedean valued field.
What sort of background should one have to read the book? Basically
some algebra (one should certainly know what prime and maximal ideals are),
some topology (having taken a year course in it somewhere along the line
should suffice), and some functional analysis including some things about
Banach algebras. As to the functional analysis, the elementary properties
of locally convex spaces plus their duality theory should do; as for Banach
algebras, not much is required per se, but the generalization of Banach
algebra results to locally m-convex algebras will be more meaningful if
something about Banach algebras is known. Essentially it is the same
argument that would be given as to the desirability of some acquaintance
X
PREFACE
with metric spaces before studying general topological spaces: It's not
absolutely necessary, but it's nice to have.
Our notational conventions do not require any special comment. The
only one that is unusual in any way is the use of V to mark the end of a
proof, but the reason for that choice had best remain a mystery. Except
for Chapter 6, each chapter has a large number of exercises attached to it.
The earliest ones are routine and meant for practice as well as informa-
tion; the reverse is true of the later ones. References are given as well
as extensive hints, many of which are really proofs written in telegraphic
style. These later exercises are meant mainly to provide additional
information about or information tangential to topics developed in the
text. As simply "exercises", they should be approached with extreme
caution.
Before commending our lucubrations to you, there are many people we
would like to thank for many different reasons: St. John's University for
providing facilities and a reduced teaching load to Narici; the National
Science Foundation for providing a summer grant to Suffel; Elizabeth Suffel
for doing some onerously difficult typing; to many other friends, some for
having made comments about the book which were directly helpful to it,
others for just being there. As for errors that might appear in the text,
we shudder, apologize now, and vigorously stipulate that any that might
remain are all the fault of the first-named author.
CONTENTS
Chapter 0 Fundamentals
0.1	Topologies defined by families of functions	]
0.2	Uniformities defined by families of functions	b
Chapter 1	Algebras of Continuous Functions	7
1.1	The Stone-Cech compactification	"
1.2	Zero sets	'
1.3	Maximal ideals and z-filters	'3
1.4	Maximal ideals and the Stone-Cech compactification	17
1.5	Replete spaces	20
1.6	Characters and uT	23
1.7	0-1 measures, ₽T, and Ulam cardinals	34
1.8	Shirota's theorem on repleteness	44
Chapter 2	Topological Vector Spaces of Continuous Functions	6 1
2.1	Metrizability of C(T,F,c) and hemicompactness	62
2.2	Completeness and к -spaces	63
2.3	к-spaces, к -spaces and pseudofinite spaces	65
2.4	Continuous dual of C(T,.F,c) and support	73
2.5	Barreledness of C(T,F,c)	93
2.6	BOrnologicity of C(T,F,c)	99
2.7	Separability of C(T,F,c)	107
2.8	The bornology of C(T,F,c)	110
Chapter 3	Lattices and Wallman Compactifications	135
3.1	Lattices	136
3.2	Lattices and associated compactifications	142
3.3	Wallman compactifications of topological spaces	147
3.4	₽T and Wallman compactifications	152
3.5	A class of Wallman-type compactifications	155
3.6	Equivalent Wallman spaces	161
Chapter 4	Topological Algebras	175
4.1	Topological algebras	176
4.2	Multiplicative sets and multiplicative seminorms	181
4.3	Locally m-convex algebras	184
4.4	Final topologies and quotients	189
4.5	The factor algebras	192
4.6	Complete LMCH algebras and projective limits	197
4.7	The spectrum	201
4.8	Q-algebras and algebras with continuous inverse	204
4.9	Topological division algebras and the Gelfand-Mazur theorem	210
4.10	Maximal ideals and homomorphisms	220
4.11	The radical and derivations	236
4.12	Some elements of Gelfand theory	241
4.13	Continuity of homomorphisms	268
xi
Chapter 5	Hull-Kernel Topologies	299
5.1	Hull-kernel topologies	300
5.2	Regular algebras and normality	conditions	302
5.3	Condition hH	306
5.4	Я as a Wallman compactification	of	M	308
5.5	The X-repletion ofM	C	311
5.6	Frechet algebras	312
Chapter 6	LB-Algebras	329
6.1	LB-algebras	329
6.2	Some properties of	LB-algebras	332
6.3	Complete IMC LB-algebras	340
References	349
Index of Symbols	363
Index
365
ZERO
Fundamentals
THIS SHORT CHAPTER contains some things which are basic for what follows
and makes certain things explicit, such as "completely regular" not
including "Hausdorff." Mainly it deals with topologies and uniformities
determined by families of functions and how the two are related; a few
facts about uniform spaces are listed, with references to Bourbaki for
proofs, in Sec. 0.2. These latter facts are put to use right away in Sec.
1.1 where the Stone-Cech compactification PT of a completely regular
Hausdorff space T is obtained as a uniform space completion of T with re-
spect to the uniformity induced by the space C(T,R) of real-valued continu-
ous functions on T. In Sec. 1.5 the repletion (real compactification) uT
of T is obtained similarly.
0.1 Topologies defined by families of functions. We assume familiarity
with topology and the theory of uniform spaces and choose Bourbaki's
General Topology, Parts 1 and 2, hereinafter referred to as Bourbaki 19b6a
and 19bbb respectively, as our standard reference on these subjects.
In dealing with topological spaces, Hausdorff separation is not includ-
ed in any instance unless specifically indicated. By saying that two sub-
sets A and В of a topological space T are separated by open sets, we mean
that disjoint open sets U and V exist containing A and В respectively. If
a continuous function x:T -• ГО, 13 exists which maps A into {o] and В into
{1], we say that A and В are separated by a continuous function. Thus a
completely regular space is one in which each point t and the complement of
any neighborhood of t may be separated by a continuous function. Occasion-
ally "Tihonov space" is used as a synonym for "completely regular Hausdorff
space. "
Unlike Bourbaki, Hausdorff separation is not included in "compact."
By "locally compact" we mean that each point in the space possesses a
neighborhood whose closure is compact. A space is а-compact if it is a
countable union of compact subsets, Lindeltif if every open cover contains a
countable subcover.
Call a set clopen if it is closed and open and a topological space
zero-dimensional if it possesses a base of clopen sets. An example of a
zero-dimensional space follows.
Example 0.1-1 VALUED FIELDS A field К together with a real-valued map
| |:K — R such that for all a, bCK
2
0.1 TOPOLOGIES BY FUNCTIONS
(a)	|a | > 0 and =0 iff a=0;
(b)	|ab | = |a | |b |
(c)	|a+b | < |a | + |b |
is called a valued field; the map | | is called a valuation on K. If | |
satisfies (c1) below instead of (c), then |	| is a nonarchimedean valuation
and К a nonarchimedean valued fie Id:
(c ' ) |a+b | < max( |a |, |b |.
In either case d(a,b) = |a-b | is a metric on К and when К carries the metric
topology, К is a topological field. If the valuation is nonarchimedean, it
is straightforward to verify that spheres, open or closed, in K,
{afK | |a] < r or < r], r > 0, are clopen in K. Thus any nonarchimedean
valued field is a zero-dimensional topological space.
NOTATIONS R and C stand for the real and complex numbers respectively
carrying their usual topologies. £ denotes R or C without specifying
either. N, Z,, and Q denote the natural numbers, integers, and nationals
respec tively.
If S and T are topological spaces, C(S,T) stands for the set of all
continuous maps of S into T. If there is a notion of "bounded" set in T,
if T was a topological vector space, for example, then C^(S,T) denotes the
collection of all bounded continuous maps from S into T, i.e. all continu-
ous maps whose range is a bounded subset of T. If T is a topological field
then C(S,T) and 0^(3,!) are each T-algebras with respect to the pointwise
operations: (x+y)(t)=x(t)+y(t), (xy)(t)=x(t)y(t), and (ax)(t)=ax(t) for
x and у in C(S,T) or C^(S,T) and afT. It is always to these operations
that we refer when we speak of spaces of continuous functions as algebras,
rings, or linear spaces.
(0.1-1) INITIAL TOPOLOGIES
set, ((T , J )) a family
p p€M
map from T into T . The topology у generated by the sets LJ
4	-1	'
i.e. the topology having U x
pfM p
ogy for T with respect to which each of the maps x is continuous,
called the initial topology determined
it is given by finite intersections of
and G€7 •
Li
Initial topologies are a means to
(Bourbaki 1966a, p. 30, Prop. 4). Let T be a
of topological spaces, and, for each pfM, x a
(7 ) as a subbase, is the coarsest topol-
7 is
by the maps (x ) „ and
p рем
sets of the form x-1(G)
transport topologies in
a base for
where peM
ranges ot
functions back to the domain. Final topologies, discussed next, are a
vehicle for the reverse direction.
(0.1-2) FINAL TOPOLOGIES (Bourbaki 1966a, p. 32, Prop. 6). If T is set,
0. FUNDAMENTALS
3

family of topological spaces and x a map from T into T for each
p,eM then there is a finest topology J for T with respect to which each x
is continuous. «7 is called the fina 1 topology for T (determined by the
maps (x )) and ,7 consists of those subsets U of T sucl, that x (U) is open
|1	|_L
in T for each p,€M.
If (T,J7) is a topological space and C(T,R) the class of all continuous
real-valued functions on T then certainly the initial topology 3 deter-
mined by C(T,R) on T is coarser than «7. The same is true of the initial
topology 3, determined by C. (T.R) on T. Moreover «7, CL «7 CL «7. When T
b	b	b c
is a completely regular Hausdorff space, however, the three topologies
coincide ((0.2-5)).
Example 0.1-2 COMPACT-OPEN AND POINT-OPEN TOPOLOGIES ON C(T,F) If К is a
compact subset of the topological space T then the map
p„:C(T,F) - F (=R or C)
К ~	~	~
x -» sup |x(K) |
is a seminorm on the linear space C(T,I?). Viewing C(T,F.) as an algebra,
p is a multiplicative seminorm on C(T,F) in the sense that
К
p (xy) < p (x)p (y). The initial topology determined by the maps p as К
К	К К	к
runs through the compact subsets of T is the compact-open topology for
C(T,£). When C(T,F) carries the compact-open topology it is denoted by
C(T,F,c). The compact-open topology is a locally convex Hausdorff topology
for the linear space C(T,F) and a locally m-convex Hausdorff topology for
the algebra C(T,F.) (see (4.3-2)). A neighborhood base at 0 in C(T,l?,c) is
given by the collection of all positive multiples of sets of the form
V„ = {x€C(T,F) Ip (x) < 1} or V = {xfC(T,F) Ip (x) < 1}
p	IK	P	IK
as К runs through the class of compact subsets of T. If T is compact, then
a base at 0 is given by positive multiples of just and in this case
C(T,F,c) is a Banach algebra, its norm being simply tEe sup norm.
Another name for the compact-open topology is the topology of uniform
convergence on compact sets, the reason being that a net p, -> x in the
compact-open topology iff ц, — x uniformly on each compact subset of T.
If, instead of taking the class of all compact subsets of T, we take
the collection of all singletons, the ensuing weaker locally m-convex
Hausdorff topology for C(T,F) is the point-open topology (topology of
pointwise convergence or simple convergence) .	C(T,I?,p) denotes C(T,£)
4
0.1 TOPOLOGIES BY FUNCTIONS
endowed with the point-open topology.
Example 0.1-3 a(X.X') AND TVS CONVENTIONS "Topological vector space"
(TVS) does not include Hausdorff; when desired we will say Hausdorff TVS
(HTVS). "Locally convex topological vector space" and "locally convex
Hausdorff topological vector space" will occasionally be abbreviated to
LCS and LCHS respectively. All TVS 1s are assumed to have R or C as their
underlying field unless otherwise specified. If X is a TVS, X' denotes the
linear space of all continuous linear functionals on X. The weak (or
weakened) topology a(X,X') for X is the initial topology generated by the
maps
P , :X - F
x ~
x -> | < X ,x' > |
as x' runs through X'. It is clearly the coarsest topology for X with
respect to which each x'fX' is continuous, i.e. also the initial topology
generated by the family of maps X' on X. a(X',X) is the initial topology
determined on X' by the maps x' — | < x,x' > | as x runs through X. These
weak topologies, being determined by families of seminorms, are clearly
locally convex topologies.
The polar S° of a subset S of a TVS X is the collection of all x'fX'
such that sup|x'(S) | < 1; the dual consideration applies to polars of
subsets of x'.
Example 0.1-4 G?-TOPOLOGIES. t(X.X') AND B(X.X') If & is a collection
of a(X,X')-bounded subsets of the LCHS X, then the collection ©° of polars
S° of sets ScG is a set of absorbent balanced convex subsets of x'. Hence
the collection of positive multiples of finite intersections of sets from
&° forms a neighborhood base at 0 for a locally convex topology for X'
called the Q-topology. Dual considerations apply to (5”topologies for X.
Another way to view ©-topologies is as follows. If & is a collection
of a(X',X)-bounded subsets of X1, then < x,S > is a bounded set of scalars
for each Sf©
and
each xpX. The maps
ps:X - £
sup |
of S°) on X. The (initial) topology
the ©-topology mentioned above.
Taking (~p to be the class of all balanced convex a(X' ,X)-compact sub-
are seminorms
determined by
(ps
the
is actually the gauge
seminorms (p„)„^ is
о bfto
sets of x', or all a(X',X)-bounded subsets of X', the ©-topologies
0. FUNDAMENTALS
5
generated are respectively the Maскеу topology t(X,X') and the strong
topology p(X,X').
0.2 Uniformities defined by families of functions
INITIAL UNIFORMITIES (Bourbaki 1966a, p.
a family of uniform spaces, and,
coarsest uniformity U for T, called the
the maps (x ) , . with respect to which
p pEM
A fundamental system of entourages for Ц
10.2-1)
be a set, (T )
p pfM
from T into T . Then there is a
176, Prop. 4). Let T
for each PEM, x a map
initial uniformity determined by
each x^ is uniformly continuous,
is given by sets of the form
U(V1,...,Vn) = {(t,t')ET»T|(x (t),x
In	I p, |
where {p. ,. . . .p^] is a finite subset of M and each is an entourage in
If у is a map from a uniform space S into the uniform space (T,U),
у is uniformly continuous iff x • y:S — T
pEM.
If each T is a Hausdorff uniform space and
P
separates points in T (i.e. for each pair (t,t')
then there is some pEM such that x (t) x (t1))
p	P
uniform
T
then
each
s tructure.
initia1
s uniformly continuous for
if the family (x )
p pEM
of points from T, if t ^t1
then U is a Hausdorff
As stated in (0.2-2) below, the topology determined by Ц is just the
topology determined by the maps (x )
INITIAL UNIFORMITIES VS. INITIAL TOPOLOGIES (Bourbaki 1966a,
determined by an initial uniformity
coarsest topology with respect to which
the initial topology determined by the
i.e. it is
with a topology if the topology deter-
A topological space
The central
compatible
T is uniformizable if a compat-
uniformiza-
characterization of
p. 144, Theorem 2).
A topologi-
(0.2-2)
p. 177, Corollary). The topology
determined by maps (x ) is the
7 H 4 p pEM
each x is continuous,
P
maps (x )	.
p pEM
A uniformity Ц is
mined by Ц is just <7^.
ible uniformity Ц exists on T.
bility is:
(0.2-3) UNIFORMIZABILITY (Bourbaki 1966b,
cal space T is uniformizable iff it is completely regular.
(0.2-4) THE UNIFORMITY OF A COMPACT HAUSDORFF SPACE (Bourbaki
p. 199, Theorem 1). On a compact Hausdorff space T there is exactly one
uniformity compatible with the topology of T. The entourages of this
uniformity are all neighborhoods of the diagonal in T»T.
1966a,
6
0.2 UNIFORMITIES BY FUNCTIONS
Two uniformities of special interest are the initial uniformities
determined by the real-valued continuous functions C(T,R) and C^(T,R) on a
topological space denoted by J^(T,R) and t^(T,R) respectively and sometimes
shortened to simply and	If T is completely regular and Hausdorff,
C(T,R) and C^(T,R) each separate points in T and so determine Hausdorff
uniform structures for T. Letting -J and denote the topologies
determined by and on T and letting xT be T's original topology, it is
clear that	When T is completely regular, the three topologies
coincide.
(0.2-5) COMPLETE REGULARITY AND THE INITIAL TOPOLOGY DETERMINED BY
C(T,R) ON T (T,J ) is completely regular iff K/' = •cZ •
Proof Suppose that T is completely regular and let V be a neighborhood of
sfT. There is some XfCb(T,R) such that x(s)=0 and x(CV)={l], and
ftfT||x(t) | < 1/2} is a basic neighborhood of s in the topology determined
by that is contained in V. It follows thatJ’C-
Conversely,	then T is uniformizable, hence completely
regular by (0.2-3). V
Concerning completions of uniform spaces, we need the following re-
sults, the upshot of which is that Hausdorff uniform spaces are densely
embedded in an essentially unique complete Hausdorff uniform space.
(0.2-6) COMPLETION (Bourbaki 1966a, p. 191, Theorem 3).
(a)	For any uniform space T there is a complete Hausdorff uniform
space T, called the Hausdorff completion of T, and a uniformly
continuous map i:T -• T which the property:
(P) Given any uniformly continuous map f of T into a complete
Hausdorff uniform space S, there is a unique uniformly continuous
map g:T — S such that f=g • i.
If (ij,T^) is another pair consisting of a complete Hausdorff uniform
space T^ and a uniformly continuous map i^:T -> T^ having property (P) then
there is a unique uniform space isomorphism h:T — T^ such that i^=h» i.
(b)	(Bourbaki 1966a, p. 194, Corollary). If T is a Hausdorff uniform
space then the canonical map i:T -• T is a uniform space isomor-
phism of T onto a dense subspace of T. In this case, T is called
the completion of T.
ONE
Algebras of Continuous Functions
AS P. SAMUEL has remarked, there are two principal methods of
investigation in point set topology. The first — the "internal" method —
refers to the topological space alone. Separation axioms, compactness, and
connectedness, for example, are usually expressed solely in terms of the
topological space. The second method — the "external" — uses the real
numbers as an analytic tool. Here they usually appear via the channel of
real-valued continuous functions on the topological space T. At times the
entire class of such functions is called upon, as in the definition of
complete regularity; in other instances a subclass such as the continuous
pseudometrics is singled out, as with uniform spaces. From the external
standpoint, if one wants the topological space T described accurately by
its continuous functions, C(T,R), one must have enough of them. And to
guarantee a good supply of nontrivial continuous functions, T is assumed to
be completely regular and Hausdorff throughout.
A goal of this chapter is to develop an algebraic external method —
interplay between the topological structure of T and the algebraic structure
of the algebra C(T,R). In Chapter 2 the more complex interactions between
the topological space T and the topological vector space (or algebra)
C(T,R,c), C(T,R) with compact-open topology, are investigated. But here con-
sideration is restricted to topologicoalgebraic interactions. Generally
what can be said of the algebraic structure of algebras of continuous func-
tions C(T,g)? How sensitive is C(T,R) to T? I.e. can essentially different
T's determine the same space of continuous functions? The converse question
is clearly trivial: homeomorphic spaces do have the same space of con-
tinuous functions. The former question is answered (negatively) in
Section 1.6. The machinery to answer it, and others like it, is developed
along the way. To answer that particular question, the idea of a
"repletion" UT, a certain type of completion of a completely regular
Hausdorff space T, is needed. This leads to the related issue of "replete"
spaces. (If replete spaces produce the same algebra of continuous
functions, they must be homeomorphic.)
The Stone-Cech compactification 0T of T is also cultivated as a
completion (Sec. 1.1) and later, in Section 1.7, as a space of finitely
additive 0-1 measures. The kinship between gT and UT, both being
7
8
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
completions, emerges in the measure-theoretic setting with UT appearing as
the countably additive members of gT.
Because of our interest in replete spaces, for reasons which will
become apparent as the chapter develops, questions concerning which par-
ticular spaces are replete (R is, for example) or the broader issue of what
classes of spaces are replete are unavoidable. In trying to answer the
latter question we are led to certain fundamental set-theoretic questions —
a prospect apt at first to strike terror in the hearts of many mathema-
ticians (us, in particular) — such as: Can Ulam cardinals exist within the
framework of Zermelo-Fraenkel set theory? Some discussion of these matters
is given in Section 1.7 before going on to Shirota's theorem in Section 1.8
governing repleteness of complete uniform spaces.
1.1 The Stone-Cech Compactification
Very early on, with respect to the definition of uniform spaces, Weil
recognized that every compact Hausdorff space — hence every subspace of a
compact Hausdorff space — could be viewed as a uniform space. He did this
(Weil 1937, p. 24) using only internal methods. In order to prove a
converse statement — namely that every uniform space could be viewed as a
subspace of a compact Hausdorff space — he had to use the real numbers. An
internal construction of this fact, similar to the way in which the Wallman
compactification is obtained in Chapter 3 as a space of ultrafilters, was
given by Samuel (1948).
That every uniform (= completely regular) Hausdorff space T possesses
a compactification to which every bounded continuous function on the space
may be extended is the subject of this section. The reason for our
interest in it lies in the close connection between the maximal ideals of
C(T,R) and the points of the compactification, the Stone-Cech compactifica-
tion, as elaborated on further in Section 1.4. The construction of the
Stone-Cech compactification given here, as a uniform space completion —
hence as a space of ultrafilters — is due to L. Nachbin.
Existence and uniqueness of the Stone-Cech compactification 0T were
first proved by Stone (1937), using the methods of Boolean rings. Kakutani
(1941) has a construction using Banach lattices. Cech (1937) simplified
Stone's original proof while Wallman-type compactifications (cf. Chapter 4
A compactification S of a topological space T is a compact space con-
taining a dense homeomorphic image of T. A Hausdorff compactification is
a compactification which is a Hausdorff space.
1.1 STONE-CECH COMPACTIFICATION
9
and Wallman 1938) yield ST if T is normal. An approach using Banach
algebra techniques, due to Gelfand and Silov, is discussed in Section 4.12
A realization as a space of measures due to Varadarajan is given in
Section 1.7 while still other avenues to gT appear in the exercises.
In this section T is a completely regular Hausdorff space, C(T,R) is
the algebra (with pointwise operations on the functions of C(T,R)) of con-
tinuous functions taking T into R, and C, (T,R) will be the subalgebra of
C(T,R) consisting of all bounded functions.
The protean Stone-Cech compactification of T appears as the maximal
ideals of C, (T,R) (appropriately topologized) , as a subspace of a product
b
of closed unit intervals, as the z-ultrafilters of T (appropriately
topologized), a space of measures (Sec. 1.7) and as the completion of T
with respect to a certain uniform structure. We examine the last construc-
tion and show (Theorem 1.1-1) that in some sense ST is unique.
Let x С C, (T,R) and consider the sets V(x,e) =
b
|(s,t) € Txt| |x(s) - x(t) | < e [ for e > 0. Then the collection of entourages
{v(x,e)|x€0^(T,R),e> o} form a subbase for a uniform structure on T
compatible with the topology of T.
Definition 1.1-1. THE STONE-CECH COMPACTIFICATION. The completion gT of
the completely regular Hausdorff space T with respect to the uniform struc-
ture C is called the Stone-Cech compactification of T.
b
Theorem 1.1-1. ELEMENTARY PROPERTIES OF gT. Let T be a completely regular
Hausdorff space. Then:
(a)	gT is a compact Hausdorff space.
(b)	Each x e C (T,R) can be (uniquely) extended to
b
x C C(gT,R) = C, (gT,R) .
~	D
(c)	If is a compactification of T which is a Hausdorff space and
й	л	-
each x С C (T,R) can be (uniquely) extended to x € C(gT,R), then
b	~	_
gT and gT are equivalent compactifications of T (i.e. gT and gT
are homeomorphic under a mapping which extends the "identity"
on T) .
Proof (a) To show that gT is compact, it is sufficient to show that T
with the uniform structure C, is totally bounded for then gT is complete
b
and totally bounded and (Bourb. 1966a, p. 202) therefore compact. For each
x 6 C^(T,R), there exists a closed interval [a,b] C R such that
As R is a Hausdorff space, the extension x must be unique.
10
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
x(T) C [a,b] . Consider a finite set of points s = a < s, < . .. < s <b = s
о	1	n	n+1
such that I S£+j_ _ S£ I < e/2.
Now if [s.,s. , ] О x(T) 0, there exists t. £ T such that
1 1+1 -1 x
x(t^) £ tsi'si+]J and follows that x tsi,Si+]J c v(x»£) [t^] where
V(x,e) [t±] = |t £ t| (t^t) £ V(x,e) } . If [s±,s	] П x(T) = 0, then
-1 П -1
x [Sj/Si+]J = 0* Hence, since T = U x	it follows that
i=0
T = Uv(x,e) [t^] where the union is taken over those i such that
[s^s^] П x(T) 1 0.
(b)	Since x(V(x,e) [t] ) C S£(x(t)) for each t £ T, S£(x(t)) denoting
the open sphere of radius e about x(t), it follows that x is uniformly con-
tinuous on T and, since R is complete, x may be uniquely continuously
extended to x : 6T + R (Bourbaki 1966a, p. 190, Th. 2).
(c)	Let ST be a Hausdorff compactification of T with the property that
every x £ C^tTjR) can be extended to x £ C(gT,R) . As ST is a completely
regular Hausdorff space, a subbase of entourages for a uniform structure
generating the topology on ST is given by the sets V(x,e) =
|(s,t) C STxST I | x(s) - x( t) | < e }. Since V(x,e) О T*T = V(x,e) , we see that
we may view T as a subspace of the uniform space ST. Since ST is complete,
we see that ST is a completion of T as is ST, and by uniqueness of
completions, ST is equivalent to ST. V
1.2 Zero Sets
In this section, as in the preceeding, T is a completely regular
Hausdorff space. Letting x denote a function belonging to the algebra
C(T,R) , z(x) will denote the set of points in T at which the function x
vanishes. It is not difficult to see that the set z(x) does not determine
the function x. Yet the collection z(M) = |z(x) |x£m|, where M is a
maximal ideal in C(T,R), uniquely determines M. Moreover the properties
of z(M) will enable us to tell whether M is the kernel of a homomorphism
of C(T,R). In this section we begin the study of the zero sets z(x) of
functions x £ C(T,R) and their relationship to the algebraic structure of
the ring C(T,R). In particular we go from T to C(T,R) back to T again in
showing that the zero sets z(x) of continuous functions x on T determine
the topology of T.
A subset of T of the form x 1(0), x £ C(T,R), is called a zero set and
is also denoted by z(x). For collections E of functions from C(T,R), z(E)
stands for |z(x)|x£El. Some elementary properties of zero sets follow.
1.2 ZERO SETS
11
(1.2-1) ELEMENTARY PROPERTIES OF ZERO SETS. For continuous real-valued
functions on the completely regular Hausdorff space T:
2	2
(a)	z(x +y ) = z(x) О z(y);
(b)	z(x) = z( |x|) ;
(c)	z(xy) = z(x) и z(y) ;
(d)	For any x £ C(T,R), there exists у £ C(T,R) such that 0 $ у 1
and z (x) = z (y) ;
CO
(e)	О z(x ) = z(x) for some x £ C(T,R);
n=l П
(f)	x 1([r,“)) is a zero set for any'real number r and any
x £ C(T,R) .
Proof (a), (b), and (c) are evident. To prove (d) we let у be the
continuous function defined by y(t) = min||x(t) |,l}. To prove (e) we
assume 0 = x =1 for each n and consider x = S 2 nx . Since S 2 Пх is
n	n	n
CO
uniformly convergent, x £ C(T,R). Clearly z(x) = О z(x ). To prove (f),
n=l П
we consider z(y) where у = min|x,r|-r. V
The result established in (1.2-3) below generalizes the result of
(1.2-1)(f). First we need the following topological result.
(1.2-2) G„ SETS AND ZERO SETS. Let H be a closed subset of the normal
о
space T. A necessary and sufficient condition that H be a zero set is that
H be G„.
о
Remark Thus if T is a metric space, then every closed set is a zero set.
Proof If H = z (x) , then H = Cl }1 £ T | |x(t) | < 1/n}. Conversely suppose
H = CIU , U open. By Urysohn's lemma there exist functions x £ C(T,R),
n n	n	-
0 = x Si, such that x (H) = lof and x (CU ) =11}. Similar to the proof
n	n * ’ n n ' 1
of (1.2-1)(e), we consider x = S 2 x^. By the uniform convergence of this
series, x £ C(T,R). A simple argument shows that z(x) = Cl U = H. V
n
(1.2-3) A CHARACTERIZATION OF ZERO SETS. For any completely regular
Hausdorff space T, the collection of zero sets z(C(T,R)) is given by
{x 1 (K) |x £ C(T,R) ,К C R is closed}.
Proof Since К C R and К is a closed set, by the remark after (1.2-2) there
exists g : R -> R such that z(g) = K. Then for each x £ C(T,R) , gx £ C(T,R)
and z(gx) = |t£T|g(x(t))=o| = |t£T|x(t)£K}= x”1 (К) . V
We now prove that the zero set neighborhoods of points in T generate
the topology of T. This result is useful in proving the Gelfand-Kolmogorov
theorem (Theorem 1.4-1), in proving that ST can be realized as a lattice
12
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
(Wallman) compactification of T (Chapter 3), and is generally useful in the
study of algebras of continuous functions.
(1.2-4) ZERO SET BASES. In any completely regular Hausdorff space T, the
zero set neighborhoods of any t £ T form a base for the neighborhood filter
at t.
Proof Let N be an open neighborhood of t in T. There exists an open
neighborhood N of t in gT such that N = N О T. As gT is a regular space
there exists a neighborhood U of t in gT whose closure is contained in N.
By the normality of gT there exists x £ C(gT,R) such that x(cl U) = |o| and
x(CN) = {1}. Let x be the restriction of x to T. Then clU О T C x 1(0) Г)
T = x-1(0) C N П T = N. V
Definition 1.2-2. COMPLETELY SEPARATED SETS. Two sets А,В С T are
completely separated if there exists x £ C^tTjR) , 0 <_ x < 1, such that
x(A) = {1} while x(B) = |o}.
In the result that follows, we show that the sets which are completely
separated in T are those whose closures in gT are disjoint.
Theorem 1,2-1. COMPLETELY SEPARATED SETS.
For any two subsets A and В of
the completely regular Hausdorff space T, the following conditions are
equivalent.
(a)	A and В are completely separated.
(b)	cl A О cl В = 0.
P	P
(c)	A and В are contained in disjoint zero-sets. In particular note
that if A and В are zero sets, then they are disjoint if and only
if theV are completely separated.
Proof To
prove that
separated.
Letting x
(a) implies (b) suppose that A and В are completely
Then for some x £ C, (T,R), x(A) = {of while x(B) = 11}.
continuous extension of x to gT, x (cl A) = 4 0 f
g 1	>
= 11}. Thus cl A О cl В = 0. Conversely, if cl A О cl В =
1	g л g	g g
lemma there exists x £ C(gT,R) , 0 <_ x <_ 1, such that
The function x when restricted to
denote
the
while
x (cl B)
P
Urysohn's
x(cl A)
p
clearly
(I	g-
separates A and B. Thus (a) and (b) are equivalent.
To
while x(B)
prove that (a) implies (c),
Then AC x 1(0),
let x £ C, (T,R) be such that x(A)
-1 b ' -1
BCx (1). By (1.2-3) x (1) is
0i by
T
a
zero set and (c) follows. To prove that (c) implies (a) let A C z(x),
2	2 2
B,_z(y) for x,y £ C, (T,R) and z(x) Г) z (y) = 0. Taking w = x / (x +y ) , we
b
see that w £ C, (T,R) , w(A) = 10 }, w(B) = 11}, and therefore that A and В
b	' ’
are completely separated. V
1.3 MAXIMAL IDEALS AND z-FILTERS
13
In general, if A and В are subsets of T, clfc П b)C cl A О cl В. We
“	P	P
now show that if A and В are zero sets, then equality prevails. This
result will be useful in proving the Gelfand-Kolmogorov theorem
(Theorem 1.4-1).
Theorem 1.2-2. CLOSURES OF INTERSECTIONS OF ZERO SETS. Let T be a
completely regular Hausdorff space. If x,y £ C(T,R), then
cl z(x) О cl z(y) = cl (z(x) О z(y)).
P	P	P
Proof
Let p e
hood of
Clearly we
need only show that cl z(x) О cl z (у)
P	P
cl z(x) n cl z(y) .
P	P
С с1^г(х) П z(y)).
Applying (1.2-4) let V be a zero set neighbor-
p in gT and V = V О T. Then V О z(x) and V О z(y) are zero sets
in T. Since V is a neighborhood of p in gT, it follows that p £ cl
(v Oz(x)). Similarly p £ cl^(v О z (y) ). Thus by Theorem 1.2-l(b)
6
the
zero sets V О z(x) and V О z(y) are not completely separated. By
Theorem 1.2-l(c),
V О z (x) П z (y)
the neighborhoods
it follows that V О z(x) О z (у) / 0 and thus
0. Since by (1.2-4) the neighborhoods V form a base for
of p, p £ clfc(x) О z(y)). V
P
1.3 Maximal Ideals and z-Filters
In this section we consider a notion which lies at the foundation of
an approach to the study of algebras of continuous functions — the z-filter.
The principal result, Theorem 1.3-1, establishes a connection between
z-filters and maximal ideals.
Definition 1.3-1. z-FILTERS. A z-filter F is a nonempty collection of
nonempty zero sets, closed under the formation of finite intersections, and
such that if a zero set A contains some В £ F, then A £ F. A z-ultrafilter
is a z-filter which is not properly contained in any other z-filter.
Thus to define z-filter, we simply cast "filter" in the setting of
zero sets (of a completely regular Hausdorff space) rather than in the
class of all subsets. The usual Zorn's lemma argument confirms the
following result.
(1,3-1) z-ULTRAFILTERS. Every z-filter may be embedded in a z-ultrafilter.
Some of the standard properties of filters are shared by z-fliters, as
the following result shows.
(1.3-2) z-FILTERS. (a) If the zero set A meets each member of the
z-filter F then there is a z-filter containing F and A.
(b) The z-filter F is a z-ultrafilter if and only if the only zero
sets which meet every member of F actually belong to F.
(c) If a union, A U B, of zero sets belongs to a z-ultrafilter F, then
A £ F or В £ F.
14
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
Proof (a) The collection H of zero sets which contain an element of Ffl A
or F is a z-filter containing F and A.
(b) This follows directly from (a) and the definition of z-ultrafilter.
(c) Suppose that neither A nor В belongs to F. Then, by (b), there
are sets A1,B' £ F such that А1 П A = В1 П В = 0. Thus A U В fails to meet
А1 О В1 which is contradictory. V
Theorem 1.3-1. z-ULTRAFILTERS AND MAXIMAL IDEALS. The map
z : I -+ Z
I z(I) = (z(x) |x e Ij
establishes a (not necessarily 1-1) correspondence between the collection I
of proper ideals I of C(T,R) , T a completely regular Hausdorff sp'ace, and
the set Z of all z-fliters. The restriction of this map to M, the
collection of all maximal ideals, produces a 1-1 correspondence between M
and the collection of all z-ultrafliters. Specifically, if F is a
z-ultrafilter, then z 1(F) is the maximal ideal associated with F.
Proof First we show that z(I) is a z-filter. Since I contains no units,
2 2
0 z(I) . Since z(x) П z(y) = z(x +y ) for each pair x,y e I, z(I) is
stable under finite intersections. If x £ I and z(x) C z(w) for some
w £ C(T,R) then xw € I and, by (1.2-1), -
z(w) = z(x) IJ z (w) = z(xw) £ z (I) .
Next suppose that M is a maximal ideal. To see that z(M) is a
z-ultrafilter, suppose that z(w) is a zero set which meets every element of
z(M) and consider the ideal J generated by M and w. A typical element of
J is of the form xw+m where x £ C(T,R) and m £ M. Since z (w) ("I z(m) / 0,
it follows that z(xw+m) / 0. Thus no element of J is invertible, J is
proper, and it follows from the maximality of M that w £ M. By (1.3-2)(b)
it now follows that z(M) is a z-ultrafilter.
Conversely suppose that F is a z-filter. We contend that
I = z 1(F) = |x|z(x) £ F} is an ideal. Utilizing the relations
z(xy) = z(x) U z(y) and z(x+y) Z) z(x) О z (у) , it is clear that I is closed
under addition and multiplication by elements from C(T,R). Thus I is an
ideal and clearly z(I) = F. It remains to be shown that z|^ is 1-1 and
maps M onto the class of all z-ultrafilters. To demonstrate the surjec-
tivity just mentioned, we show the ideal z 1(F) to be maximal whenever F is
a z-ultrafilter. It у ( z 1(F), then z(y) ( F; if F is a z-ultrafilter,
2 2
there exists z(x) 6 F such that z(x) О z(y) = 0. Thus z(x +y ) =0 and
1.3 MAXIMAL IDEALS AND z-FILTERS
15
x +y is invertible in C(T,R). It follows
and у is C(T,R) which proves that z \F) is
For any ideal I, z '*'(z(I)) is a proper
z	= M for maximal ideals M, and z|^
that the ideal generated by I
a maximal ideal.
ideal containing I. Thus
is seen td be injective. V
The main characteristics of the Stone-Cech compactification are
summarized in Theorem 1.3-2.
Theorem 1.3-2. CHARACTERIZING gT. Let S be a Hausdorff compactification
of the completely regular Hausdorff space T. Then the following statements
are equivalent.
(1)	S and gT are equivalent compactifications.
(2)	For any pair x,y £ C(T,R) , if z(x) О z(y) = 0 then
clgZ(x) n clgZ(y) = 0.
(3)	For any pair x,y £ C(T,R) , clg(z(x)О z (y) ) = clgz(x) О clgz(y).
g
(4)	Each X € C, (T,R) can be extended to some x £ C(S,R) .
b
Proof By Theorems 1.1-1 and 1.2-2 we already know that (1) and (4) are
equivalent and that (1)	(3) =*> (2) . Thus it only remains to show that
(2) => (3) and (3) =*> (4) .
(2)	(3) . Clearly clg(z(x)Оz(y) ) Cclgz(x) DclgZfy). Suppose that
s £ clgZ(x) О clgZ(y). As S is a compact Hausdorff space, there is a base
of zero set neighborhoods at s. Certainly each	О T is a zero
set in T so that z(x) О V and zfy) О V are zero sets. Let Vs be an
p	p	a
arbitrary zero set neighborhood of s in S and note that
(z(x)PV ) П Vs = z(x) П fvSnvS') .
v u' a	\ p a/
In the same way s £
clg(z(y)CIV^) and therefore
Hence s £ clg(z(x)ОV ).
cls(z(x)PV ) П clg(z(y)nv|j) / 0 .
Thus, by (2), z(x) P, z(y) P 0 and, as is an arbitrary zero set
neighborhood of s, it follows that s £ clg(z(x)Оz(y)).
(3) => (4) . To prove (4) we must extend an arbitrary x £ Cb(T,R) up to
s
S so that the resulting function x is continuous. For each s £ S we wish
g
to define x (s). To accomplish this we show that there is a unique
z-ultrafilter F on T which converges to s and then unambiguously define
s s
x fs) as the limit of a z-filter on R derived from F . To that end let F
s	/ s\ S
be any z-ultrafilter on T containing the z-filter base (V ) where is
16
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
a base of zero set neighborhoods of s in S and	О T for each p.
Since (V ) C F it follows that the filterbase F -+ s.
p s	s
If F is another z-ultrafilter convergent to s and distinct from F we
s
may choose z (w) f F and z(y) £ F such that z (w) О z(y) = 0. Bv (3) , or
s
more fundamentally by (2), cl^z(w) О cl^z (у) = 0. Since s f cl^z (y) — i.e.
s is an adherence point of F^ - s f clgZ(w) and s is not an adherence point
of F thus contravening the convergence of F to s.
g
Now, to obtain the definition of x (s), let [a,b] be any closed
interval containing x(T) and В be the class of all closed subsets E of
s
[a,b] such that x (E) £ F . Clearly В , a z-filter of subsets of [a,b] is
s	s	S
a filterbase and, as [a,b] is compact, there is an adherence point x (s)
of В .
s
To see that B^ actually converges to s we show first that is prime,
i.e. if A and В are closed subsets of [a,b] whose union belongs to В then
s
one or the other of A and В belongs to В . Indeed if AU В £ В then
-1 -1-1 S	S
x (A) u x (B) = x (AUB) £ F . But F is a z-ultrafilter and it follows
s	s
“1 “1
that x (A) or x (B) belongs to F and В is seen to be prime.
S	S S
To show that В -+ x (s) , let V be an arbitrary zero set neighborhood
s s
of x (s) in [a,b]. As [a,b] is completely regular there is a zero set
U C [a,b] such that xS(s) f [a,b] - U С V. Now U UV = [a,b] f В and,
g	®
since В is prime, either U or V belongs to В . But x (s) t U and there-
s	s s
fore U ? В . Hence V < В and it follows that В * x (s) .
s	s	s s
All that now remains is to show that x is continuous. To this end
S
let V be any zero set neighborhood of x (s). The job now is to exhibit a
g
neighborhood W of s € S such that x (W) С V. By the complete regularity of
g
[a,b] there exists a zero set V1 C [a,b] such that x (s) f [a,b] -V1. Thus
VUV' = [a,b] and x”1(V) Ux"1(V) = T, so cl x 1 (V) U cl x 1(V') = S.
-1 . Ь -1
We contend that s / cl x (V*)» If it did, if s f cl x (V ) , then,
as s adheres to each set in F , cl x (V1) Cicl_z(w) / 0 for each z(w) c F .
s S	Ь	ь
By (3) it now follows that x-1(V') О z(w) / 0 for all z(w) e Fg. Since Fg
is a z-ultrafilter and x ^(V1) is a zero set, x ^(V1) f F . By the
s s s
definition В , V e В . On the other hand В + x (s) and x (s) V — a
ss	s
contradiction. Thus s clgX (V).
The foregoing argument shows that s belongs to the open set
S - cl X-1 (V) = W C <11 X-1 (V) . If p t W C cl x 1 (V) then as W is a
s	s	s
neighborhood of p, every neighborhood of p will intersect x (V) and
1.3 MAXIMAL IDEALS AND z-FILTERS
17
”1	s
therefore x (V) f F . Thus V f В -+ x (p) so, as V is closed in [a,b] ,
S	P	g P
x (p)f V. This proves that x (W) С V. V
1.4 Maximal Ideals and the Stone-Cech Compactification
It is clear that if t is a point of the completely regular Hausdorff
space T and M^_ = [x( C(T,R) |x(t) = o], then M^_ is a maximal ideal of C(T,R) .
In fact it is the kernel of the nontrivial homeomorphism x -+ x(t) from
C(T,R) onto R. As is verified in (1.4-2) below, if T is compact, then
these "fixed" ideals M^_ constitute the set of all maximal ideals of C(T,R).
Even if T is not compact tho, each of the maximal ideals of C(T,R) is
"fixed" on a point p of the Stone-Cech compactification gT of T, a fact
which is the essential content of Theorem 1.4-1, the main result of the
section.
Definition 1.4-1. FIXED MAXIMAL IDEALS. Let T be a completely regular
Hausdorff space. A maximal ideal M of C(T,R) is fixed if there is some
t f T such that M = M^_ = [xeC(T,r) |x(t) =0}. Otherwise M is called free.
(1.4-1) FIXED IDEALS. The maximal ideal M is fixed iff F3 z(x) 0.
--------------------- xfM
In this case Oz(x) = ftl for some t in the completely regular Hausdorff
XfM------------------*• '
space T.
Proof If M = M then surely t f П z(x). If s t then, since T is
-----	t	xfM
completely regular, there is some x f C(T,R) such that x(t) = 0 — hence
xfM,— while x(s) = 1 so si О z(x) . Thus П z(x) = {t}.
t	r xfM	xfM
Conversely suppose that M is maximal and П z(x) / 0. If
xfM
t f Г1, z(x), then x(t) = 0 for all xfM and M is seen to be a subset of
xfM
M^_. The maximality of M then implies that M = M^. V
For any T, completely regular and Hausdorff as usual, C(T,R) possesses
fixed maximal ideals — namely those of the form M^_ = (x|x(t) =0]. As
(1.4-2) shows, if T is compact, then all maximal ideals of C(T,R) are of
this type. Conversely if T is not compact, C(T,R) always possesses free
ideals.V
(1.4-2) MAXIMAL IDEALS ALL FIXED WHEN T IS COMPACT. If T is a compact
Hausdorff space then the maximal ideals of C(T,R) are all fixed.
Proof We show that there are no free ideals in C(T,R) when T is compact:
we show that if M is free then M contains a unit.
If M is free then for each t f T there is some x f M such that
x^(t) / 0. Since each x^_ is continuous there must be open neighborhoods U^_
18
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
for each t on which x^ does not vanish. Since T is compact, finitely many
of the U^'s, U , ...,U say, must cover T. It is now easy to see that the
1	n
function
2	2
у = xt + • • • + x f M
1	n
never vanishes on T and is therefore invertible. V
As a consequence of (1.4-2) we may determine the form of the maximal
ideals of C^(T,R) whether T is compact or not. First consider the map
cp : Cb(T,g) + C(gT,g)
x -> x
where x denotes the unique extension of x to gT. Through the algebra
isomorphism ® the maximal ideals of C, (T,R) are seen to be in 1-1 corre-
b
spondence with those of C(gT,R). Since gT is compact, the maximal ideals
of C(gT,R) are all of the form
Mp = {x( C(gT,R) |x(p) = 0} pe gT .
And M corresponds (i.e. m(M ) = M ) to the maximal ideal M in C, (T,R)
P	P P	pb~
consisting of those x С C, (T,R) such that x(p) = 0.
b
In the case when T is not necessarily compact. Theorem 1.4-1 below
characterizes the maximal ideals of C(T,R). Before proving it, note that
(1.4-2) may evidently be recast as follows:
If T is compact then all maximal ideals of C(T,R) are of the form
M = {x£C(T,R) Ipe cl z(x)} pf gT.
P	P
Theorem 1.4-1. gT AND MAXIMAL IDEALS. (Gelfand-Kolmogorov) The mapping
ВТ + M
p + (xf C(T,R) I PC cl z(x)} = M
p	p
establishes a 1-1 correspondence between the class M of maximal ideals of
C(T,R) and the points p of gT.
Proof .First we show that M is an ideal in C(T,R) . If x ( M and
P	P
у f C(T,R) , then z(x) c z(xy) , so p c cl z(x) c cl z(xy) . For x,y f M ,
P	P	P
p f cl z(x) О cl.z(y) = cl [z(x)O z(y)] c Cl„z(x+y), the equality
P	P	P	p
concerning closures of zero sets following from Theorem 1.2-2. Thus M₽ is
an ideal.
1.4 MAXIMAL IDEALS AND STONE-CECH COMPACTIFICATION
19
To prove that M is maximal we first show that z(M ) is a
P -1	P
z-ultrafilter and then that z (zCM^)) = (Theorem 1.3-1). Let у be such
that z(y) П z (x)	0 for all x С M .
P
that у С M , i.e. that p € cl„z(y).
P	6
of p, we claim that x = xj^ belongs
M
P
We show that z(y) C z(M ) by showing
P
If z(x) is any zero set neighborhood
to M . To see this let p f int z(x).
P л	P
Then for each neighborhood V of p , VO int z(x) is a nonempty neighborhood
P
T is dense in PT, V О int z(x) О T is also nonempty. Thus V
p
= z(x) О T for each neighborhood V of p and it follows that
meets
As
z(x)
of p .
P c
int z(x) c cl z(x) .
P	P
x e
Hence
M . We have shown
P
neighborhoods of p (i.e.
that
the
zero
every member of a basic family of
set neighborhoods; see (1.2-4)) meets
z(y) , so p f cl z (y) and
-1 &
z ( z (M ) ) = M .
' D	p
such
4 P
some mfM
P
therefore w
To see
M .
P
this let
It only
we z 1(
remains to show that
Now let
for some p f
such there must be
z(M )). As
P
p £ cl z(m)
P
The desired equality now follows.
that z(w) = z(m). Hence
= cl_z(w) and
p
M .
P
M be
PT.
If no M
P
that p 0 cl z (x ) .
P P
a maximal ideal in C(T,R). We wish
To prove this we show that M С M for some p C PT.
P
then for each pt PT there is
contains M
Thus
PT = U
PC PT
PT =
for some p ,
Pn
C PT.
Thus
to show that M = M
P
C(cl z (x )) . Since
P P
some x e M such
P
PT is compact.
C (cl z(x
P - 1
у e
e
n
u

n
О clz(x
i=1	8 P;
/ n
cl„/ n
z
0
since closure and intersection may be interchanged for zero sets by
n
Theorem 1.2-2; it follows that О z(x ) = 0. But x ,...,x r M and
i=l pi	pl Pn
n
therefore О z(x ) = 0 is impossible by Theorem 1.3-1. It now follows
i=l Pi
that there is some p С рт for which M = M .
P
Finally we show that if p q then / M^. If p and q are distinct,
there are disjoint zero set neighborhoods U and V of p and q respectively
in PT and functions x,y c C(T,R) such that z(x) = U О T = U and
z(y) = V П T = V. As U Ov = 0, x and у cannot belong to the same
maximal ideal. V
20
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
Theorem 1.4-1 enables us to characterize compactness externally,
namely: The completely regular Hausdorff space T is compact if and only if
each maximal ideal in C(T,R) is fixed. To see the sufficiency of the
condition, we need only note that if T is not compact, then for any
p f gT - T, M is not fixed.
P
The algebra C(T,C) of complex-valued continuous functions does not
differ markedly from C(T,R) if one takes preservation of the results of
this chapter as the datum. The С(T,R)-completion of T yields gT; so does
the С(T,C)-completion. The maximal ideals of C(T,C) are in 1-1 corre-
spondence with the maximal ideals of C(T,R) under the mapping
M -* re M + i re M where re M denotes the collection of real parts re x of
functions x in M. The inverse of this map is the map sending the maximal
ideal J of C(T,R) into J +ij. The Gelfand-Kolmogorov theorem then pairs
the point p f gT with the maximal ideal M + iM of C(T,C) where
P P
Mp = {x (C(T,R) |p f clgZ(x)} . By similar considerations, a marked
propinquity is exhibited between the subalgebras of bounded continuous
functions, whether real- or complex-valued. Even if R is replaced by the
4
topological division ring H of quaternions with euclidean (i.e. R )
topology, and "ideal" by "two-sided ideal," the similarities persist: The
collection H (T,H) of horn omorphisms of C, (T,H) onto H endowed with the
b	b
initial topology generated by C, (T,H) again yields gT, for example, so the
b
maximal two-sided ideals of C(T,H) are in 1-1 correspondence with the
maximal ideals of C(T,R) . It is worth noting here that R, C and H are the
only connected 1эса11у compact topological fields; if a locally compact
field is not connected, then it must be totally disconnected (Bourbaki
1964, Ch. 6, §9, no. 3, Cor. 2).
Some discussion is devoted to algebras of continuous functions taking
values in a topological field or a topological ring in the exercises at the
end of this chapter and scattered in a few other places throughout the book.
1.5 Replete Spaces
The Stone-Cech compactification gT of T enabled us to characterize the
maximal ideals of C(T,R) in a certain way (Theorem 1.4-1, the Gelfand-
Kolmogorov theorem): Each p f gT corresponded to the maximal ideal
M = [xf C(T,R) |pc cl z(x)} .
P	P
When is M the kernel of a horn omorphism of C(T,R) into R? A sufficient
p	~
condition is that p С T for then M is the kernel of the horn omorphism p
p
sending x into x(p). As we shall see later (	(1.6-1)and (Theorem
1.5 REPLETE SPACES
21
1.5-	1), the "super adherence points" of T-those points pfgT such that for all
countable families (V^) of neighborhoods of p in gT,Ov must meet T —
determine all maximal ideals which are kernels of homomorphisms. The
collection UT of all such points is the repletion of T and it is to this
collection that this section is devoted. Once the maximal ideals which are
kernels of homomorphisms have been characterized as above, we also know
which spaces T have evaluation maps as the only homomorphisms of C(T,R):
namely those and only those for which UT = T. Spaces T with this latter
property are called replete and constitute the only spaces for which
C(T,R,c) — C(T,R) with compact-open topology — is bornological, as is shown
in Chapter 2. A surprising fact about replete spaces is that nonreplete
spaces are so hard to come by. The space [0,0) of ordinals less than the
first uncountable ordinal 0 is not replete (Example 1.5-1) but any subspace
of Rn is (Theorem 1.5-3), as is any complete uniform space T, provided only
that T is of non-Ulam (nonmeasurable) cardinal (Theorem 1.8-1), to name
just two broad classes of replete spaces.
Before starting the definition of repletion we recall the following
result on extension by continuity; for a proof see Kelley 1955, page 153 or
Dugundji 1966, page 243.
(1.5-1) EXTENSION BY CONTINUITY. Let T be a completely regular Hausdorff
space and R U [“} be the one-point compactification of R. Any function
x f C(T,R) can be uniquely extended to a function xf C(gT,RU{“})-
Using the extensions whose existence were just noted, we now define
that subspace of gT called the repletion of T.
Definition 1.5-1. REPLETION. The repletion (real compactification) UT of
the completely regular Hausdorff space T consists of those p f gT for which
each x(p) is finite, i.e. x(p) », for every x c C(T,R). If T = UT, then
T is called replete.
Since T is dense in gT, the only way to extend a function x f C(T,R)
to a continuous real-valued function on any subspace S, TC SC 8T, is by
continuity. Thus uT is the largest subspace of gT to which each x c C(T,R)
possesses a real-valued continuous extension. Any time a subspace T of a
topological space S is such that each x C C(T,R) may be extended to a
continuous real-valued function on S, T is said to be C-embedded in S. A
similar meaning is attached to the expression C^-embedded, meaning that all
bounded continuous functions are continuously extendible. Thus in this
nomenclature, UT is the largest subspace of gT in which T is C-embedded.
22
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
Note also that gT is the only compact Hausdorff space in which T is
C -embedded by Theorem 1.1-1.
b
It is easier to exhibit spaces which are replete than spaces which are
not. The reals, for example, are replete, as is any separable metric space
(Theorem 1.5-3); so is any compact space and, more generally, any Lindelof
space. As Shirota's theorem (Theorem 1.8-1) shows, any complete uniform
space whose cardinal is non-Ulam is replete. The space [0,Q) of ordinals
less than the first uncountable ordinal Q is not replete as is established
in Example 1.5-1.
In our next theorem we show that UT may be obtained as the completion
of T with respect to the initial uniformity C determined by C(T,R) on T,
paralleling the way in which gT was defined. Among other things this shows
repletions UT to be themselves replete: U(UT) = UT.
Theorem 1.5-1. CHARACTERIZATIONS OF THE REPLETION. Let T be a completely
regular Hausdorff space. Then:
(a)	(Nachbin) UT is the completion of T with respect to the initial
uniformity C determined by C(T,R) (i.e. C is the weakest uniform structure
for T with respect to which each x f C(T,R) is uniformly continuous).
(b)	UT consists of those points p e gT such that for all sequences
(V ) of neighborhoods of p in gT, (CIV )О T 0. Thus p ? UT if and only
n	n
if there is some G. set (in gT) containing p which does not meet T.
о
Proof (a) Let C, denote the initial uniformity determined on T by C, (T,R)
-----	b	b
and C the initial uniformity determined on UT by C(UT,R), i.e. the initial
uniforitlity determined on UT by the functions x for x f C(T,R).
We show that UT is the C-completion of T. We do this by observing
that (T, C) is a dense uniform subspace of (UT,C) and proving that UT is
C-complete. (That T is dense in UT follows from the fact that T is dense
in gT.)
To show that UT is C-complete, it suffices (see for example Bourbaki
1966a, Prop. 9, p. 186) to show that any С-Cauchy filterbase В on the dense
subset T converges to a point of UT. Since C, is weaker than C, any such
b
В will also be C, -Cauchy. Hence there is some p e gT such that В + p. It
b
remains to show that p f UT, i.e. that each x(p) is finite. By the
continuity of any such x, x(B) = x(B) converges to x(p) in the Hausdorff
space R (J {°°}- The fact that В is С-Cauchy however implies that x (B) is
Cauchy in R, hence convergent to a point of R so that its unique limit,
namely x(p), is finite.
1.5 REPLETE SPACES
23
(b) We show that p t UT if and only if there is a denumerable family
(V ) of neighborhoods of p in gT such that (OV ) О T = 0.
n	n
If p f1 UT then there is some x С С (T, R) such that x(p) = °°. The
neighborhoods
vn = {te бт| |x(t) | > n] (n=i,2,...)
have the desired property: indeed
Ov = x-1({~}) c gT- T .
n
Conversely suppose that p possesses a sequence (V ) of neighborhoods
n
in gT whose intersection does not meet T. Clearly we may assume that (V )
n
is a nested sequence of closed neighborhoods. Since gT is normal there is
a nested sequence of neighborhoods of p in gT such that C Un C lint
and continuous functions у on gT such that 0 = у = 1, у (U ) = 1 and
n	n	n n
yn(C(int V )) =0. As we verify next, the function
x : T R
t + Г yn(t)
n=l
is continuous on T and x(p) = Since (V ) is nested and CIV does not
n	n
meet T, (T-V^) is an ascending open cover of T. If t f T-V then
у (t) = 0 for all m = n so
m
m-1
*(t) = Г	•
n=l
A straightforward argument now reveals the continuity of x on T.
Before showing that x(p) = °°, note that in any topological space, if
A is a dense subset and V a neighborhood of t then V П A is also a
neighborhood of ts int V = int VOA and int V C int VOA С V О A for if
W is any neighborhood of s € int V then W О (int VOA) = (WOint V) О A 0.
Since each of the functions у , m < n, is continuous and equal to 1
m —
everywhere on U , then x(t) 1 n at any t f U ОТ. Therefore x(t) = n
n	_______ n
everywhere on the neighborhood U^O T of p and since this holds for every
n, the desired result follows. V
Clearly C(T,R) and C(UT,R) are indistinguishable as algebras. And, as
part (a) of the above theorem shows, (T,C) is a dense uniform subspace of
24
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
(UT,C). Moreover, since uT is C-complete, it is seen that repletions are
replete — i.e. that U(UT) = UT.
If S and T are homeomorphic completely regular Hausdorff spaces under
a homeomorphism h : T -> S say, then the map
f : C(T,R) + C(S,R)
x -* x'
where x'(s) = x-h 1(s) , s f S, shows C(S,R) and C(T,R) to be algebra-
isomorphic, a hardly surprising fact whose converse for replete spaces S
and T we shall establish shortly. Using this we can demonstrate the
topological invariance of repleteness, something a bit more striking — a
bit more striking because it is closely related to a particular uniform
structure of a space and homeomorphisms do not necessarily preserve uniform
properties. The homeomorphism t -+ t/(l+|t|), for example, maps the
complete uniform space R onto the incomplete space (0,1). Note too that
repleteness was only defined for completely regular — hence
uniformizable — spaces.
(1.5-2) REPLETENESS IS A TOPOLOGICAL INVARIANT. Let S and T be
homeomorphic completely regular Hausdorff spaces. Then'if S is replete,
so is T.
Proof Let f be as in (1) above and let C(S) and C(T) denote the initial
uniformities induced on S and T by C(S,R) and C(T,R) respectively. Our
goal is to show that T is С(T)-complete and to this end let В be a
C(T)-Cauchy Silterbase in T, and note that this is equivalent to saying
that x(B) is a Cauchy filterbase in R for each x C C(T,R). With h as in
(1), consider h (B) . For any x1 € C(S,R)
x'« h (B) = x (B)
by the way x' was defined, so h(B) is C(S)-Cauchy — hence convergent to
some s f S. It follows that В -+ h ''"(s). V
Before getting to more specialized examples, the С(T,R)-completeness
characterization of replete spaces T yields the following embeddability
theorem for them, a property which is useful in proving many subsequent
theorems.
Theorem 1.5-2. EMBEDDABILITY IN PRODUCTS OF R. A completely regular
Hausdorff space T is replete if and only if it is homeomorphic to a closed
subspace of a cartesian product of real lines. Thus closed subspaces,
intersections, and products of replete spaces are replete.
1.5 REPLETE SPACES
25
Proof Necessity. Let T be replete — i.e. C-complete — and consider the
evaluation map
m C(T,R)
e : T R
t -> ( X (t) )	.	.
yxec(T,R)
Since T is completely regular and Hausdorff, e is injective and exe (i.e.
the map (s,t) -+ (e(s),e(t)) maps the subbasic C-entourage V(x,e) =
|(s,t)€ Txt| |x(s) - x(t) | < e| into the relative subbasic product entourage
{((y(s)) , (y(t)) yeC(T(?) | Ix(s) - x(t) I < e] .
Hence e and e are uniformly continuous (e is a unimorphism). Thus the
C-completeness of T implies the completeness of e(T) in the product
uniformity. Completeness is a Hausdorff uniform space however implies
closedness, so the necessity of the condition is established.
Sufficiency. Since repleteness is a topological invariant by (1.5-2)
A
it suffices to prove that closed subspaces T of a product R are replete.
Since products of complete spaces are complete (Bourbaki 1966a, p. 186,
Prop. 10), any closed subspace T is complete in the product uniformity.
By the definition of the product uniformity each projection pr^ (a( A) of T
into R is uniformly continuous, hence certainly continuous, i.e.
[pr |a< A} c C(T,R)
so the product uniformity is coarser than C, the initial uniformity induced
by C(T,R) on T. Since С-Cauchy filters must therefore also be Cauchy
filters with respect to the product uniformity, the desired result is seen
to follow. V
The above result shows closed subspaces of replete spaces to be
replete but the property of repleteness is obviously not passed on to every
subspace — provided there are nonreplete spaces, i.e. provided there is a
space T for which T UT for then T is a nonreplete subspace of the replete
space UT. But where to look for a nonreplete space? Theorem 1.5-3 below
eliminates a wide class of spaces. Among other things it shows every
subspace of R and Rn to be replete.
Theorem 1.5-3. CLASSES OF REPLETE SPACES (Cf. Theorem 1.8-1) If the
completely regular Hausdorff space T possesses any of the following
properties, then T is replete.
26
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
(a)	Lindelof (i.e. open covers possess countable subcovers);
(b)	а-compact (T possesses a countable cover consisting of compact
sets);
(c)	second countable (the topology possesses a countable base);
(d)	separable metric.
Proof Certainly every а-compact space is second countable and every
second countable space is Lindelof, as is any separable metric space. So,
the only thing that needs to be shown is (a). To do this we use the
characterization of repleteness of part (b) of Theorem 1.5-1. For
p C gT - T, let U(p) denote the collection of closed neighborhoods of p.
Since CIU(p) = [p], gT being Hausdorff, then
T C C( Пи (p) ) = U CU .
ueu(p)
If T is Lindelof (CU) , . possesses a countable subcover (CU ) or,
UCU(p)	n
equivalently, (CIU ) О T = 0. Thus p ( UT. V
n	~
Though completely regular Hausdorff Lindelof spaces are replete, the
converse is false. To see this, let T denote R with the topology
generated by the half-open intervals (a,b]. T is completely regular,
Hausdorff and Lindelof (see below), hence replete, hence so is txt. txt is
not Lindelof however, since the closed subspace {(t,-t) 11 f T] is discrete
and nondenumerable.
Proof that T is Lindelof. Let G = (G^) be an open cover of T and let I
denote the set of all points x c R belonging to an open interval (a,b) such
that (a,b) is contained in some covering set G^. As I is second countable
in the Euclidean topology, a countable collection of the intervals (a,b)
covers I; thus a countable subset of G covers I. Next we claim that
E = R - I is a countable set. If x ( E then, since some G contains x,
P
there is some a c R such that (a ,x] C G . Furthermore distinct x's in E
x	x	p
yield disjoint intervals (a^jx]: indeed if x < у and (ax,x] П.(а^,у]	0,
then x f (a^,y] which contradicts the fact that x / I. Now by associating
a rational number r e(a ,x] with each x € E, it follows that E is
X	X
countable. It remains to add one covering set G^ for each x f E to the
countable set of G which covers I to obtain a countable subcover of T. V
Having just generated a replete discrete space, it is natural to
inquire about conditions under which discrete spaces are replete. In
Section 1.7 it is seen that precisely those not of Ulam cardinal are
replete (see discussion before and after Definition 1.7-1).
1.5 REPLETE SPACES
27
Before giving an example of a nonreplete space we set forth two
topological notions: countable compactness — every countable open cover
has a finite subcover — and pseudocompactness — every real-valued
continuous function is bounded <=> C (T,R) = C, (T,R). That the former
implies the latter is clear since the open cover {t€ t| |x(t) | <n} ,	nSJj.,
has a finite subcover for each continuous function x. Tho the two
notions are generally distinct, they coincide in normal Hausdorff spaces
(Dugundji 1966, XI, 3.7 and XI, 3.9, pp. 231-2).
Generally we can make the following identifications:
Cb(T,B) = C(gT,B) and C(T,B) = C(UT,B) -
If T is pseudocompact, all four may be identified for then
C(T,R) = C, (T,R) . Moreover:
- b
(1.5-3) T PSEUDOCOMPACT => COMPACT= REPLETE. If T is a completely
regular Hausdorff space, then T is pseudocompact if and only if gT = UT.
Thus in the class of pseudocompact spaces T, compactness is equivalent to
repleteness.
Proof If there is a point p € gT - UT then, as constructed in the proof
of Theorem 1.5-1 (b), there is some x f C(T,R) which is unbounded on T.
Thus if T is pseudocompact, gT = UT. Conversely if UT = gT then
C(T,R) = C(UT,R) = C(gT,R) = C, (T,R)
-	-	- b
and T is seen to be pseudocompact. V
The problem of exhibiting a nonreplete space is now reduced to that
of producing a space which is pseudocompact, but not compact.
Example 1.5-1.	[0,Я) IS NOT REPLETE. The completely regular Hausdorff
ordinal space [0,Q), where Q is the first uncountable ordinal, is countably
compact — hence pseudocompact — but not compact. To see that it is not
compact, view it as a subspace of the Hausdorff space [о,й] and note that
it is not closed. We prove countable compactness by showing each sequence
to have a cluster point. If (t ) is a sequence from [0,Q), it can be
n
rearranged so that it is increasing. The rearranged sequence then has its
supremum, t say, as a limit. Clearly t is countable — i.e. t f [0,Q) — and
is a cluster point of the original sequence. It now follows from (1.5-3)
that [0,Q) is not replete.
Later conditions under which spaces C(T,R,c) of continuous functions
with compact-open topology are barreled and bornological are obtained:
28
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
(Th. 2.5-1) C(T,R,c) is barreled if and only if unbounded continuous
functions x|E exist on each closed noncompact subset E of T.
(Th. 2.6-1) C(T,R,c) is bornological if and only if T = UT.
Taking relative pseudocompactness to mean that the restriction of each
x € C(T,R) to E is bounded, the barreledness result becomes:
C(T,R,c) is barreled if and only if each closed relatively
pseudocompact subset of T is compact.
This version together with the characterization of relative
pseudocompactness of our next result is helpful in proving that
bornological spaces of continuous functions must be barreled (see (2.6-1)).
Note too that saying a subset E of T is relatively pseudocompact is
different from saying that E is pseudocompact. If E is closed and T is a
normal Hausdorff space, however, then each x ( C(T,R) possesses a
continuous extension to C(T,R) and the two notions do coincide.
(1.5-4) RELATIVE PSEUDOCOMPACTNESS. Let T be a completely regular
Hausdorff space.
Then E С T is relatively pseudocompact if and only if
cl E C UT.
P
equivalent
(Thus
to the
as already noted in (1.5-3) T being pseudocompact is
equality UT = 6т.)
Proof We
show that E is not relatively pseudocompact if and only if
cl.E <Z UT.
P
Suppose that p € cl E - UT, so that there is some
P
= ”. Thus for each n C N there is a point t € E
Conversely if E is not relatively pseudocompact,
x(p)
x C C(T,R) such that
at which x is > n.
then a positive
function x c C(T,R) exists which is unbounded on
sequence (t ) of points from E
n
cluster point p
E.
Thus there is a
As for the
of (t ) in the
n
last statement
such that x(t ) >
n —
compact space Вт,
we need only note
n.
Consequently at any
x(p)
that cloT = 6т. V
P
1.6 Characters
and UT
Homomorph!sm of C(T,R) means algebra homomorphism of C(T,R) into R;
character means nontrivial homomorphism and each character h determines an
ideal, its kernel ker h, of codimension one, hence a maximal ideal. (Note
that being of codimension 1 is only a sufficient condition for maximality
of an ideal. For maximal ideals M of C(T,R), if codim M 1, then
codim M = ”.(Gillman and Jerison, 1960, p. 173a) Codim M = 1 is a
necessary and sufficient condition for a maximal ideal to be the kernel of
a character, the canonical map.x -+ x + M c X/M = R being the character going
with the maximal ideal M. This latter type of maximal ideal is also called
1.6 CHARACTERS AND uT
29
a real maximal ideal. We denote the set of characters of C(T,R) by H or
H(T), where necessary, and often identify characters and their kernels.
It has already been established (Theorem 1.4-1, the Gelfand-Kolmogorov
theorem) that points p f 6T determine maximal ideals
Mp = {x f C(T,R) |p€ clgZ(x)")
and that all maximal ideals of C(T,R) are of this type. Here we show that
codim M = 1 iff p f UT ,
P
in other words that M is the kernel of a character if and only if p f UT.
P
Once we have this, another property of real maximal ideals emerges — namely
that M is real if and only if z(M ) is stable with respect to countable
P	P
intersection — which is useful in proving Shirota's remarkable result
(Theorem 1.8-1) that every complete uniform space of non-Ulam cardinal is
replete.
(1.6-1) UT AND CHARACTERS. Let T be a completely regular space and let
x° be the unique extension of x € C(T,R) to C(UT,R), and let M be as in
P
the Gelfand-Komolgorov theorem. Then there is a 1-1 correspondence between
UT and H, determined by
a : UT H
P + P*
with
p* : C(T,R) -> R
U, .
x x (p)
and ker p* = M . In other words M is the kernel of a character if and
P	P
only if p £ UT.
Proof Two things are clear from outset: for p £ UT, the evaluation map
p* is a character, and different p's determine different characters. Tc
show that ker p* = M , it suffices to show that ker p* = M . In turn this
P	P
is equivalent to showing that for each x c C(T,R)
xU(p) = 0 => p f cl z(x) .
p
Since there is a neighborhood base of zero sets at p, by (1.2-4), it
g fi
suffices to show that any neighborhood z(y₽), y₽ € C(6T,R), of p meets
z(x) if x°(p) = 0. Clearly у = У^1т Скег p* so x^+ y^ € ker p*.
2 2
Consequently there exists t € z(x +y ) = z(x) П z(y).
30
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
P
Hence there must be a net (t )
3	6
continuity x (t ) -> x₽(p), but
Let e denote the identity of C(T,R), the mapping sending each t j T
into 1. To see that the map p -> p* is onto H, let h be any character.
There exists p e ST such that ker h = M and it remains only to show that
P
p ( UT, for then ker h = ker p* = M , whence h = p*. To see that p € UT,
6
consider any x ( C(T,R) and its continuous extension x₽ ( C (ST, R(J {«°D ; we
о
show that x₽ (p) ( R. As x-h(x)e ( ker h = M
P
€ cl„z (x-h (x) e ) .
P
from (z(x-h(x)e) converging to p. By
Q
for all p, x₽(t ) = h(x), so
о	и	P
x (p) = h(x) f R. V
It is now clear that the spaces T for which evaluation maps x -+ x(t)
are all the characters of C(T,R) — i.e. H = T* — are precisely the replete
spaces. As a special case, one sees that compact spaces have this
property. The spaces T for which all the maximal ideals of C(T,R) are of
codimension one are the pseudocompact spaces, i.e. those for which
6T = UT ((1.5-4)).
In the discussion immediately preceding (1.5-2) the hardly surprising
fact that homeomorphic spaces S and T produce the same algebra of
continuous functions was noted. The converse is certainly false for UT can
differ^ from T (Example 1.5-1), yet C(UT,R) is always the same as C(T,R).
Still, there is a result going in the converse direction. If C(S,R) is
isomorphic to C(T,R), then the characters of the algebras are also
identifiable, hence so are US and UT — at least in the sense that they are
in 1-1 correspondence. But this is rather weak. Are they homeomorphic?
Is the 1-1 correspondence alluded to above a homeomorphism? As it happens,
it is. Part of the vehicle we use to show it is the following result which
provides for a topological identification of UT and H.
(1.6-2) UT IS HOMEOMORPHIC TO H(T). Let T be a completely regular
Hausdorff space and H and M the sets of characters and maximals ideals
respectively of C(T,R). In virtue of the Gelfand-Kolmogorov theorem,
M may be topologized by reflecting 6T's topology onto it. Viewing H as a
subspace of M then the bijection of (1.6-1)
Since repleteness is
the possibility of T
a topological invariant by (1.5-2), T UT precludes
being homeomorphic to UT.
1.6 CHARACTERS AND uT
31
a : UT -> H
p -> p*
(p* (x) = xU(p) for each x ( C(T,R)) is a homeomorphism. Moreover the
topology H inherits as a subspace of M is the initial topology generated by
C(T,R) on H. A typical subbasic neighborhood of p‘ f H is given by
V(p*,x,e) = h| |q* (x) - p* (x) | = |xU(p) - xU(q) | < e]
x c C(T,R) , e > 0.
Proof of (1.6-2) is routine and is omitted. We turn next to settling
the questions raised prior to it concerning the "sensitivity" — in Gillman
and Jerison's phrase — of the space of continuous functions to the
underlying topological space. Generally the space of continuous functions
is not sensitive enough to distinguish between spaces, but in the broad
class of replete spaces, it is.
(1.6-3) RECOVERING T FROM C(T,R). Let S and T be completely regular
Hausdorff spaces. If the algebra C(S,R) is isomorphic to C(T,R) then the
repletions of S and T are homeomorphic. Consequently in the class of
replete spaces, the spaces are homeomorphic if and only if their algebras
of continuous functions are isomorphic.
Proof If f : C(T,R) -+ C(S,R) is an isomorphism, then
f' : H(T) -+ H(S)
p* -> p*.f 1
establishes a 1-1 correspondence between the characters of C(T,R) and
C(S,R). With a as in (1.6-1), then
c, "I
a f' a
ut —> H(T) —> H(S) —> us
establishes the desired homeomorphism: with notation as in (1.6-2), the
subbasic neighborhood V(p,x,e) of a point p £ UT is transformed first into
V(p*,x,e), then into V(p*‘f \f(x),e) by f, and finally into the subbasic
neighborhood V(a 1(p*-f 1),f(x),e) of a 1(p*-f	= (a 1,f'«<7)(p) C US. V
As sets, C(T,R) and C, (T,R) = C(6T,R) obviously differ if and only if
-	b
T is not pseudocompact. But, tho different, could they be isomorphic
for non-pseudocompact T? If so, then UT = u(6T) = 6t — hence T is
pseudocompact.
A forerunner of (1.6-3) is Banach's result (Banach 1932, p. 170,
Theorem 3) that in the class of compact metric spaces S and T, S and T are
32
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
homeomorphic if and only if C(S,R) is isometric to C(T,R) when each space
carries its sup norm metric. Stone (1937, p. 469) then generalized this
isometry-homeomorphism to compact Hausdorff spaces. Gelfand and Kolmogorov
(1940) turned their attention to C(S,R) and C(T,R), S and T compact
Hausdorff, as rings alone and showed them to be ring-isomorphic if and only
if S and T were homeomorphic. By taking x >_ у if and only if x(t) >_ y(t)
for all t, C(T,R) becomes a distributive lattice, and Stone (1941) showed
that, as a lattice-ordered group, C(T,R) characterizes the compact
Hausdorff space T; Kaplansky (1947) showed that as a lattice alone, C(T,R)
characterizes T. Kaplansky's result has been extended to noncompact T in
Shirota 1952b and Henriksen 1956.
Theorem 1.6-1. COUNTABLE INTERSECTIONS AND CHARACTERS. Let T be a
completely regular Hausdorff space and p С 6т. Then the following are
equivalent:
(a)	M is the kernel of a character (namely p*);
P
(b)	z(Mp) is stable under countable intersections;
(c)	z(M ) has the countable intersection property.
P
Proof It is clear that (b) implies (c). To show that (a) implies (b) we
first show that (a) implies (c).
Let (x ) be a countable family of functions from M . Certainly the
n	-n	₽
function inf( x ,2	) has the same zero set as x so we may assume that
1 n1	n
0 < x < 2 П. Thus x = S x € C(T,R) and z(x) =O z (x ) . To show that
— n —	n	n n
not (c) not (a), suppose thatn z (x ) = 0 so that x is invertible in
— 1 6
C(T,R). We s*how that p t UT by showing that (x )₽(p) = °°.
m	m
As p e О cloz(x ) = clo П z(x ) by Theorem 1.2-2, for each m f N,
.pnp	n
n=l	n=l
m
there is a net (t ) С О z(x ) converging to p in 6T. For e > 0 then by
V .. •, n
choosing m sufficiently large we may guarantee that
Г x
, n
n=m+l
S 2 n < e on T. Hence x (t )	> 1/e for each index p and, by the
n=m+l
continuity of (x 1)^, | (x 1)® (p) | >_ 1/e. The desired result that
— 1 в
(x ) (p) = 00 now follows.
To prove (a) => (b) it remains to show that z(x) ( z(M₽). Suppose
that z (y) ( z (Mp) . Then z (у) О z(xj e z (M₽) for each n and so, by the
previous argument,
1.7 MEASURES, gT AND ULAM CARDINALS
33
z (у) О z (x) = О z (у) О z (x ) / 0 .
n	n
As z(x) meets each set in the z-ultrafilter z(M ) ,
P
it follows by (1.3-2)(b)
that z (x) € z (M ) .
P
Last we show that (c) => (a).
such that x®(p) = By (1.2-1) (f) ,
g
n f N. As x₽ is continuous at p and
n f N. Thus each x ^([n,°°)) € z (M )
P
If p £ UT then there is some x t C(T,R)
x ( [n,°°) ) is a zero set for each
g	“1
x₽(p) = ”, p C cl x ([n,”)) for each
-1 P
and О x ([n,”)) =0. V
n
1.7 0-1 Measures, ST, and Ulam Cardinals
There is a 1-1 correspondence between z-ultrafliters on T and points
of St, as well as with maximal ideals of C(T,R). Associated with each
z-ultrafilter F is a 0-1 measure (on the algebra A of sets determined by
z
the zero sets Z) defined by taking m(E) = 1 if E f F, 0 otherwise.
Moreover associated with each t f T is the 0-1 measure m^_ concentrated at t
(m. (EC A ) = 1 if and only if tC E), thus providing a natural embedding of
t z
T in the space M of 0-1 measures on T. By suitably topologizing M , the
о	о
map t -> m actually embeds T homeomorphically in M , as shown in
t	о
Theorem 1.7-1. So topologized, not only is M compact, but functions
о
x f C. (T,R) may be continuously extended from T to M . The vehicle by
b	о
which this is done is defining the extension x to be Jxdm at m, i.e.
x : M + R
о
m -+ / xdm
This is what Varadarajan (1965) did and this result is presented here.
Converting the z-ultrafilters into a space of measures — a topological
vector space of measures in fact — has the advantage of making a host of
results from measure theory available.
A difficulty which this leads us into is confrontation with some
problems in set theory. The measures m^_ concentrated at points of T are
countably additive set functions which may be defined on the class P(T) of
all subsets of T, with m^_(T) = 1. But are these the only such set
functions? Could there be a countably additive {0,1}-valued set function p
defined on P(T) such that p (T) = 1 but p ({t}) = 0 for each t € T? The
question of whether such a function — an Ulam measure — can exist on a set
T has antecedents as far back as 1904 when Lebesgue asked: Does there
exist a measure m defined on P([0,1]) such that m([0,l]) = 1 and such that
34
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
congruent sets* have the same measure? Since all one-point sets must have
the same measure under these conditions, each singleton must have measure 0.
The question was answered in the negative in 1905 by Vitali: No such
measure could exist. Moreover, as shown by Banach and Kuratowski in 1929,
even if the congruence requirement is dropped, no such measure can exist on
[0,1] if the continuum hypothesis is	assumed. If	the	generalized	continuum
hypothesis (GCH) is	assumed, then no	such measure	can	exist on any	set E,
as Banach showed in	1930. Calling a	cardinal |t|	an Ulam cardinal	if an
Ulam measure can be	defined on P(T) ,	the question	may	be phrased:	Are
there sets of Ulam cardinal? Within the framework of Zermelo-Fraenkel (ZF)
set theory, the existence of such cardinals cannot be proved — at least if
one assumes ZF to be consistent, for their nonexistence has been shown to
be consistent with ZF. Their nonexistence is even consistent with
ZF + Axiom of Choice (ZFC) (Shepherdson 1952, Ulam 1930). In ZF + GCH, as
previously mentioned, their nonexistence can be demonstrated. Possibly
their nonexistence can even be proved in ZF or ZFC.
For us a finitely additive measure on T is a finitely additive
nonnegative real-valued function m defined on the algebra A^ of sets
generated by the zero sets Z of the completely regular Hausdorff space T
for which the following "regularity" condition holds:
For each AC A , m(A) = sup{m(Z) |Z C Z,	ZCA] .
(Thus knowledge of m on Z is sufficient to determine	its	behavior	on	A^.)
The difference of two finitely additive measures on T is called a finitely
additive signed measure on T. In the event that the	set	function	is
countably additive, i.e. m(lJE ) = J)m(E) whenever	(E )	„ is a	pairwise
n	n	n nfN
disjoint collection from A^ with union in A^, it is called either a measure
or signed measure, as the case may be. A 0-1 measure has {0,1} as its
range. The collection of all finitely additive 0-1 measures is denoted by
M , the finitely additive measures by M+, and	the finitely additive	signed
measures by M. For each m €	M,	there are m+,	m C M+ such £hat m =	m+- m
where m+(A) = sup jm (В) | BC A,	B€	A } and m~ (A.)	= -inf {m (В) | ВС A, В €	] for
A f A . The total variation	of	m is I ml (A) =	m+(A) +m (A) . The function
z
m -+ ||m|| = |m| (T) defines a norm on the linear space M with pointwise
X and Y are congruent if there is some rf [0,1] such that for all у f Y
there is some x € X such that у = x + r (mod 1) , and for each x € X there
is some у C Y such that у = x + r (mod 1) .
1.7 MEASURES, gT AND ULAM CARDINALS
35
operations with respect to which M is a Banach space. It is not difficult
to see that
|m| (A) = supf Л |m(E^) I |, E^C A, (E^) pairwise disjoint ]
(see Dunford and Schwartz 1958, p. 137). Moreover a bounded finitely
additive real-valued set function m is a finitely additive signed measure
if and only if for each A € A there are zero sets Z , Z such that
z	12
Z^C Ac CZ^ and |m| (CZ^-Z^) < e so that the elements of M are just the
"regular" finitely additive bounded real-valued functions on T, "regular"
in the sense that for each A ( A , and e > 0, there is a closed set CC A
z
in A and an open set G D A in A such that m (G-C) < e (Dunford and
z	z
Schwartz 1958, p. 137). In (2.4-5) it is established that the space M is
the continuous dual of the Banach space C, (T,R) equipped with the uniform
b
norm — a fact that we will have need for later in this section
(Theorem 1.7-1 (a)).
Using the definition of integral given in Dunford and Schwartz 1958,
Chapter III, or Alexandrov 1940, we now define a topology on M, the vague
topology, by specifying a subbasis at each m^ f И. A typical subbasic
neighborhood of m is given by
{ m e M
xdm- xdm <el = V(m , x,e)
I о •> о
(T,R) and e > 0. In other words
the vague topology is the
where x f
initial topology determined by the maps [ Jxd* | x f C^ (T, R) ] on M. A net
from M converges to m f M in the vague topology if and only if Jxdm^ ->
for each x € C, (T,R) . In our first result we present an alternate
b
(mp)
Jxdm
characterization of "vague" convergence to be used in the sequel.
(1.7-1) VAGUE CONVERGENCE. A net (m^) from M+ converges to m f M+ in the
vague topology if and only if
(1) m (T) -> m(T) and
P
(2) lim sup^m^ (Z) £m(Z) for each zero set Z.
Condition (2) may be replaced by
(21) lim inf m (CZ) > m(CZ) for each zero set Z.
P P -	+
Thus a net converges in the vague topology of M whenever it converges in
The proof of (2.4-5) of course does not depend on anything following
Theorem 1.7-1.
36
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
the product topology of M (a collection of functions mapping A into R) so
+	z ~
that the product topology of M is at least as strong as the vague topology.
Proof First suppose that (m ) is a net from M+ converging to m f M+ in
the vague topology and let Z f Z, Z denoting the zero sets of the
completely regular Hausdorff space T. By the regularity of m, given e > 0,
there exists Z‘ ( Z such that Z C CZ1 and m(CZ') < m(Z) +e. By
Theorem 1.2-1 (c) , we may choose x € C, (T,R) such that 0 < x < 1,
x(Z) = [1] and x(Z') ={0}. For this x and each p f L
m (Z) = xdm < xdm .
P Jz P - J P
On the other hand
xdm = xdm <_ m(CZ 1) < m(Z) + e .
J Jcz'
Since + m, Jxdm^ + Jxdm so that
lim sup m (Z) = lim sup xdm S lim xdm = xdm < m(Z) + e .
P P	P Jz P J P J
As e is arbitrary, lim sup^m^ (Z) £m(Z)
Conversely, suppose that the conditions
is clear that mp(T) + m(T) .
hold. We claim that for each
x ?C^(T,R), lim supp Jxdm^ <_ Jxdm. Furthermore it suffices to prove this
inequality for those x's for which 0 < x < 1, for by appropriately choosing
scalars a > 0 and b we can always force ax+b to satisfy the condition
0 < ax + b < 1. Assuming the result to hold for 0 < у < 1 then yields
lim sup (ax+b)dm < (ax+b)dm .
P J P - J
But J (ax+b) dm = a Jxdm + bm(T) and
lim sup (ax+b)dm = a*lim sup xdm + b’lim m (T) .
P J	P	p J p	p
Since b-lim m (T) = bm(T), it follows that
P
lim sup xdm < xdm .
P J P _ J
To prove the inequality for 0 < x < 1, note first that for each к j N,
T
= U x 1([(i-l)/k,i/k)) .
i=l
1.7 MEASURES, gT AND ULAM CARDINALS
37
Letting	Z^ be the zero set x 1([i/k,”)), this becomes к т = U (z. -Z.) . i=l 1"1 1
Now the	simple functions x and x defined at t f Zi-i~ Zi x(t) = (i-l)/k	and	x(t) = i/k	(i = l,...,k)
satisfy	the inequality x <_ x <_ x. Thus Г	f	к
(1)	xdm < xdm = 1/k + 1/k- v m (Z.) J P - J P	i=l P 1
and	к	f	f
(2)	1/k £ m (Z.) = xdm <_ xdm . i=l	J	J
As lim sup m (Z.) < m(Z.) for each i,
P P !	-	!	k
limits in (1) and replacing (1/k) Z
i=l
and using (2),
it follows that, taking superior
к
lim sup m (Z.) by (1/k) Z m(Z.),
1	i=l 1
lim sup xdm < 1/k + xdm
P I P - I
for each к ( N. Letting к establishes the desired inequality.
Applying the inequality for -x yields
lim inf xdm > xdm
P J P - J
and we may conclude that Jxdm^ -> Jxdm for each xf C^(T,R). V
Now the measures concentrated at the points t in T, defined to be
1 at the sets AC to which t belongs and 0 otherwise, evidently
constitute an injective image of T in M via the map
о
C₽: T -> M C M+
о
t -> mt
Indeed if t / s and x ( C(T,R) maps t into 0 and s into 1 then т^(г(х)) = 1
while m (z(x)) = 0. It is equally evident that the elements of co (T) are
s
countably additive.
38
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
Theorem 1.7-1. gT - M Let M denote the collection of all finitely
_______________________о	о
additive 0-1 measures on the completely regular Hausdorff space T. Then:
(a)	M with vague topology is the Stone-Cech compactification of T,
о
and
(b)	the repletion UT of T is the collection of all countably additive
elements of M , i.e. the measures in M .
о	о
Proof (a) First we shall show that the 1-1 mapping cp given above is a
homeomorphism. Using the fact that there is a base of zero sets at each
point t € T, one may show that for each x c C^tTjR) and t С T that
Jxdm^ = x(t) . Since Jxdm^ = x(t),
cp({t ? t| |x(t)-x(tQ) | < e}) = {mt
= M OV(m , x,e)
о t
о
and cp is seen to be a homeomorphism.
Next it is shown that ср (T) is dense in M by showing that, given
о
m ( Mq, there is a net from tp(T) which converges to m (in the vague
topology).
Let Z denote the collection of all zero sets Z for which m(Z) = 1.
m
Z becomes a directed set with respect to the ordering
m
Z1 — Z2 iff Z2CZ1-
Since each Z C Z is nonempty, we may choose some element t from Z. We
m	z
contend that the net (m ) converges to m in the product topology,
\ z4ez
m
therefore also in the vague topology by (1.7-1).
Suppose that A f A . If m(A) = 1 then there is a zero set Z C A such
z	о
that m(Z ) = 1. Thus for each Z f Z , if Z > Z (i.e. Z C Z ) then
о	m	— о	о
m (A) = 1 so that m (A) -+ m(A) . If m(A) = 0, then m(CA) = 1 and some
z	hz
zero set Z С CA must exist whose measure is 1. Hence each Z C Z greater
о	m
than or equal to Z , being a subset of Z , fails to meet A and m, (A) =0.
°	°	bZ
Thus m (A) * m(A) in both cases and ц/т) is seen to be dense in Mq.
ZZ	°
The next step in verifying that is a compactification of T is to
show that M is compact and Hausdorff in its vague topology. As for the
о
separation, let m, and rri be distinct points of M . Then m, - rri is a
12	о	1	2
1.7 MEASURES, gT AND ULAM CARDINALS
39
nontrivial finitely additive regular set function. Clearly the total
variation	of - m2 is either 1 or 2.
Choose pairwise disjoint sets A^,...,A^ ( A^ with the properties that
Z | (m -m )|(A.) = |m -m | (T) and m (A ) / m (A ) . Assuming that
1 z? J-	J- £	J- J-	& Л.
m^(A^) = 1 and т^(А^) = 0, choose zero sets Z and F such that Z and F are
each subsets of A , m^(Z) = 1, and (F) = 0. Then for the zero set
Z^ = Z u F we have Z,F CZ^CA^ and it follows that m^tZ^) ~ m (A^ and
m2(Z1) = m2(A1) . For each i, 2 <_ i £ n, choose zero sets Z^C A^ such that
(Z^) =	(A^) in the same manner as Z^ was chosen. As the Z^'s
are pairwise disjoint, so are their closures in 0T. Thus there is an
x® €C(T,R) such that x^(cl„Z.) ={1} and x® ( U clo(Z.)) = 0 . Letting
~	61	i=2	6 1
x denote xPI , it follows that
1 T
xdtm^-m^) = (m^-m^) (Z.^ = 1 .
Finally then { m (	| | Jxd (m-m^) | < 1/2 and {m €	| | Jxd (m-m,,) | < 1/2] are
disjoint neighborhoods of and m^ in the vague topology.
Since a typical continuous linear functional x1 on C, (T,R) with the
b
uniform norm is of the form x1(x) = Jxdm for some unique finitely additive
signed measure m f И by (2.4-5), the vague topology of M is in fact the
weak-topology a(C (T,R) 1 ,C (T, R) ) . Thus, by Alaoglu's theorem, a subset
b	b
of M is compact in the vague topology whenever it is closed in the
vague
of each
topology and norm-bounded. As the norm, i.e. the total variation,
element of M equals 1, it only remains to show that M is (vague)
~	о
a net from M convergent to m f M.
о
If not, then there is some A ( A such
z
0 and choose Z^,Z2 ( Z such that
m(Z^) and m(CZ2) are
so are their closures in
< 1, x6(cl6Z2) = {0],
of x® to T and consider
о * "	’
in M. To this end let (m ) be
P P«L
First we claim that m € M*.
closed
that m (A) < 0. Let d = -m(A)/3 >
Z^C AC CZ2 and |m|
both less than -2d.
It follows that
Вт and we may
and x® (cl Z )
P -L
choose
[1].
(CZ2-Z1) < d.
Since Z^ and
x6 € C(0T,R)
Let x denote the restriction
Z are disjoint,
Z	о
such that 0 < x₽
xdm = xdm
Z2
xdm + xdm =	xdm +
CZ2-Z1 JZ1 Jcz2-Z1
m(Z )
40
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
As m(Z^) < -2d and
xdm
CZ2"Z1
< |m| (CZ^Z^ < d
it follows that Jxdm
nonnegative since it is the integral of a
hand, for each у J L, Jxdm^ is
nonnegative function with respect
to a finitely additive measure. Thus Jxdm / Jxdm which contravenes
choice of (m^) and m. We conclude that m € M . Furthermore as each
m belong to M"4 and m -> m, the net m
P	P
for each A € A .
z
that 0 < m (A) < 1
On the other
the
0 < m(A)
< m(T) = 1
contrary
assumption
zero sets Z, and Z„
1	2
(T) -> m(T) by (1.7-1) so that
m and
P
To see that m ( M we make the
о
A € A . Then there must be
z
0 < m(Z ) < m(CZ ) < 1.
disjoint sets are disjoint,
= fO} and x®(cl Z ) = fl}.
1	p z
for some
such that Z, С A C CZ„ and
1	2
Once again, by the fact that the g-closures of
g	g
we may choose x₽ f C(0T,R) such that x₽(cl_Z_)
pl
g
Now let x denote the restriction of xp to
T, and choose zero sets Z*
Z** (by (1.2-3)) such that
CZ* = {t€ t||x(t)| <1/4}
and
Z** = {tf t| |x(t) | <1/4} .
It follows that Z^ C CZ* C Z**C CZ2> By (1.7-1) lim sup m^(Z**) <_
m(Z**) < 1, so that an index p f L exists such that m (Z**) < 1 for each
о	p
p > p . As each m ( M , m (Z**) = 0 for each p > p . Using condition (21)
— о	pop	— о
of (1.7-1) and a similar argument it follows that an index p^ f L exists
such that m (CZ*)	= 1 for	each p	> p,.	Choosing p	>	p ,p. we see that
p	—	1	—	о 1
m (Z**) = 0 and m	(CZ*) =	1 even	though	CZ*C 'Z**,	a	contradiction. Hence
P	P
m € M , M is closed, and therefore compact in the vague topology,
о о
Having shown that M is a compactification of T the one thing
remaining to do is to prove that each x € C (T,R) has a continuous
b
extension to M (Theorem 1.1-1 (c)) . We effect the extension of x f C, (T,R)
о	b
to M by taking
о
x(m)
xdm
(m€ M ) .
о
Clearly x is continuous on by the very definition of the vague topology,
but does x extend x? Since there is a base of zero set neighborhoods at
each t € T, we may conclude that х(т^) = Jxdrn^ = x(t) at each t f T, and it
1.7 MEASURES, gT AND ULAM CARDINALS
41
is seen that x is an extension of x. It now follows that /4 with
о
vague topology is the Stone-Cech compactification of T.
(b) Recall (Theorem 1.6-1) that a point m in M (= ST) belongs to the
о
repletion UT of T iff (1) its associated maximal ideal
M = (xfC(T,R) |m Ccl.z(x)} has the property that z(M ) is stable under
m	~ 1 p	m
countable intersections. [Or (2) the codimension of M is 1, i.e.
m
C(T,R)/M is isomorphic to R. ]
m
Prior to showing that each m ( UT is countably additive via statement
(1) above, we establish the following technical fact:
z(M ) = {z c Z|m(Z) = 1} = Z .
m	m
As Z is clearly a z-filter and z(M ) is a z-ultrafilter, it is only
m	m
necessary to show that z(M ) C Z to establish the equality of the two sets,
m m
A point in z(M ) is of the form z(x), x( C(T,R), where m € cl_(z(x)).
m	p
Thus there must be a net (t ) from z(x) such that m^ -> m. Therefore, by
p	p
(1.7-1), lim sup (z(x)) £m(z(x)). By the way the m are defined, each
P	P
m^_ (z(x)) = 1 however. Hence m(z(x)) = 1, and the desired inclusion is
P
established.
Suppose now that m € UT, so that Z is stable under countable
m
intersections, and that m is not countably additive. This being the case.
there must be a sequence (A ) of sets from A decreasing to 0 such that
n	z
each m(A ) = 1. And we may choose zero sets F C A such that each
n	n n
m(F ) = 1.
n
n
Now each of the zero sets Z = П F ( Z and ClZ = ClF cdA = 0.
n , m m	n n n
m=l
This brings us to the contradictory conclusion that m(Az ) = 0, i.e.
n
nz Z .
n m
Conversely, suppose that m с M is countably additive. To show that
о
m f UT we show that z(M ) = fZ fZ|m(Z) = 11 is stable under countable
mi'	1
intersections. To this end, let (Z^) be a countable family of sets from
z(M ). Thus, for each n ( N, m(Z ) = 1 and, since a countable intersection
m	- n
of zero sets is a zero set by (1.2-1) (e),
m(c П Z )) = m( U CZ ) < Г m(CZ ) = 0 . V
nc N П	nf N П nC N П
To view Part (b) of the above theorem
in a somewhat different light,
note that the correspondence
42
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
M = 6т -> Z
о	u
m -> z (M )
m
(see Theorem 1.3-1) between ST and the z-ultrafilters on T pairs the
elements of UT with the 6-z-'ultraf liters, those z-ultrafilters stable under
the formation of countable intersections: In the theorem it was
established that zfM^) = {z (x) |m(z (x)) =1} . Thus, by Part (b) of the
theorem, there is a 1-1 correspondence between the measures (i.e. countably
additive members) of M and the 6-z-ultrafliters.
о
In proving that UT consists of the measures in M we made no mention
о
of how an x ( C(T,R) is extended up to a continuous function on UT. We did
see, however, how to continuously extend bounded functions to ВТ: For
x с C, (T,R) , take
b
x : gT = M -+ R
о
m -> Jxdm
Thus it is not unreasonable to suspect that this same method might be used
to extend the functions of C(T,R) up to elements of C(UT,R). Indeed if
x € C(T,R), then x 1([-n,n]) is an increasing sequence of zero sets
converging to T. Hence for any 0-1 measure m, m(x ^([-n,n])) = 1 for
some n. Thus
xdm
xdm
x 1([—n,n])
as |x| is bounded by n on x ([-n,n]). To see that the real-valued
function
x • UT -> R
m -> Jxdm
is continuous on UT, let (m^) be a net from UT converging to m€ UT in the
vague topology. As x 1((-n,n)) is the complement of a zero set Z (see
—1 n
(1.2-3)) and (x ((-n,n))) increases to T, there is an index N such that
m(Z„) = 0. Thus, by (1.7-1) , lim sup m (Z„) < m(Z„) = 0 and m (Z,,) -+ 0.
N	p p N — N	pN
Consequently there must be an index Pq
Setting xN = min(x,N), it follows that
such that m (Z ) = 0 for all p > p
p N	— о
1.7 MEASURES, PT AND UIAM CARDINALS
43
xdm - xdm = x„dm - x„dm
P J J N p J N
for each p >_ p°. Hence Jxdm^ -+ Jxdm and x is seen to be continuous.
At this juncture we examine Part (b) of Theorem 1.7-1 above for the
class of discrete spaces. Our purpose is to provide a foundation for
Shirota's result presented in the next section. There it is essentially
shown that a completely regular Hausdorff space endowed with a complete
compatible uniform structure is replete if and only if each closed discrete
subspace is replete.
If T is discrete then it is evident that Z and A coincide with the
z
collection P(T) of all subsets of T. Thus a discrete space T is replete if
and only if each (countably additive) 0-1 measure on P(T) is concentrated
at a point of T.
Definition 1.7-1. ULAM CARDINALS. A (countably additive) 0-1 measure
defined on the collection of all subsets of a set T which is not
concentrated at a point of T is called an Ulam measure. Since the
existence of such a measure is clearly a property of the equipotence class
of T rather than just T, the cardinal numbers |t| for which Ulam measures
exist are called Ulam (measurable) cardinals.
In this terminology Part (b) of Theorem 1.7-1 yields the result that:
A discrete space is replete if and only if it is not of Ulam cardinal.
A complete uniform space (T, U^) remains complete when equipped with
a finer uniformity provided Uj possesses a fundamental system of entou-
rages which are closed in	(Bourbaki 1966a, p. 185, Prop. 7). Be-
cause of this and the facts that products and closed subspaces of replete
spaces are replete, we have:
(1.7-2) NON-ULAM CARDINALITY IS HEREDITARY AND PRODUCTIVE. (a) If T is
not of Ulam cardinality then neither is any subspace of T. (b) If (T ) is
a family of sets none of which is of Ulam cardinal, then ttT is not of Ulam
p,
cardinality either.
Furthermore |N| is not an Ulam cardinal. Indeed if m is a 0-1 measure
on (N) then	m({n}) so that an integer n exists for which
m({n})=l. Hence m=m , i.e. m is concentrated at n. This fact together
with (1.7-2) yields the conclusion that every cardinal less than or equal
to a cardinal of the form
2 |n |
44
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
is not an Ulam cardinal, a fact which indicates that if Ulam cardinals
exist at all, they must be very large.
1.8 Complete Implies Replete
The fact that broad classes of spaces are replete was mentioned in
Section 1.5. In Theorem l'.5-3, for example, it was shown that each
Lindelof space is replete. In the principal result of this section another
broad class of such spaces is established: each space whose cardinality is
not Ulam and whose topology is given by a complete uniform structure. For
the sake of that demonstration it is helpful to single out the following
notion of "discreteness."
Definition 1.8-1. d-DISCRETE. Let d be a pseudometric defined on the
set T. A family (T ) of subsets of T is d~discrete of gauge p > 0 if
d(T ,T,) = inf{d(s,t)|(s,t)f T xt 1 i p for all distinct pairs of indices
Л,р. A set S С T is d-discrete if ({s})gfg is d-discrete.
The basic properties of d-discreteness follow.
(1.8-1) d-DISCRETENESS. Let d be a continuous pseudometric on the
completely regular Hausdorff space T. Then
(a)	Any	d-closed set A	С	T is a zero set (viz.	the	map	t * d(A,t));
(b)	the	union of any d-discrete family	(T )	of	d-closed	(i.e.	closed
with respect to the topology induced by d) sets is d-closed,
(c)	any	d-discrete set	S	is closed;
(d)	any	d-discrete set	S	is discrete.
Proof (a) Clear.
(b)	Suppose that t is a d-adherence point of (JT^. For each e S p the
set {s€T|d(s,t) < e] meets just one of the sets T^; it follows that
t f cl.T = T .
d P P
(c)	By (b) the set S = U cl.fsl is d-closed. If t is an adherence
„ d' J
SfS
point of S it is also a d-adherence point of S so that t ( cl^fs] for some
s e S. If we assume that t s and if V is a neighborhood of t in the
Hausdorff space T excluding s, then the neighborhood (in T),
V A [re T|d(r,t) <p] fails to meet S. Thus t = sf S.
(d)	If s ( S where S is d-discrete, the neighborhood {t|d(s,t) <pl of
s meets S only in the point s. V
Theorem 1.8-1 (Shirota) COMPLETE IMPLIES REPLETE. A completely regular
Hausdorff space T is replete if and only if there exists a complete
compatible uniformity on T and no closed discrete subspace of T is of Ulam
1.8 COMPLETE IMPLIES REPLETE
45
cardinality. Thus any complete Hausdorff uniform space which is not of
Ulam cardinality is replete.
Proof From our earliest result on repleteness, Theorem 1.5-1, and the
comment right after it, we know that the initial uniformity C = C(T,R)
induced by C(T,R) on T is a compatible complete uniformity for replete T.
As to the cardinality of closed discrete subspaces of T, (1) any closed
subspace of a replete space is replete by Theorem 1.5-2, and (2) for
discrete spaces, repleteness is equivalent to not being of Ulam cardinality
by the discussion before and after Definition 1.7-1. So much for necessity.
To prove sufficiency, i.e. that the conditions imply UT С T, we choose
p ( UT and consider z(M ). Assuming U to be a complete compatible
P
uniformity on T, we show z (M ) to be a U-Cauchy filterbase. As such it
P
converges to a point of T. Generally, however, the filterbase z (M₽)
converges to p. Since Вт is Hausdorff, it follows that p c T.
We must show that z(M ) contains sets that are U-small for any U f U.
P
Since U is generated by the family of all U-uniformly continuous
pseudometrics on T, there exists a pseudometric d on T and a p > 0 such
that d(s,t) < 2p => (s,t) € U. By the well-ordering principle there is a
well-ordering on T. For each n € N we now define by transfinite
induction a d-discrete family (Z ) , of U-small zero sets as follows:
ns s€T
Z s = {tCT|d(s,t)£ p-p/n and d (Z^, t) к p/n for all r < s } .
If u is the first element of T then Z / 0 for each n € N: any other Z
nu	ns
may be empty.
The next stage of the proof consists of showing that some nonempty Z
is a (U-small) member of z(M ). To do this we introduce the sets
P
and show that some Z € z(M ) . We first contend that U Z = T.
n p	_n n
For tf T let s be the first element of the set { r f T | d (r, t) < p] . Now
choose nf N such that d(s,t) = p-p/n. If r < s then for any qf Z
d(q,t) d(r,t) - d(r,q) p- (p-p/n) = p/n
so
d(Znr,t) > p/n .
It follows that t € Z and consequently that T = U z - Next we show that
ns	n n
46
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
some Z meets each member of z(M ). If not, then for each n f N there is
n	p
an F € z(M ) such that F О Z = 0. This leads to the following
n p	n n
contradictory statements:
ПР € z(M )
n n p
but
pF = (П F ) П T = (PF ) П (U Z ) = 0
'nn nn	nn mm
Since z(M ) is a z-ultrafilter it follows that the zero set Z belongs
P	n
to z(M ) .
P
To show that Z r z(M ) for this n and some s € T, we introduce an
ns p
auxiliary set S С T to which the hypothesis may be applied. First single
out the nonempty members Z of Z ; let the family of such sets denoted by
nt n
(Z ) ..By the Axiom of Choice one element s may be selected from each
np pfM	p
Z and we may form the set S = fs IpCMl. Since (Z ) , . is d-discrete,
p	1 p1 J	np ptM
S is a closed discrete subspace of T.
The family	consists of disjoint sets so the map
f : P(S) -> Z
from the power set P(S) of S into Z, the zero sets of T, is injective.
Since z(M ) is a z-ultrafilter, it now follows that
P
f-1(z(M )) = { Acslf (A) € z(M ) }
P	P
is an ultrafilter on S. Since p € UT, z(M ) is stable under the formation
F	— j_
of countable intersections, and it follows that f (z(M )) too is stable
P
under countable intersections. As such — since |s| is not Ulam — the 0-1
measure associated with f 1(z(M )) must be concentrated at some s € S. In
P	P
other words,
fs } e f-1( z (M )) .
1 P	P
Therefore
Z - f (fs }) c z (M ) . V
np ' P	P
47
Exercises 1
1.1	Complete Regularity and Algebraic Generality
For all topological spaces S there is a completely regular Hausdorff
space T and a continuous surjection f : S -> T such that the map x -> x*f is
an isomorphism of C(T,R) onto C(S,R). To see this, first prove (a) and
(b) below.
(a)	If T is a Hausdorff space whose topology is the initial topology
T
determined by some subfamily of R , then T is completely regular.
(b)	If A is a subfamily of C(T,R) which induces T's topology, then a
map f from a topological space S into T is continuous if and only if x*f is
continuous for each x f A.
Now define s - t in S to mean that x(s) = x(t) for all x C C(T,R).
Let T be the collection of all equivalence classes determined by this
equivalence relation, and define f : S + T to be the canonical map sending
s f S into the equivalence class s determined by s. For each x € C(S,R)
consider the map
x' : T -> R
t -> x(t)
Let T carry the initial topology determined by the family A of all such
maps x'. The continuity of f now follows from (b), the complete regularity
of T from (a) . Last, show that A = C(T,R) .
1.2	Properties of gT
Let T be a completely regular Hausdorff space and gT be its Stone-Cech
compactification. Then:
(a)	If T C S C gT then Вт = gS.
(b)	If S is a clopen (i.e. closed and open) subset of T then cloS and
P
clo(T-S) are complementary open subsets of gT.
В
(c)	An isolated point of T is isolated in gT.
(d)	T is open in gT if and only if T is locally compact.
1.3	The Stone-Cech Compactification of N
With N denoting the discrete space of natural numbers and gN its
Stone-Cech compactification we have:
(b)	If M C N then M is open in gN.
(c)	, If E denotes the even positive integers, then gE, g(N-E) and gN
are all homeomorphic.
48
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
(d)	There is a homeomorphism h : gN -> 0N such that h(N) = N, h is its
own inverse and h(p) / p for each p € 0N - N.
1.4	Zero Sets
If I is any ideal of C(T,R) then z(I) = {z(x) |x? l] is a filterbase.
For any point p f ST and = {x € С (T, R) | p € cl^z (x) ] as in the Gelfand-
Kolmogorov theorem (Th. 1.4-1), show that z (M ) -> p.
P
1.5	Rings of Integer-Valued Continuous Functions (Pierce 1961)
Here T is any topological space, Z the discrete ring of integers, and
C(T,Z) and C, (T,Z) the rings of continuous and bounded continuous maps from
~	b
T into Z respectively. Some of the basic properties of C(T,Z) and C, (T,Z)
-	-	b
are set forth below.
(a)	Adjoints. Let S be a topological space and f : T -> S a continuous
map. The map
f' : C(S,Z) -> C(T,Z)
x -> x*f
is the adjoint of f. The map f' is (i) a ring homomorphism, (ii) maps
C, (S,Z) into C. (T,Z), and (iii) is injective if and only if S contains no
b	b
nonempty clopen set disjoint from f(T).
(b)	Analog of 6т. There is a continuous map f of T into a compact
totally disconnected (the component of any point is the point itself)
Hausdorff space 6т such that f' is an isomorphism of C, (0T,Z) onto C, (T,Z) ;
b	b
moreover, up to homeomorphism, 6т is the only compact totally disconnected
space for which C, (T,Z) is isomorphic to C, (0T,Z) . [Hint: Given a subset
b	b
S of C(T,Z), let ₽s denote the topological product of the discrete spaces
x(T), x £ S. Let f be the map sending t £ T into (x(t))	€ P and note
S	X( S о
that f is continuous. Let S = C, (T,Z) and define 6т to be the closure
S	b
of fg(T) in Pg. Since x(T) is finite for each x € S, P$ must be compact.
As for the uniqueness, let B(T) and B(6T) denote the Boolean algebras 6f
clopen subsets of T and 6т respectively. Since the isomorphism f' maps
idempotents into idempotents, the Boolean algebras B(T) and B(6t) are
isomorphic. By a result of Stone (1937), since 6т is compact and totally
disconnected, 6т must be homeomorphic to the Boolean space of B(6t).]
(с)	T in 6т. The natural injection of T into <5T is 1-1 if and
only if the clopen subsets of T separate points. It is bicontinuous if and
only if T is a O-dimensional T space.
EXERCISES 1
49
(d)	ST Versus бт. If T is a completely regular Hausdorff space then
ST is homeomorphic to St if and only if each pair of subsets of T which are
separated by C(T,R) are separated by a clopen subset of T.
1.6	Stone's Theorems Rings A for Which C, (T,A)/M = A
b
If T is a completely regular Hausdorff space and M a maximal ideal in
c, (T,R), then C, (T,R)/M is isomorphic to R as follows readily from the
results of Section 1.4. Due to its appearance in Stone 1937 (Theorem 76),
we refer to this result as Stone's Theorem. If R is replaced by the
complex numbers or the quaternions, Stone's theorem still holds, as
remarked in Section 1.4, but what if R is replaced by a topological
division ring A? I.e. after introducing a notion of boundedness in A, does
Stone's theorem still hold in C, (T,A)? The answer is no and necessary and
b
sufficient conditions on A (essentially local compactness) for Stone's
theorem to hold are developed in (b) below.
(a)	Boundedness. Analogous to the way boundedness is defined in
topological vector spaces, a subset S of a topological ring A is bounded
if for any neighborhood W of 0 in A there are neighborhoods U and V of 0
such that US c W and SV c W. If A is a metric ring, metric boundedness
implies this notion of boundedness.
If A is a valued field, the two
notions coincide. The closure of a bounded set is bounded, as is any
convergent sequence bounded. If x -> 0 and (y ) is bounded, then
n	n
lim x у = lim у x =0. The collection of all continuous maps from a
n n n n n n
topological space T into A whose range is a bounded subset of A is denoted
by C (T,A)-
b
0 (i.e. for
Division rings A where subsets H which are bounded away from
some neighborhood V of 0, HOV = 0) must be such that H is
bounded are said to be of type V.
(b)	Stone Rings. (Kowalsky 1955, Theorems 4 and 5) A topological
division ring К is a Stone ring if, given any topological space T,
C (T,K)/M is isomorphic to К for any maximal ideal M of C, (T,K) . A Stone
b	b
ring of type V must be locally compact, and any locally compact topological
division ring is a Stone ring. Other approaches to this result appear in
Goldhaber and Wolk 1954 and Correl and Henriksen 1956; among other things
the latter paper makes some corrections to Goldhaber and Wolk's results.
The question of which rings are Stone rings was first raised in Kaplansky
1947, page 183, this paper also deals with other results about rings of
ring-valued functions.
50
1- ALGEBRAS OF CONTINUOUS FUNCTIONS
(c)	Functions with Relatively Compact Range. (Correl and Henriksen
1956) Let T be a completely regular Hausdorff space, let К be a totally
disconnected topological division ring, and let C (T,K) denote the
c
collection of all functions in C(T,K) with relatively compact range in K.
Then, for any maximal ideal M of С (T,K), C (T,K)/M is isomorphic to K.
c	c
(d)	Integer-Valued Continuous Functions. (Pierce 1961, page 381,
Cor. 3.2.4) Let T be any topological space, Z the discrete space of
integers and C(T,Z) the ring of continuous maps from T into Z. For any
maximal ideal M in C(T,Z), C(T,Z) is isomorphic to the integers modulo p
for some prime p.
1.7	A Construction of ВТ and UT Using Extensions (Chandler 1972)
The spaces Вт and UT are C, - and C-extensions of the completely
b
regular Hausdorff space T (i.e. T is dense in ВТ and UT and functions from
C, (T,R) and C(T,R) can be
b
Moreover, as will be seen
continuously extended to ВТ and UT respectively).
from what follows, Вт and UT are the "largest"
C, - and C-extensions of T
b
disjoint union of all the
in the sense that each is a quotient of the
C, - or C-extensions of T.
b
Since each C, - or C-extension of T has cardinality smaller than or
I । b
2 1 1
equal to 2	, each such space can be viewed as a subspace of the power
2
set P (T) = P(P(T)) with an appropriate Hausdorff topology. We obtain ВТ
and VT by taking the disjoint union of all C, - and C-extensions,
2	b
respectively, of T from P (T) and then (to insure that the resulting spaces
will be Hausdorff, among other things) identifying points for which all the
extensions agree: specifically, for C-extensions of T, if Y is the
union of all such S , s, t are elements of Y, define s - t if xa(s) = x^(t)
for each x in C(T,R) where x and xP are the appropriate continuous
extensions of x; then take UT to be the set of all such equivalence
classes. For ВТ use the same equivalence relation on the class of all
C, -extensions. To each x € C(T,R) there corresponds a unique x° : VT R
b	и _
defined by x (t ) = x (t ). Let UT carry the initial topology determined
и “	“
by all the x ; ВТ is topologized similarly.
(a)	ВТ and UT are C - and C-extensions of T respectively,
b
(b)	ВТ and UT are completely regular Hausdorff spaces.
(с)	ВТ is compact. (It suffices to show that each maximal ideal of
C, (T,R) is fixed at a point of ВТ; cf. Theorem 1.4-1.)
b
(d)	UT is replete. (Assume the existence of a proper C-extension
of T.)
EXERCISES 1
51
1.8	Compactifications and Semicontinuous Functions
(Nielsen and Sloyer 1970)
As shown in Section 1.4 the Stone-Cech compactification of a
completely regular Hausdorff space T is given by the collection of all
maximal ideals of C(T,R). For T^-spaces T a similar sort of thing can be
done — i.e. a compactification pT can be obtained — using semicontinuous
functions, rather than continuous ones. A lower semicontinuous function
x : T + R is one for which x 1(a,“>) is open for each a € R. The collection
L(T) of nonnegative lower semicontinuous functions with pointwise
operations is not algebra however. Additively and multiplicatively it is a
semigroup with identity) it is closed under multiplication by nonnegative
scalars.
(a)	Ideals and Filters. A subset I of L (T) is an ideal if 1+ I С I,
L(T)I С I, and for each x € I there is an open set U such that k^x = x
where denotes the characteristic function of U. Letting с(T) denote the
closed subsets of T, filters from с(T) are called c-fliters. Under these
conventions the map x -> z (x) maps ideals of L(T) into c-fliters.
(b)	Fixed Ideals. An ideal I in L(T) is fixed if Oz(I)	0)
otherwise I is free. The fixed maximal ideals of L(T) are precisely the
sets I = {x€L(T) |x(t) =o}. If T is compact, then every ideal is fixed.
(c)	The Stone Topology. A base of open sets for the Stone topology
T on the space M(T) of maximal ideals of L(T) is given by sets of the form
U (x) = {мсМ(Т)|х f mJ as x runs through L(T).	(M(T),T) is homeomorphic to
T if T is compact; consequently, for compact T^-spaces S and T, L(T) is
isomorphic to L(S) if and only if S and T are homeomorphic. (The map
t -> I takes T injectively — since T is T^ — into M(T) .)
(d)	Compactification. The map u sending t into I embeds T
homeomorphically as a dense subset of M(T). M(T) moreover is compact in
its Stone topology.
1.9	Generalized Stone-Cech Compactification (Bachman, Beckenstein,
Narici, and Warner 1975; Narici, Beckenstein, and Bachman 1971)
Let S be a Hausdorff space containing at least two points. For any
topological space T, let Cc(T,S) denote the set of all continuous maps of
T into S with relatively compact range. (If S is a locally compact field,
C (T,S) is just the set of all bounded continuous functions.) A Hausdorff
c
space T is S-completely regular if the initial topology determined by
C (T,S) on T is T's original topology. Thus in this terminology the
c
completely regular Hausdorff spaces are precisely the R-completely regular
52
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
spaces. The Hausdorff space T is ultraregular (or zero-dimensional) if
there is a neighborhood base at each point consisting of clopen sets. (Any
ultrametric space — i.e. any metric space in which d(a,b) <
max (d (a, c) ,d (c ,b) ) holds instead of the triangle inequality — is certainly
ultraregular.)
(a)	For any ultraregular Hausdorff space S containing at least two
points, T is ultraregular if and only if T is S-completely regular.
(b)	If T is S-completely regular then the initial uniformity C (T,S)
c
determined by Cc(T,S) on T is compatible with T's original topology. (The
uniform spaces into which the functions x € C (T,S) map T are the compact
		c
spaces x(T).) The C (T,S)-completion В T of T is a compactification of T
c	S
and each x € C (T,S) may be continuously extended to ST, the range of the
c ___________________ S
extension being x(T). Moreover each continuous function from T into a
compact S-completely regular space W has a continuous extension to BgT.
(с)	T is S-completely regular if and only if T is homeomorphic to a
A
relatively compact subset of the cartesian product S for some index set A.
Consequently cartesian products of S-completely regular spaces are
S-completely regular.
(d)	Let G be the algebra of clopen subsets of the ultraregular space
T and let p be a finitely additive measure mapping G into {0,1}, addition
being performed modulo 2 in {o,l|. In addition assume that if'S,S’ € G,
S э S’ and p(S) = 0, then p (S’) = 0. (This last property would
automatically be fulfilled if addition were not performed modulo 2 — if,
say, {o,l^ were a subset of a field whose characteristic was not 2.) Let
M (T) (or just M ) denote the collection of all such 0-1 "measures"
о	о
endowed with the topology generated by the sets V(p ,S,,...,S ) =
о 1 n
{p € M p (S .) = p (S .) , j = 1, . . . , n}. M (T) is also a compactification of T
1 О 1 О J	J	J о
and, for T and F ultraregular, B'T is homeomorphic to M (T).
F	o
(e)	If T is an ultraregular space and F an ultraregular topological
field, C (T,F) is an F-algebra with respect to the usual pointwise
c
operations. Characters of F-algebras are nontrivial homomorphisms of the
algebra into F. Let H (T,F) denote the characters of C (T,F) endowed with
c	c
the weakest topology with respect to which each of the maps h * h(x) ,
h € H (T,F), x б C (T,F), is continuous. Show that H (T,F) is homeomorphic
c	c	c
to B_T. (Cf. (1.6-2) .)
F
(f)	A clopen partition of a space T is a pairwise disjoint cover of
consisting of clopen sets. Letting U denote the collection of all finite
EXERCISES 1
53
clopen partitions of the ultraregular space T, the sets U(th xU.) , (U.) € U,
form a base of entourages for a compatible uniformity for T. The
completion BQT of T with respect to this uniform structure is a
compactification of T which coincides (is homeomorphic) with BgT whenever S
is ultraregular. Thus it is superfluous to write BgT in the class of
ultraregular spaces S.
(g)	A Hausdorff space T is ultranormal if, given any two disjoint
closed subsets C and D, there is a clopen set U containing C and disjoint
from D. If T is ultranormal, S is any Hausdorff space containing at least
two points, then T is S-completely regular and & ,(T',S)-(J (T,R). Con-
sequently, for ultranormal T, 5T=|3gT for any Hausdorff space S containing
at least two points.
1.10	The Banaschewski Compactification (Banaschewski 1955).
Let T and F be ultraregular spaces. For each cover U of T let
W(U) = Uxu. The sets {w(U)|U is a finite open cover of tJ form a
fundamental system of entourages for a compatible uniformity В(T) on T.
B(T) is precompact since each finite open cover of T consists of W(U)-small
sets, so the В(T)-completion В T of T is a compactification of T. We refer
b
to B^T as the Banaschewski compactification of T. If T is ultranormal
then C (T,R) = B(T) (where C (T,R) is the initial uniformity determined by
c	c
the class C (T,R) of continuous maps from T into R having relatively
c
compact range in R) so ВТ = B^T. (In an ultranormal space every
open cover is refined by a finite open cover.)
For any ultraregular space F, C (T,F) = В(T) so, with B„T as in
c	F
Exercise 1.9, B„T = B,T for any ultraregular space F containing at least
F b
two points. Moreover, as noted in Exercise 1.9 (f), this is another way to
see that the "F" in B_T is superfluous in the class of ultraregular
F
spaces F.
Another way to get В, T is as the C (T,Z)-completion of T where Z
be-	-
denotes the discrete space of integers (Pierce 1961). Still another way is
as the "E-compactification of T" (Exercise 1.11) where E is a two-point set.
1.11 E-Complete Regularity and E-Compactness (Mrowka 1956 and
Engelking and Mrowka 1958)
Let T and E be Hausdorff space, let С(Т,ЕП) denote the class of all
continuous maps from T into the n-fold cartesian product En, and let
(T,E) = (J C(T,En). The space T is E-completely regular if for any closed
nt N
54
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
subset A of T and point t 4 A there is some x f (T,E) such that
x (t) i. cl x (A) .
(a)	E-Complete Regularity. (Mrowka 1956) E-complete regularity is
hereditary and productive; T is E completely regular if and only if T can
В
be embedded in some cartesian product E for some index set B.
(b)	E-Compactness. (Engelking and Mrowka 1958) An E-completely
regular space T is E-compact if there is no E-completely regular space S
which contains T as a dense subspace and is such that each x € C(T,E) may
be continuously extended to some x 6 C(S,E). T is E-compact if and only if
T is homeomorphic to a closed subspace of a cartesian product EB for some
index set B. Thus closed subspaces and cartesian products of E-compact
spaces are E-compact.
(i)	With the interval [0,1] carrying its usual topology, a space T is
[0,1]-compact if and only if T is compact.
(ii)	T is R-compact if and only if T is replete.
(iii)	T is |0,lj-compact if and only if T is 0-dimensional (in the
sense that there is a base of clopen sets for the topology on T)
and compact.
(iv)	With N carrying the discrete topology, T is N-compact if and only
if T is 0-dimensional and replete.
(c)	E-Compactification. (Engelking and Mrowka 1958, Theorem 4) If T
is E-completely regular then there is an E-completely regular extension в T
E
of T such that (i) S T is E-compact and contains T as a dense subset and
E
(ii) each x f C(T,E) possesses an extension to some x t C(BeT,E) .
If S is any E-compact space, then each x f C(T,S) possesses an
extension x € C(B T,S) .
E
The extension 6 T is uniquely determined by (i) and (ii) in the sense
E
that if X is any extension of T satisfying (i) and (ii) then there is a
homeomorphism h of ВТ onto X such that h(t) = t for each t f T.
E
1.12	Repleteness and the Repletion
Throughout this exercise T denotes a completely regular Hausdorff
space.
(a)	Properties of UT. (i) If X is replete then each continuous
function x:T -> X has a continuous extension to UT. (ii) If a sequence (Z )
n
of zero sets of T has empty intersection then so does (cl^Z ) in UT.
(iii) If (Z ) is a sequence of zero sets in T then cl (f) Z ) =Q cl (Z ) .
n	и n n n и n
EXERCISES 1
55
If X is a replete space containing a dense homeomorphic image of T and
satisfying one of the properties given above, then there is a homeomorphism
h taking X onto T which reduces to the identity on T.
(b)	Intersections of Replete Spaces. An arbitrary intersection of
replete subspaces of a given space is replete.
(c)	Continuous Inverse Images of Replete Spaces. If f is a
continuous function taking the replete space T into the completely regular
Hausdorff space X and F is a replete subspace of X then f 1(F) is replete.
(Note that not every open subset of a replete space is replete as evidenced
by (0,П) in [0,Q] with the order topology.) Furthermore if every subspace
of a replete space is replete, then all the one-point subsets of the space
are G.-sets (cf. (g) ) .
о
(d)	Unions of Replete Spaces. In any completely regular space, the
union of a replete subspace with a compact subspace is replete. On the
other hand, it is not necessarily the case that the union of two replete
subspaces is replete (Mrowka 1958) .
(e)	Ultrafilters and Replete Spaces. T is replete if and only if
each ultrafilter F on T for which F Г) 1 (Z denotes the zero sets of T) has
the countable intersection property converges to a point of T.
(f)	Continuous Functions with Compact Support on a Replete Space.
Let C (T,R) denote the collection of all continuous functions on T which
К ~
vanish outside some compact subset of T, the subset depending on the
particular function. If T is replete then C (T,R) = Г~| M where M is
K ~ p«8T-T p	P
as in the Gelfand-Kolmogorov theorem (Th. 1.4-1).
(g)	When are all Subspaces Replete? Each subspace of T is replete if
and only if (i) given a completely regular Hausdorff space X for which a
continuous map f:X -> T exists such that f '*'(^t}) is compact for each t in T
then X must be replete or (ii) each 1-1 continuous image of T is replete or
(iii) T-{ tj is replete for each t in T.
1.13	A Generalized Repletion (Bachman, Beckenstein, Narici and
Warner 1975)
Let F denote a complete Hausdorff space containing at least two points
and let T be an F-completely regular topological space as defined in the
preceding exercise. Let C(T,F) denote the initial uniformity determined by
the continuous functions C(T,F) on T. The completion и T of T with respect
F
to C(T,F) is called the F-repletion of T and has a number of properties in
common with the repletion UT of a completely regular (or R-completely
56
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
regular in this context) Hausdorff space T. T is F-replete if T = и T.
 	F
For example, и T is the largest subspace of B_T (see preceding exercise) to
F	В
which each x € C(T,F) may be continuously extended and this property
characterizes T up to homeomorphism in the following sense: If T is a
dense subset of an F-completely regular space S such that each x e C(T,F)
may be continuously extended to S, then there is a unique homeomorphism h
mapping и T onto и S such that h(t) = t for all t 4 T. In fact и T also
F	F	F
admits the following characterization.
(a)	According to part (b) of the preceding exercise, each x 4 C(T,F)
has a unique continuous extension to x e C(B_T,B_F). In this notation
F F
U T = < t e В„тI x(t)4 F for all x 4 C(T,F)[ .
F i F	i
(b)	If F is F-completely regular, then F is F-replete. Thus R is
R-replete or, simply, replete.
(c)	If T is F-completely regular then T is F-replete if and only if T
g
is homeomorphic to a closed subset of the cartesian product F for some
index set B. Consequently cartesian products and closed subspaces of
F-replete spaces are F-replete.
(d)	If F is replete then every F-replete space T is replete.
(e)	In analogy with the characterization of UT given in
Theorem 1.5-1 (b), referring to the space В T defined in part (f) of
о
Exercise 1.9 , we single out the subspace и T of В T when T is ultraregular
° °
consisting of those points t 4 В T such that for each sequence (V ) of
о	n
neighborhoods of t in В T, (Qv ) A T 0. If T is ultranormal then
о	n
u°T = u^T. Otherwise we have (f) and (g) below.
(f)	If T is ultraregular and F is replete and ultranormal then
и T = и T.
о F
(g)	If the idempotent entourages (V such that V“ V = V) form a base
for F's uniformity — as happens for example when F is ultrametrizable — and
F is not compact, then u^TCl UqT.
(h)	If T is an ultraregular Lindelof space then T = и T (cf. the
о
analogous result for UT in Th. 1.5-3). If the idempotent entourages form a
base for F's uniformity and F is not compact, then every ultraregular
Lindelof space T is F-replete, i.e. T = и T.
F
(i)	With reference to part (d) of the preceding exercise, for T
ultraregular and F an ultraregular topological field, и T is homeomorphic
о
to the subcollection M (T) of countably additive members of M (T).
о	о
EXERCISES 1
57
(j)	Let F denote a complete ultraregular topological field and let T
be ultraregular, and let x denote the continuous extension of x « C(T,F) to
и T. For each p t и T, let p denote the map
F	F
p : C (T,F) -> F
x -> x(p)
The map p + p is a homeomorphism from и T onto the space H(T,F) of all
F
characters (nontrivial homomorphisms into F) of C(T,F) endowed with the
weakest topology with respect to which the maps
H(T,F) -> F
h -> h(x)
are continuous for each x € C(T,F). (Cf. (1.6-2).)
(k)	If F is a complete nonarchimedean valued field with nontrivial
valuation, then F is replete if and only if F has non-Ulam cardinality.
1.14 Pseudocompactness
In any topological space T, the following conditions are equivalent:
(a)	T is pseudocompact.
(b)	The continuous image of T, in any topological space S, is
pseudocompact.
(c)	The continuous image of T in any T^ space (sets A and В such that
neither meets the other's closure may be separated by open sets) is
countably compact.
(d)	The continuous image of T in any metric space is compact.
(e)	Every real-valued continuous function x on T attains the values
inf x(T) and sup x (T) .
(f)	Every countable collection of nonempty zero sets in T has
nonempty intersection.
(g)	Every locally finite collection U of complements of zero sets of
T — i.e. such that each point of T has a neighborhood which meets only
finitely many members of U — is finite.
In the class of weakly normal Hausdorff spaces — Hausdorff spaces in
which closed sets disjoint from countable closed sets may be separated by
open sets — pseudocompactness and countable compactness coincide. In the
class of completely regular Hausdorff spaces, pseudocompactness is
equivalent to the uniform closure of each ideal of C(T,R) again being an
ideal. (Nanzetta and Plank 1972)
58
1. ALGEBRAS OF CONTINUOUS FUNCTIONS
The product of pseudocompact spaces need not be pseudocompact. In
fact (Comfort 1967) if (T^) is a countable family of completely regular
spaces,Пт is not necessarily pseudocompact even tho the product of any
n
finite number of them is. Under additional hypotheses such as sequential
compactness for example on the factors, pseudocompactness of product can
be guaranteed.
Clearly not every subspace of a pseudocompact space need be
pseudocompact (consider (0,1) in [0,1] for example). If T is a completely
regular pseudocompact space, then the closure of every open subset of T is
pseudocompact. Cf. also Stephensen 1968, Glicksberg 1959, and Bagley,
Connell and McKnight 1958.
1.15 Products
Let S and T be infinite completely regular Hausdorff spaces.
(a)	Stone-Cech Compactifications. В (Sxt) = BSXBT if and only if Sxt
is pseudocompact (first proved in Glicksberg 1959, reproyed in Frolik 1960).
(b)	Repletions. So far it is known (Comfort 1968) that the following
conditions suffice for u(Sxt) to = uSxut.
(i)	SXT is not of Ulam cardinal, T is a к-space (each subset of T
which has a closed intersection with each compact set must itself
be closed (Section 2.3)), US is locally compact.
(ii)	SXT is not of Ulam cardinal, T is a к-space and S is
pseudocompact.
(iii)	Sxt is not of Ulam cardinal and uSxt and usxut are k-spaces.
(iv)	Each compact subset of T and each pseudocompact subset of US is
not of Ulam cardinal, T is a к-space and US is locally compact.
1.16 Connectedness and the Stone-Cech Compactification
Let T be a completely regular Hausdorff space.
(a)	An open subset U of ВТ is connected if and only if U О T is
connected. (Easy to show U О T to be disconnected if U is. If U О T is
disconnected there are nonempty disjoint open subsets V and W of T such
that
then
If p
U Г1 T = V U W.
Now cl.(UpT) = cl.V (Jcl.W DU and if cl V О cl_W = 0
В	В В	BP
= (cl.vPu) (J (cl WOU) is disconnected and the proof is complete.
В	В
cl.V IP cl W, choose x e C(BT,R) such that x(p) = 0 and x(BT-U) = 1.
В В
Define у on T by taking x(t) = y(t) except for t « W and y(t) < 1/2; let
y(t) = 1/2 otherwise. The continuous extension
6	6
coincides with x on cl_V so у (p) = 0, yet у
p
yB of у 4 Cb(T,R) to Вт
1/2 on cl W. This
p
U
c
contradiction shows U to be disconnected.)
EXERCISES 1
59
(b)	At each t 4 T, ST is locally connected at t if and only if T is
locally connected at t. (Use (a) and the fact that a family (V ) of open
neighborhoods of t e T in ST is a base of open neighborhoods at t in ST if
and only if (V^O T) is a base of open neighborhoods of t in T.)
The next result serves as a lemma to (d).
(c)	UT is not locally connected at any point not in UT. (For
p « ST - UT there exists x 4 C(T,R) such that x(p) = “ where x denotes the
continuous extension of x to C(ST,RU{°°}) mentioned in (1.5-11 For
i = 0,1,2,3, let Z. be the set of all t 4 T such that n < x(t} < n+1 for
i	—	-~
some integer n Hi (mod 4). The sets Z^ cover T so p is in one of their
closures in ST, p 4 cl-Z. say. Now p / cloZ_ since Zq and Z, are
pl	p J	1	J
completely separated, so there must be a neighborhood U of p disjoint from
Zj. If ST is locally connected at p,
neighborhood U' of p. By (a), U' О T
integer n such that 4n < x(t) < 4n+ 3
fact that x(p) = “.)
(d) ST is not locally connected
then U contains a connected open
is connected, so there must be some
for all t « U' От, contradicting the
unless T is locally connected and
pseudocompact. (Recall that T is pseudocompact if and only if ST = UT
((1.5-4)).)
Rather than give hints to the remaining parts, we simply list the
results and refer to Henriksen and Isbell 1957 for proofs.
(e)	Let uT denote the completion of T with respect to the finest
compatible uniform structure on T. Clearly uT C UT. T is locally
connected if and only if uT is locally connected.
(f)	ST is locally connected if and only if T is locally connected and
pseudocompact.
(g)	ST is locally connected if and only if every (completely regular
Hausdorff) space S containing T as a dense subspace is locally connected.

TWO
Topological Vector Spaces of Continuous Functions
IN CHAPTER ONE exclusive attention was devoted to the algebraic prop-
erties of the algebra C(T,R) of real-valued continuous functions on a
completely regular Hausdorff space T, with special focus directed at how
these algebraic properties were the progeny of topological properties of
T. Here we switch to F-valued functions on T, F=R or C, and endow C(T,F)
with the compact-open topology, the weakest topology for C(T,F) with
respect to which each of the seminorms p^(x)=sup |x (К) |, К a compact subset
of T, is continuous. C(T,F) so topologized is denoted by C(T,I?,c) and is
a locally convex Hausdorff topological vector space; some brief discussion
of its most fundamental properties is given in Chapter 0. The point of
this chapter is the development of the interaction between topological
properties of T and topological vector space properties of C(T,F,c).
Since this is the compact-open topology, it is to be expected that
the structure and abundance of the compact subsets of T will play a central
role in determining certain properties of the topological vector space
C(T,B^,c). This is very much the case in characterizing completeness and
metrizability of C(T,F,c), as shown in Secs. 1.1 and 1.2 where necessary
and sufficient conditions on T for C(T,F,c) to be metrizable and complete,
respectively, are obtained. Tho the structure of the compact subsets
of T figures less prominently in subsequent results, they still play a
significant role.
A way of representing the elements of the continuous dual C(T,F,c)'
of C(T,F,c) as integrals with respect to certain Borel measures is obtained
in Sec. 2.4. The representation is used in developing the idea of support
of a continuous linear functional on C(T,F,c) — a compact subset К of T
related to the continuous linear functional x' in such a way that if the
continuous function x vanishes on K, x' must vanish on x— which is used
to develop necessary and sufficient conditions on T for C(T,F,c) to be
barreled.
Among the other results obtained are necessary and sufficient condi-
tions on T for C(T,F,c) to be infrabarreled, bornological, and separable.
These characterizations enable us to give an example (Example 2.6-1) of a
barreled space which is not bornological. In Sec. 2.8 such things as when
C(T,F,c) is Montel or reflexive are considered. Both of these in particu-
61
62
2. SPACES OF CONTINUOUS FUNCTIONS
lar have a very simple answer: T must be discrete.
2.1	Metrizability of C(T,F,c) and Hemicompactness
Metrizability and completeness of C(T,F,c), the algebra of continuous
functions mapping the completely regular Hausdorff space T into j? (=R or C)
with compact-open topology, naturally enough have to do with the abundance
and structure of the compact sets in T. If T is compact, for example,
C(T,I?,c) is complete and metrizable: it is a Banach algebra in this case.
More generally completeness and metrizability of C(T,F^,c) are equivalent
to T being a к -space and T being hemicompact respectively.
R
Definition 2.1-1. HEMICOMPACTNESS. A topological space T is hemicompact
if there is a countable family (K^) of compact subsets of T such that each
compact subset of T is contained in some К .
Since one-point sets are always compact, the sets (K^) must cover T
if T is hemicompact.
Hemicompactness is clearly stronger than а-compactness which merely
requires that the space be a countable union of compact sets. In the
presence of local compactness however, а-compactness is enough to guarantee
hemicompactness. Indeed suppose that T=UnKn> where each is compact.
By covering with relatively compact open sets and using the compactness
of , a relatively compact open set U^is seen to exist which contains K^.
Similarly a relatively compact exists containing cl U^U^- Thus, by
induction, a countable family (U ) of relatively compact open sets exists
such that cl U , C U for each n and T=U U  Now if К is an arbitrary
n-1 n	n n
compact set, there is a finite collection U.,...,U. which covers К and it
к	Ik
follows that КС U . ,cl U.=cl U, . Since T= U cl U , T is hemicompact.
1=1 i к	n n
Thus a fortiori any locally compact second countable space is hemicompact.
Lest it be thought that local compactness is unduly strong for the
purpose of forcing a а-compact space to be hemicompact, we note that in a
first countable Hausdorff space, hemicompactness implies local compactness
as we now show. To this end let (K^) be an increasing cover of compact
subsets of T such that each compact subset of T is a subset of some K^ and
let (V ) be a decreasing base of open neighborhoods of a point t f T. If
no cl V is compact, then V Ф К for any n. For t f V -K , then t -»t
n	nn	J	nnn	n
and { t^Jljft] is compact. Thus { t^} (J {t} C K^ for some m which contradicts
the way in which t was chosen. Therefore T is locally compact,
m
The space C) of rational numbers with Euclidean topology is clearly not
locally compact. As Q is metrizable, it is first countable; thus, as a
2.1 METRIZABILITY AND HEMICOMPACTNESS
63
result of what we have just shown, it is not hemicompact. Hence g is a
c-compact space which is not hemicompact. Some of these conclusions are
listed in (2.1-1) below.
(2.1-1) LOCAL COMPACTNESS AND HEMICOMPACTNESS (a) A locally compact
ст-compact space is hemicompact. (b) A hemicompact first countable Hausdorff
space is locally compact.
Theorem 2.1-1. METRIZABILITY OF C(T,F,c). C(T,F,c) is metrizable iff the
completely regular Hausdorff space T is hemicompact.
Proof Suppose that T is hemicompact and let (K^) be a family of compact
subsets of T such that each compact subset К of T is a subset of one of
them. For p (x)=sup|x(K) I , x C C(T,F,), then p (x) < p (x) for some n.
К	I I	~	к
It follows that the compact-open topology is generated by the countable
family (p ) of seminorms and is therefore metrizable.
^n
Conversely if C(T,F,c) is metrizable, it must have a coilntable neigh-
borhood base at 0 of the form a V =a fx
n p^ nl
is a sequence of positive numbers and the
Thus for each compact set К of T there is
f C(T,F)|p (x) < 1} where (an)
Kn
K^ are compact subsets of T.
an integer n such that
a V С V • Now suppose
n PKn PK
Because T is a completely
that K<£k , i.e. that there is a t e K-K .
n	n
regular Hausdorff space, there is an x ( C(T,F)
such that x(t)-l and x(Kn)={0}.
contradiction implies that КС
Hence x с a V while x { V  This
n 4 Pk
and T is seen to be hemicompact.v
2.2	Completeness of C(T,F,c) and k -Spaces
— *'		•	--— 1 R----	" —
In this section we show that C(T,F,c) is complete iff T is a "k -space!'
R
Other facts about k -spaces in general, as well as the related notion of
R
"k-space", are discussed in the next section.
For C(T,F,c) to be complete, each Cauchy filterbase must converge.
A filterbase ® of functions is Cauchy in C(T,F,c) iff =Гв1„|в С "I,
where Bl„={xl |x fВ }, is a Cauchy filterbase in C(K,F,c) for each compact
subset К of T. By the completeness of C(K,j?,c) then (B is seen to converge
to a continuous function on each compact subset of T. Certainly then if T
is a topological space with the property that an F-valued function is con-
tinuous whenever its restriction to each compact subset is continuous, then
<£> will converge to a function continuous on all of T and C(T,F,c) will be
complete. Completely regular Hausdorff spaces on which continuity on com-
pact sets implies continuity are called k -spaces so that a sufficient con-
dition for C(T,F,c) to be complete is that T be a k-space. As it happens
64
2. SPACES OF CONTINUOUS FUNCTIONS
this condition is
For ECT, p
h
PE(x)=sup |x(E) |;
also necessary, a fact which we prove in our next result,
denotes the seminorm defined at each x f C(T,F) by
Vp ={x f C(T,F)|pE(x) < 1] and VpE=fx c C(T,F)|PE(x)<l)-
Theorem 2.2-1. COMPLETENESS AND к -SPACES  Let T be a completely regular
R
Hausdorff space. Then the following statements are equivalent:
(a)	C(T,F,c) is complete.
(b)	C(T,F,c) is quasi-complete (i.e. every closed and bounded subset
of C(T,F,c) is complete).
(с)	T is a к -space.
К
Proof. The implication (c) -> (a) has already been established; (a) ->
(b) is obvious.
(b) -♦ (c) . Suppose that C(T,F,c) is quasi-complete and let x be a
bounded F-valued function on T with continuous restriction to each compact
subset. It remains to be shown that x is continuous.
Since @T is compact
to a continuous F-valued
each compact subset К of
we obtain a net (x ) in p (x)V
К	1
of T, directed by taking K<K'
and Hausdorff - hence normal - x| can be
IК
function у on pT such that p (у )= p (x I ) for
К	pl ik К 1 lx
T. Letting x denote the restriction of у to T,
К	К
extended
indexed by the family of compact subsets К
i¥f KCK'. It is easy to see that (x ) is
К
Cauchy in C(T,F,c) and pointwise convergent to x.
Thus by the complete-
ness of the closed and bounded set p^(x)Vp , x
that x is an unbounded real-valued function on
C C(T,F). Next suppose
T with continuous restric-
tion to each compact set K.
x (t)
n
For each integer n > 0 we define
Гx(t) if |x(t) | < n
S n if x(t) > n
L-n if x(t) < -n
The restriction of x^ to К is certainly continuous for each compact К so,
by the previous argument, x^ f C(T,.F). Clearly (x ) converges pointwise
to x on T. Moreover if К is compact then, as x is continuous, and there-
fore bounded, on K, x I =x|„ for all sufficiently large n. Thus x x
n IK IK	j-ь	n
uniformly on each compact set and (x ) is Cauchy in C(T,j?,c). Letting В
denote the closure of (xn|n f n] in C(T,j?,c), it follows by the quasi-
completeness of C(T,F,c) that В is complete; thus (x^) has a limit у in
C(T,.F,c). It follows that x -> у pointwise and x=y C BCC(T,£). In the
event that x is an unbounded complex-valued function on T with continuous
restriction to each compact set, the above argument may be applied to the
2.3 k-SPACES
65
real and imaginary parts of x. V
2.3 k-Spaces, k„-Spaces and Pseudofinite Spaces
R
What kinds of spaces are k-spaces? If sequential continuity implies
R
continuity, for example, then the space is a к-space, for then continuity
R
on compact sets yields continuity on the compact set {t^jUftjfor any t^-" t.
Thus any first countable space (completely regular and Hausdorff) is a
k -space.
R
Another class of kR-spaces is the к-spaces, spaces in which a set is
with each compact subset is open in the compact set.
and x is an F-valued function which is continuous
of T, then for each open subset G of F, x \g)OK =
Thus each
R
open if its intersection
Indeed if T is a k-space
on each compact subset К
x I \g) so that x \g) is open in T and x is continuous on T.
IK
k-space is a k -space. Familiar classes of spaces that are also k-spaces
R
include the locally compact spaces and the first countable spaces (see
Exercise 2.2(b)). It is not the case however that every completely regular
Hausdorff k^-space is a k-space.
Example 2.3-1. A k^-SPACE WHICH IS NOT A k-SPACE. Let A be an uncountable
set, let W denote the nonnegative integers with discrete topology, let W^
carry the product topology, and let
wi th
that
tha t
S = {t e W^ |t(a) = 0 for all
its subspace topology. W^ is a
W* is a k^-space we make use of
its repletion uS is W^, both of
(a)
but countably many a f a}
In proving
and
k^-space but not a k-space.
the facts that S is a к -space
R *
which follow from (a) below.
If x is a sequentially continuous real-valued function on
x(t^) -< x(t) whenever t_ -• t in S, then x is continuous,
over
S, i.e.
More-
ls a
tion
— x(t) whenever t
n
associated with each such sequentially continuous x there
countable subset C of A and a continuous real-valued func-
Q
x^ on W such that x “ x • pr^ where pr^ is the projection
о
Prc-S - W
t
C
Once the existence of x
follows from the
X = x„ • pr„
C C
has been established the continuity of
continuity of x^ and pr^,. Furthermore, as
It is also true that S is a k-space (Exercise 2.2(g)) so that S is an
example of a k-space whose repletion is not a k-space.
66
2. SPACES OF CONTINUOUS FUNCTIONS
will be seen, the representation x=x^-pr^ is a consequence of the existence
of a countable set CCA with the property that for each t f S, x(t)=x(tc)
where t =tk , к denoting the characteristic function of C. Indeed if such
С С С»
a set C exists, then for each s f W we define x^,(s) to be x(t) where t is
chosen to be any element of S such that t|^=s. The sequential continuity
of x^ follows from the sequential continuity of x and thus continuity of
x follows
strate the
t e s.
C
from the second countability of W . Hence it remains to demon-
existence of a countable set CCA such that x(t)=x(t^,) for each
Prior
to doing this however we digress somewhat
in establishing three
preliminary technicalities, the last one of which
the existence of C.
is
used in the proof of
(i) If (A ) is an uncountable collection
b dCB
of A such that any one
a from A belongs
of
to
countable subsets
at most a countable
there is an uncountable subset В CB such
о
elements from
b and c are distinct
be the collection of
all с С В for
В .
о
which
tha t А (") A 0
bi-l bi
there are only a
number of the A. , then
b
that A. 0 A =0 whenever
b c
Proof (i): Let b f В and let U,
-------4—b
there is a finite sequence b=b ,b,,...,b =c in В such
о 1 n
< n. As A, is countable and for each a f A.
=	b	b
there are only a countable
such sequences for n=l. It follows by induction that the
countable number of c such that afA^ClA
number of
cardi-
tive integer n so
b,c from В and we
element from each
the class of sequences Ь=Ь^,...,b^=c is
that U, is countable. U, Г1U =0
b	be
can produce an acceptable Bq by
of the distinct U's.
b
(ii) If (A, ), „ is an uncountable collection
4 '	4 b'bfB
each containing at most к elements, then there exists a finite
countable for any
posi-
pair
or U, =U for each
b c
choosing exactly one
of finite subsets of A,
subset ZCA and an uncountable subset В CB such that A, 0 A -Z
о	be
for distinct b,cfB .
о
Proof (ii) : Let m be the largest integer such that a set ZCA having m
elements and an uncountable subset B'CB exist such that ZCA, for all bfB1
b
(m may be 0). Then for fixed a f A, a C A^-Z for at mos t a countable
number of elements b from B1 for otherwise a could be adjoined to Z. Thus
by (i) there is an uncountable set B^cB' such that (A^-Z) О (Ac~Z)=0 (or,
equivalently, A^OA^Z) for each pair of distinct elements b,ceBQ.
Let к be a positive integer and be the collection of all t c
that vanish at all but at most к values of A.
2.3 к-SPACES
67
(iii)
If <4>ьев
there is a
is an uncountable family of functions from then
sequence (b^) of distinct elements from В such that
lim t =t. Furthermore there is some finite set ZCA and b f В
bn
such that (t^)=t (for any t c S recall that t^tk^ where k^ is
the characteristic function of Z) .
Proof (iii) : Let A^={a f А|г^(а)^о]. Each A^ contains at most к elements
so by the previous result there is an uncountable set BqC В and a finite
set ZCA such that А. ПА =Z for all distinct elements b, с г В . If Z = 0
be	о
choose any sequence (b^) of distinct elements from Bq and any b f В . Now
each a belongs to at most one A. so that tb (a) — 0=(t ) (a). If Z i1 0
n	n	° Z
then consider the family of restrictions (t, I )	. Since each t can
D |Z bCB	d
assume only integer values at each of the finite number of a's in Z, there
is an uncountable collection of t^'s that agree on Z. From this collection
choose some t, and a sequence of distinct elements b . Then, since t = t
b	n	’	bn b
on Z, it follows that t^ (a) — t^(a) for each a € Z; tb (a) — O=(tb)z(a)
whenever a 0 Z. This establishes the result.
Given a sequentially continuous map x:S -> R we are now ready to prove
that a countable set CCA exists such that x(t)=x(t^) for each t C S. For
each positive integer к let A, be the set of all a f A for which there is
к	*
at c S such that x(t ) x(t ') where t l=(t ). r ,. We contend that
a	a' 4 a '	a a'A-[a]
each A^ is countable. Suppose that this is not the case for some k. De-
noting the diagonal of R x R by D, it follows that for each a C A^,
(x(t ),x(t '))fR x R-D. But R x R-D=U ,,F where each F is closed in
a a ~	~	~	~ mfN m	m
R x R. Thus at least one of the F 's, say F , contains an uncountable
~	~	m	mo .
number of the pairs (x( t^) ,x (tg 1)) , i.e. there is an uncountable set BCA^
such that (x(t ),x(t '))eF for each a f B. By the previous result there
з a
is a sequence of distinct elements b С B, b f B, and a finite subset ZCA
m
such that t, -• (t, )„. Since the b 's are distinct, for each a e A there
bm b Z	m
is a point in the sequence beyond which b ^a so that tb ' (a) = t, (a)_,(t, ) (a),
m	Dm	bm	b Z
Thus both sequences (t ) and (tb ') converge to the function (t ) which
bm	m	b Z
belongs to S . It follows that lim (x(tb ),x(tb ’)) = (x((t,) ),x((tb)7)) is
contained in the closed set Fm and Fm О D=0;we have reached a contradic-
tion - a contradiction which arose from the assumption that A^ was uncount-
able. Hence C= U, A, is countable and we claim that this is the desired set.
к к	к	к
First we verify that x(t)=x(tc) for t f S . If t C S let a^,...,ar be
the elements of A not in C at which t does not vanish. Then, as each
a^ i and
68
2. SPACES OF CONTINUOUS FUNCTIONS
it follows that
ЬС= ((tA-{a1})A-{a2})"-)A-{ar}’
X(t) = X(tA-{ai}> = x((tA-{a 1})A-{a2} > = "• = x(tC>’
Next suppose that t C S and let t, — t where t, f S^. Clearly lim (t. ) =t
к	к	к С С
so that
x(t) = lim x(tk) = lim x((tR)c) = x(lim(tk)c) = x(tc)
and this completes the proof of (a).
(b) S is a k-space and US = w\
R
That S is a k-space follows from (a) once we observe that any real-
R
valued function on S which is continuous on each compact subset of S must
be sequentially continuous.
The product W^ is homeomorphic to the closed subspace of consisting
of all functions assuming nonnegative integer values and is thereby a re-
д
plete space (Theorem 1.5-2). Since S is dense in W , it follows by (1.6-3)
that US=W^ once it has been noted that each continuous real-valued function
actually a restriction of a continuous function on W^. This fact
from the representation given
be viewed
on S is
follows
prc may
Is a
as a continuous map
k -space.
in (a) once it has been observed that
on all of IjA rather than just S.
To show that
suffices to show that C(W^,R,c) is
is a Cauchy net in
complete (Theorem
C^,R,c) ;
W is a k - space it
2.2-1). Hence suppose that (x^)
we must show that it converges uniformly on each compact subset
A
to a function which is continuous on W . As compact subsets of S are also
is Cauchy in C(S,R,c). Because
is the repletion of
C(S,R,c). We
-> x pointwise
and pointwise
Let t f w\ If^ de-
compact in the product, the net (x^ |$)
S is a k^-space, so that C(S,R,c) is complete, and
S, there is an xfC(W^,R,c) such that x. I x I in
д	~	1 IS IS
x^ — x in C(w ,R,c). It is enough to show that x^
since a net of functions which is uniformly Cauchy
on a set must be uniformly convergent on that set.
notes the collection of all finite subsets of A directed by set inclusion,
then (t ) .is a net from S convergent to t in w\ Furthermore each ele-
F	A
ment of this net and its limit lie in the compact set K=/7^^[fl, t(a) 1C	.
Now consider
claim that
A
on W
convergent
|x.(t)-x(t) | < |x.(t)-x.(t) | + |Xj(t)-Xj(tF) |
+ |x.(t^-x(tF) | + |x(tF)-x(t) |.
2.3 k-SPACES
69
Since	is uniformly Cauchy on each compact subset of , there is an
index i such that |x.(t)-x.(t) I < C/4 whenever i, j > i . As (x.). T is
O	ll	j l	—	<->	V ' 1 z-T
uniformly Cauchy on К it is uniformly Cauchy on
pointwise on S and therefore on {tp |Fe/i; thus (x^)
to x on {t |F€j] . Consequently there is an
x(tp) | < f/4 whenever j > j° and any Ff^.
F such that
Hence
о
Moreover x. -» x
converges uniformly
index j such that
о
Fixing j > iQ> jo,
|x . (t)-x . (tp,) | and |x(tp,)-x(t) | are less than f/4.
J J	*
|Xp(t)-x(t) | < f whenever i > 1° and we have shown that C(w ,R,c) is com-
plete .
A
(d) W is not a k-space.
We shall exhibit a non-closed set QCW^ such that QOK is closed in К
IW
we may choose
for each compact set К thereby establishing the fact that W is not a k-
space. Let Q' be the collection of all t C W^ such that for some n c W,
t(a)=n for all but at most n values of a and 0 otherwise. To begin with
we claim that Q1 is closed. Indeed if the net ( с-ц )-ц j from Q' converges
to t then for each a c A, t^(a) -> t(a). Thus each t(a)fW. If t=0 then
t C Q'. If t 0 then there is some b c A such that г(Ь)=п^О. Let BCA
be the collection of elements at which t is 0 (B may be empty). If we
assume that В contains a subset B1 having n+1 elements then, since t^ -» b>
there is an i such that U=t on B'U{b}. But this implies the contradictory
facts that t.(b) = t(b)=n and tp(a)=O for at least n+1 values of a. Thus В
has cardinality less than or equal to n. Now suppose that a^b and a t В.
Then for some i,	on {a,b}.
t(b)=n and t f Q1 .
Since t.(a)=t.(b)=n, it
follows that t(a) =
Next we prove that 0 is a limit point of Q1 so that Q=Q'-{0] is not
closed. The collection of all finite subsets of A is directed by set
F
inclusion. For each F f v let t be that element of Q which is 0 on F
F ,
and m on A-F, where m is the number of elements in F. Clearly t C Q
for all F/Й and tF -> 0.
A
It remains to show that QOK is closed in К for each compact set KcW .
Since each К is contained in a compact set of the form
L^ is a finite subset of W, and Q'=QU[o} is closed in 1
n.L , where
afA a
it suffices
each
to
prove that 0 is not an element of the closure of Q О n.L in П L .
afA з	afA з
To do this, first choose an integer n such that max, k=n for an infinite
a
let F be a finite subset of A with more than n
number of indices a and
elements. If we assume
I1L then it follows
a€A a
that 0 is a member of the closure of Q О aJ^La in
that an element tfQ О П L exists such that t=0 on F.
a
70
2. SPACES OF CONTINUOUS FUNCTIONS
Now since t ( Q,t is not identically 0 and therefore F1, the set of indices
at which t vanishes, is finite and contains F. This leads directly to the
fact that there is an a € A-F1 for which max, , k=n. Now, since t f Q and
keLa
vanishes on a set containing more than n elements, t(a) > n which contra-
dicts the fact that t f П and it follows that 0 is not a limit point of
Q nriL^-v
Even if T is not a к-space, the compact subsets of T "generate" a
topology xj'k which renders T a к-space. Indeed the collection of all sub-
sets UCT that meet each compact subset К of T in an open subset of К is a
topology called the k~extension topology of T. It is the strongest topology
on T for which all the injection maps i :K -» T, t -• t, К compact, are con-
tinuous. Clearly the к-extension topology is at least as fine as the
original topology. This, combined with the fact that each i :K -> (T) is
continuous, implies that the original topology and the k-extension topology
have the same compact sets. Thus T is a k-space in the k-extension topol-
ogy. In addition we make note that both the original and k-extension
topologies induce the same subspace topology on each compact subset of T.
With the notion of the k-extension topology at our disposal we are in
a position to describe
the
circums tances
under which a k^
-space
is a k
space.
(2.3-1) WHEN IS A k„-SPACE A k-SPACE? Let T. denote the space T equipped
R	к
with the k-extension topology.
iff T, is com'
k
pletely regular.
Proof.
Since the k-extension topology coincides with the
original topology
when T is a k-space, T^=T is completely regular.
Conversely suppose that
1^ is completely regular,
as the original topology,
As the k-extension topology is at least as fine
C(T,F)C C(Tk>F). On the other hand, if xfC(Tk,F)
then x is continuous on each compact subset of T^. Since T and T^ have the
same compact sets, induce identical subspace topologies on the compact sub-
sets, and T is a kR-space, then x € C(T,F) and C(T,F)=C(Tk>l?) . Now, since
T and T^ are completely regular, then T=Tk and it follows that Tisak-
space . v
Though it is not generally true that a completely regular Hausdorff
space T is a k-space when C(T,F,c) is complete, there are categories of
spaces for which it is true - two such are the hemicompact spaces and the
pseuofinite spaces (defined below).
2.3 к-SPACES
71
Definition 2.3-1. PSEUDOFINITE SPACES. A completely regular Hausdorff
space T is pseudofinite if each compact subset КС T is finite.
Certainly every discrete space is pseudofinite. Every P-space (i.e.
those completely regular Hausdorff spaces T for which each prime ideal in
C(T,R) is maximal) is pseudofinite and there is a plentiful supply of non-
discrete P-spaces (Gillman and Henriksen, 1954). Most of the remainder of
this section is devoted to characterizing completeness of C(T,R,c) for
pseudofinite spaces and hemicompact spaces. First we characterize pseudo-
finite spaces.
(2.3-2) CHARACTERIZATIONS OF PSEUDOFINITE The following are equivalent
for a completely regular Hausdorff space T.
(a)	T is pseudofinite.
(b)	The compact-open and point-open topologies coincide on C(T,F).
(c)	1^ is discrete.
Proof. Since the compact subsets of T are finite, the implication (a) — (b)
is clear. To obtain the converse, i.e. the implication (b) — (a), let К
be a compact subset of T. By (b) there is a finite set FCT such that
V С V where
PF PK
vp = {xeC(T,F) |pK(x) = sup |x(K) | < 1}
К
etc.. Now suppose that K^F, i.e. that there is a point t ( K-F. Since T
is completely regular there is an x€C(T,F) such that x(t)=2 while x(F) = {0]
so that x € V and x V . Thus КС F and К is finite.
PF	PK
As 1^ (as in (2.3-1)) and T have the same compact sets, the implication
(c) — (a) is clear. On the other hand if T is pseudofinite, each compact
(hence finite) subset of T is discrete. Therefore any subset U of T meets
each compact set К in a relatively open set and T^ must be discrete.?
It is now clear by (a) — (c) of (2.3-2) and (2.3-1) that any pseudo-
finite k-space is a k-space. However, as we
R
pseudofinite k-spaces are the discrete ones.
R
(2.3-3) PSEUDOFINITE к -SPACES ARE DISCRETE
R
shall presently see, the only
If T is pseudofinite, then
the following statements are equivalent.
(a)	C(T,F,c) is complete (i.e. T is a k -space, by Theorem 2.2-1).
(b)	Tfc is discrete.
(с)	T is a k-space.
Proof. Since T is pseudofinite, by (2.3-2), T^ is discrete. Thus the
equivalence of (b) and (c) is clear. To see that the implication (a) — (b)
72
2. SPACES OF CONTINUOUS FUNCTIONS
holds, it is enough to note that - by (2.3-2)(b) and the fact that C(T,F)
F	F
is dense in the product space T~ - C(T,F)=T~. Finally (c) — (a) follows
by Theorem 2.2-1 and the fact that к-spaces are к -spaces, v
R
As we mentioned earlier the class of hemicompact spaces is also a class
of spaces
in which к -spaces
R
are k-spaces.
(2.3-4) HEMICOMPACT k_~SPACES Let the completely regular Hausdorff space
R
T be hemicompact. Then C(T,F,c) is complete (or, equivalently, T is a
k^-space)
iff T is a k-space.
Proof. The
a k^-space.
implication
Since T is
к -• к is obvious.
R
a k-space whenever
(2.3-1), it suffices to prove that
Furthermore T^ must be hemicompact
sets as T. Thus T, is LindelBf so
к
regular. To this end let t(T and
is hemicompact, T may be written as
Conversely, suppose that T is
1^ is completely regular by
hemicompact, then T^ is normal.
T is, because T^ has the same compact
is only necessary to show that T^ is
U be an open neighborhood of t. As T
the union of an increasing sequence of
if T is
if
it
compact sets K^ having the property that each compact subset of T is con-
tained in some Kn- We may assume without loss of generality that t ( Kp
Our first contention is that there exists an increasing sequence (W^)
of closed neighborhoods of t in Kn such that (i) WnC UHK^ for each n, (ii)
W ПК =W for all m < n, and (iii) (int W )ПК =int W for all m < n where
n mm	—	nn mmm	—
int denotes the interior taken in К . Since K, is a compact Hausdorff
n	n	1
space and therefore regular there is a closed neighborhood W of t in K^
contained in UCIKj. Let us assume that closed neighborhoods W^,...,Wn_^
of t exist satisfying the three properties stated above. As Wn_^ is a
closed subset of иПК in the normal space К there is an open subset V of
n	n	n
К such that W ,C V Cel V CUflK . Since W . is a neighborhood in К ,
n	n-1 n n n	n-1	n-1
an open set U CK exists which meets К , in the set int _W , . Hence
r	n n	n-1	n-1 n-1
cl U ПК =cl(int ,W ,)CW , and it follows that the closed neighbor-
n n-1	n-1 n-1	n-1
hood of t in К , W =(cl U Пс1 V )UW . meets К , in the set W , and is
n n n n n-1	n-1	n-1
a subset of иПК . Furthermore the relations int W DV Пи , К .Пи =
n	n n n n’ n-1 n
int ,W ,, and V ПК Dint .W . imply that int W ПК .Dint W ..
n-1 n-1	n n n-1 n-1 J	n n n-1	n-1 n-1
The reverse inclusion is obvious and the contention has been established.
Next we claim that W=	is a closed neighborhood of t in T^ con-
tained in U, thereby proving that T^ is regular. It is clear that WCU.
That W is closed in the hemicompact k-space T^ follows immediately from
the relation ВПК =W for each n. Certainly WD. (Lint W . Since
n n	J ntN n n
U int W ПК =int W for each m, и int W is an open subset of W in T,
n n n m m m	n n n	k
2.4 CONTINUOUS DUAL OF C(T,F,c)
73
containing t. Thus W is a neighborhood of t and the proof is complete, v
Thus it is natural to inquire when C(T,F,c) is fully complete. In
the event that the completely regular Hausdorff space T is hemicompact or
pseudofinite, C(T,F,c) is fully complete iff T is a k-space. Indeed if T
is a hemicompact k-space, C(T,F,c) is a Frechet space by Theorem 2.1-1 and
(2.3-4). As any Frechet space is fully complete (Husain, 1965, 4.1, Prop.
3) the desired result follows. The converse is trivial. As for the case
T
of pseudofinite spaces, we saw in the proof of (2.3-3) that C(T,F)=F and
the compact-open and product topologies agree whenever T is a pseudofinite
k-space. This, combined with the fact that an arbitrary product of real
lines is fully complete in the product topology (Husain, 1965, 5.5, Prop.
14), leads to the conclusion that C(T,F,c) is fully complete.
Unfortunately it is not generally the case the C(T,F,c) is fully com-
plete when T is a k-space, and necessary and sufficient conditions on T
for C(T,F,c) to be fully complete are not known. Ptak (1953) has shown
that T is a k-space whenever C(T,F,c) is fully complete and has given a
counter example for the converse (see Exercise 2.3(b)).
Some of the interrelationships between к-spaces and k^-spaces, etc.,
are summarized in Table 1 below. T denotes a completely regular Hausdorff
space and K, with or without subscript, a compact subset of T.
2.4 The Continuous Dual of C(T,F,c) and the Support
A matter of considerable importance in the theory of locally convex
spaces X is characterizing the continuous dual x', the space of continuous
linear functionals on X. The considerations of this section have primarily
to do with the continuous dual C(T,F,c)' of C(T,F,c) where T is a completely
regular Hausdorff space. C(T,F,c)1 is characterized here in two ways, one
involving the evaluation maps T* = [t*|teT] (t*(x)=x(t) for XfC(T,F)) and
the other, more important for our purposes, represents the elements of
C(T,F,c)' as integrals with respect to certain set functions defined on the
Borel subsets of T. This result has predecessors as far back as 1909 when
Riesz first characterized С(Г0,1],К)' in terms of Riemann-Stieltjes inte-
grals. Subsequent generalizations were made which culminated in the well-
known characterization of C(T,F,c)' for compact Hausdorff spaces T in terms
of integrals with respect to regular set functions defined on the Borel
subsets of T (see Theorem 2.4-1). As the elements of C(T,F) are all
bounded when T is compact, it was natural for others to consider the space
С^(Т,Н) (with sup norm) and integral representations of the elements of
74
2. SPACES OF CONTINUOUS FUNCTIONS
property	definition	implied by	equivalent to
hemi- compact	(Def. 2.1-1) A countable family (K ) exists such that each К К n	n for some n.	loc. compact + CT-compact	C(T,F,c) me trizable
k-space R	(Sec. 2.1) For x:T - R if x 1 ~	IK is continuous.	seq. conti- nuity -» con- tinuity or k-space	C(T,F,c) complete
k-space	(Sec. 2.3) For each K,GO.K open in К -> G open in T.	к +pseudo- finite or k^+ hemicompact	
cr-compact C(T,F,c) +	metrizable compact hemicompact к	C(T,F,c) +hemicompact or pseudofinite
Table 1
2.4 CONTINUOUS DUAL OF C(T,F,c)
75
C^(T,R)‘ were obtained for T satisfying various normality conditions (see,
for example, Dunford and Schwartz, 1958 and Alexandrov (1940, 1941, 1943)).
By abandoning the space of regular set functions on the Borel subsets of a
topological space T for the larger class of regular set functions on the
Baire subsets of T it was possible to establish integral representations
of the elements of C(T,F,c)'. We follow this approach in obtaining a rep-
resentation of Cb(T,F)' (where Cb(T,F) carries the sup norm) for completely
regular Hausdorff spaces T, but then take an alternate route for C(T,F,c)'-
one which utilizes knowledge of the dual of C(T,F,c) for compact T. In so
doing we obtain integral representations in terms of set functions on the
Borel subsets. These representations are then used to prove the existence
of the "support" of an element x1fC(T,F ,c)', a minimal compact subset S of
T with the property that if yfC(T,F) vanishes on S, then x1 must vanish on
y. The notion of support is put to use in the next section where necessary
and sufficient conditions on T are obtained for C(T,F,c) to be barreled and
bornological respectively.
The following result is useful later on.
(2.4-1) T* AND THE DUAL OF C(T,F,c) Let X denote the family of compact
subsets of the completely regular Hausdorff space T, K* = [t*lt€K], and
K. the balanced convex hull of K.
be
Then C(T,F,c)-=[UK€Hcla(x^x)(K*)bcl=
H where X=C(T,F,с), X* the algebraic dual of X of all linear functionals on
X, and the square brackets denote linear span. If T is compact, then
C(T,F,c)'=[cla(x,x)(T*)bc].
Proof Since t*fH for each t f T, then (X,H) is a dual pair. By the
Mackey-Arens theorem we need only show that the compact-open topology is a
topology of uniform convergence on a collection of balanced convex a (H,X)
compact subsets of H to prove that H is the dual of X. To do this we show
first that each set E=cl .	. (K*), is absolutely convex and o(H,X)-
o(X'c,X} be
compact. And to do this it is sufficient to show that E is o(X*,X)-bounded
for it is already a(X*,X)-closed, and the closure of an absolutely convex
set is absolutely convex; o(X*,X)-boundedness, in turn, is shown by show-
ing tha t for each x € X, sup | < x ,E > | < co. To this end le t t^, . . . , t^jK
and Up . . . , Lin?F be such that E ||u | < 1. Then for any x с X
|2d.t.\x) | = |Epix(ti) | < (S |U. |)pK(x) < pR(x)
76
2. SPACES OF CONTINUOUS FUNCTIONS
where p*,(x) =sup ]x (K) |. It follows that E is bounded and that for each
f " CW,X)(K*)bc> lf« I < PK«-
The next and final step is to show that
(clc(X*,X) (K*)bC) = Vp = ^xeX IPK(X) -
К
Since lf(x) I < p (x) for each x«X and each f € c 1 ,v. V.(K*). , it is clear
1	1 К	Л.) DC
tha t
% C (clo(X*,X)(K*)bc) ’
K.
To obtain the reverse inclusion, consider any x f (cl . (K*), )° and
'	o(X*,X) be'
any tfK. Then |x(t) | < 1 and it follows that p^(x)=sup |x(K) | < l.V
In Sec. 1.7 we had occasion to consider the class of finitely additive
signed measures (differences of finitely additive nonnegative regular real-
valued set functions) defined on the algebra of sets generated by the zero
sets of a completely regular Hausdorff space. We have need of some of the
measure-theoretic notions set forth there but somewhat more general ver-
sions of them must be obtained first. The algebras generated by the zero
sets Z. and the closed sets of T are denoted by and ac respectively
while the c-algebras they generate are denoted by<8^ and®, called the
Baire and Borel sets respectively. The total variation |y | of a finitely
additive set function у defined on any of these algebras is the set function
defined at each set A in the algebra to be the sup of the sums S |n(E.) |
taken over all finite pairwise disjoint collections (E.) of subsets of A
taken from the algebra. In the event that у is a nonnegative finitely addi-
tive set function defined on an aIgebra of sets, it is bounded. Thus a
difference of. two such nonnegative finitely additive set functions is also
a bounded set function. Conversely, a bounded real-valued finitely addi-
tive set function у defined on an algebra of sets has a finite-valued total
variation |y | [Dunford and Schwartz, 1958, p. 97] and can be decomposed
[ibid., p. 98, Th. 8] into a difference of nonnegative additive set func-
tions y+ and у such that p=y++y . Moreover a complex-valued additive set
function у defined on an algebra of sets is bounded iff | is finite-valued
and, in this case, may be written in the form p=yr+iy^ where y^ and y^ are
bounded real-valued additive set functions.
A finitely additive set function у defined on CL or <8 is regular if
Z a
for each A on which у is defined and C > 0 there are Z,Z'fZ such that
ZCACCZ' and |y| (CZ'-Z) < C. Similarly if у is defined on CL c or в у is
2.4 CONTINUOUS DUAL OF C(T,F,c)
77
regular whenever В is in the domain of definition of u and ( > 0 there are
closed and open sets C and U such that CCACU and |u,|(U'C) < (.
Our first goal is to characterize C^(T,F)' and, as might be expected,
we begin with the real-valued case. As it happens the elements of Cb(T,R) 1
can be decomposed into a difference of "positive" components. More pre-
cisely, if x'fC, (T.R)1 then x'=x '-x ' where x 1 and x 1 are positive
J	b ~	p n	p	n
linear functionals, i.e. x '(x) and x '(x) are > 0 whenever x > 0. Prior
’	p	n	—	—
to establishing such a decomposition we verify that positive linear func-
tionals are always continuous.
(2.4-2) POSITIVE — CONTINUOUS Any positive linear functional defined on
C, (T,F) where T
b
is a completely regular Hausdorff space is
continuous.
Proof To see that a positive linear functional h defined on Cb(T,F) is
continuous when C^(T,F) carries the sup norm topology, let x€C^(T,F).
Clearly then
' PT(x)l < x < pT(x) 1
where l(t)=l for each t С T. Thus, since h is positive,
- pT(x)h(l) < h(x) < PT(x)h(l)
and it follows that |h(x)| < |h(l) |p^,(x). If x is complex-valued, it fol-
lows that |h(x) | <2h(l)p^,(x) so that h is continuous.?
(2.4-3) Cb(T,R) ' AND POSITIVE LINEAR FUNCTIONALS If T is a completely
regular Hausdorff space and Cb(T,R) carries the topology induced by the sup
norm, then corresponding to each x'fCb(T,R)' there are positive linear
functionals x ' and x ' such that x'=x '-x
p	n	p n
Proof First we define x ' on the nonnegative elements of C, (T,R) : if
p	b ~
x > 0 then
X ' (x) = supfx' (y) |y € Cb(T,R) , о < у < x}-
Clearly Xp'(ax)=aXp'(x) for a > 0 and x > 0. Next we claim that Xp'(x+y) =
Xp'(x)+Xp'(y) whenever x, у > 0. To see this suppose that 0 < w < x+y.
Clearly* 0 < wA x < x and 0 < w-(wA x) < y. Thus
x ’ (w) = x ' (w A x) + x' (w- (wA x) ) < Xp' (x) + x₽ ' (y) ,
and, taking the supremum over all such w, we obtain
The meet хЛ у and join x\J у
a set T are the functions t
of real-valued functions x and у defined on
-> min (x (t) ,y (t) ) and t -> max(x(t) ,y(t)) .
78
2. SPACES OF CONTINUOUS FUNCTIONS
xp'(x+y) < xp'(x) + xp'(y).
On the other hand, if О < v < x and 0 < w < у then 0<v + w<x + y and
x'(v) + x'(w) = x' (v+w) < xp'(x+y)
from which it follows that xp'(x) + xp'(y) < x '(x+y).
We can now extend x ' to all of C, (T,R) as follows: if xfC, (T,R)
write x=x -x for some x+, x >0 and x+, x”fCb(T,R) (e.g. x=x V 0-(-x) V 0) .
Then define x '(x) = x ' (x+) - x '(x ). To see that x 1 is well-defined,
PPP	P
suppose that x = u-y = v-w where u,y,v,w > 0. Then u+w = y+v and since
u, У, v, w, u+w, y+v > 0,
xp'(u)+xp'(w) = Xp'(u+w) = Xp'(y+v) = Xp'(y) + Xp'(v).
Hence x '(u)-x '(y) = x '(v)-x '(w). Next we show that x ' is linear on
P P	P P	P
C^(T,R). Let x,yeCh(T,R) and consider
xp' (x)+xp' (y)=[xp’ (x V 0)-Xp' ((-x)v 0) 3+ГХр ' (у V 0)-Xp' ((-у) V 0) J
=x ' (x V 0+y V 0)-x ' ((-x) V 0+(-y) V 0) .
Now x+y = (xVO + yVO) - ( (-x) V 0 + (-y)VO) and (x V 0) + (у V 0) , (-x) V 0 +
(-y) VO > 0; thus
Xp'(x) + Xp'(y) = Xp'(x+y).
To establish the homogeneity of x ' we note first that x '(ax) = ax '(x)
P	P	P
for a > 0 and xfC^(T,R) follows directly from the fact that it is valid
whenever x > 0. On the other hand for xfC^(T,R) we have
Xp'(-x) = xp'((-x)+ - (-X)')
= Xp ’ (x -x+)=Xp’(x )+Xp ' (-x+)=Xp' (x )-xp'(x+)=-Xp'(x).
Hence Xp' is a positive linear functional. We complete the proof by observ-
ing that хд' = Xp' - x' is a positive linear functional, v
Now how are positive linear functionals represented?
(2.4-4) POSITIVE LINEAR FUNCTIONALS ARE "GENERATED" BY REGULAR ADDITIVE
NONNEGATIVE SET FUNCTIONS If T is a normal (completely regular)
Hausdorff space and h is a positive linear functional on С^(Т,Н), then
there exists a finite nonnegative regular additive set function ц defined
on the algebra Л (A^) generated by the closed subsets й (zero sets Z. )
of T such that h = Г dp (P du is taken here in the same sense as in Sec. 1.7).
2.4 CONTINUOUS DUAL OF C(T,F,c)
79
Proof Both situations - where T is normal and where T is completely regu-
lar - are dealt with similarly. In fact if the terms "zero set" and
"complement of zero set" are substituted for "closed" and "open" respec-
tively in the argument given below for the case where T is normal, we
obtain the proof for completely regular T.
We begin by defining a real-valued set function on the class of all
subsets of T which we prove to be subadditive. We subsequently show that
this set function when restricted to the appropriate algebra is, in fact,
regular and additive and it is finally established that h and the regular
additive set function у are related by the formula given above.
Suppose that T is a normal Hausdorff space.
(a)	Definition of у onP(T). If U is an open subset of T we define y(u)
to be the supremum of the values h(x) where x C C^(T,R) and 0 < x < k^, k^
denoting the characteristic function of U. If A is an arbitrary subset of
T, 4(A) is defined to be the infimum over all open UDA of the values y(U) .
Clearly у is a nonnegative monotone set function and у(Й)=О. Further-
more if U is open and 0 < x < k^, then 0 < x < 1 and 0 < h(x) < h(l) so
that y(U) < =>. Hence it follows that у is finite-valued on all of the
power set P(T).
(b)	у is subadditive. To see that у is subadditive, first consider any
two open sets U and V and let x € C^(T,R) be such that 0 < x < k^^^ and
h(x) > y(UUV)-f where 0 < ( < 1/2. Next consider the closed set
H=x 1([e, 1]) Псис V and choose u, v ( Cb(T,R) such that u(H)={0"(, u(CV) =
{ 1}, v(CV)={o}, and v(u ^(0))={l}. Since v(t) = l whenever
2	9	2
u(t)=O, we may set y=xv /(u^+v£) and obtain a continuous function on T such
that 0 < у < x, y=x on H, and y(CV)={0}. Hence у < k^. As у < x, the func-
tion w'=x-y satisfies 0 < w' < 1. Furthermore we claim that the closed set
F=w' \C2f,l])CU for if it is not then, for some t€F, we have tfCU. But
w' (t) > 2C certainly implies that x(t) > f so that t€H. This however is a
contradiction for we know that w'(H)={o). Thus FCU and we may choose
reCb(T,R) such that 0 < r < 1 and r(F)={l] while r(CU)={o) so that
w=rw' < ky. Recalling that у < k^, it follows that h(y)+h(w) < p(V)+y(U)
so that the desired conclusion - p(UUV) < p(U)+y(V) - follows once we show
that x < w+y+2c for then
p(UUV) < h(x) + ( < h(w) + h(y) + (2h(l) + 1) €
< 4(U) + y(V) + 2(h(l) + l)f
80
2. SPACES OF CONTINUOUS FUNCTIONS
for each 0 < j < 1/2. To see that x < w+y+2f first note that on F, where
w=w', we have x=w+y. On the other hand if t / F, then x(t)-y(t)=w'(t)<2f
and
x(t) < y(t) + 2f < w(t) + y(t) + 2e
Having just established subadditivity for open sets, let A and В be
arbitrary subsets of T and UDA, VDB be open subsets such that
p(U) < u(A)+f and p(V) < p(B)+€ where € is some pre-assigned positive
number. Then UUVDAUB so that
p(AUB) n(UUV) < ix(U) + d(V) < |X(A) +11.(3)+ 6
and subadditivity of ix on P(T) follows.
Thus |X is "outer regular" on the class of all subsets of T. By re-
stricting p to ft - the smallest algebra of sets containing^- we wish
to obtain a regular additive set function.
all subsets
regular and additive. To prove
this
of T on which ix is "inner regular
consider
e.
the class й of
a = {Аст|р(А) = sup{p,(F) |ADF, FetJ}}
We show that p is a (perforce nonnegative and regular) additive, set func-
tion on (the algebra)6L and thatdD ac-
First we show ц, to be additive on the closed subsets of T. To this
end suppose that F and H are disjoint closed subsets of T and, correspond-
ing to the positive number c, let W be an open set containing F(JH such
that p(W) < p(FUH)+<. Since T is normal, disjoint open sets UDF and
VDH exist . Choose u, v ( Cb(T,R) such that 0 < u, v^l, 0 < u < k^p^’
° < v £ к , h(u) > p(unw) - e/2, and h(v) > n(VPW) - f/2. Then
— Vf iw
|X(F) + |X(H) < n(unw) + u(vnw)
< (h(u) + (/2 + h(v) + f/2
= h(u + v) + (
<	ix((unw)u(vnw)) + c
<	L ix(w) + c
<	H(FUH) + 2f
Consequently p(F)+p(H) < u(FUH). Equality follows from the subadditivity
of ll. Next suppose that the disjoint sets A, В (0. so that closed sets F
and H exist such that ADF, BDH, p(F) > p(A) - (.12., and p(H) > p(B)- e/2.
Thus
2.4 CONTINUOUS DUAL OF C(T,F,c)
81
ц(А) + ц,(В) < |j,(F) + ц(Н) + € = ui(FUH) + C £ u(AUB) + e
and additivity on Ob follows once we establish the fact that AUB (Q. •
Since ц, is subadditive on P (T) we can say that
H(AUB) g ц(А) + ц,(В) < n(FUH) + e .
Hence sup[u(E)|e € ijr , ECAUB] = ll(AUB) and AUB (Q,. There is just one
more thing to prove before we can say thatQ. is an algebra containing Or -
namely CAfd whenever A ((2 . A prerequisite for doing this is the knowledge
that each open U f Ct . Indeed if p,(U)=O then, as 0CU and p,(0)=O,
sup{ ll(F) I FC U, FC 6 } 2ru(U). The reverse inequality follows from the defini-
tion of u and U (Ob  If p,(U) > 0 and 1/2 > ( > 0 then there is some
x e Cb(T,R) such that 0 < x <	and h(x) > p,(U) - (. Clearly
V=x \[2f,l].)C F=x 1(Ef,l])C U. Now x-2f < 0 on CV so the continuous
function y=max(x-2f, 0) k^ and therefore
h(x) - 2fh(l) = h(x - 2f) < h(y) g p,(v).
It follows that 11(F) > U(V) > h(x)-2fh(l) > u(U) *(2h( l)-l-l)f, whence we con-
clude that U f 6L. Now suppose that A € Ob. № wish to show that CA F Ob •
To this end let F be closed and U be open such that FCACU, u(F) > u(A)-f/2
and u(A) > u(U)-f/2. Hence u(U)-u(F) < C. As 11 is additive on Ob , U~F is
open and belongs to Л, and U=FU(U-F), it follows that d(U-F)=u(U)-U(F)<C.
But
|1(CA) - H(CU) < |1(CF) - u(CU) = n(CF-CU) = |1(U-F) < f
and, since CU is closed, y,(CA)=sup{u(E) |ECCA, E closed}. Thus CA efl, and
Q. is an algebra containing the closed sets 6 . In view of our previous
remarks then u. is a regular additive nonnegative set function on
It only remains for us to prove that И(х)=^ xdq, for each x€C^(T,R);
in fact we need only establish the inequality h(x) < j* xdp. for each nonzero
xfC^(T,R) for the inequality for x=0 is trivial and the reverse inequality
follows from this one with the substitution of -x for x. We obtain these
inequalities by constructing a simple integrable function F with the prop-
erties that J'Fdu be less than or equal to J’xdu + €p,(T) and h(x) be less
than or equal to j’Fdu plus a function of c (c > 0) which tends to zero as
€ - 0+.
(d) h(x) < Jxdp, for xfC^(T,R); Let x be a nonzero element of C^(T,R) and
consider 0 < ( <||x|| . First we partition T into a finite number of dis-
joint sets B, f Ob on each of which the oscillation of x is less than €/2.
J	к c
82
2. SPACES OF CONTINUOUS FUNCTIONS
To do this realize that x (T) c | |x | I’ | |x | | + 1) anc* choose n large
enough so that (2 | |x | | + l)/n < f/2. Then partition [- | |x | |, | |x | | + 1)
into the n subintervals of the form
Ik = С-| Iх I I + (k-D(2||x|| + l)/n, - | |x | | + k(2 | |x | | + 1Уп ]
and set B, =x \l, ) for k=l,...,n. It follows that the B, 's are pairwise
к к	к	r
disjoint elements о f а. (в, ,C CL for each k) , T=UB , and x has an
j	~c \ k	к
oscillation less than f/2 on each	B, .	Let c,	be the	right endpoint of	I,
к	к	о	-	k
plus f/2 and F=Ec, к . Then x(t)	< c	= F(t)	< x(t)	+ e for each teB,	and
К	К	К
k=l, .. . ,n. Hence
Fdp < С xdu + fp(T).
In the remainder of the proof we show that h(x) is less than or equal to
J1 Fdp plus a function € with limit equal to 0 as € — 0+. Since x < on
B. , W- =x \-oo,c, )DB, . Recalling that each B, (CL and that U is regular on
к к	к к	к с
<2. , we see that an open set U, DB, exists such that p(U,) < p(B.) + C/n
с	к к	к	к
for 1 < к < n. Setting V, =*W, О U, we obtain x < c, on V, and u(V, ) < u(B, ) +
—	—	к к к	к к	к	к
f/n for each k. Thus
Ec n(V. ) < Ec p(B ) + (Sc )e/n < p Fdp + ( II x|| + 1 + e)f.
Now
there is a continuous partition of unity [xp...,x] subordinate to
(V,,...V } so that 0 < x, < к
1 nJ	- к - Vk
and each V. is the complement of
that Vi=C(yi 1(0)) by (1.2-1) (d)
(In the case where T is completely regular
a zero set, we may select 0 <2 y. < 1 such
so that the functions x^=y^/Sy. (i=l,...,n)
sum to 1 on T and 0 < x^ < k^ for each i.) It follows that h(x^) < p(Vk)
for each k. Furthermore 0 < xx, < c, x, for each k: thus
— к к к
Let T be a topological space and x:T -> R be nonnegative and continuous.
The support of x, supp(x), is cl^f tfT |x(t)^O]. A (continuous) partition
of unity subordinate _tq a cover (A.) of T is a family F of continuous
nonnegative-valued functions (x^) on T such that for each i, 0 < x^ < 1,
£hx^(t) = l for each t€T, and supp(xp C A^ for each i. It is not difficult
to show that if T is a normal Hausdorff space and (Up...,U } an open cover
of T, then there is a partition of unity subordinate to {u^>.-.,Un}.
2.4 CONTINUOUS DUAL OF C(T,F,c)
83
h(x) = h(£xxk) = E h(xxk) < £ h(cRxk)
= E ckh(xk) < E ck^(Vk).
Combining the above inequalities, we obtain
h(x) < [ xd|j, + c (|Jl(T) + | |x | | + 1 + f)
for each € > 0 and the proof is complete. V
It is clear now that each element of C^(T,^)' is representable in
terms of an integral with respect to a bounded regular additive real-
valued set function on
az if T is completely regular and Hausdorff or a
bounded regular additive real-valued set function on if T is a normal
Hausdorff space. The result is equally clear for the complex case.
Certainly if ц, is a given bounded real- or complex-valued regular
additive set function then
IJ xdn | < J’|x |d ||i | < | |x|| |U |(T)
so that a linear correspondence ц x1 is established between the bounded
F-valued regular additive set functions and C^(T,F)1.
It follows from the inequality above that ||x' || < |ц,|(Т). To obtain
the reverse inequality, let ( > 0 be given and let	(or й)
be pairwise disjoint sets such that |p.(A^) | > ||i|(T)-€. As ц, is regular,
closed sets (zero sets) C. and open sets (complements of zero sets) U^ exist
such that C^C A^C U^ and |ц,|(и^-Ср < €./n, for 1=1,...,n. Now Cp...,Cn
are pairwise disjoint so that pairwise disjoint open sets (complements of
zero sets) V exist such that С С V. : This is clear if T is normal. When
i	it
T is completely regular and ц, is defined on CL, we demonstrate the exist-
ence of the for the case n=2; the general case follows with the aid of
the observation that a finite union of zero sets is a zero set [ (1.2 -1) (c) j.
By Theorem 1.2-2, cl C^
function xP, 0 < x^ 1
x₽(cl C )={!}.
p	-1
by (1.2-3) that Vj=x (
Now let W.=U.O V.;
and cl Cn are disjoint so that there is a continuous
3 2	R
defined on [3T such that x₽(cl C,)={0] while
₽ i
Thus, denoting the restriction of x^ to T by x, it follows
’,1/4) and V2“x ^(3/4,“>) serve the desired purpose.
W. is open when T is normal and is the complement
is completely regular. As |ц. | is monotone,
1=1,...,n. By the normality of T (or the normality of
on <2Z ) continuous functions 0 < x^ < 1 defined on T
Letting a.€F be such that
of a zero set when T
PT when p. is defined
exist such that x^(C^)={l) and x^(CW^.)={0).
84
2. SPACES OF CONTINUOUS FUNCTIONS
a p(A )=|p(A )| and setting х=2\а^х^, it follows that
Ij xdp, - E.|p(A.) || = lE.a.j' w^x.dp - | p.(A±)| |
= lEiEai.f C.xidtl ‘ MAi>H +	„.-c.x.d^l
= |E.a.EH(C.) -^(A )] + E.a.Jw._c/ d^|
11
< |ц |(a -с ) +	|u|(w -c.) < 2e.
Since the W 1 s are pairwise disjoint, ||x|| < 1 and so ||x’|| > | p | (T) - 3c.
Thus | | x11 | =| p| (T) and we may s ta te :
(2.4-5) CONTINUOUS LINEAR FUNCTIONALS ON Cb(T,F) AND REGULAR ADDITIVE
SET FUNCTIONS If T is completely regular (normal) and Hausdorff,
then Cb(T,F).',is isometrically isomorphic to the linear space of all bounded
finitely additive regular F-valued set functions defined on ^z(<2c)
equipped with the total variation norm. (When F=R these are the elements
of KI, the finitely additive measures of Sec. 1.7.)
Certainly both results of (2.4-5) apply to the compact Hausdorff
situation. Indeed in this case we may say even more. As a consequence of
our next result the set functions may be considered to be countably addi-
tive set functions defined on or @ as the case may be.
(2.4-6) (Alexandrov) A BOUNDED REGULAR ADDITIVE SET FUNCTION IS
COUNTABLY ADDITIVE If p is a bounded regular, real- or complex-
valued, additive set function defined on CL^, the algebra generated by the
closed subsets of a compact Hausdorff space T, then p is countably
additive on &c> i-e. p.(UA^)=I p(A^) w'lenever (An) a countable family
of pairwise disjoint sets from Q, with union in Q, Moreover p has a
unique regular countably additive extension to the c-algebra 63 of Borel
subsets of T, i.e. the cr-a Igebra generated by С» .
Remark As in previous results, e.g. (2.4-4), the theorem remains valid if
the closed sets "С» are replaced by the zero sets Z. , Q, by and ® by
ф the cr-algebra generated byjZl (sometimes called the Baire sets).
Proof First we show that the total variation |p| is countably additive on
CL c- Let f > 0 be given and let (A^) be as in the statement of the theorem.
Choose E ((/ such that EC A and |p | (A-E) < C and for each n S 1 let U^ be
an open subset of T containing A^ such that |p|(Un*An) < f/2n. As E is
compact, there are U^,...,Un that cover E so that
2.4 CONTINUOUS DUAL OF C(T,F,c)
85
VW-Wk) -e-Ek=i-e •
Since |p | is a nonnegative finitely additive set function, it is monotone
and finitely subadditive and it follows that
^kCN 1 M'1 (A1P E k=l 1 ^l(Uk> ' C lkL|(Uk=lUk) ' C
> I ml (E) - e > |p| (A) - 2f.
Thus |il (A|P > lU-l(A). On the other hand
Ek=llpJ (Ak> = Ml (Uk=l V |Ц,|(А)
for any n so ipl is seen to be countably additive.
As ipl is a f ini te-va lued set function on an algebra, it is bounded,
so that ml (A) = ^|p I (A^.) < 00 and
l^l(Uk inAk> ° ^k in (Ak> °
as n -• Thus
lm(A) - n(Ak)I = мик>пАк)| < mi (uk >nAk) " о
and p is countably additive on d,^.
Since a bounded real-valued additive set function is regular iff its
positive and negative components with respect to its Jordan decomposition
are regular and a complex-valued additive set function is regular iff its
real and imaginary parts are regular, we may reduce the problem of exist-
ence of an extension of p to 0 to the case where p is a regular non-
negative additive set function on 0, . Given such a set function p first
we extend it to an outer measure (i.e. a nonnegative monotone countably
subadditive set function which takes 0 into 0) p" on P(T). To this end
let
p"(s) - inf{zkeNp(Ak) |Ake CLc, scukAkl
for each SCT. Clearly p takes 0 into 0 and is monotone. To see that it
is also countably subadditiva, suppose that S=UkSk and f > 0 is given.
For each к > 1 choosfe a sequence (A, ) from CL, such that S, C U and
k	kn'	с	к nflST kn
p (S, ) > E .,p(A	) - f/2 . Now SC(J U A, so that
к nfN kn	к n kn
86
2. SPACES OF CONTINUOUS FUNCTIONS
^(S) Vn ^кЛ
On the other hand
Ek ^<Sk> > Vn ^Akn> - f
it	if
and it follows that ц, (S) <	(S,<).
To obtain a class of sets on which ц, is countably additive, we single
it	if
out the family of ц, -sets: A subset RCT is а ц iset if (this condition
sometimes being referred to as the Caratheodory condition)
Z(S) = /(SOR) + n*(SnCR)
*
for each SCT. First we claim that the class of ц, -sets is an algebra on
it	ic
which u is additive. Clearly T is a p -set, as is the complement of any
u'-set. It only remains to show that the intersection of two p -sets is
again a p'-set. Let X and Y be u -sets so that for each SCT
if
(1)	p (S) = (SOY) + u, (SflCY)
(2)	/(SCIY) = ki*((snY)("lX) + L?((SA Y)("l CX)
and
(3)	/(snc(xnY)) = p*(S("l C(Xfll)fl Y) + H*(Snc(xnY)n CY)
= Z(sncxnY) + u*(snCY).
Now by using Eqs. (2) and (3) to substitute first for ц (SflY) in Eq. (1)
it	1
and then for u- (SCI CKOY) in the resulting equation we obtain
i?(s) = u."(sn (xOy)) + u"(snc(xnY)) (SCT)
it	tV
so that XdY is а ц -set. To see that Ц. is additive on the ц -sets, first
it
observe that for disjoint U -sets X and Y
Hsa(xuy)) = p*(sn (xtjY)nx) + j'(sn (XU Y)A CX)
= /(SAX) + /(SAY).
Thus, by induction,
(a) n‘(snuk=1 xk) = /(snxk)
for any finite class of disjoint ц, -sets X^,...,Xn and any SCT. Finite
additivity follows by substituting T for S in (4).
What about countable unions of u'-sets? Are they u, -sets? Let (X. )
,	П	К
if
be a sequence of p -sets with union X; then for each n > 1, Uk=i X^ is a
U -set and for each SCT
2.4 CONTINUOUS DUAL OF C(T,F,c)
87
n	n
ц, (S) = ц, (sn Uk=1 Xk) + u, (snc(Uk=1 Xk)).
Then, by (4),
n *	-v	n
i?(s) = zk=1 /(snxk) + 4(snc(uk=1 xk))
for each nfN. By the monotonicity of ц,"
* n *
p (S) > Ek=1 4 (SOI) + 4 (ЗПСХ)
and, letting n — jj, it follows that
As) > ^eN I?(snxk) + Z(sncx).
As ц, is countably subadditive,
к	к	к	к	к	к
p (S) > 2^ p (snxk) + p (sncx) > p (snx) + p (sncx) > p (S)
so that two things follow; X is а p -set and
u*(snx) = ^kfN /(snxk)
for each SCT. By the substitution of T for S in this equation, p is seen
to be countably additive on the p -sets.
Next we show that p and p agree on Q and each element of Cl is a
j.	c	c
p -set so that p is a countably additive nonnegative extension of p to the
Borel sets ®. If A C Q-c then АСА and, by the definition of ц ,
4 (A) < p(A) . On the other hand, if (A^) is a countable family offsets
from Q, and AC U A then the sets В where В =A, and В =A *U, , A, for
c	nn	n	11 nn к=1 к
n > 2, are pairwise disjoint elements of Q, with union equal to UA^.
Hence by the monotonicity arid countable additivity of p on CL^
P(A) = Ц(АП(ипВп)) = и(ип(АПВп)) = En ц(АПВп)
< E ц(Вп) < £ p(A ).
n	n
Thus p(A) < p (A) and p |	= p. To see that each A f Q.^ is a p -set, let
SCT, f > 0, and choose a sequence (A ) from CL such that SCU A and
n	c	n n
C + p (S) > “n 4(An).
Hence, by the definition of U and its subadditivity
88
2. SPACES OF CONTINUOUS FUNCTIONS
C + MS) > Sn n(An) = Еп(ц(ЙЛА) + ц(АпПСА))
> /(SPA) + p''(SO>CA) > p*(S).
As C is arbitrary, it follows that A is а p -set.
Having demonstrated the existence of a nonnegative countably additive
extension of ц to Й we now prove its uniqueness. Let \ be another such
extension,
Since X is
В c(B, and (A^) be any sequence from Q,
countably additive and nonnegative, it is
such that BcU .A .
nflT n
monotone and countably
subadditive so that
ЧВ) М(ЦАп) < Шп) =Sn^n)-
Thus, by the definition of p , X(B) < p (B). As CB f® we also have
к
X(CB) < p (CB). Consequently
H(T) = X(T) = X(B) + X(CB) < p*(B) + p*(CB) = |/(T) = p(T).
Hence )(B)+X(CB)=u (B)+p (CB) so X(B)=p (B) for each В f (0.
Finally then it only remains to show that p is regular on . Let
В C® and choose a sequence (A^) from such that
|/ (B) >	^(An) ‘ f •
As p is regular on Л. there exist open sets U A such that
p(A^) > p(U ) - f/2n for each n. Thus
ц*(В) >'En H(Un) - 2f > u(Un'Jn) - 2f
where the last inequality follows by the countable subadditivity of p on
a .
c
<S . v
As
U r„U is
nfu n
open, p*is seen to be regular on
As was mentioned earlier there is a characterization of
C(T,F,c) for
completely regular Hausdorff spaces T in terms of set functions defined on
the Baire sets. For an outline of this result, see Exercise 2.4. At this
time we present a characterization of C(T,F,c)' in terms of "Borel set
functions" with "compact support." A set function p defined on the Borel
subsets of a completely regular Hausdorff space T has compact support if
there is a compact set КС T with the property that | p |(T-K)=O.
Theorem 2.4-1 A MEASURE THEORETIC CHARACTERIZATION OF C(T.F.c)1 If T is a
completely regular Hausdorff space then the continuous dual C(T,F,c)' of
C(T,F,c) is algebraically isomorphic to the space of countably additive
regular F-valued set functions with compact support on the Borel subsets
fl of T via the map p -• [*-dp.
2.4 CONTINUOUS DUAL OF C(T,F,c)
89
Proof Certainly the collection of all countably additive regular .F-valued
set functions with compact support is a vector space over £ and the mapping
p -* f.dp is a linear map into the algebraic dual (i.e. the collection of
all linear functionals) of C(T,F). To see that pdP is a continuous linear
functional, let К be a "supporting" compact subset for p, i.e. |p|(T-K)=O.
Then for each x f C(T,F)
IJ xdp | = (j* xdp | < J |x |d |p| < |p|(K)p (x)
К	К
and J•dp is continuous on C(T,F,c). As for 1-1-ness, suppose that P^O so
that |p|(K) > 0 where К is a supporting compact subset of T. Let
B,,...,B f ® be a pairwise disjoint decomposition of К with the property
1 n
that E|p(B.) | > (3/4) |p|(K). Since p is regular, there are closed sets C
and open sets U. such that C.CB^CUj (1=1,...,n) and
|p|(U.-C.) < (l/4n) |p | (K) for eaclb i, 1 < i < n. Utilizing the fact that
C and CU^O К are compact, it follows by the complete regularity of T that
there are continuous functions x^ on T, 0 < x < l,such that x^(CU.O K)={0}
while x.(C.)={l}. (One could find such a continuous function defined on
•3T and restrict it to T.) Choose a^=|p(C.)|/p(C^) provided р(С^)^0 and 0
otherwise, and let x=S.a.x.. Now consider
ill
jxdp - J xdp = E.a. j* x.dp = E. a . Г x.dp
к 1 1 к 1	1 1 и. 1
1
=	x. dp + j* x^dp) = n^a.J x^dp + £р(С^).
и. - c. c,	и. * c.
11 1 11
But | J" x^dp | < |p |(U^*C^) < |p|(K)/4n for each i so it follows that
Ci
|£а.^ x^dp | < |p| (K)/4. Furthermore
U.-Cf
E |H(C.) | = E |4(B. ) | - E( |p(B . ) | - | p(C. )| )
>	(3/4) |p |(K) - E |p |(B.-C.)
>	(3/4) |p|(K) - (1/4) |p |(K) = (1/2) |P|(K).
Hence | j* xdp | > (1/4) |p |(K) > 0 and the mapping is seen to be 1-1.
Finally we contend that the mapping is onto. If x1CC(T,£,c)1, we shall
show first that there is a compact set КС T and a continuous linear func-
tional f defined on C(K,F,c) such that x'(x)=f (x I ) for each xCC(T,F).
К	~	К К
90
2. SPACES OF CONTINUOUS FUNCTIONS
Since x1 is continuous there is a neighborhood of the origin V =
PK
{xfC(T,F) Ip (x) < 1} on which x1 is bounded. But the ideal p '^(O)CV
~ l К	К	Pj£
so that x1(p (O))={o}. Thus if yfC(K,F) we may continuously extend it up
since if y^ and yg are any two extensions of у to T, y^ -y^ f p^ (0) so
that x1 (y^1 )=x 1 (ye). As x'(Vp ) = fK(S^(O)) (where S^(0) is the open unit
ball in C(K,F,c)) f -is continuous.
*** к
Focusing attention on f we see by (2.4-5.6) tha t
К
additive regular F-valued set function p defined on
~	к
of К such that f^( • )=^-dp^. It onIy remains to show
there is a countably
the Borel subsets ®
К
that a countably addi-
tive regular F-valued set function p can be defined on the Borel subsets
Ф of T by the equation P(B)=P (ВПК) (Be®) for then it follows that p has
К
compact support and
x'(x) = f (x| ) = Lx | dp = J xdp
Jx. IJX. w Jx. IJX. JX. v
for each xfC(T,F). First it is necessary to show that ВПК f (53^ for each
В f®. Let JB denote the collection of all subsets of T of the form
EU(B-K) where E f ® and В f®. It is straightforward to verify that JB
is a a-algebra. Furthermore if C is closed in T then COK t ф and C -
К
(спк)и (c-k)c<S. Hence, as JO contains all the closed subsets of T, <8 э ©.
Thus <СПК={оПКр fX>] D ®ПК={ВПК|3 f®}. But Лпк= ®K so that
contains ©OK and the set function p has a meaningful definition. Obvi-
ously p is countably additive, but what about regularity? Let В f (B. As
ВПК f ® given any C > 0 there is a closed set CCK and an open subset U
К
of К such that CCBOKCU and Ip | (U-C) < f. Let U be any open subset of
К
T such that IJDCK and IJOK=U. If B^,...,B^ are pairwise disjoint Borel
subsets of T with union contained in IJ-C then U(B^OK)CU-C. Thus
Z|P(B.) | = Z ^(В.ПК) | < e
and, by taking the supremum over all such sums, it follows that |p|(U-C)^f.
Since CCBCU, p is regular and the proof is complete. V
Prior to presenting the result which both defines and establishes the
existence of "support of a linear functional" it is necessary to introduce
the notion of support for set functions.
(2.4-7) THE SUPPORT OF A SET FUNCTION If p is a nontrivial regular count-
ably additive set function on®, the Borel subsets of the compact Hausdorff
space T, then there exists a unique compact set KCT called the support of jj.
2.4 CONTINUOUS DUAL OF C(T,F,c)
91
such that
j* xdp = J* xdp
for each xfC(T,F) and no proper compact subset of К has this property. К
is denoted by supp(p).
Proof Let K=T-UxT where ^7 denotes the collection of all open subsets U
of T such that |p|(u)=O. Since p is regular then for each kCN there exists
a compact set and an open set such that K^CKCV^, |p |(V^-K^) < 1/k.
Now V^'KCV^’K^ so |p |(Vfc*K) < 1/k and it follows that
|^|(Vk) = jl-1 |(vk-K) + |p ](K) - Ipl(K).
Consider the compact set T-V^.; since K=T-Uj*, it is contained in Uj'
and, therefore, a finite collection U,,...,U exists such that
n	In
T-V^C Ch _^1L • Hence |p |(T*Vk)“O and consequently |p | (T)= ||X | (V^) for each
k. Clearly then |p | (T)= |p | (K) .
Next we note that
I Tm xd^ ' L xd^l^ Гт „ Ixl dlpl< sup|x(T)| iM-r(T-K) = 0.
v 1	' Jx.	u 1 К
Thus J" xdp=J" xdp for each xfC(T,F,c). To see that К is minimal in the
class of compact sets with respect to this property, suppose that a proper
compact subset K1 of К exists such that Г, xdp=Г xdp for each xfC(T,F,c).
We claim that a contradiction can be reached if we show that lpi(K-K')=0.
Indeed if this is so then
|p|(T-K') = |p|(T*K) + |p |(K-K') = 0.
But T-K1 is open and T*K' containsЦ<Г properly.
To see
that |p|(K-K')=0, let E^,...,Enf(B be a pairwise disjoint de-
of K-K'. By the regularity of p, compact sets C^C Ek exist
such that |p|(E^-C^) < e/n^2. Since T is normal, there are continuous func-
tions x„ (1 < i,j < njii/j) such that 0 < x^j< 1, xy(C.) = {0} and
Setting x.=S,,.x.±/n-l we see that 0 < x. <1, x.(C.)=
1 JFl ij	- 1 -	1 J
let x=E^d^x^ where d^= |p(r\) |/p(C^) if р(С.)^0 and 0 <
composition
6. .
otherwise.
Now
Consider
IVK1
xdd* = Il IP xdd* = 2 Гр xd^ = 2 ''c Xd^ + Z Г n xdP
JUEk	JEk	uCk	JE,4
k
= z dkxkd^ + z Те, -c, xd^
к	k k
=	+ s jEk-^xd^
92
2. SPACES OF CONTINUOUS FUNCTIONS
Thus Elki(C^) I = E _ -xdp. and it follows
к Tt
2 2
EIH(Ck)l < Elul (Ek-Ck)n < (e/2n )n
tha t
Cons ider
E|p(Ek)|
<	E |ix(Ck) | + E |n(Ek-Ck) |
<	E |n(ck) | + E jpl^-cp < e.
Since f is arbitrary, E|p(Ek)| =0. Finally, taking the supremum of all
such sums, we obtain |p|(K'-K)=0.
As for uniqueness suppose that C is another such set. Consider the
open set T-C. It suffices to show that |p|(T-C)=0 for it follows first
that T-C C xT . This implies that CCK and, by the minimality of K, we see
that C=K. To show that |p |(T-C)=0, we proceed in exactly the same way as
we did to prove the minimality of К and we leave the remaining details to
the reader. V
(2.4-8) SUPPORT OF A CONTINUOUS LINEAR FUNCTIONAL If T is a completely
regular Hausdorff space and x1fC(T,F,c)1 then there exists a compact subset
supp(x') of T, called the support of x1, with the properties:
(a)	whenever xfC(T,Fj vanishes on supp(x'), x'(x)=0;
(b)	if E is a closed subset of T having property (a), i.e. x'(x)=0
whenever x(E)={o], then EDsupp(x').
Proof In the proof of Theorem 2.4-1 it was established that given x1 there
is a compact set KCT and countably additive regular ^-valued set functions
p and p^ defined on the Borel sets ® of T and (Ц, of К respectively such
that pK(BOK)=p(B) for each Be® and x'(x)=[’K xdp=J X|K f°r eac'1
xeC(T,F). We claim that the support of u , whose existence was demonstrated
As Х'(х)"Т8иРР(рК)х1К 4=
x vanishes on supp(p^) and
(b), let E be closed in T with
in (2.4-7), is actually the support of x1.
Г .	. xdp for xeC(T,F), x'(x)=0 whenever
Jsupp(pl
supp(p) is seen to satisfy (a). To prove
the property that x'(x)=0 whenever x(E)={0l. It will follow that
EDsupp(pK) if we can show that ju^. | (supp (p^) О CE)= |P | (supp(p^) О CE)=0 for
then the minimality of supp(p^) ((2.4-7)) implies that supp(p^)=supp(p^)ОE.
To this end suppose C > 0 has been given and let	be a pairwise
disjoint decomposition of supp(p^)OCE. By the regularity of there are
pairwise disjoint compact sets Kp...Kn such that B^DIC and | p |(В^-К^)<е/2п^
2.5 BARRELEDNESS OF C(T,F,c)
93
for 1=1,...,n. Thus
E|h(b.) | < £|u(k )| + e/2n.
Since K.
i
is compact and disjoint
from the closed set EUU.,.K
J Fl i
there is
a continuous real-valued function x. defined on the
i
Hausforff space T such that 0 i x^ < 1, x^(K^)={l},
Let x=Zd^x. where d^= |p(IC ) |/p(IC ) if ц,(К.)^О and 0
completely regular
and x. (E J Uj5tlK ) = f0-|.
otherwise. As x
vanishes on E we have
0 = x1 (x) = Г ,	. xdp - Г , , _ „„xdp = £ Г xdp
•IsuppCu^) JsuppCp^jn CE p JBi
= S J xdp t E Г xdp = E!p(K )j t S Г xdp, .
Ъ.Х ,	«J D , К .	1	11 D, К
1 11 11
It is clear that 0 < x(t) < n for each tcB^-IC and each 1=1,...,n; hence
E ||J.(K ) | < C/2 and it follows that Е|н(В^)| < f. Since the disjoint decom-
position (B^) and C are arbitrary, |p |(supp(p^)ОCE)=0 and the proof is
complete. V
2.5 Barreledness of C(T,F,c) The notion of barreled space, introduced in
Bourbaki (1953a), has its roots in a desire to determine a wider class of
spaces for which a Banach-Steinhaus principle holds, taking as that prin-
ciple the statement: If H is a family of linear maps of the Banach space X
into the normed space Y such that H(x)={h(x) |hCH] is a bounded subset of Y
for each xfX, then IIHII is bounded. (In function-theoretic language, H is
uniformly bounded.) Thus, in the situation just mentioned, pointwise
boundedness implies equicontinuity. The intrinsic properties of spaces
yielding this kind of connection are principally what was sought. In view
of this desire it is natural to define barreledness externally, via condi-
tions on the space of continuous linear maps of such spaces: The locally
convex space X is barreled if for any locally convex space Y, any pointwise
bounded family H of continuous linear maps of X into Y is equicontinuous.
The external description is equivalent to the internal description: A
locally convex space X is barreled if every barrel in X is a neighborhood
of 0.
Obviously Banach spaces are barreled; incomplete normed spaces, how-
ever, needn't be. Thus if T is compact and Hausdorff, C(T,F,c) is barreled,
but there should certainly be completely regular Hausdorff spaces T for
which C(T,F,c) is not barreled. Necessary and sufficient conditions on T
94
2. SPACES OF CONTINUOUS FUNCTIONS
for C(T,£,c) to be barreled were
obtained independently by Nachbin (1954)
and Shirota (1954) - C(T,F,c) is barreled iff for each closed non-cotnpact
subset E of T there exists xf(T,F^) which is unbounded on E - and this sec-
tion is devoted to proving that result. A later ramification, due to
Warner (1958), concerning necessary and sufficient conditions for C(T,.F,c)
to be infrabarreled is also presented: C(T,.F,c) is infrabarreled iff for
each closed non-compact subset E of T there is a nonnegative lower semi-
continuous function у on T which is unbounded on E and bounded on each
compact subset of T.
Theorem 2.5-1 BARRELEDNESS OF C(T,_F.,c) If T is a completely regular
Hausdorff space then C(T,£,c) is barreled iff for each closed non-compact
subset S of T there is some xfC(T,£) which is unbounded on S. We refer to
such spaces T as NS-spaces.
Proof To see that the condition is necessary, suppose that C(T,F^,c) is
barreled. Let S be a
[xCC(T,F)| sup )x(S)l <
which is unbounded on
closed non-compact subset of T and
The requirement that there be
consider Vp -
some xfC(T,^)
being absorbent.
Since V„ is almost a
PS
the necessity of the
neighborhood of 0 in
S is clearly equivalent to V„ not
PS
barrel, being closed and absolutely convex, to prove
neighborhood of 0 so
condition, it is enough to show that V„ is not a
PS
the barreled space C(T,F,c). Suppose that VD is a
PS
that a compact set К and a positive number C exist
This, however, implies that SCK for if tfS-K
such that eVn C V_
PK ps	, ,
there must be some xCC(T,£) such that x(t)=2 while x(K)=(0}. Hence
but x^V which is contradictory. Thus CV С V
P$ pK ps
means that S is compact, contrary to assumption.
no basic neighborhood of 0. The necessity of the
implies that SC К
then
*e(ev )
PK
which
contain
Therefore V can
pS
condition is now evident.
To establish the sufficiency, we shall prove
indirectly that an arbi-
trary barrel V is a neighborhood of 0. First we use the barreledness of
the Banach algebra (with sup norm) of uniformly bounded continuous ^-valued
functions on T to construct a closed subset К of T with the property that a
scalar multiple of V is contained in V. Then, to show that К is compact,
P К
we assume that it is not and, by using the condition, we obtain a contradic
tion to the fact that V is absorbent.
To this end let Y be the subalgebra of C(T,F.) consisting of all uni-
formly bounded continuous functions. Equipped with the uniform norm, Y is
a Banach algebra, and, as such, it is barreled. Now VOY is certainly
absolutely convex and absorbent. That it is closed in Y and therefore a
2.5 BARRELEDNESS OF C(T,F,c)
95
barrel in Y follows from the facts that V is closed in C(T,F,c) and that
the norm topology is finer than the induced compact-open topology of Y-
Thus VOY is a Y-barrel so a d > 0 exists such that
dV C VPYC V.
PT
Since 2y
d so
and x
Next we establish the important technicality that for a subset SCT if
all the elements of C(T,F) that vanish on S belong to the absolutely convex
set V, then aV„ С V for some a > 0, that is if the elements which vanish
PS
on S must belong to V, then so must all sufficiently small elements on S
belong to V. First we consider the case Fy=R. Our choice for a is d/2.
Suppose that Pg(x) < d/2 and let y=max(x,d/2)+min(x,-d/2).
vanishes on S, 2yfV. Moreover, it is easy to see that p^(2(x-y)) <
2(x-y)fV. Hence, by the absolute convexity of V, it follows that
x=(1/2)(2y)+(1/2)(2(x-y))fV which is the desired conclusion.
Next let £=£. In this case a=d/4 will suffice. If Pg(x) < d/4
is the real part of x, then Pg(xr) £ d/4. It follows as in the real case
(by choosing y=max(x^,d/4)+min(xr,-d/4)) that 2x^fV. Similarly, letting
denote the imaginary part of x, 2x^CV. Thus by the absolute convexity of v,
x=(l/2)(2xr) + (i/2)(2xi)fV.
In view of the result just established, it is now a matter of producing
which has the property that each of the elements vanishing
V. Furthermore V=V°°, .and for x to belong to v°° it cer-
for each x'fV° to vanish on x. We also know that supp(x'),
has the property that x'(x)=0
natural choice for К is the
a compact set К
on it belong to
tainly suffices
the support of x' (as defined in (2.4-8)),
whenever x vanishes on supp(x'). Thus the
sure of Ux,pyO supp(x'), since whenever x
x ' fV° and xfV°°=V.
vanishes on it, x'(x)=0 for
It remains only to show that K=cl ((j , o supp(x')) is compact;
1 X tv
in doing this that we use the hypothesis for the first and only time.
pose that К is not compact. Then,
yfC(T,F) which is unbounded on K.
tute a.decreasing sequence such that О cl U =0 and U (W0 for each nCN.
n n	n	~
Since cUn supp(xn'), it follows
vanishes
n
Since the sets
clo-
each
is
Sup-
by the condition, there is an element
Thus the open sets U = |y| l(n,co) consti-
Choose x 'fV° such that U Osupp(x ')^0.
n	n	n
that a function x fC(T,F) exists such that x
n ~	i
Moreover we may assume
creasing sequence and
of complements of the
tha t x '(x )=1.
n n
П „cl U =0, it
nfN n
is increasing
on CU and x '(x )^0.
n n n
(cl U ) are a de-
n
follows that the family (C(cl U ))
and (j C(cl U )=T. As
neN 4 n
96
2. SPACES OF CONTINUOUS FUNCTIONS
x (c(cl U ))=ГО] for tn > n then, for any sequence (c ) of complex numbers,
mv v n 1 J	n
х=Г c x reduces to a finite sum of continuous functions on each of the
nfjl n n
open covering sets C(cl U ) and it follows that xfC(T,F). Ey (2.4-8) each
of the sets supp(xm') is compact so, for m fixed, cl Un fl зирр(х^')=0 for
all but finitely many values of n and, by going to a subsequence of the
U 's if necessary, w’= may assume that cl U О supp(:; ' )=0 for all n > m.
n	7	n	m
Thus, for each mCN , c may be chosen so that
~ m
x ' (x) = Ecx ' (x ) = c +Em_}c x ' (x ) = m.
m	nnmvn' m n=l n m 4 n'
To complete the proof we show that V cannot absorb x. Let a be an arbi-
trary positive number and suppose that xfaV. Since aV=(a ^V°)° the absurd
conclusion follows that (a ^)x 1(x)=(a ^)m < 1 for each m> 1. V
The one and only time that the condition was used in proving its suffi-
ciency for C(T,F.,c) to be barreled was in establishing the compactness of
the set cl(Ux> ?o supp(x’))
for the arbitrary barrel V.
Therefore it
fices to substitute the condition:
cl(U supp(x'))
X tD
is compact for
suf-
each
weakly bounded set BcC(T,F,c)' for the condition stated in the preceding
theorem to obtain the conclusion that C(T,£,c) is barreled. Conversely, if
C(T,F.,c) is barreled, then the condition of Theorem 2.5-1 holds and it fol-
lows from the proof of the
each barrel V of C(T,F,c).
theorem that cl(Uxi^^o supp(x')) is compact for
Since a set BCC(T,F,c)' is weakly bounded iff
it is contained in the polar of some barrel, cl(Ux,fj supp(x')) is compact
for each weakly bounded B. Thus we have established another equivalent con-
dition for C(T,_F,c) to be barreled and it is recorded in our next result.
If BC C(T,F,c)1 the set cl((Jxi supp(x')) is called the support of В
and is denoted by supp(B).
(2.5-1) SUPPORT AND BARRELEDNESS Let T be a completely regular Hausdorff
space. C(T,F,c) is barreled iff supp(B) is compact for each weakly bounded
set BC C(T,F,c) ' .
As every barreled space carries its Mackey topology, C(T,F,c) carries
the Mackey topology whenever the support of each weakly bounded set
BCC(T,F,c)' is compact. Actually for C(T,F,c) to carry its liackey topology
this condition need only hold for absolutely convex weakly compact B. We
prove this and its converse next.
(2.5-2) SUPPORT AND MACKEY TOPOLOGY If T is a completely regular Hausdorff
space then C(T,F.,c) carries Mackey topology iff supp(B) is compact for each
weakly compact absolutely convex set BCC(T,F,c)'.
2.5 BARRELEDNESS OF C(T,F,c)
97
Proof Since a locally convex Hausdorff space carries its Mackey topology
iff every weakly compact absolutely convex subset of its dual is equicon-
tinuous, we shall show
equicontinuous iff its
a compact set КС T and
that supp(B)C К and is
that the weakly compact absolutely convex set В is
support is compact. If В is equicontinuous there is
a positive number a such that BC(aV )°. We claim
Pj£
therefore compact. Suppose that xfC(T,F) vanishes
on K.
Then p (mx) < a
К
for each integer m and it follows that
|x ' (mx) | < 1
for each such m and any x'fB. Thus x'(x)=0 and, by the minimality of
supp(x'), supp(x')CK for each x'fB. Hence supp(B)c K.
Conversely, suppose that supp(B) is compact. Since В is weakly compact,
the set B(x) = {x'(x) |x ' CB} is bounded for each xfC(T,F). Corresponding to
each x'fВ we define a linear functional x" on C(supp(B),F) by the formula
x"(y)=x'(y') where ypC(supp(B),F) and y' is any continuous extension of у
up to T. That x" is well-defined follows immediately from the observation
that any two extensions agree on supp(B), therefore on supp(x'). To see
that x"eC(supp(B),F,c)', it is necessary to show that x" is bounded on
[yeC(supp(B),F) | sup | у (supp(B)| < 1]. Thus it suffices to show that x' is
bounded on V	={xCC(T,jj)lsup (x(supp(B)[ < 1]. To this end, suppose
psupp(B)
that x is real-valued and sup | x(supp (B) )| < 1. Let w=max(x, l)+min(x, - 1)
Then w vanishes on supp(B), which contains supp(x'), so x'(w)=O. By the
continuity of x' there is a compact set К such that x' is bounded on V ,
___________________________________________________________________ Pj£
i.e. there is a positive number M such that x'(z) < M for all zCV .
PK
Since x-w is clearly bounded by 1 on all of T, it follows that [x'(x-w)|<
Having shown that each x"fC(supp(B) , F, c) ' , we set B"={x" |x ' рв] . As
B"(y)=B(y') for each yfC(supp(B),F) and each continuous extension y' of у
T, we see that B" is a pointwise bounded collection of linear functionals
M.
to
on
the Banach space C(supp(B),F,c). Hence, by the Banach-Steinhaus theorem,
B" is equicontinuous, i.e. for each a > 0 there is a positive d such that
sup B"(y) < a whenever sup |y (supp(B) )l < d. By the definition of B" it fol-
lows that sup|B(x)[< a whenever sup |x (supp(B) )| < d and В is equicontinuous.?
In proving that the weakly compact set В is equicontinuous whenever
supp(B) is compact, the weak compactness of В is used only to guarantee that
B(x) is bounded for each xfC(T,F). Consequently the implication is also
valid for weakly bounded B.
(2.5-3) SUPPORT AND EQUICONTINUITY A weakly bounded subset В of C(T,F,c)',
T completely regular and Hausdorff, is equicontinuous whenever supp(B) is
c ompa c t.
98
2. SPACES OF CONTINUOUS FUNCTIONS
We now make use of (2.5-3) in establishing a necessary and sufficient
condition for C(T,J^,c) to be infrabarreled, a condition analogous to the
one given in Theorem 2.5-1.
(2.5-4) INFRABARRELEDNESS OF C(T,F,c) Let T be a completely regular
Hausdorff space. C(T,^F,c) is infrabarreled (i.e. every bornivorous barrel
is a neighborhood of 0) iff for each closed noncompact subset S of T there
is a nonnegative lower semicontinuous function у defined on T which is un-
bounded on S and bounded on each compact subset of T.
Proof Recall that a locally convex Hausdorff space is infrabarreled iff
each strongly bounded subset of the continuous dual is equicontinuous
(Horvath 1966, p. 217, Prop. 6).
Now, to prove necessity, let S* be the homeomorphic image in the weakly
topologized space C(T,F,c)' of the closed non-compact set S, i.e. the set
of evaluation maps t* as t runs through S. Since S* is not weakly compact,.
3" is not equ Lcontinuoits; moreover since the space
C(T,F,c) is infrabarre .ed, S* is not strongly bounded (Horvath 1966, p. 217,
Prop. 6). Thus a weakly bounded set A C(T,F,c) exists such that A(S)=S*(A)
is unbounded. Consider the nonnegative function t — у (t )=sup |A (t )| . As A
is weakly bounded, t*(A)=A(t) is bounded for each tfT and у is real-valued.
Moreover each function t -» jx(t)j is continuous on T so the map
t -> sup^f A>x (t )1 is lower semicontinuous (Dieudonn^ 1970, (12.7.6), p. 25).
Finally sup iy (S)| =sup |A (S)| = 00 while for each compact KCT, sup|y(K)(=p (A)<
К
since A is bounded and у is the desired function.
Conversely suppose that the condition holds. We must prove that each
strongly bounded subset HC2C(T,F,c)' is equicontinuous. Since H is weakly
bounded it suffices by (2.5-3) to show that supp(H) is compact. Assuming
that supp(H) is not compact, we use the condition to contradict the strong
boundedness of H. Indeed if the closed set supp(H) is not compact there
is a real-valued nonnegative lower semicontinuous function у defined on T
and unbounded on supp(H). Thus у ^(n,“) meets supp (H) for each n. As
у \n,m) is open and supp (H)=cl ((_) , supp(x')) then, for each nfN, there
t H	~
is an x 'fH whose support meets у (n,m). Next we construct a weakly
n
bounded subset {y^} of C(T,F,c) on which H is unbounded, thereby establish-
ing the desired contradiction. First note that supp(xn')^.y ^(-°=,п] so
that by (2.4-8)(b) a function xn€C(T,F) exists such that x^ vanishes on
у ^(-®,n] and x^1 (x ) = 1. We define a weakly bounded sequence of functions
yn=£^_^a^x^ such that xn'(yn)=n for each n. To see that the set {y^} is
bounded in C(T,F,c) - hence weakly bounded - let К be any compact subset
2.6 INFRABARRELEDNESS OF C(T,F,c)
99
of T. Since у is bounded on К, у (n,«’)Ok=0 for all n larger than or
equal to some tn. Thus for each tfKC у	xn(t)=O for n > tn so that
У (K)=y (K) for all n > tn. Hence with p„(w)=sup |w(t)| , [p (y ) | nCN} is
a bounded set for each К and {y^} is bounded. As for the specific values
of a , they are easily established by the equations E ,П ,a, x 1 (x, )=n (n=l,
n	k=lknk
2,...). This establishes the contradiction and completes the proof. V
2.6 Bornologicity of C(T,F,c) A linear map between normed spaces is con-
tinuous iff it maps bounded sets into bounded sets. A wider class of
spaces for which boundedness‘implies continuity is the bornological spaces,
first defined by Mackey (1946). The external description of such spaces is
that a locally convex space X is bornologica1 if, for any locally convex
space Y, a linear map A :X -> Y which maps bounded sets into bounded sets
must be continuous. An equivalent internal description is that a locally
convex space X is bornological if each balanced convex bornivore in X is a
neighborhood of 0.
There are incomplete normed spaces which are bornological (since they
are metrizable) but not barreled. In the converse direction, each com-
plete bornological space must be barreled and, as will be shown in this
section ((2.6-1)), if C(T,F,c) is bornological, it must be barreled. Quite
early on Dieudonne (1953) raised the question of whether there are barreled
spaces which are not bornological. Shortly afterwards the question was
answered in the negative by Nachbin (1954) and Shirota (1954) who determined
necessary and sufficient conditions on T for C(T,R,c) to be barreled (Theo-
rem 2.5-1; C(T,F,c) is barreled iff T is an NS-space) and to be bornologi-
cal (Theorem 2.6-1): The bornologicity of C(T,R,c) is equivalent to T
being replete. Thus to exhibit a barreled space which is not bornological,
it suffices to produce an NS-space which is not replete, and this is done
in Example 2.6-1.
The subspace X of the normed space consisting of the sequences (Pn)
which are eventually 0, being dense in > has as its cont:i-nuous dual x'.
Since X is metrizable, X is bornological. But there are weakly bounded sets
in X' - {(U'n)e^2 llkLnl- n’ f°r example - which are not strongly (=in
the norm here) bounded, hence not equicontinuous. Hence the Banach-Stein -
haus theorem does not hold and X is not barreled.
100
2. SPACES OF CONTINUOUS FUNCTIONS
Theorem 2.6-1 C(T,F,c) IS BORNOLOGICAL IFF T IS REPLETE For C(T,F,c) to
be bornological it is necessary and sufficient that the completely regular
Hausdorff space T be replete, i.e. uT=T-
Proof Necessity: Suppose that T is not replete, so that there is some
t еьТ-Т. It follows that the map t ”, x -> x“(t ), xv denoting the unique
о	о	о
continuous extension of xfC(T,F) to oT, is a nontrivial ^F-valued homomor-
phism of C(T,F). Only those t*'s from T can produce continuous homomor-
phisms of C(T,F,c), so t* must be discontinuous.^ To conclude that C(T,F,
c) is not bornological, we need only show that t" maps bounded sets into
bounded sets. If t does not have this property then there is a bounded
set BCC(T,F,c) end a sequence (x^) of points of В such that xn^(tQ) =
t“(x ) -» ro. Now consider the open sets Vn={tfPT| x u(t) > xnwito)
As the V 's are neighborhoods of t euT, there is a point sf(Ov )ПТ by
n	о	n
Theorem 1.5-l(b). But SfClV implies that s*(x )=x u (s) -> ® which contra-
il	n n
diets the fact that s’" (which must be continuous since SfT) must take В
into a bounded subset of F. Hence t^(B) is bounded and the necessity of
the condition has been demonstrated.
Sufficiency: In proving that C(T,F,c) was barreled in Theorem 2.5-1, an
arbitrary barrel V was shown to be a neighborhood of 0 in three steps.
First the existence of a d > 0 was established such that dV С V. It
PT
was proved next that for any subset S of T with the property that x?V
whenever x(S)=[o}, there was an a > 0 such that aVp С V. It then only
remained to produce a compact subset К of T with the above property. A
similar approach is used here. For C(T,F,c) to be bornological, it is
necessary and sufficient for each absolutely convex set that absorbs all
bounded sets to be a neighborhood of 0. We shall prove that a stronger
condition actually holds, i.e. that every absolutely convex set V that
absorbs all bounded sets of a certain type (order segments) is a neigh-
borhood of 0. First we define an order segment and prove it to be
bounded. If x and у are real valued continuous functions on T such that
x < y, then the order segment [x,y] consists of all wfC(T,R.) such that
x < w < y. If К is compact then p (w) < max (p (x), p (y)) for each
К	к к
wffx.y] so that [x,y] is bounded.
^An apology is perhaps due the reader here for this result is proved in
Example 4.10-2. Our desire to place Theorem 2.6-1 near to closer rela-
tives motivated us to locate it here, rather than in Chap. 4.
2.6 INFRABARRELEDNESS OF C(T,F,c)
101
Proceeding in the fashion outlined above we show that a d > 0 exists
such that dV С V. Since V absorbs
PT
such that [-bl,bl] С V. Now choose
all order segments there is a b >
d—b/2 and suppose
that pT(x) < d.
it follows that xp[-bl,bl]С V. In the event that F=C we see that
0
If
2x
r
and 2x^ belong to [-bl,blj where x and x are the real and imaginary parts
of x respectively. Thus, by the absolute convexity of V, x=(l/2)(2x^)+
(i/2)(2x^)eV. Now, as was shown in the proof of the previous theorem, for
any subset S of T with the property that x belongs to the absolutely convex
set V whenever x(S)={0}, there exists a
Thus it is just a matter of producing a
positive number
compact S with
a such that aV CV.
PS
the above property.
To do this we begin by defining the
notion of a support set of V.
closed subset К of gT is a support set of V if the continuous function xfV
whenever x^(K)=0 (here x₽ denotes the unique extended real-valued extension
of x€C(T,R) to gT which exists by (1.5-1)). An example of such a set is gT
Itself. The intersection of all such support sets is called the support of
V and is denoted by K(V). After showing that K(V) is a support set of V we
shall complete the proof by showing that K(V)С T.
The fact that K(V) is a support set will be established with the aid of
two facts:
(1) A closed subset К of gT is a support set of V iff xfV whenever x^
vanishes on a neighborhood of К (i.e. a superset of К in gT whose interior
contains K).
(2) Any finite intersection of support sets is a support set of V.
One half of (1) is trivial. To obtain the other half, suppose that the
condition holds, i.e. that xfV whenever xP vanishes on a neighborhood of K.
To see that К is a support set of V, suppose that xP vanishes on K. It re-
mains to show that x^ vanishes on some neighborhood of К for then x will
belong to V. First suppose that F=R. and let G={tfgT| x^(t) < d/2 for d
such that dvn C v}. G is
PT
y=max(x,d/2)+min(x,-d/2)
clearly an open neighborhood of K.
and claim that (2y)^ vanishes on G.
Next we define
By the unique-
ness of the extension w -* w^, it now follows that y^-max(x^, (d/2) 1) +
min(x^,(-d/2)1). Hence (2y)^ must vanish on G and 2yfV- Furthermore
2(x-y)edVp CV and therefore x=( 1/2) (2y+2 (x-y) )ev. If F=C, we define G to
be {t€ gT | x £(t) , xi₽(t) < d/4}, yr=max(xr, (d/2) l)+min(xr, (-d/4) 1) , and
y1=max(x1,(d/4)l)+minx(x1>(-d/4)1). After observing that 4(xr-yr) and
4(x.-y.) belong to dVp С V, it follows as in the real case that 2x^ and 2x^
belong to V- Thus x=(I/2)(2x )+(i/2)(2x )ev.
A
102
2. SPACES OF CONTINUOUS FUNCTIONS
To establish (2) it certainly suffices to show that the intersection
of two support sets of V, X and Y, is a support set of V. Set K=XOY and
sct 'ose that xP vanishes on W, an open neighborhood of К in gT. Now X and
Y-W are closed and disjoint in the normal space gT so there are disjoint
open sets U and P such that XCU and Y-WC P. Since gT is normal, open
sets W. and W, exist such that КС W, C cl W.C U and Y’WCW.Cc 1QW„ CP. By
R Л i
the normality of gT there exists zfC^(T,R.) such that zP(cl	and
z^(cl W-)=[o}. Clearly 2xz vanishes on (WUW„)OT. Since WUW„ is open
g	Z	z
in gT and T is dense in gT,
WUW, = (WUWJOcl TCcl ((WUWo)nT).
2	2 g g 2
Thus by the continuity of (2xz)P, (2xz)P vanishes on cl (WUW9)OT) and
therefore also on WUV^. Now we can use the facts that Y is a support set
of V and WUW£ is a neighborhood of Y together with (1) to conclude that
2xzfV. In the same way the extension (2x(l-z)) vanishes on and, there-
fore, 2x(l-z)fV. Finally x=( 1/2) (2xz)+(1/2) (2x( 1-z))eV and it follows by
(1) that К is a support set of V.
Having established (1) and (2), we are now ready to prove that K(V),
the support of V, is a support set of V. To this end suppose that x^ van-
ishes on an open neighborhood W of K(V) in gT. Since gT is a compact
Hausdorff space and K(V) is the intersection of all support sets of v,
there ari support sets K^,...,K such that CIKCW. By (2), О К is a
support set of V so that x must belong to V since xP vanishes on the open
neighborhood W of ("IfC. Thus (1) may be invoked again to conclude chat
K(V) is a support set of V.
The final thing to be shown is that K(v)C T. Let tf gT_T; we shall
prove that t£ K(V). By Theorem 1.5-l(b) and the repleteness of T, there is
a decreasing sequence (Wn) of closed neighborhoods of t such that (^nwn)^T
=0. We claim that at least one of the sets 0T - int W^ is a support set of
V- As tr int W , it follows - after establishing the claim - that t^K(V).
n
Suppose that none of the sets gT-int Wn is a support set of V. Then, for
each nfN, there is an element х^ссСТ,£,c) such that xnP(gT-int Wn)=0 and
x^ ( V. Let y=supnn|xn(. To see that у is continuous on T, fix a positive
integer m and consider any n > m. Clearly T-W^C gT-int Wn and, therefore,
x vanishes on T_W . Thus y=max( Ix, 1 ,2 lx„| , . . . ,m |x | ) on T_W so that
n	m	12	m	m
у I ,, is continuous on T-W . As (T-W ) is an increasing sequence of open
IT-Wm	m	m
sets, whose union is T, we see that у is continuous on T. Since V absorbs
all order segments, there is a positive number к such that [-y,y]ckv.
2.6 INFRABARRELEDNESS OF C(T,F,c)
103
Now if ,F=R. then for each n, nx ([-у,у]Ckv and x^fV for all n > k. On the
other hand, if £=£, then 2п|х£п1^ у for each n and it follows that the real
and imaginary parts of 2x£n belong to V for all n >k. Hence, by the abso-
lute convexity of V, x. fV for all n > k. Thus in both cases, the con-
J	2n
tradiction that x fV for some pfN has been established. We conclude that
P
one of the sets gT"int W^ is a support set of V from which it follows that
t/K(v).
In summary we have shown that K(V) is a support set of v, contained in
T so, for some a > 0,
Recall (Sec. 1.5)
aV„ С V and the proof is complete. V
PK(V)
that a subset E of the completely regular Hausdorff
space T is relatively pseudocompact if the restriction x I of each x^C(T,R)
is bounded and ((1.5-4)) that E is relatively pseudocompact iff cl ECUT.
В
If T is not an NS-space, then there is some closed noncompact subset E of
T such that each xfC(T,R) is bounded on E, i.e. E is relatively pseudo-
compact, or, equivalently, cl ECuT. If T were replete, then cl ECT which
В	в
implies that E is compact, contrary to assumption. Thus if T is not an
NS-space, T is not replete. In view of Theorems 2.5-1 and 2.6-1, then:
(2.6-1) C(T,F,c) BORNOLOGICAL-» C(T,F,c) BARRELED Let T be a completely
regular Hausdorff space. If C(T,£,c) is bornological then it is barreled.
As was noted at the beginning of the section the characterizations of
barreledness and bornologicity of С(Т,£,с) given in Theorems 2.5-1 and
2.6-	1 make it possible to construct an example of a barreled space which is
not bornological.
Example 2.6-1 AN NS-SPACE WHICH IS NOT REPLETE Let p be the initial ordi-
nal of the fourth class and [0,p) the collection of all ordinals smaller
than p equipped with the interval topology. T is defined to be .lie sub-
space of [0,p) obtained by deleting from [0,p) all non-isolated points
(i.e. limit ordinals) with a countable neighborhood base.
(a)	T is an NS-space. We claim that for any non-compact subset S of
T there is an x€C(T,R.) which is unbounded on S. As any such S must be
infinite, a countable sul set C of S must exist. The existence of the de-
sired unbounded function will follow once we show that C is closed and
*The finite ordinals comprise the first class, the denumerable ordinals the
second class, those with the same cardinality as 0, the first uncountable,
ordinal, constitute the third class, and the fourth class consists of all
those with the same cardinality as the first ordinal larger than all those
of the third class.
**The interva 1 topology of [0,Q) has as a base intervals of the form (a,b]
where a,bf[0,fi) and a < b.
104
2. SPACES OF CONTINUOUS FUNCTIONS
discrete and T is normal, for then an unbounded continuous real-valued
function on C exists which can be extended to T. The normality of T can
be obtained by virtually the same argument used to prove that any of the
ordinal spaces [0,a) is normal (see, for example, Dugundji 196b, Chap. VII,
Sec. 3, Ex. 2). The closedness as well as the discreteness of C will fol-
low once we show that an arbitrary countable intersection of open subsets
of T is open in T. Indeed if this is the case, then any countable union
of one-point sets in the T^ space T is closed. Furthermore, writing C=
[t } there are open subsets U of T with the property that t fU and
1 n	nm	n nm
t (U for n^m. Thus the open set О . U contains t and excludes each
m nm	m^n nm	n
t for n#n. Thus C is discrete in the subspace topology provided count-
m
able intersections of open sets are open.^"
Let (G ) be a sequence of open sets and bfOG . If b is isolated,
n	n
then it is an interior point of OG . If, on the other hand, b is not
n	’
isolated then there is an increasing sequence (a^) of ordinals a f[0,b)
such that (an>b]OTCGn for each nfN. Letting a=sup a^, it follows from
the fact that there is no countable neighborhood base at b that a < b and
so (а,Ь]Г)ТС ClG^. Thus О Gn is open and, by the above discussion, T is
an NS-space.
(b)	T is not replete. Let S=TU{p] inherit the subspace topology of
Co, ц] with the interval topology. Since a+1 is isolated and belongs to
(а,ц,]ОТ for each a f[0,p), p, is a limit point of T in S and therefore T is
dense in S. Once it has been shown that each xfC(T,R.) can be extended to
an x'ec(S,R), it will follow, by Theorem 1.1-1, that 0T and |3S are equiv-
alent compactifications of T. In fact we may then identify fBT and gS and
view S as a topological subspace of (ЭТ.
We contend that each xfC(T,R.) is actually constant on a tail of T,
i.e. on a set of the form [а,ц)ОТ, and is therefore extendible to S. In
proving this, a few technical results are needed which we establish now.
A subset F of T is said to be cofina1 in T if for each SfT there is an
a?F such that a > s.
(i)	If the cardinality #F of F is	t'ie cardinality of the first
uncountable ordinal Q, then F is not cofinal in T.
tThe condition on T that each countable intersection of open sets is open
is equivalent to requiring that each prime ideal of C(T,R.) be maximal, i.e.
that T be a P-space, (Gillman and Henriksen 1954, p. 345; cf. also Gillman
and Jerison 1960, p. 63).
2.6 INFRABARRELEDNESS OF C(T,F,c)
105
Proof (i) : It follows from the fact that a+1 fT for each а сГО.ц,)
that T is cofinal in [0,ц,); thus any set F which is cofinal in T is also
cofinal in [0,p.).
cardinality of [0,a)
and (i) follows.
Hence if F is cofinal, [0,p,) = Ua^p[0,a).
and the cardinality of a are the same,
Since the

(ii)	The supremum of each subset E of T of order type 0 (i.e. order-
isomorphic to be [0,Q)) belongs to T.
Proof (ii): If E is such a set and a=sup E, then a since [0,0)
does not contain sup [O,C2)=C2. Thus there is an order-isomorphism of
EUfa) onto [0,Q] mapping a into 0. The assumption that a (T is equiva-
lent to assuming the existence of an increasing sequence of elements from
E which converges to a. Thus if a ^T, there is an increasing sequence of
ordinals from Co,fl) convergent to 0 which is contradictory.
(iii)	If A and В are closed disjoint subsets of T then one of them
must be bounded, i.e. not cofinal.
Proof (iii) : Under the assumption that A and В are closed, disjoint,
and cofinal, we use transfinite recursion to construct an element in АО В
to reach a contradiction. Suppose that f is a map taking [0,a) into AUB
where a < 0. Then, as f([O,a)) is not cofinal in T by (i) while A and В
are cofinal in T, there must be elements of A and elements of В larger
than sup f([O,a)). Let a^ and b^ be the smallest elements of A and В
larger than or equal to sup (f ( [0,a) )O A) and sup(f ( [0, a) )O B) respectively.
Since AOB=0, a^b^ so that one must be larger than the other. If a^ > b^
then define h (f) to be the smallest element of В larger than ar; in the
a	f
event that b^ is the larger, ^^(f) is to be the smallest element of A
larger than 1\.. Now by the principle of transfinite recursion, there is
a unique function F:[0,C2) -<AUB such that F(a)=h (F.r„ .) for each a<0.
a | Lu>a1
Now we contend that F is an order-isomorphism. To see this, let P be the
*The principles of transfinite recursion and induction are (Dugundji 1966,
p. 40, 5.2 and 5.1):	(a) Recursion. If W is a well-ordered set and E an
arbitrary class such that for all xfW there is a rule R^ that associates
with each map f:[y€W|y < x] - E a unique Rx(f)eE, then there is a unique
map F:W -> E such that F(x)=R (F.r .
x ty^W у < x
If Q is a subset of the well-ordered set W
) for each xfW. (b) Induction
such that [yfW|y < x]cQ — xEQ
for each xfW, then Q=W.
106
2. SPACES OF CONTINUOUS FUNCTIONS
collection of all a€[0,o]
use transfinite induction
such that F,r„ , is an order-isomorphism. We
|C°.a)
to show that Р=[0,П]. Suppose that [0,a)c P,
i.e. F,r is an order-isomorphism for each a'<a, and let b and c
|L°>a )
elements of ГО,а) such that b < c. Then F(c)=h (Flr„ .). Set a . and
L	c [0,c)	f
equal to the smallest elements of A and В respectively larger than
sup[F([0,c))O A] and sup[F([0,c))CIB]. As F(b) is < sup(F ([0 , c) )O A)
sup(F(ГО,с))ПB), it follows that'F(c) is larger than F(b). Hence F . r„ .
| Lu>al
is an order-isomorphism, afP, and P=[0,Q]. By transfinite induction and
be
bf
or
the definition of h it follows that F is an order-isomorphism. Thus, by
a
(ii), sup F(C0,C2))fT.
Next we claim that sup F(C0,fi))eAOB. Since F is a net defined on the
directed set [0,0) and sup F([0,Q)) is its limit, it is only necessary to
show that the net is frequently in each of the sets A and B. L'et se[0,C2).
If F(a)ltA, then F(a) is the smallest element of В which is larger than the
first element of A larger than sup(F(Co,a))dA). To see that F is fre-
quently in A, consider any af[0,Q). We wish to show that there is some
a'eCa,C2) such that F(a')fA. If F(a)^A, then it will be shown that
F(a+l)eA. If F(a)^A, then sup(F([0,a)) 0 A)=sup (F([0,a+l))0 A) . Thus x ,
the smallest element of A > sup(F (Co,a)) 0 A) , coincides with x^, the small-
est element of A > sup(F([0,a)) 0 A). Moreover since F(a)fB, then F(a) is
the smallest element b of {bfB lb > sup(F([0,a+l))0B)}. Hence b =
ат!	'	d’l 1
F(a) > x =x and, by the definition of h , , F(a+l)fA. Thus F is fre-
a a+1	a+1
quently in A and the same is true of B.
Returning to the task of showing that each function xfC^(T,R.) is con-
stant on a tail of T, we note that for each tfT the set cl[x(a) |a >t] is a
nonempty compact set. Consequently nt^cl[x(a) |a > t] is nonempty and an
element r can be extracted from it. It follows that Gn=[afT | fx (a)-r | < 1/n]
is closed and cofinal in T for each nfN. Since the set Fn=[afT |ix(a)-rj>l/n]
is closed and disioint from the closed cofinal sets G„ for each nfN, F
must be bounded by (iii). Let a^sup Fn and choose apT such that a>sup a^.
Certainly then x(t)=r for each t > a (otherwise tfF^ for some n).
Appealing to the discussion at the beginning of (b), we may conclude
that negT-T. To see that цеиТ, it is enough to show that (ClVn)OT^0 for
any sequence (V ) of neighborhoods of ц, in fJT. Let (b ) be an increasing
sequence from T such that (b , ц,] C Vn П S. Since T is cofin-sl in [0,ц,) and
{b^} is not cofinal in T, (sup bn>p,]QT^0 and (sup b^, ц,] c ClnVn. This com-
pletes the proof, v
2.7 SEPARABILITY OF C(T,F,c)
107
2.7 Separability of C(T,F,c) M. and S. Krein (1940) were the first to
mention the characterization of separability of C(T,F.,c) for compact T
that appears in (2.7-1). Warner (1958) generalized this to the general
completely regular Hausdorff space T and this result, as well as some of
its consequences, appears below.
(2.7-1) SEPARABILITY OF C(T,F,c) WHEN T IS COMPACT If T is a compact
Hausdorff space, then C(T,F.,c) is separable iff T is metrizable.
Proof Suppose that C(T,F,c) is separable so that a countable dense subset
{xnlneN} exists in C(T,F,c). Let U(y,a)=[(s, t) | |y(s)-y(t)| < a) be a
typical subbasic entourage in the uniformity	generated by the con-
tinuous F-valued functions on T. One of the x must be such that the uni-
~	n
form p (x -y) < a/3, given yeC(T,F). Clearly U(x ,r)CU(y,a) for any
Tn	'	n
positive rational r < a/3. Thus the uniformity has a countable subbase,
and T is metrizable.
Conversely, suppose that T is metrizable. Then there exists a count-
able base of entourages (U(x,n,...,x, П,а )) .a > 0, for the unique
°	1	kn n r.cN n
(since T is compact) uniformity <-(T,F). Since T is Hausdorff, the uni-
formity is separating, i.e. given (t,s)fTxT, t^s, at least one of the
basic entourages fails to contain (t,s). Thus, for some nfN, and some
1 < i < к , x n(t)^x n(s) so that the collection S= U fx,n,...,x, n]
-	- n’ j j	nfN* l 1	kn
separates the points of T. Hence, by the Stone-Weierstrass theorem, the
collection of all linear combinations of monomials of elements from S with
rational or Gaussian (i.e. of the form a+ib where a and b are rational) co-
efficients - depending upon whether £=!< or C - is dense in C(T(iF,c). v
As an immediate application of this result, we remark that for a com-
pletely regular Hausdorff space T, C^(T,F), with sup norm topology, is
separable iff T is compact and metrizable. Indeed, if C, (T,F) is separa-
ble, then so is C(gT,F,c). Thus by (r|.7-l), gT is metrizable and there-
fore T is compact. (Indeed if gT is metrizable and pegT-T, then there is
a sequence (t ) of distinct points from T convergent to p. Set Zg=
{t2n|n€N} and zo={ с2п+1 |пей5 • As zecj{p} and ZQU{p} are closed in gT,
the disjoint sets Zg and Zq are closed in T. ^ut, by (1.2-2), the class
of closed sets is the same as the class of zero sets in any metric space
so that Z and Z are disjoint zero sets in T whose closures are not dis-
e о
joint, each containing p, in gT, in contradiction to Theorem 1.2-1. Thus
g.T=T and T is compact.
Before going on to the more general result on separability of С(Т,£,Г
108
2. SPACES OF CONTINUOUS FUNCTIONS
for non-compact T, let us examine more closely what follows from assuming
C(T,F,c) to be separable. Let [x |nfN} be dense in C(T,£,c). The initial
uniformity generated by [x^} is certainly weaker than the uniformity
£>(T,j?), so the former topology must also be weaker than the latter. Let
T' denote T endowed with the weaker topology. Since {x } is dense in
C(T,£,c), these functions must separate points in T. Under the map t-*t
where t(n)=xn(t), T' is homeomorphic to a subspace of the product space R~.
This countable product of real lines is separable and metrizable; hence so
is T1. Thus whenever C(T,£,c) is separable, T admits a separable metrizable
topology which is weaker than the original topology. As we shall now see,
this condition is also sufficient.
Theorem 2.7~1 SEPARABILITY OF C(T,F,c) Let T be a completely regular
Hausdorff space. Then C(T,F.,c) is separable iff T admits a separable
metrizable topology which is weaker than the original topology.
Proof It only remains to verify suffic'ency. Let T1 denote T equipped
with a weaker separable metrizable topology. We assert that C(T',F.,c) is
separable. By Urysohn's metrization theorem (Kelley 1955, Theorem 4.17,
p. 125) T' is homeomorphic to a subspace M of the compact metrizable prod-
uct space [O,1]~=S. Certainly it suffices to show that C(M,F.,c) is separa-
ble. Since C(S,F.,c) is separable by (2.7-1), it contains a countable dense
subset fzn}- Let Wn=Zn|M’ F is comPact i° anc* W?C(M,F.) then, by
Tietze's extension theorem, there is a function ZfC(S,F) that agrees with
w on F.
Now for each a > 0 there exists
z such that p (z -z)
n	HFk n '
w -w=z -z on F so p (w -w) < a and [w ] is dense in C(M.F,c).
n n	F n	nJ
< a. But
It follows
that C(T',iF,c) is separable.
To see that С(Т,£,с) is separable, let [x^} be dense in C(T',iF,c) and
let К be a compact subset of T. Clearly К is also compact in T1. Next we
claim that T and T1 induce the same topology on K. Indeed the injection
map taking
the compact space К into the Hausdorff
•riace T'
is
1-1 and con-
tinuous and therefore a homeomorphism.
xfC(T,F.) can be extended, by the Tietze
Thus each
extension
restriction x. of
theorem, to T . Letting
у denote the extension,
it follows
that p„(x -y) < a for some
К n
nfN corre-
sponding to the given positive number a. Since x fC(T,F) and p (x -x) =
n	K. n
P„(x *y) < a, [x ] is also dense in C(T,F,c). V
К n	n 
An immediate consequence of
this theorem and Urysohn's metrization
theorem is
that the cardinality
of T must be less than or equal to that of
the continuum whenever С(Т,£,с) is separable.
To illustrate the converse
2.7 SEPARABILITY OF C(T,F,c)
109
is false, we may take T to be the compact ordinal space	where Q is
the first uncountable ordinal. Now the cardinality of T is < that of the
continuum while T is not metrizable since no countable base exists at Q so
that C(T,£,c) is not separable by (2.7*1).
For any discrete space T whose cardinality is < that of the continuum,
C(T,F.,c) must be separable; for In this case there must be a 1-1 map from
T into R so, by transferring the Euclidean topology of R on the Image of T
in R, a separable metric topology is obtained for T which is weaker than
the discrete topology and it follows that C(T,£,c) is separable.
(2.7-1) showed that for compact T, metrizability of T was enough to
guarantee separability of C(T,£,c). We show next that C(T,£,c) Is separa-
ble even when T is a countable union of compact metrizable spaces.
(2.7-2) o-COMPACTNESS OF T AND SEPARABILITY OF C(T,F,c) If the complete-
ly regular Hausdorff space T is a countable union of compact metrizable
sets, then C(T,£,c) is separable. If, in addition, T Is locally compact,
then C(T,£,c) is also metrizable.
Proof Let T=	where each is a compact metrizable subspace of T.
Since the union of a neighborhood-finite collection of closed metrizable
subspaces is metrizable (Dugundji 19bb,p.l94f. , and Ex. 9.4, p. 207), any
finite union of compact metrizable spaces is compact and metrizable. Thus
we may assume that (K^) is an increasing sequence. Since C(Kn,£,c) is
separable by (2.7-1) there is a countable dense subset [x^ |ifN}c C(K ,£,c)
for each nCJJ. By the Tietze extension theorem, each x^ has a continuous
extension y^ to all of T. To see that {y^ |(n, i)fNxN} separates points
in T, let t and s be distinct points of T. Choosing n large enough so
that t, sfK , it follows from the fact that К is Hausdorff that у (s) +
n	n	?ni
У ^t) for some ifN. Thus the uniformity generated by the countable family
(Уп1 | (n, i) f NxN} induces a separable metrizable topology on T which is
weaker than the original topology of T. The separability of C(T,F,c) now
follows from Theorem 2.7-1.
To see that C(T,£,c) is also metrizable when T is locally compact it
is enough to recall that a locally compact ^-compact space is hemicompact
by the discussion following Def. 2.1-1. v
What about the converse of (2.7-2)? Consider T=R with discrete topol-
ogy. As we have already mentioned, prior to (2.7-2), C(T,R,c) is separa-
ble. On the other hand, only finite subsets of T are compact so that T is
not even о-compact. Thus the converse of (2.7-2) is false. A partial
110
2. SPACES OF CONTINUOUS FUNCTIONS
converse is obtainable however when C(T,F,c) is metrizable.
(2.7-3) SEPARABILITY OF C(T,F,c) WHEN T IS HEMICOMPACT Let T be a com-
pletely regular Hausdorff space. C(T,£, c) is separable and metrizable iff
T is a hemicompact space in which each compact set is metrizable.
Proof It is only necessary to prove the necessity of the condition. If
C(T,F,c) is separable, then T admits a separable metrizable topology which
is weaker than the original topology. Thus the injection map of a typical
compact subset of T into T with the weaker separable metrizable topology is
a continuous injection, and therefore a homeomorphism, v
In the table below the arrows denote implication, T a completely regu-
lar Hausdorff space, "weakly" separable metrizable indicates the existence
of a separable metrizable topology weaker than the original topology, and
"о-metrizably compact" means that the space can be written as a countable
union of compact metrizable subspaces.
Summary
T compact T weakly	T o-metrizably
\ separable	compact
metrizable	/	\
/	\ +T locally compact
X	C(T,F.,c) separable
C(T,F,c) separable	and metrizable
2.8 The bornology of C(T,F,c) The bornology of a topological vector space
X is the collection Ф of all bounded subsets of X. A base for the bornol-
ogy is a collection Q. of bounded sets such that each Be® is a subset of
some A e(L. This section deals with the bornology of С(Т,£, c) . In particu-
lar, the following questions are investigated:
(1) Under what conditions on T are all the bounded sets of C(T,F,c)
precompact, relatively compact?
(2) When is there a countable base for the bornology of C(T,£,c)?
The first question is dealt with in our first two results, while the
second is answered in (2.8-3). In answering (1) the critical topological
properties of T are pseudofiniteness and discreteness. As will be shown,
the pseudofinite spaces are the spaces for which all the bounded subsets of
C(TJ(F,c) are precompact, while the necessary and sufficient condition for
С(Т,Б),с) to be Montel is that T be discrete. Questions concerning the
countability properties of the bornology are less sharply answered in that
2.8 BORNOLOGY OF C(T,F,c)
111
they are not characterized solely by topological properties of T: There is
a countable base for (0 on C(T,£,c), for example, iff T is pseudocompact
and C(T,£, c) is sequentially complete.
(2.8-1) WHEN DOES C(T.F.c) CARRY ITS WEAK TOPOLOGY? Let T be a completely
regular Hausdorff space, and let X=C(T,.F,c).
Then the following statements
are equivalent:
(a)	Each bounded subset of C(T,.F, c) is precompact;
(b)	T is pseudofinite;
(c)	C(T,F.,c) carries its weak topology o(X,X').
Proof (a) — (b) : Let К be a compact subset of T. By (a), the bounded set
V is precompact so that
PT
x CC(T,
П	 -j,
most n-1 distinct points.
F) such that V_ c
functions Xp,...,
claim that К can contain at
number of
there are a finite
U ; =i (x;+"p ) • We
tit К
Indeed if t,,...,t are distinct points of К
I n
then by the complete regularity of T, there is a function xfV such that
,	-	PT
x(t.) = -lif Re x.(t.) > 0 and x(t.) = l if Re x.(t.) < 0 for i = l,...,n. But
R
then p (x-x.) > 1 for each i, 1 < i < n, so that xfU. , (x.+Vn ). Thus
К i —	—	—	1 t=l i
each compact set К must be finite.
(b) - (c):
If each compact subset of T is finite,
the point-open and
compact-open topologies
coincide on C(T,£) .
Since all the evaluation maps
t,f (where t*(x)=x(t) as x runs
through C(T,F.)) are continuous linear func-
tionals on C(T,£,c)=X, a(X,X') is generally finer than the point-open topol-
ogy. Thus, given that T is pseudofinite, all three topologies coincide.
(c)	— (a): It is generally true that the weakly bounded subsets of any
locally convex Hausdorff space are weakly precompact, v
In addition to answering the question of when C(T,iF,c) carries its weak
topology, the proof that (b) — (c) reveals that if the compact-open and
weak topologies coincide on C(T,£) then the common topology must coincide
with the point-open topology.
Reflexivity of C(T,£,c) was first characterized for compact T by S.B.
Myers (1949). Our next result, due to Warner (1958), subsumes Myers'
result.
(2  8-2) REFLEXIVITY OF C(T.F.c) Let T be a completely regular Hausdorff
space. Then the following statements are equivalent':
* Let X be a locally convex Hausdorff space. X is semireflexive if each
strongly continuous linear functional on X1 is generated by an element of X.
X is reflexive if the natural map between X and X" is a surjective topologi-
cal isomorphism when X" carries its strong topology. Equivalently, X is
reflexive if it is semireflexive and barreled. X is semi-Montel if each
bounded subset of X is relatively compact, Montel if it is semi-Monfel and
infrabarreled.
112
2. SPACES OF CONTINUOUS FUNCTIONS
(a)	C(T,£,c) is a Montel space;
(b)	C(T,F.,c) is reflexive;
(c)	С(Т,£,с) is semireflexive;
(d)	T is discrete.
Proof The implications (a) — (b) -> (c) are generally true, for any locally
convex Hausdorff space.
(c) — (d): First we show that T is pseudofinite. Suppose that an in-
finite compact set КС T exists and let (t ) be a countable collection of
distinct points of K. Since К Is compact, the sequence (t ) has a cluster
point t which we may assume to be different from each t . By the complete
regularity of T, there are functions х^еССТ,^) such that xn(tm) = l for m <n,
x (t)=0, and p (x ) < 1. Thus fx 1 is bounded. Since a locally convex
n	T n	—	1	nJ
Hausdorff space is semireflexive	iff	it Is	semi-Montel in	Its	weak topology
(Horvath 196b, Chap. 3, Sec. 8, Prop. 1) it follows that the weak closure
of {xnl is compact,	so	that (x^)	has	a weak cluster	point	x.	Since	x lies
in the weak closure	of	{x^}(n > mzfor each	positive	f and	mfN	there	is an
integer n > m such that
|t*(x ) - t*(x) I = lx (t ) - X(t ) I < e .
I tn n m l I n m	m l
Thus x(t^) = l. Since x is continuous, it follows that x(t) = l. But x (t)=0
for each n?N and x is a weak cluster point of (x ), so x(t)=O. It follows
~	n
that each compact set is finite.
Having just shown that T is pseufofinite, (2.8-1) may be invoked to-
gether with the remarks immediately following it to conclude that C(T,F,c)
carries Its weak topology (which, In this case, coincides with the point-
open topology). In light of this and the semireflexivity of С(Т,£,с), we
see that С(Т,£,с) is semi-Montel and, therefore, quasi-complete. By Th.
(2.1-2), С(Т,£,с) is complete. Since C(T,£,c) carries the point-open
topology - the subspace topology induced by the product topology - and
T	T
C(T,Fj is dense in £ , C(T,£)=£ . It follows that T is discrete.
T
(d) — (a): Since T is discrete, C(T,£,c) coincides with F carrying
the product topology. Since a product of Montel spaces is a Montel space,
the proof is complete, v
(2.8-3) WHEN IS THE BORNOLOGY OF C(T.F.c) COUNTABLY GENERATED? Let T be a
completely regular Hausdorff space. Then the following statements are
equlvalent:
(a)	There exists a countable base for the bornology of C(T,F,c);
(b)	Each Cauchy sequence from C(T,£,c) converges uniformly on T;
2.8 BORNOLOGY OF C(T,F,c)
113
(с)	T is pseudocompact and C(T,F,c) is sequentially complete;
(d)
C(T,F,c)
Proof (a)
The family of sets (nV„ )
_	4 ₽t neN
(i.e. Vn is bornivorous).
PT
(b) : Suppose that (x ) ,
is a base for the
x CC(T,F), is not
bornology of
uniformly conver
gent on T. Then (x ) also fails to be uniformly Cauchy on T. Thus for
some c > 0 there are strictly increasing sequences of integers (s ) and
(r ) such that r < s < r and p (x -x_ ) > f. As the functions x„
n	n n n+1 T sn rn	sn
and x are contint ius, open neighborhoods V of some t exist so that
rn	n	n
|xs (t)-xr (t) | > f for each tfV^. By using (a) we shall show that (x )
is not Cauchy in C(T,£, c) • Furthermore we claim that in order to do this,
it is only necessary to show that there is
a compact set К meeting an in-
finite number of the V 's.
n
Indeed if such a compact set К exists then
there is a strictly increasing sequence of integers
(n^) such
0. Thus p (x -x ) > C tor each к and (x ) is not Cauchy
К s „	t	n
nk nk
To prove that such a compact set exists, let us assume to
that КП7п +
nk
in C(T,F,c).
the contrary
i.e. that for each compact set К there is an integer n^ such that VnOK=0
for all n > n^. By (a) there is a countable base (A^) for the bornology
of C(T,F,c). Certainly C =sup{|x(t )| IxfA } is finite fo’- each n- Since
~	n	n I ' n
T is completely regular, for each nfN, an element у CC(T,F) can be found
which vanishes on T-V and evaluates о C +1 at t . Now the set В=Гу 1
n	n	n	n
cannot be contained in any one A as у (A . On the other hand we claim
n n n
that В is bounded, which would contradict the fact that (A ) is a base for
n
the bornology of C(T,£,c). To see that В is bounded it is enough to note
that sup p„(y )=max, _	P„(y ) for each compact K. This contradiction
n К n 1 < n < n^ К n
establishes the result.
(b) — (c); Certainly C(T,,F,c) is sequentially complete whenever (b)
holds. To see that T is pseudocompact, we need to show that each xfC(T,R)
is bounded. Let
In if x(t) > n
x(t) if (x(t)| < n
-n if x(t) < -n
If К is compact then, for some mfN, |x(t)|< m for each tfK and therefore,
x (t)=x. (t) for each tfK whenever n, к > m. Thus (x ) is Cauchy in C(T,F,c)
ПК	n	~
and, by hypothesis, converges uniformly on T. Since x is the pointwise
limit of (x ) on T. it is
n
large n, p^,(xn~x) < 1 and
also the uniform limit. Hence, for sufficiently
the boundedness of x follows from that of x .
n
114
2. SPACES OF CONTINUOUS FUNCTIONS
(c) -• (d) : Suppose that (d) does not hold, i.e., that there is a
bounded set В which is not contained in any nV . Then for each n there is
3 PT
an x CB and a t CT such that |x (t )i> n . We claim that the sequence of
n	n	n n ~
_ 2
partial sums of the series E. к | x. | is Cauchy in C(T,F,c) so that the
Kt N к	~
This follows from the obser-
I) < M S^=mk where M=
x is an unbounded continuous
series converges to an element x of C(T,£,c).
vation that for each compact K, p„(E,P к x,
_2	K k m	к
sup npv(x). As x(t ) > n n =n for each n,
xfВ К	n
function on T and we conclude that (c) fails to hold.
The proof that (d) -> (a) is trivial, v
The situation examined in the previous theorem is considerably simpler
for the class of replete spaces. Indeed if T is replete then T is pseudo-
compact only if T is compact ((1.5-3)) and thus it is necessary for T to
be compact whenever the bornology of С(Т,Б),с) is countable. Conversely,
whenever T is compact, both conditions of (c) are satisfied so that the
bornology of С(Т,£,c) is countably generated.
We conclude the chapter with a consequence of (2.8-3), a characteriza-
tion of those pseudocompact spaces T which are compact in terms of topo
logical vector spaces properties of C(T,F,c) (cf. (1.5-3)).
(2.8-4) PSEUDOCOMPACTNESS VS. COMPACTNESS A pseudocompact (completely
regular Hausdorff) T is compact iff C(T,F.,c) is infrabarreled and sequen-
tially complete.
Proof The necessity of both conditions follows immediately from the obser-
vation that C(T,iF,c) is a Banach space whenever T is compact. To see that
they are also sufficient, we note first that by the previous result, each
bounded set is contained in some integral multiple of VD . Thus since
T
C(T,F,c) is Intrabarreled, Vn is a neighborhood of the origin and the norm
~	PT
p^ generates the compact open topology. C(T,iF,c) is thereby a Frechet
space so that by Theorem 2.1-1, T is hemicompact. Consequently T is a
Lindelb'f space and it follows by Theorem 1.5-3 that T is replete. The
result now follows by an application of (1.5-3) which asserts that a pseu-
docompact replete space is compact. V
EXERCISES 2
115
2.1 C(T,F,c) BORNOLOGICAL IFF ULTRABORNOLOGICAL (de Wilde and Schmets
1971). A locally convex Hausdorff space X is bornological iff it is an
inductive limit of normed spaces (X ) i.e. iff
ц'ц,€М’
A :X "* X such that the linear
Li IX
is a neighborhood of 0 iff for
span of U A (X )
LXfM |X IX
each LXfM A ^(0) is
LX
there exist linear maps
is X and a set UC X
a neighborhood of 0 in
X is ultrabornologica1 if the spaces X can be chosen such that they
V
are Banach spaces.
The point of tnis exercise is to show that C(T,F,c) is bornological iff
it is ultrabornologica1.
(a)	Bornivores in C(T,F,c) For хеС(Т,£, c), x > 0, let N(x) =
{yfC(T,F) | |y| < х]. For replete T, show that an absolutely convex subset
of C(T,£,c) is bornivorous (absorbs all bounded sets) iff it absorbs each
N(x). (Hint: Cf. proof of Theorem 2.6-1.)
(b)	Bounded linear functionals on C(T,F,c) For T replete, show that
a linear functional f on C(T,£,c) is bounded iff f(N(x)) is bounded for
each nonnegative x in С(Т,£,с).
In (c) and (d) some technicalities needed for the proof of (f) are
discussed.
(c)	If every absolutely convergent series in a normed space X is con-
vergent, then X is a Banach space. (Hint: Let (x ) be a Cauchy sequence
in X and let (m^) be an increasing sequence of positive integers such that
llx -x II < 2 к for all m > m, . The series x +E, __„(x„ -x ) is conver-
m -	N - *	™k+1 m/
gent, and (x ) = (x +E, , (x —x )) is a subsequence of (x ).)
mN mr "k=l^ mk+1 m^'	’
Any absolutely convergent series S N xn In a normed linear space
X can be written E „с у where у -’О and E .,1c i converges. (Hint:
nfN n n	n	nfN n
Let (N^) be an increasing sequence of positive integers such that
S », IIх II < 2 for each i > 1. For each m > 0 such that N. < m < N. ,
m > N,- 11 m" —	.	i- — i+l
— l	i	- m	- i
let c =max(2'|lx |l , 2	).) Let x =c у . As II у |l < 2 for some i and each
m	m	n n n 1 n —
n, and i — as n -» =>, it follows that у — 0. Also T |c I < E 2 m +
n	nfN n — mfN
S. „2 E „ |lx II and therefore E „ |c | is convergent.
ifN mfN 1 m	nfN n
Return attention now to C(T,£,c). Given N(x) , let X denote the space
spanned by N(x) in C(T,_F) and || the gauge of N(x). If yp(l/n)N(x) for
each nfN then clearly y=0. Hence || Ц* is a norm on X^.
(e)	X^ is a Banach space. Hint: As || у || ^=inf[r > 0 | |y| f rN(x) 1 =
supx(t)^o	> N<x)=[yexx |ll ydx < !}• Now by parts (c) and (d) it is
sufficient to show that given any sequence (xm) from X^, with x^ -> 0, y=
^mfN^ ^m eC(T>£) • Clearly the series converges pointwise to у and we show
116
2. SPACES OF CONTINUOUS FUNCTIONS
that yfC(T,F) as follows. For e > 0, choose a neighborhood V of t^fT such
that у is bounded on V, and a positive integer M such that
sup IE 2 m(x (t) —x (t )) I < e/2. Now there exists a neighborhood V1
4fVl m > M m mol —
of t such that V'c V and suptfV, |^m=]2 m(xm(t)	|* Thus suPLf„, |y(t)“
y(t ) | < f anc* yfC(T,F). Now it remains to show that yfX^.
(f)	If T is replete, then C(T,F.,c) is ultrabornologica 1. Hint: It is
clear that C(T,£)= (Jx > qX and as X^ is a Banach space we need only show
that an absolutely convex absorbing set UC C(T,£,c) is a neighborhood of
0 if and only if UOX is a neighborhood of 0 in X^ for all nonnegative func-
tions xfC(T,F). Using (a), this follows iff U is bornlvorous. But C(T,_F,c)
is bornological whenever T is replete so that the bornivore U must be a
neighborhood of 0.
2.2	k-Spaces A topological space T is a k-space if a set UCT is open
whenever UAG is open in G for each compact subset G of T. Clearly any
function x mapping a k-space T into a topological space Z is continuous
whenever each of its restrictions to a compact subset of T Is continuous.
A space T having the property that a Z-valued function defined on T is con-
tinuous whenever each restriction to a compact subset is continuous is re-
ferred to as
-space (cf. the definition of a k-space in Sec. 2.2).
R
Thus If T is
a h-
a k-space then it is a k^-space for each topological space
Z.
The converse is also true:
(a)	к <—>k„ FOR ALL Z A topological space T is a k-space iff It is a
k^-space for all topological spaces Z. Hint for sufficiency: Consider the
function I:(T,;f) — (T,J^) where -T is the original topology of T and Is
the к-extension topology (defined before (2.3-1)).
(b)	LOCAL COMPACTNESS OR 1st COUNTABILITY -» к All locally compact
spaces and all 1st countable spaces are k-spaces. As there are locally
compact spaces which are not 1st countable and 1st countable spaces which
are not locally compact the class of k-spaces is strictly smaller than
either the class of locally compact spaces or the class of 1st countable
spaces.
Though a k-space need not be locally compact it must have a locally com-
pact "ancestor":
(c)	A k-SPACE IS A QUOTIENT OF A LOCALLY COMPACT SPACE A topological
space Is a k-space iff it is a quotient of a locally compact space. Hint.
Necessity: Let^7 denote the class of all compact subsets of T, Gl =
{(t,G) |tfG} for each Gf , and Tl= {g1 |Gc4?} • If equipped with the final
topology generated by the Injection maps
EXERCISES 2
117
i„:G - T1	(Getf) ,
t - (t,G)
T1 is referred to as the free union of , i.e. if the topology of G is
transferred to G1 in the natural way a subset of T1 is open iff its inter-
section with each G' is open in G' and so T1 is clearly locally compact.
The relation ~ defined on T1 by:
(t,G) ~ (s,H) iff t = s,
is an equivalence relation on T1 and the mapping
h:T - T1/R
t - h(t) = {(t,G) |teG}
is a bijection. It remains to show that h is a homeomorphism when Т'/R
carries the quotient topology, i.e. the final topology generated by the map
p:T'- T1/R
(t,G) - h(t) = {(t,H) |teH}.
The result follows from the observation that for each UCT, p \h(U) =
Ufu'ClG1 |Gf }, and the fact that U is open in T iff UOG is open in G for
each GC.
Sufficiency: If there exists a locally compact space S, and equivalence
relation R and a homeomorphism h taking S/R onto T then the mapping
*
f:S - S/R - T
s -* Rs — h(Rs)
has the property that a set UCT is open iff f \u) is open in S. Thus to
prove that T is a k-space it suffices to show that f \u) is open whenever
UClG is open in G for compact GCT. Since S is locally compact so that
there exists a covering of S by relatively compact open sets fv }	, the
a OfSA
desired conclusion follows from the sequence of observations:
(1)	UClf(clV ) is open in the compact set f(clV^),
(2)	there exists an open set WCT such that UOf(clV )=WOf(clV ),
-1-1	»	O'
(3)	f (U)dVff=f (W)PVff .
(d)	SUBSPACES A subspace of a k-space need not be a k-space. Indeed
if T is any completely regular Hausdorff space which is not a k-space, e.g.
д
T=W where A is uncountable (discussed in Example 2.3-1), then T is certain-
ly a subspace of the k-space ₽T. What about closed subspaces of k-spaces?
They are also k-spaces.
118
2. SPACES OF CONTINUOUS FUNCTIONS
(e)	QUOTIENTS A quotient space of a k-space is a k-space.
Hint: Use (a).
(f)	PRODUCTS (Bagley and Young 196b). We already know by Example
2.3	-1 that an infinite product of k-spaces needn't be a k-space. What
about finite products? Unfortunately a product of two k-spaces need not be
k^ even if one of the spaces is
(fl) Let T be a completely
metrizable.
regular Hausdorff hemicompact
k-space which
is not locally compact. Then C(T,R,c) is metrizable (and a k-space by (b))
while C(T,R,c)X T is not a k-space.
Hint: It suffices to show that the evaluation map
e: C(T,R,c) X T -> R
(x,t) - x(t)
is continuous on each compact subset of C(T,R, c)x T but is not continuous
on all of C(T,R,c)X T. To prove the first assertion it is enough to con-
sider e on compact subsets of the form FxK where F is compact in C(T,R,c)
and К is compact in T. Fix (x^.t^f FxK. As F |^={x |xf F} a compact
subset of C(K,R,c) and thereby equicontinuous, for c > 0 there exists a
neighborhood U of t in T such that |x(t)-x(tQ) | < f/2 for all tfUOK and
all xfF. It follows that |x(t)-xo(tQ) | < f whenever (x,t)f ((x^+f/2	)
XU)AFxK.	K
To establish discontinuity of e we produce a net (x ,t ) in C(T,R,c)
а а аел	~
x T convergent to (0,1^) (where 0 denotes the function on T which sends
each tfT into OfR and t is fixed in T) such that x (t )=1 /» 0=0(t ). As T
~	°	a a	°
is not locally compact there exists tQfT with no relatively compact neigh-
borhood. Let W(tQ) denote the neighborhood system at t°. Since T is hemi-
compact we may choose an increasing sequence (K ) of compact sets such that
rl/n Vp IneN} is a nelghbornood base at 0 in C(T,R,c). For each nfN and
К	~	~
U?U(t ) ?here exists t tU such that t (K )=0 and x (t	) = 1. It
о	n,u	n,u n	n,u n,u
follows that (x ,t ) are the elements of a net on N x U.(t ) convergent
n,u n,u	4 о	°
to (0,t ).
Thus requiring that one of a pair of k-space be 1st countable is not
enough to guarantee that the product be a k-space. However
(f2) a product of two 1st countable spaces is a k-space;
(f3) a product of two Hausdorff k-spaces, one of which is locally com-
pact, is a k-space.
Hint: So that (a) may be used in establishing this result some preliminary
results are needed. Let C(S,Z,c) denote the space of continuous Z-valued
functions defined on S with the compact-open topology, i.e. the topology
EXERCISES 2
119
having as a subbasis all sets of the form [K,vl={xfC(S,Z) |x(,K)CV} where К
is a compact subset of S and V is open in Z.
Let T, S, and Z be topological spaces and xfC(TxS,Z). If t is fixed
in T then x(t) (s)=x(t,s) (sfS) defines an element of C(S,Z). Thus x maps
elements of T into continuous Z"valued functions on S. Conversely suppose
we are given a function x:T — C(S,Z). Then x(t,s)=x(t) (s) ((t,s)eTxS) de-
fines a Z-valued function on TxS which is continuous in the variable s for
each fixed tfT.
(f4) If x is a given continuous function then $ is also continuous.
Hint: Suppose x(t)e [G,V]={yfС (X, Z) |y(G)CV} where G is compact in S and
V is open in Z. We must find a neighborhood U of t in T such that
x(Ut)cCG,v], or equivalently, x (t1 , g) =x (t1) (g) f V for all t'eU^ and gfG.
Now x(t, g)=x(t)(g)fV for each gfG so by the continuity of x,open sets
U CT and W C S exist such that x(U X W )C V. As G is compact and is
t.g	g	t,g g	n
covered by finitely many W 's, .say W , . . . ,W , U =.f\u serves the de-
g gf gn ь ^ = 1 ь>ё1
sired end.
(f5) If x: T -> C(S,Z,c) is continuous and S is locally compact and
Hausdorff then the associated function x (defined above) is also continu-
ous .
Hint: Suppose (t ,s ) is a net in TxS convergent to a point (t,s)f TxS
a a afA
and V is a neighborhood of x(t,s) in Z. We shall prove that x(t ,s ) is
л	a a
eventually in V. As x(t)fC(S,Z) and S is locally compact Hausdorff there
exists a relatively compact neighborhood W^ of s such that x(t) (clW^CV.
Thus x(t)e [clWs,V]. Now x is continuous so there exists a neighborhood U
of t in T such that x(U )c[clW ,V]. Since (t ,s ) — (t,s) there exists an
t	s	a a
afA such that for a < B, (t ,s )f U xW . Hence x(t ,s )=x(t ) (s )fV for
n6ss	₽B ₽g
oj.1 g > a.
Returning to the proof of (f3) we claim that x: TxS -> Z is continuous
whenever x I is continuous for each compact set KCT. To see this sup-
I Kxb
pose that xl is continuous. Then xl is also continuous by (i). As T
iKxS	IK
is a k-space it follows that x is continuous and so by (ii) x is continuous.
Thus it remains to show that x „ is continuous whenever x I is continu-
IKxS	IKxG
ous for all compact GCS. But this fact follows by exactly the same argu-
ment since К is locally compact Hausdorff and S is a k-space (by part (b)).
(g) (Noble 19b7) T A k-SPACE -A uT A k-SPACE In Example 2.1-1 it was estab-
lished that the collection of all non-negative integer-valued functions
defined on the uncountable set A, is not a k-space. Moreover we saw that
the set S consisting of all elements of w” which vanish at all but at most
120
2. SPACES OF CONTINUOUS FUNCTIONS
a countable number of elements
from A is a k-space with US=W^.
R	~
Here we
claim: S is a k-space whose repletion is not a k-space.
Hint: To see that S is a k-space let
taining s which is not a neighborhood
К such that CUCIK is not closed in K.
s be fixed in S and U be a set con-
of s. We shall exhibit a compact set
Since U is not a neighborhood of s
there exists an element s^fS such that s^U.
countable subset A,=fa, . lifN} (if A, is finite a Q
1 L	1	lj 1J
Let Fj={a^}; then the neighborhood { tfS 11 (a ^)=s (a ^) } i/U so we
S£^U such that (a 11)=s (а ^) . Let A2=fa2j |1?й] denote the ele-
A on which S£ and s differ and set с'о={а^ |1 < i, j < 2}. Then
the neighborhood {tfS |t(a)=s(a) for each af^j^U so there exists s^U
such that s3=t on F^• Continuing by induction we obtain a sequence (s^)
elements lie in CU and, denoting the collection of ele-
As s^, sfS they differ on
=a, T for some J and all
can find
ments of
from S all of whose
mentu of A on which
Oij |1 < i>J < n}.
s and s differ by A ={a . lifN‘1, s ,=s
n	n 1 nj I	n+1
Set C=UF^. If a^C then Sn(a)=s(a) for
sn(a) - s(a) .
If afC there exists m > 0 such that afF for
n
on F =
n
each n so that
each n > m.
a
Thus s ,(a)=s(a) whenever n > m and s (a) -» s(a). Therefore s -1 s in S,
n+1	-	n	n
K=[sn |nfN}(J{s} is compact and CU П K={sn |nfN} is not closed in К as s^CUClK.
(h) ASCOLI THEOREMS (Kelley 1955, pp. 223-2+9; Bagley & Young 19bb)
The point of this exercise is to obtain generalizations [(iii) and (iv)
below] of Ascoli's Theorem as presented in Kelley 1955 (Theorem 21, p.23b),
as well as the version in Kelley's Theorem 7.17 (p. 233).
If F is a family of maps from a topological space S into a topological
space T, then any topology -S' for F which makes the evaluation map e sending
(x,s) into x(s) from FxS into T is called jointly contini ous  Two results
on joint continuity are needed for the Ascoli Theorems.
(i)	If S and T are Hausdorff spaces, FC C(S ,T) is a topology for F
which is finer than the compact-open topology and which makes (Fj)xS
a k-space, then -J is jointly continuous.
(Hint ' Let C be a closed subset of T, К a compact subset of (FjJJ'jxS,
and (x,s) be a point outside of M=Kf~le \c) . If (x,s) |(K then obviously
(x,s) ( cl M. Suppose (x,s)eK and (x,s) i e \c) . Let U=T-C and let Kg
be the projection of К into S. There is a compact neighborhood N of S
relative to К such chat x(N)CU and e([N,u]xN)CU where [N,u] =
“1
[y€F |y (N)c;u}e^cr "J . Thus ( [n, u] xN)He (C)=0. It follows that (x,s) is
not in the closure of M relative to (F,^7)xKg. But since MCKCJFxKg, it
follows that (x,s)^ cl M. Since FxS is a k-space, e ^(c) is closed and the
proof is complete.)
EXERCISES 2
121
(ii)	If (F^T^) is locally compact, S a Hausdorff k-space, and T a
Hausdorff space then is jointly continuous.
(Hint: Use (f3) and (i) above.)
Using (ii) , the generalizations of Kelley's Ascoli Theorems mentioned
above (Kelley's Theorem 7.17 and 7.21 respectively) may now be obtained
using the same proofs that are in Kelley.
(ill)	ASCOLI THEOREM Let S be a Hausdorff k-space, T a Hausdorff uni-
form space and FCC(S,T). Then (F^T.) is compact iff (1) (F,<T) is closed
in C(S,T,c) where C(S,T,c) denotes C(S,T) with compact-open topology, (2)
F(s) has compact closure for each sfS and (3) F is equicontinuous.
(iv) ASCOLI THEOREM Let S be a Hausdorff k-space, T a regular Haus-
dorff space and FCZC(S,T). (F,<7^) is compact iff (1) (F^T^,) is closed in
C(S,T,c), (2) F(s) has compact closure for each SfS and (3) F is evenly
continuous. (To say that F is evenly continuous means that for each SfS,
each tfT, and each neighborhood U of t there is a neighborhood V of s and
a neighborhood W of t such that x(V)C.U whenever x(s)fW.)
2.3 k-SPACES AND FULL COMPLETENESS OF C(T,F,c) (Ptak 1953; see also com-
ment following (4.12-8).) Let T be a completely regular Hausdorff space.
C(T,F,c) is complete iff T is a k-space, whenever T is a hemicompact space
((2.3-4)), This equivalence is not general for, as we have seen in Example
2.3-1, there are completely regular Hausdorff spaces which are not k-spaces
but for which C(T,,F,c) is complete. What condition on the LCHS C(T,F,c)
will force T to be a k-space? Ptak (1953) has shown that T is a k-space
whenever C(T,X,c) is fully complete. Unfortunately the converse is not
true. Recall that a LCHS X is fully complete if every continuous linear
transformation A taking X onto a LCHS Y which is almost open (i.e. cl A(V)
is a neighborhood of 0 in Y for each neighborhood V of 0 in X) is an open
map.
(a) If C(T, F, c) is fully complete then T is a k-space
(i)	The image of a fully complete space under a topological homomor-
phism (i.e, a continuous and open linear map) is fully complete. (Suppose
the topological homomorphism A takes the fully complete LCHS X onto the
LCHS Y and let В be an almost open continuous linear map from Y onto the
LCHS Z. If U is a neighborhood of 0 in Y then A \u) is a neighborhood of
0 in X. Therefore, because X is fully complete and В-A is almost open,
B(U)=(B‘A) (A \u)) is open in Z,)
(ii)	If S is a closed subset of T and C(T,F,c) is fully complete, then
so is C(S,F,c). Furthermore if C(T,F,c) is fully complete, then T is normal.
122
2. SPACES OF CONTINUOUS FUNCTIONS
Consider the restriction map
R: C(T,F,c)---»-C(S,F,c)
x -------^х|д
R is clearly linear. Next we claim that the range of R is dense in
C(S,£,c). Indeed, if yfC(S,F) and KG.S is compact then у ] has a continu-
ous extension у defined on all of T. Thus 9 |„=y |„ and so in any basic
к. K.
neighborhood of y, say y+fV = {y+z |z€C(S ,F), p (z) < there exists an
element of the range of R.
It only remains to show that R is a topological homomorphism for then,
by (i), the range of R is fully complete. As R is certainly continuous
our concern is to establish openness of R. But C(T,F,c) is fully complete
so it suffices to show that R is almost open. Let К be compact in T and
consider the set cl R(Vn )
cl R(Vp )3[yfC(S,F) |P]
0 in C(I„E,c).
compact. Now
that u agrees
joint from H.
. In the event that КП3^0 we contend that
(y) < 1/2} so that cl R(Vp ) is a neighborhood of
’Kns(y) - 1/2 and HCS is
' PK
Suppose that yfCCSj^F) such that p
by what has already been shown, there exists uCC(T,£) such
with у on (КПЗ)иН. Thus the set W-{tfK | |u(t)l > 1} is dis-
Choose ZfC(T„E) such that 0 < z < 1, z(H)={l], and z(W)={0}.
Set x'=zu and x' |g=x- If tfK then there are two possibilities. First sup-
pose that tfW. Then z(t)=O and |x'(t)|=O < 1. If tfV then |x’ (t) | =
|u(t) | |z(t) | <1. As x agrees with у on H (since z(H)=[l} and u=y on H),
yecl R(V ).
PK „
If КП S=0 then we may choose
a function ZfC(T,F) such that 0 <z < 1
z(K)={0}, and z(S)={l}. Now if у is any element of C(S,,F) and HCS is com-
pact there exists ufC(T,_J) with the property that u I =y then x'=zueV and
H
x=x’ I agrees with у on H. Thus cl R(V )=C(S,^) is a neighborhood of 0 in
S	Prr
C(S,F).
Hence R is a topological homomorphism onto a fully complete dense sub-
space of C(S,X) so that R is, in fact, surjective, and 0(3,^) is fully com-
plete. Moreover, by the ontoness of R, each continuous real-valued func-
tion defined on 3 has a continuous extension to all of T. Consequently T
is normal by the Tietze extension theorem.
(iii)	If C(T„E,c) is fully complete then T is a k-space Suppose that
ECT and EfiK is closed for each compact KCT. Show that S=cl E=E. To
this end consider the topology on
seminorms pR^ E< • )=sup | (• ) (КП E) |
C(S,F) generated by the collection of
as К runs through the compact subsets of
EXERCISES 2
123
S for which Kf)E^0 and let C(S,_F,Cq) denote C(S,^F) with this topology. To
see that this space is Hausdorff, let XfC(S,X), x/O . Then there exists sfS
such that x(s)^O and so there exists a neighborhood of s in S in which
x never vanishes. Now E is dense in S so EflVs^0 and p^tj(x)^O for any
tfEClV .
s
As the compact-open topology is finer than the c° topology of C(S,£)
the bijective mapping
I; C(S,F,c)
C(S,F,c )
~ о
(by
prove that the compact-
(where cl
о
.e origin for
(ii)) it is only nec-
is continuous. Since C(S,F,c) is fully complete
essary to show that I is almost open in order to
open and c topologies coincide. To see this show that cl V
о	о Pj
denotes closure in the c° topology) is a c^-neighborhood of t
each compact KGS. Indeed, if К П E/0 we claim that 1/2 V _________ _ .
PKOE ° PK
Let Xf 1/2V and suppose that HCS is compact and H("IE^0. Using the
procedure ou^flned below, show that there exists a function x'fC(S,£) such
that x* agrees with x on HHE and p (x* ) < 1 to establish the desired
К
elusion. Let W={teK | |x(t)| > 1}; note that W is compact and, since
suplx(Kf) E)1 < 1/2, Hf)E is also compact so there exists ZfC/S,^) such
x'=x.z. If tfHPE then z(t)=l
Next suppose that tfK. If tfW
If t/W then lx(t)l< 1 and, since
°-neighborhood of 0 provided КП E
argument similar the one above show
in-
thac
0 < z < 1, z(H(lE)=(l} and z(W)=[O}. Set
so x’(t)=x(t) and x' and x agree on Hf>E.
then z(t)=O and therefore lx'(t)|=O < 1,
Thus cl V is .
° PK
using an
any event cl V is c -neighborhood of 0.
° PK °
complete, I is a topological isomorphism and
^0. What if КП E=0? Then by
that cl V =C(S,F) so that in
° PK
Thus, since 0(3,^,c) is fully
the compact-open and c°-topologies agree on C(S,j?).
Now show that E=S. If teS-E then the compact-open neighborhood V
being also a c°-neighborhood of the origin contains a neighborhood V
where К is compact in T and КПЕ^0. Since KflE is closed and t/KOE
exists yeC(S,F) such that 0 < у < 1, y(t)=l, and у(КЛЕ)=[о]. Hence
yfV while y0V - a contradiction. It follows that S=E and T
, РКЛЕ	P{t}
k-space.	1 J
(b) There are (completely regular Hausdorff) k-spaces T for which
C(T,_F,c) is not fully complete.
Hint: Consider the ordinal spaces [O,tej and	where ш is the first
infinite ordinal and О the first uncountable ordinal. Then [0,U)]X[0,q] is
РКЛ E
: there
is a
x
x
a
124
2. SPACES OF CONTINUOUS FUNCTIONS
a compact Hausdorff space and the open subset T=^0,ш]Х[О, Q] - { (ш, Q) } is a
locally compact completely regular Hausdorff space. Hence T is a k-space
by Exercise 2.2(b). On the other hand T is not normal (see Dugundji 1966,
p. 145, Ex. 4) so, by (ii) of part (a), C(T,F.,c) is not fully complete.
2.4 C(T,R)' AND BAIRE MEASURES Throughout this exercise we take T to be
a completely regular Hausdorff space. We know from (2.4-4) that each pos-
itive linear function h on С^(Т,Д) has the form h(.)=J"(.) dp, where p is a
regular additive non-negative set function defined on Q-2, the algebra
generated by Z , the zero sets of T. Recall how p was defined: for
any ZfZ, P(Z)=sup{h(x) |xe С^(Т,Н), 0 < x < k^^} and for arbitrary AfR(T),
P(A)=inf (p(CZ) |zeZ ,AC C Z}. In (a) below we show that p is, in fact,
countably additive on the o-algebra generated by^ (the Baire sets)
and that, if h is obtained as the restriction of a positive linear func-
tional on C(T,R) the representation as an integral given above holds for
all xeC(T,R).
(a) POSITIVE LINEAR FUNCTIONS ON C(T,R) AND BAIRE MEASURES Let h be a
positive linear function on C(T,R) and p be the set function defined above.
Then P is a regular measure on <53 and
a
h(x) = J* x dp (xeC(T,R)).
For each xfC(T„R) there exists EC <3a such that x is bounded on E and P(CE)
=0.
Sketch of Proof Let P denote the collection of complements of zero sets.
First show that
(i) for each GeP and € > 0 there exists Hep such that cl HCG and
P(G)-e < P(H): Choose wfc (T,R) such that 0 < w < к and h(w) > p(G)-e/2.
_ 4	b	G
Now H=w (e/2,l]eP by (1.2-3) and cl HCG. If p(H) < p(G)-f then
h(w)-e/2 - h(w-(e/2)l) < h((w-(€/2)i)V 0) < P(H)
< p(G)-e < h(w)-f/2
-a contradiction.
To establish countable subadditivity on P prove:
(ii) if (G ) is a sequence from P such that GOG , for all n and
n	n n+1
f) G =0 then lim p(G )=0: Consider first the situation where flcl(G )=0.
n	n n	n
As p is monotone on (p(G )) is a decreasing sequence of real numbers;
as such it possesses a limit a > 0. If we assume a to be positive then for
each n > 0, there exists x eC, (T,R) such that 0 < x < к and h(x ) > a/2.
n b ~	— n — G	n
Let x=I^ xn> Since f"\Gn=0, each tfT can belong to only anfinite number of
EXERCISES 2
125
the sets G , so that x is a real-valued
n
over, since Qcl Gn=0, each tfT belongs
is open and only a finite number of the
continuous at t for each teT. Applying
non-negative function on T. More-
to U =C(cl G ) for some n. But U
n	n	n
x, ’s are non-zero on U so x is
к	n
h to x we obtain the contradiction:
h(x) = Г h(xR) + h(v xk) > У (n > 0)
k=l	k>n
Next consider a sequence (G ) for which Qg =0. For any given e > 0, by
n	n
(i), there exists VfP such that cl(V )CZG and „(0, )-«72n < >j,(V ) for
n	n n	n	n
n > 0. Thus if we set H =V, and inductively define H =V П H , it follows
11	n n n-1
that cl H C2G and H О H , for each n > 0. Furthermore, with the aid of
n n n n+1
the observation:
G -H C. (G -V )U(G -H ) (n > 0)
nn nn^n-ln-1
and the fact that u is subadditive on Q, (since it is additive and mono-
Z	n к
tone on (X ), it is readily established that u(G ) < u(H )+e Г 1/2 < p,(H )
z	n	n , .	n
for each positive n. As Del H Cl I"|G =0, so that lim u(H )=0, it fol-
n n	n n
lows that lim u,(G ) < 6, Hence lim p(G )=0,
n n —	n n
(iii)	p is countably subadditive on P, i.e, if G fP for each n > 0
n
then h(UG ) < у P(G ): First we claim that for each G there exists a se-
n — n n	n
quence (Z ) of zero sets such that G = CJ Z , Indeed, by (1.2-1(d) there
nm	n m nm i
exists a 0 < y^ < 1 from C(T,JR) such that Gn=yn (0,1]. Then choosing
Z =y 1 Г1/т,1] - a zero set by (1.2-3) - the claim follows. Now G=U G
n,m n	n n
- tj W where each W equals some Z . Set M, = VJ W . Since GfP by
mm	m	nm	к m < m
(1.2-l(e)) and each	by (1.2-l(c)) the sequence (GdCM^) satisfies the
hypothesis of (ii). Thus for anv positive € there exists k^ such that
p(G Cm ) < e. But	is contained in some finite union (Jn №n so
that
u(G) < u(Un NGn) + d(GnCMk ) < Zn N U(G ) + f.
—	о	—
(iv)	ц, is countably subadditive onp(T): Let A= (J^A^ and choose
GOA such u(G ) < p(A )+£/2П for each n > 0. Set G= U G and note that
nn	n	n	nn
GOA. Thus
ц,(А) < u,(G) < p(Gn> Zn +
In summary then ц, is non-negative, monotone, countably subadditive and
p(0)=O; i.e. p is an outer measure on P(T). Hence the collection of p-
measure sets, i.e. the sets Ef P(T) for which ц,(А) ц,(А Л Е)+Ц(А ПСЕ) for
each A€ P(T)> form a ст-algebra of subsets of T on which и is countably
126
2. SPACES OF CONTINUOUS FUNCTIONS
additive (see Dunford and Schwartz 1958, III 5.4, p. 134).
(v)	u is a regular measure on - the Baire sets: To see that u is
a measure on *3 it is enough to show that each GfP is ц-measurable. Let
ACT, f > 0, and choose НС P such that HOA and ц(А) > ц(Н)-р. Then, since
u is additive on Q. and monotone on P(T),
z
ц(А) + e > ц(Н) = u(HflG) + u(HPCG) > u(AfiG) + u(aDcg)
and the result follows. As for regularity, given	there exists GJA
and Н2ЭСА such that u,(G)-u(A) < f/2 and ц(н)-И(СА) < e/2. But H-CA-A-CH
so that
ц(А) - ц(СН) = U(A-CH) = ц(Н-СА) - ц(Н) - ц(СА) < f/2.
Hence CHCACG and p(G-CH) < f.
To finish the proof of (a) it remains to show that
(vi)	for each xeC(T,R) there exist real numbers a, b such that
Ц(x '’[a b])=p.(T): It suffices to prove the result for x > 0. Suppose that
ц(x l[0,r]) < ц(Т) for all r > 0. It follows then that a sequence of real
numbers
0 < a < b < a < b < ... < a <b <...
112	2	n n
exists such that lim a =lim b and ц/х^Га ,b "]) > 0 for each n > 0.
_^nn nn	nn
Letting с =1/ц(х Га ,b ]) for each n > 0 we define geC(R,R) as follows:
n	n n	~ ~
g(t) =	/ 0	t < 0 1 (c1/a1)t	teto^] \,2. c	tf[a ,b ], 1 k-«l к	u n n 1 Ec / (a -b )] (t-b ) + £ \	n+1 n-bl n	П K=1	nfN	
		c, tf[b , a 3 к 	n n+1	, nfN
Then the gn(t) = so that у	function yeg*xeC(T)R). Let g(t)	t < an n & Ck	b an =g .xeCb(T,R), yn < y, and yn	m (t)= I ck for	t₽x Va ,b "], and m m
m < n. As h is positive		k=l	
EXERCISES 2
127
Ь(у> > ь<Уп> - J-, r x.lt »,
U	m m
m=l
dp.
n m
2 (2 c )
m=l k-1
for each n > 0. This
contradiction
As a consequence of (vi) we see
^X'lfam’bm])
x -• h(x) = J X dp,
> n
establishes the result.
that the functional
is real-valued on C(T,R). Furthermore suppose that x > 0. Then, again by
(vi), there exists r > 0 such that u.(x b[0, r])=p,(T) so that ft(x)«=Pxdp;, Let
x^=min(x,r). Clear ly S (x)=h (x ). On the other hand x^eC^(T,R^ s£ tfidt
h(x) > Ь(х^)=^ xrdp=h(xr)=h(x). We may therefore conclude that h-h is a
positive linear functional on C(T,R) which vanishes on C, (T,R). Thus to
finish the proof of (a) we need only prove that
(vii)	a positive linear functional x on C(T,R) which vanishes on
Cb(T,R) vanishes identically on C(T,,R): Since each x«c(T,R) may be ex-
pressed in the form x=(xV0)-((-x)V 0) it is enough to prove that х'"'(х)=0
whenever x > 0. If xeC (T,R) there is nothing to show, so let us assume
—	b ~
that x is unbounded on T. Choose an increasing sequence (a ) of real
n n > 1
numbers from x(T) and let t eT be such that x(t )=a for each n > 0. Next
n	n n
define the bounded functions
(	0 for tex 1 ГО,a ,]
i	n-1
x (t) •= \ x(t) - a , for tex l(a ,,a 1
n	I	n*1	n-1 n
( c -1/ \
\ a - a , for tex (a ,
n n-1	n
where a =0 and neN. Each such function is continuous on T. If x(t)e(a . ,
о	~	n-1
a ] then x, (t)=a -a, , for к < n, x (t)=x(t)-a	,, and x, (t)=0 for к > n.
nJ	k kk-1	n	n-1	к
Thus, for any such t, x(t)=£^ x^(t). If x(t)=0 then x^(t)»=0 for all к so
that once again x(t)=£^ x^(t). Therefore	converses pointwise to x on
T. Next set у =a x for each positive n. Clearly Г у (s)=0 for all :
n^nn	J n J n
the open set x [O,apso that y(s)=£n y^Cs) exists and is continuous on
x ЬГо,ал). For sex~l(a ,, a ) for some n > 1 we see that
1	n-1 n+1	—
s in
128
2. SPACES OF CONTINUOUS FUNCTIONS
.	< n-1 ak(ak‘ak-l) + an(x(t) - an-P for SeX'1(an-l’aJ
Ws) -	1
(	< n ak(ak‘ak-l) + an+l(x(t) - V for sex’ ^n’W
so that y(s)=IL y. (s) exists and is continuous on x \a _ , a	Now
к к	n-1 n+1
У - ^k < n yk “ ^k > n yk -	> n ak Xk^ > n an xk “ an(x -
and, as x" vanishes on C^(T,£,)>
Z(y) - x*(y - £k £ n yk) > an x,r(x - n xk) - an x*(x)
for each positive n. Thus x (x)=0.
In (2.4-2) and (2.4-3) we saw that a linear functional оп:С^(Т,^) is
continuous in the uniform norm iff it is the difference of two positive
linear functionals. Close scrutiny of that result reveals that we actually
proved that a linear functional is order-bounded (i.e. it sends sets of the
form [xp x^ ]-(x |x^ < x < x^} into bounded sets of numbers) iff it is a dif-
ference of two positive linear functionals. The same proof may be used to
obtain the corresponding result for linear functionals on C(T,R).
(b) ORDER-BOUNDED LINEAR FUNCTIONALS A linear functional x* on C(T,R) is
ft ft ft	ft	ft
order-bounded iff x »x -x where x and x are positive on C(T,R).
p n	p	n	~
Now it is a simple matter to see that (a) may be extended to
(c) ORDER-BOUNDED LINEAR FUNCTIONALS ON C(T,R) AND REGULAR SET FUNCTIONS
If x is order-bounded on C(T„R) then there exists a unique regular count-
ably additive set function p. defined on 0 such that
x (x) = j1 x dy, (xeC(T,R)).
Furthermore for each xfC(T,R) there exists Ее such that x is bounded on
E and |u|(CE)=O (where |p | represents the total variation of p).
Proof of uniqueness If and	are two regular countably additive set
functions defined on yielding the same finite integral for each
xeC(T.R) then	is regular and countably additive on ® and Jxdp=O
for each xeC(T,R). Let Ae 0 and choose G, HeP such that CHCAC.G and
~	a
|ul|(G-CH) < e. By the normality of T there exists a function xeC^d,^)
such that 0 < x < 1, x(Ch)={1} and x(CG)={0). Then
|u(A) | = |p,(A) - Jx du. | = | J’ k^ du - jx du |
“ | /H<kA ' x)dd I “ I / x dp. | < P (GflH)
EXERCISES 2
129
Thus u(A)=O for all Aj^.
Since our aim is to obtain a representation theorem for the elements
of C(T,R,c)' it is natural at this point to inquire as to the circumstances
under which all order-bounded linear functionals are continuous.
(d) T REPLETE ~»(0RDER-BOUNDED-*» CONTINUOUS): If T is replete then the
linear functional x* on C(T,^) is order-bounded iff x is continuous in the
compact-open topology. Thus, in this case, the elements of C(T,R,c)' are
characterized in (c).
Proof As any order segment [x,y]={zfC(T,R) |x < z < y} is bounded in the
compact-open topology (see proof of Theorem 2.6-1), continuity implies
оrder-boundedness. To obtain the converse we recall that in the proof of
Theorem 2.6-1 it was shown that any absolutely convex set V which absorbs
all order-segments is a neighborhood of the origin in C(T,R,c) provided T
is replete. Since x ([x,y])CMx'	((-€,e)), e > 0, x is continuous,
(e) BOUNDED CONTINUITY FOR T NOT REPLETE Of the homomorphisms of C(T,R)
only the evaluation maps T ={t |tfTj are continuous in the compact-open
topology. Any homomorphism fixed to a point of uT-Т, i.e. a homomorphism
of the type p (x)=x (p), XfC(T,R), pevT-T, is discontinuous. On the other
hand any such homomorphism is bounded as the following argument shows: If
x < у then by the denseness of T in uT, xu < yu. Thus for any x < z < у
it follows that
p'(x) = xv(p) < p (z) = zu(p) < p (y) = yb(p)
and p of any order segment is a bounded set of numbers.
What then do the elements of C(T,R,c)' look like when T is not replete?
(f) C(T,^,c)' AND RESTRICTIONS OF REGULAR SET FUNCTIONS Let T be a com-
pletely regular Hausdorff space. If x'fC(T,R,c)' then there exists a set
function ц.“Ц^-|Л2 defined on p(T), where ц and are outer measures which
when restricted to 63^ are regular measures, and a compact set KC1 T such
that
x'(x) = f x du. for xeC(T.R)
and ц. is countably additive on К=[АЛ К |Ae .
Proof Just as in (2.4-3) we may decompose x' into a difference of positive
linear functionals x'=x '-x ' where for x > 0. x ' (x)=sup(x* (y) lyfС. (T,R),
pn	— p	b ~
0 < у < x} and for arbitrary x=(xV 0)-(xA 0), x 1(x)=x '(xVO)-x'(xAO).
As x' is continuous there exists a compact set KCT and a constant M > 0
130
2. SPACES OF CONTINUOUS FUNCTIONS
such that |x1 (x) | < M p^,(x).
It follows that
|x'p(xA 0) | < 2M PK(X) so that x ' and x^' are
measures u
P
' P	P
both continuous. Thus by (a)
u defined on P(T) which
n
' (x)= Гх du and
P - P
Next we claim that compact sets К and
P
' If this isn't the case,
n
(К) < u (T). Thus there exists
P
As К and clCG„ are disjoint
p К	J
XK - 11	and
x„ du =1. On the
К p
and
and its proof there are outer
when restricted to are regular measures such that xj
x ' (x)= Гх du for each xeC(T,R).
n J n	~
К exist such that u (К )=p. (T) and p, (K )=u (T):
n	"	"
say for Up>
G,,f P such
p '" 'p' ’"p ' ' nn'
then for each compact KC.T, Ц, i
that Gt,2)K and u (GIZ) < ц (T).
К	p К p
exists x fC(T,R) such that 0 <
Hence x
P
in gT there
{1/d (CG ) }.	л	|Л	|
1 p К	p К J К p -J CG^
the compact subsets of T are ordered by set inclusion,	is a net in
C(T,R, c) clearly converges to the zero function, thereby contradicting the
other hand if
x
continuity of x* . Letting K=K UK it is seen that u (K)=u (T) and u (K)=
P	p n	p p	n
P (T).
n
At this point it is clear that it suffices to consider positive x', i.e
x'=x ' and P=u . The next thing to do is to transfer x' over to C(K’,R).
P	P	~
Each xeC(K,R) has a continuous extension to T (extend x to gT and then re-
strict it to T) so that it is natural to attempt to define x^(x)=x'(y)
where у is some continuous extension of x up to T. To see that this defi-
nition is meaningful we need only show that x'(z)»=0 for any zfC(T,R) which
vanishes on K. If z is such a function then G=z	)fP and contains K.
Thus p(T-G )=0, p.(G )=u(K) and therefore
e	f
„ z du + J
4
G
e
z dP. | < e u(K)
for each e > 0.
an outer measure on
Now Xj,' is clearly positive so there exists
P(K) which is a regular measure on the Baire subsets of К such that
x ’(x)=[x du for each xec(K,£). We contend that (iii) the Baire subsets
К Р К
of К and 65 ClK coincide: Denoting the class of all complements of zero
3
sets in К by P observe that P.Ok={gOК |Ge P }C3 P . Conversely, if
In.	I'**
Gf P then there exists a non-negative xfC(K,R) such that x (0,co)=G.
k	- 1
Thus, if у is a continuous extension of x to Tzy (0,<»)D K=GfPDK. Now
ig is the er-ring generated by p so it follows that	is the c-ring
generated by P = Pfj К (Halmos, 1950, p. 25, Theorem E).
К
The next thing to be shown is that u and agree on the power set
P(K). As the first step in that direction we claim that (iv) p. and U
К
agree on P = POK: Let Gf P so that for each f > 0 there exists yeC(K,R)
EXERCISES 2
131
such that 0 < у < к (on К) and u,K(G)-e < x^' (y). As GC1T there also exists
ah HeP such that u(H)-e < p,(G). Let F denote the compact set (tf К 11 у (t) |>
e}. Since cl^CH and F are disjoint in gT there exists UCC(T,R) such that
0 < u < 1, u(F)=[l} and u(CH)={0}. Let у be any continuous extension of у
to T, x=u у and x | =x. It follows that у < x + f and 0 < x < к . Thus
К	H
4„(G) - e < x ' (y) < x ' (x + e) = x ' (x) + e ц (К)
К	к	к	к	к
“ X’ (х) + е UL-(x)
к
< ц,(н) + е Р^(х) < ii(g) + е(1 + ик(Ю).
As f is arbitrary ^(G) < 11(G). To obtain the reverse inequality choose
JeP such that JCIK'G. For e > 0 choose ZfC(T,R) such that 0 < z < k^ and
u(J)-e < x' (2). Setting z=z |z it follows that 0 < z < k on F and
К	G
PK(G) > x*,' (z) = x' (z) > ul(J) - € > 4(G) - e.
Hence 11(G) > u(G) and finally u =p, on P .
К	к	к
(v) u and u^ agree on P(K): Let ACK, and choose Ge P such that
H„(G)-e < p. (A). If Hep and H(1K=G then ll(H)=u(HO K)=p(G) so that
К	к
и (A) - e < 4(H) - e - U(G) - e - p. (g) - e < u. (A).
К	к
Thus u.(A) < Uj,(A). On the other hand for f > 0 there exists JeP such that
J3A and p(J)-e < 11(A). Then, noting that i(J(lKpii(J), JHKePIZ, and J f~) К
К
A, we see that
p^(A) - e < pK(jf)K) - € = n(j) - e < 4(A)
for each e > 0. Hence 4 (A) < u(A).
К
Since u is countably additive on the Baire subsets of К (=®Г)КЬу
К	a
(iii)), и, is countably additive on €a()K. Furthermore for each xeC(T„R)
x’ (x) = x^ (x |K) = j* x|K dp^= du. V
2.5 FIELD-VALUED CONTINUOUS FUNCTIONS (Cf. Exercises 1.9 and 1.13). Let
T be an ultraregular space and К a complete nonarchimedean nontrivially
valued field. The main point of this exercise is to characterize when
C(T,K,c) is "barreled" and "bornological" (parts (e) and (f)) after intro-
ducing suitable meanings for those terms.
A bounded measure p, on T is a finitely additive g-valued set function
on the algebra© of clopen subsets of T such that for some positive number
M, |u(S) | < M for all Sf©. The linear space (usual operations) of all such
measures 4 is denoted by B(T,K). If 4 assumes only the values 0 and 1 and
132
2. SPACES OF CONTINUOUS FUNCTIONS
is monotone (i.e. if u(S)=O, then jj,(S')=O for all S'CS), then ц is called
a monotone 0-1 measure. The set of all such measures is denoted by M (T).
We note that any 0-1 measure is monotone if the characteristic of J<, ch j<,
is not 2; if ch )C=2, then there are non-monotone 0-1 measures.
(a)	C^ (T.K.)1 and B(T,K) Let C^ (T,K) be the subalgebra of functions
in C(T,K) with relatively compact range, endowed with the topology of uni-
form convergence on T, and let C (T,K)' denote the continuous dual of C^
(T,K). Show that there is a 1-1 correspondence between B(T,K) and С (T,K)'»
(Hint: If ffC (T.K)1, let uc(S)=f(kr,) where k is the characteristic func-
tion of Sf(S. Conversely, let D(T,K) be the linear span of the functions
(kQ)c,~ • For deB(T,g) and 2 a k eD(T,K) let f (2 a k )=£ a u,(S ). Show
О	X О -j	11.	1 o	i 1
that f is continuous on D(T,K) and extend it to C (T,K) by continuity.)
1Л	c
For UfB(T,K) let be the collection of sets Sf€i such that for all
clopen subsets S' of S, u,(S')=O. The support, supp ц.» °f 11	(2.4-7))
is defined to be C((jD ). If supp u, is compact, u is said to have compact
support, and the collection of all such measures is denoted by Bc(T,K).
The next result establishes a 1-1 correspondence between Bc(T,K) and C(T,K,
c)'.
(b)	В (T,K) and C(T,K,c)' Let T be compact, u,fB(T,K), P=(S^) be a
finite partition of T into disjoint clopen sets, and dp=max^ |ll(S^) |. For
t rS, and xrC(T,K), lim., „Ex(t.)u(S ) exists and is denoted by fxdp,.
Show that J • du is a continuous linear functional on C(T,K,c). Conversely,
if feC(T,K,c)' then there exists u^eB(T,K) such that f(x)=jxdp,^ for each
xeC(T,K). Moreover, letting j|u, || =sup |u,F(S) |, ii f К - 1| u,, |j .
For frC(T,K,c)', let Df denote the collection of clopen sets S such that
f(xks)-0
C(U Df).
for all xeC(T,K)- The support of f, supp f, is defined to be
Next, it is shown that there is a bounded measure u such that supp
u,f=supp f and that this notion of support of a continuous linear functional
has essentially the same properties as support does in the classical case
(cf. (2.4-7) and (2.4-8)).
(c)	Bounding sets and vanishing sets A bounding set for ffC(T,K,c)' is
a compact subset К of T for which there is some NR > 0 such that |f(x) | <
N ,p (x)=N sup lx(K) I for each xfC(T,K). A vanishing set for f is a compact
set К with the property that if x vanishes on K, f must vanish on x.
(i) Show that a compact set is a bounding set iff it is a vanishing
set. (Hint: Let К be a vanishing set for f, xrC(T,K), and x* the restric-
tion of x to K. Note that each xfC(K,g) can be extended to an xfC(T,I<) as
EXERCISES 2
133
follows. Let sup |x(K) |=M, and construct a finite disjoint cover (W^) of К
of clopen subsets of T such that sup {I x(t)-x(t' ) I |t, t1 eW. f| K} < M/2 . For
t.fW.DK, let x =S? .x(t,)k .
1	1	1	1=1	1 VL
In similar fashion construct a
while sup (x^CT) | < M/2, and so
Then sup |(x-x^) (К) | < H/2 while sup]x^(T)| <M.
function x^ such that sup |(x-x^-X2 ) (K) | < M/4
on. Consider the map
f':C(K,£,c)
X*
f(x)
Show that f is a well-defined
continuous linear functional on C(K,K,c) and,
for f. )
thus, that К is a bounding set
(ii)	For each ffC(T,K,c)', show that is a ring of sets.
(iii)	Show that supp f=C(l_)D^) is compact and nonempty iff f is nontriv-
ial. (Hint: Let К be a vanishing set for F. If Se6> and SCCK, then SfD^.
Hence СК C LjDf and supp fCZK.)
(iv)	Show that if SfGJ and S 0 supp f^0, then there is some xeC(T,K)
such that x(CS)=[o} and f(x)=l.
(v)	If T is compact, then supp f is a vanishing set for f. (Hint:
Suppose that supp f is a subset of the clopen set
CS is covered by finitely many S^fD^. Thus CSfD^
Thus any clopen superset of supp f is a vanishing
supp f is a vanishing set. Indeed, if x vanishes
{teT |jx(t)|< l/n}, then supp f C. S^ for each neN and each S
set. Thus f(x)=f(xkg )+f(xkcg )=f(xkg ).
n	n	n
compact subset К of T, by the definition of S^ it follows that if (x)| <
N^(l/n) for all nfN and f(x)=O.)
and ц,
. Show that supp u^=supp f and
to set inclusion) vanishing set
S. Since CS is compact,
and S is a vanishing set.
insures that
set; this
on supp f
and
S =
n
a vanishing
is
n
As |f(x) I < N P (xk ) for some
К К sn
be the bounded measure defined by taking
that supp f is
for f,
measure
(Hint:
defined
That every
(vi) Let feC(T,K,c)'
uf(S)=f(kg) for each SfG>
the minimal (with respect
Let К be a vanishing set for f and let be the bounded
on the clopen subsets SflK of К by taking u (Sf) K)=u^(S )=f (kg).
clopen subset of К is of the form SЛ К for some clopen subset S of T is
guaranteed by the compactness of K. To see that цf1 is well-defined, note
that if X, Ye<2> and Х,ПК=УГ)К,
thus f (kx)=f(ky)=f (kx/^ y). Let
let f'(x1)=f(x)=jx'dUf'. (Also
f=supp f', supp Pf=supp Uf', and supp
f
then (Х-ХПу)ПМ so that f(k Yf_v)=0;
A- Alli
x' be the restriction of xeC(T,K) to K, and
note that (ц,^)'=Ul^, у) Now show that supp
supp f'. Then use (b) and (v)
134
2. SPACES OF CONTINUOUS FUNCTIONS
above to show that f'(x')=f(x)=lx'du '=Г	.x'dp ' = f rix'durl =
•J f J supp p,' f J supp f' f'
Г .x'dp )
•J supp f f
A topological vector space X over Ji is called local ly K-convex if its
topology is generated by a family P of nonarchimedean seminorms (i.e. for
each peP, p(x+y) < max(p(x), p(y)). Letting C denote the closed unit disc
about 0 in K, a subset U of X is called absolutely К-convex if CU+CUCLU.
If p is a nonarchimedean seminorm, all sets of the form {xfX|p(x) <
are absolutely К-convex. X is К-barreled if every absolutely K-convex
closed absorbent subset of X is a neighborhood of 0. Similarly X is K-
bornological if every absolutely К-convex bornivore is a neighborhood of 0.
The space C(T,K,c) is a nonarchimedean locally К-convex space.
К is called spherically complete if every totally ordered collection of
closed discs in К has nonempty intersection and an analog of the Hahn-Banach
theorem holds for X when Ji has this property (Ingleton 1952, Narici, Becken-
stein and Bachman 1971, p. 78).
(d)	Relative K-pseudocompactness (Cf. (1.5-4)) A closed subset L of
T is relatively K-pseudocompact if every xeC(T,ji) is bounded on L. Show
that a closed set L is relatively K-pseudocompact iff cl LCu T where p T
~ ₽o ° °
and и T are as in Exercises 1.9(f) and 1.13(e). Thus the "K" in "relatively
о	~
K-pseudocompact" is superfluous and may be omitted.
(e)	K-barreledness of C(T,K,c) If К is spherically complete, show that
C(T,K,c) is J^-barreled iff every relatively K-pseudocompact subset of T is
compact.
(f)	t(-bornologicity of C(T,K,c) If Ji has non-Ulam cardinal, then
C(T,JC,c) is I<-bornological iff vqT=T.
(g)	If J is a spherically complete field of non-Ulam cardinal, then
K-bornologicity of C(T,^,c) implies K-barreledness of C(T,K,c).
THREE
Lattices and Wallman Compactifications
Lattices, in general, possess two descriptions: As an ordered space and a
more algebraic description as a set equipped with two operations. The
equivalence of these descriptions is discussed in the first section. Ex-
amples of lattices abound; a few are of special concern to us. (The pro-
totype lattices are collections of subsets of a given set.) Later we fo-
cus particular attention on the lattice of closed subsets of a topologi-
cal space. Another of special interest is the lattice of zerosets of a
topological space.
Using various lattices £ of subsets of a topological space T one may always
construct a nearby compact space w(T,£) which is a compactification of T
under certain conditions. More generally, there is a compact space wL
associated with any lattice L and this is discussed in Sec. 3.2; the com-
pact space wL is the collection of all L-ultrafiIters endowed with a cer-
tain topology. Given a topological space T we have already seen an in-
stance where a compactification has been obtained from an associated space
of ultrafiIters: In Chap. 1 it was seen that the Stone-Cech compactifica-
tion gT of T was the collection of all ultrafilters from the lattice Z of
zerosets of T.
We specialize to the setting of main interest co us in Sec. 3.3 and re-
strict consideration to lattices £of subsets of a topological space T.
Conditions on £ are developed which ensure that the map
t -	= {A € Z |t € A]
is a 1-1 map of T into w(T,£). It must be guaranteed that is an Z-
ultrafilter and that distinct t's produce different ultrafiIters. When
these conditions are satisfied
we can ask whether the compact space w(T,£)
is a compactification of T. It
a homeomorphism iff £ is a base
is always dense in w(T,£).
turns out ((3.3-2)) that the map t -> is
for the closed sets in T, while |t j T]
In Sec. 3.4 sufficient conditions are obtained for w(T,£) to be gT. If T
is any completely regular Hausdorff space e.g. and one takes the lattice £
to be the latticed of zerosets of T, it happens that w(T,Z) = ₽T.
Wallman compactifications constitute a more general breed of compactifica-
tion than the Stone-Cech. Every Stone-Cech compactification may be realized
135
136
3. WALLMAN COMPACTIFICATIONS
as a Walltnan compactification, as mentioned in the proceeding paragraph,
and so can any one-point compactification of a locally compact Hausdorff
space be realized as a Wallman compactification. One-point compactifica-
tions, however, are certainly only rarely Stone-Cech compactifications,
(bounded continuous functions on R for example may generally not be contin-
uously extended to the one-point compactification of R). This raises the
question: Perhaps any compactification is of the Wallman type? This ques-
tion, which has to do with felicitously choosing a lattice of subsets,
hasn't been answered yet. Even so, some progress has been made and in Sec.
3.5: Given a Hausdorff compactification S of a topological space T and a
lattice £ of subsets of T, necessary and sufficient conditions are obtained
for w(T,£) to equal S in Theorem 3.5-1.
The subject of Sec. 3.6 is'when' two lattices/-/ and £ of subsets of a topo-
logical space T determine the same Wallman compactification. These results
are used in Chap. 5 (principally in Sec. 5.4) where we investigate how
spaces// of maximal ideals of certain kinds of topological algebras X may
be realized as Wallman compactifications w(.A/c,£) of the space//, of closed
maximal ideals of X.
3.1 Lattices A partially ordered set containing the supremum and infimum
of each finite set is a lattice. Though we introduce lattices here - ac-
tually distributive lattice with 0 and I - we scarcely investigate them,
providing only what is needed for use in the sequel. Some of our nomencla-
ture is unconventional - "filter" for "additive ideal without 0" for exam-
ple - because of the desire to use language which is topologically sugges-
tive. For more complete information about lattices per se, the reader is
referred to the standard work of Birkhoff (1948) and to Gratzer (1971).
The power set P(T) of a set T, i.e. the collection of all subsets of T,
equipped with the relation
A <: В iff AC В
is a partially ordered set in which each pair of elements A,В has a greatest
lower bound дПв and a least upper bound AUB. Motivated by this example
we define a lattice to be any partially ordered set (L, <) in which each
pair of elements a,b€L has a greatest lower bound aAb, the "meet" of a and
b, and a least upper bound avb, the "join" of a and b. L is a dis tributive
la ttice provided
3.1 LATTICES
137
aA(bvc) = (аЛЬ) v (аЛс)
and
av(bAc) = (aVb) A (aVc)
for all a,b,cfL- Actually these rules are equivalent to each other. The
power set lattice P(T) is certainly distributive. Moreover P(T) has a
least element 0 and a greatest element T. When a lattice L possesses a
least or greatest element it is unique and is denoted by 0 and 1 respec-
tively. Corresponding to each element AeP(T) there is an element CA, the
complement of A, having the properties:
АЛСА = A 0 CA = 0, AVCA = A U CA = T.
If each element a of a general lattice L with 0 and 1 possesses a comple-
ment b, i.e. aAb=O, aVb=l, L is said to be complemented and a complemented
distributive lattice is referred to as a Boolean lattice.
A more algebraic descrip.ion of these ideas is also useful. First we
replace the symbols V and A by + and . respectively, i.e. for a, b in the
lattice L we define a+b=avb and ab=aAb. Then it clearly follows that L is
a commutative semigroup with respect to each of these operations. Further-
more if L is a lattice with 0 and 1 then these semigroups possess identi-
ties. Some things concerning these new operations are apparent; in par-
ticular the idempotent properties: For any afL
a + a = aa = a.
Moreover
a < b iff a + b = b iff ab = a.
In a distributive lattice the distributive laws take the form:
a (b + c) = ab + ac
a + be = (a + b) (a + c),
while in a complemented lattice to each element a of the lattice there
corresponds and element b such that
a + b = 1 and ab = 0.
Having just seen how to convert a < formulation of a lattice into a
(+,.) phrasing, we now consider the reverse question: Given a set L pro-
vided with operations + and ., when is it possible to recover the ordering?
What conditions must + and . satisfy?
138
3. WALLMAN COMPACTIFICATIONS
(3.1-1) ALGEBRAIC DESCRIPTIONS OF LATTICES Let L be a non-empty set pro-
vided with two operators -I- and •. If
(1)	L is a commutative semigroup with respect to each operation,
(2)	a + a = aa = a for each aCL, and
(3)	a + b = b iff ab = a
then the relation
a < b iff a + b = b
defines a partial order on L with respect to which L is a lattice. If a,
bfL then avb=a+b and aAb=ab. Moreover if each of the commutative semi-
groups possesses an identity then the identities serve as 0 and 1 for the
lattice. L is a distributive lattice if
(4)	a(b + c) - ab +cC and ab + c = (a + c)(b + c) for all a,b,cfL.
Finally if L contains 0 and I and
(5)	for each a€L there exists bfL such that a + b - 1 while ab = 0
L is a Boolean lattice.
Proof Clearly < defines a partial order on L. To see that aVb=a+b it is
necessary to show that a, b < a+b. Indeed
a + (a+b) = (a+a) + b = a+ b
so that a < a+b. In the same way b < a+b. Suppose that a, b < c and con-
sider
(a + b) + c = a + (b + с) = a + c = c.
Thus a+b < c and a+b is therefore the join of a and b. An analogous argu-
ment yields aAb=ab. The remaining claims are obvious, v
Example 3.1-1 THE POWER SET BOOLEAN LATTICE Let T be a set and L the
collection of all subsets of T. For A,BfL let
(1) A • В = A fl В
(2) A + В = AU В.
L with these operations is, as was previously noted, a Boolean lattice.
Rather than consider all subsets of T, it is of course possible to restrict
attention to subclasses of P(T) which contain a least upper bound and a
greatest lower bound of each pair of its elements with respect to the in-
clusion ordering. It is not necessary for АЛВ and AVB to be АО В as hap-
pens for example if L consists just of sets 0, T, A, and В where AU B^T,
АП В^0, and A^B.
Example 3.1-2 THE LATTICE OF CLOSED SETS Let T be a T^ topological space
and L be the collection of closed subsets of T. Then L is a distributive
3.1 LATTICES
139
lattice with 0 and 1 with respect to IJ and 0 . It is clear that the com-
plement of any closed set is closed iff the topology on T is discrete.
Hence, L cannot be a Boolean lattice unless T is discrete.
A different-looking example of a lattice is given next.
Example 3.1-3 THE LATTICE OF CONTINUOUS FUNCTIONS Let T be a topological
space. Then C(TJ(R) is a lattice with respect to the partial order
x < у iff x(t) < y(t) for each tfT (x, y€C(T,R.)).
Since min(x,y) = %((x+y)-| x-y|)) and max(x,y) = %(x+y+| x-y | ) are continuous
and
(хЛу) (t) = min(x(t), y(t)),
(xvy) (t) = max(x(t), y(t))
хЛу and xVy belong to C(T,J1). It is evident that the distributive laws
hold. On the other hand C(T,R) contains neither a least nor a greatest
element. If Instead of taking all of C(T>(R) we choose L=[x,y3=
{zfC(T,R) |x < z < y}, where x and у are continuous real-valued functions
on T such that x < y, then L is a distributive lattice with 0=x and l=y.
In most cases however, L is not complemented, e.g. if x and у are the con-
tinuous functions assuming the real number values 0 and 1 respectively,
the only elements in L with complements are x and y.
In Chap. 1 it was shown that in the class of replete spaces T, the
algebra C(T,j<) completely characterizes T, i.e. if T and S are replete
spaces and C(T,R.) and C(S,R) are isomorphic algebras, then T and S are
homeomorphic ((l.b-3)).
As noted ia Sec. l.b the lattice properties of C(T,R) are often enough
to characterize T, e.g. if T and S are compact Hausdorff spaces and C(T,R)
and C(S,R.) are order-isomorphisms (there exists a 1-1 mapping ф taking
C(T,_R) onto C(S,R) such that x < у implies ф(х) < ф(у))1Ьеп T and S are
homeomorphic (Kaplansky 1947).
Since the distributive function lattices of the previous example differ
so drastically in appearance from the set lattices given in Examples 3.1-1
and 3.1-2 it is perhaps surprising to find that any lattice can be "repre-
sented" by a set lattice. More precisely, if L is a lattice then there
exists a set T and a collection of its subsets which,when supplied with the
ordering defined by set inclusion,is order-isomorphic to L. Indeed by
taking T=L and £ to be the collection of all subsets of T of the form
La = [b f L|b < a] (a C L)
140
3. WALLMAN COMPACTIFICATIONS
the mapping a -• L is seen to be an order-isomorphism between L and
Even more can be said - if L is a distributive lattice then L is order-
isomorphic to a set lattice in which the meet and join of a pair of sets
are given by Q and U respectively (Nobeling 1954, p. 28, and Hermes 1955,
p. 10b).
The lattice of the following example is quite similar in form to the
lattice of closed sets mentioned in Example 3.1-2. After linking lattices
and compactifications, it wi1 1 be seen (Example 3.3-3) '.hat the lattice of
all closed sets leads to the Wallman compactification. Sublattices of the
lattice of all closed sets yield other compactifications. The lattice of
zero sets of e completely regular Hausdorff space T, for example, produces
«Т ((3.4-3)).
Example 3. 1-4 THE ZERO SET LATTICE Let T Oe a completely regular Hausdorff
space andZ the collection of all zero sets of functions in C(T,JR). By the
results of Sec. 1.2, Z. is a distributive lattice with 0 and 1 with respect
to U and 0 :
2	2
z(x) u z(y) = z(x + y )fZ
z(x) П z(y) = z(xy)fZ
The notions to be considered next are abstractions of concepts partic-
ular to the power set lattice. First we have a generalization of "filter".
Definition 3.1-1 LATTICE FILTERS AND ULTRAFILTERS A subset F of a distrib-
utive lattice L with 0 and 1 is a lattice filter (or simply "filter") if
(1) 0 { F,
(2) F is stable under finite products (i.e. if a, bfF then abfF) and
(3) (the "upper bound" condition) if afF and b > a then bfF.
A lattice filter F is a lattice ultrafilter (or just "ultrafilter") if no
other filter contains F as s proper subset.
Analogous to the power set situation, ultrafil :ers can be characterized
in terms of a "primality" condition.
(31-2) ULTRAFILTERS Let F be a lattice filter, in the distributive lat-
tice L with 0 and 1. Then if F is an ultrafilter and a+bfF, either a€F or
b?F. Conversely, if L is a Boolean lattice and
a+bfF->aeForbtF (a, b C L)
then F is an ultrafilter.
Proof Suppose that F is an ultrafilter, a+bfF, and b^F. Then, letting
H={c€L Ic+bfF], it follows that H is a filter: Indeed O^H as bf(F. If c,d€H
3.1 LATTICES
141
then, since cd+b=(c+b)(d+b)CF, cdfH. Moreover, if c€H and d > c then, as
d+b > c+b, it follows that d+bfF and d€H. Clearly afH and FCH so the
maximality of F implies that a€H=F.
Conversely, if F is a filter in the Boolean lattice L and F contains a
or b whenever a+bfF, suppose that HDF and afH while a^F. As IfF, a^F,
and l=a+a ' where a' is the complement of a, it follows that a'€F. But then
both a and a' belong to H so that O=aa'fH - a contradiction. Thus H=F and
F is an ultrafilter, v
Example 3.1-5 LATTICE FILTERS VS. SET FILTERS In a power set lattice P(T)
a collection of subsets is a lattice filter iff it is a filter in the ordi-
nary set - theoretic sense. In a lattice £ of subsets of a set T (with
respect to и and 0 ) a lattice filter is not, in general, a set filter,
as it need not contain all supersets of elements of the filter - just those
belonging to X.
The primality condition given in (3.1-2) completely characterizes ultra-
filters in Boolean lattices. Below we obtain a characterization of them in
any distributive lattice with 0 and 1 (Theorem 3.1-1) by essentially
strengthening condition (3) of the definition of a lattice filter. First,
however, we define a notion which is somewhat weaker than that of a lattice
fiIter.
Definition 3.1-2 LATTICE FILTER SUBBASE A subset S of a distributive lat-
tice L with 0 and 1 is a lattice filter subbase if it has the finite inter-
section property, i.e. all finite products of elements of S are non-zero.
Every lattice filter subbase S is contained in a lattice filter for
clearly the collection F(S) of all elements b of L which are greater than
or equal to some finite product of elements from S is a lattice filter and,
in fact, is the smallest lattice filter of L containing S. As promised, we
now show that by strengthening the third condition of Definition 3.1-1, a
characterization of ultrafilters is obtained.
Theorem 3.1-1 ULTRAFILTERS AND THE "MEET" CONDITION Let L be a distribu-
tive lattice L with 0 and 1. Then FC L is a lattice ultrafilter iff
(1)	F is a lattice filter subbase,
(2)	F is stable under finite products, and
(3)	(the "meet" condition) for any cfL, if ca-^O for each a€F then c€F.
Re ma гк Clearly conditions (1) and (2) are equivalent to conditions (1) and
(2) of the definition of a lattice filter (Definition 3.1-1). Moreover the
meet condition is stronger than the upper bound condition of Definition
3.1-1: Whenever b > a for some a€F then, for each dfF, bd-^O - indeed if
142
3 WALLMAN COMPACTIFICATIONS
bd=O then ad=bad=a(bd)=O - a contradiction. Hence, by the meet condition
the upper bound condition of Definition 3.1-1 is fulfilled; b€F.
Proof Clearly (1) and (2) are necessary for F to be a lattice ultrafilter.
As for the meet condition suppose that ca^O for each afF. As the smallest
lattice filter H generated by the lattice filter subbase FU[c] contains
the ultrafilter F, H=F and, therefore, c€F.
Conversely, as was noted in the remark above, conditions (1), (2) and
(3) force F to be a lattice filter. If H is a lattice filter D F and cfH
then ca^O for each afF. Thus c€F by (3), H=F, and F is therefore a lattice
ultrafilter. V
The collection of all lattice filter subbases of a given distributive
lattice L with 0 and 1 Is partially ordered by set inclusion and, via the
Zorn's lemma argument, lattice filter subbases which are maximal with re-
spect to set inclusion can be shown to exist. Such objects are referred
to as maxima 1 lattice filter subbases. It is perhaps surprising that max-
imal lattice filter subbases are in fact lattice ultrafilters viz. ;
(3.1-3) MAXIMAL FILTER SUBBASE ULTRAFILTER In a distributive lattice
L with 0 and 1 a lattice filter subbase is maximal iff it is a lattice
ultra filter.
Proof If H is a lattice ultrafilter and S is a lattice filter subbase con-
taining H then the smallest lattice filter F(S) containing S contains H.
Thus by the maxima 11 ty of H,
HC SC F (S) = H
and S=H thereby proving that H is a maximal lattice filter subbase.
Conversely, suppose that S is a maximal lattice filter subbase. Then,
as S1 - the collection of all finite products of S - is a lattice filter
subbase containing S, S' =S . By Theorem 3.1-1, it remains to show that cfS
whenever ca^O for each afS. But this is clearly the case as SU{c]=S'U {cj
is a lattice filter subbase containing S. V
Section 3.2 LATTICES AND ASSOCIATED COMPACTIFICATIONS In this section we
show that given a distributive lattice L with 0 and 1 there is an associ-
ated compact space wL: wL consists of all lattice ultrafilters. As ob-
served in Sec. 3.1 given a topological space T there are certain lattices
which one naturally tends to consider, e.g. the lattice (S> of all closed
subsets of T and, for T completely regular and Hausdorff, the lattice
of all zero sets of T. Determining the properties of wL in these special
cases is what this section is directed toward.
3.2 LATTICES AND COMPACTIFICATIONS
143
To begin with we single out a collection of subsets of the family of
all ultrafilters of a distributive lattice L with 0 and 1 to serve as a
base of closed subsets for a topology. For each a€L we define the basic
set	generated by a to be the collection of all ultrafilters to which a
belongs. As no ultrafilter can contain 0,	Thus	for the basic sets
to be a base rather than a subbase of closed sets for a topology it suf-
fices that they be stable with respect to the formation of finite unions.
Indeed, as we now show, ® U	for each pair a, bfL. If Ff ©
a b a+b	a
then, since for any bfL a a+b, a+bfF and FC ®	. Thus (S,® C (g
J	a+b	a7 b a+b
and so ® U ®. C® . On the other hand, as ultrafilters satisfy the
a b a+b	J
pritnality condition of (3.1-2)
Fc ® ,, - a+bfF - afF or bCF - Fc(3 U <S, .
a+b	a b
Hence <S U	and it follows that ) is a base of closed sub-
a b a+b	a ael
sets for a topology on the family of ultra filters. This topological space
is referred to as the Waliman space genera ted by L and is denoted by wL.
If I is a lattice of subsets of a topological space T with respect to
union and intersection the associated Wallman space is denoted by w(T,£).
Cer ta inly
man space
the family (c®A)Aex
w(T,&> ) , О being the
is a base of open sets of w£. In the Wall-
collection of all closed subsets of the
space T, these open sets are in a sense generated by the open subsets of
Example 3,2-1 BASIC OPEN SETS IN w(T,£) Let X be a lattice of subsets of
the set T with respect to union and intersection. Then (a) a base for the
open sets of w(T,£) is the collection of sets of the form
Уц = ^€w(T,£) |AC^ *АПи 5* 0]
where U runs over all complements taken in T of sets in T; in other words
У consists of all ultrafilters whose traced 0 U on U is a filterbase.
Moreover the sets VIT have the property that UC W — C y^.
Ff£. If^CC®F then
HC.CF. As АПН^О for
u	 	
Proof We show that V =C®„ for each
~	Cr F
an HC^5 exists such that H("1F=0 and so
meets each element of and
pose that eVCp- Then as FOCF=0, F^^ and	The truth of the
assertion about the is obvious, v
open subsets of a topological space T generate an open base
Will a base (S for the topology of T generate an open base
Although not generally true (see Exercise 3.3(c)) this will
To obtain the reverse
so that
each A€^ , CF
inclusion sup-
monotonicity
Thus the
for w(T,X^ ) •
for w(T, C* ) ?
144
3. WALLMAN COMPACTIFICATIONS
be the case if® satisfies the normality condition of (b) below.
(b) If ® is a base of open sets for T which separate the closed subsets of
T, i.e. for each disjoint pair F^, F2?^ there exist disjoint U^, such
that F^CUj and F2CU2’ t'ien tami-^y	is an °Pen base for w(T^r).
Proof Let WCT be open and V^=C®^ . Then there exists an F€^ such
that FClCW=0. Choosing UC® such that FCUCW, it follows tha t ^7 c V c -V
U W
This last result hints at the existence of an interrelationship between
the topological natures of T and w(T,'C»). Indeed, the interrelationship is
a strong one.
Returning to the
distributive lattice
Т^-пезз we show that
wL. As c1w(f}= 0 [l®a
Certainlv if GC (")
a€F
general (i.e. lattice) situation we claim that for any
L with 0 and 1, wL is a compact T^ space. To establish
cl [f1=[F1 for each FfwL where cl denotes closure in
wl 1 1 1	w
|Fc ®a}=	^Е®а’ we neec* onFy show that Cl ®a={F}>
then FCG and therefore, by the maximality of F,
G=F. In order to establish compactness another fundamental property
basic sets is needed: Namely stability under finite intersections;
fact 63 Cl Ф ® , for each pair a, bfL. First, if Ff @ Cl then
a b a b
so that abcF and FC Ф , . On the other hand, if abCF
ab
both a and b belong to F and FC (3 Cl . It clearly
a b n
stable property that for any a1,...,a^cL, Q ф= =0
wL i s s hown
of the
in
First, if Ff ® Cl
a b
then, since a, b
follows from this
a, b€F
some subset
1’’‘’~n
to be compact. A closed subset of
Л C L in that it be written in the form «А
be a family
of closed subsets of wL with the
that the family |aCUA } also
property. Thus, as	ai’a2’
and (JA is a lattice filter subbase,
containing UA, iu rollows that Fc
and Cl<fA ?t0. Hence wL is compact.
The above discussion is summarized
It is clear
iff a,....a =0. Next
1 n
uniquely determined by
.	= Г1 ® . Le t (?. )
Л аел a
finite intersection property.
enjoys the finite intersection
for any choice of a,,...,a CUA
1 n
"ai
wL is
. .a ^0
n
Letting F be any lattice ultrafilter
for each af UA" whence FC 0 J",
A
in
the two following results. In the
first, (3.2-1), we list those fundamental
properties of the basic sets used
and in
in defining the Waldman topological space
Tj-ness for ease in future reference.
(3.2-1) ELEMENTARY PROPERTIES OF THE BASIC SETS
la ttice
with
obtaining
Let L be
compactness and
a distributive
(1)
(2)
for
for
0 and 1 and ((P) ) ,, the
Ja a€L
each FfwL, Cl ® ={f],
afF a 1
each pair a, bfL
®aU ®b=®a+b and ®a°
associated
family of
® = (B , and
b ab
basic sets. Then
3.2 LATTICES AND COMPACTIFICATIONS
145
(3) for any choice of a^,...,an€L
n £ a =0 iff van=0-
i=l L
Theorem 3.2-1 wL IS A COMPACT T^ SPACE If L is a distributive lattice
with 0 and 1 then the Wallman space wL is a compact T^ space.
At this point the reader may be wonde"li when wL is Hausdorff. ' 3 is
certainly Hausdorff if for each pair of distinct elements F, G€wL there
exists a pair of disjoint complements of basic sets CC^i and ®a ' to
which F and G belong. Corresponding to each such pair F and G there exists
at least one pair of elements afF and bfG such that ab=O. For С(£>а । and
С , to be disjoint, or equivalently for (3a	।= ®a ®^i=wL,it Is nec-
essary and sufficient that a'+b' belong to each ultrafilter. To be certain
that Fee®, and G€C® , it suf fees that a'b=ab'=O. Thus if to eac-li pair
b	a
of nonzero elements a, bfL such that ab=O there corresponds a pair a', b' fL
such that a'b=ab'=O and a'+b' belongs to each ultrafilter then wL is Haus-
dorff. Lattices of this type are called normal because of the similarity
of the lattice condition to	the topological notion of normality.
Definition 3,2-1 NORMAL LATTICES A distributive lattice L with 0 and 1 is
norma 1 if corresponding to each pair of nonzero elements a, b€L such that
ab=O there exists a pair a', b'fL such that
(1) a ' b = ab' =0 and
(2) a' + b'fS’ for each FfwL
If in a normal lattice we choose а. у pair of ultrafilters F and G con-
taining a and b respectively then Gl£®a, and	Sin:: ®a iU =
®a'4l-l’wI" Ff-^a ' and ®b'" H?nce aa'-^O and bb'^O. Moreover the ele-
ments a'+a and b'+b are > a and b respectively and the elements a, a'+a,
b, b'+b satisfy conditions (1) and (2) of the definition. Thus we may state
an equivalent but stronger looking definition of normality: To each pair
of nonzero elements a, b of L such that ab=0 there exists a' > a and b' > b
such that (1) and (2) hold.
Now the discussion preceding the definition served to prove that wL is
Hausdorff whenever L is normal. As it happens the converse is also true.
We have :
Theorem 3.2-2 L IS NORMAL IFF wL IS	Let L be a distributive lattice
with 0 and 1. Then wL is Hausdorff iff L is a normal lattice.
Proof It remains to prove that L is normal when wL is Hausdorff. Since wL
is compact it is also normal. If a, beL are nonzero,ab=0,then ® fl ® =0.
II	a b
By the normality of wL there exists a pair of disjoint open sets and V
146
3. WALLMAN COMPACTIFICATIONS
containing (6 and respectively. Since cU and CV are both closed,
subsets R and S of L exist such that C U. = П ® and сУ =	More-
cfR C	dfS
over
n ®cn <Ba = n Sdn <0b = 0.
c€R	dfS
Now (53 and , being closed in the сопгаг~t space wL, are compact and so
a	b
.,c CR and d,,...,d cS such that
’ -	1 m4
there exists c^s
n
n
1=1
m
<8C П ®a - П <Sb = 0>
1 i=i j
n
If we set b’=c ,..
n
and © ,=	(3
j = l '
n
and,
d.
J
b' a
a' dr
therefore
and
. . ,d then by (3.1-1)(3)	,=
m	d
= 0.
Next we claim that
ter subbase and,
that Е€<Вь1Г1 ©a
which it follows
Indeed if b'a-^O then S = {a,b'}
as such is contained in' some ultrafilter F. It
- a contradiction. Finally CUCS^j and cYG.©al from
b1a=a1b=0.
is a fil-
follows
tha t
® ,n®
a b
a
, fi,U ,Z3CltUcV = wL.
+b a b
Hence a'+b’ belongs to each ultrafilter and L is a normal lattice. The
converse has been proved in the discussion prior to Definition 3.2-l.V
In the remainder of this section we present
latt
some examples of normal
L be a Boolean lattice,
Example 3.2-2 BOOLEAN LATTICES ARE NORMAL Let
i.e. a complemented distributive lattice with 0
b€L are nonzero and ab=0. Choosing a'=a and b'
and 1, and suppose that a,
to be the cor. plement of a
we see that a'b=ab'=O and a'+b'=l belongs to each ultrafilter of L.
Hence
L is normal.
Example 3.2-3 .Z IS NORMAL Let T be a completely regular Hausdorff
and2.be the lattice of all zero sets in T (see Example 3.1-4).
that Z. is a normal lattice let and Zj be disjoint zero sets.
Theorem 1.2-2 cl Z, and cl Z_ are disjoint so that there exists
M . ₽2
ous function 0<x^___
[0} while x^(cl Z )={1}. Thus, denoting the restriction of x
it follows by (1.2-3) that CZ^ =x (-a:,l/4)and CZ^=x (3/4,“=)
complements of zero sets containing Z^ and Z£ respectively.
space
To
see
Then by
a contlnu-
P< 1 defined on the normal space ?T such that x^(cl zp =
P to T by x,
are disjoint
Thus Z ’ 3 Z
3.3 COMPACTIFICATIONS
147
and ZjJZj, and Z^UZ^, being equal to T,
In the next example another reason for
belongs to each z-ultrafilter.
using the terminology "normal
lattice" comes to light.
Example 3.2-4 T IS T;-» (T IS NORMAL	C IS NORMAL) Let T be a
topological space. Then if T is normal, in essentially the same way as the
lattice of zero sets was shown to be normal, it can be shown that is a
normal lattice. Conversely suppose С» is normal. Then by the stronger
form of normality mentioned after Definition 3.2-1 if F and E are disjoint
closed sets there exists closed sets F'Z)F and E'DE such that
F'nE = Е'П F = 0
and F'UE1 belongs to each C»-ultraf i Iter. Next we claim that T =f'|_JE'-
If not then there exists tjfF'U E1. Since T is T^, the set {t} is closed.
Letting^" be any x$,-ultrafilter containing the (^-filter [Ae<£|{t}cz A] , it
follows, since {t}P) (E'(J F' ) =0, that E'UF'jf^?-. As E'UF1 must belong to
each (J-ultrafiIter no such t can exist and T=E'LJF'. Thus CE1 and CF'
are disjoint open sets containing F and E respectively so that T is a
normal topological space.
3.3 WALLMAN COMPACTIFICATIONS OF TOPOLOGICAL SPACES In the representa-
tion of the Stone-Cech compactification of a completely regular Hausdorff
space T as the collection of all z-ultrafilters T is imbedded in [3T via
the mapping
tp : T ---»- gT
t ----= {ZeZ| t€Z} •
At this time we wish to imbed a set T in the collection w(T,£) of all £-
ultrafilters where I is a distributive lattice with 0 and 1 of subsets
with respect to U and П . Motivated by the above we consider the map-
ping cp taking t€T into ^}:t=[Ae£ |teA]. Our purpose now is to determine
conditions on £ such that cp is a 1-1 mapping of T into w(T,£). If this is
to be the case then certainly each -3^ must be an ultrafilter. But is
always an £-filter, so by the meet condition	in order for to
be an £-ultrafliter (for each teT) it is necessary and sufficient that for
each tCT and lattice element	there exists an element Be<?t such that
АГ)В=0 (Theorem 3.1-1). In other words for each tfT and Ae£ such that t(A
there exists Be £ such that teB and A(~)B=0. A lattice £ satisfying this
condition is called an g- la ttjce. Thus cp maps T into w(T,£) izf £ is an
Of-lattice. What about 1-lness of cp? Suppose that tp is 1-1; i.e. if t^t'
148
3. WALLMAN COMPACTIFICATIONS
then	1 . As and ' are unequal there exists Af^t such that
A^«^?t' or Ae^.' and kt Thus for each pair of distinct elements t, t'fT
there exists an element Af£ containing exactly one of the points. Such a
lattice is called a B~ la ttice. Hence if cp is 1-1 then Z is a g-lattice.
Conversely if £ is a g-lattice and t-^t' then the element A belongs to exact-
ly one of the filters or 1 so that <7^' •
We summarize the icregoing discussion in the next definition and result.
Definition 3.3-1 g AND1g-LATTICES Let Z, a collection of subsets of a set
T, be a distributive lattice with 0 and 1 with respect to LJ and О . Then
(a)	Z is an g-lattice if for each tfT and ACZ such that ttfA there exists
BC£ such that t€B and AflB=0.
(b)	I is a g-lattice if for each pair of distinct points there exists
Ac£ containing one of the points and not the other. If Z satisfies both
conditions we refer to it as an gg-lattice.
(3.3-1) IMBEDDING T IN w(T,J) Let T be a set, J!CP(T) be a distributive
lattice with 0 and 1 with respect to U and П , and tp be the mapping of T
into the collection of ^-filters taking t into ^У?^=[АсТ ItfA}. Then
(a)	tp(T) C. w(T,£) iff Z is an g-lattice, and
(b)	cp is 1-1	iff X is a g-lattice.
Example 3.3-1 .Z	IS AN gg-LATTICE Let T	be a	completely	regular	Hausdorff
space and Z the	lattice of all zero-sets	of T. Suppose	that tfT and ZeZ
such that t£z. Then, as Z is closed and	T is	completely	regular	there ex-
ists an xfC,T,R) such that 0 < x < 1, tfx \o) and ZCx 1(1). Thus
-1 -1 ~ -1
x (0) (Z, x (0)П Z =0 and tfx (0) so thatX is an g-lattice. Given dis-
tinct points t, t'cT we may apply the same argument to the pair t and ft'}
to obtain an element x \o) C.Z. such that tfx \o) and t1 ^x \o) (equivalent-
ly [t'Jflx \o)=0). Hence Z is also a g-lattice.V
Example 3.3-2 IS AN gg-LATTICE-О T IS T; Consider the lattice "C of
all closed subsets of the topological space T.
(a)	is a g-lattice if and only if T is T .
Proof If t and t1 are distinct points in the Tq space T then elftj^clft1}.
Thus either clft] or clft1} is an element of £ containing exactly one of the
points. Conversely suppose there exists Af & containing one of the points,
say t, and not the other. Then t' belongs to the open set CA while t does
not and, as t and t1 are arbitrary, T is T . V
о
(b)	is an gg-lattice iff T is T^.
Proof If T is Tj then T is Tq and, by (a), is a g-lattice. Let A? and
3.3 COMPACTIFICATIONS
149
t^A.
and is
As T is Tp [t}<£,.
Thus {t} is an element of which contains t
disjoint from A, and is therefore an а-lattice. Conversely, sup-
pose thatt$- is an a₽-lattice. Then as 6 is a p-lattice, if t and t' are
distinct points of T there exists Ac containing one of the points and not
the other. Suppose that t^A and t'fA. Since & is also an а-lattice there
exists ве& such that tfB and АЛМ. Hence tfCA and t'^CA while t'fCB
and t^CB and, there Tire, T is Tp V
Since our ultimate goal is to force w(T,£) to be a compactification of
T when T carries a topology, our next result is concerned with conditions
on £ under which cp(T) is a dense homeomorphic image if T in w(T,£).
(3.3-2) IMBEDDING THE TOPOLOGICAL SPACE T IN w(T,£) Let £ be ag-lattice
of subsets of the topological space T so that the mapping
T -----w(T, £)
t -----1-
t
is 1-1. Then
(a)	cp Is continuous iff all the sets in £ are closed in T;
(b)	tp is a homeomorphism iff £ is a base of closed sets of the space Г,
(c)	cp(T) is dense in w(T,£). Thus w(T,£) is a T^ compactification of
T whenever £ is a base for the closed subsets of T. Moreover w(T,£) Is the
"smallest" T^ compactification of T in the sense that if " is already com-
pact then <p(T)=w(T,£) .
Proof (a) Recall (Example 3.2-1) that the sets ={?€w(T,£) |AC</ ->
A ПСЕ?М} for Fe£ constitutes an open base for w(T,£). It is easy to see
that cp is continuous iff each Ff£ is closed in T.
(b) Suppose that {F }	Then it follows that <?(/") ffFff) =
Л (Ъ (T). Thus if £ is a base for the closed subsets of T, cp is a
O' Ff,
relatively closed mapping. Since cp is also continuous by (a), cp is a ho-
meomorphism. Conversely, if Ф is a homeomorphism and F is a closed subset
of T then cp(F) =	,, F CP(T) for some family [f } , л of elements of £.
v сгСЛ a	a a( A
As cp Is 1-1 and cp(F)=cp( ^F^), F= Fff and £ is a base of closed subsets of T.
* If S is a Hausdorff compactification of T and T is compact then, as T is
thereby closed in S, S=T. This needn't be the case if we merely assume
that T and S are Tj for if S is an infinite set equipped with the cofinite
topology (^ set is open iff it is the complement of a finite set) and T is
a proper Infinite subset of S with the cofinite topology both T and J are
compact while T is dense in S.
150
3. WALIMAN COMPACTIFICATIONS
(c) That cp(T) is dense in w(T,£) follows from the observation that
У for each teCF and each Ff£. Thus each basic open subset of
t Cr
w(T,£) contains points of cp(T) and cl^cp (T)=w(T,£). In the event that
T is already compact and is an £-ultrafiIter then, as^, a collection
of closed subsets of T has the finite intersection property, ^0. If
we choose tf it immediately follows from the maximality of<^ that
^=^ . Hence ф(Т)“ч(Т,Х) ,V
Compactifications of type generated by the theorem are called "Wallman-
type" compactifications. We reserve the name "Wallman compactification"
for the special case discussed in our next example.
Example 3.3-3 THE WALLMAN COMPACTIFICATION OF A 7 SPACE If T is a
space then the collection of all closed subsets of T is an 07?-lattice
(Example 3.3-2) and the mapping
c₽:T ----w(T,-£)
t------>- _?
v t
is homeomorphic as "tx is certainly a base of closed sets ((3.3-2)(b)).
Thus w(T,?x) is a T^ compactification of T by (3.3-2)(c) and is referred
to as the Wallman compactification of the space T. Since <x is normal
iff T is normal (Example 3.2-4) and w(T,'<x) is Hausdorff iff (x is normal
(Theorem 3.2-2), the Wallman compactification is Hausdorff iff T is normal.
Example 3.3-4 w(T,Z ) IS A HAUSDORFF COMPACTIFICATION OF T If T is a
completely regular Hausdorff space then we recall thatJZ. is a normal ag-
lattice (see Example 3.2-3 and Example 3.3-1). It remains to show that_Z
is a base for the closed subsets of T in order to conclude that w(T,Z) is
a Hausdorff compactification of T. To s- e this, recall that rhe topology
of T is the weakest topology with respect to which each x€C. (T,R) is con-
fa ~
tinuous.
Thus a subbase for the closed subsets of T consists of all sets of the
form x L(C) where x€C^(T,R) end C is closed in n. As this collection coin-
cides wlth)Z ((1.2-3)) and z. is closed with respect to the formation of
finite unions Z. is a base
for the closed sets.V
In general if X is dense in Y and ZCY then it need not follow that
Xf|Z be dense in Z.
In fact this need not be the case even if Z is closed
in Y. If this is so for closed sets i.e. if cl(XHZ)=Z for all closed sub-
sets Z of Y, then X is said to be very dense in Y.
Observe
that Y being
precludes the existence of a very dense proper subspace X for if ye Y-X
3.3 COMPACTIFICATIONS
151
then {у }^с 1 (хГ) [у } )=с 10=0. Thus Т (or rather а?(Т) ) cannot be very dense in
w(T, £) .
Our next result has a twofold purpose. The first part shows that when
T is a topological space, T is fairly close to being very dense in w(T,£),
while the second part indicates in a sense just how far off T can be from
being very dense in w(T,£). The symbol cl denotes closure in w(T,£).
(3.3-3) T IS "NEARLY" VERY DENSE IN w(T,£) Let £ be an ^-lattice of sub-
sets of the set T.
(a)	Then for each Ff£, cl m(F)=cl (® Dtp(T)) = -3
W	W г т	Г
(b)	If T is T-j space, £= £, and [F^jo'CA }c£> then
ci (Q ® n<p(D) = F •
W O'	' Iff
Proof (a) Certainly <B Л tp(T)=co(F) . Observe that cl ( 0 Q tp(T) ) C- ® •
F	‘	w г	F
To obtain the reverse inclusion let,? € ф, and be a typical basic
F	CH
open neighborhood of in w(T,X) where	Then, as	and FC,/,
Ln
РЛСН^0. Choosing any teFQCH it follows that Ff.^ and therefore	.
Since t also belongs to CH, CH meets every set in and ^CH’
v meets 6„ П ф(Т) for such Hf£ so that ?fcl (fl f)tp(T)).
Ln	г	w г
(b)	Since Fa e we have
<p(t) = c₽(Q f^) =	®>n^F Лф(т)
Thus by (a), cl (О A О tp(T))= 6_	. V
W n Г	( I ГУ r
a	a
Remark As a result of (a) of the preceding result the collection of clo-
sures of the basic elements Af£ forms a base of closed sets for w(T,£).
Recall that in the Stone-Cech compactification ₽T of a completely reg-
ular Hausdorff space T the closures of disjoint zero sets of T are disjoint
(Theorem 1.2-1). An analogous result for Wallman-type compactifications
is true and is obtained from part (a) of the previous result.
(3.3-4) DISJOINT LATTICE SETS HAVE DISJOINT CLOSURES IN w(T.J) Let T be
a topological space and f be a base of closed subsets of T which forms an
ag-lattice. Then for any pair F, Hf£,
FflH = 0 - cl F(lcl H = 0.
w w
Proof By (3.3-3) (a), cl cp(F)= ®_ and cl tp(H)=S„. Since Ff)H=0,
	W	Г	W	H
(cl cp(F)) Л (cl C₽(H)) = ft ft © = 0.V
W	W	Г П
152
3. WALIMAN COMPACTIFICATIONS
A similar property of w(T,£) is one of the properties that distin-
guishes the Hausdorff compactifications of completely regular Hausdorff
spaces which are of the Wallman-type, (see Theorem 3.5-1).
3.4 gT AND WALLMAN COMPACTIFICATIONS For w(T,£) to be a Hausdorff com-
pactification of the topological space T it is necessary and sufficient
that I be a normal Ofg-lattice which serves as a base of closed subsets of
T (see Theorem 3.2-2, (3.3-1), and (3.3-2)). Viewed from a different
point of view this amounts to saying that a topological space has a Haus-
dorff Wallman-type compactification iff there exists a base of closed sub-
sets of T which forms a normal ag-lattice of subsets of T. Such topolog-
ical spaces are singled out in Definition 3.4-1. After that we develop a
sufficient condition for coincidence to occur between gT and w(T,£).
Definition 3.4-1 SEMI-NORMAL SPACES The topological space T is called
semi-normal iff there exists a base of closed subsets of T which forms a
normal ag-lattice with respect to U and (Л . Such a base is referred to
as a norma 1 base for T.
We already know that the completely regular Hausdorff spaces are pre-
cisely the spaces with Hausdorff compactifications so it is reasonable to
suspect that the semi-normal and completely regular Hausdorff spaces are
related. Indeed, as we now show, in the class of T^ spaces, the classes
of semi-normal and completely regular spaces coincide. The notion of a
completely regular space is defined in an external fashion utilizing the
elements of C^(T,R). Our next result provides an internal characteriza-
tion of complete regularity via the notion of a normal base.
(3.4-1) INTERNAL CHARACTERIZATION OF COMPLETE REGULARITY A topological
space T is a completely regular Hausdorff space iff it is semi-normal.
Proof If T is a completely regular Hausdorff space then as we have shown
in Example 3.3-4 the lattice of zero sets Z is a normal ag-lattice which
serves as a base for the closed subsets of T.
Conversely, as we have already noted, a semi-normal topological space
T has a Hausdorff Wai Iman-type compactification w(T,£) where £ is a normal
base for T. Since w(T, £) is a compact Hausdrrtf space then the subspace T
must be completely regular and Hausdorff. v
How does a Wallman compactification w(T,£) of a completely regular
Hausdorff space T compare with gT? Certainly they could only be equiva-
lent when w(T,£) is Hausdorff - i.e. iff £ is a normal base for T. In
this case then, by Theorem l.l-l(c), w(T,£)=gT if T is C^ - imbedded in
w(T,£). In the following discussion we determine those bounded continuous
3.4 WALIMAN AND STONE-CECH
153
real-valued functions which are continuously extendible to a given Hausdorff
Wallman compactification w(T,X) of T.
Definition 3.4-2 £-UNIFORM CONTINUITY Let £ be a normal base for the com-
pletely regular Hausdorff space T. Then the function xfC(T,R) is J-unl-
formjy continuous if for each c > 0 there exist sets	(£ such that
n	In
T= U CA. and, using "o" for "oscillation",
1=1 1
o(x, CAp = sup{|x(s) - x(t)| |s, tCCA.} < C
for each i = l,...,n.
Certainly any ^-uniform continuous function is bounded. Furthermore,
the collection of all such functions forms a closed subalgebra of C^(T,R)
in the uniform norm. In our next result we show that any xfC^(T,R) that
pulls closed subsets of £ back into members of £ must be Х-uniformly con-
tinuous .
(3.4-2) A CLASS OF X-UNIFORMLY CONTINUOUS FUNCTIONS Let T be a completely
regular Hausdorff space and £ a normal base for T. If xfC^(T,R) and
x 1(C)eX for each closed CCR then x is X-uniformly continuous.
Proof Let f > 0 be given. As x(T) is bounded in R there exists a finite
n ~
number of real numbers a^,...,an such that (_) (a.-f/4, a.+(/4)3x(T) .
Thus each set A.=x \c(a.-f /4, a.+r/4))fX anti 1= CA.. Furthermore, as
-1	1	1	1	i=l 1
CA^=x (a.-f/4, ap-e/4), o(x,CA^) < (/2 < ( for each i and, since ( is
arbitrary, x is X-uniformly continuous. V
The X-uniformly continuous functions are precisely those which are
continuously extendible to w(T, X) •
Theorem 3.4-1 CONTINUOUS EXTENDIBILITY TO w(T,X) Let T be я completely
regular Hausdorff space and £ a normal base for T. Then xfC(T, () has a con-
tinuous extension to w(T,X) iff x is X-uniformly continuous,
w
Proof Let x be the continuous extension of xfC(T,R) up to w(T,X)- Then,
as JJ (x^'^x^)- (.12., xW(^)+f/2) )=w(T, X) and w(T,X) is compact there
^fw(T,X)	w 1 w
exists a finite subfamily Uj=(x ) (x (^^)-f/2, xW(^p+c/2)) , i=l,...,n,
covering w(T,X). Each of these sets, being open in w(T,X)> is a union of
.	w
basic sets of the form	where AfX. Now the oscillation of x on each
such set is less that or equal to f. Moreover w(T, X) is the union of all
the basic sets making
VCA coverlng W(T>£>
cp as in (3.3-1), cp(T)
1 < j < m then
up the U., 1=1,...,n, so that there exists
such that o(xW, V ) < ( for j=l,...,m.
С U V,,. ,T= iYcA.. If s, tfCA. for
i-i “j jY, J	J
^CAj’ ’ ’ • ’
Since with
s ome
154
3. WALIMAN COMPACTIFICATIONS
|x(s) - x(t)| = | xW(5's) - xW(^t)|< €
s ince A^CA , and x is ^-uniformly continuous.
Conversely, suppose that x is X-uniformly continuous and let fw(T, £).
As is a filterbase on T, x(^')={x(A) jAe^} is a filterbase on R. Fur-
thermore, as x(T) is bounded, x(?) has a adherence point in R which we de-
note by xW(^) . We shall show that x(^) -> xW(J) by proving x^) to be a
Cauchy filterbase with the aid of the ^-uniform continuity of x. Indeed
n
for a given C >0 there exists A, , . . . ,A such that T=l_I CA. and
1 n n	1=1	1
o(x,CA.) < C for each 1 < i < n. Since fl A.=0 there exists 1 < j < n
1	i=l 1
such that	. By the maximality of there exists A?^- such that
A^flA=0 or, equivalently, AC CA^ . Thus for any pair t, s€A, |x(t)-x(s) | <€
and it follows that xQ^) is Cauchy.
w
To complete the proof we must do two things. We must show that x ,
w
as defined above, extends x and Is continuous on w(T,£). To see that x
extends x, let t€T and cons Ider . We claim that xW(^)=x(t). As
x(<^ ) -» xW ) there exists AC^t such that for each s€A, |x (s) -xW(^.) | <€
Thus, since tCA,
|x(s) -x(t)|<|x(s) - xW(^t) |+ |x(t) - xW(4J't) |< 2f
and so x(^.y- ) -» x(t). Hence x )=x(t) . It remains to show that x is
t	t	n
continuous on w(T, Z) • Le t A, , • . • jA €£ be such tha t TSU CA. and
1 n	1=1 t
o(x,CA.) < C for 1=1,...,n. We contend that given any	fw(T, X) there
exists	for some 1 < 1 < n such that J- f V and |xw(i/)-xw(^)|<3e
7"i	n	^i
whenever W C V . . Since fl A.=0 if -J- Cw(T,J‘) there exists AC# and an
CAi	1=1 1
A. , 1 < i < n such that AflA.=0. Thus ACCA. and V , . Now suppose
i” ~	1	1	CA.
that 4/ also belongs to V . As x(^) -• xW(^) and	— xW(4J )
there, exists Af4t and B€ 4/ such that
|x(s) - x“(^) | < € for all s€A
and
|x(s) - xW(4J ) j < ( for all s?B.
Let tfAflCA^ and sCBClCA^. Then
|x(t) - x”(^) | < C, |x(s) - xW(4J) I < «
and, since t, s€CA^,
|x(t) - x(s) | < €.
3.5 A CLASS OF WALIMAN-TYPES
155
Thus it follows that jxw(-^ )-xw (^ ) [ < 3€ and the proof is complete, v
As a result of this theorem together with (3.4-2) examples of Hausdorff
Wallman-type compactifications which are equivalent to gT may be given.
(3.4-3) gT IS A WALLMAN-TYPE COMPACTIFICATION Let T be a completely reg-
ular Hausdorff space, z. be the collection of zero sets of T, and A* be
the collection of all closed subsets of T. Then
(a)	gT and w(T,X ) are ecuivalent compactifications of T;
(b)	gT and w(T,'C») are equivalent compactifications of T iff T is
norma 1.
Proof (a) Since Z is a normal base for T, w(T,Z) is a Hausdorff com-
pactification of T. By (1.2-3), x \c)fZ for each closed subset C of R
and x€C. (T,R). Hence, by (3.4-2) and Theorem 3.4-1, all the elements of
C^(T,R) are continuously extendible to w(T,2i) so, by Theorem l.l-l(c),
w(T,Z.) and gT are equivalent compactifications.
(b) As is a normal lattice if the Tj—space is normal (Example
3.2-4) and & is an Ofg-lattice whenever T is T. (Example 3.3-2) it only
_ 1 *
remains to note thac x (C)e£ for each xfC^(T,£) and closed subset C of R
for then (3.4-2) and Theorem 3.4-1 allow us to conclude that w(T,£>) and
gT are equivalent compactifications. Conversely, if w(T,C>) and gT are
equivalent compactifications, w(T,£j ) is Hausdorff and T is normal (Exam-
ple 3.3-3) . V
3.5 A CLASS OF WALLMAN-TYPE COMPACTIFICATIONS We have seen that any com-
pletely regular Hausdorff space T possesses a Wallman-type Hausdorff com-
pactification. In particular the Stone-Cech compactification is equivalent
to the Wallman-type compactification w(T,.Z) where Z is the normal org-
lattice of zero sets. Frink (1964) raised the question of whether all
Hausdorff compactifications of completely regular Hausdorff spaces are
Wallman-type compactifications. In the main result of this section, due
to Brooks (1967) and Alo and Shapiro, 1968a and 1968b, necessary and suf-
ficient conditions are given for a Hausdorff compactification of a com-
pletely regular Hausdorff space to be of the Wallman type. This result is
then used to show that the one-point compactification of a locally compact
Hausdorff space is a Wallman-type compactification. Further results along
these lines can be found in Exercise 3.7.
Theorem 3.5-1 WALLMAN - TY PE ? Let T be a completely regular Hausdorff
space, S a Hausdorff compactification of T, and £ a distributive lattice
with 0 and 1 of closed subsets of T with respect to U and Л . Then S
and w(T,£) are equivalent compactifications of T via a homeomorphism a:
156
3. WALIMAN COMPACTIFICATIONS
w(T,£) - S where	)=o( [AC £ |ceA})=o(tp( t) )=t for each tfT iff
(1)	for each pair A, Bf£, c 1 (AQ B) =clgA П c l^B,
(2)	for each AfX and sfS-cl^A there exists Bf£ such that sCcl^B and
АП B=0, and
(3)	for each pair of distinct points s, t€S there exists A, Be £ such
that stfclgA, t^clgB, and A|JB=T.
Remark We shall write S=w(T,X) when they are equivalent in this way.
Proof Suppose that S is a Hausdorff compactification of the completely reg-
ular Hausdorff space T and there exists an onto homeomorphism a: w(T,£) S
such that a(s^'t_)=a(cp(t) ) = t for each tCT. Then from the fact [(3.3-3) (a)]
that cl (cp(A))- ®. = {/ew(T, £) lAC^} for each Af£ it follows that
w	A	I
(*) o((B ) = a(cl (cp(A)) = cl (A)
AW	b
for each A€£. Thus for A, B€£,
cl Ariel В = a(® )Па( <8 R) = a(©.n® ) = a( ft) = cl (АЛВ),
So	A	d	Ad	A f 1 d	b
thereby establishing (1). As the sets AeX, constitute a base of closed
sets in w(T,£) we see by (*) that the sets cl^A are a base of closed sets
for S. Therefore, if A€£ and sCS-cl^A we may write the closed set [s] =
(]cl В where В €£ for each a.
a S a	a
Since dgA is
compact
and fails
to meet
flcLB there
a s a n
cl AQ( П cl
b i=1 n
are a finite number of sets
В )=0. Since (1) holds so
O'-
B„, , . . . , В such tha t
ffi “s
that cl (Г) В )=f|cl„B ,
S i = l а/ S af
if we
As for (3) we may use the argument
choose в= Cl в e£ property (2) follows.
i=l ai
similar to the one used to prove (2) to obtain the existence of a pair of
disjoint sets A, BeX such that teclgA and sCclgB. Now it may happen that
AJJB^T. However, since T is a homeomorphism it follows that w(T,£) is a
Hausdorff compactification of T with respect to the imbedding tp: T -» w(T,£),
t —	Thus £ is a normal OfS-lattice and corresponding to A and В there
exist supersets A 1 , B'fi such tha t A'U В ' =T and A' Л B=B ' Л A=0  Thus by (1)
cl A'H cl B=cl B'H cl A=0 so that t(cl B' and siclA1. This establishes (3).
Ь	Ь Ь	Ь	Ь	о
Conversely, suppose that conditions (1), (2) and (3) hold. First we
claim that I is a normal ag-lattice. Indeed properties (2) and (3) imply
that £ is an a5-lattice. We now show that £ is normal. If C, De£ with
criD=0 then, by (1), clgCd clsD=0. With r, s€C and t€D we may use (3) to
construct sets A , В e£ such that cl CCcl A . , cLDCcl В ; r, s/cl В ,
st st	S S st b S st	b st
t(clA and A US =T- Since П. .cl A CC(clD), there exist (s., t.)
S st	st st	(s,t) S st S	it
i=l,...,n, such that Л cLA . CC(cl D). Now (C).A )U(fVB )=T,
1 b S « E .	Ь	1. S • E •	L S . E ,
3.5 A CLASS OF WALLMAN-TYPES
157
Ar= Plj.As t CcD, and U-Bs t =B^ for each r€C. We may repeat the same
argumentLon the B^ to form sets A, Bf£ with the properties АЛС=0, B(3D=0,
and AL) B=T.
Next we define a mapping a: w(T,£) S which turns out to be an onto
homeomorphism. Let^? €w(T,£).
Then as </ has the finite intersection prop-
erty so does
the collection of closed sets
n
f<P
cl F/0.
Moreover this intersection
we make the assumption to the contrary,
[clsF jFG^ } •
consists of just
then there exist
Since S is compact
one point for if
distinct s,
tefpe^_ clgF. Thus by (3) there exists A, B€£ such that s/clgA, tffclgB,
and AlJB=T. Since S=clgT=clgA L)clgB, sCcl^B so that clgB meets clgF tor
each F(^ . Hence 0^с lgВ О clgF=c 1 (ВQ F) for each Ft-J- and it follows that
В meets all elements of the ultrafi Iter J- ; hence B€L? • However since
Bc^?, t must belong to clgB and we have arrived at a contradiction,
fore Л -jcl F is a singleton and we define
Ft^- S
oGy-) = s where
There-
clsF’
To see that a is 1-1 suppose that £ and are distinct ultrafilters so
that contains an element G not in . As is an ultrafilter there
exists F(^ such that FHG=0. Thus ) i.cl^G while f(-^/)CclgG so that
Next we contend that о is onto. To this end, suppose that sfS and let
d~ ={f€£ |s CcI^f} . Now is a filter subbase; in fact if F, then
sCc IgFfl clgG=c lg (Ffj G) by (1), whence FCIGC^. To see that<^- is an ultra-
filter suppose that , i.e. s^clgB. By (2) there exists Af£, s^clgA such
that АГ)В~0- Hence we have produced an element AG? which fails to meet В
and it follows by Theorem 3.1-1 that J- is an ultrafilter. Clearly <}(</) =
s and, as s is arbitrary, a is onto.
The space S is known to be compact,
and a
is Hausdorff
If we show that w(T,£)
is continuous then it follows that a is a homeomorphism,
lish the continuity of a F it is enough to show that cr(6^)=clgA.
then A€<f so that	,^cl FCcl A. On the other hand if
A	Ft^/- S S
t ® then A^L? and there exists Ff^ that fails to meet A. Thus
" -1
c IgA Pl clgF=c lg (A Fl F)-0 and	) tc IgA . Hence cr is continuous.
It is interesting to note that this theorem provides us with an alter-
To estab-
Indeed if
nate argument for the fact that gT and w(T,Z.) are equivalent whenever T is
a completely regular Hausdorff space. Recall first that a Hausdorff com-
pactification S of T is equivalent to gT iff c lgz (y)c IgZ (y) =clg (z (x) Пz (y) )
158
3. WALIMAN COMPACTIFICATIONS
for any pair z (x) , z(y)fZ (Theorem 1.3-2). As w(T,Z.) is a Hausdorff com-
pactification of T (Theorem 3.2-2 and Example 3.2-3), it follows by (1) of
the previous result that cl z(x)(lcl z(y)=cl (z(x)Az(y)) whenever z(x),
w	w	w
z(y)fZ whence w(T,Z.) and gT are equivalent.
The one-point compactification of a locally compact Hausdorff space is
of the Wallman type as Theorem 3.5-2 (and the discussion after it) below
shows. In order to more easily describe the lattice used in the creation
of the appropriate Wallman compactification of Theorem 3.5-2, it is useful
to single out the class of "locally constant" functions named in Definition
3.5-1.
Definition 3.5-1 LOCALLY CONSTANT FUNCTIONS Let S be a Hausdorff compacti-
fication of the completely regular Hausdorff space T and let E denote the
collection of continuously extendible (to S) xfC^(T,R) for which the exten-
3
sion x is constant on some neighborhood of each point pfS-T. Elements of
E are called locally constant.
Theorem 3.5-2 A CLASS OF WALLMAN-TYPE COMPACTIFICATIONS Hausdorff compacti-
fications S of locally compact Hausdorff spaces T are of Wallman-type if
S-T is O-dimensional, (i.e. there is a base of clopen (-closed+open) sets
for the topology of S-T). Specifically S=w(T,z(E)) where z(E) denotes the
lattice (distributive with 0 and 1 with respect to U and Г) ) of zero sets
z(x) as x runs through E.
Proof We show that the lattice conditions (1) - (3) of Theorem 3.5-1 are
satisfied by z(E): namely,
(1)	for all z(x), z(y)€ z(E), c lg (z (х)П z (y ) )=c lgz (x) A c lgz (y) ;
(2)	for each z (x) and s^clg(z(x)), there exists z(y)f z(E) such that
s€clgZ(y) and z(x)A z(y)=0 ;
(3)	pairs of distinct points s, tfS there exists x, yjE such that
tlfcl z(x), s(cl z(y), and z(x)(Jz(y)=T.
S	Ь
Condition (1) That cig (z (x)fl z (y)) d clo z(x)f]clg z (y) is clear.
Conversely, if sfclg z(x)f|clg z(y), there are two cases to consider: sfT
and s^T. If sfT, then Sfz(x)n z(y) since z(x) and z(y) are closed subsets
S S
of T. Thus sfclg (z (х)Г1 z (y)) . If s^T, then, by continuity, x (s)-y (s)=0.
Since x and у are each constant in some neighborhood U of s in S,
U A TCz(x)fl z(y) . Since T is dense in S, then it follows that each neigh-
borhood of s in S meets z(x)Oz(y); hence sfcl (z(x)flz(y)).
3	°
Condition (2) We show that if s(cl z(x), x (э)^0. If s€T, then s£z(x).
S	°	3
If s^T, and x (s)=0 then there is a neighborhood U of s in S on which x is
0. Then, as in the proof above that z(E) satisfies condition (1), the
3.5 A CLASS OF WALIMAN-TYPES
159
contradictory result that sfclg z(x) follows. With у =x -x (s), we have
yS(s) = 0, yjE, and z(y)f|z(x) = 0-
The first two imply, as in the argument for condition (1) above, that
seclg z(y).
Condition 3 Here we use local compactness for the first time. We use it
to show that S-T is a closed subset of S.
For S-T to fail to be closed, there must be a point tfT which belongs
to cls(S-T).
that U(1T is
For such a t, let U be an open neighborhood of t in S such
the interior (in T) of a compact neighborhood U of t in T.
The compact neighborhood U is closed in S, so that its complement CUt is
open in S and contains S-T. Now we have the contradictory implications:
since teclg(S-T)
0 s4 (s-T)n uc:cutn U;
while, since UПCU is a nonempty open subset of S, it must meet T, i.e.
0 s4 си л (ипт)с cutn U = 0-
Thus S-T is closed in S.
As for condition (3) proper, we now consider three cases:
(a)	s, tfT;
(b)	tfT, s^T;
(c)	s, t<T.
Case (a): s, tfT. Since S-T is closed, there are pairwise disjoint open
neighborhoods U , Ut and V of s, t and S-T respectively in S. Consequently
there exists functions x, y€C(T,R) such that:
x(t) - 1,	xS(S-Ut) = {0}
y(s) = 1,	№(S-U ) = {0}
s 1
Since S-U and S-Ug each contain V and V3S-T, both x and у belong to E.
If tfdg z(x), then, since tfT, x(t) would have to be 0. Thus t^cl^ z(x)
and, similarly, s^clg z(y). Finally, since
T= (T-Us)U(T-Ut)Qz(x)iJ z(y),
condition (3) is seen to be satisfied in this case.
Case (b): tfT, s^T. If S-T={s], as would happen for the one-point com-
pactification for example, then an argument similar to that used in the
proof in case (a) demonstrates that condition (3) is satisfied.
If S-T contains more than one point then, as S-T is 0-dimensional, there
are clopen subsets F, G of S-T which are closed in S, such that sfG and
160
3. WALLMAN COMPACTIFICATIONS
S-r=FL)G. The required functions x, yfE shall be constructed such that
S	S	S
x (S-T)={0] while у (F) = {0} and у (G) = {1}. As {t}, F and G are pairwise
disjoint there exist a pairwise disjoint open neighborhoods U, V, and W
containing t, F, and G respectively. By the normality of S, an additional
open set N exists such that GCNCcl NC.W. Now we choose x, yCC, (T,R) with
S S	5	b ~
continuous extensions x , у jC(S,R) such that
x(t) = 1, xS (S-U) = {0}
and
yS(clsN) = {1}, yS(S-W) = {0j.
Certainly t^clg z(x) and s(cl^ z(y). To see that xfE we need only note
that S-IDS-T and x is thereby locally constant on S-T.
As for у being locally constant, it clearly suffices to show that
S	S
У (V)={0] while у (N)={1}. Indeed, since S-W13VOF and clgNZ)G, these
facts follow. Finally we claim that z (x) (J z (y)=T. But this is an immedi-
ate consequence of the relations (S-U) Л TC. z(x) , (S-W) Л TCz(y) , and UDW-
0 so that case (b) has been established.
Case (c) : t, s^T. Again using О-dimensionality there exist disjoint clopen
subsets, F and G, of S-T (which are therefore closed in S) such that tfF
and sfG. Thus we may choose disjoint neighborhoods U1DF and V2G, N and V
such that UIDclgNTjNOF and V Зс^'л'З'л'ЗС, and x, y€Cb(T,R) such that
xS(clsN) = {1}, x(S-U) = {0}
and
yS(clgW) = {1}, yS(S-V) = {0}
Now S-UZJG so that xS(G)-[0] and xS is locally constant on S-T. Similarly
yS is locally constant on S-T. That condition (3) is satisfied follows as
in case (b). V
Remark As is well known [Narici, Beckenstein, and Bachman 1971, pp. 151-
153] a locally compact Hausdorff space is 0-dimensional iff it is totally
disconnected, i.e. the only connected sets are singletons. In the proof of
the previous result we saw that S-T was compact. Thus the result can be
restated to say that a Hausdorff compactification S of a locally compact
space T is of Wallman type whenever S-T is totally disconnected.
We mention previously that the case of a one-point compactification of
a locally compact Hausdorff space T is handled by the previous result. In-
deed, as T is locally compact, S is Hausdorff and S-T, being a singleton,
is zero-dimensional so that S-W(T,z(E)) where z(E) is the collection of all
zero sets of the continuously extendible functions (each such function is
3.6 EQUIVALENT WALLMAN SPACES
161
locally constant on S-T). Another situation which is subsumed by Theorem
3.5-2 is the case of zero-dimensional Hausdorff compactifications S of a
locally compact zero-dimensional space T. The question of whether a zero-
dimensional Hausdorff compactification of a not necessarily locally compact
O-dimensional Hausdorff space is of Wallman type has been answered in the
affirmative by Alo and Shapiro (Exercise 3.7(d)).
3.6 EQUIVALENT WALLMAN SPACES In this section the notion of equivalent
Wallman spaces is defined and then a sufficient condition that two lattices
of subsets of T produce equivalent Waliman spaces is given in Theorem
3.6-1. Theorem 3.6-1 is used in proving Theorems 5.A-1 and 2, concerning
characterization of the space of maximal Ideals of a topological algebra
as a Wallman compactification of the space of closed maximum ideals.
Definition 3.6-1 EQUIVALENT WALLMAN SPACES Let £ and H be ^-lattices of
subsets of a set T, let cp and ф be the canonical maps of T into w(T,£) and
и(Т,Я ) :
ф:Т - w(T,H)	, c₽:T - w(T,£)
t -	= {Ae^|t€A}, t -	= {Ae£|tfA}
w(T,£) and w(T,X ) are equivalent if there exists a homeomorphism a
w(T,X )
such that а-ср=ф. We also say that the lattices £ and X are equiva lent 
£ and X are required to be of-lattices in the above definition, so that
the mapping cp and ф can be defined ((3.3-1) (a)) . It is not required that
cp and ф be 1-1, i.e. that £ and H be (3-lattices (cf. (3.3-l)(b)). We also
note that T needn't be a topological space in Definition 3.6-1, so w(T,£)
and w(T,H ) certainly needn't be compactifications of T. Should it be the
case that T is a topological space, w(T,£) and w(T,X ) are compactifica-
tions of T, cp and ф are homeomorphisms of T into w(T,£) and w(T,/{ ) respec-
tively, as in (3.3-2), and Definition 3.6-1 reduces to the fact that w(T,£)
and w(T,X ) are equivalent compactifications of T.
Below we state and prove a pair of necessary and sufficient conditions
for the Wallman spaces w(T,£) and w(T,?() to be equivalent under the as-
sumption that £C?((cf. Theorem 1.3-2).
Theorem 3.6-1 EQUIVALENT WALLMAN SPACES Let £ and 7/ be а-lattices of
162
3. WALLMAN COMPACTIFICATIONS
If £CH , then the
equivalent:
If F, Hf)(, then F(1H = 0 iff c 1 rp(F) П c 1 tp(H) = 0 where cl, de-
XX	x
in w ( T, £) .
(2)
(3)
Proof
subsets of T with tp and i|r as in Definition 3.6-1.
following are
(1)
notes closure
If, F, НеЯ> then cl tp(FC)H) = c 1 rp(F)П c 1 tp(H) .
£ £ £
w(T,£) and w (T,H ) are equivalent.
(1) -• (2): We consider two cases.
Case (a) : Suppose Ff£ and HC X •
Suppose J (!cl (.tpfFn H) . By (3.2-l)(l), [Л=Л т С
•L t	К. t	К.
cl^(FOH) is an open subset of the compact space w(T,£)
J- . Thus CUfl	® K)=0 so there exist K^, . . .	such that
СиЛ(Г)?8 )=0. Since ®n R =Qi(BK , by (3.2-l)(2), and К=Лдс it
follows that ®KC'U. Thus Л c l^tp(F(l H)=0 and therefore Kf)Ff)H=0.
Hence, by (3.2-1) (2), (3.3-3) (a), and (1),	®F Л cl tp(H) =
йкПгП cli j,tp(H)=c l^_cp(K Л F) Л c l^tp(H)=0. Since К€ч/, then .$ € as
Del ф(Н)=0, 4? t <8 0 Cl Ф(Н)=С1 <P(F)f|cl tp(H). Thus
К. г X	г X	X	X
cl^cp(F)H cljP(H)C2 cL^FHh) and the result for case (a) is proved.
Case (b) : Suppose F, H?X> and again choose sJ2 ^c l^(FC) H) . Since^- =
C|	once again there exists KcJ2 such that ^KCiw(T,£) - c1^cP(fC|h) =
U and therefore КЛ FCIH=(KH F)fl (кЛн)=0.
By (1), cl^tp(KflF)CI cl^a(Kf|H)=0. Since Kf£, from part (a) it follows
that с1^ф(К)/1 cl^tp(F)(3 cl^cp(K)C| cl^cp(H)=0 , or more simply,
cl „Ф(К)Л cl „cp(F) Л cl cp(H)=0. As cl cp(K)=®t, and KGjC , e (Й, and there-
X	X	X	X •k-	&
fore once again j- £cl cp(F)Cl cl tp(H) . Thus cl tp(F)Ocl cp(H)Ccl CP(FCIH).
xx	xxx
(2) -* (3): We must construct a homeomorphism a between w(T,£) and w(T,Z{)
such that сг*С|>=ф. Let fw(T, £) and define
Certainly с1^,ср(ГЛн)С cl^tp(F)C) cl^p(H).
Фи. Now U=w(T,£) -
containing the point
a(^) = [веЯ |?ес1£Ф(в)}
Note that^Ca(^) for if	«'clj,co(F)= We prove:
(a) ct(^)€w(T,/{ ) and therefore a(w(T, X))C w(T,/{.) •
By (2) it follows that a(^) is a filter. Suppose Af/( and A^ctG^2). Then
0cl^co(A). As cl^co(A) is the intersection of basic closed subsets
there exists K€ £ such that cl^!p(A)Cf5^ and { <J^. Thus KV </- and, since
is an £-ultraf liter, there exists Fc^ such that FC.CK. Thus since
cp(A)C.® then AC.K and therefore FdA=0. Since Ff^-CIa(^-) we have found
К
a set in the filter a(^-) disjoint from A and therefore is an ultra-
filter (Theorem 3.1-1).
3.6 EQUIVALENT WALLMAN SPACES
163
f1(1f2=0.
variance
Next we claim that a is onto. Suppose that ^7 €w(T,X )• Then as the
sets in satisfy the finite intersection property, the family of closed sets
[c1£c₽(d) pe3} satisfies the finite intersection property and since w(T,X)
is a compact space, о	Let &	we contend that
c1£C?^D^ = '•'J’-’ and CT(’?)=	• Suppose л/р ^2e^De>jClj. cp(D) and
Then by Theorem 3.3-1 there exist	and F2e'^2 suc'1 b*1311
We shall now show that both Fp	~ a conclusion which is at
with the fact that is a filter. To prove this, note that
for all Df and <3^, =cl^cp(F^). If for some De 4/, F^DD=0,
, this is a con-
Thus FjflD^0 for all DC-Jf and, since F^fX^H and 0 is an
F e^f . Similarly F.j 4/ and П cl r;>(D) = f<f]. As
1	ъ X
for all D€</ i it follows by the definition of ст(х/ ) that
Thus 4/	) by the maximality of 4-f . Our next contention
</ i ч x и xwi a x x	—t , anu t/ i ' '•««A-. x wkl - / » x x x»_»u о viu
1 £ 1 £ 1
then, by (2), cl tp(F.)Ocl D=0 and, as ec 1 <p(F.) Л c 1 D,
i- 1	£	1	£1	£ ,
tradiction.
ultrafilter,
3 ecl <p(d)
^Ca(/).
is that :
(c)	d is 1-1.
Recall the definition of a(^) and the observation that	if
3 ,^<?9, there exists F.e^., F„e such that F.riF9=0. But then if
XX,	X X X. X.	X x.
ст( лУ^)=ст(^2) > xt p 2^-CT(^p so that the absurd conclusion that 0=
Fl^ I?2CCT<''^’p follows.
We prove that:
(d)	ст-ф = ф.
Let teT and ={Ae£ltfA}. Suppose Bea(^/ ). Then «У ecl <p(B). If t^B
t	t	t £
then, as /( is an а-lattice, there exists Ct/-fsuch that tfC and С(ЛВ=0.
But t€C so that л? tec l^cp(C) . Hence x/fc f| cl^cp(B) . By (2),
</'(_Сс1^с:(С)/| clj,tp(B) = clj,cp(BG C)=0 and we have a contradiction. Thus tfB
for each Вест(^/^_) and therefore a(x/t)C2 ^={Ве>( [tfB]. But both ct(^ ) and
1 are ultrafilters; hence ст(.У^) = Ц and а'ср=ф. To finish the proof
that (2) -> (3) it only remains to show that:
(e)	a is bicontinuous.
To see that a is continuous, let Be^f and consider the basic closed set
= $ew(T, X) |Be4/} of w(t, X).
Then
°-1(<2B) = ^^fJewd.X ) |Bf^} = {7е«(т,£)
= С?|веа(,?)} = {/ l^ec 1£ф(в)} = cl^cp(B)
and a 1(61 ) is a closed set. Therefore о is continuous.
D
164
3. WALLMAN COMPACTIFICATIONS
To show that a is continuous, let AfrcX • We recall that /С ct(/)
for each fw(T, £) . Thus а(<Зд)=ст([У’|Ае<^ }С.б2д . However, if ^Уе£2д then
«7 =a(-3^) for some €w(T,£) and, as CL ct(.jc) =	, A meets each element
of . Thus A must actually belong to the ultrafilter . So that <F
"1
and 64 Сст((Я). Thus 0. =a((8.) and ст is continuous.
A A	A A
(3)	— (1) ; Let F, be such that FflH=0. If tfF then CT(^ )= kJ
and, as (Cl„, -f 1(64„). Consequently (p(F)C'CT	As a is a
t Г t	Г	г
homeomorphism, а	is closed in w(T,£) and cl tp(F)C ст (£Z. ). Simi-
_ F	X	F
larly clj,tD(H)Ca (6tH). As FDH=0, ДрПб1н=0 and, therefore,
cl£cp(F) Л cl^cp(H)=0. V
EXERCISES 3
165
Exercises 3
3.1 SEPARATION IN LATTICES Let L be a distributive lattice with 0 and 1.
L is a dis jointed lattice if a, bsL, a/0, b^O, then there exists ceL such
that ac^O and bc=O or bc^O and ac=O.
(a)	If L is a disjointed lattice then for all Cw L (Sec. 3.2)
iff a=l.
(b)	The lattice of all closed subsets of a topological Tq space T is a
disjointed lattice iff T is a T^ space.
(c)	If T is a topological space and L the lattice of all closed sub-
sets of T, then if L is an а-lattice, L is a disjointed lattice. Conversely
if L is a disjointed ^-lattice, then L is an Of-lattice.
3.2 BOOLEAN LATTICES; THE STONE REPRESENTATION THEOREM In this exercise
some results are presented about Boolean lattices (complemented distribu-
tive lattices with 0 and 1) including the Stone representation theorem (f)
which can be generalized to distributive lattices (Stone, 1936), as can a
number of the parts of this exercise.
All lattices are assumed to contain 0 and 1; a' denotes 1-a. A non-
zero element a of a lattice L is an atom if when bfL and b < a then b-0 or
b-a .
(a)	If L is a finite lattice then for every cfL there exists an atom
afL such that a < c.
(b)	If L is a lattice and a an atom in L, then for every bfL, ab=G or
ab=a .
(c)	Let afL where L is a lattice and let R(a) be the set of all atoms
that are < a. Then R(ab)=R(a) Q R(b) , R(a ' ) = R( 1)-R(a) . If L is finite,
then R(a)=R(b) if and only if a=b. Let a ,...,be atoms in a Boo
algebra L. Then R(max^a^)= [a . . . .a^}
Boolean algebras L^ and L2 are said to be isomorphic if there ex-
ists a 1-1 correspondence f between them such that f(a')=f(a)' for all afL^
and t^a^A a2)=f(a1)Л f(a2) for all a^, a2fL .
(d)	If L is a finite Boolean lattice, then L is isomorphic to the
Boolean lattice of all subsets of the set of atoms of L.
Hint: Let X=fa,,...,a 1 be the atoms of L and consider
----	1 nJ
f:L----w-P(X)
a----► R(a)
166
3. WALLMAN COMPACTIFICATIONS
(e)	Two finite Boolean lattices with the same number of elements are
isomorphic.
Statement (d) is true for infinite Boolean lattices as well. However,
in proving this, use is made of Zorn's lemma. As is often the case, the
lemma is used in showing that every nonzero afL can be embedded in an
ultra fiIter.
(f)	Every Boolean lattice L is isomorphic to a Boolean lattice of sub-
sets of a set X.
Hint; Let X be the set of ultrafilters of L. Let
T:L —*-P(X)
a—>(JfX |af^} for a t 0
and T(O)=0.
A Boolean lattice is called complete if for every WGLL, sup W exists.
L is called atomic if every element in L is > some atom.
(g)	A Boolean lattice L is isomorphic to P(X) under the mapping T of
(f) if and only if L is complete and atomic.
3.3 ZERO-DIMENSIONALITY OF w(T,6») AND ULTRANORMALITY In this exercise T
is a Tj space. It is shown in (a) that w(T,C*) is zero-dimensional iff T
is ultranormal.
(a)	Let "(j be the lattice of all closed subsets of T. Show that w(T,{D
is zero-dimensional iff T is ultranormal.
Hint: If w(T,^) is ultraregular then, being compact, w(T,^») is ultra-
normal. Hence if <8 and <8 are disjoint basic closed sets in w(T,C>),
F	К
there exists (see Example 3.2-1) a clopen set L? V y (where T-U^f £ for
iEI) such that ® C(J V. and	(J Vv. =®- 1Thus with tp as in Sec.
F ifl ui	K ifl ui
3.3,	F=®^Ht(T)CU U., K(1Uu.= ®„Аср(Т)П UVtt< and UU. is
F	tn 1 ia 1 K	ifl ui	1
clopen.
Conversely, if T is ultranormal, the clopen subsets of T separate the
closed sets. Letting & denote the clopen subsets of T, by Example 3.2-l(b)
{Уц|и€®} is a base for the topology of w(T,l£?). As each Uf(2> is clopen,
it follows that for eachUfG? is clopen.
(b)	A normal T^ space T is ultranormal iff gT is ultraregular and in
this ease g'f=w(T,xJ?).
Hint: If gT is ultraregular then show that gT is ultranormal. Utilizing
normality of T, show that the disjoint closed sets in T have disjoint clo-
sures in gT.
EXERCISES 3
167
Conversely, suppose that T is ultranormal and sfgT. Let sfU where U is
open in pT and take x6fC(BT,R) such that z(x^)GU where z(x^) is a neigh-
borhood of s. As T is ultranormal, let V be clopen in T such that z(x) =
z(x\ifl TCVCUflT. Observing that V and CV are zero sets in T show that
cl V and cl CV are disjoint clopen sets in RT with sfcl VCU.
P Ё	₽
(c)	Let T be an ultraregular space which is not ultranormal. Show
that (У |UfG^ is not a base for the topology of w(T,<£).
Hint: The sets Vjj are clopen in w(T,£) and w(T,^>) is not ultraregular.
(d)	AN ULTRAREGULAR SPACE WHICH IS NOT ULTRANORMAL Let Q be the first
uncountable ordinal and ш the first infinite cardinal. Let S=[0 ,ш]х[О ,Q]
and T=S-{ (ш,(1)}. Show that S and T (carrying the product of the order
topology on the ordinal numbers) are both ultraregular but that T is not
ultranormal.
Hint: T is not normal as the closed sets К={(а,ш) |0 < a < Q] and F=
{(n,Q) |0 < n < w] cannot be separated by disjoint open sets (Dugundjii
1966, p. 145).
Given a T^ space T and a lattice £ of closed subsets of T, it was found
that under certain conditions the collection w(T,£) of all ultrafilters
can be topologized to obtain a T^—compactification of T ((3.3-2)). In
Exercises 3.4 - 3.6 we consider a collection J(T,£) of prime filters con-
taining w(T,£), topologized in such a way as to make J(T,£) a ^-compactif-
ication of T.
3.4 VERY DENSE SPACES Let Y be a topological space.
A subspace X is said
to be very dense in Y if, when F is any closed set in Y
then cly(Ff)X)=F,
i.e. X is "dense" in any closed subset.
(a)	X is very dense in Y iff for any
(b)	If X is very dense in Y then for
there exists a unique open set G'CLY such
yfY, уfcly (cly [y}QX) .
each relatively open subset G of X
tha t G 'f| X=G. The set G is com-
pact iff G' is compact.
Hint: Show that a closed subset F of X meets G iff clyF meets G'.
(c)	If X is very dense in Y and (B ) is a base for the closed sets of
a
X, then (cl В ) is a base for the closed sets of Y.
Y a
(d)	If X is very dense in Y, then if (F ) is a family of closed sub-
01
sets of X, then cl,(f| F )= (Л cl F .
y a a a Y a
(e)	If X is very dense in Y, then X is normal iff Y is normal. X is
a T -space iff Y is a T -space,
о	о
3.5 VERY DENSE SPACES AND IRREDUCIBLE CLOSED SETS A set F in a topologi-
cal space T is irreducible if when F=F^U F2 with F^ and F2 closed sets,
168
3. WALLMAN COMPACTIFICATIONS
then F=F^ or F=F£.
(a)	Let X be very dense in Y. Then a closed subset F of X is irre-
ducible in X iff clyF is irreducible in Y. If y€Y, then cly{y] is irre-
ducible in Y and cly{y}('lX=Fy is irreducible in X.
(b)	Let X be very dense in a TQ-space Y and Irr X denote the collec-
tion of all irreducible closed subsets of X. Then
I:Y--->-Irr X
У----#-с1у{у}ПХ
is a 1-1 mapping.
If the mapping I is onto, then Y is referred to as a Jacobson comple-
tion of X. If X=Y, X is said to be Jacobson complete. In the event that
X (and therefore Y) is compact, with the compact open subsets of X forming
a base for the topology closed under the formation of finite intersections,
then Y is called a spectra 1 completion of X and if X=Y, then X is called
spectrally complete.
If F is an irreducible closed subset of a space X, then xfX is a ge-
neric point of F if cl [x}=F. If X is very dense in Y, then we say that
X
y€Y is a generic point of F in Y if cly{y}flX=F.
(c)	An ultraregular space is spectrally complete. Part (d) is in-
cluded to make the more general case of (e) clearer.
(d)	Let F be a closed subset of a T^ space X with no generic point.
Then a T space Y=XU[y] can be formed in which X is very dense and у is a
generic point of F in Y.
Hint: Let & ={G |G is an open neighborhood of some x<f}. As F is irre-
ducible, $ is a filter among the open sets of X. For y^X, define neighbor-
hoods in Y=X(J [y] as follows.
(1)	If yfNQY and NOG for some G€<8, N is a neighborhood of y.
(2)	If xCNCZX and xCGCLN for some G€<S, then Nu(y] is a neighbor-
hood of x.
(3)	If N is a neighborhood of xfX and the condition of (2) does
not hold, then N is a neighborhood of x.
(e)	If X is a Tj—space, a Jacobson completion of X exists.
Hint: Let W be a set whose cardinality is strictly bigger than the cardi-
nality of Irr X. Let Z=X(JW and(Z={sCZ|S is a TQ-space with respect to
some topology^T and X is very dense in S}. Order the objects (S ) with
iff SjCS2 and	By Zorn's lemma there ex-
ists a maximal element (S^</ ). Show that is a Jacobson completion of X
EXERCISES 3
169
according to the following scheme: If PCX and F is irreducible with no
generic point in SM, let yfZ-SM and define a topology on S^[y} as follows.
If (Sis the filter among the open subsets of X associated with F as in the
hint to (d), let£8 be the open subsets of S^ associated with the open sub-
sets in by Exercise 3.4(b). Letting N denote a subset of S ,
(1)	If there exists G'e® such that G tZ N, then NU[y] is a
neighborhood of y;
(2)	If sCS^ and there exists GCN such that SCGCZ.N, then NL)[y]
is a neighborhood of s;
(3)	If sCS^ and N is a neighborhood of s which fails to satisfy
the condition of (2), then N is a neighborhood of s.
Show
that S U fy) violates the maximality of S, .
M	M
(f)	Let X be a compact T^-space whose compact open
sets form a base
for the topology which is closed under the formation of finite intersec-
tions. Then X is spectrally complete iff for any filterbase® of compact
open subsets of X, Г) <0 /0.
Hint: If there exists such that f)0=0 , then assume that $3 is an
ultrafilter in the class of compact open sets of X. A compact open set
belongs to S iff it meets every set in (0 . Define neighborhoods in Y=
Xij[y} for y/X as follows:
(1)	If NCX and there exists В C® with BCN, then N(j{y} is a
neighborhood of y.
(2)	If xfX and NCZX is a neighborhood of x in X with xfBCIN for
some B€(&, then N(j{y] is a neighborhood of x.
(3)	If xfX and NCTX is a neighborhood of x and the condition of
(2) is not satisfied, then N is a neighborhood of x in Y.
Y is a TQ~space and X is very dense in Y. Note
neighborhoods of x are of type (2)}. Show that
3.6 JACOBSON FILTERS IN A LATTICE AND JACOBSON
that F^=cl^[y}nx={xeX | all
yeclYFy-
COMPLETIONS In this exer-
cise L is a distributive lattice with 0 and 1. A prime filter in L is a
filter such that if a, bfL, and a+bfP , then a or bfP . A Jacobson filter
is a prime filter which is equal to the intersection of all the ultra-
filters containing it. The set of all ultrafilters, Jacobson filters, and
prime filters will be denoted by W(L), J(L), and P(L) respectively, If L
is a lattice of subsets of a set T, these sets will be denoted by W(T,L),
J(T,L), and P(T,L) respectively. We topologize P(L) by taking as a base
for the closed sets the sets
170
3. WALLMAN COMPACTIFICATIONS
= (Ap(L) Jac/’}
with afL.
(a)	clp(L)[P}=[p'eP(L) |ЛС.'р'} and P(L) is a TQ-space.
(b)	Show that C^af)W(L) = y:eW(L) | there exists with ab=0) but
that in P(L) this relationship does not hold.
(c)	Show that if W(L)CS, then w(L) is very dense in S iff SCJ(L).
(d)	Show that for every S such that W(L) CSC1 J(L), S is a normal
space iff L is a normal lattice.
For the remaining parts of this exercise we assume that T is a T^~
space and tj» the lattice of all closed subsets of T.
(e)	If FfO, then the filter P ={Kc</’ IfCk} is a prime filter iff
FC IrrT.
(f)	If T is compact, a filters? is an ultrafilter	=
[Kc£|tcK] for some tCT.
(g)	When T is compact, a filter is a Jacobson filter iff
for some FC IrrT.
(h)	Let T be compact and very dense in the T^-space S. Then letting
Fs=clg [s } f| T, the mapping
<7:S—*-J(T,£ )
s—
F
s
is a homeomorphism such that
a(T)=w(T,£>) and a is onto J(T,£>)
iff S is
a Jacobson completion of T.
Note: It now follows that a
is unique and topologically
Jacobson completions
equivalent to J(T,t^).
of a compact Tj- space
3.7	WHEN COMPACTIFICATIONS ARE OF WALLMAN TYPE (Alo and Shapiro 1968a
and 1968b, Brooks 1967a; cf. also Sec. 3.5). In this exercise Y denotes
a Hausdorff compactification of the completely regular Hausdorff space T;
otherwise the notation of Secs. 3.2 - 3.4 is assumed to be in force. In
(b), (c), and (d), conditions are obtained which make Y a Wallman type
compactification of T.
(a)	Let I be a normal base (Definition 3.4-1 and (3.4-1)) for T.
If ^Cw(T,£) and G is open in w(T,£) with </CG, then there exists Ac£ such
that ,^:eclwtp(A)= 'ЗдСЮ and '23^ is a neighborhood of .
Hint: Since w(T,£) is a compact Hausdorff space there exists ACX such
that CGd®^ while is disjoint from some open neighborhood U .
Hence	CI G.
CA
EXERCISES 3
171
(b)	If £ is a normal base for T then Y and w(T,£) are equivalent com-
pactifications of T iff the class of sets [cl^A |Af£} contains a neighborhood
base for Y.
Hint: Let = L IJ C S> . } and В <,.?) = {c 1 CA i Ъ CV&) } . The sets of Bfc?)
L^A I	CA	i	CA
satisfy the finite intersection property and the condition of (b) yields
!ftB(?) = {y}. Define y=f(^). Clearly if^-=tp(t) for some t€T, then f(/) =
f(cp(t)) = t. Also B(/) -• y;hence if V is a neighborhood of y, then there
exists iB-.fVC/') such that cl CACv. Thus if С , f ((7) ecl CAC V,
CA	1	CA	Y
f($CA)CV and f is continuous. The fact that both spaces w(T,£) and Y are
compact leads to the completion of the proof.
(c)	(cf. Theorem 3.5-1). Y is equivalent to w(T,£) for some lattice £
of closed subsets of T iff £ satisfies the following two conditions:
(1)	с1у(АЛВ)=с1уАПс1уВ for all A, Bf£.
(2)	If V is a neighborhood of у in Y, there exists Ae£ such that
y€clyAC.V.
Hint: Note first that statement (2) is weaker than the condition of (b) in
that the sets clyA need not be neighborhoods of y. Let yfY and =
[Af£ |yec lyA]. Then by (1) and (2) is an ultrafilter. Since Y is a com-
pact Hausdorff space, as У is an ultrafilter there exists yfO cl A and
_	AG? I
У Cx?" • Hence У’ = J- and the mapping
g:Y—»-w(T,£)
У У
is 1-1 and onto. Let g(y)=-^y€^
BCCA. If уес!^, then as Bf«^,
tradiction. Hence yCCCd^) . If
&СД. Then there exists B&^ such that
у fc lyA Cl clyB=c 1^ ft B) and we have a con-
wcCCcIyA), then obviously A^^ and
and g is continuous.
(d) Show that any ultraregular compactification Y
w cA'
CA
of (the necessarily
ultraregular space) T is equivalent to w(T,£) for some lattice £ of closed
subsets of T.
Hint: Let £={GftT|G is clopen in Y}.
3.8	WALLMAN COMPACTIFICATIONS AND E-COMPACTIFICATIONS Let T be a T^space
and E be a Hausdorff space.
(a)	E-closed sets A subset A of T is E-closed if for some nfN and
xfC(T,En) , A=x \f) for some closed subset F of the product ЕП.
Finite unions and finite intersections of E-closed sets are E-closed so
that the E-closed sets form a ring of sets. T is E-completely regular
(Exercise 1.11) iff for each closed subset F of T and each t not in F there
172
3. WALLMAN COMPACTIFICATIONS
are disjoint E-closed subsets A and В of T such tfA and FC.B.
(b)	Separating families (Steiner 1968). A family-c? of closed subsets
of T is separating if for each closed subset F of T and each t not in F
there are disjoint sets A and В in^ such that tjA and FC.B. If^is a
separating family then the Wallman space w(T,^/") is a compactification of
T. Conversely, for any topological space T, if w(T,^ ) is a compactifica-
tion, then T must be T^ and the ring of sets generated by^ is separating.
(c)	E-closed sets and Wallman compactifications (Piacun and Su 1973) .
By Exercise 1.11(a), if T is E-completely regular there is a map h which
В -	t
homeomorphically embeds T in the product E for some index set B. If 4 de-
notes the family of all E-closed subsets of E^ and J1 ={ FC T |F=h \f') for
some F' (J } then w(T,^? ) is a compactification of T. (Show that^ is a
separating family and use (b).) If^? is the family of all E-closed subsets
of T then w(T,^? ) is a compactification of T iff T is E-completely regular.
If the closed interval [0,1] in its usual topology and T are each E-
completely regular and^ denotes the collection of all E-closed subsets or
T, then w(T^ ) is an E-compactification (Exercise 1.11) of T.
(d)	(Piacun and Su 1973) If the zero-dimensional and normal space T
has more than one point, each closed subset of T is E-closed, and^ denotes
the ring of all closed subsets of T, then w(T,^ ) is an E-compactification
of T.
If^ is the family of all closed subsets of the discrete space T then
w(T,^ ) is an E-compactification of T.
3.9	(Brooks 1967a) HAUSDORFF COMPACTIFICATIONS ARE "WALLMAN" QUOTIENTS
It is well known that each Hausdorff compactification S of a completely
regular Hausdorff space T is a quotient of the Stone-Cech compactification
gT (Dugundji 1966, XI, 8.2). In this exercise it is established that each
such compactification S is also realizable as a quotient of the Wallman-
type compactification W(T,2&) obtained by utilizing the lattice of zero
sets of continuous real-valued functions which are continuously extendible
to S.
(a)	W(T;Zel is a T^-compac ti f ica tion of T: )7,e is an a8-lattice,(see
Definition 3.3-1 and Example 3.3-1) of subsets of T*that serves as a base
of closed subsets of T. Thus, by (3.3-1) and (3.3-2), w(T,Ze) is a T^-
compactification of T.
Before proceeding any further we settle the question of whether S and
w(T, £ ) are equivalent compactifications (a conjecture motivated perhaps
by the fact that 3т and w(T,Z) are always equivalent ((3.4-3))).
EXERCISES 3
173
(b)	S and w(T,Z ) are not always equivalent compactifications
Hint: Let T=N with the discrete topology and S be the one-point compactifi-
cation of N. In this case Z-= P(T) so that w(T, Z^)“w(T,Z ) = 0T.
Let C (T,R) denote the continuous real-valued functions of T which are
s
continuously extendible to S and C (T,R) those continuously extended to
w(T,Ze).
(с)	C..(T,R)<Z:C (T,R) Each continuous real-valued function of T which
is continuously extendible to S is Z^-uniformly continuous (Definition
3.4-2) and is therefore continuously extendible to w(T,Ze) by Theorem
3.4-1. (Note that this theorem is valid even if the requirement that the
lattice be normal is dropped).
Hint: Let xfCs(T,£) and choose a, bcR such that x(T)C(a,b). Given ( > 0
choose a partition a=t < t, < ... < t =b of Га.bl such that t.-t. , < r/4
for i=l,...,n. For each 1 < i < n-2 set yi=(ti+2 " mi-n(ti+2,x)) (max(t.,x)
-t.). Clearly each y.€C (T,R) and z(y.)=x \-«5,t.]Ux ^[t	,“>)=T-x \t.,
1	S	1-1-	l_"i £.	1.
t. n). Hence Q.z(y.)=0 or, equivalently, T=U-Cz(y.)* Furthermore,
14-2	i i	i i
since Cz(yp=x	°(x > Cz (xp )=sup{ f x (t)-x (s)f | s , tf Cz (yp } < f and
x is 2 -uniformly continuous.
Thus each x€Cs(T,Il) can be extended uniquely to a continuous function
xe: w(T,Ze) -• R such that xe(^t)=x(t) for all tfT. Then /?={(^,^7)e
w(T, Z e)x w(T,^e) |xe (^ )=xe (47 ) for all xfCs(T,R)} is an equiva-
lence relation and we consider w(T,Ze)/7? with the quotient topology.
(d)	w(T,Z^)/R and S are equivalent compactifications of T
Hint: Certainly the canonical map K: w(T,Z-e) _ w(T, 7 )IR is continuous
so w(T,Z )/R is compact. As each xe is constant on the elements of ,
x (K^/-)=x (# ) is a well-defined continuous real-valued function on
w(T,2Te)//^ (see Dugundji, 1966, pp. 123-124). The family U of all such
functions x' is a uniformly closed subalgebra of C (w (T, Z e)/7^ , R, c) which
contains the constants and separates the points of w(T,Z)/R . Thus
w(T,Z )/Д is Hausdorff and by the Stone-Weierstrass Theorem (Dunford and
Schwartz 1958, Vol. 1, IV 6.16, p. 272), IX =C(w(T,Ze) //? ,R). Consider the
ma pping
f: C(S,R) - С (Т,£) - C(w(T,Ze)/R ,R>)
У - У |т - (-v lT)'
is a surjective algebraic isomorphism. Thus it follows that there exists
a homeomorphism h taking w(T,_Z )/R. onto S such that (fy) (K^)=y(h(K-J^))
for each y€C(S,R) and Kj fw(T,Ze)/i( (see (1.6-3)). Finally, since
174
3. WALIMAN COMPACTIFICATIONS
K^'t={^t] for each tfT
y(t) = ye(#t) = (y jT) ’ (K^t) = (fy) (K^t) = у(ЬЦЗ)
for each yeC(S,R) and t€T. Thus с=Ь{^} for each CfT and w(T,Z.e)/and
S are equivalent compactifications.
FOUR
Cmmutative Topological Algebras
The Theory of Banach algebras began in 1939 with the pub-
lication of Gelfand's striking paper "On normed rings" follow-
ed by his "Normierte Ringe" in 1941. From then on it gained
momentum and tho its development abated somewhat in the mid-
fifties, it remains an active area. Considering the develop-
ment of the theories of topological groups and topological
vector spaces throughout the forties, the investigation of top-
ological algebras was ineluctable. For the study to begin in
earnest it was desirable that the theory of Banach algebras
mature further so that its most prominent distinguishing fea-
tures showed more clearly. But still another factor had to go
into the mix: there had to be some reason to examine topolo-
gical algebras as algebras and not merely topological rings,
that is, some way to get the scalars to play a more significant
role.
During the forties the theories of topological algebras
and topological rings began to undergo simultaneous develop-
ment. Topological rings were extensively treated in Kaplansky
1947b and 1948 and topological algebras received the feature to
distinguish them from topological rings in Arens lr946b with the
introduction of local m-convexity, a subset U of a topological
algebra being called multiplicatively convex, or m-convex for
short, if it was convex and UU was again a subset of U. A
topological algebra is locally m-convex if it possesses a
base at 0 of balanced m-convex sets. The next major devel-
175
176
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
opment in the theory of topological algebras, or more accurate-
ly locally m-convex algebras, came with the publication of
Michael's 1952 Memoir on the subject.
We begin the study of commutative topological algebras in
this chapter. The algebras C(T,F) of continuous functions
of Chapter 1 are endowed with the compact-open topology and
drawn upon to motivate certain results as well as to serve as
examples and counterexamples. Banach algebras too of course
serve in a similar way. And altho we consider it unlikely that
a reader of this book would be completely unfamiliar with
Banach algebras, no formal knowledge of them to speak of is re-
quired. In any event the standard references of Rickart 1960
and Naimark 1960 contain whatever background material one might
want for the sake of comparison of Banach algebras with topo-
logical algebras. Chapters 5 and 6 continue what has been be-
gun here. In Chapter 5 the primary thrust is devoted to the
space of maximal ideals of a locally m-convex algebra, espec-
ially in regard to its relationship to the space of closed
maximal-ideals as a compactification of it. In Chapter 6 a
special type of topological algebra—essentially inductive lim-
its of Banach algebras—is investigated.
ALL OUR ALGEBRAS ARE ASSUMED TO BE COMMUTATIVE AND TO
POSSESS AN IDENTITY. THE UNDERLYING FIELD IS R OR C, DE-
NOTED GENERICALLY BY F. For some results the underlying field
must be C. When this is necessary the more special hypothesis
is explicitly indicated.
4.1 Topological Algebras
Definition 4.1-1 BASIC NOTIONS A linear space X equipped
4. TOPOLOGICAL ALGEBRAS
177
with an additional binary operation, called vector multiplica-
tion and denoted by xy for x,y € X, is called an algebra
if X is a ring with respect to vector addition and vector
multiplication, and a(xy) = (ax)у = x(ay) for all scalars a
and all x,y € X. A linear map A:X->Y, where X and Y are
algebras is called an (algebra) homomorphism if A(xy) =
(Ax)(Ay) for all x,y € X. If A is also 1-1 it is referred
to as an (algebra) isomorphism or an embedding. The null space
of an algebra homomorphism A, i.e. A 1(0), is renamed the
kernel of A and is denoted by ker A. "Homomorphism of X"
unmodified signifies a homomorphism whose range is in X's un-
derlying field which (as is mentioned below) is always F = R or
C; such homomorphisms are also called real or complex homomor-
phisms, as the case may be.
A topological algebra is an algebra endowed with a non-
trivial topology xZ which is compatible with X's linear
structure and such that the map X*X->X ,	(x,y) +xy is con-
tinuous. The algebra X and the topology xZ are compatible
or xT is compatible with the algebraic structure of X when
(X,<7) is a topological algebra. Topological isomorphism in
the context of topological algebras means an algebra isomor-
phism which is also a homeomorphism.
For the sake of simplicity, all algebras considered are
assumed to be commutative, and to possess an identity. But how
much is lost by specializing to commutative algebras with iden-
tity? As regards the presence of an identity: not much; as
regards commutativity: a good deal. Little is lost through
the identity requirement because any algebra X may be embedd-
178
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
ed (via a topological isomorphism) in an algebra which has
an identity* so that X may be viewed as a subalgebra of an
algebra with identity. Results proved for X^ are then "re-
stricted" to X. Commutativity, however, is a different mat-
ter: there is just no way to identify a noncommutative alge-
bra with a subalgebra of a commutative algebra and generally
the theory of commutative algebras is quite different from that
of noncommutative algebras.
An immediate consequence of the definition of a topologi-
cal algebra is that the map x^wx is continuous for any
w с X; it is a homeomorphism if w is a unit. In fact con-
tinuity of the map (x,y)->xy is equivalent (using the fact
that X is a TVS) to the continuity of each of the maps x+wx
and continuity of (x,y)+xy at 0. This is easily verified
using the identity xy-xQyo = (x-x )(y-yQ) + (x-xQ)yo + xQ(y-yol
Theorem 4.1-1 BASES FOR COMPATIBLE TOPOLOGIES The filterbase
® in the algebra X determines a base at 0 for a compatible
topology for X iff
(a)	® is a neighborhood base at 0 for a topology which
is compatible with X's linear structure
(b)	For each V € <Q there exists a В g (Q such that
BB с V.
Proof. We prove only the sufficiency of the conditions. Let
V(0) denote the filter of neighborhoods of 0 determined by ®.
* To embed a topological algebra X without identity in a
topological algebra X^ with identity, consider F * X with
the product topology and pointwise operations. The element
(1,0) then serves as an identity for Xj.
4.1 TOPOLOGICAL ALGEBKAS
179
To prove that (x,y)+xy is continuous, consider a neighborhood
of xy,xy + U where U € V(0), and choose a balanced neigh-
borhood V of 0 such that V+V+Vc U. By (b) there exists a
В € & such that BBCV. Now choose a real number a, 0 < a < 1,
such that ax € В and ay € B. Since
2
(x+aB)(y+aB) = xy + axB + ayB + a BB
2
c xy + BB + BB + a V
cxy + V + V + V
c xy + U ,
it follows that X is a topological algebra.
Corollary 4.1-1 SUBBASES If is a collection of subsets of
the algebra X which is a neighborhood subbase at 0 for a
topology which is compatible with X's linear structure and
such that for each S € there exists a V g such that
WcS, then the filterbase generated by is a base at 0 for
a compatible topology for the algebra X.
The prototype topological algebra is the normed algebra:
An algebra X is a normed algebra if it is a normed space and
ПХУ II < II xll II У|| • A normed algebra which is also a Banach space
is a Banach algebra. The space of n-tuples Fn with sup norm
and pointwise operations is a Banach algebra, as is the space
C(T,£,c) of Example 0.1-2 of scalar-valued functions of the
topological space T with pointwise oeprations when T is
compact (i.e. (xy)(t) = x(t)y(t) and sup norm).
If T is not necessarily compact, the compact-open and
point-open topologies (Example 0.1-2) are each compatible with
the algebraic structure but we defer proving this until Section
4.3. Other examples which include such things as products and
180
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
quotients are given throughout the chapter.
As an immediate application of the basis theorem we have:
(4.1-1) INITIAL TOPOLOGIES Suppose that X is an algebra, Y
is a topological algebra with neighborhood filter at 0 denot-
ed by V (0) , and A:X-*Y a homomorphism. It is easy to verify
that the filter A 1(V(0)) determines a topology which is com-
patible with X's linear structure. To see that it is compat-
ible with the algebraic structure as well, we first note that
for any V € V(0), there is a Be V(0) such that BBc V.
Hence А "’"(В) А ‘’’(В) cA "’ (BB) cA (V) . The topology deter-
mined by A 1(V(0)) is called the initial (inverse image,weak)
topology induced by the homomorphism A. One of the conse-
quences of this result is that any subalagebra M of a topo-
logical algebra X is a topological algebra in its relative
topology as its relative topology is that induced by the homo-
morphism x+x of M into X.
(4.1-2) FINAL TOPOLOGIES Suppose X is a topological alge-
bra with filter of neighborhoods of 0 denoted by V(Q), Y an
algebra, and A:X+Y a homomorphism. It is easy to see that
the collection ф of subsets U of Y such that A "’"(U) € V(0)
forms a base at 0 for a topology compatible with X's linear
structure. For any U	we may select В g V(0) such that
BBcA-1(U). Thus A(B)A(B) =A(BB) cA(A-1(U)) c U. Since
A-1(A(B)) о В €V(0), it follows that A-1(A(B)) €V(0), i.e.
that A(B) e fi, so that <8 is a base at 0 for a topology
which is compatible with Y's algebraic structure. The topo-
logy generated by (£> is called the final topology for Y de-
termined by the homomorphism A.
4.2 MULTIPLICATIVE SETS AND MULTIPLICATIVE SEMINORMS
181
4.2 Multiplicative sets and miltiplicative seminorms
For most of our subsequent results, we shall specialize
our investigation to a particular type of topological algebra--
"locally m-convex algebras." We prepare the ground for that
discussion now.
Definition 4.2-1 MULTIPLICATIVE CONVEXITY. A subset U of an
2
algebra X is called multiplicative (idempotent) if U = UUc U.
It is called multiplicatively-convex or m-convex if it is con-
vex and multiplicative, absolutely m-convex if it is balanced
and m-convex.
An immediate example of multiplicative sets is afforded by
the spheres-open or closed-of radius 1/n, n € N. about 0 in
any normed algebra. As is apparent, each such sphere is ab-
solutely m-convex as well.
Preservation of multiplicativity is the subject of our
next result.
(4.2-1) PROPERTIES OF MULTIPLICATIVE SETS. Let X be an al-
gebra. If Uc X is multiplicative, then so is
(a)	its convex hull U ;
c
(b)	AU if U is balanced and |A| < 1;
(c)	its balanced hull U, ;
b
(d)	its balanced convex hull U, ;
be
(e) any direct or inverse homomorphic image;
(f)-------its closure cl(U) if X is a topological algebra.
Proof, (a) Consider any two elements of U : x = la x and
----------	c	n n
у = Eb у (a ,b >0, la = Zb =1) and their product xy =
J nrm n m— n m	J
£ a b x у . Since, for all n and m, a b >0,
n,mnmnm	n m —
2
£ a b =1, and x у € U, it follows that (U ) c U .
n,m n m	n m	с	c
182
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
(b)	Clear.
(c)	If Лх, цу € 4^ = C^OJU, then Ах-цу € C1(0)U.
(d)	This follows from (a) and (c) , since U, = (U, ) .
be b c
(e)	For any homomorphism A, A 1(U)A 1(U) cA 1(UU) c
A-1(U) and A(U)A(U) = A(UU) cA(U).
(f)	Similar to what was done in (3.1-11), it follows that
2
cl(U)cl(U) ccl(U ) ccl(U) by continuity of multiplication.?
By well-known results on convexity parts (e) and (f) re-
main true if you substitute "m-convex" for "multiplicative."
In making the transition from normed spaces to locally
convex spaces, one goes from topologies determined* by a norm
to topologies determined by families of seminorms. The analog
for normed algebras is from a topology determined by a single
norm to that determined by a family of multiplicative semi-
norms .
Definition 4.2-2 MULTIPLICATIVE SEMINORMS. A seminorm
p on an algebra X is multiplicative if p(xy) < p(x)p(y) for
all x,y € X.
We note that for a multiplicative seminorm p to be non-
trivial on an algebra X, it is necessary and sufficient that
p(e) be non-zero. The trivial seminorm (i.e. identically
zero) is multiplicative and generates the trivial topology.
(4.2-2) MAXIMA OF MULTIPLICATIVE SEMINORMS. If p. ,...,p
are multiplicative then max ajPj is a multiplicative semi-
norm for any collection of non-negative numbers a^,...,an>
*"Determined" in the sense that the "norm topology" is the
weakest topology with respect to which the norm is a continu-
ous map; a similar meaning is attached to "determined by a
family of seminorms."
4.2 MULTIPLICATIVE SETS AND MULTIPLICATIVE SEMINORMS
183
Proof. Since max.a.p. is clearly a seminorm we show that
-----	3 J J
max ajPj is multiplicative. To do this it suffices to consi-
der two multiplicative seminorms q and r, and show that p =
max(q,r) is multiplicative. To this end consider p(xy) =
max(q(xy), r(xy)). Without loss of generality we may suppose
that p(xy) = q(xy). Thus p(xy) = q(xy) < q(x)q(y) <
max(q(x), r(x)) max(q(y), r(y)) =p(x)p(y).V
It is straightforward to verify that a base at 0 for the
initial topology generated by a family P of seminorms is given
by positive multiples of finite intersections of sets of the
form Vp = {x|p(x) < 1}, peP. To enable us to be rid of fin-
ite intersections and be able to use simply positive multiples
tive seminorms is introduced.
Definition 4.2-3 SATURATED FAMILIES. A family P of seminorms
(multiplicative or not) is saturated if max^p^eP for any
finite subset {p.,...,p } of P.
1	n
It follows immediately from (4.2-2) that there is never
any loss in generality in assuming a family of seminorms to be
saturated. As a particular example of a saturated family of
multiplicative seminorms, consider the algebra C(R,R) of con-
tinuous real-valued functions endowed with the topology genera-
ted by the multiplicative seminorms pn,neN, defined at each
xeC (R, R) by: pn(x) = su₽te [-n n] । • These seminorms gen-
erated the compact-open topology for C(R,R) (cf. Example
4.3-1).
If X and Y are algebras and A:X+Y is a homomorphism
then p-A determines a multiplicative seminorm on X for any
184
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
multiplicative seminorm p on Y. Any normed algebra norm is a
multiplicative seminorm. Further examples of multiplicative
seminorms are given in the examples in Section 4.3.
(4.2-3) MULTIPLICATIVE SEMINORMS AND GAUGES. (a) If p is a
multiplicative seminorm, then = {x|p(x) < 1} is absolutely
m-convex and absorbent. (b) If U is absolutely m-convex and
absorbent then its gauge рц, рц(х) = inf{a>0|x g aU}, is a
multiplicative seminorm.
Proof.	(a) If
proves that
the gauge of an
x,y € V , then p(xy) < p(x)p(y) < 1, which
is multiplicative. (b) It is well-known that
absolutely convex absorbent set is a seminorm,
so it only remains to show that рц is multiplicative. To
this end, suppose x,y € X and a,b >0 are such that x g aU
2
and у € bU. Then xy € abU c abU so рц(ху) < ab. The re-
sult now follows from the arbitrary nature of a and b.V
4.3	Locally m-convex algebras
A topological algebra (X,^) is a locally m-convex alge-
bra (LMC algebra) if there is a base of m-convex sets for
V(0). We also say that •J' is locally m-convex or is an LMC-
topology. X is a locally convex algebra if X is a topological
algebra which carries a locally convex linear space structure.*
If, in addition to being locally m-convex,J is Hausdorff, we
say that X is an LMCH algebra, and «Г to be LMCH. An LMC alge-
bra which is a complete metrizable topological space is a
Frechet algebra.
Clearly each normed algebra is an LMC algebra and each
* Clearly any LMC algebra is a locally convex algebra. The
converse is not true, however, and a counter-example may be
found in Exercise 4.7 (c).
4.3 LOCALLY m-CONVEX ALGEBRAS
185
Banach algebra is a Frechet algebra.
Our first result about LMC algebras shows that we can do
a little better than just say that there is a base of m-convex
sets at 0. Our second, (4.3-2), shows LMC topologies to in-
evitably be generated by families of multiplicative seminorms.
(4.3-1) BASES AT 0 FOR LMC TOPOLOGIES. The following condi-
tions are equivalent on any algebra X:
(a)	X is an LMC algebra;
(b)	X is a locally convex TVS and there exists a base of
multiplicative sets at 0;
(с)	X is a TVS and there exists a base of absolutely m-
convex sets at 0;
(d)	X is a TVS and there exists a base of closed abso-
lutely m-convex sets at 0.
Proof. (а)^=ф-(Ь) : X is clearly a locally convex TVS if it is
an LMC algebra and it possesses a base at 0 of m-convex sets.
(b)Z^>(c): Since X is a locally convex TVS it has a base
(£,' at 0 of balanced convex sets. It also has a base Ф at 0 of
multiplicative sets. Thus each B' € ©' contains a multipli-
cative set В c ® and the collection of absolutely m-convex
sets B, also serves as a base at 0.
be
(c)rzz^>(a) : Let <8 be a base of absolutely m-convex sets at
0. <B clearly satisfies the conditions of the basis theorem for
topological algebras, Theorem 4.1-1, from which (a) follows.
(c)=^(d) : By (с) X is a TVS and has a base at 0,8, of
absolutely m-convex sets. Since X is a topological algebra
(since (с)	>(a)), the closure of a multiplicative set is also
multiplicative [(4.2-1)(f)] while absolute convexity is pre-
186
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
served by closure in any TVS.
Now for any V € V(0) there exists U € V(0) such that
cl(U) с V, since any TVS is regular; cl(U), in turn, contains
some В € It follows that V contains the closed absolutely
m-convex set cl(B).
(d)=^(c) : Clear, v
(4.3-2) LMC ALGEBRAS AND SEMINORMS. A topological algebra X
is locally m-convex iff its topology is generated by a family
of multiplicative seminorms.
Proof. If X is locally m-convex, it has a base (3 at 0 of ab-
solutely m-convex neighborhoods by (4.3-1). The gauges of the
sets В € (В are multiplicative seminorms by (4.2-3) which ob-
viously generate the topology. The converse follows immediate-
ly from (4.2-3) (a).V
Our assumption that a topological algebra not carry the
trivial topology implies that if the topology is generated by
a family P of multiplicative seminorms, then there exists some
p € P which is not trivial.
Some examples of LMC algebras follow.
Example 4.3-1 COMPACT-OPEN AND POINT-OPEN TOPOLOGIES. Consider
the linear space C(T,F) of all scalar-valued continuous func-
tions on the topological space T with pointwise linear opera-
tions. For x,y € C(T,F) define the pointwise product of x
and у, xy, at any t € T as (xy)(t) = x(y)y(t). With these
operations, C(T,F) becomes an algebra whose identity is the
map which takes each t g T into 1. The compact-open topology
for C(T,F) ((Example 0.1-2 ) was generated by the maps p_
where
4.3 LOCALLY m-CONVEX ALGEBRAS
187
G is a compact subset of T and p„(x) = suplx(G)I. Since each
G	1
such seminorm is clearly multiplicative with respect to point-
wise multiplication, it follows that C(T,F,c), C(T,F) with
compact-open topology, is locally m-convex.
With C(T,F) as above then C(T,F,p) - C(T,F) endowed
with the point-open topology (Example 0.1-2)-is generated by
the seminorms {pfc|t g T] where pfc(x) = |x(t)|. Each p is
multiplicative and so C(T,F,p) is a locally m-convex algebra.
Example 4.3-2 INFINITELY DIFFERENTIABLE FUNCTIONS. Consider
the algebra <£ of infinitely differentiable functions on [0,1]
(or [a,b]) with pointwise operations. The seminorms pn(x) =
supt [.g	| x (t) |?(п=0,1, .. .) determine a first-countable
T2-hence metrizable-topology for JB . These seminorms are not
2
multiplicative, however, [if x(t) = t, then p^(x ) =2 while
2
P1(x) =1 so that p1(x ) X Pj_ (x) •p1 (x) ] but the topology is
locally m-convex nevertheless. To see this, note that the same
topology is generated by the seminorms rn
1 < j < nPj'
A straightforward computation using Leibniz's rule for comput-
ing the j-th derivative shows that for each n rn(xy) <
2nr (x)r (y). Thus the seminorms q = 2nr are multiplica-
n n -1	^n	n
tive as qR(xy) = 2nrR(xy) < 2n(2nrn(x)rn(y)) = qn(x)qR(y) and
(qn) also generates the original topology. Thus is a met-
rizable LMCH algebra. Actually Ж is a Frechet algebra, i.e.
oB is complete. To see this, suppose (xn) is a Cauchy se-
quence in o&. It follows that (xn) is a Cauchy sequence for
each Pj,j=O,l,... . In particular for j=0, we see that
(xn) is a uniform Cauchy sequence, so there is some x such
that x->x uniformly in the usual function-theoretic sense on
188
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
[0,1]. Is x infinitely differentiable? Since (xn> is Cauchy
with respect to p., then the sequence of derivatives (x ')
1	n
is also a uniform Cauchy sequence on [0,1]. Hence x is diff-
erentiable and x' is the uniform limit of the xn'* Continuing
in this fashion, x is seen to be infinitely-differentiable and
xn(k) -> x(k) uniformly on [0,1] for each k=0,l,... . In
turn this is the same as saying that p, (x -x) -+ 0 for each
К n
k=0,l,... and therefore xn“>x in c6's topology.
JQ is not normable, however, as it cannot contain a bounded
neighborhood of 0: Any neighborhood of 0 will contain sequen-
ces (xn) whose subsequently derived sequences will achieve
arbitrarily large values, thus violating the condition that a
set В is bounded iff for each seminorm p generating the topo-
logy, (p(x )) is a bounded set, for each sequence (xn) from
B. The sequence determined by taking xn(t) = Ю 1^sin 10nt
serves as a fairly dramatic example of this phenomenon for
V
po
Example 4.3-3 INITIAL TOPOLOGIES; INITIAL LMC TOPOLOGIES. Let
(xp)p0j be a family of topological algebras, let X be an alge-
bra, and suppose for each p g M there is a homomorphism
A^sX+X^. The initial topologyxT determined by the family
for x ((0.1-1)) has a neighborhood subbase at 0 of
sets of the form A^ 1(B ), where B^ is a basic neighborhood
of 0 in X • That«J4 is compatible with X's algebraic structure
is evident by virtue of the results of (4.1-1) (on initial topo-
logies) and the subbase theorem. Corollary 4.1-1.
Moreover, since the inverse homomorphic image of a multi-
plicative set is multiplicative [(4.2-1) (e)] and finite inter-
sections of multiplicative sets are multiplicative, it follows
4.4 FINAL TOPOLOGIES AND QUOTIENTS
189
that is LMC if each X is LMC.
If (Xp)
is a family of topological algebras
then fix =X
with the product topology and pointwise operations
((V(yp) =
(x^y^), etc.) is also a topological algebra: The projection
maps pr^ are homomorphisms of X into X^ and the product to-
pology is the initial topology determined by the family (pr^)•
Thus by Example 4.3-3 the product topology is compatible with
the algebraic structure of X. Moreover by the same example:
(4.3-3) PRODUCTS OF LMC ALGEBRAS. A product of topological
algebras is LMC iff each component space is LMC.
4.4	Final topologies and quotients. In (4.1-1) we observed
that any subalgebra of a topological algebra is a topological
algebra in its relative topology. To deal with quotients of
topological algebras, we use (4.1-2). The linear subspace I of
an algebra X is an ideal if xlc I for each x с X. If X is a
topological algebra and X/I carries the final topology induced
by the canonical homomorphism x+ x+I of X onto X/I, then X/I
is a topological algebra by (4.1-2). We denominate this topo-
logy for X/I the quotient topology. (The term "factor topo-
logy" will be reserved for a different object, considered in
the next section.) Moreover, since homomorphisms preserve mul-
tiplicity, X/I is seen to be LMC if X is.
The topology of the following example facilitates the
passage from consideration of final topologies generated by a
single map to final topologies determined by a family of maps.
Example 4.4-1 SUPREMUM TOPOLOGY. Let X be an algebra and let
(<Л1J U eM be a family of compatible topologies for X. By the
supremum (sup) topology -J for X we mean the topology genera-
190
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
ted by the sets U „<7 ; we denote ч/ by v <T •
*	цеМ u	J vu u
By letting denote X topologized byand letting i
denote the canonical injection of X into X^,<J is readily iden-
tified as the weakest topology for X with respect to which each
i is continuous-. It follows from Example 4.3-3 that J is com-
patible with the algebraic structure of X. By the same reason-
ing it follows that is locally m-convex if^^ is.
A neighborhood base at 0 for^/ is given by finite inter-
sections of the form ,	where each is a-neighbor-
1-	Uf
hood of 0, 1=1,...,n.
We define the final topology on an algebra X determined by
a family (A^)of homomorphisms A^zX^+X, where each X^ is a topo-
logical algebra, as the supremum of all compatible topologies
for X with respect to which each A^ is continuous. The trivial
topology is one such topology for X, so the family of all such
topologies is not empty. Similarly we obtain the final LMC
topology for X determined by the (A^) as the supremum of all
LMC topologies for X with respect to which each A^ is continu-
ous. This topology has a neighborhood base at 0 given by the
collection of all absorbent absolutely m-convex subsets V of X
for which A^ '’’(У)
is a neighborhood of 0 in X^ for each index
Ц. It immediately follows that any homomorphism A taking X
equipped with the final LMC topology determined by (A^) into an
LMC algebra Y is continuous iff A*A^ is continuous for each
index Ц.
As the final LMC topology discussed above is clearly lo-
cally convex, it is coarser than the final locally convex topo-
logy for X determined by the A , the supremum of all locally
4.4 FINAL TOPOLOGIES AND QUOTIENTS
191
convex topologies making each A^ continuous. These two topolo-
gies do not always coincide (see Exercise 4.7). In the discus-
sion to follow we present two instances in which the final lo-
cally convex and final LMC topologies coincide.
Example 4.4-2. If X is a normed algebra and (X^) is a collec-
tion of ideals of X such that X = и X^ then the final locally
convex and final LMC topologies determined by the injection
maps i^iX^+X coincide.
Proof. It is only necessary to show that a typical neighbor-
hood of 0 in the final locally convex topology contains a
neighborhood of 0 in the final LMC topology. To this end let
V be an absorbent absolutely convex subset of X such that
V fl X^=i^ '’’(У) is a neighborhood of 0 in X^ for each p. Then
for each p there is a positive number < 1 such that
W = {xeX lli x || < e } с V П X . Setting W = и W it follows
ц	Ц	jj	jj	JJ
that W is absorbent. To see that W is multiplicative, let
x,yeW, so that xeW^ and yeW^ for some indices p and g. X^
is an ideal so xy e X^. Since || xyll < II xll-Hy|| <	1 = e ,
xy e c W, and W is multiplicative. The balanced convex hull
W^c of W is an absorbent absolutely m-convex subset of V.
Moreover since i 1(W. ) = X П W. contains W , a neighbor-
p be p be	p
hood of 0 in X^ for each p,	is a neighborhood of the ori-
gin in the final LMC topology generated by the maps (i^) con-
tained in V.
Example 4.4-3. If X is an LMC algebra containing an increasing
sequence of ideals Xn, each carrying the subspace topology,
such that X = U X^, then the final locally convex and LMC
topologies generated by the canonical injection maps i coin-
192
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
cide.
Proof. Let P be a family of multiplicative seminorms generat-
ing the LMC topology of X and let V be an absorbing absolutely
convex set such that, for every n,in 1 (V) = V n Xn is a
neighborhood of 0 in Xn- Then for each к e N there is a fi-
nite subset P с P and a positive number e, < 1 such that
К	К
V, = {хеХ, |p(x) < e, ,peP,} с V П X, . For each neN, let
К	К	К К	К
Q = U?nP1zW = {хеХ |р(х) < min. , е. ,peQ }, and W=U W .
n k=l k' n	n1^	l<k^n к n	n
As in the preceding example, once it has been shown that W is
absorbing and multiplicative, it follows that	is a neigh-
borhood of 0 in the final LMC topology and W^c с V. It is
clear that W is absorbing; to see that it is multiplicative as
well let x,yeW. Then there are n,meN such that xeWn and
yeWm- Assuming that n > m and noting that xy e X^ since
X is an ideal in X, it follows that for each peQ c Q ,
m	m n
p(xy) < p(x)p(y) < min. , e, min, , e. < min, , e, . Thus
l^k<n k	k ~ l<k<n k
xy e Wm c W and W is multiplicative.
4.5 The Factor Algebras. It is well-known that any locally
convex Hausdorff space X is embedded in a product of Banach
spaces. It is easily seen that the analogous statement for
LMCH algebras also holds [(4.5-1)] and this fact has a number
of ramifications such as Arens’ invertibility criterion of the
next section [Theorem 4.6-1 (e)]. Generally the effect of
(4.5-1) is to produce theorems of the form: If each factor al-
gebra (defined below) of the LMCH algebra X has property (*),
then so does X.
If p is a multiplicative seminorm on an algebra X, then
the null space or kernel of p, N = p ^(0), is an ideal since,
4.5 THE FACTOR ALGEBRAS
193
for any
x € Np and у € X, p(xy) < p(x)p(y)
0. Let p de-
note the factor norm: p(x+N ) = inf p(x+N ) = p(x). In view
of the multiplicativity of p and the above relationship it is
clear that X/N is a normed algebra with respect to p. This
normed algebra is the pre-factor algebra (associated with p);
its completion is the factor algebra associated with p. The
extension of p from
X/Np
to the completion will still be de-
noted by p.
Definition 4,5-1.
When X is an LMC
algebra and (p ) is
a sat-
urated (Definition 4.2-3) family of seminorms generating X's
topology, the completions X of the normed algebras
X/N are
called a set of factor algebras for X.
(4,5-1) AN LMCH ALGEBRA IS A SUBSPACE OF A PRODUCT OF BANACH
ALGEBRAS. Let P be a saturated family of multiplicative semi-
norms for the LMCH algebra X determining a set
(Xp’p € P
of
factor algebras for X.
Then the map x+(x+N )	„ embeds X
P P € P
in the topological algebra
Proof. It is readily seen that the given LMCH topology of X
coincides with the initial topology ((4.1-1)) generated by the
П
P
X .
€ P P
linear map
А:Х->П X
P P
x+(x+N ) so that A is continuous and
relatively open (i.e. A maps open subsets of X into open sub-
sets of A(X)). That A is 1-1 follows immediately from the
fact that X is Hausdorff.V
We note that the assumption that the generating family of
seminorms be saturated involves no loss of generality because
of (4.2-2), i.e. any family of seminorms generating the topo-
logy may be extended to a saturated family of seminorms genera-
ting the same topology.
194
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
We might also add that though we can't speak of the set of
factor algebras for an LMC algebra, it doesn't make any real
difference for our purpose: one set is as good as another for
the applications that follow.
Another feature of the Banach algebras worth noting is
that each of the canonical homomorphisms к :X+X , x+x+N
is continuous as к ({xfX[p(x)<l})=V	=
Ц	Ц ‘ Ц	pp
{x|p^(x) < 1}. In fact the original topology on X is the same
as the initial topology for X induced by the family (к ).
Some examples of factor algebras follow. In the first,
there is a natural set of factor algebras, each of which is
complete to begin with. Using some of the information about
the factor algebras of these two algebras gleaned here some in-
formation about the spectrum of elements in these algebras is
obtained in the next section.
Example 4.5-1. FACTOR ALGEBRAS OF C(T,£,c) ARE SUPREMUM NORMED
ALGEBRAS. Let C(T,F,c) be the LMC algebra of Example 4.3-1:
continuous scalar-valued functions on the topological space T
with compact-open topology. Only this time assume that T is
a completely regular Hausdorff space so that continuous func-
tions on compact subsets may be continuously extended to the
whole space*. Letting denote the family of compact subsets
of T, the topology on C(T,_F,c) is that generated by the semi-
norms pG(x) = sup|x(G)| as G runs through . We denote the
null space of p_ by simply N .
Lj	Lj
* Indeed, if G is compact in T it is closed in the compact
Hausdorff space gT. Thus, as gT is normal, any continuous
function x on G may be continuously extended to gT.
4.5 THE FACTOR ALGEBRAS
195
We observe that (p„) is a saturated family of seminorms.
G
We now show that each of the normed algebras C(T,F,c)/N„ is
isometrically isomorphic to the Banach algebra C(G,F) of con-
tinuous complex-valued functions on the compact set G with su-
premum norm (	= C(G,_F,c)). In particular consider the con-
tinuous linear maps AG:C (T,£) -+C (G,F) , x+x | (G €^). Cer-
tain things about A are evident: A„ is a multiplicative, on-
to map since T is a completely regular Hausdorff space. Hence
C(Tr<i;)/NG is (algebraically) isomorphic to C(G,F) the iso-
morphism being the map x+N +xI„. Since p„(x+N„) = p„(x) =
p (x| ), C(T,F)/N is seen to be isometrically isomorphic to
C(G,F,c).
Example 4.5-2. ANALYTIC FUNCTIONS ON THE UNIT DISC. Let H de-
note the set of analytic functions on the open unit disc in the
complex plane (2. With respect to pointwise operations and the
compact-open topology, H becomes an LMC algebra with topology
generated by the saturated family of multiplicative seminorms
Pn(x) = sup|x(Cn)| where cn= tu € C||ц| < 1 - ^}. By the
analytic identity theorem, the only function analytic on the
open unit disc which vanishes on any of the Cn is the function
which is identically 0 on the disc. Thus pn ^(O) = {0} or,
equivalently, each pn is a norm. In this example too we wish
to ascertain some facts about a set of factor algebras for H.
Unlike the situation of the preceding example, however, the
pre-factor algebras are not complete, so the completions must
be determined. Noting that H is a first countable Hausdorff
space, it follows that H is metrizable. Thus Cauchy sequences
are adequate to describe completeness. Moreover, in view of
196
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Pn С (C
Weierstrass's theorem on uniformly convergent sequences of
analytic functions* it follows that every Cauchy sequence con-
verges and H is a Fre'chet algebra.
Similar to the approach used in the preceding example, we
consider the map A :H+C(C ,C,c) , x+xI_ where C(C ,C,c) de-
n n *	1 C	n
n
notes the sup norm algebra** of continuous functions on Cn-An
is a homomorphism whose kernel is N = p ^(O), which is
pn
{Or, so each A is an isomorphism. Since p (x) = p (x+N ) =
n	n	n Pn
.(x), H/N is seen to be isometrically isomorphic
.. c	pn
to H| = {h|c |h € H} in C(C ,C,c).
n	n
* A.I.Markushevich, Theory of Functions of a Complex Variable,
Vol. 1, Prentice-Hall, Englewood Cliffs, N.J., 1965,pp.327-330.
(Weierstrass's theorem on uniformly convergent sequences of
analytic functions). If the sequence (xn) is uniformly conver-
gent on each compact subset of a domain D (i.e. open and conn-
ected) and if each xn is analytic on D, then the function x,
determined at each Ag D according to x(A) = lim xn(A) is also
(k)
analytic on D. Moreover each sequence of derivatives (xn )
(k)
converges uniformly to x on each compact subset of D.
** The compact-open topology on C(T,C) coincides with that ge-
nerated by the sup norm when T is compact, so the notation
"C(C ,C,c)" is consistent with topologizing C(C ,C) by the sup
n ~	n ~
norm.
4.6 COMPLETE LMCH ALGEBRAS AND PROJECTIVE LIMITS
197
If H|c were a complete subset of C(C ,C,c) it would
n	n
certainly have to be closed, but this is not the case for in
00
C(C ,C) there is a function§ x(A) = E a An which is analy-
n ~	n=0 n	J
tic throughout Cn except for at least one point on the bound-
ary. Hence x is certainly not the restriction of a function
which is analytic on the entire open unit disc. The function x,
however, is clearly the uniform limit on C of the polynomials
n
x (A) = .Z a. A-J and each x is a restriction of a function
n 3=0 j	n
which is analytic on the open unit disc. Thus h|c is not
n
closed, hence not complete. Next we determine the completion.
If x € H, then x is expressible as a power series about
0 which converges to x uniformly on any compact subset of the
open unit disc. Thus, the completion of H| in C(C г<С,с)
Cn	n ~
is seen to be the subalgebra P(C ) of those x € C(C , C)
n	n ~
which are uniform limits of polynomials on Cn-
4.6 Complete LMCH Algebras and Projective Limits.
One characteristic ((4.5-1)) of complete LMCH algebras X
is that they may always be imbedded in a product of Banach al-
gebras, namely a set of factor algebras for X. Another proper-
ty of complete LMCH algebras is that they are projective limits
of their factor algebras, and that is proved here. First some
elementary discussion of projective limits is set forth. Some-
what dual to the notion of projective limit is the notion of
CO
§ The function in mind is x(\) = djd/j2) ( А/ (1-1/n) ) -1 . The
result follows from Pringsheim's theorem and the Cauchy-Hada-
mard formula. cf. Markushevich, Theory of Functions of a Com-
plex Variable, Vol. 1, Prentice Hall, Englewood Cliffs, N.J.,
1965, p. 390.
198
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
inductive limit which is discussed in Chapter 6 in relation to
the notion of LB-algebra.
Definition 4.6-1 PROJECTIVE SYSTEMS AND LIMITS. Let (X )
	 a
be a family of sets indexed by the preordered set Л. If for
each pair (a,g) g ДхД with a < g there is a mapping
h О:Х+Х such that h is the identity function for each
ag 8 a	aa	J
a € Л and h R ° h„ = h whenever a < g < у then the fam-
otp Pi	oi Y	—
ily of sets together with the maps constitutes a projective
system. The subset X of ПХ^ consisting of all (xj with the
property that h R(xR) = x whenever a < g is called the
projective limit of the system. In the event that each Xa is
a topological space and each hag for a < g is continuous
the projective system is called a topological projective sys-
tem. The term topological projective limit is used to denote
the associated projective limit X equipped with the product
topology. Going one step further if each Xa is a topological
algebra and the maps hag(a<g) are continuous homomorphisms
then we obtain a projective system of topological algebras and
a projective limit of topological algebras, X. Certainly X is
a topological algebra as each hag is an algebra homomorphism
and so X is closed with respect to all the algebraic operations
defined on Пх„.
a
Part (e) of the following result, first proved by Arens
(1952) later reproved by Michael (1952) using projective lim-
its, is of great importance in proving some of the results to
follow. The force of it is that it enables us to convert cer-
tain questions allied to invertibility (such as spectral ques-
tions) in topological algebras into similar questions in Banach
4.6 COMPLETE LMCH ALGEBRAS AND PROJECTIVE LIMITS
199
algebras. Since the latter theory has been heavily mined, this
is quite desirable.
Theorem 4.6-1 A COMPLETE LMCH ALGEBRA IS A PROJECTIVE LIMIT OF
ITS FACTOR ALGEBRAS. Let X be a complete LMCH algebra with
topology generated
by the saturated
family of non-trivial semi-
norms P
Then
(p )
a a€A
set
The
with associated
factor algebras
Л is directed by
the relation: a
(X )
a a€A
g iff
Pa
3'
(b)
g then the
continuous extension
of the mapping
h a:X +X
ag g a
a nontrivial homomorphism
a
g
a
with
the collection (X )
a
(h Q
ag
a€A
constitutes a projective
together with the mappings
system.
(d)
if Y denotes the projective limit of the system of
equipped with pointwise multiplication and the induced
product topology, then the mapping T:X->Y , x->(x+Na) is an onto
topological (algebra) isomorphism.
(e)	ARENS INVERTIBILITY CRITERION, the element x g X
is invertible iff x+N is invertible in X for each a f A.
a	a
Proof, (a) This follows directly from the fact that P is sat-
urated.
(b)	Clearly ha^ is well-defined. For xg X and a < g,
p(x+N ) = p (x)< p„(x) = p_(x+N ). Hence h . restricted to
1 a a — g	g g	ag
X/N. has norm less than or equal to 1; thus so does h _.
p	0!* P
(c)	Clearly, if a < g < у then (h °h_ )(x+N ) =
up P У	у
h „(x+N„) = x+N = h (x+N ) and h (x+N ) = x+N . Since the
ag g	a ay у	aa a	a
maps (hag) are continuous it follows that the collection is
200
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
a topological projective system.
(d)	First we note that by virtue of the fact that the
mappings hag are homomorphisms, pointwise multiplication is
a binary composition on Y. It is clear that T is a homomor-
phism. Suppose that x / 0; since X is Hausdorff, pa(x)/0 for
some a £ Л and T is seen to be 1-1. Finally we show that T is
onto. To this end let у = (y^) € Y. Since each Xa is the
completion of the corresponding normed linear space X/Na there
exists sequences (xna) c x such that for each a,
P„ (Xn„+VYJ < for n > 1. We prove that (x ) is a
Cauchy net in the complete space X and that its limit is the
pre-image of у under T. The set of pairs (n,a) is clearly
directed by the ordering: (n,a) > (m,g) iff a > 3 and n>m.
To see that (x a) is Cauchy it is enough to show that
p (x -x .)->0 for each fixed y. Indeed if a,g > у and —,
^y na mg	1	' — 1	n
— < e/2 then p (x -x .) = p ((x +N -y ) - (x _+N -y )) <
m	^y na mg y na у Jy	mg у Jy —
p (x +N -y ) + p (x a+N -y ) < p (x +N —у ) + p.(x n+N--y_)<
*y ' na у ‘у у mg у J у — na a J a ^g' mg g 1 g
— + — < e. Let x = lim x ; it only remains to show that
n m	na	-1
x+Na=ya for each a. Fix a and suppose that g > a. Then
Pa(x+Na-ya’ = ₽a( (x-Xng+Na} + ^ng^a^a’’ < ₽a (x-Xng+Na> +
p (x a+N -y J. Since z+z+N is a continuous mapping for each
a 'ng a ‘a	a	rr s
a, we have that x _+N —(n'	x+N . Thus there exists <5 c
ng a	a
and an integer N such that p (x a-x+N ) < e/2 whenever n>N
a' ng a	—
and g > 6. Furthermore, since hag <1 it follows that
p (x _+N —у ) < p.(x „+N.-y.) < — . Hence, if we take g > a,6
^a ng a Ja — g ng g J g n	—
and n > N, 2/e we obtain p (x+N -y ) < e. Since e is arbi-
01	01	01
trary, p (x+N -y ) = 0 and x+N = у .
1’ ^a a J a	a 1 a
(e) Due to the fact that x+x+N is a nontrivial homo-
4.7 THE SPECTRUM
201
morphism, the invertibility of x implies that of x+Na for
each a. Conversely, if each x+Na is invertible in X~, then,
as T is an isomorphism, we need only prove that (x+N^) is in-
vertible in Y. Moreover, since ((х+Nj is the inverse of
(x+Na) in ПХа, we are only required to show that
((x+Nj 1) € Y. To this end, recall that, for a < g, hag
maps the identity of onto the identity of X&. Thus
hag[(x+V’1] = [hae(x+Ne)]-1 = (x+Na)-1-v
4.7 The Spectrum.
In Banach algebras, the notion of the spectrum of an ele-
ment plays an important role. The fact that no element in a
complex commutative Banach algebra with identity has an empty
spectrum, in particular, is the cornerstone of the Gelfand
theory of commutative complex Banach algebras. To define the
spectrum one doesn't even need a topological algebra; our no-
menclature on the subject is set forth in Definition 4.7-1 be-
low.
Definition 4.7-1 SPECTRAL NOTIONS. Let X be an algebra. The
element x с X is a unit (is regular) if x is invertible in X,
i.e. there exists an element у с X such that xy=e. (When
such a у exists it is unique and is denoted as x 1.) Other-
wise x is singular. The scalar ц is a regular point of x € X
if x-pe is regular. Otherwise p is a singular point of x. The
set a(x) of all singular points of x is the spectrum of x.
Its complement-the set of all regular points of x-is denoted by
p (x) . The extended real number r (x) = sup., r , lai is called
°	a€o(x)'
the spectral radius of x and U(o) denotes the set of x с X
with rQ(x) < 1.
202
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
As has been remarked, when we say "Y is a subalgebra of X"
this includes the requirement that the identity of X belongs to
Y. In this context, it is necessary to distinguish between
a (x) , those ц such that x-pie is singular in X, and ov(x),
those ц for which x-pe is singular in Y (for x's in Y, of
course). Generally as the population of the algebra becomes
larger, the opportunity for x-pie to have an inverse increases
so that:
(4.7-1) if x € Y where Y is a subalgebra of X, then
ox(x) c cjy (x) .
We compute the spectrum of an element of C(T,F,c).
Example 4.7-1 SPECTRA IN C(T,.£,c). Consider the LMCH algebra
C(T,F,c) of continuous F-valued functions on the completely
regular Hausdorff space T with compact-open topology (Examples
4.3-1 and 4.5-1). An element yeC(T,F,c) is invertible iff
y(t) / 0 for each teT. Thus pieo(x) iff y(t) = ц for some
teT and o(y) = y(T).
Arens' invertibility criterion, Theorem 4.6-l(e), enables
us to characterize spectra of elements in complete LMCH alge-
bras in terms of spectra of elements in certain Banach alge-
bras-a situation of some desirability because of the informa-
tion already available about spectra in Banach algebras.
(4.7-2) A CHARACTERIZATION OF THE SPECTRUM. Let P = (p )
-	—	— _	s s Cv
be a saturated family of multiplicative seminorms for a com-
plete LMCH algebra X. For each s € S, let Ng = pg ^(0). In
the factor algebra Xg (Definition 4.5-1) let og(x) denote the
spectrum of x+N . Then a(x) = U a (x).
S	S tv s
Proof. The result follows immediately from Theorem 4.6-l(e).V
4.7 THE SPECTRUM
203
We use Arens' invertibility criterion (i.e. Theorem 4.6-1
(e)) in this form to characterize spectra of elements in the
algebra described in Section 4.5.
Example 4.7-2 SPECTRA IN H. In Example 4.5-2 we showed that
a set of factor algebras for the Frechet algebra H of functions
analytic on the open unit disc was given by the subalgebras
P(Cn)—uniform limits of polynomials on Cn—of sup norm alge-
bras of continuous functions C(C ,C,c), n=l,2,..., where C
n ~	n
denotes the closed disc of radius 1-1/n about the origin.
Identify x+Nn with its isometric isomorphic image x|
n
in P(Cn). In the case at hand it is easy to compute the spec-
trum o(x|„ ) of x|_ e P(C ) in С(C ,C,c): It is just
'С	'C	n	n	J
n
x(Cn). Thus x(Cn) C,"(xlc )• We contend that equality holds,
n
To see this, suppose ц / x(Cn)• Since x-ц is never 0 on Cn
and Cn is compact, ) х-ц) must have a positive lower bound, a
say, on C . Thus 1/(х-ц) is an analytic function on C , but
the question is: Can it be expressed as a uniform limit of
polynomials on Cn?
Since x|c c P(CR) there is a sequence (f ) of Poly-
nomials which converges to x-ц uniformly on C . Clearly we
may assume that inf If (C )| > b for some b > 0. The set of
J	m1 m n 1 —
zeros of each f , therefore, has a positive distance from C ,
m	c	n
and as a result for each m f N, 1/fm is expressible as a
power series whose radius of convergence is > 1-1/n. More-
over since (l/fm) converges to 1/(х-ц) uniformly on C , it
follows that 1/(х-ц) is expressible as a uniform limit of
polynomials on C . Hence if ц / x(Cn)' then ц is a regular
point of x |c in P(Cn), i.e. ц/ a(x | ) which establishes the
n	n
204
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
equality.
Now for any x € H, a(x) = Цх(Сп) = x(Sj(0))- Thus, as
was the case in the proceeding example, the spectrum of an ele-
ment is just the set of values it assumes.
The results of the proceeding two examples make it clear
that the spectrum of an element in a complete LMCH algebra need
be neither closed nor bounded. In a Banach algebra, by con-
trast, a(x) may be expressed as the continuous image of a
compact set and so is closed and bounded (cf. Section 4.8); it
is always contained in the disc of radius lixi| about 0. Since
the algebra H is a Frechet algebra, metrizability is seen not
to be the critical ingredient.
4.8 Q-algebras and algebras with continuous inverse.
As is well known maximal ideals in Banach algebras are
closed sets. This feature is unfortunately lost in the gener-
al case. It is recovered, however, in the special type of al-
gebra we define below although we defer proving this until
(4.10-1).
Definition 4.8-1 Q-ALGEBRAS. A topological algebra in which
the set of units is open is called a Q-algebra.
Any Banach algebra is a Q-algebra (4.8-2); so is the space
^0 of infinitely differentiable functions on [a,b] as is
proved after (4.8-1) below. An example of a topological alge-
bra which is not a Q-algebra is given in Example 4.8-1 now.
Example 4.8-1	C(T,^,c) IS NOT GENERALLY A Q-ALGEBRA. Let
C(T,F,c) be the topological algebra of continuous scalar-valu-
ed functions on the non-compact completely regular Hausdorff
space T of Example 4.5-1, with compact-open topology. To show
4.8 Q-ALGEBRAS ANQ ALGEBRAS WITH CONTINUOUS INVERSE
205
that C(T,F,c) is not a Q-algebra, we show that each neighbor-
hood of the identity e contains a non unit. To this end con-
sider a typical basic neighborhood of the identity e+W(G,e) =
{x € C(T,F,c)|sup |x(t)-l| <e}, where G is a proper compact
subset of T and e > 0. By the Tietze extension theorem there
is a continuous function у on T such that y(G) = {1} and, for
some fixed t XG, y(t) = 0. The non—unit у then is clearly a
member of e+W(G,e).
(4.8-1) Q-ALGEBRA IFF INT Q / 0. The topological algebra X is
a Q-algebra iff its set Q of units has nonempty interior.
Proof. Clearly only the sufficiency of the condition need be
proved. To this end, let x g int Q and let у be any unit.
Consider the map w+yx ^w. As remarked in Section 4.1 such a
map must be a homeomorphism which maps Q onto Q. Hence for
V € V(x) such that V c Q, yx 1V is a neighborhood of у which
lies in Q.V
As an application of (4.8-1)we now show that the space
of infinitely differentiable functions on [a,b] with the top-
ology generated by the seminorms p (x) = sup	|x^n^ (t)I
n	t€[a,b]
(n=0,l,...) (first discussed in Example 4.3-2) is a Q-algebra.
By (4.8-1) we need only show
and to this end consider any
and | x(t)-l | <1 for each
function l/x(t) is defined
[a,b] thereby implying that
(4.8-2) BANACH ALGEBRAS ARE
that the neighborhood e+V c Q
po
x с e + V . Since e(t) = 1
po
t c [a,b] , it follows that the
and infinitely differentiable on
x is invertible.
Q-ALGEBRAS. If X is a Banach al'
gebra then it is a Q-algebra. In particular if II e-x|| < 1 then
is a unit.
206
4. COMMUTATIVE TOPOLIGICAL ALGEBRAS
Proof. Let X be a Banach algebra with set of units Q. To show
that Q is open it suffices by (4.8-1) to show that S^(e) c Q.
To accomplish this we note that ||e-x|| < 1 implies that the
00	П
sequence of partial sums of Z lle-x|| is Cauchy. Hence the
n=0
sequence of partial sums of 1 (e-x) is Cauchy in X and the
n=0
series therefore converges to у say. To see that xy=e, write
x as e-(e-x) and consider (e-(e-x))(e+(e-x)+...+(e-x)n) =
e-(e-x)n+l from which the desired result follows.V
As was mentioned in the previous section a pleasing fea-
ture of Banach algebras—the fact that all elements have com-
pact spectra—is lost in the general case. Our next two re-
sults combine to recover this property in Q-algebras.
(4.8-3) IN Q-ALGEBRAS ALL ELEMENTS HAVE BOUNDED SPECTRA. For
any topological algebra X, (a) X is a Q-algebra iff U (a )
(the set of all x with r0 (x) < 1) has non-empty interior;
(b) If X is a Q-algebra then the spectrum of each ele-
ment is bounded.
Proof.	(a) First we assume that x is an interior point of
Up), or equivalently, that there is a neighborhood W c V(0)
such that x+W c U (a ) . Now rQ (y) < 1/2 for each у € (1/2) x +
(1/2)W so for each such у, e+y is invertible. Thus
e + (l/2)x (1/2)W cQ and it follows, by (4.8-1), that Q is
open.
Conversely	if Q is open, choose	a balanced neighborhood W
of 0 such that	e + W	c Q. Then for	each	x € W and A € F
with |A| >1,	it is	clear that Ae	+ x =	A(e + А ^x) c A(e+W)
c Q so that -A	/ a(x). Thus r0(x)	<1,	W c U(o) , and 0 is
an interior point of U(a).
4.8 Q-ALGEBRAS AND ALGEBRAS WITH CONTINUOUS INVERSE
207
(b) Since there is a neighborhood W € V(0) such that
W c U(o), it follows that for each x с X there is a A g F,
with sufficiently small absolute value, such that A x € W c
U(cj). Thus since ro(Ax) = |A|r (x) and r^fAx) <1 then
rQ (x) < °0. V
(4.8-4) COMPACTNESS OF SPECTRA IN Q-ALGEBRAS. If X is a Q-al-
gebra then for each x с X (a) p(x) is open, and (b) a(x) is
compact.
Proof. By (4.8-3(b)) each x с X has bounded spectrum so we
need only prove (a). To do this, let A g p(x); then Ae-x g Q
and, by the openness of Q, there is a neighborhood W € V(0)
such that Ae-x+W c Q. Clearly for any x с X the map
F -+ J£e + X , a + ae + ae-x is continuous and it follows that
there exists an e > 0 such that S£(A)e-x c Ae-x + W c Q.
Thus S£(A) c p(x).V
Since Banach algebras are Q-algebras [(4.8-2)], each ele-
ment in a Banach algebra X has compact spectrum. Moreover,
since || e-(е-х/р)ц = Цхц/|р| < 1 whenever |p| > ||x||, it al-
so follows by (4.8-2) that each such p € p(x). Thus a(x) c
< A e F | | A | < llx|] } .
Yet another property of Banach algebras—continuity of the
map x+x 1 — is lost in the general case. In fact it is not
generally the case that inversion is continuous in Q-algebras
(see (Exercise 4.7(c)) although LMC algebras do possess the
property (we prove this in (4.8-6)). As mentioned in Example
4.8-1, C(R,F,c) is not a Q-algebra but x+x 1 is continuous
anyway by (4.8-6).
Definition 4.8-2 CONTINUOUS INVERSE. A topological algebra X
20'8
4. C0IB1UTATIVE TOPOLIGICAL ALGEBRAS
in which the map x+x is continuous at e is an algebra with
continuous inverse. We also say that X has continuous inverse.
(4.8-5) THE RESOLVENT MAP IS ANALYTIC. In an algebra X with
continuous inverse: (a) the map x+x 1 is continuous every-
where on Q, the set of units of X; (b) if X is a complex alge-
bra then for any x с X, if p(x) is an open set (e.g. if X is
a Q-algebra) the resolvent map rx:p(x)+X , X+(x-Xe) 1 is an-
alytic on p(x)*.
Proof. (a) Let ® be a filterbase in Q convergent to x € Q. We
show that 1 = {в 1|b c <13} converges to x 1. As has been
observed previously, if у is a unit the map x+xy is a homeo-
morphism of X onto X, for any topological algebra X. Thus, for
any x € Q,y->x 1y is a homeomorphism. If the filterbase <£> in
Q converges to x, then x ^->e. By the assumed continuity of
the map w+w at e, it follows that x’S^-’-e, from which it
readily follows that <3 1->-x 1.
(b) Let rx(A) = (x-Ae) 1. The key observation in prov-
ing the analyticity of the resolvent map A+rx(A) is that
(*) rx<A) “ rx<P) = (A-p)rx(A)rx(p) (A,U€P(x)). To see this
consider rx(A) *'rx(p)= (x“^e)rx^) = (x-pe+pe-Ae)rx(p) from
which (*) follows. By part (a), the resolvent map p(x)+Q-*-Q ,
A+x-Ae+(x-Ae) is continuous. Thus the analyticity of rx(A)
on the open set p(x) now follows from (*).V
*
The set p(x) is open in any Q-algebra ( (4.8-4 (a))). We note,
however, that it need not be connected. For example, consider
the function x(t)=t in C(T,C,c) where T is a closed annulus in
C. Clearly cj(x)=T and p(x), being the complement of T, is
not connected.
4.8 Q-ALGEBRAS AND ALGEBRAS WITH CONTINUOUS INVERSE
209
At the outset of the discussion on continuity of inversion
we mentioned that not all topological algebras have continuous
inverse. It is the case, however, that a very large class of
topological algebras containing the normed algebras, namely the
LMC algebras, possess the property.
(4.8-6) LMC ALGEBRAS ARE ALGEBRAS WITH CONTINUOUS INVERSE. If
X is (a) a normed algebra, or, more generally, (b) an LMC al-
gebra, then X has continuous inverse.
Proof. (a) To see that the map x+x 1 is continuous at e,
suppose (xn) is a sequence of units convergent to e. To show
that xn l-’-e, we need only note that ||xn l-e|[<llxn ^ll lle-xnl|,
the boundedness of (llxn 1,1) being apparent from the relation
-1	00	к
[in the completion of X] x = E (e-x ) for ||x -e||< r < 1
k=0
(see (4.8-2)) and the convergence of (xn) to e.
(b) Let® be a filterbase of units convergent to e. We
prove that ® 1 = {В 1|b	converges to e.
Let P be a family of multiplicative seminorms generating
the topology on X and for each
$+e and the map x+x+N^ from
continuous, it follows that
Banach algebra, so inversion is
p € P, let Np=P 1(0). Since
X into the factor algebra X₽ is
+N ->e+N in X . But X is a
PPP	P
continuous and it follows that
® 1+Np= ((3+Np) 1->-e+N . In other words £ ('£> 1-e+Np) =p (® 1-e)->-0.
Since p is arbitrary, it follows that (3 1->-e.V
Since the algebra C(T,F,c), T non-compact, completely
regular, and Hausdorff, of Example 4.8-1 is an LMC algebra, it
follows that inversion is continuous in C(T,£,c); C(T,F,c) is
not a Q-algebra however, as is pointed out there.
210
4. COMMUTATIVE TOPOLIGICAL ALGEBRAS
4.9 Topological Division Algebras and the Gelfand-Mazur Theorem
A division algebra is a not necessarily commutative alge-
bra in which each nonzero element has an inverse. The Mazur-
Gelfand theorem,Theorem 4.9-1, for algebras with continuous in-
verse shows that there is essentially only one complex locally
convex Hausdorff (LCH) division algebra with continuous in-
verse. Both parts of the theorem remain true when our general
assumption of commutativity is lifted and one needs only to
carefully preserve the order in which things are written down
to see that this is so. Something which is essential to the
validity of the theorem, however is the fact that the algebras
be complex. The real LCH topological division algebra R in its
usual topology is certainly not topologically isomorphic to C,
for example. The quaternions too constitute a real topologi-
cal division algebra—in fact a Banach algebra, hence a topo-
logical algebra with continuous inverse by (4.8-(j(a)—which is
distinct from C.
It was first established by Frobenius that any finite-di-
mentional division algebra over the real numbers is (topologi-
cally) isomorphic to R,C, or the quaternions. Subsequent gen-
eralizations of this result were made by S. Mazur (1938), G.
Silov (1940), and I. Gelfand, for normed algebras. In addition
to establishing the result for complex LCH algebras (Theorem
4.9-l(b)), Arens (1947a) also proved that essentially the only
real LCH topological division algebras with continuous inverse
are R,C, and the quaternions, part of which is established in
Theorem 4.9-2.
The classical Liouville theorem states that a bounded en-
4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM 211
tire function must be constant. With suitable analogs for
"bounded" and "entire" we prove a version of Liouville's theo-
rem for entire vector-valued functions х:С+Х. This version
((4.9-1)) is needed to prove our main result, Theorem 4.9-1.
If X is a topological vector space and G a subset of C
then x:G+X is bounded if x(G) is a bounded subset of X,
'bounded' subset of X in the sense that it is absorbed by any
neighborhood of 0.
Definition 4.9-1 ANALYTICITY. Let G be an open subset of the
complex plane and X be a topological vector space. The map
x:G+X is analytic in G if the limit lim	x (ц)—X	=
----------------- ^o
x'(dq) exists at each DQeG.
In the version of Liouville's theorem to follow, it is im-
portant that the vector space possess sufficiently many contin-
uous linear functionals. In particular we want the topological
vector space X to have enough continuous linear functionals so
that the information that each continuous linear functional
vanishes on a certain vector is enough to guarantee that the
vector is 0. Whenever a subspace S of linear functionals on a
vector space has the property that f(x)=0 for all feS im-
plies x=0, then the subspace is called total. In particular
the set X' of all continuous linear functionals on a locally
convex Hausdorff space is always total, as shown by the Hahn-
Banach theorem.
(4.9-1) LIOUVILLE'S THEOREM. Let X be a topological vector
space and suppose that its dual X' is total. If x:C->X is
entire and bounded, then x must be constant.
Proof. Since the continuous linear image of a bounded set is
212
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
bounded, if x is bounded, then so is fx for any feX'. Thus
the standard Liouville theorem applies to each of the entire
functions fx and we conclude that fx is constant for each fgX'.
Hence for any y,AgC, f(x(u)) = f(x(A)), so f(x(u)-x(A))=0.
Holding у and A fixed and noting that the last equality holds
for every f in a total set of linear functionals, we conclude
that x(n) = x(A). Since u and A are arbitrary, the constancy
of x is proved.V
Theorem 4.9-1 COMPLEX LCH DIVISION ALGEBRAS WITH CONTINUOUS IN-
VERSE. (Gelfand-Mazur) Let X be a complex LCH algebra with
continuous inverse. Then (a) for any x € X, a(x) / 0 and (b)
if each non-zero element in X has an inverse, X is topologic-
ally isomorphic to C.
Proof. (a) For x с X, suppose a(x) = 0. Then for each Age,
letting rx(A) = (x-^e) 1, the map A->rx(A) is seen to be en-
tire by part (b) of (4.8-5). We show that г^(А) is bounded.
To this end, consider the filterbase formed by the sets Bn =
{A c c||A| > n}, n=l,2,... . For any seminorm p € P, where
P generates X's topology, A g Br, and any x € X, p(A 1x) =
|А Г1p(x) <n 1p(x). Thus, letting <3 1 = (B 1) , it follows
that (since p is arbitrary) (0 ^х+О. Hence е-Ф^х^е. Since
X has continuous inverse and since e-A 1X must be regular
for any A / 0, it follows that (е-Ф^х) 1->-e 1=e which im-
plies pHe-fS^x) l)->p(e) for any p € P.
Thus, for e > 0, n sufficiently large and |A| > n,
(*) p((x-Ae)-1)=|A|-1p((e-x/A)-1) < |A|-1p(e)+e < (l/n)p(e)+e,
the upshot of it all being that p(rx(A)) is bounded for |A|
sufficiently large. Since rx is a continuous function,
4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM 213
p(rx(X)) is certainly bounded for |A| < n. Now, since p is
arbitrary, it follows that r is bounded, i.e. r (C) is a
X	X ~
bounded subset of the LCHS X. That is r is bounded entire
x
function, so it must be constant by Liouville's theorem (4.9-1).
Certainly P'rx is a bounded function too for any p € P
and (*) implies that p(rx(A))=0 for each Л € C. Since p is
arbitrary and X is Hausdorff, it follows that rx(A)=0 for
each A which is absurd for how can 0 be the inverse of any-
thing? Thus the assumption o(x)=0 has led to a contradiction
and the desired result, (a), follows.
(b) According to the result just proved in (a), given any
x € X, x-Ae is singular for some A. But if each non-zero ele-
ment is regular, it must be that x-Ae=0 for some A, that is,
x=Xe for some A. Thus X=Ce and the map x=Ae+A is clearly
an algebra isomorphism from X onto C. As for its being a ho-
meomorphism as well, suppose P is a saturated family of semi-
norms which generate X's topology. Since P is saturated, basic
neighborhoods of 0 are of the form a where a > 0 and pc P.
Now x = Aec aVp iff p(Ae)=|A|p(e) < a. That is, for non-
trivial seminorms p, aVp={Ae||A| < a/p(e)} and the fact that
the above map is a homeomorphism is now clear.V
As a consequence of Theorem 4.9-1 and the fact that any
LMC algebra has continuous inverse [(4.8-6)], the only complex
LMCH division algebra is C. We can go slightly further and say
that the only complex LMC division algebra is C. To see this,
it suffices to show that o(x)/i)3' for any x in a complex LMC
division algebra X (with nontrivial topology). As the topology
on X is not the trivial topology there exists a proper balanced
214
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
m-convex neighborhood of 0 in X. Consequently there exists a
nontrivial multiplicative seminorm p on X, and an associated
factor algebra (See Sec. 4.5)
Since X is a Banach alge-
X .
P
bra, o(x+Np) / 0 for any x € X by the previous theorem.
Since the map x-t-x+N^ is a nontrivial homomorphism from X into
Xp, then ц € o(x+Np) implies и € a(x).
For a time it seemed that there were probably no complex
topological division algebras (cf. Kaplansky, 1948, p. 811)
other than C. Williamson (1954) however showed that this was
not the case by exhibiting a topology for the algebra C(t), the
quotient field of the polynomial algebra £[t] of polynomials in
t with complex coefficients, which is compatible with the alge-
braic operations. We present this construction in our next ex-
ample .
Example 4.9-1 A COMPLEX TOPOLOGICAL DIVISION ALGEBRA DISTINCT
FROM C. Let M(0,l) be the space of all Lebesgue measurable
functions on (0,1) that assume finite complex values almost
everywhere. The quotient field of the algebra of polynomials
in t, C(t), may be identified with the class of functions of
M(0,l) consisting of ratios of polynomials with complex co-
efficients. Let 6 be the filterbase of sets of the form
B(k,e) = {f€M(0,l)|m(I(|f| > k) ) <e} where m is Lebesgue
measure, the numbers e and к are positive, and I ( |f| > k) =
{t € (0,1)||f(t)| > k}.
First we claim that ft is a neighborhood base at 0 in
M(0,l) for a compatible topology for the algebra M(0,l) called
the topology of convergence in measure. A filterbase^ on
M(0,l) converges to 0 (in the topology of convergence in meas-
4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM 215
measure) if for any B(k,e) ۩ there exists F such that
F с В(к,e), i.e. for each f € F,m(I(|f| > к) < e.* To show
compatibility, we show that the conditions of the Theorem 4.1-1
are satisfied by®. Changing the order of things slightly we
consider first the condition that to each B(k,e) € 63 there
corresponds a neighborhood B(k^,e^) with the property that
1/2
B(ki'ei) ВСкд^ер cB(k,e). If we choose k^=k ' and £^=£/2
then |f(t)g(t)| > к implies either |f(t)| > k^ or |g(t)| >
кду where f and g €M(O,1), and it follows that I(|fg| > k)
cz I ( | f | > k^J U I ( | g | > k1> . Thus if f, g € Bfk^^^^^) both
sets of the right side of the foregoing inclusion are of meas-
ure less than and fg € B(k,e). As for compatibility with
the linear structure, we note first that ® is closed with re-
spect to multiplication by positive scalars. The conditions
that the sets of® be balanced and that to each B(k,e) there
corresponds a neighborhood B(k2,£2) such that В ^2,62) +
B(k2,£2) cB(k,e) are established with the aid of the rela-
tions I ( I AfI > k) <z I ( | f | > k) for f € M(0,1) and | A | <1 and
I ( I f+g I > к) c I ( I f I > |) и I ( I g I > y) for f and g € M(0,1).
To see that the elements of ® are absorbent, let B(k,e) €
and f € M(0,l). Then the sets I(|f| > n), n > 1, form a de-
creasing sequence of measurable sets, each with finite measure,
and having an intersection of zero measure. Thus m(I(|f| > n))
+0 and, for A with large enough absolute value, we have
* In the case of a sequence (fR) from M(0,l) , f^O in the
topology of convergence in measure iff fn~>0, "in measure" in
the classical measure-theoretic sense of the phrase.
216
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
|A|к > n where n is a fixed integer with the property that
m(I(|f| > n) ) < e. Consequently for all such A
m(I(| (1/X)f| > k)) = m(I(|f| > |X|k)) < m(I(|f| > n)) and
(1/A)f € B(k,e).
Now jC(t) with the subspace topology (also called the topo-
logy of convergence in measure) is a topological algebra and we
contend that it is Hausdorff. To see this let r(t) =
P(t)/q(t)g B(k,e) for each к > 0 and e > 0. Then for any
к > 0, it follows that m(I(|r| > k) =0. Thus m({t € (0,1)|
r(t) / 0}) < £ m(I(|r| > 1/i)) = 0. Hence p(t) = 0 almost
i=l
everywhere on (0,1) so by the continuity of p(t) on (0,1), p(t)
is identically zero.*
The topology of convergence in measure, however is not
locally convex. To prove this it is sufficient to prove that
inversion is continuous at e (the function identically equal to
1 on (0,1)) for if we assume that C(t) is locally convex then
the absurd conclusion follows, by Theorem 4.9-1 (b) that C_(t) is
isomorphic to C. To show that C(t) has continuous inverse let
*On all of M(0,l), the topology of convergence in measure is
not a Hausdorff topology, because there are non-zero measurable
functions that are zero almost everywhere. To avoid this prob-
lem one usually defines M(0,l) to be the class of all equiva-
lence classes of measurable functions on (0,1) generated by the
equivalence relation of equality almost everywhere. The equiva-
lence classes containing a rational function contain exactly
one rational function andC(t) may still be identified with a
subalgebra of M(0,l). Furthermore, if f€ B(k,e) then so does
every other function in the equivalence class generated by f.
4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM 217
B(k,e) we shall exhibit k' >0 such that 1/f € e +
B(k,e) whenever f € e + B(k',e). This will be accomplished if
k' satisfies (*) {t € (0,1) |f(t) / 0}П I(|l/f-e|> к) c
I(|f-e| > k'). Treating k' as an unknown and assuming that
f(t) / 0 while |f(t) - 1| < k' it follows that |l/f(t) - 1|=
|(1/f(t))(1-f(t))| = f1/f(t)||f(t) - 1| <|l/f(t)|k' . To in-
sure that the above inclusion holds we set |l/f(t)|k' < k.
Since |l/f(t)|к' < к iff k' < |f(t)|k and |f(t)| > 1-k' we
have k' < k/(l+k). Any such k' satisfies (*).
One might wonder: could there exist locally convex complex
division algebras other than C? There can, and this is dis-
cussed in Exercise 4.6.
For multiplication to be continuous in C(t) endowed with a
linear topology -O', x/can be neither too coarse nor too fine: As
is discussed in Exercise 4.7, if<7' is locally convex, it cannot
be a weak topology—i.e. there can be no linear space Y such
that (C(t),Y) is a dual pair for which 41 = a(C(t),Y)—and
multiplication is discontinuous when C(t) carries the finest
locally convex topology.
In addition to Theorem 4.9-1 being interesting in its own
right it plays an important role in the development of "Gelfand
theory", roughly the consequences of topologizing the set of
maximal ideals of an algebra in a certain way (see Sec. 4.10
and Sec. 4.12)).
*Con't: Thus the sets of ® may be considered as being com-
posed of equivalence classes rather than just functions, there-
by making M(0,l) a Hausdorff topological algebra in the topo-
logy of convergence in measure.
218
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
The analog of Theorem 4.9-1 for real algebras is presented
next. As usual, consideration is restricted to commutative al-
gebras. If the (real) algebra is non-commutative however, it
must be the quaternions, as discussed in Exercise 4.8.
Theorem 4.9-2 REAL LCH FIELDS WITH CONTINUOUS INVERSE. Every
real LCH algebra X with continuous inverse in which each non-
zero element has an inverse is topologically isomorphic to
either R or C,. Furthermore X is topologically isomorphic to R
2	2
iff for all x,y e Xz x +y =0 implies x=y=0; in this case X
is called formally real.
Proof. We consider separately the cases when X is formally
real and when it is not. When X is formally real, we introduce
operations to XxX (identical in form to the operations one
introduces to RxR to form	which make it a complex LCH al-
gebra with continuous inverse and then apply Theorem 4.9-1 to
conclude that Xxx is topologically isomorphic to C; the re-
striction of this topological isomorphism to X x {0} shows X
to be topologically isomorphic to R. If X is not formally real
then X is shown to be topologically isomorphic to C.
Suppose that X is formally real and let Y = Xxx. For
(x,y), (w,z)eY and a+ib e C we define (x,y)+(w,z)=(x+w,y+z)
(a+ib)(x,y)=(ax-by,ay+bx) and (x,y)(w,z)=(xw-yz,yw+xz). With
these operations and the product topology Y is easily seen to
be a complex LCH algebra. (To verify that complex scalar mul-
tiplication is continuous, it is helpful to observe that the
map (x,y)->(y,-x) is continuous.) To see that Y is a field,
note first that (e,0) is the multiplicative identity of Y and
22	22-1
suppose that (x,y)/(0,0). Then x +y / 0, (x +y ) exists in
4.9 TOPOLOGICAL DIVISION ALGEBRAS AND THE GELFAND-MAZUR THEOREM
219
22-1	22-1
X and (x,y) [ (x,-y) ( (x +y )	,0)] = [(x,y)(x,-y)]((xZ+yZ) ,0)
22	22-1
= (x +y ,0)((x +y )	,0) = (e,0) so that (x,y) is invertible
and Y is seen to be a field.
In order to be able to call upon Theorem 4.9-1 to conclude
that Y is topologically isomorphic to C, it only remains to
show that Y has continuous inverse. To this end suppose that
<(х^,у^)) is a net convergent to (e,0). Then x^->-e and y^->-0.
-1	22-1
As	=	> '°) for each and x has
continuous inverse, it follows that
: , у ) 1
U Jir
as well
Thus Y has continuous inverse and is topologically isomorphic
to C as established by the map a+ib-*- (a+ib) (e,0) = (ae,be) of
Theorem 4.9-1. By restricting the map to X—more precisely to
Xx{0}—it follows that X is topologically isomorphic to R.
What if X is not formally real? i.e. there are nonzero
2 2	-1
x,yeX such that x +y =0. If so, let j=xy and extend mul-
tiplication to multiplication by complex scalars as follows:
(a+ib)z = az + b(jz) (ze X). It is easily verified that X
is now a complex LCH algebra. Theorem 4.9-1 may now be applied
and it follows that X is topologically isomorphic to С.. V
Following the proof of Theorem 4.9-1 we remarked that the
only complex LMC division algebra with nontrivial topology is
C. An analogous statement can be made for real algebras: the
only real LMC fields with nontrivial topology are R and C_.
Just as in the complex case, the fact that the topology on the
real LMC field X is nontrivial guarantees the existence of a
proper balanced m-convex neighborhood of 0 in X. Thus there is
a nontrivial multiplicative seminorm p on X and an associated
factor algebra X₽. Now X/Np=F(e+Np) where F=R or C and the
220
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
mapping у(e+N₽)	is a topological isomorphism. It follows
that the mapping h: X->-X/Np->-F , x->-x+Np=y (e+N^) ->-ц is a nontriv-
ial homomorphism. Since all nonzero elements of X have in-
verses, h 1(0)={0} and h is an isomorphism. That h is con-
tinuous is clear; that it is also open follows from the open-
ness of the canonical homomorphism x->-x+Np.
4.10 Maximal ideals and Homomorphisms.
A proper ideal which is not properly contained in any
other proper ideal is a maximal ideal. If I is a proper ideal,
a straightforward Zorn's lemma argument shows that there ex-
ists a maximal ideal containing I. In particular, if x is a
singular element then (x)=xX is an ideal containing x called
the principal ideal generated by x. Thus any singular element
is contained in a maximal ideal. In this section we prove two
basic topological results about ideals in certain types of top-
ological algebras and then investigate the connections between
maximal ideals and homomorphisms. We also look at some exam-
ples .
(4.10-1) IDEALS IN Q-ALGEBRAS. If X is a Q-algebra then (a)
the closure of a proper ideal is a proper ideal; (b) maximal
ideals are closed.
Proof. Clearly we only need to prove (a). If I is an ideal in
X, x,y € cl I and z € X, it is easy to show by considering
filterbases on I convergent to x and у that x+y and xzf cl I.
The important part is to show that cl I/X. Since X is a Q-
algebra, however, there exists a V c V(e) such that V con-
sists entirely of units. Thus VC1 1=0 and e cl I.V
In the algebra C(R,F,c) as in Example 4.8-1, the set I
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
221
of continuous functions x which vanish outside some compact set
Gx constitutes a proper ideal which is clearly dense in
C(R,J?,c). Thus no maximal ideal containing I can be closed.
In the result which follows, we consider (maximal) ideals
I of an LMC algebra X such that the quotient topology on X/I is
not the trivial topology. If I is a closed ideal (as is the
case for maximal ideals in Q-algebras), then the quotient topo-
logy on X/I is Hausdorff, so the quotient topology is certainly
not trivial in this case.
A weaker sufficient condition for nontriviality of the
quotient topology when X is LMC is that I not be dense in X.
More generally, if H is a linear subspace of the locally con-
vex space X and cl H/X, then the quotient topology on X/H is
not trivial and we now outline a proof of this fact. Indeed if
H is not dense in X, then there is a convex neighborhood of the
origin V and an element x С X such that (x+V) Cl H = 0. If
the quotient topology on X/H is trivial, then (x+H) + V = X =
-x-V+H (since (x+V)+H is a neighborhood of x+H in X/H). It
follows that there exist elements v,w e V such that x+v/2+
w/2 e H n (x+V), and this is a contradiction. As was mention-
ed after (4.10-1), C(R,F,c) is an algebra containing dense
ideals.
(4.10-2) QUOTIENTS OF MAXIMAL IDEALS IN LMC ALGEBRAS. In the
event that X is a (commutative) real LMC algebra and M is a
maximal ideal in X such that the quotient topology on X/M is
not trivial then X/M is topologically isomorphic to either R or
C. In particular if X is a Banach algebra then X/M is topolo-
gically isomorphic to R or C if X is a real algebra or just C
222
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
if X is a complex algebra. Furthermore the quotient topology
on X/M is induced by the inf norm: ||x+M || = infmeM |jx+m ||.
Proof. As regards the main assertion all we need do is apply
the remarks following Theorem 4.9-1 and Theorem 4.9-2 to the
LMC division algebra X/M. If X is a Banach algebra then it is
LMC and, being also a Q-algebra [(4.8-2)], the maximal ideal M
is closed in X from which it follows that the quotient topology
on X/M is Hausdorff, hence nontrivial. It is easy to verify
that the quotient topology is induced by the inf norm.v
As mentioned before (4.10-2) it suffices for the maximal
ideal M to be closed to guarantee that the quotient topology be
nontrivial. In a Banach algebra, all maximal ideals are
closed but this is not generally true for LMC algebras. There
may be non-closed maximal ideals even in Frechet algebras, as
shown by the discussion of C(R,F,c) after (4.10-1). This
property—existence of non-closed maximal ideals—constitutes,
therefore, a major difference between LMC algebras and Banach
algebras.
Notation. "Homomorphism" here means "complex or real homomor-
phism", depending on whether the algebra is real or complex,/'/
(orM(X)) denotes the maximal ideals of an algebra Х,Л/с» (or
ЛС(Х)) denotes the closed maximal ideals of a topological al-
gebra X and X^ denotes the nontrivial continuous homomorphisms
of the topological algebra X. Note that X^ с X', the continu-
ous dual of the topological vector space X.
Our first result connecting the notions of maximal ideal
and homomorphism is obvious.
(4.10-3) MAXIMAL IDEALS AND HOMOMORPHISMS. The kernel of any
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
223
non-trivial homomorphism of any algebra X is a maximal ideal.
Clearly the kernel of any non-trivial continuous homomor-
phism is a closed maximal ideal in any topological algebra. It
follows immediately from (4.10-2) and the remark following
(4.10-2) that any closed maximal ideal in an LMC algebra is the
kernel of a continuous homomorphism. We state these facts in
our next result.
(4.10-4) Mc IN LMC ALGEBRAS. If X is a complex LMC algebra
then a maximal ideal Mcz X is closed iff M is the kernel of
some continuous complex-valued homomorphism. If X is a real
LMC algebra then M is closed iff M is the kernel of a real-val-
ed homomorphism or a complex-valued homomorphism. Thus for
complex LMC algebras there is a 1-1 correspondence between Xc»
the closed maximal ideals, and x\ the continuous homomor-
phisms, namely that established by pairing M with the homomor-
phism x->-x+M.
Frequently it will be convenient to identify Xc and
when X is a complex LMC algebra. Thus, for example, we speak
of Д "with о(X’,X)-topology". An examination of the basic
neighborhoods of 0 shows that this is the weakest topology on
Ж with respect to which each of the Gelfand maps
xsA( -*C, M-+X+M, (or X^-+Q, f+f(x)) continuous; the topology
induced by o(X',X) on Xh (or Ж ) is called the Gelfand top-
ology.
In Q-algebras all maximal ideals are closed [(4.10-1)], so
in complex LMC Q-algebras each non-trivial homomorphism is
continuous by (4.10-4). Furthermore, if X is a Banach algebra,
real or complex, and f is a non-trivial homomorphism (real-val-
224
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
ued if X is real, complex-valued if X is complex) on X, then
for any xg X we have f(x-f(x)e) = 0 which implies that
f(x)eo(x). Thus, by the remark following, (4.8-4), |f(x)| <
||x |j for every x and it follows that ||f || < 1. We summarize
these facts in our next result.
(4.10-5) HOMOMORPHISMS OF LMC Q-ALGEBRAS ARE CONTINUOUS. If X
is an LMC Q-algebra, all homomorphisms are continuous. Fur-
thermore, each homomorphism on a Banach algebra is not only
continuous, it has norm less than or equal to one.
Questions concerning the continuity of homomorphisms of
topological algebras into topological algebras are treated in
Section 4.13. In particular a result similar to (4.10-5) is
established in Theorem 4.13-1 where it is shown that any homo-
morphism of a complete barreled LMCH Q-algebra into a strongly
semisimple fully complete LMCH algebra is continuous.
Unfortunately (4.10-5) does not remain true for an arbi-
trary LMC algebra. Our next result is of critical importance
in Example 4.10-1 where we construct a whole class of LMCH al-
gebras on which discontinuous homomorphisms exist.
(4.10-6) DISTINCT HOMOMORPHISMS ARE LINEARLY INDEPENDENT. Any
collection of distinct non-trivial homomorphisms of an algebra
X is linearly independent.
Proof. Suppose {f^,...,fnJ is a linearly dependent set of
distinct non-trivial homomorphisms of X into ^F which is mini-
mal in the sense that the removal of any one homomorphism from
the set leaves a linearly independent set. It is clear that
n/1. Suppose that Za^f^=0 where no a^=0. Since
there exists ye X such that fn<Y) / f1(y). Holding у fixed
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
225
and permitting x to be any vector in X we see that
Zn .a.f.(xy) = Zn na.f.(y)f.(x) = 0, and f (y)Zn na.f.(x) =
(У)(x) = 0- Subtracting the second equation from the
first, we have	(У) “fn(У) )	(x) = 0 for all xe X.
Since fj(y)_f (У) / 0 the last equation implies that
{f , fn_]_J is linearly dependent which condradicts the
minimality of {f.}. V
Example 4.10-1 DISCONTINUOUS HOMOMORPHISMS. Let X be an alge-
bra and H be a family of non-trivial homomorphisms on X with
the property: (*) there exists f € H such that j.A ker f =
°	f e Ho
{O}, where H =H-{f }. In this case f will be a discontinu-
o о	о
ous homomorphism when X carries a(X,HQ).
There are many algebras which satisfy (*). In particular
if X is the complex algebra of continuous functions on [a,b],
C([a,b],F,c) with sup norm topology, then let H be the collec-
tion of homomorphisms determined by the evaluation maps
t* :C [a,b]-C (te [a,b]) , x->-x(t) and f to be t *. To see
~	о	о
that (*) is satisfied it is enough to note that a continuous
function x which vanishes for all te [a,b]-{t } must also van-
o
ish at t . Since the essential feature of t here is that it
о	о
is not an isolated point of [a,b] it is easy to see that,
more generally, we can take x to be any B-* algebra, i.e. an
algebra of continuous complex-valued functions C(T,C,c) on a
compact space T with sup norm topology, and t to be any point
of T which is not an isolated point.
Letting [H ] denote the linear span of Hq in X*, it
follows by (*) that (X, [HQ]) is a dual pair. Furthermore,
since the seminorms p^ defined at each xe X by p^(x)=|f(x)|,
226
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
f€ HQ, which generate o(X, [HQ]), are clearly multiplicative,
(X,о(X,[H ])) is an LMCH algebra. In view of that fact that
[HQ] is the continuous linear dual of (X, a(X, [H ]) ) , it fol-
lows that any non-trivial continuous homomorphism g of X must
belong to [HQ]. If we assume that g HQ itself then
HQ U {g} is linearly independent by (4.10-6) and, therefore,
[HQ] is a proper subset of [H и {g}]. But this is ridicu-
lous, so gc Hq. Thus X^=Hq and, consequently, f X^.
As it happens the topological algebras of Example 4.10-1
are rarely, if ever, complete (see Exercise 4.5). An example
of a complete LMCH algebra on which discontinuous homomorphisms
exist is afforded by C(T,R,c) where T=[0,R) and fi is the
first uncountable ordinal. T is not replete (Example 1.5-1) so
there are homomorphisms of C(T,R,c) which are not evaluation
maps by (1.6-1). But all the continuous homomorphisms of
C(T,R,c) are evaluation maps as shown in Example 4.10-2, so
discontinuous homomorphisms exist on C(T,R,c). As T is an
open subset of the compact space [0,R], it is locally compact
and therefore also a k-space (Exercise 2.2(b)). Hence by Theo-
rem 2.2-1, C(T,R,c) is complete. Other examples of complete
LMCH algebras on which discontinuous homomorphisms exist are
given in Exercise 2.2(g). Nevertheless there is still a close
relationship between M. and in complex LMCH algebras; e.g.
uM = uAc (see (4.10-9) and Г(М = ClA<c (see (4.11-1)).
Our next result is another example of how information a-
bout an LMC algebra can be gleaned from the factor algebras.
It asserts that the continuous homomorphisms of an LMCH alge-
bra can be "obtained" from the (continuous) homomorphisms of a
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
227
set of factor algebras. First we state some conventions about
certain things in topological vector spaces.
If X is a topological vector space then the continuous
dual X' is the linear space of all continuous linear function-
als on X. The polar B° of a subset В of X is the set of x'eX'
such that sup|<B,x'>| < 1. The adjoint A' of a linear map
A:X->-Y where X and Y are topological vector spaces is defined
at y'eY' according to <Ax,y ,> = <x,A'y'> .
(4.10-7) X11 FOR LMCH ALGEBRAS. Let X be a LMCH algebra with
topology generated by the saturated family (p^) of seminorms
with associated factor algebras (X^). Then: (a) For each index
U, 1(X h)=Xhn V/ where is the canonical homomorphism
x~>-x+N^ (see Sec. 4.5) and V^={x£ x|p^(x) < 1}; (b) X^=
цк^' (X ; (c) each K^'(X^h) is compact in the Gelfand topology
of X^ (i.e. the relative o(X',X)- topology) and к ' (X^*1) and
X are homeomorphic via the correspondence	when
each space carries its Gelfand topology.
Proof. (a) Let f€ k^' (X^) so f=K^'	=	for some
f С X h. Then for each хе X we have f(x)=f (к (x))=f (x+N ).
Thus |f(x) | = |f^(x+N^)|	<; ]5^(x+N^) = p^ (x) since any homomor-
phism f on a Banach algebra is continuous with norm one. Cer-
tainly then |f(x) | <: 1 whenever xe and so ft V^°.
Then, as N^cV^, f must vanish on N^ and we may unambiguous-
ly define the nontrivial homomorphism g^:X/N^+F by the equa-
tion g^(x+N^)=f(x). Since p^ (x+N^)=p^(x) for each xf X,
the boundedness of f on implies the boundedness of g^ on
V. ={k (x) e X/N Ip (к (x)) < 1}. Thus g is a continuous non-
Рц U c D lhD D
trivial homomorphism of X/N^; hence it can be extended by
228
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
continuity to a nontrivial homomorphism fp e X^\ For апУ x€ X
we have f(x) = д^(к^(х)) = f^(K^(x)) = (^(f ))(x) and it
follows that f = к' (f ) € к' (X h) .
У У li li
(b) It sufficies to show that = и X^n V °. If f£ J?
li li
then f is bounded on for some ц. Suppose that |f(x)| > 1
for some xf V . Then the contradictory facts follow that
{ | f (x11) | | n e N} is unbounded while x11 e for all n e N.
Hence If(x)I < 1 for each xf V	and fg Q V °.
1	li	li
(c) It is easy to see that X^ is closed in the continuous
dual X' of X so that the о(x1,X)-compactness of V^° [the polar
of a neighborhood of 0 in any topological vector space X is al-
ways о(X1,X)-compact by Alaoglu's theorem] and part (a) com-
bine to show that (X^) is о (X1 , X)-compact. As X is
Hausdorff in its Gelfand topology it suffices to show that
к 'is well-defined to conclude that it is a homeomor-
phism (it is clearly injective). Thus suppose that к '(f ) =
к '(g ) so that f (x+N ) = (k 1(f ))(x) = (k 1(g ))(x)=g (x+N )
li li	li li li li	lili	lili
for each x+N e X/N . As X/N is dense in X , f =g and the
li li	li	li li li
proof is complete.?
As an application of (4.10-7), in our next example we de-
termine all the continuous homomorphisms of C(Tf<F,c) when T
is a completely regular Hausdorff space. Note that the result
is trivial if T is replete since in that case all the homomor-
phisms of C(T,F) into F are evaluation maps by (1.6-1) and
evaluation maps are always continuous when C(T,F) carries the
compact-open topology.
Example 4.10-2 C(T,£,c)h=T* FOR COMPLETELY REGULAR HAUSDORFF T.
Since T is a completely regular Hausdorff space, a set of fac-
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
229
tor algebras for C(T,F,c) is the set of Banach algebras
C(G,F,c), where G is a compact subset of T [see Example 4.5-1}
We claim that C(T,F,c)^=T* follows from (4.10-7) provided we
can show that C(G, F,c)^=G* for each compact set G. To see
this let f€ C(T,F,c)h; then by (4.10-7) and the statement
"C(G,F,c)h=G* for each compact G", there is a compact set Gc T
and a point tc G such that for any xf C(T,F,c) f(x) =
(к '(t*)(x)=t*(к (x))=t*(xI )—X(t).
b	b	b
To show that C(G,F,c)^=G*, when G is compact, it suffi-
ces to note that G* constitutes all the homomorphisms, for all
homomorphisms on a Banach algebra are continuous, [(4.10-5)].
That G* constitutes all the homomorphisms follows immediately
from (1.6-1).V
Example 4.10-3 H^=D*. Consider the LMCH algebra H of analytic
functions on the open unit disc D of the complex plane carry-
ing the compact-open topology (See Examples 4.5-2 and 4.7-2).
230
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
We have already seen (Example 4.5-2) that H is a Frechet alge-
bra with a set of factor algebras (P(Cn)) where p(cn) con-
sists of all uniform limits of polynomials on Cn={teC||t|<
1-1/n] endowed with sup norm topology. Thus, as in the pre-
vious example, once we show that the continuous homomorphisms
of p(cn) are just the evaluation maps t* for teCn it follows
that H^=D*. Moreover, if t/s, then t*/s* for they differ on
the polynomial x(u)=u-t. Hence is in 1-1 correspondence
with D.
To see that the continuous homomorphisms of P(Cn) are just
evaluation maps let heP(Cn)^ and set t=h(y) where the poly-
nomial y(u)=u. Since yeP(Cn), ,||y|^l-l/n, and || h ||=1 since
P(Cn) is a Banach algebra, then t=h(y)eCn« For x(u) =
^v_navuk we see that x = a yk and h(x) = a, (h(y))k=
п к
Z^_Qa^t =x(t)=t*(x). Thus h=t* on polynomial functions on Cn>
Furthermore if xeP(Cn) there is a sequence (xn) of polynomi-
als which converges uniformly to x. Hence by the continuity of
h, h(x)=lim h(x )=lim x (t)=x(t) and h=t*.V
' n n n n
It is well known that in any complex Banach algebra the
spectrum of an element x is just the set of values f(x) as f
runs through the (continuous) homomorphisms or, put another
way, the range of the Gelfand map x (see the remark immediately
following 4.10-4). With the aid of the Arens invertibility
criterion [Theorem (4.6-1)] we generalize this in our next re-
sult to complete complex LMCH algebras.
(4.10-8) a(x) = Xh(x) IN COMPLETE COMPLEX LMCH ALGEBRAS. If X
is a complete complex LMCH algebra and x is any element in X
then a(x) = {f(x)|fe X^}.
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
231
Proof. For any xg X and f€ X*1, f (x-f (x) e) =0. Hence for any
such f, x-f(x)e is not invertible in X, and, therefore,
{f(x)|fe X^Jc o(x). To obtain the reverse inclusion let
Ag a(x); then for some factor algebra X^, the canonical image
(x-Ae) is not invertible in X^ by Theorem 4.6-1. Consider
a maximal ideal M с X containing к (x-Ae). As X is a Ban-
11 U	U	U
ach algebra, M^ is closed (by(4.8-2) and (4.10-1)) and is in
fact the kernel of a continuous nontrivial homomorphism f of
X by (4.10-4). Thus f (к (x-Ae))=0. Surely f=f к is a
continuous nontrivial homomorphism of X and, as f(x-Ae)=0,
A=f (x) . Thus o(x)c {f(x) | f € X^ }. V
We already know if x is a singular element then x lies in
some maximal ideal. Our next result shows that in complete
LMCH algebras, every singular element lies in some closed maxi-
mal ideal.
(4.10-9) uM = иЛ(с IN A COMPLETE COMPLEX LMCH ALGEBRA. If X
is a complete complex LMCH algebra, then uM = uXc-
Proof. Clearly иД(сс uM- If xf иЖ then x is not inverti-
ble and 0e o(x). Thus by (4.10-8) there exists an fe Xh such
that f(x)=0. Since the kernel of f is a closed maximal ideal,
x € UMC-V
(4.10-9) shows that any proper principal ideal (z) =
{xz|xeX} in a complete complex LMCH algebra X can be embedded
in a closed maximal ideal. Some further information on when an
ideal can be embedded in a closed maximal ideal is given in
(4.10-10) below.
(4.10-10) EMBEDDING IDEALS IN CLOSED MAXIMAL IDEALS. Every
proper nondense ideal I in a complex LMCH algebra X can be em-
232
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
bedded in a closed maximal ideal.
Proof. Suppose I
of multiplicative
is not dense in X and P is a saturated family
seminorms generating the topology on X. Since
I is not
dense in
for some
I+N in
P
e > 0 where V :
P
If xel and
X.
X, there is a peP such that in (e+eVp)=0
Consider the ideal
Hence (I+N ) n (e+eV )=0
p	p
noting the factor algebra
yeNp then
and J=I+N
P
(Def. 4.5-1)
p(x+y-e)=p(x-e)> e.
is not dense. Now de-
associated with p
it is clear that clv (k(J)) is
p
canonical homomorphism of X into
an ideal in X
P
(where к
by X ,
P
is the
it is proper.
Indeed if we make
Xp). Moreover
the
assumption
we claim
that
to the contrary
then e+Np e
cl (к (J)). Thus, with
X
p
elements x el and yeN such that
e	P
e as above, there must be
p((x£+y+Np)-(e+Np)) < e.
However p(x£-e)=£>((x£+Np)-(e+Np))=p((xe+y+Np)-(e+Np))< e and
this is a contradiction. Since cl (к(J)) is a closed proper
XP
ideal in the Banach algebra X₽ there is a (perforce closed)
maximal ideal M z>cl (k(J)). We contend
p Л
p
the desired maximal ideal. Certainly M is
that М=к 1(M ) is
P
a closed ideal con-
taining I; it only remains to show that it
is maximal. Letting
h be the surjective complex homomorphism of X₽ determined by
Mp, it immediately follows that M=k ^(h ^(0)) and M is the
kernel of the nontrivial homomorphism h*K. Thus M is maximal
and the proof is complete.?
As mentioned earlier, an immediate consequence of (4.10-9)
is that any proper principal ideal in a complete complex LMCH
algebra X can be embedded in a closed maximal ideal. It is
natural to inquire if this is true for any finitely generated
ideal (z^,...,zn)={Zx^z^|x^e X,i=l,...,n}. In other words,
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
233
when is (z^,...,z ) proper? With the added condition of me-
trizability—i.e. if X is a complex Frechet algebra--then this
is so, as is proved in (4.10-12). First, however, we must es-
tablish the following technical fact.
(4.10-11). Let h be a homomorphism between the topological al-
gebras X and Y such that h(X) is dense in Y, and h(e)=e. Given
elements z.,...,z e X, suppose that elements u, ,. ..,u e X
I n	I n
exist such that Zu^z^=e. It follows that there exist
У1,...,УП€ Y (y,=h(u^) for example) such that Zy^h(z^)=e.
Then for any such	an^ neighborhoods □	of
0, there exist v,,...,v € X such that Zv.z.=e and h(v.)€
I	n	11	i
y.+U. for i=l,...,n.
j i i
Proof. Suppose that Zu^z^=e for u^,...,u e X. Then setting
v^=w^+ui (e-lWjZj) for i=l,...,n, it follows that Ev^z^e
regardless of the choice of the w's. By the hypothesis
then, choosing У]/''''УП such that Zy^h(z^)=e, (a); h(v^)-y^=
(h(w^)-y^)+h(u^)h(e-ZjWjZ^) for each i. Now let V and W be
balanced neighborhoods of the origin such that V+Vc LL for
each i and Z? .WcV. Note that h (u . ) h (e-Z .w . z .) =
1=1	i' j J j
h(u^)(e-Zjh(Wj)h(Zj)). Since e=Zy^h(z^), the right-hand side
becomes
Zjh(u±)h(Zj)(yj-h(Wj)).
Thus, as multiplication is
continuous and h(X) is dense in Y we can
such that h(u^)h(Zj)(Yj-h(Wj)) e W and
choose w.,...,w
1	n
h(w.)-у. e V, for
i,j=l,...,n. By (a) and the choice of V and W it follows that
h (vp -yi e LL. V
(4.10-12). EMBEDDING FINITELY GENERATED IDEALS IN CLOSED MAXI-
MAL IDEALS. Let X be a complex Frechet algebra with topology
generated by the saturated family (p^) of seminorms (where it
234
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
is assumed without loss of generality that p^< P^+j for each
k) and with associated factor algebras (X^). Then (a)
(z.,...,z ) is proper in X iff (к z ,...,k z ) is proper in
X^ for some k; (b) any proper finitely generated ideal can be
embedded in a closed maximal ideal.
Proof. (a) If (z^,...,zR)=X then Ex^z^=e for some
x.,...,x s X. Hence Ек, (x.)к, (z.)=e+N, and it follows that
X	П	JC 1 JC 1	JC
(Kkzl'••'Kkzn>=Xk for each Conversely suppose that
(Kkzl'’’’' KkZn^=Xk f°r each ВУ an induction process we con-
struct a Cauchy sequence in X^ for each fixed i, 1 < i < n,
convergent to an element u^(k) such that (и^(к))к is an
element of the projective limit of the X^1s and E^u^(k)k^(z^)=
e+N^ for each k> 0: We conclude the proof by invoking Theo-
rem 4.6-1 to obtain elements u.eX such that k, (u.)=u.(k)
i	к i i
for each i and к as x^=u^ turns out to be a solution of
^ixizi=e"
We remind the reader that hrg(s> r) denotes the exten-
sion by continuity to X of the mapping x+N ->-x+N (see Theo-
S	SI
rem 4.6-1) and Kr the canonical homomorphism of X into Xr. We
choose y,°,...,y ° e X such that Е.у.°к (z.)=e+N . Pro-
J1 ' n о	i о i о
ceeding inductively we find that elements yim,...,ynm e xm
exist by (4.10-11) such that Е.у.тк (z.)=e+N and
m 1 m
₽m-l(hm-l,m(yim)-yim-1)< 1/2m (i=l,---,n)- m> k then,
since h, ,. h _	=h. and IJh. , || < 1,
k,m-l m-l,m km	Irk,m-1H —
( > ₽k(hkm(yi -hk,m-lyi ’’=₽k(hk,m-l(hm-l,m(yi ’“yi ”
pm-l(hm-l,m(yi >-yi ’ <
Set u^(k,m)=hkm(у^т) for m> k and 1 < i < n. Substituting
in (*) we obtain
4.10 MAXIMAL IDEALS AND HOMOMORPHISMS
235
p^_(1K (k,m)-tu (k,m-l))< l/2m for m> k. We now claim that
(и^(к,т))т> is a Cauchy sequence in X^.
Indeed if m> j> к
p (u.(k,m)-u.(k,j))< Z® p (u.(k,t)-u.(k,t-l))
Jy -L	X	Ц.- J “г ± К 1	1
we have
^™=j + ]_2	Since X^_ is complete there exists an element
m
u^(k) e X^_ such that u^(k,m)	u^(k). To see that the
"tuple" (u.(k)),	_ is in the projective limit of the X, 1s
1 К > и	К
for each i we observe that for m> к h, , , .(u.(k+l,m)) =
к,k+l i
h, , , . (h, , . (y m)) = h, (y.m) = u. (k,m) . Thus as m-«°, using
k,k+l k+l,m '' km J i ' i '	3
the continuity of h, . ., we see that h ,	(u.(k+1)) =
JC f JC“r J.	JC f Kt J. 1
u. (k) for each i so (u. (k) )
is indeed an element of the
projective limit for each i. The fact that Z^u^(k)(z^) =
e+N^ for each к also follows by taking a limit with respect to
m in the equation (recall that Km=K]< and the waY which
that y.m were chosen) Z.u.(k,m)K. (z.)=h. (Z.y.mK (z.) =
J i	ii'ki km iJ i m i'
h, (e+N )=e+N. . Finally elements u, ,...,u e X exist by
km m к	J	1'	' n	J
Theorem 4.6-l(d) such that K^(u^)=u^(k) for each k> 0, so
p (Z . u. z . -e) =p. (Z . u. (к) k. (z . ) -e+N. ) ) =0 for each k> 0 and the
rk ill к ii'ki к	—
concluding statement—Z^u^z^=e—follows since X is Hausdorff.
(b) If (z^,...,zn) is proper in X then, by (a), there is a
k> 0 such that ( k. z. , . . . , k. z ) is proper in X. . Since X, is
—	к 1'	к n	г г	к	к
a Banach algebra there exists a closed maximal ideal
M. zj ( k, z	k, z ). Clearly M=k, ^(M. ) is a closed ideal in
к к 1 к n	-1 к к
X containing (z,,...,z ). Since X, is a complex Banach alge-
J.H	К
bra there is a complex homomorphism h of X^ such that is the
kernel of h. Thus M is the kernel of k, *h and the maximality
к	-1
of M follows.?
236
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
4.11 The Radical and Derivations
The radical of an algebra is an ideal of great utility in
the study of the structure of topological algebras. Especially
sharp statements can be made when the radical is the zero ideal
For example, if the complex algebra X has {01 as its radical,
there is at most one topology on X with respect to which X is
a Banach algebra.
In this section we use the properties of the radical and
derivations to produce an example of a Frechet LMC Q-algebra
which is not a Banach algebra.
Definition 4.11-1 THE RADICAL AND SEMISIMPLICITY. The radical
Rad X, of an algebra X is the intersection of all maximal
ideals in X. If Rad X = (о), X is referred to as semisimple
while a topological algebra is called strongly semisimple when-
ever = {o).
Thus in a strongly semisimple complex LMC algebra where
there is a 1-1 correspondence between c and X*1, if every
continuous homomorphism vanishes on an element x, then x must
be 0. Moreover in such algebras, since the continuous homo-
morphisms seperate the points of X (i.e. if x/0, there is some
f C X^1 such that f(x) / 0) the topology on X must be Hausdorff;
Thus "strongly semisimple LMC" is the same as "strongly semi-
simple LMCH," for complex algebras.
Example 4.11-1 A STRONGLY SEMISIMPLE ALGEBRA. Consider the
algebra C(T,F,c), where T is a completely regular Hausdorff
space of Example 4.5-1. Since the continuous nontrivial homo-
morphisms of C(T,F,c) are just the evaluation maps on the
points of T (See Example 4.10-2) if хе П,МС then x(t) = 0 for
4.11 THE RADICAL AND DERIVATIONS
237
all te T and x=0.
Clearly X is semisimple whenever it is strongly semisim-
ple. We now show that in complete complex LMCH algebras, semi-
simplicity implies strong semisimplicity.
(4.11-1) RAD X = c IN COMPLETE COMPLEX LMCH ALGEBRAS. If X
is a complete complex LMCH algebra then Rad X = Г1)ИС- Thus a
complete complex LMCH algebra X is semisimple iff X is strongly
semisimple.
Proof. Certainly Rad Xc AMC* Suppose that xf ПМС* As
f(x)=0 for all f£ X^, f(e-xy)=l for each f£ X^ and ye X,
so by (4.10-8) e-xy is invertible in X for each ye X. If we
suppose that x^ M for some Me/4< then since, X is a commu-
tative ring with identity, X/M is a field and there exists ye X
such that (x+M) (y+M) = e+M. Thus e-xye M contradicting the
invertibility of e-xy. Thus xe Rad X and ClMccRad X.V
The notion of "derivation" defined below is purely alge-
braic. For the sake of the definition X needn't be commutative
or possess an identity.
Definition 4.11-2 DERIVATIONS. Let X be an algebra. A linear
map D:X->-X is a derivation on X if for all x,ye X, D(xy) =
xDy + (Dx)y.
Clearly the trivial linear transformation is a derivation
and so is the differentiation operator on spaces of infinitely
differentiable functions. Let X be a linear space and <^£(X,X)
the noncommutative algebra of all linear maps taking X into X
[where (AB)x = A(Bx)]. Fix Be3C(X,X) and define Dg(A)=AB-BA
for any Af^(X,X). The transformation AB-BA is called the
commutator of A and В and the derivation D is referred to as a
— - 	—  -	о
238
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
commutator operator. Finally, let E be a field and X the alge-
bra of all formal power series in a single variable t with co-
oo
efficients from E. That is, X = { E a tnI a g e) where addi-
n=0 n n
tion and scalar multiplication are performed componentwise
while multiplication is taken to be the Cauchy product.* Then
OO	oo
£ a tn * S na tn 1 is a derivation on X.
n=0 n	n=0 n
(4.11-3) CONTINUOUS DERIVATION MAPS A COMPLEX BANACH ALGEBRA
INTO ITS RADICAL. If X is a (not necessarily commutative) com-
plex Banach algebra and D a continuous derivation on X, then
D(X)c Rad X.
Proof. As usual L(X,X) denotes the complex Banach space of
continuous (=bounded) linear transformations on X. Since D is
bounded and L(X,X)
is absolutely convergent in L(X,X) for any age. We denote
aD
the sum of this series by e . Let f be any nontrivial homo-
.	аэ
morphism of X and define f to be f • e . Since f must be con-
tinuous (see the discussion proceeding (4.10-5)), f is a con-
tinuous linear functional on X. To see that f is in fact a
homomorphism we first note that since D is a derivation, a
"Leibniz rule" holds: for any positive integer n.
Dn(xy) _ 7 D^'x Р^у
n! i+j=n i! j!
Thus, since f is continuous.
l а» I t<lAx> {(D^y)
n=0 i+j=n i!	j!
* The Cauchy product
у = X . . a . g ..
n i+i=n 1 1
of
Sa tn and S3 tn
n	n
is
Sy tn where
n
4.11 THE RADICAL AND DERIVATIONS
239
o° i i °0 i “i
_.	. „a f(D x) v aJf(DJy) .	.
Since f (x) f (y) = 1 — r-: — T, -------------J-:—•LL- and the two
a a i=o i! j=o □!
series above converge absolutely, the product of these two
series equals the Cauchy product(*) and f (xy) = f (x)f (y).
01	01	01
Since ||f jj < 1 (see the discussion proceeding (4 .10-5))
anf(Dnx)
and D is	bounded the	series	fa(x)	=	  converges ab-
solutely	for each	fixed xf	X and	any	a. Thus the mapping
a->-fa(x) (with x held fixed) is an entire function of a. As
was the case for	f,	||fjj < 1/ and	it follows that ||fa (x) || <
||x |j for	each а €	C.	Hence	fa(x)	is	a bounded entire func-
tion of a and therefore is a constant by Liouville's theorem
[(4.9-1)]. It follows from the identity theorem for power
series that f(Dnx)=0 for all n>l; thus, in particular,
f(Dx)=0. Since f was any homomorphism we conclude that Dx C
Rad X.V
Next we state an immediate corollary of the proceeding
proposition for ease of reference.
(4.11-4) CONTINUOUS DERIVATIONS ON COMPLEX SEMISIMPLE BANACH
ALGEBRAS ARE TRIVIAL. If X is a semisimple complex Banach al-
gebra then the only continuous derivation on X is the trivial
one.
The fact that the commutator of a pair of bounded opera-
tors on a complex Banach space is never the identity transfor-
mation was first proved by H. Wielandt (1949-50). An indepen-
dent proof can be given for matrices using the elementary pro-
perties of the trace function on matrices.
(4,11-5) COMMUTATORS AND THE IDENTITY.! If X is a complex
+ The validity of the proof below depends on Theorem 4.6-1,
(4.10-7), (4.10-8) and (4.11-3), all of which remain true with-
out the assumption of commutativity as the reader may verify.
240
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Banach space and A,Be L(X,X) then AB-BA cannot be the iden-
tity map x-»-x on X.
Proof. Consider the derivation D :L (X, X) ->-L (X, X) , H-+HB-BH.
—	в
It is readily seen that DB is bounded ( |[DB(H) |[< 2 ||B || ||H ||) .
Since L(X,X) is a complex Banach algebra, it follows by
(4.11-3) that D (L(X,X))c Rad L(X,X). But 1^ Rad L(X,X), so
В
AB-BA = D„(A) / l.V
The results just obtained now enable us to exhibit a com-
plete barreled semisimple Q-algebra which is not a Banach alge-
bra.
Example 4.11-1 A BARRELED SEMISIMPLE FRECHET Q-ALGEBRA WHICH IS
NOT A BANACH ALGEBRA. Consider the algebra Z of all infinite-
ly differentiable functions on [a,b] of Example 4.3-2 with
the LMCH topology generated by the seminorms pn(x) =
sup, r u1|x(n,(t)|	(n=0,l,...). As was established in Ex-
ample 4.3-2, Si is a Frechet algebra. Since any complete metric
space is a Baire space any LCS which is a Baire space is bar-
reled (see Horvath, 1966, pp. 213-214), the algebra aS is bar-
reled. In a remark following (4.8-1) we proved that 5B was a Q-
algebra by showing that V = {x€jtj|p (x) < l}cQ. The fact
po
that <£} is semisimple is apparent from the observation that any
element xg Rad X must vanish on [a,b] as the evaluation
maps t :X> ^C, x->-x(t) are continuous homomorphisms on .
To show that is not a Banach algebra we have only to
exhibit a nontrivial continuous derivation on by (4.11-4).
Clearly the differentiation operator D:^->-aO, x*x' is a non-
trivial derivation and it is continuous since D ^(V	)=V
pn	pn+1
for each n> 0.
4.12 SOME ELEMENTS OF GELFAND THEORY
241
We can also observe that (4.11-4) is no longer true if
the condition that X be a Banach algebra is relaxed: It is not
even true for barreled Frechet semisimple Q-algebras as this
example illustrates.
Following the proof of (4.11-3) (Singer and Wermer (1955))
the suspicion grew that perhaps continuity of the derivation
was not a necessary ingredient in the hypothesis of that result
and this suspicion has been somewhat borne out. Curtis (1961)
*
proved that every derivation on a regular commutative semi-
simple Banach algebra with identity is continuous. This was
subsequently generalized to semisimple Frechet algebras by
Rosenfeld (1966). B.E. Johnson (1969)	proved that every
derivation on a semisimple commutative Banach algebra is con-
tinuous and hence trivial by (4.11-4).
Miller (1970) and Gulick (1970) have considered higher
order derivations and some of Gulick's results subsume Rosen-
feld's generalization of Curtis' theorem.
A number of the results just mentioned may be found in
the Exercises for Chapter 5.
4.12 SOME ELEMENTS OF GELFAND THEORY - THE TOPOLOGIZING OF Xh
AND THE MAPPING y.
In this section we consider a mapping useful in analyzing
* Let X be a commutative Banach algebra and Л( its space of max-
imal ideals. If xe X, define х:Д(->> C by the formula x(M)=h(x),
where M=ker h, and provide Л( with the weakest topology with re-
spect to which each x is continuous. If the functions x, xf X,
separate points and closed sets inA(, i.e. for each closed Fcz/И
and MeM there is an x such that x(M)=l and x(F)={0}, then X is
called regular. For further discussion see Sec. 5.2.
242
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
the structure of LMCH algebras—the mapping V sending xf X into
the corresponding evaluation map on taking a topological al-
gebra X into C(X^,F) the algebra of continuous functions on
X^ equipped with the relative о(X',X)-topology (see Definition
4.12-1). As we have seen it is frequently possible to reduce
questions pertaining to the structure of LMCH algebras to re-
lated questions about the factor algebras with the aid of the
canonical homomorphisms and some of their properties. In this
section we shall see that in the event X^ is о (x1 ,X) -compact,
V can at times be expected to behave in a similar way in re-
ducing questions concerning the LMCH algebra X to related ques-
tions about the Banach algebra C(X^,F,c). In order for V to
be useful in transforming questions from X into C(X^,F,c) it
is sometimes desirable for V to be continuous—our third result
deals with this. We conclude the section by establishing cer-
tain conditions under which a topological algebra is a Banach
algebra, by proving that V is a topological isomorphism.
Michael (1952) has referred to LMCH algebras X for which
V is an isomorphism onto C(X^,F) as full algebras (Definition
4.12-1). In Michael (1952) the question was raised: Is V a
topological isomorphism when X is a full Frechet algebra?
Warner (1958) answered the question in the affirmative and his
proof appears in (4.12-8). (In fact the analogous statement
also holds for many topological algebras over nonarchimedean
valued fields.) Thus V can also be used to identify certain
topological algebras as function algebras.
Definition 4.12-1. THE MAPPING V. Let X be a complex topolo-
gical algebra or a real topological algebra for which X^ / 0
4.12 SOME ELEMENTS OF GELFAND THEORY
243
and X carry the Gelfand topology, i.e. topology induced on X
by a(X’,X) (see the discussion following (4.10-4)*). Then we
define the mapping V: X->-C (X^ ,F) , x->-x by the rule: x(f)=f(x)
for each fe x\ As X^ carries the induced weak-* topology,
the set {f € X*11 | 'Fx (f) - Vx (f ) | < e} is open in X^ for any
f € X^ and e > 0, and, therefore, Vx is indeed an element of
о
C(X ,F). Although V is seen to be a homomorphism, it is not
generally 1-1 or onto. In the event that X is a LMCH algebra
and V is 1-1 and onto, X is referred to as a full algebra.
In order for V not to be 1-1, there must exist a nonzero
xeX such that 'Fx=0--in other words an x for which f(x)=0 for
each fcX^. Thus V is 1-1 iff X is strongly semisimple (Def.
4.11-1). In case X is a complete complex LMCH algebra so
that Rad X = ПAtc ((4.11-1)) and each Me is the kernel
of some fcX^ ((4.10-4)), it follows that V is 1-1 iff X is
semisimple. We record these findings in our first result.
(4.12-1) SEMISIMPLICITY=> У 1-1. If an LMCH algebra is (a)
strongly semisimple or (b) semisimple, complete, and complex,
then <F is 1-1.
As a consequence of (4.12-3) we see that V takes any com-
plete complex LMCH Q-algebra into a Banach algebra: We show
that for such algebras X^ is weakly compact. The following
technicality, (4.12-2), is a convenience.
(4.12-2) In any complete complex LMCH algebra (X^)° = U(o).
Proof. In a complete complex LMCH algebra X, o(x)={h(x)|he Xh}
for each xe X by (4.10-8). Thus
* If H is any collection of nontrivial homomorphisms on X then
the topology induced on H by o(X*,X) is also called the Gelfand
topology (See Exercise 4.4).
244
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
(Xh)° = (xf X|sup , |h(x)|< 1} = {x|r (x) < 1} = U(o).V
h€ ХП	°
(4.12-3) Q-ALGEBRAS AND COMPACTNESS OF Xh. Let X be a complete
complex LMCH algebra. If X is a Q-algebra, then (Xh)° is a
neighborhood of 0 in X and Xh is о(X1,X)-compact. Conversely,
if X is barreled and X^ is о (X1 , X)-compact, then X is a Q-al-
gebra.
Proof. Suppose that X is a Q-algebra. Then, by (4.8-3), the
set U(o) = {хе Х|гд(х) < 1} has non-empty interior; in fact,
0 e int(U(a)) (see the proof of(4.8-3) and U (a) € V(0) . By
(4.12-2) U(a) = (Х^)°; hence (Х^)° is a neighborhood of 0 in
X. Now X^c (X11)00, so we can utilize the fact that the polar
of a neighborhood of 0 in X is о(X1,X)-compact together with
the fact that X^ is a(X1,X)-closed in X' to complete this part
of the argument.
Conversely, suppose that X is barreled and X^ is o(X',X)-
compact. Then X^ is о(X1,X)-bounded so (Х^)° is a barrel in
X. Since X is complete and barreled, it follows that U(o) =
(X^)° is a neighborhood of 0 in X. Thus X is a Q-algebra by
(4.8-3).V
(4.12-3) can be used to obtain still another realization
of the Stone-Cech compactification (3T of a completely regular
Hausdorff space T, namely as the space of continuous nontrivial
homomorphisms of the Banach algebra С^(Т,С,c) of bounded con-
tinuous C-valued functions on T with sup norm. The space
С^(Т,С,с)^ of continuous homomorphisms is compact in its Gel-
fand topology by (4.12-3). So, identifying T with the space T*
of evaluation maps on С^(Т,С,с), a possibility afforded by
the complete regularity of the Hausdorff space T, we see that,
4.12 SOME ELEMENTS OF GELFAND THEORY
245
in some sense, С^(Т,С,с) is a compact Hausdorff space con-
taining T. If (a) T and T* can be identified as topological
spaces, (b) T is dense in (T,C,c)\ and (c) each
хеС^(Т,Е,c) can be extended continuously to a function on
Cfc (T,C, c) h, then Cb(T,C,c)h must be ВТ by Theorem 1.3-2. We
now verify that these three conditions are indeed met.
By (4.10-4) we see that С^(Т,С,с)^ may be identified
with the spaced of maximal ideals of С^(Т,С,с). Moreover the
Gelfand topology (first discussed after (4.10-4)) is the weak-
est topology for С^(Т,С,с)^ with respect to which the maps
xiC^ (T,C, c)h-4Z , f->-f(x) are continuous for each xeC^TjC,).
Since T is a completely regular Hausdorff space, its topology
is the initial topology determined by С^(Т,С) on T ((0.2-5));
a basic neighborhood of a point t in T is therefore a set of
the form V(t ; x. , . . . ,x ,e)={teT||x.(t)-x.(t )| < e,i=l,...,n)
о 1 n	i i о '
where x.,...,x eC, (T,C) and e >0.
1 n b ~
A typical basic neighborhood of the evaluation map
t*eC, (T,C,c)h in the relative Gelfand topology on T* would be
ob
{t*e T* | | x± (t*)-x± (t*) | < e, i=l, . . . ,n}= {t*eT | | x± (tQ) -x.^ (t) | <
e,i=l,...,n} where x, , . . . ,x eC, (T,C,c) and e > 0.
1	n b
Hence the map t*t* embeds T homeomorphically in
C, (T,C,c)h and condition (a) is seen to be satisfied,
b
As for (c) , given any xeCj3(T,R), x is a continuous ex-
tension of x to Cfa (Т,С,с)\ i.e. xeC (C^ (T, C, c) ,C) .
Last, it must be shown that T* is dense in С, (Т,С,с)\
b
If T* is not dense in C, (T,C,c)h there is some feC, (T,C,c)h
b	b
and a basic neighborhood V=V(f;x^,...,xn,e) of f such that
Vn T* = 0. Consider the functions y^=x^-f(x^)eC^(T,C),1< i<n.
246
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
For each tQeT there is some i such that | y. (t )| > e. Now,
_______________	2
with z = £д_Уд_Уj_' it is clear that |z(t) |> e for all teT:
hence z is invertible in С^(Т,С,с). Yet, for each i, f(y^)=0,
so f(z)=0. In other words the maximal deal ker f contains the
unit z which cannot be. We conclude that T* is dense in
С^(Т,С,с)^ and therefore also that рт=С^(T,C,c) or, equiva-
lently, that рт=Д( .
Next we establish a criterion for continuity of ’F. An
instance in which ¥ is discontinuous is given in Example 4.12-1
(4.12-4) CONTINUITY OF У. If X is (a) a complete complex LMCH
Q-algebra or (b) a barreled complex LMCH algebra then V is con-
tinuous .
Proof. (a): By (4.12-3) we know that X^ is о (X1 , X)-compact;
consequently C(X^,C,c) is a commutative complex Banach alge-
bra with identity with the compact-open topology •J' generated
by the supremum norm || || over x\ A neighborhood base at 0 for
xj' is given by sets of the form {ge C(X^)|sup h|g(f)| < e} =
c	f e X
{ge C(X^) | ||g || < el (e> 0) . Since (X^)° is a neighborhood of the
origin by (4.12-3), and v”1 ( {ge C (Xh) | ||g || < 1} ) =
{xe X|sup|< x,X^>|< 1} = (Xh)°, it follows that V is continu-
ous .
(b) If К is а о(X1,X)-compact subset of X^, then,
since X is barreled, K° is a neighborhood of 0 in X. But
k° = {xex||h(x)|<i, he kJ = {xe x||vx(h)| < i, he к} =
Y-1({ge C(xh)|supfe K|g(f)I <1).V
In the event that V is 1-1—e.g., if the LMCH algebra X
is strongly semisimple or semisimple, complete and complex
((4.12-1))—under what conditions will V 1 be continuous?
4.12 SOME ELEMENTS OF GELFAND THEORY
247
(4.12-5) CONTINUITY OF У 1. If X is a LMCH algebra and its
Gelfand map V: X^C(X^,F,c), x->-x is 1-1, then V 1 is continuous
whenever each equicontinuous subset of X’ is contained in a
multiple of the о(x1,X)-closed absolute convex hull of some
compact subset of x\
Proof. Let Ес X' be equicontinuous and choose nelj and a
compact set Kc X^ so that Ecn(cl V.K ) = nK°°. Then
о(x ,X) be
E° z> (l/n)K° and therefore ^(E0)^ (l/n)V(KO). But НЦК0) =
{'Fxlp (Vx) < 1} = V П V(X) so that V is a relatively open map. V
K	pk
Actually we are most interested in determining conditions
under which V is a topological isomorphism. The following
well-known result moves us in that direction.
(4.12-6) У IS AN ISOMETRIC ISOMORPHISM <=> ||( )2||=||( ) ||2. If X
is a complex Banach algebra, then V is an isometric isomorphism
into C(^,C,c) iff ||x2 || = ||x ||2 for each xeX.
Proof. We first establish the formula гд (x) =limn ||xn Ц^11 (xeX) .
In any complex algebra with identity о (x11) = o(x)n =
{рП|рео(х)} for each neN and each xeX since the polynomial
хП-ре=ПП , (x-ц.е) where ц,,...,ц are the n-th roots of u.
1=1 i	1 n	и
Thus гд(хП) = гд(х)П for each neN and xex. Now, by the
discussion following (4.8-4), гд(у)< ||y |j for each yeX and
so гд(х) = ^(x11)!/11 < ||хП ||1//П (neN,xeX). To conclude the
proof we show that lim supn |^<n Ц1/11 < гд(х)  Let a be an arbi-
trary real number larger than гд(x). We shall see that
||xn ||ly<n < a for all but at most a finite number of indices n
which will yield the desired conclusion.
Recall that the resolvent map r ,u-*(x-ue) 1, is analy-
tic on p(x) by (4.8-5). But whenever |ц| >гд(х),р e P(x).
248
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Thus, as was noted prior to (4.9-1),	is analytic on
{ ц||и|> гд(x)} for each feX'. On the other hand, r^(u) =
-U	ПхП for each |u|> ||x||. By the continuity of feX',
f(rx(u)) = (~u 1)J-n>QlJ П£(хП) for each |u|> ||x ||. As Laurent
expansions are unique, this expression is valid for all |u|>
гд(х). Hence if гд(х)<Ь< a then En>Qf(xn/bn) converges
and f(xn/bn)->-0 for each feX'. But a weakly convergent se-
quence in a Banach space is bounded so that a positive integer
N exists such that ||xn/bn |[< N for all neN. Therefore
||хП ||l/n < (N'*'//n)b for each n and by choosing n sufficiently
large, it follows that цхп |р-/п < a.V
2	2
Thus in Banach algebras where ||( ) || = ||( ) || , V is a
topological isomorphism. In complex LMCH algebras what happens
if the topology is generated by seminorms which satisfy this
condition? As we shall see, V is a topological isomorphism for
such algebras when C(x\c) carries a certain topology which
is generally weaker than the compact-open topology. For ease
of reference, we state some definitions, the first of which de-
scribes the aforementioned topology of C(X^,C).
Definition 4.12-2 THE WEAKENED COMPACT-OPEN TOPOLOGY. Let X
be a complex LMCH algebra and let X^ carry the o(X',X)-sub-
space topology. The topology for С(Х^,£) generated by the
seminorms PE,PE(f)=sup|f(E)|, feC(Xh,£), where E runs over
the closed equicontinuous subsets of x\cX') is referred to
as the weakened compact-open topology of C(X^,C) and is de-
noted by ’Jwc-
As X^ is о (X1 , X)-closed, each closed equicontinuous sub-
set E of X^ is о (X1 , X)-compact, so the weakened compact-open
4.12 SOME ELEMENTS OF GELFAND THEORY
249
topology is weaker than the compact-open topology of C(X ,C).
That this inclusion may be proper is demonstrated in Example
4.12-1.
Example 4.12-1 ^7* f COMPACT-OPEN TOPOLOGY. Let T be an un-
countable compact Hausdorff space and X=C(T,C) carry the LMCH
topology generated by the family of seminorms p_ where G is a
countable compact subset of T. Just as in Example 4.5-1, it
can be shown that the typical factor algebra X =C(T,C)/N_
(NG={xeC (T,jC) | x (G) ={ 0 } } ) and is isometrically isomorphic to
C(G,C,c) via the map x+N_->-x|_. Thus, as in Example 4.10-2,
it follows by (4.10-7) that X^=T*. Furthermore we claim that
the 1-1 map t*t* from T onto T* is a homeomorphism. Since T
is compact and T* is Hausdorff in the о(X1,X)-topology, it
suffices to demonstrate continuity. Certainly if (t^) is a net
from T converging to t then for each xeC(T,C), t* (x) =x (t^ )->-
x(t)=t*(x) so that the map t+t* is a homeomorphism. To de-
termine the weakened compact-open topology of C(X^,G)=C(T*,C)
we must decide which of the compact subsets of T* are equicon-
tinuous. Suppose К is compact and K* is equicontinuous. There
is a
pact
that
neighborhood eV^ where
e> 0 and G is a countable com-
subset of T such that sup|K*(eV )|< 1.
If we assume
К is uncountable then there is some teK-G. But T is com-
pletely regular and Hausdorff and G is compact so the contra-
dictory conclusion follows that a function xeC(T,C) exists
such that t*(x)=x(t)=2 and x(G)={0}. Hence К must be count-
able in order for K* to be equicontinuous and the weakened com-
pact-open topology is generated by the seminorms pK* where К is
a countable compact subset of T. In other words the mapping
250
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
V :C (T, С) ->C (T* , C) , x+x is a topological isomorphism when
C(T*,C) carries the weakened compact-open topology. It fol-
lows by an argument similar to the one used to show that K* be-
ing equicontinuous implies К is countable that V is not a
weakened compact-open neighborhood of 0 so that '^wc does not
coincide with the compact-open topology. It is also worth not-
ing that I is not continuous when C(T*,C) carries the compact-
open topology.
The topology of the complex LMCH algebra of the preceding
example is generated by a family of square-preserving multipli-
2	2
cative seminorms p , i.e., p (x )=p (x) , and, altho X and
К	к к
С(х\£,с) are not topologically isomorphic, X and (C(X^,C),
^Tvc) are. The theorem to follow, Theorem 4.12-1, generalizes
the conclusion of the example to the extent that for algebras
whose topology is determined by a family of square-preserving
seminorms, V is always a topological isomorphism into C(X^,C)
with the weakened compact-open topology.
Definition 4.12-3, SQUARE ALGEBRAS. A square algebra is a
complex LMCH algebra whose topology is generated by a family P
of square-preserving multiplicative seminorms.
It is simple to show that no loss of generality results
from assuming the family P to be saturated, i.e., that P is
closed with respect to the formation of finite maxima.
Theorem 4.12-1 WHEN IS У A TOPOLOGICAL ISOMORPHISM? Let X be
a complex LMCH algebra whose Gelfand map V is 1-1 (e.g. if X is
complete and semisimple or strongly semisimple). Then the fol-
lowing are equivalent.
(i)	X is a square algebra;
4.12 SOME ELEMENTS OF GELFAND THEORY
251
(ii)	there is a topological isomorphism of X into a pro-
duct algebra П^£МС(G^,C,c) where each is a compact Haus-
dorff space;
(iii)	there is a topological isomorphism of X into an al-
gebra C(T,C,c) where T is a locally compact Hausdorff space;
(iv)	there is a topological isomorphism of X into an al-
gebra C(T,C,c) where T is a completely regular Hausdorff
space;
(v)	V 1 is continuous when C(X^,C) carries the weak-
ened compact-open topology;
(vi)	V is a topological isomorphism when C(X^,C)
carries the weakened compact-open topology.
Furthermore if V is continuous when C(X^,C) carries the
compact-open topology (e.g., if X is a complete Q-algebra or a
barreled algebra) then we may replace (v) and (vi) by:
(v1) V:X+C(X^,C,c) is a topological isomorphism.
Proof, (i)=>(ii) Let P be a saturated family of square-preserv-
ing multiplicative seminorms generating the topology on X. Re-
call that the factor algebra X is the completion of
(Np=p 1(0))
equipped with the norm
£>(x+N )=p(x)
X/N
P
so that if
we also denote the norm on Xp by p, then £ is square-preserving
too. Thus, by (4.12-6), Xp is isometrically isomorphic to a
subspace of	Since (Theorem 4.6-1) (d)) X is topo-
logically isomorphic to the projective limit of the X
(a sub-
algebra of ПрХр), it follows that X is topologically isomor-
phic to a subalgebra of ПрерС(Хр\е,с) .
(ii)=>(iii); Set G^'={d}xG^ and transfer the topology
of G^ to	let G=UpeMGp'’ If G carries the topology J in
252
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
which a subset U of G is open iff U П G^ 1 is open in G ’ for
each peM, then G, referred to as the free union of the G^
(Dugundji 1966, 131-133), is clearly locally compact and Haus-
dorff. Moreover since a function x defined on G is continuous
iff each of its restrictions x|^ , is continuous, the map
U
П ,.C(G , С, c )->C (G, C, c) , (x )->x where x(n,t)=x (t)	for
(p,t)eG 1, is a surjective topological isomorphism.
(iii)=>(iv): Clear.
(iv)=^(v); Let W be a topological isomorphism of X into
C(T,C,c). Then for each teT the functional t*»WeX^. To see
that the map h,t^t*-W is continuous when X^ carries the
о(X1,X)-topology, fix xeX and let (t^) be a net convergent to
t. Then h(t ) (x)=t*(W(x))=W(x) (tp)-»-W(x) (t)=t*(W(x))=h(t) (x)
for each xeX, and h is seen to be continuous. Thus if К is a
compact subset of T then h(K) is compact in X^. Since W is a
topological isomorphism, U={xeX I p (W(x) ) < e} where KcT is
compact and e >0, is a typical basic neighborhood of 0 in X.
But e(h(K))° = {xex| | h (t) (x)| <e,teK} = U so that a basis for
the neighborhood system at 0 consists of multiples of polars of
compact equicontinuous subsets of x\ Now V (e (h (K) ) °) =
{’Fxlp, (Vx) < e} i.e., V is a relatively open map when
1 h (K)	—
C(X^,C) carries the weakened compact-open topology.
(v)=»(vi); Since V is an algebra isomorphism, it only
remains to show that ¥ is continuous. Let К be a closed equi-
continuous subset of Xh and consider the subspace neighborhood
V=eV ГН'(X) = {'Px| p (Vx) < e} . Clearly У-1 (V) ={xeX| p„ (Vx) < e} =
PK	К -	к
{xex||f(x)|< e, feK} = eK°, a neighborhood of 0 in X, and V is
seen to be continuous.
4.12 SOME ELEMENTS OF GELFAND THEORY
253
(vi)=^-(i); Since ¥ is a topological isomorphism, the
LMCH topology of X is generated by the family of multiplicative
seminorms of the form p • V where К is a closed equicontinuous
lx
subset of Xh. But (p - V) (x2)=p ((Vx)2)=p (Vx)2=((p V)(x))2
lx	lx	lx	lx
so that X is a square algebra.
(i) =>(v'): Finally suppose that V is continuous when
C(X^,C) carries the compact-open topology. If (i) holds, then
so does (v) . But x<7'wC is weaker than the compact-open topo-
logy, so V 1: C (Xh ,C, с) П V (X)->-X is continuous and V:X •*
С(х\с,с) is a topological isomorphism.
On the other hand, if (v1) holds, then by essentially the
same argument as in (vi)=>(i), X is seen to be a square alge-
bra. V
Returning to the phenomenon observed in Example 4.12-1,
we see that even tho the map V : X-+C (X^ ,C, c) may fail to be a
topological isomorphism, it is still possible for X to be alge-
braically and topologically embedded in an algebra C(T,C,c).
Indeed if X is the square algebra of Example 4.12-1, then, by
(iii) of the preceding theorem, there is a topological isomor-
phism of X into an algebra C(T,C,c) where T is a locally com-
pact Hausdorff space. In Theorem 4.12-2 we focus attention on
those complex topological algebras which are embeddable as com-
plete subalgebras of algebras C(T,£,c) which separate points
in the compact Hausdorff space T. In this case V :X->-C(X^,_C,c)
must be a topological isomorphism.
Let X be any topological algebra and suppose that Kc X^
is о(X1,X)-bounded so that K° is absorbent. We denote the
gauge of K° by p . Note that X^ itself is о (X1 , X)-bounded if
К
254
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
X is a complete complex LMCH Q-algebra by (4.12-3).
(4.12-7) CONTINUITY OF THE SEMINORMS pT, AND r . If X is a
barreled complete complex LMCH algebra then each of the semi-
norms p.., where К is а о (X1 , X)-bounded subset of Xh, is con-
tinuous. Thus if X is also a Q-algebra, then the gauge of
(X^)° is continuous: in this case it equals the spectral ra-
dius seminorm r .
a
Proof. As K°, the polar of the weak -* bounded subset К of x\
is a barrel in the barreled space X, it is a neighborhood of 0
in X, and therefore, p is continuous. In the event that X is
also a Q-algebra, X^ is a (X ’, X)-compact by (4.12-3) so the
gauge of (Х^)° is continuous. Thus it only remains to prove
that rQ is the gauge of (Х^)°. Denoting the gauge of (Х^)°
by p, we recall that p(x) = inf{a> 0|xe a(X^)°} for each xeX.
If гд(х)=0, clearly then p(x)=гд(x)=0. Hence suppose that
ra(x)^0. Since ra((ra(x)) 1x)=l and (Xh)°=U(a) by (4.12-1)
it follows that (r (x) ) ^xf (Х^)° and р(гд(х) ''’x)< 1. Thus
p(x)< гд(x). However as a(X^)°={x|гд(x)< a}, we have
ra(.x)< a for each xe a(X^)° which, in turn, implies that
rQ(x)j< inf{a> 0|xe a(X^)°} = p(x) and the proof is complete.?
In the next result we focus attention on complete sub-
algebras of function algebras on a compact set, with the aid
of the mapping V. Our next definition is a prerequisite for
that result.
Definition 4.12-4 UNIFORM ALGEBRAS. If X is a complex topolo-
gical algebra then X is a uniform algebra* provided that there
* Some authors, Browder 1969b for example, designate such al-
gebras as function algebras.
4.12 SOME ELEMENTS OF GELFAND THEORY
255
exists a compact Hausdorff space T such that X is topologically
isomorphic to a closed subalgebra of C(T,C.,c) which separates
the points of T, i.e. if t^ and t2 are distinct points of T
there is a function x in the closed subalgebra such that x(t^)^
X (t2) 
In our next result, among other things, we obtain a nec-
essary and sufficient condition for V to be a topological iso-
morphism when X is a semisimple barreled complete complex LMCH
Q-algebra. Warner proved that V is a topological isomorphism
when X is a full Frechet algebra ((4.12-8)).
Theorem 4.12-2 CRITERION FOR WHEN X IS A UNIFORM ALGEBRA. If X
is a semi simple barreled complete complex LMCH Q-algebra then
the following statements are equivalent:
(a) X is a uniform algebra,
(b) V is a topological isomorphism,
(c)J , the topology generated by the spectral radius norm гд,
is a topology of the dual pair (X,X').
Proof. Since X is a semisimple complete complex LMCH algebra,
X is strongly semisimple by (4.11-1). Thus (Vx)(f)=f(x)=0 for
all fg X^ iff x=0, i.e. iff V is an isomorphism. Furthermore
as X is a barreled complex LMCH Q-algebra, it follows by (4.12-
5) that гд is a continuous norm on X with respect to its ori-
ginal topology and so	With these observations in mind
we now proceed to establishing the necessary equivalences.
(c )^>(b)z»(a) ; Since the implication (b)=5>(a) is obvious
we prove only that (c) implies (b) . Suppose that is a top-
ology of the dual pair (X,X'). Since is a locally convex
metric topology it follows that =i(X,X') (the Mackey topo-
256
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
logy). But X is barreled, so sZ = T(X,X') and J = Vr  Hence T
is a topological isomorphism.
(a)=> (c); Suppose that X is a uniform algebra; then it
may be viewed as a closed subalgebra of a function algebra
C(T,C,c) where T is compact Hausdorff. As such it is a
Frechet algebra and -J = T(X,X'). Having already observed that
, it will follow that kZ, is a topology of the dual pair
if it can be shown that a (X,X')	We accomplish this by
proving that each ff X' is continuous in the <Zr topology. If
f£X' then there is a continuous linear functional f on
C(T,C,c) extending f. By (2.2-1), C(T,£,c/is the space span-
ned in the algebraic dual C(T,C)* of C(T,g) by the o(C(T,C)*,
C(T,C))-closure of the balanced convex hull of the set T* =
{t*|te T} of evaluation maps. First consider an evaluation
map t* С T*. As C(T,C,c) is a complete complex LMCH algebra
and the (continuous) homomorphisms of C(T,C,c) are just the
elements of T* (Example 4.10-2), we have |t*(x)| <
sup{t*(x)|te T} = r (x) for each xf X by (4.10-8). Clearly
n
it follows that |(.^a^tt)(x)|< ro(x) for each хе X, when-
ever 2||< 1 and t^C T, 1 < i < n. Furthermore, any con-
tinuous linear functional g in the o(C(T,C)f C(T,C))-closure
of (т*)^с is the limit of a filterbase 0 c (T*)j3c: (x)-kj(x)
for each хе X. Thus |g(x)| = lim|®(x)|< гд(х) for each
хе X. Hence the continuous linear functional ?, being a linear
combination of elements g from the weak-* closure of (Т*)^с,
must satisfy an inequality of the form |f(x)|< К гд(х) for
A
each хе X, where К is some positive constant. Since f=f on
X, the <7 -continuity of f follows.V
4.12 SOME ELEMENTS OF GELFAND THEORY
257
Michael (1952) proved that the Gelfand-topologized space
X^ of nontrivial continuous scalar-valued homomorphisms of a
full Frechet topological algebra X is hemicompact (Def. 2.1-1)
and he raised the question as to whether it is a k-space (Sec.
2.3 ). Warner (1958) answered this question in the affirmative
by showing that for such algebras the mapping V is a topologi-
cal isomorphism. We present these findings in our next result.
(4.12-8) FULL FRECHET ALGEBRAS. If X is a full Frechet alge-
bra then the algebraic isomorphism V, taking X onto C(X^,F,c),
is a topological isomorphism. Furthermore X^ is a hemicompact
k-space.
Proof. Let Y denote C(X^,F) carrying the complete metrizable
locally convex Hausdorff topology -J" with neighborhood base at 0
of sets 'P(u) where U is a neighborhood of 0 in X. We shall
show that X^ is a completely regular Hausdorff space in its
Gelfand topology and Y=C(X^,F,c). This suffices to prove the
theorem as the contention that X^ is a hemicompact k-space
follows by Theorem 2.1-1 and (2.3-4).
As the Gelfand topology of X^ is the relative o(X',X)-
topology, Xh is a completely regular Hausdorff space. The pro-
cedure for showing that Y=C(T,F,c) is as follows: First we
show that •J' is finer than the compact-open topology by showing
that the seminorms p which generate the compact-open topo-
lx
logy are continuous on Y. Thus the identity mapping I taking
Y onto C(X^,F,c) will be a continuous isomorphism. Next we
prove that C(Xh,F,c) is barreled so that I is almost open*.
* A linear map A:X->-Y, X and Y topological vector spaces, is
almost open if for each neighborhood U of 0 in X, cl A(U) is
a neighborhood of 0 in A(X).
258
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Finally we shall conclude from the fact that Y is fully com-
plete that I is open, i.e. a topological isomorphism.
To see that>J~ is finer than the compact-open topology,
consider the collection (X^)* of evaluation maps on Y. If
heX^ then h*(Vx)=Vx(h)=h(x) so that h* is continuous on Y.
h	h
Hence (X )*cY'. Furthermore (Xn)* equipped with the rela-
tive o(Y',Y) topology is homeomorphic to X^. Therefore if К
is compact in X^ then K* is equicontinuous on the barreled
space Y as K* is a(Y1,Y)-compact. Since p (U)=sup|U(K)| =
sup|К*(U)| for each neighborhood U of 0 in Y, the continuity
of p follows from the equicontinuity of K* and is stronger
than the compact-open topology.
By our previous discussion it only remains to show that
C(Xh,F,c) is barreled. Since C(X^,F,c) is barreled whenever
it is bornological ((2.6-1)) it suffices by Theorem 2.6-1 to
show that Xh is replete. Thus, by Theorem 1.5-3(b) it suffices
to show that X^ is o-compact. Letting (Un) be a countable
base at 0 for the metrizable space X, it follows that X1 =
{0}° = (f^un)O=U un°- Thus X' as well as the closed subspace
are o-compact and the proof is complete.?
By scrutinizing the proof it is seen that a locally con-
vex Hausdorff topology of C(T,F) which renders C(T,F) fully
complete * and barreled and with respect to which all the
evaluation maps are continuous, must be the compact-open topo-
logy. A somewhat more general result along these lines is
(4.13-2).
* A locally convex Hausdorff space X is fully complete (or
Ptak) if for each locally convex Hausdorff space Y, every
4.12 SOME ELEMENTS OF GELFAND THEORY
259
In Banach algebras with involution (Def. 4.12-5) the Gel-
fand map V has more sharply defined properties. For example,
for A*-algebras X, V(X) is dense in C(Xh,C,c) and if X is a
B*-algebra, V maps X isometrically isomorphically onto
С(х\с,с). In the discussion to follow these notions are
broadened and the results mentioned generalized.
Definition 4.12-5 ALGEBRAS WITH INVOLUTION. A complex topolo-
gical algebra X is called an algebra with involution if there
is a map x+x* of X into itself such that for x,yeX and peC
(i) (x*)*=x, (ii) (x+y)*=x*+y*, (iii)(px)*=px*, and (iv) (xy)*=
x*y*. (It follows from (i) and (ii) that x+x* is bijective.)
X is a symmetric algebra with involution if 'Px*= Vx for each
xeX; sometimes "symmetric algebra with involution" will be
shortened to "symmetric algebra." An algebra with involution
X is a star algebra (or *-algebra) if there is a family P
of multiplicative seminorms generating the topology on X such
2
that for each peP and xeX, p(xx*)=p(x) . As was the case
for square algebras, no loss of generality is entailed by
assuming P to be saturated.
If X=C(T,C,c) where T is any completely regular Haus-
dorff space, then X is an algebra with involution with respect
to complex conjugation. Such an X is clearly also a *-algebra.
By Example 4.10-2 and the definition of the Gelfand topology,
X^ may be topologically identified with T. Moreover, identify-
ing T and X*1, Vx is just x for each xeX, from which it follows
* Cont. continuous almost open map A:X-»-Y is relatively open
in the sense that neighborhoods of 0 in X are mapped into nei-
ghborhoods of 0 in A(X). Every Frechet space is fully complete
(Horvath 1966,p.299,Prop.3(a)). See also Exercise 2.3.
260
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
that X is symmetric.
In the class of complex Banach algebras, the symmetric
algebras with involution are the A*-algebras and the star alge-
bras are the B*-algebras.
The definition given above for a symmetric algebra with
involution is motivated by our desire for such algebras to sat-
isfy the condition that V(X) be dense in C(X^,C,c). Indeed
to obtain this sort of density for complete X (Th. 4.12-3 (c))
we make use of the Stone-Weierstrass theorem which makes it
necessary for V(X) to be closed under complex conjugation.
Hence a natural condition to impose on X is that Vx* = Ух.
(4.12-9) SYMMETRIC ALGEBRAS. Let X be a complete LMCH algebra
with involution and V be the Gelfand map. Then the following
are equivalent:
(a)	For any xeX, xx* + e is invertible.
(b)	If x is self adjoint (i.e. x=x*) then a(x)cR;
(с)	X is a symmetric algebra.
Proof.	(а) =ф.(Ь) : Suppose that while (a) holds there is a
self-adjoint element x whose spectrum a(x) is not wholly real.
Choose u=a+ib e a(x) where b / 0 and consider
2	2	2	2
(x-(a+ib)e)(x-(a-ib)e) = (x-ae) + b e = b ((x-ae/b) + e).
Now w = (x-ae)/b = (x*-ae*)/b = ((x-ae)/b)* = w*. Since (a)
2
holds, ww* + e = ((x-ae)/b)	+ e has an inverse, say y. But
2
then (x-(a+ib)e)((x-(a-ib)e)(y/b) ) = e which contradicts the
assumption that u=a+ib e a(x).
(b)^-(c) : Clearly if xeX then, by taking u and v to be
the self-adjoint elements (x+x*)/2 and (x-x*)/2i respective-
ly, x=u+iv and x*=u-iv. Thus, for heX^, h(x)=h(u)+ih(v)
4.12 SOME ELEMENTS OF GELFAND THEORY
261
while h(x*)=h(u)-ih(v). Since h(u)e o(u) and h (v) e cj(v)
and h(u) and h(v) are real, it follows that h(x*)=h(x) and
therefore (fx*)(h) = h(x*) = h(x) = ух(h) for each xeX.
(c)=^(a): If heX^ and xeX then h(xx*+e) =
h(x)h(x*) + h(e) = (Ух) (h) (Ух*) (h) + 1 = (Ух) (h)TVTRh) + 1 =
h
|h(x)|	+ 1 > 0. As X is complete, o(xx*+e) = (h(xx*+e)|heX }
by (4.10-8) and therefore 0 / o(xx*+e). Hence xx*+e is in-
vertible. V
Prior to discussing the connection between star algebras
and symmetric algebras we record some elementary properties of
star algebras.
(4.12-10) PROPERTIES OF STAR ALGEBRAS. If X is a star algebra
2
with P a family of seminorms such that p(xx*) = p(x) for
each peP and generating the topology on X (cf. Def. 4.12-5)
then
(a) for each xeX and pep, p(x) = p(x*) and
(b) X is a square algebra.
2
Proof. (a) Certainly for any xeX (p(x))	= p(xx*) <p(x)p(x*)
2
and p(x*)	= p(x*x**) = p(xx*) <p(x)p(x*). Thus p(x)=0 iff
p(x*)=0. If p(x) and p(x*) are not zero then it follows from
the equations above that p(x) = p(x*).
2 2	2	2
(b) For xex and pep consider p(x )	= p(x (x )*)=
p(x* 2 *(x*)2) = p((xx*)2) = p((xx*)(xx*)*) = (p(xx*))2 = p(x)4.
2	2
Thus p(x ) = p(x) and X is a square algebra.V
(4,12-11) COMPLETE STAR ALGEBRAS ARE SYMMETRIC. A complete star
algebra is symmetric.
Proof. Suppose that X is a Banach algebra, i.e., P ={ ||||} and
(X, i|||) is complete. By (4.12-9) (b) it suffices to show that
262
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
each self-adjoint element has a real spectrum. Let x be self-
adjoint; we claim that i / a(x). As the spectrum of an ele-
ment yeX coincides with ('Ey) (X^1) by (4.10-8), it follows
that for any polynomial p with complex coefficients, o(p(x)) =
p(o(x)) = {p(u)|u e o(x)}. Hence if we assume that i e a(x),
then, for any aeR, (l+a)i e o(x+aie). Since e is self-adjoint
an element yeX is invertible iff y* is invertible. Therefore
since (x+aie)* - (-(l+a)ie) = (x+aie - (l+a)ie)* and
(l+a)i e o(x+aie), - (l+a)i e o((x+aie)*). In the proof of (4.
12-6) it was shown that гд (у) = limn ||yn ||ly<n for any yeX.
Since X is a square algebra by (4.12-10), i.e., ||y2 || = |jy jj2,
2
it follows that гд(у) = ||y || for each yeX. Thus (1+a)	<
2
гд(x+aie)гд((x+aie)*) = ||x+aie || ||(x+aie) * || = [jx+aie || =
||(x+aie) (x+aie) *|| = j[x2+a2e || < ||x2 || + a2 since ||e |j = гд(е) =
sup { | h (e) | | heX^} = 1 by (4.10-8). But then l+2a < ||x ||2 for
each aeR—an obvious contradiction. We conclude therefore
that i / a(x). Next suppose that y=a+ib e a (x) where b^O.
Then, for p(c)=(c-a)/b, i=p(n) e o(p(x)) = o((x-ae)/b) and
(x-ae)/b is self-adjoint. Thus each self-adjoint element of
X has real spectrum and X is a symmetric algebra.
Suppose now that X is a star algebra with P as in Def.
4.12-5 and
Then the norm p(x+Np) = p(x) (Np=p 1(0))
2
satisfies the condition ]5((x+Np)(x*+Np)) = p(x +Np) for each
xeX. If we define (x+N^)
x*+Np then X/Np is an algebra
with involution.
Moreover it is also clear from the fact that
p((x+Np)*) = £>(x+Np)
that this involution has
sion to X
P'
the completion of
X/Np,
so that
a unique exten-
2
£>(zz*) = (p(z))
for each
zeX .
P
Hence
(Xp,f>)
is a symmetric algebra for each
4.12 SOME ELEMENTS OF GELFAND THEORY
263
peP. Suppose that xeX is self-adjoint. Then for each peP,
x+N	is self-adjoint in X and a(x+N )c R. Since X is com-
P	P	P ~
plete, we know by (4.7-2) that a(x) = Upepa(x+N₽)c R. Thus X
is symmetric.?
In our next result some well-known results about A*- and
B*- algebras are generalized.
A homomorphism H between algebras with involution X and Y
is a star homomorphism (or *-homomorphism) if Hx* = (Hx)* for
each xeX.
Theorem 4.12-3. X"=" C (X*1, £, ^wc) • Let x be an LMCH algebra
with involution. Then
(a)	if X is a symmetric algebra, V(X) is dense in
C(Xh,C,c);
(b)	If X is a star algebra, V is a topological star iso-
morphism of X into С(X^,C,</wc) ;
(c)	if X is a complete star algebra then X is a symme-
tric algebra and V is a topological star isomorphism of X onto
C(Xh,C,<7' )
wc
Proof, (a)	?(X) is a subalgebra of C(X^,£, c) containing the
constant functions, separating the points of x\ and closed
under complex conjugation. The desired result follows from the
Stone-Weierstrass theorem (Dugundji 1966, pp. 282-293): A sub-
algebra of C(T,C,c), T a Hausdorff space, closed under conju-
gation, separating points of T, and containing constants is
dense in C(T,C,c).
(b) The result of (4.12-10)(b) already shows that a star
algebra is a square algebra. Hence V is a topological isomor-
phism into C(Xh,C,c7' ) by Theorem 4.12-1. It remains to
** wc
264
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
show that V is a star isomorphism. In the proof of (4.12-11)
it was argued that each of the factor algebras is a star al-
gebra (B*-algebra) with respect to the involution obtained by
(uniquely) extending the involution defined by (x+N^)* = x* +
Np on X/Np. Let Vp be the Gelfand map taking xj^ into
С(Хр\с,с). It was also established in the proof of (4.12-11)
that the norm in a B*-algebra coincides with the spectral radi-
us rQ. Thus Vp is an isometry and ^p(Xp) is complete, hence
symmetric by (4.12-11). Hence we may conclude by (a) that
V (X ) = С (X \c,c). We also claim that V is a star isomor-
p p	p ***	p
phism. Indeed
V z for each
P
V (X )
P P
zeX .
P
is a symmetric algebra so that	=
Finally, recalling that X11= Up^pKp (Xph) ,
where is the canonical map
follows that for any h = к'(h )
p p
x->-(x+Np) by (4.10-7), it
eKp(Xph), (Тх*)(h)=Ух*(Kphp)=
Kp(hp (x*) =hp (x*+Np) =hp ((x+Np) *) =hp (x+Np) =Kp (hp) (x) = ('Fx) (h) .
Hence V is a star isomorphism and the proof of (b) is complete.
(c) We know from (4.12-11) that any complete star algebra
is a symmetric algebra. V is a topological star isomorphism by
(b) and the surjectivity of V follows from (a) and the fact
that the compact-open topology is stronger than the weakened
compact-open topology.V
It is well known that for spaces, paracompactness is
equivalent to the existence of partitions of unity (defined in
a footnote to a part of the proof of (2.4-4)) subordinate to
any given open cover of the space. When V(X)=C(X^,C,c), for
example, figuring that V will be reasonably kind to partitions
of unity on X, it may be that existence of partitions of unity
on X play a role in characterizing the algebras X for which X^
4.12 SOME ELEMENTS OF GELFAND THEORY
265
is paracompact. But first: What is a partition of unity for
an LMC algebra?
Definition 4.12-6 PARTITIONS OF UNITY. Let X be an LMC alge-
bra with topology generated by a saturated family P of multi-
plicative seminorms. A family (xp)peP of elements
a P-partition of unity if (1) for
called
each
qeP
xp from X is
there is a
finite
each q
subset F of P such
q
vanishes on all but
that q(xp)=0
finitely many
iff
and
x ' s)
P
for each
peP, q(Xp)=0
(3) for each qeP and xeX, q(x-£	_ xx
p q P
The force of (3) is that the net of elements
for all peP; (2) for each
0.
indexed by the directed (by
2 nXX
peF p
set inclusion) family of finite
p i
F
q
P -
q i Fp;
subsets F of P converges to x so that we may sensibly write
Z rxx = x for each xeX. In particular 2 x = e.
peP p	P P
(4.12-12) WHEN IS Xh A LOCALLY COMPACT PARACOMPACT SPACE? Let
X be a complete LMCH algebra with involution and V be the Gel-
fand map taking X into C(x\c,c). Then V is a surjective top-
ological star isomorphism and X^ is locally compact and para-
compact iff X is a star algebra whose topology is generated by
a family P of multiplicative seminorms (as in Def. 4.12-5) for
which a P-partition of unity exists.
Proof. Suppose that V is a surjective topological star isomor-
phism and X^ is a locally compact paracompact space. Then X is
a star algebra generated by the family of seminorms p -V
lx
where К is a compact subset of x\ As X^ is locally compact,
each heX^ has a relatively compact open neighborhood U^. By
the paracompactness of x\ a neighborhood finite refinement of
(U, ). h exists consisting of relatively compact open sets.
П П £ A
266
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Let 4 be the family of closures of the sets in the refinement.
H is clearly a neighborhood finite cover of X^. Furthermore
we may assume that # is closed with respect to the formation of
finite unions without altering the fact that if is a neighbor-
hood finite cover of x\ Now paracompactness of X^ also guar-
antees the existence of a partition of unity (f_)_ и subordi-
G G £</
nate to ,	i.e., for each f_> 0, f_(h)=0 whenever h/G and
G—	G
fG iS identitY element of C(X^,£).
Let К be an arbitrary compact subset of x\ Since iff is
neighborhood finite, only finitely many G’s meet K. But
covers К and is closed with respect to the formations of finite
unions. Thus К is contained in some Ge^ and therefore the
saturated families {p_lGe#} and P = Ip’llGeJ} generate the
G 1	G
topologies of C(X^,C,c) and X respectively. Moreover if for
each Ge# we set F = (G'e#|G'nG / 0} then p (f ,) = 0
G	G G
for all G / F„, so p„(f-2 , _ ff„.) = 0 for each	and
G	G G ~ G
h
f eC(X ,C). Hence it clearly follows that the family X =
{V ^(fg)|Ge#} serves as a P-partition of unity in X.
Conversely suppose that X is a star algebra with topology
generated by P and (xp)pEp is a P~partition of unity for X.
Then V is a star isomorphism of X onto C(Xh,C,J^c) by Theorem
4.12-3 (c). First we show that X^ is locally compact. Let
Gp = {heXh|h(xp) / 0} = C((Yxp) 1(0))
each Gp is open. Since UpepGp =
for each peP. Clearly
(if hex^ vanished at each
vanish on e and be trivial) it suffices to
Xp, then h would
show that each Gp has compact closure to conclude that Xh is
locally compact. Indeed, GpC X^ = UqepKq(Xqh) by (4.10-7)
(b) . Now
q / F , then q(x )=0 and therefore x e N .
P	P	P 4
4.12 SOME ELEMENTS OF GELFAND THEORY
267
Thus for each hq e X^h, (Ухр) (hq. Kg) = <hq’Kq) (xp) =
h (x +N ) =0 and k'(X c (Vx ) ^(O). Therefore
q p q	q q p
G, П к'(X h)=0 whenever q/F . This means that
p q q	p
G с и „ к' (X h)-a compact set by (4.10-7) (c).
p qef q q
'
To see that X is a paracompact, we begin by establishing
the
fact that each Gp meets only finitely
Gpn Gg * 0
heK'(X^) for some
s s
p, suppose
there is
an
in G . Now
q
Thus neither
seP
many Gq's. For fixed
heG which is also
P
h(Ng)
whence
{0}.
x nor x belong to 1
P q
finite, if we assume that F П F
P q
then there must be an element seF
P
finitely many Fq's. This means that s fails to vanish on in-
finitely many members of
N
s
so
F . As
q
F
P

0
seF П
P
for infinitely many q's
which also belongs to in-
:p)peP whi-ch is contradictory.
Hence there are only finitely many Gq's that meet Gp.
Now suppose that U is an open cover of X^. For each peP
choose C. ,...,C eU
1,P	np,p
cl Gp is compact). Next
To see that X^
G .
P
for
some peP and
=G П C.
। p 1
are a neighborhood
U.
so
, n and
P
U.
i,q - - q
Then, for each
which meet and cover G
P
? np, let
let heXh
for each
UpeP U i=lUi
therefore to C.
for some
(recall that
u.
=C .
Then
П
heG
P
heu.
Finally we
claim that the
P
U.	’ s
But
refinement of LZ
f G meets G , it
P	q
On the other hand
. G meets each
P
is possible for Gp
U.
to meet
. , U. c G
q i,q q
Thus Gp can meet only finitely many
arbitrary U.
suppose that GpП Gq = 0-
U. П G = 0.
i,q p
Next, consider an
so that
U. ' s
By definition U.
e
U . Hence U has a
neighborhood finite refinement and X^
is paracompact
By Theorem 4.12-3(c)
the final thing to establish is
268
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
that ~-^wc and the compact-open topology coincide,
that К is a compact subset of X*1. We know that the
neighborhood finite, they form an open cover of X*1,
Suppose
Gp1s are
and each G
P
meets only finitely many G 's.
Hence К can meet only finitely
many G ’s and is therefore contained in some finite union of
the G 's.
On the other hand we saw that each G meets only
finitely many of the covering sets Kq(Xq^)'
contained in some finite union of k'(X ^) ,
qeP. Thus К is
each one of which
is equicontinuous by (4.10-7Xa). In conclusion К is equicon-
tinuous and the weakened compact-open topology coincides with
the compact-open topology.V
4.13 Continuity of Homomorphisms.
In Section 4.10 we proved that any (complex) homomorphism
in a complex LMC Q-algebra is continuous (4.10-5) and gave ex-
amples of complex LMCH algebras on which discontinuous homomor-
phisms exist (Ex. 4.10-1). In this section we discuss continu-
ity of homomorphisms between complex LMCH algebras. We prove
that any homomorphism taking a complex barreled Q-algebra into
a strongly semisimple fully complete (See Sec. 4.12) complex
LMCH algebra is continuous. From this we obtain the fact that
there is at most one topology on a complex algebra with respect
to which it is a fully complete barreled LMCH Q-algebra—a re-
sult which subsumes the well known fact that there is at most
one Banach algebra topology for a complex semisimple algebra.
In proving the theorem on the continuity of homomorphisms
mentioned above it is first established that the given homomor-
phism is continuous when the range space carries a special top-
ology, called the homomorphism topology. We define this topo-
4.13 CONTINUITY OF HOMOMORPHISMS
269
logy now.
Definition 4.13-1 HOMOMORPHISM TOPOLOGY. Let X be a topologi-
cal algebra and denote the nontrivial continuous homomorph-
isms of X. The initial topology on X generated by the linear
span [X^] of Xh in X’ is denoted by o(X,X^) and referred to
as the homomorphism topology. A base of neighborhoods of 0 for
o(X,X^) consists of sets of the form V(0,f.,...,f ,e) =
1 n
fxf x| |f^(x)| < e, f e x\l < i< n} where e > 0. If the linear
transformation A maps X into the topological algebra Y and A
is continuous when X and Y carry their homomorphism topologies
then A is called homomorphically continuous.
Suppose that X is a complex LMC algebra. Clearly if it
is strongly semisimple then X^ separates the points of X and
о(Х,Х^) is Hausdorff. Conversely, if a(X,X^) is Hausdorff
then for any non-zero xf X there is some ge [X^] where
g = Ea^f^ and f € X*1, such that g(x)^0. Thus f^(x)^0 for
some i and the kernel of f^, a closed maximal ideal, does not
contain x. Hence X is strongly semisimple. We summarize these
observations below.
(4.13-1) HAUSDORFF HOMOMORPHISM TOPOLOGIES. If X is a complex
LMC algebra then о(Х,Х^) is Hausdorff iff X is strongly semi-
simple.
The main result of this section is in three parts. The
first two parts are concerned with continuity of a homomorphism
when one or both spaces carry their homomorphism topologies.
The third, which is established with the aid of the closed
graph theorem is the result mentioned in the introductory re-
marks of this section on continuity of homomorphisms.
270
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Theorem 4.13-1. CONTINUITY OF HOMOMORPHISM. Let A be a homomor-
phism taking the complex topological algebra X into the complex
topological algebra Y. Then:
(a)	A is homomorphically continuous if it is continuous
when X and Y carry their original topologies;
(b)	If X is a complete LMCH Q-algebra and Y is a topolo-
gical algebra then A is continuous when Y carries its homomor-
phism topology and X its original topology;
(c)	If the complete LMCH Q-algebra X is barreled and the
LMCH algebra Y is strongly semisimple and fully complete, then
A is continuous.
Proof, (a) For any f^,...,fnC Y*1, note that f^A,...,fnA e
X*1. Clearly for each e > 0,	A(V(0,f^A,...,f A,e) c
V(0,fp...,f e), so A is homomorphically continuous.
(b) Since X is a complete LMCH Q-algebra, it follows
from (4.12-3) that (Х^)° is a neighborhood of 0 in X. Con-
sider any finite collection f^,...,f € Y^; then by (4.10-5) we
have that f^A,...,fnAg X^u(0). Thus for any 6> 0 and
xc 6(Xh)°,|f^A(x)|< 6 for i=l,...,n and we see that
A(6(Xh)°)c V(0,f1,...,fR,6). Hence A is continuous when Y
carries o(Y,Yh).
(c) By (b) A is continuous when Y carries o(Y,Y^).
Since Y is strongly semisimple, u(Y,Yh) is a Hausdorff topo-
logy, so the graph of the continuous homomorphism A is closed
in the product topology on X*Y when X carries its original
topology and Y carries а(У,У^). [Any continuous map f: S-+T
where S is a topological space and T a Hausdorff space has a
closed graph in SxT]. Now this product topology is clearly
4.13 CONTINUITY OF HOMOMORPHISMS
271
coarser than the one induced by the original topologies of X
and Y, so the graph of A remains closed in the product of the
original topologies. Finally, since X is barreled and Y is
fully complete, the closed graph theorem implies that A is con-
tinuous when X and Y carry their original topologies.?
As an application of (c) above, suppose that X is a semi-
simple fully complete, barreled, complex LMCH Q-algebra when X
carries either of the topologies zF or •zT'’ . Then the identity
homomorphism (X^Z) *(X,zj'' ) , x->-x is bi-continuous by part (c)
of the previous theorem and tT =кГ'. Hence we may state:
(4.13-2) UNIQUENESS OF THE TOPOLOGY OF A SEMISIMPLE, FULLY
COMPLETE BARRELED COMPLEX LMCH Q-ALGEBRA. If X is a complex
semisimple algebra then there is at most one topology with re-
spect to which X is a fully complete, barreled, LMCH Q-algebra.
Thus there is at most one topology making such an X a Banach
algebra.
272
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
Exercises 4.
4.1 WEAKLY TOPOLOGIZED ALGEBRAS. Let X be a real or complex
algebra and X' a total subspace of the algebraic dual X*.
(a)	COMPATIBILITY OF a(X,X'). The following are equivalent:
(1)	(X, o(X,X’)) is an LMC algebra.
(2)	(X, o(X,X’)) is a topological algebra.
(3)	For each x’g X’, x' 1(0) contains а о(X,X')-closed
ideal of finite codimension.
(4)	The map (x,y)->-xy is continuous at 0 when X carries
o(X,X') and Xxx its product topology.
Hint. We sketch proofs of two preliminary statements, A and B.
(A)	Let x’e X', W be а о(X,x')-neighborhood of 0 and
Wo Y a subspace of X. If Wu W^c {x1}° then Y, xY, Yx c
-1	3
x’ (0) for each xf X. Moreover if W is also contained in
{x1 }° then xYxcx' 1(0) for all xf X. To see that Yx c
x1 1(0) let n be a positive integer and ye Y, хе X and con-
sider пух = ц Iny(ux) where pxf W for small enough p. Thus
пухе {x'}° for each such n and it follows that yxe x' 1(0).
(B)	If V is а о(X,X1)-neighborhood of 0 then
L = n{x' 1(0)|x'e V°} is а о(X,X1)-closed subspace of finite
codimension. (To prove the finite codimensionality of L choose
{x',...,x'}с X’ such that {x',...,x1}°с V. It follows that
In	in
L э Ql-1Xi 1(0) is of finite codimension.)
We now provide a hint to the proof that (1) implies (3).
For x'e X’ there exists an absolutely m-convex closed
о(X,X’)-neighborhood of the origin V such that Vc {x'}°. Let-
ting L = n{y' 1(0)|у'€ V°}, a a(X,X’)-closed subspace of
EXERCISES 4
273
finite codimension by (B), it follows that Lc V°° = V. To see
that L is an ideal, consider nxy for a positive integer n,
-1	2
xe X, and ye L. Then nxy = ух(пу y) e VLcV с V for suffi-
ciently large since n is arbitrary, xye L. Consider the
о(X,X’)-closed subspace of finite codimension Y=LП x' 1(0).
Since YcLcV = Vu V2cx,-1(0), it follows by (A) that YX,XY
ex' 1(0)—thus YX,XYcY. We now provide a hint to proving
that (3) implies (1). Given x' e X' it suffices to obtain a
multiplicative neighborhood V of 0 such that Vc {x'}°. If I
is a a(X,X')-closed ideal of finite codimension, then it fol-
lows that X/I is a finite-dimensional normed algebra. Let
x'(x+I) = x'(x), since x' is continuous, choose a multiplica-
tive neighborhood U of 0+1 in X/I such that Uc {x'}°. Then
V = к 1(U) (к the canonical homomorphism of X onto X/I) is
the desired neighborhood.
Finally we provide a hint to the proof that (4) implies
(3) . By (4), the composite map (x,у, z) -> (xy , z) ->-xyz is contin-
uous when X is endowed with the o(X,X') topology and the pro-
ducts with their product topologies. It follows that there is
an absolutely convex о(X,X')-closed neighborhood of 0,W, such
that Wc Wu W2 U W2 c{x'}°. If L=n{y' 1 (0) | у' € W°} then
LcW°° = W and L is a о (X,X' )-closed subspace of finite co-
dimension. Since the ideal I generated by L coincides with the
subspace spanned by Lu XL и LXu XLX, it follows by (A) that
lex' 1(0). Once again if к is the canonical homomorphism of
X onto X/I, the proof is completed by observing that к(I) is a
closed subspace of X/I and I=I+L = к 1(к(1)).
In the discussion to follow it is shown that multiplica-
274
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
tion in an LMCH algebra X with continuous dual X' is separately
continuous in the o(X,X') topology but may not be jointly
continuous with respect to that topology.
(b)	SEPARATE VERSUS JOINT WEAK CONTINUITY OF MULTIPLICATION.
(i) If X is an LMCH algebra with continuous dual X' then
multiplication is separately о(X,X1)-continuous, i.e., the maps
x->-yx and x->-xy are continuous. (ii) If X is a complex semi-
simple A*-algebra (see after Def. 4.12-5) with continuous dual
X', then (X,o(X,X')) is a topological algebra iff X is fin-
ite-dimensional.
Hint to (ii): Suppose that a(X,o(X,X’)) is a topologi-
cal algebra and X is infinite-dimensional. Since X is semi-
simple, {0} = {f 1(0)|f€ X^}, and X is infinite-dimensional,
then X^ is infinite. Now X^ is a subset of the unit sphere of
the Banach space X' [(4.10-5)]; so f = 2 ..2 nh С X' where
r	neN n
(hn) is any distinct sequence of elements of x\ By (3) of
(a), f 1(0) contains а о(X,X1)-closed ideal I of finite co-
dimension. For each xf I, x* xf I and it follows that
h (x) = 0 for each n. Since each h can be written as h . к
n	n	n
where к is the canonical homomorphism of X onto X/I with
hn(x+I)=hn(x) for each x, it follows that (X/I)1 is infinite-
dimensional which is a contradiction.
4.2 Д =MC IN COMPLETE COMPLEX ALGEBRAS WHENEVER X c IS FINITE.
If M is finite in a complete complex LMCH algebra, then/{=<4fc
and all homomorphisms of X are continuous.
Hint: Let /Ц = {M1,...,Mn) and suppose that M / Atc
n
for some Mg^ . By (4.10-9), M = и £=1МПМ^. Assuming that
МП M. /f M(] M^ for any j / i we may find x^ e MQ M^ such
EXERCISES 4
275
that x.^ МП M.. Let у = x. + П x.. As ye Mf) M, for some
1	3	1	3	к
к, consideration of the possibilities i^k and i=k leads to
a contradiction.
4.3	THE CLOSED IDEALS OF C(T,£,c). If A is a closed subset
of the Hausdorff space T, then I = {x€ C(T,F,c)|x(A) = 0} is
a closed ideal in C(T,F,c). Thus the mapping А->-1д estab-
lishes a correspondence from the class of all closed subsets of
T into the class of all closed ideals of C(T,.F, c) .
(a) If T is Hausdorff then the closed subsets of T and the
closed ideals of C(T,F,c) are in 1-1 correspondence via the
mapping А->-1д iff T is completely regular.
Hint: Consider A = ^0 ^.x ^(0) and note that Ic 1Д.
If G is any compact subset of T then I' = l|G = {x|G|xe 1} and
I' = fair are ideals in C(G,F,c). I' is dense in I' and
thus I is dense in 1д.
4.4	FUNCTIONALLY CONTINUOUS COMPLEX ALGEBRAS. A complex topo-
logical algebra X is functionally continuous if X^ constitutes
all of the nontrivial homomorphisms of X. In the exercises to
follow, X^ carries the Gelfand topology o(X’,X) restricted
to Xh) .
(a)	If X*1 is not compact and there exists a closed (in X^1)
non-compact о(x*,X)-bounded subset of X*1, then X is not func-
tionally continuous.
(b)	A CLASS OF ALGEBRAS WHICH ARE NOT FUNCTIONALLY CONTINUOUS.
Let T be a completely regular Hausdorff topological space which
is locally compact and pseudocompact but not compact (e.g. the
ordinal space [0,0) where 0 is the first uncountable ordinal).
Then C(T,C,c) is a complete LMCH algebra which is not func-
276
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
tionally continuous.
(c)	A COMPLEX ALGEBRA X WHERE Xh IS COMPACT AND DISCONTINUOUS
HOMOMORPHISMS EXIST. Let X be the algebra of polynomial func-
tions on [0,1] with sup norm topology.
Hint: In this case X^ = [0,1]*; consider also homomor-
phisms defined by evaluation at points not belonging to [0,1].
(d) A CLASS OF FUNCTIONALLY CONTINUOUS COMPLEX ALGEBRAS X FOR
WHICH X^ IS COMPACT. Let X be a complex topological algebra
for which X^ is compact in the Gelfand topology. If X is a
symmetric algebra (Def. 4.12-5) and uM = UM then X is
c
functionally continuous.
Hint: As any homomorphism f on X induces a homomorphism
f on 'F(X), (Sec. 4.12) via the formula f('P(x)) = f (x) , it
suffices to show that the nontrivial homomorphisms of У(X)
are just the evaluation maps determined by the points of x\ To
do this first note that the hypothesis implies (1) if Fg Ч'(Х)
then F€ V(X) where F(h) = F(hT for he Xh; and (2) if
V(x)(h) f 0 for each he Xn then l/У(x)e ¥(X). Now use (1)
and (2) and imitate the proof that C(T,C,c)h = T* when T is
compact (see Example 4.10-2).
4.5	COMPLETENESS OF INITIAL TOPOLOGIES GENERATED BY HOMOMOR-
PHISMS . Let X be a complex algebra and H be a collection of
nontrivial complex homomorphisms on X with Gelfand topology
(see footnote to Def. 4.12-1). Letting F(H) denote the algebra
of all complex-valued functions on X, the map ,F:X-»-F(H) , x+Vx
has its range in C(H,C), the continuous complex-valued func-
tions on H. [H] denotes the linear span of H in X*.
(a)	The algebra X is о (X, [H] )-complete iff 'P(X) = C(H,C) =
EXERCISES 4
277
F(H) .
Hint: First note that X is о(X,[H])-complete iff: V(X)
is a complete subspace of F(H,p) where F(H,p) denotes F(H)
endowed with the point-open topology, as discussed in Example
4.3-1. Note that F(H,p) is complete. Show that V(X) is
dense in F (H,p) .
(b)	X is o(X,[H])-complete iff the Gelfand topology on H is
discrete and 'i (X) = C(H,C).
(c)	If X is a Banach algebra and H=X^, then X is a(X, [H])
complete iff H is finite. Furthermore if X is also semisimple
then (X,o(X,[H])) is complete iff X is finite-dimensional.
(d)	Let H be a set of nontrivial homomorphisms of X separating
the points of X and containing at least two non-isolated points
f and f^ (when H carries the Gelfand topology). Then X is not
a(X,[H ]) complete with HQ = H - {fQ}.
Hint: Note that x=0 if f(x)=0 for all f€ Ho=H-{fQ}
and recall from Example 4.10-1 that f is a а(X,[Hq])-discon-
tinuous homomorphism. Use the presence of f^ in HQ to show
that C(HQ,C) / F(H ).
4.6	A LOCALLY CONVEX TOPOLOGY FOR THE ALGEBRA C(t). Let C(t)
be the quotient field of the polynomial algebra C[t] (Example
4.9-1) of polynomials in t with complex coefficients. Let
re C(t) and consider r as a complex-valued function of a com-
plex variable. As such it has a Laurent series expansion
oo
r(t) =	2 a ts. Let S be the collection of all double se-
s= -°° s
quences wn, where n>l and wn(s) = (_s+l)n^ s+-*-) if s < -1;
1 if s=0; and (s+1)“<s+1)/n if s > 1. We define pR(r) =
2 la |w (s).
s= -°° * 1 2 s i n
278
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
(a)	The metric locally convex topology generated by the family
of norms P = (Pn) is compatible with respect to the algebraic
structure of C(t).
Hint:
gers r ans s
n £ 1.
Prove that w4n^r)w4n^s) wn(r+s) for all inte-
)2c e V for each
P,	Pn
and then show that
(b)	C(t) is not complete in the above topology.
Hint: Show that the sequence (rn(t)) =
— s (s fl) - s
(sE^(s+l)	t ) is Cauchy but does not converge.
4.7	BOUNDS ON COMPATIBLE TOPOLOGIES FOR C(t). Let C(t) be as
in the previous exercise. In order for multiplication to be
continuous with respect to a locally convex topology it can
neither be too weak (part (a)) or too strong (part (b)).
(a)	IftT is a compatible topology for C(t) then for each nei-
ghborhood of the origin there exists a neighborhood U such that
Uc V and sup {|a| |ar e u} <°° for each re C (t) . Thus if
is locally convex there exists a base of continuous norms gen-
erating and any weak topology o(C(t),N), where (C(t),N) is
a dual pair, fails to be compatible.
(b)	Multiplication in C(t) is separately continuous with the
finest locally convex topology applied, but this topology is
not compatible.
Hint: First observe that if ф is a real-valued function
on a complete metric space which never vanishes, then the set
of A such that lim ф (A) =0 is nowhere dense. Next real-
°	^Ao
ize that each re C(t) has a unique representation (by partial
fractions) in a finite sum of the form ^b^ gtS + s^sr>ibx,r
(t-A)-r. Let В be the set of all linear functionals f such
EXERCISES 4
279
that f((t-AQ) r) = 1 for a particular Aq and all r g 1,
f(tS)=0 for each s	0, and f((t-A) r) = 0 for each A/Aq
and r> 1. Then V=B° is a neighborhood in the finest locally
convex topology on C(t) and |b^	| <; 1 for all r(t) € V and
all A. Assume multiplication is continuous so that there ex-
2
ists a neighborhood of the origin U such that U с V. Define
Ф(A) = sup {|a||a(t-A) и} . Since ф(A) is never zero it
follows that there exists A and 6 > 0 such that for each e> 0
there exists A/A^ with |A-A^ | < C and |ф (A) |6. Using
.	-1	-12
this, show that 2	ф(А^)ф(А) (t-A^) C U eV but the coeffic-
ients b^ i of these rational functions are not bounded.
(c)	C(t) endowed with the topology of Exercise 4.6 is a local-
ly convex Q-algebra which is not an LMC algebra.
Hint: Show that inversion is not continuous.
4.8	NONCOMMUTATIVE DIVISION ALGEBRAS WITH CONTINUOUS INVERSE.
(Arens 1947a). In Sec. 4.9 it was seen that the only commuta-
tive real or complex locally convex Hausdorff division algebras
with continuous inverse are R and C. Here it is established
that if X is a real noncommutative LCH division algebra with
continuous inverse, then X is topologically isomorphic to the
quaternions.
To show this, proceed as follows. The center Z of X is
a commutative division algebra over R with continuous inverse
and so by Theorem 4.9-2, Z is isomorphic to R or C_. Since X is
not commutative, X possesses an element x which does not belong
to Z. The subalgebra Y generated by x and Z is a commutative
real division algebra with continuous inverse. As such, Y must
be isomorphic to R or C. Z, moreover, cannot be isomorphic to
280
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
C since x is outside Z. Furthermore each xeX must satisfy a
real polynomial equation of degree not greater than two.
Clearly X is at least a two dimensional space over R. if
{e,y^} is linearly independent in X, then у satisfies an ir-
2	2
reducible polynomial of the form t +at+b. Since 4b-a >0 we
2 1/2	7
may set x^ = (2/(4b-a ) ' )(y^+ae/2). It follows that x^ =-e.
Furthermore this relation makes it impossible for {e,x^} to be
a basis for X for in this event X is seen to be isomorphic to
C. Let {e,x1Zy2} be linearly independent in X such that
2
y2 =-e. Then Х^+У2 and xl-y2' neither of which is a mul-
tiple of e, must have minimal polynomials of degree 2. Hence
2
there must be real numbers r,s,u and v such that (x^+y2)	=
2
r(Xi+y2) + se = x-^2 + У2Х1 “ 2e and (x1~y2)	= u^xl-y2^ +
ve = -(х^у2+у2х^) - 2e. By adding these equations and using
the linear independence of {e,x^,y2}, we obtain r=u=0 and
s+v=4. Thus (1) Х^У2 + У2Х1 = (s+2)e = -(v+2)e. Since the
minimal polynomials of Х^+У2 and xl~^2 are irreducible, s
and v must be negative. It follows that 4< s< 0, so that
-s2-4s> 0. Let x2 = ( (s+2 ) / (-s2-4s)1//2) x1+(2/(s2-4s)1//2) y2
so that {e,x^,x2} is linearly independent, x22 =-e, and
x.x- = -x-x.. Finally, select x_=x.x_. To see that
J_	J_	J J-
{e,x^,x2,x2} is linearly independent, suppose that x^fe+gx^t
2	2
hx~. Then x-x. = x. x9 = -x_ = fx.+gx. + hx.x_ =
h(fe+gx^+hx2) so that f=g=h=0. Thus {e,x^,x2,x^} is lin-
2	2	2
early independent. Furthermore x. =x =x =x x„x = -e. It
J. £* О J- J
only remains to show that e,x^,x2, and x^ span X. Suppose
that xeX and x / Re. Then there is no loss in generality in
2
assuming that x =-e so that equations analogous to (1) may be
EXERCISES 4
281
obtained for the pairs (x^,x), (x^,x}, and (x^,x), i.e. there
are real numbers k,m, and n such that x^x=xx^=ke,x2x=xx2=me,
and x.x=xx =ne. Since x =x x=-x_x , it follows that
•J	j	О J-	J-
xxq = (xx.)x. = kx„ - x.(xx9) = kx9-mx. + (x.x9)x = kx9-mx. +
x_.x = kx9-mx. + x e - xx. which implies that 2y = -(kx.+mx9+
nx^). Hence {е,Х1,х2гХз} spans X and the proof is complete.
4.9 A-NORMED ALGEBRAS. Let X be a complex algebra which is a
LCHS. A seminorm p defined on X will be called an A-seminorm
if for each xex and all ye X there exists a real number
m
p,x
such that
p (xy) < m p (y) .
P » x
If the topology on X is
> 0
generated by a single A-seminorm, then we refer to X as A-norm-
ed. If the topology on X is generated by a family of A-semi-
norms, then we refer to X as locally A-convex.
(a)	If X is a complete A-normed algebra, then X is a Banach
algebra.
Hint: Consider the linear space isomorphism H:X+L(X,X),
x+A^ where L(X,X) is the Banach algebra of bounded linear
transformations on X and A^y = xy for all ye X. Show that
the mapping H has a continuous inverse and that H(X) is closed
in L(X,X). Then, assuming that X is complete, apply the
closed graph theorem.
(b)	If X is an A-normed division algebra, then X is topologi-
cally isomorphic to the complex numbers.
(c)	If X is an A-normed algebra and I a closed ideal in X,
then the algebra X/I with quotient norm is an A-normed algebra.
A maximal ideal in an A-normed algebra X is the kernel of a
continuous homomorphism of X onto C iff it is closed in X.
(d)	Let C[0,l] denote the algebra of continuous complex-
282
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
valued functions on the closed interval [0,1]. Consider the
function z(t) = t, 0< t< 1/2; 1-t, 1/2< t< 1. For any
xg C[0,l] let |Ьс II = sup z(t)x(t) . Show that C[0,l] is
1 " teI0,l]
A-normed but not normed.
Hint: Consider the functions xn(t) = 1-nt, 0< t< 1/n;
0, l/n< t. Show that the functions xn->0 but the linear
2	2
transformations A /0. Consider у (t) = n t, 0< t< 1/n ;
xn	n
2	2	2	2
2-n t, 1/n < t< 2/n ; 0, 2/n < t and show that
||А (У ) || / |fy |[> 1-1/n. (e) A subset U of X is said to be an
n
А-set if for each xf X and some real number a > 0, xUc a U.
-----	x	x
Prove that all of the following are A-sets:
(a)	the closure of an A-set;
(b)	the balanced hull of an A-set;
(c)	the convex hull of an A-set;
(d)	the balanced convex hull of an A-set;
(e)	the intersection of two A-sets.
(f)	Prove that the gauge of an absolutely convex and absorbing
A-set is an A-seminorm. If p is an A-seminorm show that
{xe x|p(x)< 1} is an A-set.
(g)	Prove that X is locally А-convex iff there exists a base
of neighborhoods of 0 consisting of A-sets.
(h)	(Michael, 1952) Prove that a barreled locally A-convex
space is an LMC algebra.
Hint: Let U be an absolutely convex closed A-set which
is a neighborhood of 0 in X and a be a real number such that
a> 0 and eg aU. Consider V = {xf x|xUcU}. Show that VcaU,
2
that V is a barrel in X, and that V с V.
(i)	Let Cq(R) be the strictly positive functions in С^(Е,С)
EXERCISES 4
283
which vanish at infinity. For each xf Cq(R) let Px denote
the A-seminorm p (y) = supf|x(t)у(t)|}. Show that the family
x	t£R
of seminorms {pxJ cannot be replaced by a family of multiplica-
tive seminorms generating the same topology.
Hint: Let x^ and x? be two functions in Cq(R) and sup-
pose that for some multiplicative seminorm p the set inclusions
V eV eV hold. Let a = min{l, max x_ (t) } and let b
Px2 P S	t6R 2
be a real number such that 0< b< a. Then for some t £ R,
x2(tQ)=b > x^(tQ) and for some positive integer n, bn <
x^(tQ). Consider the function у defined by y(t) =
(t-tQ+l/x2(t), tQ-l< t< tQ; (-t+t +1)/x2(t),tQ< t< tQ+l;
0, l<|t-t |. Show that p (y) = 1 and p (y11) >
°	_	x2	X1
y(t )nx. (t ) = b П X. (t ) > 1. But as p (y) = 1,	p(y)< 1
oxo	x о
and therefore p(yn)< p(y)n < 1. But then p (yn) < 1 which
X1
is a contradiction.
(j)	Let p be an A-seminorm on X. Then the null space N₽ of p
is an ideal in X.
(k)
Let p be an A-seminorm on X and N the null
space of p.
On X/N let a norm be defined by
llx+Np II = P<x> •
Show that X/Np
is an A-normed algebra.
(1) An algebra X is a locally А-convex algebra iff it is top-
ologically isomorphic to a subspace of a product of A-normed
algebras.
(m) Let X be an A-normed algebra whose topology is generated
by a family (p )	. of A-seminorms. Suppose that the alge-
01 01 £ /1
bras X/N are complete for all a. Then show that X is an
pa
LMCH algebra.
Suppose that T is a completely regular Hausdorff space
284
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
such that to each point t of T there corresponds a Banach alge-
bra B(t). Let F(T) denote the subset of the product of the
algebras B(t) such that for each xf F(T), the function
|x|(t) = ||x(t) ||	( || || denoting the norm of x(t) in B(t)) is a
continuous real-valued function on T. If H is a subset of F(T)
and H is an algebra (operations between functions in H being
the usual pointwise operations), then we refer to H as an alge-
bra of vector-valued functions. On H the topology of conver-
gence on compact sets has as a base at 0 sets of the form (K
being a compact subset of T, g> 0) V(0;K,e) = {xf h| |x| (t) <; e
for all t€ K}. On all Banach algebras we assume that the norm
of the identity is equal to 1.
(n) Let H be defined as above. Suppose that for each te T,
= {x(t)|xe H} = B(t), and for each continuous real-valued
function f on T, fx belongs to the closed ideal generated by x
for each хе H where fx(t) = f(t)x(t) for all te T. Then
the following statements hold:
(1)	Every closed ideal in H is of the form {xf H|x(t) e I }
where I is a closed ideal in B(t) for every te T. Conver-
sely, any such collection of closed ideals I in B(t) yields a
closed ideal in H.
(2)	Every closed maximal ideal in H is of the form {xf H|
x(t)€ where I is maximal in B(t) for each t€ T.
(3)	If B(t) contains a unique maximal ideal for each tc T,
then every closed ideal in H is the set of functions that van-
ish on a closed subset of T. Conversely every closed subset of
T yields a closed ideal in H.
(4)	If B(t) contains a unique maximal ideal for each tg T,
EXERCISES 4
285
then there is a 1-1 correspondence between the points of T and
the closed maximal ideals of H where t->-M = {xe H|x(t)=0} es-
tablishes the correspondence.
In the previous four statements generalizations of theo-
rems concerning the nature of closed ideals in C(T,C) have
been presented. Indeed if B(t)=C for all t, F(T)=C(T,C).
(o)	Let P denote a saturated set of continuous A-seminorms de-
fining the topology on the locally А-convex algebra X. Let
V(p;x1F...,xr, €) = {ge p| |P(x±) - q(x^) |<	. These sets form
a base for a topology on P with respect to which P is a com-
pletely regular Hausdorff space.
(p)	Assume that for each ре P, the factor algebra
completed to a Banach algebra X
X/Np can be
is an LMC
Then show that X
P
algebra.
(q)	Assume the condition of (p) applies. Consider the algebra
H of continuous functions on P taking values in the family of
Banach algebras X where H = {f If (p) = x+N ,xf X}. Let G be
p	x1 x	P
a map taking X into H where G(x) = f for each xe X; then
show that G is an algebra isomorphism between X and H, and that
G 1 is continuous.
In the final part of this problem, a representation of X
as an algebra of vector-valued functions is obtained.
(r)	Show that if X is barreled, then G is continuous.
4.10 P-NORMED SPACES (ZELAZKO 1965). In this exercise we con-
sider a generalization of the concept of a Banach algebra. A
p-normed space X is a topological vector space X whose topology
is generated by a p-norm || ||p 0< p <; 1, where a p-norm satis-
f ies:
286
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
1)	||x||	0, ||x || =0 if	and	only if	x=0.
2)	||x+y || s ||x|| + ||y|| .
3)	||ux || = |u|P ||x || for	all	scalars	and	any xf X.
A locally p-convex topological vector space is a space
whose topology at 0 is generated by a family of balanced nei-
ghborhoods V each of which satisfies the relationship
n	n
V for any in such that | in |p 5; 1 and x^ e V.
Such a neighborhood V is called absolutely p-convex.
(a) A topological vector space X is p-normed iff there exists
a bounded absolutely p-convex neighborhood of 0.
Hint: ||x ||p = inf{rP|xe rV, r 0}.
(b) A locally bounded (there exists a bounded neighborhood of
0) topological vector space is p-normed.
Hint: For every bounded balanced neighborhood U of 0
there exists a real number k;> 2 such that U+Uc kU. The
greatest lower bound of all such numbers к is called the module
of concavity of U and the greatest lower bound k(X) of all
modules of concavity is called the module of concavity of X. Of
course k(X) 2. Let k(X) =	and consider any p< pQ.
Then there exists a bounded balanced neighborhood U of 0 such
that U+Uc 21/,pU. This relationship can be extended (Kbthe
n _.	,	n .
1966) to -£12 i'pUc U for £^12 1 <. 1. Now the absolutely
1	n	~	n
p-convex hull Г (U) = {.Z.y.x.|x.e U and |y.|p <1} of U
r	p	1=1 1 11 1	i=l1 11	=
can be shown to satisfy the relationship fp(U)c	and the
gauge ||	|^ of Гр(и) generates the topology.
(c)	Let X be a complete metrizable algebra in which multipli-
cation is separately continuous. Then the following are equi-
valent:
EXERCISES 4
287
(1)	There exists a metric p generating the topology such
that P(xy,0) < p(x,0) P(y,0).
(2)	X is locally bounded.
(3)	X is a p-normed algebra ( ||xy <; ||x[[ |[y |Jp) for
all x,y e X.
Hint: Assuming that X is p-normed (or equivalently, by
(b) and (a), locally bounded) then the topology on X is genera-
ted by a p-norm || which is not necessarily submultiplica-
tive. However, as multiplication is separately continuous,
letting denote the continuous linear map ytxy, ye X,
||yx |L
IM = suPy/0	< ”. Letting ||x	= |^x ||,	|| |Г is the
desired norm.
(d)	Show that the algebra £	(0 < p <. 1)
sequences x = (Pn) of scalars satisfying
of all two-sided
nx Up=:i -~kip'
is a p-normed algebra when multiplication is defined as Cauchy
product.
(e)	Show that the algebra X of all holomorphic functions in
OO	oo
the closed unit disc x(A) =	with ||x j|^ = n^0 I yn I ₽ is
a p-normed algebra.
In the next series of exercises we develop a concept
which leads to an analog of Gelfand theory for p-normed alge-
bras. Let X be a complex complete p-normed algebra and Kg =
{xe x|lim ||xn ||p = 0}. The spectral norm ||x ||g =
(sup{|A||Axe KgJ) P.
(f) The spectral norm has the following properties:
(1)	IIх ils< 1 ;	iff xe кд	(4)	IIMs	IIх Ils 1^ Ils
(2)	IIMs =	1 Л 1 p IIх Ils	(5)	Н*П Us =	li<
(3)	lix+y Ils	IIх lls + l|y Ils	(6)	IIх Ils	IIх lip
288
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
(7) If x^X, ||x||s> 0	(8)	||x||s = limn( ||xn|jp)1/n
Hint: To prove (3) it is sufficient to show that if
IIх lis + IHS< then llx+y||s< i- Let a > ||x ||д, B> Цу||д,
a+B< 1; as ||(х+у)П ||p = ||(a1'/,pa l/px+Bb/pB b/py)n |<
Z ,n.p n-к.кц, -1/p .n-k, -1/p ,k,,	,	Ц, -1/p ,Пц _
k=0 к a 6 H(a x) y) IIP’ by (2) H(a x) llp->0
and ||(B l/Py)П Цр+О. Letting nQ be such that if N> nQ then
||(a1//px)N k ( 31//₽y) k ||p< 1 for k=0,...,N, then || (x+y)N <;
? ,N,p N-k к N ,N. N-k.k , 1O.N ,	..... .
k50(k) а В ^к=0(к} В = (a+B) <1. Thus ||x+y ||g <1.
As in the Banach algebra case, the invertible elements of
a complete commutative p-normed algebra with identity are an
open set, and inversion is continuous on the invertible ele-
ments .
(g) If X is a complex complete commutative p-normed field and
Y is a closed subalgebra of X, then Y is a field.
Hint: Let x be a non-zero element of Y. Assuming that
x/ Ae for any scalar A we observe that A = {A|(x-Ae) Y}
is a closed and open subset of the complex plane, hence A=C.
Thus (x-(l/n)e) Y for all n and therefore x Y.
(h) A p-normed field over the complex numbers is topologically
isomorphic to C.
Hint: First prove the following statement: If X is a
p-normed field, then for every A/0, О 0, x£X and x/Ae,
there exists a sequence of polynomials Wn(x) such that
|| e-(e+xWn (x) )-1 ||p < e for large n. To do this consider the
closed algebra X(x) generated by x and e. By (g) X(x) is a
field. Now use continuity of inversion.
Now let xf X and f (A ) = ||(x b+Ae) 1 ||s - Show that as
EXERCISES 4
289
|X|->°°,	f(X)->0. Letting f(XQ) be the maximum of f, if у =
(f (XQ) ) (x-1+XQe)-1, then ||y |jg = 1 and ||(y-1+Xe) 1 ||g <; 1
for each Xe C.	Let	V (X)	= Xn+a1Xn 1+...+an	be a sequence
of polynomials.	Then	Vn(y	1) = (У 1_3^e)•••(У	^-5ne) for
suitable 3 j € C	and	||Vn (y	1) ||g <; 1. However	[Vr (y 1)] 1 =
yn[e+yWn(y)] 1 where Wn(y) = a^tc^yt...+апУП Choose Vn
such that ||Xe-(e+yWn (y)) 11|₽ <	e. Let z =	Xe-[e+yWn(y)]	1
so that ||z ||g <. ||z 1^ <	e. Then	||yn (Xe-z) ||g=	||yn (e+yWR (y))-1 ||g
< 1 and therefore 1	||Xyn-zyn	||g	| X | p ||yn	||g - ||z ||g ||yn ||g	>
( | X | p- e) ||УП ||s so that	||yn||s <	1/(|X|“€)< 1	for suitable	e
and |X | , and ||y || < 1. But this contradicts ||y ||g = 1.
From the previous result, the basic theorems of Gelfand
theory concerning maximal ideals and continuous complex-valued
homomorphisms of a Banach algebra X hold for p-normed algebras
(see Sec. 4.12).
4.11 RADICALS OF P-NORMED ALGEBRAS. In this exercise we con-
sider the relationships between the spectral norm ||	||g of a
complete commutative p-normed algebra X with identity and the
radical of X. The result of principal importance (part (c)) is
that rad X = {xe x| ||x || = 01.
(a) К = {xe x| ||x ||	<: 1} is convex and therefore ||	||* =
||	||g/p is a seminorm on X.
Hint: As К is a closed subset of X, it is sufficient to
show that if x,ye К then x+y/2e K. This is equivalent to
showing that if ||x ||g < 1 and ||y ||g< 1, then ||x+y/2 ||g s 1.
But ||x+y/2 ||s = (l/2)plimn( ||(x+y)n||p)1/n <; (l/2)plimn
(к£0(£)Р||хкуП“к||р)1/П. As ||x||g <1 and ||y|is <1, then if
n is made sufficiently large, for all 0< k<; n ||x у |L < 1
290
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
and ||x+y/2 || < (l/2)₽lim ( 2_(2П)Р) =	(y) Plim ( (2n+l)
T ~ l/2n . ,	T l/2n
(2П)Р) = 21/plimn((Jn)p) = 1.
(b) Every (necessarily continuous) multiplicative linear func-
tional f on X is continuous with respect to ||	||s and
lf <x) lp < IIхlis IIхllp-
Hint: Show that if ||x j| < |f(x) |P for some xe X then
letting у = x/|f(x)|, ye Kg (see preceding exercise) and
yn->0, violating continuity of f.
(c) Let Xh be the set of all (continuous) complex-valued mul-
tiplicative linear functionals on X. Then ||x || = sup, ft
11 HS	1 с л
|f(x) |p. Hence xe Rad X iff IIх ||s ~ 0.
Hint: From (a) ||	||* = ||	||l/,p is a seminorm on X and
as ||	||s is continuous, I = {xe X | ||x || =0} is a closed
ideal in X. Let X=X/I with a norm defined on X by ||x+I || =
infyeI l^+y II = IIх Il-
Then X is a normed algebra with the proper-
ty that all multiplicative linear functionals f on X are of the
form f (x+I) = f (x) for some fe X^. Hence all linear func-
tionals on X are continuous. From standard Gelfand theory
sup Ah lf (x+I) I = sup , |f (x) I = lim ( || (x+I)n ||)
fe x	fe x
ii. (их" Ц. >1/n - n«„<we/p>1/n= Ms/p = m*-
4.12 F-ALGEBRAS. An F-space X is a complete metrizable topolo-
gical vector space. In Dunford and Schwartz 1958 (p. 51) it is
shown that the topology of X can be generated by an F-norm,
i	.e. a mapping ||	|| of X into R satisfying
(1)	|^ |j 0, ||x || = 0 iff x=0.
(2)	||x+y || <; ||x [j + ||y|j.
(3)	||ux|| = IIх ii for |n | =1.
EXERCISES 4
291
(4)	if Un“>0, then lim ||Unx ||=0 for all хе X.
(5)	If	then lim ||yxn |[=0 for all p.
(6)	d(x,y) = ||x-y || is a metric generating the topology
of X and X is d-complete.
If X is an F-space and an algebra in which multiplication
is separately continuous, then multiplication is jointly con-
tinuous .
Hint: If Cr(xQ) = {xe	x|	||xQ-x ||<: r},	then	C]yn(0) =
Un,ne N, V is a base of neighborhoods of 0.	If	U is a neigh-
borhood of 0 in X and An =	x|xUn<zU},	then	by separate
continuity of multiplication	in	X, (JAn = X	and	as X is a
Baire space and each A is closed, there exists A such that
r	TL	П
О
int A / 0. Thus for some x£ X and r> 0, С (x)c A . If
n	r n
о	о
xfC(0,r ) then xU cU+U. Let V be an arbitrary neighbor-
°	no
hood of 0 and U be chosen such that U+UcV. Then C (0)U c
r n
2	°	°
V and also [C .	, ,	. (0) ] с V.
min(ro,l/no)
4.13 FRECHET ALGEBRAS. We consider in this exercise an alge-
bra X carrying a topology with respect to which it is a Frechet
space. Having shown in the previous exercise that if multipli-
cation in this algebra is separately continuous, then it is
jointly continuous, we show in (a) that a family of seminorms
generating the topology of X can be produced which are "almost
multiplicative". Interestingly enough, the property they do
have is sufficiently strong so that it can be said that these
algebras resemble LMC algebras quite strongly.
(a)	Let X be a Frechet space and an algebra in which multipli-
cation is separately (therefore by Exercise 4.1 jointly) con-
tinuous. Then the seminorms generating the topology of X can
292
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
be chosen so that they are increasing and p^(xy)< p^+^(x)
Pi+1(y) for all i and x,ye X.
Hint: Let (pn) be an increasing family of seminorms gen-
erating the topology and p be any continuous seminorm. Then
there exists C> 0 such that p(xy) < Cpn(x)pn(y) for some
n> 0; otherwise there exist sequences (xn) and (y ) such that
2
p(x у ) > n p (x )p (y ) >0. First we deal with the case
1 rn *n n n Jn =
where Pn^xn)Pn^n) ~ ® for infinitely many n. If infinitely
many p (x ) =0, then x +0 and in fact for any m, p (x )
-1 n n	n	m n
ultimately is equal to 0. Thus хп/р(хпУп) = Zn^®" With no
loss of generality the sequence (y ) could have been chosen so
that у +0. Now p(z у ) = 1 while z +0 and у +0. This is
n	n n	n	n
a contradiction. For the case where we can assume that p (x )
n n
and p (y ) are never zero, we simply let z = x /np (x )
n n	n n n n
and w = у /пр (у ). Once again w +0 and z +0 but
n n n n	n	n
p(z w ) >1. Now let q =p. and for the first i such that
q^(xy) < Cp^(x)p^(y), set q2 = /Ср.. Continuing in this way
we generate the desired seminorms.
A Frechet space which is an algebra in which multiplica-
tion is separately continuous (or in which there is a family of
seminorms generating the topology which satisfy the conditions
of (a)) is called a Frechet algebra.
(b)	Show that a complex Frechet division algebra is topologi-
cally isomorphic to C.
Hint; Using Theorem 4.9-1 we need only prove that inver-
sion is continuous. Since the units are open they are a G -set
о
Then Theorem 7.4 of Zelazko 1965 may be applied.
(c)	Let X be a complex Frechet algebra and suppose that there
EXERCISES 4
293
exists a closed maximal ideal in X. Show that a system of
seminorms (p^) can be found satisfying the conditions of (a)
such that рЛе) = 1 for all i.
Hint: Let (q^) be a family of seminorms generating the
topology of X and satisfying the condition of (a). If M is a
closed ideal in X then X = M + {Ae|Ag C} . If xf X then x =
m+Ae with Лe C. Define w^(x) = q^(m) + |A| for each i. If
у = m' + ye, then w^(xy) = q.(mm'+Am'+ym) + | A | | у |	q^(mm’)
+ |A|qi(m') + |y|qi(m) + |A| | у | <. qi+1 (m) qi+1 (m' ) + |A|qi+1(m')
+ |y|qi+ (m) = wi+1 (x)w	(y) . Clearly w^e) = 1 for all i
and the family (w^) generates the topology of X.
4.16 ES-ALGEBRAS. In this exercise we consider complete topo-
logical algebras X with the property that every continuous ho-
momorphism on a closed subalgebra Y of X can be extended to a
continuous linear functional on X. Such algebras are referred
to as ES-algebras. We list first a number of statements (see
Narici, Beckenstein, and Bachman 1971, Rickart 1960, and Nai-
mark 1964.)
(1)	If T is a locally compact Hausdorff space and T is
totally disconnected, then T is ultraregular.
(2)	If S is a compact totally disconnected subset of C,
then int S=0. S is totally disconnected iff it contains no
continuum.
(3)	If X is a Banach algebra and Y is a closed subalge-
bra, then ov(x)cov(x) for all xf Y and bdo (x)c bdo (x).
(4)	If X and Y are as in (3) then f6 Y^ is extendible
to fe Xh iff f is in the Shilov boundary of Y (f€ Xh is in
the Shilov boundary of X iff for every neighborhood и of f in
294
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
the Gelfand topology, there exists xe X such that the Gelfand
function Vx fails to achieve its maximum absolute value on CU.)
(5)	If X is as in (3) and ftf(X) = U U2 where	is
open and	= 0, then there exists xe X such that
2
x =x and the Gelfand function Vx is the characteristic func-
tion of U^.
(a)	Let X be a Banach algebra and Y a closed subalgebra. If
a (x) is totally disconnected for x€ Y, then c,,(x) = a,, (x).
(b)	Let X be a Banach algebra and suppose that for some xe X,
a (x) contains a continuum. Then there exists a closed sub-
X
algebra YeX and fe Y^ such that f is not extendible to
fe xh.
Hint: If there is a continuum joining A. and Ao in av(x)
let	у = х-Л^е/(A^-^i) and z =	. Then z is inver-
tible in X and there is a continuum in a (z) joining 1 and i.
X
4
Let Y be the closed subalgebra of X generated by z and e. Let
4	4
f(p(z )) = p(0) for any polynomial p in z . By the maximum
modulus principle of complex variable theory, |f(p(z^))| <
4
|jz । || hence f can be extended continuously to a homomorphism on
4	4
Y. However as z is invertible in X and f(z ) = 0, f cannot
be extended to X.
(c)	A Banach algebra X is an ES-algebra iff for any subalgebra
YeX, Y^ is totally disconnected in the Gelfand topology.
(d)	A complex Banach algebra X is an ES-algebra iff a (x)
is totally disconnected for all xe X.
Hint: If for some xe X, a (x) is not totally discon-
X
nected, taking Y to be the closed subalgebra generated by x and
e, since M(Y) is homeomorphic to о^(х), we see that X is not
EXERCISES 4
295
an ES-algebra.
(e)	If T is a compact Hausdorff space and C(T,C,c) is an ES-
algebra, then for any complex Banach algebra X such that Л((Х) =
T, X is an ES-algebra.
(f)	If G is a compact abelian topological group, then L^(G)
with convolution as the multiplication in L^(G) is an ES-al-
gebra and any multiplicative linear functional on L^(G) is of
the form f(x) = J x(t)x(t)dt where X is a continuous charac-
ter and integration is with respect to Haar measure on G.
Hint: Although L^(G) does not have an identity, it can
be complexified and an identity adjoined. The multiplicative
linear functionals on L^(G) and the algebra with identity ad-
joined can be placed in 1-1 correspondence and the spectrum of
each element in L^(G) is the same in both algebras,
(f) A Frechet algebra X is an ES-algebra iff for each x£X,
a (x) is totally disconnected.
Hint: If (Xn) is a denumerable family of factor algebras
associated with a collection of seminorms generating the topo-
logy and К is a continuum in a (x), then as a (x) =
X	X
U ..a (к (x) ) , it follows that for some n, a (к (x) ) con-
П E W X_ И	X И
n	n
tains a continuum. Thus Xn is not an ES-algebra and there ex-
ists a closed subalgebra Yn of Xn and a homomorphism f on Yn
which cannot be extended to Xn> Let Y be the projective limit
of the algebras X^ (k/n) and Y . Then Y is a subalgebra of X
and f induces a homomorphism on Y which cannot be extended to
X.
4.17 PERMANENTLY SINGULAR ELEMENTS IN TOPOLOGICAL ALGEBRAS. In
this exercise X is a complete topological algebra and we ex-
296
4. COMMUTATIVE TOPOLOGICAL ALGEBRAS
plore the nature of permanently singular elements of X, i.e.,
elements xf X such that x has no inverse in any superalgebra
of X (Arens 1958a, Zelazko 1971, Kuczma 1958, Suffel, Becken-
stein and Narici 1974) .
A topological divisior of 0 c in a normed algebra X over
a valued field is an element c€ X such that for some sequence
xr€ X,cxn+0 while xn/°. This is equivalent to requiring that
the multiplicative operator y*xy not have a continuous in-
verse. Another form of this is that a = inf Цех Ц-” 0.
x/0
(a)	An element of a normed algebra X has no inverse in any
superalgebra of X iff it is a topological divisor of 0.
Hint: Let t> 0 and X(z;t) (z transcendental over X) be
the algebra of formal power series in z with coefficients from
OO	oo
X such that for any such series f(z) = Lnx zn, llx ||tn<
2	n=0 n ' n=0 11 n 11
oo
Let ||f (z) || = Eq ||xn ||tn and J be the smallest closed ideal in
X(z;t) generated by e-cz. Let Y = X(z;t)/J be the quotient
algebra with ||f(z)+J|| = inf ||f (z) + j (z) ||. We show that for sui-
table t X is isometrically embedded in Y. Let t ||cx ||	||x ||
for all xgX (a t 1) with t > 0. Let xf X and g(z) =
x-(e-cz) L_x zn = (x-x ) + (ex -x_ ) z + (ex -x_)z +... . As
n=0 n	о	о 1	12
о
||g(z) || = ||x-xo || + ||cx0-x1||t +||cx1-x2||t + ...£• ||x ||-||xq || +
(to1Hxoll"llxlll)t +-’- = IM + (tto1} Hf(z) II' if we choose %
the proof is done.
A nonarchimedean normed algebra over a (necessarily) non-
archimedean valued field is a normed algebra such that ||x+y || s
max( ||x ||, jjy ||) for all x,ye X. A nonarchimedean LMCH algebra
is a topological algebra over a (necessarily) nonarchimedean
valued field К whose Hausdorff topology is generated by a
EXERCISES 4
297
a family P of submultiplicative seminorms such that p(x+y) <
max(p(x), p(y)) for all x,yf X, and each p€ P. The factor
algebras associated with these seminorms are nonarchimedean
Banach algebras.
(b)	Let X be a commutative nonarchimedean normed algebra with
identity over a complete nontrivially valued field. Then an
element eg X is permanently singular in X (has no inverse in
any nonarchimedean superalgebra) iff a = inf ||cx ||> 0 (c is a
x/0
topological divisor of 0 in X).
Hint: Consider t such that a > t and t such that
°	co =	°
t t 1 > 1. Apply to X(z;t) = { l-X tn I llx ||tn->-0} a nonarchi-
medean norm ||f (z) || = supn ||xntn ||. As in (a) we show that X is
isometrically embedded in Y = X(z;t)/J as follows. Let xe X.
Then x-(e-cz)f(z) = x-xq + (cxq-x1) z+. .. . If |[x || / ||x||,
then	||x-(e-cz) f (z) ||	||x||. If ||xq || / ||x || then show that
||xn_^c || = ||x || cannot occur for all n without violating the
requirement that ||cxn_^ -xn||tn^0.

FIVE
Hull-Kernel Topologies
THIS CHAPTER IS directed mainly toward investigating the question: When
may the space of maximal ideals of a complex U4C algebra X be viewed as a
Wallman compactification of the space M of closed maximal ideals of X?
As to why such a question would be considered, one could look back to alge-
bras C(T,F,c) of continuous functions with compact-open topology. The
closed maximal ideals Mc of C(T ,F,c) could be identified with the points of
T by Example 4.10-2, while all the maximal idealsM were in 1-1 correspond-
ence with ₽T. Thus, for C(T,F,c), it could be said that	, and we
have the first suggestion that the maximal ideals of a topological algebra
f
might be a compactification of the closed maximal ideals in more general
situations. Of course to begin to consider the question Я and must be
endowed with topologies and this is the subject to which Sec. 5.1 is devoted.
The ramifications of imposing normality type conditions on these topologies
are considered in Sec. 5.2.
In Sec. 5.3 a sufficient condition - called condition hH - on a topo-
logical algebra for X to equal c is discussed. The way in which this -
realizing M as SM c - is done in Sec. 5.4 is to show that/4 can be realized
as a Wallman compactification of JA. in the presence of condition hH.
Experience with algebras C(T,R) of continuous functions suggests a way
to determine which maximal ideals of a topological algebra X are kernels of
homomorphisms. For C(T,R) those maximal ideals which corresponded (under
the correspondence of the Gelfand-Kolmogorov theorem, Theorem 1.4-1) to
points of the repletion ul of T were kernels of homomorphisms. The develop-
ments of Sec. 5.4 show an analogy between T and Mc> so perhaps those maxi-
mal ideals of X which are kernels of homomorphisms are determined by some
sort of repletion of Such a repletion is defined in Sec. 5.5 and a
result of the type just mentioned is obtained there in Theorem 5.5-1.
In Sec. 5.6 the well-known result that Banach algebras are regular iff
they are normal is generalized to Frechet algebras.
5.1 Hull-kernel topologies Our goal is to relate the topological properties
of Л( endowed with hull-kernel topology to the algebraic and analytic proper-
ties of X. To define the topology we consider two operations - computing
hulls and kernels - on Ideals and then show that the one followed by the
other constitutes a closure operator.
299
300
5. HULL-KERNEL TOPOLOGIES
Definition 5.1-1 KERNELS AND HULLS The //-kernel K(S) of a subset S of
is	The -kernel k(S) of a subset S of A( c is ^gM. S=0,
then k(S)=K(S)=X. The /{-hull H(I) of an ideal I of X is [Ме/(| 1СМ}; the
-hull h(I) of I is [МеЛ(с|1СМ}.
Below are some of the basic properties of hulls and kernels.
(5.1-1) PROPERTIES OF HK AND hk The operators hk and HK defined on the
subsets S of Л( and// respectively satisfy the following relations.
(1)	hk(0)=HK(0)=0.
(2)	SCIHK(S) and Sc.hk(S).
(3)	HKH(I)=H(I) and hkh(I)=h(I) for each ideal I of X.
(4)	HKHK(S)=HK(S) and hkhk(S)=hk(S).
(5)	HK(SUT)=HK(S)UHK(T) and hk(S UT)=hk(S)Uhk(T).
(6)	HK({m})={m} and hk([M})=[M] for any MeM in the first case, any
in the second.
(7)	«([J I j)=f| H(In) and 1 j)= flh(I«here [И I ] denotes
Ц ц- Ц» ц»	М» Ц» Ц» Ц»	u> |1
the ideal generated by U^I •
Proof Due to the similarity of the arguments we only give proofs for HK.
(1)	HK(0)=H(X)=0.
(2)	If MES then, since K(S)CLM, MfHK(S).
(3)	Since IC2M for eachMEH(I), ICKH(I). Thus if MEHKH(I),
IC.KH(I)CLM and MEH(I), so that HKH(I)CH(I) . Conversely if
M EH(I), then КН(1) = П	and M EHKH(I). Thus HKH(I) =
о	MEH(I) о о
H(I).
(4)	Follows from (3) with I=K(S) .
(5)	Clear ly K(S J T) =K(S) f) K(T) . If MeHK(S (J T) , then MJK(S)f) K(T)
2)K(S)K(T). As a maximal ideal is a prime ideal, MOK(S) or
M3K(T). It then follows that MfHK(S) U HK(T) . Hence HK(S(JT)g:
HK(S)0HK(T). The reverse inclusion follows from the facts that
St) IDS and S l)TDT.
(6)	Clearly K({m}) = [m}. Since M is a maximal ideal, Н(М)={м}.
(7)	Suppose ME Л H(I ). Then M’DI for all p and thus M3[U L].
Ц Ц»	Ц,	M1
Conversely if М€Н(ГС1 I,, ]) then MIDI for each ц and therefore
u Ц» Ц»	ц,
меПцН(1 ). v
Definition 5.1-2 HULL-KERNEL TOPOLOGIES As a consequence of (5.1-1) the
operator HK defined on the collection of all subsets of At is a closure
operator and generates a topology «7 on called the /f-hull-kerne 1
HK
5.1 HULL-KERNEL TOPOLOGIES
301
topology or just the hull-kerne 1 topology. The restriction otto -Xc
is called the -hull-kerne 1 topology (or just hull-kernel topology, de-
pending upon the context) and is denoted by sT^k'
It is easy to verify that -^"^k is a^s0 determined by the operator hk
on Л'с-
By (6) of (5.1-1), (X,^ ) is a Т.-space. By the results of (5.1-1)
HK	1
(3) the НК-closed sets in Д are exactly the sets H(I) where I is an ideal
in X and the НК-closed subsets of are the sets h(I) where I is any
ideal in X.
Our next result, (5.1-2), determines a base for the hull-kernel top-
ology; H(x) and h(x) denote H([[x} ]) and h([[x}]).
(5.1-2) A BASE OF CLOSED SETS FOR THE HULL-KERNEL TOPOLOGY (a) The
sets H(x) are a base of closed sets for on M. 	(b) The sets h(x) are
HK
a base of closed sets for ,, on M. .
hk c
Proof (a) As previously noted in the discussion immediately following
Definition 5.1-2, the closed sets of (X,^/	) are all of the form H(I)
HK
where I is an ideal in X. Since H(I)=C|	H(x), (a) follows. The proof of
xel
(b) is similar and is omitted, v
(4.12-3) has to do with when M was compact in the Gelfand topology
J' , i.e. a(X ’ ,X)-compact. (5.1-3) shows thatX is always compact when it
G
carries the hull-kernel topology.
(5.1-3) M. IS НК-COMPACT (/И ,x7.„.) is a compact topological space.
HK
Proof Let (F ) be a family of НК-closed subsets of M with empty intersec-
tion. Then Q F =Г| HK(F )=Н(П! K(F )1)=0. Thus [11 К (F )]=X. Since efX
‘ P p ' P p	p P	p p
there must be indices p1,...,pn> elements	), and elements
y,,...,y ex such that e=T.y.x.. It now follows that [I 1 K(F ) "1=X and
1 n	r i i	i=l Pi J
tha t
0 = H([ ’J K(F )]	HK(F ) =d F  V7
i=l ^i	i=l ^i i=l ^i
Definition 5.1-3 HULLS OF FINITE SETS For any finite subset fx,,...,x ]
----------------- ------------------------ 'I	n-'
of the topological algebra X the /(-hull H(x,,...,x ) of fx,,...,x 1 de-
--------------------------------------------	1 n u 1 n
notes the /(-hull in the sense of Definition 5.1-1 of the ideal (x^,...,x )
generated by fx,,...,x } in X. The M -hull h(x,,...,x ) of fx,,...,x 1
L 1 n	‘-‘c--- 1 n 1 1 nJ
is defined similarly.
We note that
А £н(хг) = н(х1,... ,xn) = [ме/( |х£ем, 1=1,... ,n}
and
302
5. HULL-KERNEL TOPOLOGIES
(a) The lattice £=[h(x^, .. .	|x^ fX, nfNjis an a₽- lattice
subsets of M c generating	(b) The lattice Я =[H(X ...
nCN} is an aB- lattice of closed subsets of X generating
HK
(Th(x.) = hfxp . . . ,xn) = (МеД(с|х^еМ, i=l,...,n).
For many topological algebras X it is true that M=w(Xc , £) where £
is the lattice of hulls of finite subsets of X (see Theorem 5.4-1). We
begin the approach to that result with (5.1-4) below.
(5.1-4) LATTICES OF HULLS OF FINITE SETS Let X be a topological algebra.
of closed
• ,x ) lx.CX,
nli
Proof We prove (a) only. In view of (5.1-2) it only remains to show that
£ is an czB-lattice. To begin, since
h(x)tjh(y) = h(xy) and h(x)f|h(y) = h(x,y)
it easily follows that £ is a lattice.
To show tha t £ is an a-la ttice (Def. 3.3-1), let h(x, , . . . , x ) e£ and
1 n
Mtfh(Xp . . . ,х^) . There must be an x_. such that x . ^M. Moreover there must
be some zfX and mfM such that zXj+m=e. Thus Mfh(m) but h(m)f') h(x^, . . . ,x ) =
0 for if J£h(m)f) h (x j,. .. ,x_|) then x^, mjj which implies that efj which is
contradictory. It follows that £ is an a_lattice. The proof that £ is a
0-lattice (Def. 3.3-1) is similar. V
5.2 Regular algebras and normality conditions So far two topologies, the
hull-kernel and the Gelfand , have been considered for the space
HK	G
of maximal ideals of a topological algebra. Under certain conditions
-J"G is a compact topology foras (4.12-3) shows and, as shown in (5.1-3),
is always a compact topology for M.  These phenomena bear a strong
HK
resemblance to results of Chap. 1 dealing with maximal ideals of algebras
С(Т,£) of continuous functions: The maximal ideals of C(T,F.) were (in 1-1
correspondence with) the Stone-Cech compactification ST of T (Theorem
1,4-1). By Example 4.10-2 the continuous homomorphisms of C(T,£,c) - or
what amounts to the same thing, the closed maximal ideals of C(T,F.,c) -
are just T for any completely regular Hausdorff space T. Thus in many
cases X is the Stone-Cech compactification of M. Section 5.4 is devoted
to exploring when statements such as "X =₽Xc" and "A(=w(X , £) for some
£" hold in various topological algebras and this section is devoted to pre-
paring the ground for some of those results.
We begin by introducing the notion of regular algebra. We mention
that aside from the role regularity will play in the theorems of Sec. 5.4,
it also occupies a significant position in the theory of Banach algebras
5.2 REGULAR ALGEBRAS
303
vis-jJ-vis continuous extendibility of homomorphisms of the Banach algebra
X to superalgebras Y of X. If X is a regular Banach algebra and Y is any
Banach algebra containing X then any homomorphism (perforce continuous if
X is a Banach algebra) of X may be continuously extended to a homomorphism
of Y. Equivalently, if X is regular, any maximal ideal of X may be em-
bedded in a maximal ideal of Y (Naimark 1964, pp. 214 and 223).
For complete LMC algebras a maximal ideal is closed iff it is the
kernel of a complex-valued homomorphism. Thus, for any closed maximal
ideal M, X/M is topologically isomorphic to C and each xfX determines a
unique scalar x+M. The points x in X may now be viewed as a family of
maps x taking into C, namely M -> x+M. Viewed this way, if X separates
points and closed subsets of M (appropriately topologized) then X is reg-
ular; if X separates disjoint closed sets in Л , X is normal. A reason
for our interest in algebras which satisfy these "normality" conditions
lies in the results of Sec. 5.4: For certain normal algebras,Л1 may be
realized as a Wallman compactification of .
Definition 5.2-1 REGULAR ALGEBRAS A complex LMC algebra X is regular if
for any -J^^-closed subset F of and M^F there exists xfX such that
x(M)41i while x(F) = [o}.
A large category of regular algebras is provided by (5.2-3). An
equivalent formulation of regularity is given in (5.2-1) next.
(5.2-1) REGULAR-«-»--JThe complex LMC algebra X is regular iff
J hk=JG °П M c •
Proof We first show that -J is generally finer than
If F is a , -closed subset of M , then F=h(k(F)). Since F=
hk	c
Г) , , . Ь(х) = Г) , , ,x (0), then, since each x is continuous when Л(
' 1 xfk(F) v ' 1 'xek(F)
carries the weak-* (=«/'„) topology,
G
is therefore a -J -closed subset of
G
Suppose now that X is regular,
ogy, and M^F. Then there is an xfX
ХЙ4 while xfM' for all M'eF so that
-1
x (0) is -J -closed for each xfX. F
G
M , It follows that , C. .
c	hk G
that F is closed in the Gelfand topol-
such that x (M)={1} whi le x(F) = {0}. Thus
FCh(x) while M^h(x).
MeCF there exists x^EX such that FCh(x^) while M^h(x^).
П MECF11^ and F 1S -Jhk'C1°Sed-
Conversely if	let F be a
hk G
Then F=h (k(F) )= Axek(F)h(x) and tor M^F
Mlfh(x^) or, equivalently, x^^M. On the
Consequently ^(Mj^O while $M(F)=[0}. Now if
follows that y.,(F) = [0} while у (M)=tl| and X is
M	M
(<;='rhk)'closed
there
other
Thus for every
Consequently F=
subset of Л( •
exists x^fk(F) such that
hand x JM' for each M'fF.
M
we take У^^/^СЮ , it
therefore regular. V
304
5. HULL-KERNEL TOPOLOGIES
A stronger separation condition than regularity - normality - is de-
fined next. (5.2-2) shows it to be stronger than regularity.
Definition 5.2-2 NORMAL ALGEBRAS Let X be a complex LMC algebra. X is
Gelfand norma 1 (G-normal, weak-* norma 1) if for every pair F, К of disjoint
-closed subsets of M there exists x?X such that x(F)=[0] while x(K) =
G	c
[1}. X is hull-kerne 1 norma 1 (hk-norma1) if for every pair F, К of disjoint
-closed subsets of Л( there exists xfX such that x(F)={0] while x(K) =
hk	c
[11-
(5.2-2) GELFAND NORMAL -* REGULAR If the complex LMC algebra X is Gelfand
normal then -J _ is a normal topology on and X is a regular algebra.
G	c
Proof The normality of in the usual topological sense follows immedi-
G
ately from the Urysohn lemma characterization of normality and the fact that
the functions & are all ..J' -continuous,	„ being the weakest topology
G	G
M making them so. Since is just the topology induced
hC
X and weak—* topologies are generally Hausdorff, one-point
are J'g-closed and the desired result follows. V
Having seen
by a(Xh,X)
subsets of
for
on
tha t
it is na tura 1
what
Gelfand normality implies regularity,
relationship exists between hk-normality and regular-
regularity is not implied by hk-normality. In Sec.
to inquire as to
ity. It happens
5.6 it is shown that any Frechet algebra is hk-normal (Theorem 5.6-1);
that
M
Example 5.2-1 an example is given of a Frechet algebra which is not regular.
Conversely, does regularity imply either version of normality? "No" is the
answer to this question and this is discussed after (5.2-3).
The algebras C(T,£,c) of continuous functions are regular algebras when
T is completely regular (and Hausdorff, as usual) and normal algebras when
T is normal as the discussion below shows. For such algebras, with T a
completely regular Hausdorff space, each Mf is the kernel of a complex
continuous homomorphism and C(T,iC,c) =T* by Example 4.10-2. Thus, for any
xfC(T,£,c) and tfT, if M is the kernel of the evaluation map t*, then
(*)	x(Mt) = t*(x) = x(t).
As T is completely regular, its topology is the initial topology generated
by C(T,CI) on T, just as the topology of Aj is the initial topology gener-
ated by Я=[х |xeC(T,£)} on M c- Moreover the bijection t M between T and
л	M is actually a homeomorphism. Furthermore,
C(T,C)	X	/c
I~	I	separation of subsets of the completely regu-
,	,	।	.	lar space T by C(T,C) produces the same kind
т*>М =[m |teT]	.	. ~ u
of separation ot subsets of by X due to
5.2 REGULAR ALGEBRAS
305
the way (cf. (*) ) functions in С(Т,С1,с) correspond to functions in X.
In summary we have:
(5.2-3) REGULARITY AND NORMALITY OF C(T,C,c) The Hausdorff space T is
completely regular or normal iff C(TJ?C,c) is a regular or normal algebra
respectively.
The definition of hull-kernel normal does not suggest that -J, , need
hk
be normal topology since the functions x which separate the hk-closed sub-
sets aren't necessarily hk-continuous. To see that regularity does not
imply normality for algebras one need (in view of (5.2-3)) only consider a
completely regular Hausdorff space T which is not normal. C(T,£,c) is
then regular but not J (=^7^^)-norma 1.
Example 5.2-1 A NON-REGULAR FRECHET ALGEBRA OF ANALYTIC FUNCTIONS Let H
be the algebra (with pointwise operations) of analytic functions on the
open unit disc D of the complex plane with compact-open topology. Then
(a)	M is homeomorphic to D,
(b)	the proper hk-closed subsets of M are in 1-1 correspondence
with the countable subsets of D having no limit point in D, and
(с)	H is not regular.
Proof (a) By Example 4.10-3 the points of D are in 1-1 correspondence
with Mc- And because of this correspondence, Gelfand topology on MC=H^
is the same as the initial topology determined by H on D. Since each xfH
is continuous, the Euclidean topology of D must be finer than the initial
topology determined by H on D. But the Euclidean topology is also the
weakest topology for which the analytic map p. — p, is continuous, whence the
initial topology determined by all the analytic functions on D is finer
than the Euclidean.
(b) The basic hk-closed subset h(x) ((5.1-2)) corresponds to x \o)
under the homeomorphism t -• M . If h(x)7 M , then x70 and we may invoke
C C -1
the analytic identity theorem* to conclude that x (0) is a countable sub-
set of D having no limit point in D. As any proper closed subset of A'J
is contained in some proper hull h(x), (b) follows.
(c) By (a) and (b) it follows that 3 f so> by (5.2-1) H is not
regular .V
Since the algebra of the preceding example is not regular, it can't
possibly be G-normal. On the other hand, as we have already mentioned, it
* If x is analytic on a domain E of C, then set x ^(0) of zeros ot x does
not have a limit point in E.
306
5. HULL-KERNEL TOPOLOGIES
will be shown in Theorem 5.6-1 that all Frechet algebras are hk-normal, so
that the algebra H is hk-normal. Thus hk-normality does not imply G-nor-
mality. However every G-normal algebra is regular by (5.2-2) so that
Jj, and hk-normality follows. In summary
hk
G-normal hk-normal
//
regular
In the final result of this section the notions of G- and hk-normality
are characterized in terms of how the closures in of disjoint closed sub-
sets of Л(с behave. Prior to presenting that result we prove a prerequisite
techni ca li ty.
(5.2-4) SEPARATION IN X If E and F are disjoint subsets of then the
following are equivalent.
(а)	с 1нкЕ П c 1HKF=0.
(b)	For some xfX, x(E)=[l'] and x(F)=[0).
(c)	k(E)+k(F)=X.
Proof (а)м-и(с): As cl E П cl F=H(k(E) )Г) H(k(F))=H(k(E)+k(F)) by (5.1-1)
(7) and H(k(E)+k(F))=0 iff k(E)+k(F)=X, the equivalence follows.
(b)-«-*-(c): If xfXis such that x(F)=[0] while x(E) = [l] it follows that
xfk(F), e-xfk(E), and therefore k(F)+k(E)=X. Conversely if X=k(F)+k(E) ,
then there are xfk(F) and yfk(E) such that e=x+y. But then $(F) = [0} and
y(E)={0}. Since y=e-x it follows that x(E)=[l] and x satisfies condition
(b). V
(5.2-5) CLOSURES OF DISJOINT SETS AND NORMALITY Let X be a complex LMC
algebra. Then (a) disjoint -closed subsets of M have disjoint <Уир.-
G	c	HK
closures in At iff X is ^g-normal, and (b) disjoint J^-closed subsets of
M „ have disjoint J „-closures in M iff X is hk-normal.
C	J	HK
Proof Since (a) and (b) are established in the same way, we only prove (a).
If E and F are disjoint j -closed subsets of M, and X is G-normal, then
G	c
there is some xfX satisfying (b) of (5.2-4). Thus, by (5.2-4)(a),
cl EDcl F=0. The converse follows by simply reversing the steps. V
HK HK
5.3	Condition hH There are always maximal ideals containing a given prop-
er ideal, but it is not always true (in topologica1 algebras) that there
are closed maximal ideals containing a given ideal. In particular, there
are dense ideals; The set I of xeC(g,£,c) which vanish outside some com-
pact set G , for example, is a dense ideal in C(g,Fj, c). Under certain con-
ditions - special kinds of ideals or algebras - the existence of a closed
5.3 CONDITION hH
307
maximal ideal containing a given ideal can be guaranteed, as shown by
(4.10-10) and (4.10-12).
The subject of this section is a condition, condition hH, which says
that every finitely generated ideal is embeddable in a closed maximal ideal.
We consider equivalent descriptions of the condition and mention classes of
algebras, Frechet algebras and C(T,£,c) in particular, in which it holds.
The reason for our interest in it is that in algebras in which condition
hH holds, the spaceA of maximal ideals may be expressed as a Wallman com-
pactification of M , as shown in Sec. 5.4.
r	c
Definition 5.3-1 CONDITION hH A complex topological algebra X satisfies
condition hH if each finitely generated proper ideal is contained in a
closed maximal ideal.
The reason for calling this "condition hH" is contained in the follow-
ing result.
(5.3-1) EQUIVALENTS OF hH In a complex topological algebra X the following
statements are equivalent.
(a)	X satisfies condition hH.
(b)	With h and H as in Definition 5.1-1 and x, , . . . ,x fX then
1	n
h(x,,...,x )=0 - H(x, , . . . , x )=0.
in	in
(c)	Letting "cl " denote НК-closure in A (see Sec. 5.1), and x ,...,
HK	1
Vх’ clHKh(x1,...,xn)=H(x1,...,xn).
Proof The equivalence of (a) and (b) is clear as is the fact that (c) -• (a).
To see that (a) -• (c) , we first note that
HK 1 n	1 n	1 n	1 n
Suppose M^Hkh (x . . . ,x^) . To show that М<Н(х^,. . . ,x^) is equivalent to
showing that some x.^M. Clearly kh(x,,...,x )dtM. Therefore kh(x, , . . . ,
i	In	1
x )+M=X, so there must be zfkh(x,,...,x ) and mfM such that z+m=e. We con-
n	In
tend that h (x . . . ,x^ ,m) =0. If M ' fh(x . . ,х^ ,m) , then M'fh (x ^,. .. ,x ) and
therefore z fkh (x^ , .. . ,х^)С.М'  Since m also belongs to M' however, z+m=e€M’
which is a contradiction. Therefore h(x ,...,x ,m)=0 from which it follows
that H(x p . . . ,x^ ,m)=0 by (b) . Thus I=(x , . . . ,x^, m)=X. Since m€M, it fol-
lows that some х^£м. V
Using (5.3-1) it is evident that:
(5.3-2) HK-CLOSURES OF h-HULLS If X satisfies condition hH, then
h
clHKh(xi’-’-’xn) = ni=iclHKh(xi) for any V-’xneX‘
Some e :amples of algebras which satisfy condition hH follow.
(5.3-3) hH ALGEBRAS The class of hH algebras includes the complex Frechet
308
5. HULL-KERNEL TOPOLOGIES
algebras and the algebras С(Т,_С,с) where T is a completely regular Hausdorff
space.
Proof The result about Frechet algebras has been established in (4.10-12).
As for C(TJ?C,c) we show that if I=(x^,...,x ) is a proper ideal, then
\o)^0. If Г) \o)=0, then x=£hx |x^| ^ei but never van-
ishes. V
5.4	M as a Wallman Compactification of Af For the particular topologi-
cal algebra C(T>?C,c) the set Л( of maximal ideals is in 1-1 correspondence
with the Stone-Cech compactification ST of T (Theorem 1.4-1). By Example
4.10-2, T is homeomorphic to the space At of closed maximal ideals for
such algebras when Mis endowed with the Gelfand (=hull-kernel) topology.
Thus by just reflecting gT1 s topology over onto M , we may say that	.
LettingZ denote the lattice of zero sets of T and using (3.4-3), which
allows the convertibility of Stone-Cech compactifications into Wallman
compactifications, we may say that X=w(/(c,z.).
To what extent do results of this type for algebras of continuous
functions extend to complex LMCH algebras? By (5.1-3) we know that
(M ) is a compact Т.-space containing At - But is At a compactifica-
HK	1	C
tion cf M ? i.e. is Лс dense in Л? For algebras that satisfy condition
hH, the answer is yes by virtue of (5.3-l)(c) which asserts the equivalence
of hH to the statement cl h(x , . . . ,x )=H(x , . . . ,x ). We can go further
HK х	П x	n
and actually identify Л( as a Wallman compactification of Atin the pres-
ence of hH and results of this type are the subject of this section.
As the lattice of hulls h(x,,...,x ) of finite subsets x,,. . . ,x of
In	In
a complex topological algebra X is an a|3~ lattice of «J'^^-closed subsets of
Л( by (5.1-4), it follows by (3.3-2) that the mapping
%:	-->- w( )
n c	c
м—= [AeH |MeA]
onto a dense subspace of the compact
. w(M ,/{ ) is a Hausdorff compactification
if X satisfies condition hH then co is extend-
H
is a homeomorphism of (M^,^/
Hausdorff space w(Mc,/-{)> i.e
of M . We shall see that
c
ible to a homeomorphism of (Ж ) onto w(M > ){ } 
HK	c
Theorem 5.4-1 At - w(M ,M ) If X satisfies condition hH then there is a
homeomorphism a taking (At	onto w(/(	) such that a| =cp^.
Proof For each Mf/t we define a(M)=[Bfi( iMfcl в}.	C
I HK
We claim that a(M) is an /-{-filter. To see this suppose h(x^,...,x )
5.4 M, AS WALLMAN COMPACTIFICATION OF Д
c
309
and h(y^, . . . >Ут) belong to a(M) • Since X satisfies condition hH, (5.3-1) (c)
may be invoked and it follows that MEcl h(x,,...,x )=H(x,,...,x ) and
7	HK 1	n 1	n
MEcl„„h(y, , . . . ,y )=H(y,,...,y ). Hence x,,...,x and y,,...,y belong to
HK 1 m 1 m	1 n 1 m
M s о
мен(х1>...,хп, У1,...,ут) = clHKh(x1,...,xn, Ур-.-.У^
= с1нкН(Х1,...,Хп)Пн(У1>...,Ут).
Thus o(M) is stable with respect to the formation of finite intersections.
As supersets of cr(M) in К clearly belong to cr(M), a(M) is an /-(-filter.
To see that o(M) is an /(-ultra fi Iter s suppose that h (x^, . . . >xn)
or equivalently that
M^clHKh(Xi,...,Xn) = H(xr...,xn).
Then for some i, x^0M so, since M is a maximal ideal, there must be some
yEX and mEM such that xv+m=e. Since mfM, MEc l^h (m)=H(m) and therefore
h (m)Ed(M) . But h (m)C| h (x , .  . ,x^)=0 for if M ' fh(m) Г) h (x ^, . . . ,x^) then x^
and m each belong to M’ and the contradictory conclusion that x^y+m=eEM'
follows. Hence a(M) is an -ultrafilter.
Next we show that a is bijective. As for onto-ness, let </fw( M ).
Since the sets of J' have the finite intersection property and /f is a compact
Tj-space by (5.1-3) it follows that
^h(x1,...,xn)ejH(xl’"”xn) A(x1,...,xn)e/1HKh(xi’---’xn>	0
We contend that there is some unique ME^( such that
h(xr
. 2H(x ,...,x ) - (Ml and d(M) = <? .
X )i	n
n
If ME П, .	. -H(x ,...,x ) and M'^M
*h(x , . . . ,x )EJ- 1 n
mlfM' . Hence Mf с 1 h(m)"H(m) while M'^H(m).
HK
then there exists mEM such
Thus, to prove that
tha t
nh(X1,.
this end
_H(x ,...,xn)=[M}, it suffi ces to show that h(m)E</’. To
•>x)y l n
let. h(x ,...,x )E^-. Then, as MEH(x,,...,x ,m)=cl h(x, , . . . ,x ,m),
1 n	1 n HK 1 n
h (x . ,x^,m) = h(x1> . . . >хп)П h(m) s4 0.
Thus, since J is an ultrafilter, it follows that h(m)e^ . To show that
d(M)=<? we observe that^Ca(M) follows from the definition of a(M) and
the fact that MEc l^h(x^ , . . . ,x^) for any h(x}, . . . ,x ) E^ . But both^ and
d(M) are /f-ultrafilters ; hence ^-=a(M) .
310
5. HULL-KERNEL TOPOLOGIES
To see that a is 1-1, consider distinct elements M^, М£ЕЛ(- Selecting
such that m^|^2 it follows that M^fH(m^)=clHKh(m^) while M2^H(m^)=
cl h(m ). Thus h(m )€d(M.) and h(m )^o(M9) so that
Having shown that a is bijective, next we contend that a =cp^. In-
deed if MfXc then MfH(x^, . . . ,x ) iff Meh(xp . . . ,x ) . Thus a(M) =
[hCxp . . . ,xn) (MeHCxp . • . ,xn)}=[h(x1> . . . ,xn) (MehCxp .. . ,xn)}=9H(M) .
To complete the proof we need only show that a is bicontinuous. To
prove this it suffices to show that
a(H(x)) = ^fw(/(c,/{ ) |h(x)E^ } = ®h(x) f°r each xeX-
To this end let MfH(x). Then, as H(x)=cl h(x) by condition hH, h(x)fa(M)-
HK
{h(x1,...,xn) |MeclHKh(x1,...,xn)}. Thus &(M)f^x) and a(H(x) )C ®h 
On the other hand if	we know thatfor some M. As h(x)€^
h (x)
it follows that Mfcl h(x)=H(x) and <S . .£Zo(H(x)), that is that basic
HK	n Qx)
closed subsets of are taken onto basic closed subsets of w (л(сЛ) and
conversely. V
As we already know from previous considerations (see Sec. 3.6)// may
be replaced in the previous theorem by any lattice which is equivalent (in
the sense of Def. 3.6-1) to /-( .
In Theorem 3.6-1 we established the fact that two a-lattices X and X'
defined on a set T, for which it is true that I'd, are equivalent when-
ever two arbitrary disjoint sets A, Bf£ have disjoint closures in the
Wallman space defined by the lattice £'. Since	the lattice of all
hk-closed subsets of the T^-space ^c> Is an aP-lattice by Example 3.3-2
and the aS" lattice G^hk’ we maX аРР^У the previously mentioned result
to conclude that/-( may be replaced in Theorem 5.4-1 by iff the closures
in w(A(c,/-() of any disjoint pair of sets A, Bf^^ are disjoint. Since
w(M )=(Л( >'^7L.V) this is equivalent to requiring that for A, BE
c	HK	nk.
(*)	АЛВ = 0—-cl АПс1 В = 0.
HK HK
By (5.2-5) (b), (*) holds iff X is hk-normal. Summarizing the above discus-
sion we state (for the LMC algebra X):
Theorem 5.4-2 Л( - w( M ’ ^i,k^ ^et % satisfy condition hH and denote
all hk-closed subsets of Л(с- Then/( and w(^c,^^j are homeomorphic via
a map which reduces to cp^ : Л( — w ^c’^hk5 j M - A={Ae^hk M on
Л( iff X is hk-normal.
Going one step further, if we consider only regular algebras that sat-
isfy condition hH then/-/ may be replaced by the lattice of all Gelfand-
5.5 X-REPLETION OF M.
311
closed subsets of /f iff X is Gelfand normal for then	and Gelfand
c	hk G
normality is equivalent to hk-normality. Furthermore, in this case,w(Afc,
^G) = ₽^c by (3-4'3)’ Thus:
Theorem 5.4-3 WHEN IS A STONE-CECH COMPACTIFICATION? Let the regular
algebra X satisfy condition hH and denote the a₽-lattice of all Gelfand
G
closed subsets of At . Then At and B/t =w(X ,) are homeomorphic via a
с	с c G
map that reduces to
4- : .M ” «(A r)
$G C	c G
M	= [Af <$> iMfA}
Г1	Lr 1
on M. 
c
5.5 The X-repletion of M As shown in Chap. 1 the maximal ideals of al-
gebras C(T,R) of continuous functions were characterized by (3т. Those max-
imal ideals in C(T,R) which were kernels of (real-valued) homomorphisms
were characterized by the repletion vT of T ((1.6-1)). For certain topo-
logical algebras X the closed maximal ideals Л( behave analogously to T
in that A( = pAt as discussed in Sec. 5.4. The question naturally arises:
Can those maximal ideals of X which are kernels of homomorphisms be charac-
terized as some sort of repletion of At ? Briefly, the answer is yes (for
suitable algebras). We introduce such a repletion of in Def. 5.5-1
and show how it characterizes homomorphisms in Theorem 5.5-1.
Letting X be a complex Gelfand-normal algebra satsfying condition hH,
we may say that At =P by Theorem 5.4-3. Letting CUH denote the one-
point compactification of C, each Gelfand extension x on
a unique continuous extension x*:BAtc "*	By the results of the pre-
ceding section we may say that At =₽ A(c=w(At > H) where H is the collection
of all hulls h(x,,...,x ) of finite subsets fx,,...,x } of X. We can now
In	In
define the X-repletion of At .
Definition 5.5-1 THE X-REPLETION OF M With X a complex Gelfand-normal
algebra and notation as above the X-repletion of At c is U^(M ) =
[Jew(A<c>/{ ) |$*(J)eC for all xeX}.
As we are assuming that X is normal, the results of Theorems 5.4-1, 2,
and 3 permit us to represent At in a number of ways. In Th. 5.5-1 we have
chosen to use the representation of Theorem 5.4-1 for reasons that will
emerge in the proof of Theorem 5.5-1.
Theorem 5.5-1 THE HOMOMORPHISMS OF A GELFAND-NORMAL ALGEBRA Let X be a
complex Gelfand-normal algebra satisfying condition hH. Then MfAt is the
kernel of a complex homomorphism of X iff cr(M)fU,(A4c) where (cf. Theorem
312
5. HULL-KERNEL TOPOLOGIES
5.4-1) CT(M)={BeH |MfclHKB} .
Proof Suppose first that б(М)=^7'еих(Дс) and let I=(xeX [£*(<F)=0]. Clear-
ly I is an ideal in X. Once it has been shown that I=M it will follow that
the map sending x into £*(<^) is a complex homomorphism with kernel M.
In other words if a(M) = ^Ev (/( ) then M is the kernel of a homomorphism
X c
of X. To prove that I=M, it suffices to show that MCZI since I is a proper
ideal (e*(J) = l for all fw (, f-{ )). For xfM then MfH(x)=cl^xh (x) . Since
M is the limit of a net (M ) in h(x), then, with cp as in Theorem 5.4-1,
. Carrying
Suing to write it as x, we
x*(^)=0. Thus MCI and half the
a(M)=J?is the limit of the net a(M )=cp (M ) = ^„ =[ВеН|М fcl в].
pHpMIpHK
the function x on M. over to cp„( M ) and contiHui
с	H c
see that as M fh(x),
x (M ) =x ) =0
proof is done.
Conversely if M
X and ~ (N'} --S' , then
is the kernel of a nontrivial complex homomorphism of
for each xtX there is some -pe£ such x-pfM. Thus
MfH(x-pe)=cl^Kh(x-p.e) and once again ^c=a(M) is the limit of a net cPH(M£y) =
where M Eh(x-pe). Therefore (x-рё) (M)=0 -• (x-pe*) (^ )=0. Thus х*(У/) =
м	a
P and	(Л( ) • V
X C
5.6 Frechet algebras It is well-known that for Banach algebras regularity
and (Gelfand) normality are equivalent (Naimark 1964, p. 224). We demon-
strate the generalization of this result to Frechet algebras in this sec-
tion. Thus, as any Frechet algebra satisfies condition hH ((4.10-12)),
it follows in light of Theorem 5.4-3 that for Frechet algebras 41=5/4^.
(5.6-1) Let I and J be closed ideals in the Frechet algebra X. Then,
h(I)Dh(J)=0, I+J=X (H(I)f|H(J)=0) .
Proof Let (p ) be an Increasing sequence of multiplicative seminorms
------	n
for each n) generating the topology on X. For each n let X^
(Sec. 4.5) and let be the canon-
into X . We recall that X^= U v; '(X ^) by (4.10-7).
n	n n n
are in 1-1 correspondence under the pairing h •—* kerh
maximal ideals of a Banach algebra ar.- closed
a 1-1 correspondence also exists between
maximal ideals of X . Thus,
n
have M = U I ' ( M ) .
c n n ’1 n
if
Pn — Pn+1
note the factor algebras generated by p
ical homomorphism of X
Xh and M
c
Since all
Furthermore
[(4.10-4)].
[(4.10-1)],
/4 °f a
wi th
de-
X h, we
n
Note that neither h(I) nor h(J) can
we claim that h(I) and h(J) meet all the
M =U к ' (M ) and therefore к ' (Ц )
c n n n	n n
identi fying
X h and the collection
n	h
M with X and M
c	n
(4.10-10). Moreover
be empty by
' (/И ) past some n. Indeed h(I)
for some n. This means that
mee ts
there is a complex homomorphism f^eX^h such that the continuous homomor-
phism
‘	‘ n n
f=f • к vanishes on I.
n n
5.6 FRECHET ALGEBRAS
313
Since for any m > n we may write f=f -И =f -h • К > where h is the con1
3	3	n n n nrn m	nm
tinuous map (see Theorem 4.6-1)
Km(x) = x+Pm 1(°)—"x+₽n 1(°) = \(x)
it follows that tex ' (X ^). Thus h(I)Ol< '(M ) ^0 for m > n. As a similar
mm	mm	—
conclusion holds for h(J), there is no loss of generality in assuming that
h(I) and h(j) meet all the Kn'(^n)-
Since h( I) Г) h(J) =0 , it follows that h(xn( I) ) Qh(Kn(J) ) =0 for all n.
Indeed if this is not so, i.e. if for some Me M , к (I)+K (J)CM=ker f where
n n n
f is a nontrivial homomorphism of the Banach algebra onto C, then
I+JCker	and therefore Ь(1)Г)Ь(Л)-^0 which is contradictory. Hence
><n(I)+Xn(J) (which is an ideal in X д is not contained in any maximal
ideal in X and so the ideal К (I)+K (J) coincides with X . Also we note
n	n n	n
that by Theorem 4.6-1, I and J are the projective limits of the algebras
(К (I)) and (K (J))- Having recalled these basic facts, we begin the main
n	n
body of the proof. We will construct xfl, yeJ such that x+y=e.
Let (p ) be any sequence of positive numbers such that ZnUn converges.
Since и (I)+K (J)=X , we choose x ex (I), у ex (J) such that x +y =K (e) .
о о _______________о	____ oo	о о	О ОО
Next choose х^'ец^(1), y^'ex^(J) such that x ’+y ' =x^(e), and then choose
and w1e«1(J) such that
llhol<zl> - xollo <	H1/3max(||x1	Ц^)
and
<1’> l^ol^P ’ Uo < min<Pi/3> H1/3max(||x1’ Цу^ Ц^).
This can be done because h ,(X,) is dense in X (recall that h , (X/p, \o))=
olv 1'	о L	ol4 *1	4 '
X/po (0) so that hQpx^)OX/po (0) and that Xq is the completion of
X/p/^O)].
Let
xi = zi + x1'(x1(e) " zi " WP
and
yl = "i + У1’(«1(е) " z! "
Clearly x,+y,=x,(e). If we apply h , to x, and note that h , (x, (e))=K (e) =
1	1	J ol 1	ol 1	о
x +УО> then it follows from (1) ar.d (1') that
314
5. HULL-KERNEL TOPOLOGIES
l^ol^P ‘ Xollo < and HWyp ’ Mo <	'
By induction we may construct sequences (x ) and (у ), x fX , у fY , such
n	nnn nn
that x +y =K (e) and
n Jn n
(2)	llh (x ) - x JI , < u and llh (у ) - у II < ц, .
II n-l,nv n' n-llln-1 pn II n-l,nwn 'n-llln-1 ип
If we apply the norm-decreasing homomorphism hn_£ n_^ to the vectors in (2)
we obtain
llh „ (x ) - h ,	, (x )ll < и and llh (y )-h	(y )ll .
II n-2,n n n-2,n-l n-1 lln-2 n II n-2,n Jn n-2,n-l Jn-l lln-2 n
Continuing this process we obtain
(3)	llh (x ) - h (x )ll < ц, and llh (y ) - h (y )ll < ,,
II kn n k,n-l n-1 Hk n II knvn k,n-1 'n-1 Ilk “Ti
for all к < n.
In (3) fix к and for n > к define two sequences (x^(n)) and (y^(n))
from by taking
(4)	x. (n) = h (x ) and у (n) = h (y )
к	kn n	'k	kn vn
for n > k. Since x +y =X (e), it follows that
-	n n n
(5)	xk(n) + yk(n) = hkn(Kn(e)) = ^(e)
for all n > k. The sequences (xk(n)) and (yk(n)) satisfy conditions
(6), and (7) below.
(6)	hk,k+l(xk+l(n)) = Xk(n) and hk,k+l(Wn)) = yk(n) (n > k+l) •
Proof of (6) Recall that h, , .'h, ,,	=h, . Thus
---------—	K,k+1 k+l,n kn
h, , , , (x. , , (n) = h, , , n (h, , n (x )) = h, (x ) = x, (n) .
k,k+l k+1	k,k+l k+l,n n k,n4 n к
The third relationship, (7) below, together with the fact that Z Ц, con-
n n
verges will imply that, for any fixed k, (x (n)) and (y (n)) are Cauchy
------------------	------ К	К
sequences in *^(1) and Х^(Л) respectively.
5.6 FRECHET ALGEBRAS
315
(7)	||xk(n) - xk(n+p)||k < pn+1 + pn+2 + ... + %+p
and
||yk(n) - yk(n+P)|jk < Hn+1 + ... + (An+p
for any positive integer p and n > k.
Proof of (7) We verify the result for p=l and p=2.
laT^Zl ||xk(n) - xk(n+l)||k = ||hkn(xn) - hkjn+1(xn+1)||k < ^+1 by (3).
P=2 llxk(n) ’ xk(n+2) Ilk = Hxk(n) ’ xk(n+1) + xk(n+1) - xk(n+2)llk
< ||xk(n) - xk(n+l)||k + ||x. (n+1) - xk(n+2) ||k < pn+1 + pn+2 .
The proof is completed by putting statements (5), (6), and (7) to-
gether. By (7) there exist , for each k, vectors x(k)eHk(I) and y(k)ex
such that
(9)	xk(n) " x(k) and Ук<п) " У(к)-
By (6) h (x (n))=x (n) for n > k+1. Furthermore by the continuity of
К } Кт i Kt i	к
kk k+1 and (9) f°PPows tkat
hk,k+l<Xk+l<n» " hk,k+l(x<k+1»-
On the other hand xk(n) "* x(k) . Thus we obtain
(1°)	hk,k+l<'X<'k+1)) = x(-k) and hk k+l^y^k+1^ = y^k) •
As I and J are the projective limits of (>^(1)) and (^(J)) respectively,
there exist x€I and yfj such that
(11)	^(x) ~ x(k) and ^(y) = y(k) for aPP k-
By (5) xk(n) = yk(n)=xk(e) so, applying (9), we obtain x (k)+y(k)=xk(e) for
all k. Applying (11) we obtain
(12)	^(x+y) = ^(e) for all k.
Thus, as X is a LMCH algebra, x+y=e. As x€I and y€J, it follows that I+J-
X. V
Theorem 5.6-1 FRECHET hk-NORMAL If X is a Frechet algebra then X is
hull-kernel normal.
Proof Let F and G be disjoint hk-closed subsets of M  By (5.2-5)(b) it
suffices to show that F and G have disjoint НК-closures in At . Let I=k(F)
and J=k(G). As h(I)f) h(J) =F(1 G=0, by (5.6-1), I+J=X. Thus, using (5.1-1)0
316
5. HULL-KERNEL TOPOLOGIES
0 = H(I+J) = H(k(F) + k(G)) = Hk(F)flHk(G) = cl FDcLG. V
HK HK
Theorem 5.6-2 REGULARITY —*• NORMALITY If X is a regular Frechet algebra,
then X is Gelfand normal and
c
Proof As X is regular, the Gelfand and hull-kernel topologies coincide
by (5.2-1). As X is hk-normal, by the previous theorem, X is Gelfand
normal. To prove that Л4 = |3 Mc note that X satisfies condition hH by (4.10-
12) and apply Theorem 5.4-3. v
EXERCISES 5
317
Exercises 5
The eight exercises presented here concern themselves with continuity
of Banach algebra-valued homomorphisms on complex Banach algebras as well
as continuity of derivations on Banach algebras and certain topological
algebras. The principal results are 3(c), 4(b), 5(c), 5(d), 7(d) and
8(e).
5.1 BOUNDEDNESS OF HOMOMORPHISMS ON IDEMPOTENTS (Curtis and Bade I960)
In this exercise X and Y are complex commutative Banach algebras with iden-
tity and H is a homomorphism from X into Y. We show in (c) that H is
"bounded on the idempotents" of X - that is, that there exists M > 0 such
that
2
||H(P)|| <M||p||
for every idempotent pfX- These results can be proved without assuming
that identity is present in Y or that Y is complete through use of the
usual procedures of completion and adjunction of the identity.
(a) Let (a ) and (b^) be sequences of nonzero elements in X satisfy-
ing
(1) a b = a for all n
n n n
(2) b b =0 for m n.
n m
Then sup ||H<an)||/llanllllbnll?t” '
Hint: With no loss of generality assume that ||a ||=1. Clearly by (1) it
follows that ||bn || > 1. Let us assume that on the contrary sup||H(a^) ||/||b^|| =
eo. Select distinct elements c. . from among a and d. . from among b such
. , .	n ii	n
H-1	J	J
that ||H(c )|| > 4 J||c*£j|| where d-j corresponds to c„ through (1). Define
x .= L, 2' j c . . and consequently d. ,x . =2 c . ., H(d..)^0. Let y= £d../21||d. .11.
1 J=1 1J	1J 1 1J 1J	7 1=1.1J И ijll
Then show that x^y=c /2	J||c^j|| and ||H(y) ||||H(xp || — ||H(x^y)|| >	>
21||H(x^) ||; hence ||H(y) || > 2*- for any 1 which is a contradiction.
(b) Let H be a homomorphism of X into Y • If (x ) is a sequence of
n	2
orthogonal idempotents in X, then for some M > 0, ||H(xrl)|| < M||x || for each
nfN-
(c) Let H be a homomorphism of X into Y and let I be the set of idem-
2
potents of X. Then there exists M > 0 such that ||H(x)|| < M||x|| for all
xfl. Hence if I is bounded, H(I) is bounded.
Hint: We may assume that if e is the identity of X, then H(e) is the iden-
tity of Y for H(e) is the identity of H(X) and we may take H(X)=Y.
2
We say that z < x if xz=z. Let I^={xfl |sup||H(z) || / ||z || =“}•
318
5. HULL-KERNEL TOPOLOGIES
We assume that efl^. If xfl^ and z < x then z or x-z is in 1^ for other-
wise ||H(w) || < КЦмЦ^ for all w < z and w £ x-z.
so ||H(v) || < K(||vz||2+||(x-z)v||2) < K||v||2(||z||2+||x-z
Then if v < x, v=vz+v(x-z
2
|| ) and this contradicts
xCl .
1	2	4
Let	and choose z^ < x^ such that ||H(z Ц/||z ^|| > lb||z^|| (2+2||H
(Xi)||/||x/). Then ||H(x1-z1)||/||x1-z1||2 > [||H(z ? ||-||H(x	||] / [||x 1H||z
1ЫР > CIlHCzpilMiix^i ||Z1|| о-сцнсхрц/цх^ ] >4|N| сг+цнсхрц/нхр ].
We generate in this way a sequence of idempotents (x^) in 1^ such that
Xn+1 - Xn and llH(xn)ll/llXnl|2 >4|lXn-ll|2tn+!lH<Xn-l>||/llxn-ll|2^ Letting PnT
xn’xn+l we PnPk=° (n?tk) and llH(pn)ll/llPnl|2 П1Н(рп+1> ll/4l|pnll llPn+lll
t||H(pn) ||/||pn||	> n+1 which contradicts (b) .
5.2 HOMOMORPHISMS OF REGULAR ALGEBRAS (Curtis and Bade I960) In this ex-
ercise X is a complex commutative regular semisimple Banach algebra with
identity, Y is a complex commutative Banach algebra with identity, and H a
homomorphism from X into Y. It is established that there exists a finite
set F of points in M (the maximal ideals of X) such that if V is any neigh-
borhood of F, H is continuous on the ideal I(V)={xfX |фх(7)={0}}. Thus if
F is empty, then H is continuous. It turns out that this occurs if X=
C(M>£>C) and Y is semisimple as shown in the next exercise.
We denote by S the family of open sets ECj( such that
mE = sup||H(x)||/||x||||z|| < <»
whenever the supports of фх and i|rz lie in E and xz=x. In (a) - (e) it is
shown that the union of all sets in S is also a set in S and that the com-
plement F of this set is finite. F will be the desired finite set dis-
cussed in the introduction.
(a)	If (E ) is any sequence of disjoint open sets in , then E be-
n	n
longs to S for sufficiently large n.
Hint: If (a) is false then there exists a sequence (E ) of disjoint open
sets in X and x , z fX such that the supports of фх and фг lie in E ,
n n	’ n ’ n	n
xnzn=xn> ||x ||=1, and ||H(xn)|| > пЦг^Ц. But this contradicts 5.1(a).
(b)	Let Е^ and £2^5 • If G is an open set such that GCLE2, then
E1UGfS •
Hint: Choose x^ such that фх^ is 1 on a neighborhood of CE2 and 0 on a
neighborhood of G. Let X2=e-x^. .We may find z^, z^CX such that
ZjX^ = x^	supp фг^Лб = 0
Z2X2 ~ X2	supp фг^Л CE2 = 0
EXERCISES 5
319
Let H=E^UG and suppose supp tygCH, supp фЬСН, and gh=g. Then
supp l|i(gxi)CE.,
supp i|l(hz^)G.E^,
gx. = ghx.z. = gx.hz, and ||H(g) || < ^(gx^]] + ||Hgx21| < J|gx ^„hz +
“eJMIKII ('nE1llXlHIIZill + 'nE2llX2l|l|Z2> IWII
(c)	If Ep E2€S and G is open with GCLE^(_JE2 then G€S •
Hint: Let F=CE^H G and U be open such that FCUCUCEj. Then GCE^UL'fS •
(d)	If Ep E2fS then E^E^S.
Hint: Suppose EjU E^ S • If FCE^UEj and F is closed, then G-EjUEj -
F^ S , for if FCE1UE2 then choose open sets U, V such that FCVC?v<tl'CU
CZE1UE2- Thus UfS by (c). If GfS then E^U E2=G U VfS by (b) contrary
to assumption.
Returning to the assumption E^U E^S we can find Zp x^ such that
Xj=XjZ > supp ifrz^EjUE,? and ЦнСхрЦ > Цх^ЦЦг^Ц. Pick an open set such
that supp i|iz C. U^CU^C-E^VJ E2. Then G2=E1^E2~ U^S and there exists g2 ,
h2 such that g2h2=g2 and supp i|ih2CZG2 with ||H(g2)|| > 2 ||§2 ||llh2 II' We generate
sequences (gn) , (hn> such that gnhn=gn and ||H(gn)|| > п||§п|| ||hnl| • The sup-
ports of 'jih^ lie in disjoint open sets. This produces a contradiction of
5-l(a).
(e)	5 is closed under arbitrary unions.
' (f) There exists a finite set F of points and m > 0 such that
||H(x)|| <m||x||||z||
for all x, z?X such that supp фх, supp фгССР and xz=x. Consequently, if F
is empty, then H is continuous.
Hint: Let F=c(U3j • If F is infinite, then a sequence of points in F can
be separated by disjoint open sets. Hence F is finite.
The set F of (f) will be referred to as the singularity set of H.
(g)	If V is any neighborhood of the singularity set of H, then the
restriction of H to I(V)={xfX |x (V)={0)} is continuous and ||H(x)|| < ||x||||h||
for all xfl(V) where h is such that i|ih(CV) = [l] and фЬ is zero in some neigh-
borhood of F.
(h)	Let X be a complex commutative regular semisimple Banach algebra
with identity under || || and let || || be any other norm on X making X a
normed algebra. Then there is a' finite set FC M =X^ such that if G is any
320
5. HULL-KERNEL TOPOLOGIES
open set in M with Gf\F=0, then for some try, > 0;	< m ||x|| f°r апУ xfX
such that supp i|ixCG. Conversely if G is an open set such that ||x||^ <
m ||x|| for all x€X such that supp ijixClG, then GCIF=0.
5.3	HOMOMORPHISMS OF C(T,C,c) (Curtis and Bade I960) In this exercise we
show that((c))any homomorphism of C(T,C:,c), T a compact Hausdorff space,
into a complex semisimple Banach algebra is continuous. It is not known if
the result remains true if semisimplicity is removed.
In the parts to follow, H denotes a homomorphism from C(T,£,c) into a
Banach algebra Y, F the singularity set (Exercise 2(f)) of H, R(F) the
dense subalgebra of С(Т,)3, c) consisting of functions which are constant in
some neighborhood of each point of F, and 1(F) the ideal of functions which
vanish in some neighborhood of F (the neighborhood is not fixed).
(a)	H is continuous on R(F).
Hint: Apply t5i2)(f) to C(T,£,c) and show that for some M > 0
||H(x)|| < M||x||
for all x€I(F) .
If F={ tj,.. . , tn) let е^?С(Т,£) be such that е^е^=0 (i^j), 0 < e^(t) < 1,
for all tfT, and e.(t.)=l in some neighborhood of t.. Then consider
n	i i	i
X - iJ1x(t )e^el(F) .
We denote by К the unique continuous homomorphism of C(T,£,c) which
agrees with H on R(F). Define X=H-K; X and К are the continuous and
singular parts of H respectively. We reserve M for a constant with the
property ||K(x)l| < M||x|| for any xfC(T,£). We assume with no loss of general-
ity that clH(C(T,C) )-Y.
(b)	The range of К is closed in Y and K(C(T,£) ) Г) R={0] where R is the
radical of Y.
Hint: Let I=K \{0J). Since I is a closed ideal there exists a closed set
SCT such that I=[x€C(T,C) |x(S)={0}} (Exercise 4.3). If C(T,C,c)/I carries
the norm
||x + I|| = inf[||z|| |zfx + 1}
then C(T,£, c)/I is isometrically isomorphic to С(Х,С,с~) and ||x+I||=
sup tes |x(t) |. We may also nortn, C(T,_C)/I be defining |x+I |=||K(x) ||. By
fundamental results of Banach algebra theory (Kaplansky 1949, Theorem b.2)
||x+I||=sup teg |x(t) | < |x+I |=||K(x) ||. As К is continuous, if y€x+I, |x+I |=
|y+I |=||K(y) || <M||y||. Thus |x+I | < M-inf { ||y|| |yfx+l}=M||x+I||. Hence |x+I | and
||x+I|| are equivalent norms on C(T,C)/I. To show that the range of К is
EXERCISES 5
321
closed, let yCY and y=lim K(x^) . Since ||x -Xn+I|| < ||K(xm-xn) "* °’ th6176
exists xoeC(T,C) such that Цх^-х^ТЦ -> 0. Since ||K(xq) -K(x ) || < M||xo-xn+
I|| - 0, y=K(xo).
By the previous argument K(C(T,Cl)) is topologically isomorphic to
C(S,C,c). Since in any Banach algebra xfR iff lim ||хП11/n=0, it follows
that K(C(T,C)) Л R-[0}.
(с)	X(C(T,J2))C2 R. Hence if Y is semisimple, X=0 and H=K. Thus H is
continuous.
Hint: Let 0 be a nontrivial multiplicative (hence continuous) linear func-
tional on clH(C(T,C)). Then the well-defined functionals 0 and 0 defined
н к
by 0 (x)=0(H(x)) and 0 (x)=0(K(x)) are multiplicative, hence continuous,
H	к
linear functionals on C(T,£,c). Since they coincide on R(F) , 0^=0^. Thus
0(X(x))=O for all xfC(T,_C) and all 0.
(d)	clH(C(T,^C)) is the topological direct sum of K(C(T,/C)) and R.
Hint: Surely H(C(T, C))£K(C(T, C) ) + R so that с 1H(C(T, C) )C. K(C(T, C)) + R.
Let y=lim H(x^) . Since K(C(T,£)) is closed, унСх^-х^Л * VH(Xn’Xm)) =
r (K(x -x )). Denoting by r the spectral radius of a vector in K(C(T,C)),
(7 П tn	K.
г (К(хд-х ))=rK(K(xn-xm)) >M ||K(xn-Xm) ||- Thus there exists х^сССТ,,C) such
that K(xo) = lim K(x^). If z=y-K(xQ) then z=lim X(x^) and ZfR. Hence
clH(C(T,C)) = K(C(T,C)) + R
The topologies of H(C(T,,C)) and the topological direct sum agree be-
cause K(C(T,C)) and R are closed subspaces of a Banach space (Horvath 19bb,
p. 122).
5.4	C°°IS NOT A BANACH ALGEBRA Let ^0“ denote the complex algebra of all
comp lex-valued infinitely differentiable functions on an open interval.
There exists no norm under which c” is a Banach algebra.
Hint: Suppose there is such a norm. Let Dx=dx/dt for each xec” Consider
the bounded linear transformations L where
n
Ln(x) = (x(tQ + 1/n) - x(tQ))/l/n.
Since lim L (x)=x'(t ), then A (x)=x'(t ) is continuous for each t .
n	o	t	o	о
о
Applying the closed graph theorem, D is bounded and by (4.11-3), D=0. But
this is clearly a contradiction.
5.5	DISCONTINUOUS HOMOMORPHISMS (Curtis and Bade I960) In this exercise
we deal with the question of existence of discontinuous homomorphisms on
a complex (commutative) Banach algebra (with identity) taking values in a
(commutative) Banach algebra (with identity). In showing that they exist,
it follows therefore that the singularity set of Exercise 5.2 is not always
322
5. HULL-KERNEL TOPOLOGIES
empty.
The basic result (b) presented in this exercise is that if there is a
2	2
maximal ideal M of an algebra X such that M 5*М and M is dense in M, then
there is a discontinuous homomorphism of X into a Banach algebra Y. Ex-
amples are given in (c) and (d).
2
If X=C(T,C,c), T a compact Hausdorff space, then M =M for all maximal
ideals in X. Consequently the question of existence of discontinuous homo-
morphisms remains open in this case although Sinclair (1969) has recently
*
proved that existence ot a discontinuous homomorphism ot any C -algebra
implies existence of a discontinuous homomorphism of C([0,1],£,c).
(a)	Let X be a commutative Banach algebra with identity e and M a
maximal ideal. Let I be an ideal such that IC.M properly and I is dense
in M. Then there exists a discontinuous linear functional on X which van-
ishes on I.
Hint: Let xfM-I and f be a linear functional which vanishes on I, vanishes
on e, and such that f(x)=I.
(b)	Let X be a commutative Banach algebra with identity and M be a
2	2
maximal ideal in X such that M CL M and M is dense in M. Show that there
exists a discontinuous homomorphism of X into an algebra Y.
Hint: Let Y=M ф [Xy IXc£} where operations in Y are taken to be pointwise
***	2
addition and (nH-Xj)(m'+X ly)=mml so that the symbol у satisfies у =0, уМ=[о].
Define ||x+Xy||=||x||+|X | and
H:M---->-M ф [Xy |xec}
x----« x + f (x ) у
where f is the discontinuous linear functional of (a). Then H is a discon-
tinuous homomorphism of M into Y. H can be extended to X after the identity
is adjoined to Y.
(c)	Let X be the algebra £ , 1 p < <». Show that an identity can be
adjoined to X forming an algebra X and a discontinuous homomorphism of X
exists.
2
Hint: £ is a maximal ideal in X and £ =£ , which does not equal £ but
---- P	P P/2	4 p
is dense in £ .
P .
(d)	LetX=^ be the algebra of continuously differentiable functions
on [0,1] with norm
H = SUPtdo,l]lX(t) I + SUP«[0,l] Iх' (t) I
Show there is a discontinuous homomorphism ofa0\
EXERCISES 5
323
Hint: The structure space (space of maximal ideals of JQ is [0,1] (see
Gelfand, Raikov, and Shilov 1964, p. 23). Let M=[x^’’|x(0)=0] and N=
[xbjp’ |x ' (0)=0] . If yfN and (Pn) is a sequence of polynomials from M con-
verging uniformly to y‘, then q (t)= | p (s)ds is a sequence of polynomials
2	i П 2 n
in M with q — у in . Thus M is dense in M but not closed as it is
n	2
readily seen that y"(0) exists for every у in M .
5.6	POINT DERIVATIONS AND DERIVATIONS INTO LARGER ALGEBRAS (Singer and
Wermer 1955 A point derivation of a complex commutative Banach algebra of
X with identity is a linear functional d associated with a multiplicative
Ф
linear functional Ф such that Ь1(х,хп) = Ф(х„)Ь,(х,)+Ф(х,)Ь1 (x„) . This no-
$	1 Z Z Ф 1	1 Ф L
tion is used to explore conditions under which there exists a bounded deri-
vation of X into a superalgebra Y.
(a) If there exists a nontrivial bounded point
then there exists an extension Banach algebra Y of X
derivation dT of X
Ф
and a nonzero bounded
derivation D of X into Y. If X is semisimple, Y is semisimple.
Hint: Let Y=[(x,X) |xfX, 1(C] where (x , X p (x2 , X2) = (x jx2 ’ 1^2^ and ||(x>X)||=
max (IM, lx I). Consider the map
D:X*----->- Y
(x,0)----► (х,Ф(х))
where X*
(b)
(bounded)
is the isometric isomorphic image of X
If Ф is a continuous homomorphism and .. x
2	* 2 ~’
point derivation of X exists iff MpM$ (Mj^Mj).
in Y.
M.zt’1([°})> a
nonzero
Hint: Let f be a nontrivial linear functional annihilating m‘
can be written х=х'+Ф(х)е with x'cl-l,, then f can be
Ф
associated with Ф.
any xfX
desired
2
1Ф
shown to
and e. As
be the
point derivation of X
If XCY and if D is
a nonzero bounded derivation of X into
Y but
D(X) is not in the radical of
Y, then there exists
a nonzero bounded
point
derivation of X.
above there exists a
nontrivial homomorphism
on X by d (х)=Ф(В(х)) .
’	Ф
Hint: By the condition given
Ф whose restriction to D(X) is not zero. Define d
Ф
(d) Assume that X is semisimple. Then X has no nontrivial (continu-
2
ous derivations into a semisimple commutative extension Y of X iff M =M
(M =M) for all maximal ideals MCX.
(e) If T is a compact Hausdorff space, then C(T,C,c) has no nontrivial
derivations into any semisimple commutative extension Y.
5.7 DERIVATIONS OF COMMUTATIVE REGULAR SEMISIMPLE BANACH ALGEBRAS (Curtis
1961) In this exercise it is shown that if the Banach algebra S is regular
324
5. HULL-KERNEL TOPOLOGIES
and semisimple, then the "boundedness" requirement of (^.11-3) can be
dropped; that is, any derivation of X into itself is bounded and therefore
trivial. Moreover B.E. Johnson (1969) has shown that the "regularity"
assumption can be dropped.
Let X be a complex commutative regular semisimple Banach algebra with
identity. A 'derivation' D from X into B(X>,C>c) (bounded comp lex-va lued
functions on.M with pointwise operations and sup norm) is a linear trans-
formation such that
Dxy = xDy + (Dx)y
where
(a)	If
(b)	If denotes the chara
then letting k^=x^ with x^fX, Dx^x=0 tor xfX.
Hint: x=m^+Xe for
Хэд is idempotent.
(c)	Let D be
finite subset F of X and a continuous derivation D
x
is the Gelfand map determined by x.
is an idempotent in X, then Dx=0.
cteristic function of [м], M? and к^еЯ,
some
n^fM. As X is semisimple, x^x = x^m^+Xx^=\x^ and
a derivation of X into B(/f,£,c). Then there exists a
of X into B(Af ,£,c)
such that if D£=D-D^, then D£x(M)=0 for all xfX and MfF. If M€F then
f^(x)=D2x(M) is an unbounded point derivation. If DxfC( M >C> c)
then F=0 and D is continuous.
for all xfX
Hint: Consider a new norm on X defined by ||x||||x||+sup |Dx(M)||. Let F be
the singularity set (5.2(f)) of || || . The fact thatMf{ie linear functional
f^(x)=Dx(M) is bounded iff M€F is established as follows.
If MCF then choose a neighborhood V of F and a vector
x€X such that ^M(M)=1 while x)4(V)={0}. Take I(V)=[xex |x(V)={0}} . Let W
be an open subset of Af such that WA F=0 and if xfl(V) then the support of
5 lies in W. If x fX and x -> 0 then x xfl(V). Hence D(x x ) -• 0. But
n	n	n	n M
DxnXM(M)=Dxn(M)+xn(M)DxM(M) — 0 and ^(M) -• 0 implies Dxn(M)=fM(xn) “* °-
Thus letting H=[m |f^ is bounded) we see that CFQH and H is open. By the
principle of uniform boundedness there exists к > 0 such that |Dx(m) | <
k||x|| for all xfX and all M€H.
To show that HCF suppose M^fHOF. Let G be an open set such that
Mq(GCH and let yeX be such that y(G) = l and y(M_H) = {0). If xfX and the
support of x is contained in G, then xy=x. Thus
Dx = yDx + xDy and
EXERCISES 5
325
sup|x(M) | < sup |y(M)Dx(M) + l|x||sup |Dy(M) | < (k sup |y(M) | + sup |Dy(M) |)||x||.
меЦ	m<$
Then ||x||^ < K||x|| and this contradicts the properties of F.
If F?^0 and D is unbounded, define by
D x (M) =
Dx (M)
0
MtfF
MeF
Applying the principle of uniform boundedness, is bounded. If D£=D-D^,
D2 can readily be shown to have the desired properties.
Now show that M _F= M • For if M is isolated in M then let x CX be
M
such that	Then it follows from (b) that Dx^x=0 for all x€X. Thus
if D(X)C C(/( >,C>c) then sup |Dx(M) | < k||x||
mcM
and D is
-x(M)DxM(M)=0 and M€F. Now
continuous.
(d) If X is a commutative regular semisimple Banach algebra with iden-
tity and D is any derivation of X into itself, then D is continuous and
therefore trivial.
Hint: Consider the derivation D of X into С(И ,C,c) where tyx=x and apply
the closed graph theorem.
5.8 DERIVATIONS OF REGULAR SEMISIMPLE FRECHET ALGEBRAS (Rosenfeld 19bb)
Let X be a commutative regular semisimple Frechet algebra with identity.
We show that a derivation D:X — С(Ц ,£, c) is continuous ((e)). From this it
follows that any derivation D:X -• X is continuous by essentially the same
application of the closed graph theorem as in Exercise 5.7(d). Using these
results we show that for a certain algebra the derivations can be completely
determined. It is worth noting that unlike the situation when X is a Banach
algebra, the derivations need not be trivial.
(a)	Let H be a homomorphism from X into a seminormed algebra Y (i.e.
denoting the seminorm by p, p(x,y) < p(x)p(y), x, y€Y). Let [p |nCN] be a
family of multiplicative seminorms generating the topology on X such that
Pn < Pn+j for each nfN. Then if (x^) and (yn) are sequences in X such that
x у =x and у у =0 for n^m, it is not possible for p(Hx ) > np (x )p (v )
nJn n JnJm	n *n' n *n ’n
for all n > 1.
Hint: Modify Exercise 5.1(a).
(b)	If X is a commutative Banach algebra with identity and e is an
2	1
idempotent (e^=e^) belonging to the radical, then e^=0.
Hint: As e^ belongs to the radical, lim ||e"||^/n=0. Now use the fact that
2n n .
e^ =e^ tor any n€N.
(c)	If X is a commutative Banach algebra with identity and e^(i=l,2)
326
5. HULL-KERNEL TOPOLOGIES
-e,
are idempotents with e^-e2 belonging to the radical, then e^=e2'
Hint: (e -e„)2 £s idempotent and therefore equal to 0. Thus e +e_=2e e .
1 2Z	2 z 2	L22
Now (e =e-e^-' e ^-e2=2e2 (e i-e) > and therefore 0=(e^-e2) =4e2(e^-e) =
4e2(e-e1)--2(e1-e2) .
(d)	If X is a commutative Banach algebra with identity and M is iso-
lated in X , then there exists an idempotent e^fX such that is the char-
acteristic function of {m}.
Hint: This is a result due to Shilov (Rickart 1960, p. 168).
(e)	If D is a derivation from X into C(^|,C;,c) then D is continuous.
Hence any derivation of X into itself is continuous.
Hint: Let the factor algebras of X be denoted by (X^) and the maximal
ideals of Xn by we identify M and (JMn by (4.10-7). To prove the re-
sult it suffices to prove the following three statements.
(1)	For each n > 0 there exists F Г U such that F is finite
n u	n
and if x — 0 then Dx — 0 uniformly on cl(M -F ).
n	n	J	n n
(2)	If Ж and M is isolated, then Dx(M)=0 for all xfX.
(3)	If MCM and M is isolated in M for all m > n then M is
n	m	—
isolated in X .
Once these statements have been established then it follows that Dx^(M) "• 0
for every McX whenever x^ — 0. Statement (1) then implies that (Dxk) con-
verges to 0 uniformly on every compact set of . (If К is compact in Л/ ,
then KcMn for some n by (4.10-7) since X is barreled.)
To prove (1) fix nCN and let Y be the seminormed algebra X supplied
with seminorm
p(x) = p (x) + sup |Dx(M) I
мап
Let Fn=[M€ Mn |x — 1^(х)=0х(М) is not continuous}. Since X is a Frechet
space and for each xfX |f^(x) | < sup |Dx(M') |=k , Dx^ — 0 uniformly on
m' cM
Mn"F. If Fn is not finite then ther^1 exists a sequence (Mr) in Fn and a
sequence of mutually disjoint neighborhoods Vr of Mr for each k. Since X
is regular there are sequences (y ) and (z ) such that £ (M.) = l, у z =y
К	К	К К	К К к
and z z =0 whenever k^p. Since f is a discontinuous linear functional
k p	”k
there exists x, €X, k€N>such that
k ~
l£Mk(xk) I = lDxk<Mk) I >kpk(xk) Pk(V Pk^?'
As X is a Frechet algebra,
res tricted
discontinuous. Thus we may choose x, from
J	k
to any closed maximal ideal
Mr. Let gk=xkyk and h^.
is
Then
EXERCISES 5
327
P(gk) > sup |Dgk(M) | > |Dgk(Mk) | =
IVV ‘W + Vv Dxk<Mk> I= IW I
>kpk(xk) pk(V pk(hk) ^kpk(gk) pk(hk) for each k-
But this contradicts (a).
We next outline the proof of (3) . Suppose and M is isolated in
M for tn > n. Then by part (d) there exists for each tn > n an idempotent
e cX such that e (M) = l while e (M')=0 for all with M'cA( . Now show
tn tn	m	nr	m _______
that if n < r < s, with h as in Theorem 4.6-1, it follows that h (e )=
—	—	rs	rs' s
e and therefore h (e )=e . Thus there exists an idempotent e(X such
r	rs s r
that, with Kn as in (4.10-7), Xne=en for all n. Clear ly e (M) = l while
e(M')=0 for all m'cM with M'^M.
We use (3) to sketch (2). Let efX be an idempotent satisfying the
requirements of (3). Then by Exercise 5.7(b) Dxe'=0 for all x€X and
Dxe 1 (M)=0=x (M) De 1 (M)+e ' (M)Dx (M)=Dx (M) .
(f)	Let c”(Jp be the algebra of all infinitely differentiable com-
plex-valued functions on the real line R. Then for each yfC Qi) there ex-
ists a derivation D of с”(К) into itself such that for all x€C°°(R), Dx(t) =
x'(t)y(t) and conversely.
00
Hint: Let C (R) carry the topology defined by the seminorms
- ни
Pn k(x) = sup{ |x (t)l |te[-n,n]}
for all n,k > 0. The polynomial functions are dense in (f^R,) and for each
polynomial p, Dp(t)=p'(t)D(t). As D is continuous, the result follows.

SIX
LB-Algebras
WE HAVE SEEN that many of the features of Banach algebras are carried over
to the larger class of INCH Q-algebras, e.g., openness of the set of units,
continuity of inversion, continuity of homomorphisms, compactness of the
spectrum, etc. The central notion of this and the next two sections - the
LB-algebra - provides another class of structures containing the Banach
algebras in which some of the important properties carry over, only this
time by way of a more algebraic approach. While many of the INCH Q-algebras
namely the complete ones, are projective limits in the TVS sense of their
factor algebras (Theorem 4.6-1), an LB-algebra is an algebraic inductive
limit of a system of Banach algebras and continuous unital isomorphisms.
The LB-algebra itself need not be supplied with a topology, however.
6.1 Definition and Examples In this section we define an LB-algebra and
present some examples.
Definition 6. 1-1 BOUND STRUCTURES AND LB-ALGEBRAS" Let X be a complex
algebra. A bound structure for X is a non-empty collection $ of absolutely
m-convex subsets В of X containing e satisfying the following stability con-
dition :
For each pair B^, B2?63 there exists a set B^f® and a scalar X > 0
such that B^UB^CXB^.
The pair (Х,й) is referred to as a bound aIgebra and it is said to be
complete provided each of the subalgebras X(B) = [o(x |a€C, x€B] is a Banach
algebra with respect to the gauge p of B:
D
p„(x) = inf{a > 0 IxfaB] (xfB, BfiES) 
D	I
X is an LB~aIgebra (pseudo-Banach algebra) with respect to the bound struc-
ture 63 if (X,®) is a complete bound algebra and X=IJX(B) . If 68 i s under-
В
stood then we simply say that X is an LB-algebra (pseudo-Banach algebra).
("LB-algebra" is used in loose analogy with "LF-space", each LB-algebra be-
ing an inductive limit (not necessarily strict) of Banach algebras.)
Certainly every Banach algebra is an LB-algebra with respect to the
bound structure ® consisting solely of the closed unit ball. In the remain-
der of this section we present two examples of LB-algebras which are not
* First considered in Allan, Dales, McClure 1971.
329
330
6. LB-ALGEBRAS
generally Banach algebras.
Example 6.1-1 p-BANACH ALGEBRAS Let X be a complex algebra equipped with
a p-norm, 0 < p < 1, i.e. a real-valued function || || defined on X such that
(i) ||x|| > 0 for each x€X and ||x||=0 iff x=0,
(ii) |[X||= |X |P||x|| for each XCC and xfX,
(iii) ||x+y|| < ||x||+||y|| f°r each pair x, yeX.
If, in addition, ||xy|| < ||x|| ]|y|| for each pair x, yCX and ||e||=l, then || ||
is an algebra p-norm and X a p-normed algebra*. It is a p-Banach algebra
if X is complete with respect to the metric d(x, y)=||x-y||, x, yfX.
A p-Banach algebra X is an LB-algebra with respect to a certain bound
structure ® which we now describe. If x ,...,x are elements of X with p-
1 n
norm less than one, and M(x.) denotes the collection of all monomials in
1	i i
x,,. . . ,x (i.e. elements of the form x,Д,x n where i, , . . . , i are non-
1 n v	I n	I n
negative integers and x^=e) then $8 consists of all sets of the form
B(x^) = cl(M(xi)bc) .
Clearly each of the sets B(x^) is absolutely-convex and contains e. Since
||xy|| < ||x|[ ||У|| f°r each pair x, yfX it follows just as in the Banach alge-
bra case that multiplication is (jointly) continuous. Using this and the
fact that the sets M(x_), are multiplicative we obtain
i be
2	2	2
(B(x.))	= (cl((M(x.)bc))ZC:cl((M(x.)bcZ)
C cl(M(x.)bc) = B(x.)
Observing that B(x^) (J В(у^) С B(x^ ,у^.) we conclude that 0 is a bound struc-
ture for X. To show that® is complete it is first established that each
1
B=B(x^) is bounded in the metric space X. Indeed, if b=EX(i	)x
....x lnfM(x.) where the sum is taken over some finite collection of n-
n i be
tuples (ip...,in) and E |X(i^> • • , in) | < 1, then
ЦЪ|| < Z |x(i1,...,in)|P||x1||11....||xn||ln
If we choose 1 > r >0 such that ||xjJ| < r for each k=l,...,n then it fol-
lows from the above inequality that
i-i	i	co i,	co 1	-П
l|b|| < E Цх^ ....||xn||	< ( Eo ||x. || b...^ So IM n) <
1	n
* For more on p-normed algebras see Zelazko 1965, Chapter I.
6.1 BOUND STRUCTURES
331
so each M(x ), is bounded. Since the closure of a bounded set in a metric
i be
space is bounded, B=B(x.) is also bounded. Next we claim that p induces
1	в
a stronger topology on X(B) than the relative topology from X thereby im-
bounded there exists an f > 0 such
Thus S (O)flX(B) is a neighborhood
€
X(B), and our contention follows.
Now suppose that (x ) is a Cauchy sequence in X(B) with the topology
induced by p^. Then (x^) is certainly Cauchy in
so x x for some xfX. Given f > 0 there is an
n
x f ?B for all n, m > N. It is easy to see that
m	—
x=lim(x -x fB. Hence xfX(B), p„(x -x) < ( whenever
щ n tn	Bn
elude that X(B) is a Banach algebra.
remaining thing to note is that for each
1. Thus X=Ux(B) and X is an LB-
BeS
plying that p is a norm. Since В is
that BCSf(O)A X(B)={xfX(B) j||x)| < f} .
of 0 in the topology induced by p on
В
X, a complete metric space,
index
f В is
The only
where X€C has
the property that ||Ax|| <
aIgebra .
Example 6 1-2
X-HOLOMORPHIC FUNCTIONS
T (Def. 4.12-4)
a neighborhood
elements of
X,
N > 0 such that x
n
closed in X so x -
n
n > N, and we con-
xfX, xfX(B(Xx)),
Let
U of
i.e.
X be a uniform algebra on the
function yeC(T,C) is
t such that у can be
for each f > 0 there
X-holomor-
approxi-
exis ts
compact Hausdorff space
phic at tgT if there is
mated uniformly on U by
some xjX such that sup |(y-x) (U) | <
phic on T if it is X-holomorphic at
holomorphic functions on T, denoted
The
function yfC(T,£) is
each tfT.
The collection of
X-holomor-
all X-
by H(T,X),
gebra which contains X as a subalgebra. It is
clearly forms a complex a 1-
our contention that H(T,X)

A
is an LB-algebra.
Let l/=(Uj)j=1 be a finite open cover of T and H(T,L(,X) be the col-
lection of all yeC(T,C) such that у can be uniformly approximated on U. by
elements of X for each j=l,...,n. To see that H(T,U.,X) is a uniform al-
gebra on T it suffices to show that it is closed in C(T,C,,c). To this end
suppose that zf с 1(H(T, Ц ,X) . Then for each f > 0 there is a yeH(T,t/,X)
such that sup |(z-y) (T)|<€/2. Since yfH(T, U ,X) then for each i there is an
x. cX such that sup |(y-xp ( U ;) | < e/2. Hence, sup |(z-x.) ( и | < ( and it
follows that zeH(T,U.,X).
Next we claim that H(T,X) = UH(T,U ,X) where U runs through the collec-
covers of T. Indeed if yjH(T,X) then for each tfT
U of t on which у can be uniformly approximated
Since T= U U and T is compact, there is a fi-
tCT C
of T. It easily follows that y€H(T,LZ,X) where
tion of all finite open
there is a neighborhood
on [j by elements of X.
nite subcover, U,,...,U
1 n
U=(U.).
332
6. LB-ALGEBRAS
balls Bj^
Clearly each By is ab-
finite open
It only remains to show that the collection of closed unit
of H(T,tl,X) constitutes a bound structure for X.
solutely m-convex and contains the identity. If and are
coverings of T then so is U 12 = {U1(^ U2 |Uie 1 ’	Furthermore, be-
cause each element of Ъ(^2	contai-ne<i in elements of	and
H(T,U ,X) and H(T,U0,X) are subalgebras of H(T,L(10,X). Thus B;, U
1	Z	1Z
CB,. and this concludes the proof. In general H(T,X) is not closed
(Л12
C(T,C,c) [Rickart 1966] so H(T,X) is not generally a Banach algebra.
Neither of the algebras discussed above were endowed with topological
structure. A condition under which an IMCH algebra is LB-algebra is con-
sidered in Sec. 6.3.
Bu2
in
6.2 Some Properties of LB-Algebras In this section we indicate some of the
important features of Banach algebras that are carried over to LB-algebras.
After characterizing LB-algebras as inductive limits of Banach algebras we
prove such things as the compactness of the class of non-trivial complex-
valued homomorphisms in the Gelfand topology, that C is the only LB-algebra
which is a field, and that every maximal ideal is
the kernel of a complex
homomorphism.
to the bound structure®.
increasing limit of the
First we note that the set® is directed by the ordering:
It is clear that X(B) is a
:X(B) ->X(B’) is a continuous
Suppose that X is an LB-algebra with respect
We wish to show that X may be viewed as a sort of
aIgebras X(B).
В < В1 iff there is a \ > 0 such that BfJXB1.
subalgebra of X(B') and the injection map I , ;
В В
unital algebra isomorphism (i.e. a continuous algebra isomorphism that maps
the identity of the domain into the identity of the range), whenever В < В1.
Furthermore X, being the union of the X(B) is the limit in the ordinary set-
theoretic sense of the increasing net (X(B))B^^(reca 11 that a net of sets
(S ) converges to S if S= О U s = и n S ). The net of algebras
m	mffl t >m c rn€M t >m C
(X(B))„ _ together with the collection of continuous unital isomorphisms
B63>
defined whenever В < В1, is a particular example of an "inductive
В В
system" with X being the "inductive limit" of the system.
More generally an inductive system of linear spaces is a net of vector
spaces (all over
the same field),
defined for each
for each t and I
(Z ) ,, and a collection of linear maps
t tfM
pair (t,s) where s < t, such that I
. I =1
ut. ts US
is
tt
The induc-
I :Z -• Z , one
ts s t
the identity map
tlye limit of the inductive system is the vector
is the direct sum of the Zt's and N the subspace
ing of all elements (zt) with the property that
spacet^Zt/N, where
of the direct
E Z
ten t
sum consist-
in Z for

6.2 PROPERTIES OF LB-ALGEBRAS
333
so that z^0 implies that s < t.*
the reader to verify that the LB-algebra X is in fact
of the system given by the net (X(B))g^^gand the con-
tinuous unital (algebra) isomorphisms Ig, g (the injection of X(B) into
X(B')) for В < В'.
It is natural to ask if an inductive limit of a system of complex
each t large enough
We leave it to
the inductive limit
Banach algebras and continuous unital isomorphisms equipped with pointwise
multiplication, is an LB-algebra. The answer to this question as we shall
now show, is yes. Let (Xt)t M be a net of Banach algebras and the maps
I
ts
:X
s
t
be continuous unital (algebra) isomorphisms. The inductive limit of this
system is X= X /N where (x )eN iff S I (x )=0 whenever t is so large
SfM S	s	s < t ts s
that x ^0 implies s < t. Consider the map
x ---> <x >—> < X > + N
s	s	s
where the entry of < x > indexed by s is specified to be xg and all the
rest are 0. I is easily seen to be an algebra isomorphism so we can
s
transfer the norm of X^ to I^X^), i.e. || < x^ > + N||s=||xs||s > whereby
* One can motivate the definition of the inductive limit in the following
way. The idea is to construct a vector space Z containing the Zt's as sub-
spaces (more precisely isomorphic Images of the Zt's) such that Z^ is a
subspace of Z whenever s < t and Z is the increasing limit in the set-
theoretic sense of the Z 's. The direct sum E Z certainly contains iso-
t	tCM t	* I * 3 * s
morphic images of the Zt's but the other desired properties are lacking.
Thus we are led to consider a quotient space of the direct sum, t^Zt/N,
and the following candidates for each index s for the required isomorphism
I :Z -E Z - S Z /N
s s	t	t
z — < z >-*<z	> + N
s s	s
where < z > is the tuple with zero entries for t^s and in the entry in-
dexed by s. It readily follows that each I is an isomorphism, I (Z ) is a
subspace of I (Z ) for s < t provided N is defined as above (< Itsxs -* ~
< x > CN for each x fX ) and that E Z /N = U I (Z ).
s	ss	t s CM s s
334
6. LB-ALGEBRAS
I (X ) becomes a Banach algebra. Suppose that s < t; since < I x > -
s s	ts s
< x > fN it follows that < x >+N=<I x >+N for each x cX . With
s	s	ts s	s s
the aid of this equality it follows that I (X ) is a subalgebra of
whenever s < t. Furthermore the continuity of I implies that the norm
|| || of I (X ) restricted to Ig(Xs) is "weaker" than || ||s: There exists a
scalar X > 0 such that || ||t	|| ||g whenever s < t. Let be the
closed unit ball of I (X ) and (3=Гв 1	. We claim that® is a complete
ss	LsJsfM
bound structure for X. The elements of® are certainly absolutely m-convex
and each contains the identity < e >+ N. Moreover, if s < t then, since
s
|| ||t <	|| ||s follows that BgC Xj-gBj.- Hence if B^, we can choose
t€M such that r, s < t and obtain the conclusion that В UBC max(X ,X )B .
—	r s	tr ts t
Since X(B )=I (X ) is a Banach algebra, (X,®) is a complete bound algebra,
s s s
It remains to show that X=UX(B ). To this end consider an arbitrary
s
element (x )+N€X. There are only a finite number of indices s such that
s	n
x ^0. Denoting these indices by s,,...,s we see that (x >(-№.£ < x > +
s	In	s i=l s^
N€It(Xt)=X(Bt) for any t > s^j-.-.s^.
We summarize these results in:
Theorem 6.2-1 AN LB-ALGEBRA IS AN INDUCTIVE LIMIT OF BANACH ALGEBRAS A
complex algebra X is an LB-algebra with respect to some bound structure iff
X is the inductive limit of an inductive system of complex Banach algebras
and continuous unital isomorphisms.
Since any complex Banach algebra is a Q-algebra each non-trivial com-
plex-valued homomorphism of a Banach algebra is continuous [(4.10-5)].
(6.2-1) HOMOMORPHISMS OF AN LB-ALGEBRA ARE "BOUNDED" If X is an LB-algebra
with respect to the bound structure® then each non-trivial complex-valued
homomorphism h maps the elements of Q into bounded subsets of C.
Proof If Be© then, since efB, h restricted to the Banach algebra X(B) is a
non-trivial homomorphism of X(B). As such it is continuous. Thus it takes
B, a bounded subset of X(B), into a bounded set of complex numbers. V
In view of the fact that the bound structure ® associated with an LB-
algebra X need not arise from a topology, it doesn't make sense in general
to ask when a non-trivial homomorphism h is continuous. Even in the event
that the elements of® are bounded in some compatible topology on X, it need
not follow that each h be continuous (see Example 6.3-2).
Recall that if X is a complex Banach algebra then X^ carrying its
Gelfand topology, i.e. а(х\х) is a compact Hausdorff space. This property
is carried over to the LB-algebra case, i.e. if X is an LB-algebra, the col-
lection of all non-trivial complex-valued homomorphisms H(X) with the
6.2 PROPERTIES OF LB-ALGEBRAS
335
o(H(X),X) topology (the Gelfand topology) is a non-empty compact Hausdorff
space. We will prove this by first showing that H(X) is a "topological
projective limit" (a notion which is defined below) of non-empty compact
Hausdorff spaces followed by a demonstration of the fact that a topological
projective limit of non-empty compact Hausdorff spaces is a non-empty com-
pact Hausdorff space.
Theorem 6.2-2 H(X) IS A NON-EMPTY COMPACT HAUSDORFF SPACE Let H(X) be the
collection of all non-trivial complex-valued homomorphisms of the LB-algebra
X with bound structure ® • Then:
(a)	H(X) with its Gelfand topology is homeomorphic to the topological
projective limit (defined in the proof) of the non-empty compact Hausdorff
spaces X(B)h and continuous restriction maps R , with B> B'f® and В < B’;
BB
(b)	H(X) with its Gelfand topology is a non-empty compact Hausdorff
space.
Proof First we note that X(B)h is not empty because X(B) is a Banach
algebra.
Suppose now that hCH(X) and Bf® , the bound structure associated with
X. Let hB=h |X(B) and consider the element (Bg)gg£^ of the product B^K(B)'1.
It is clear that if В < B' (where < is as in the proof of Theorem 6.2-1)
which implies that X(B)C.X(B'), then h is the restriction of h , to X(B).
в	h в
Thus the element hfH(X) determines an element (hB)fB7^C(B) with the proper-
ty that hB=I<BB, (hg,) where R^^, is the restriction map which restricts ele-
ments of X(B')h to X(B) when В < B'. Conversely if (h )f П X(B)^ and
В B^tD
(h	,)=h whenever В < B1 we can define h:X — C via the rule h(x)=h (x)
HR	-v	В
is clear that h is a well-defined non-trivial complex homo-
Thus we have established a correspondence (which is evident-
H(X) and the subset of the product consisting of elements
(h ,)=h whenever В < B1. We also observe that since
В В
is continuous when X(B')
This subset of the product
RBB|('“B'/ “B
for xeX(B). It
morphism of X.
ly 1-1) between
(hB) for which	..B	h	h
X(B)C.X(B'), each such R^ ,, is continuous when X(B') and X(B) carry their
Gelfand topologies. This subset of the product	equipped with the
relative product topology, is referred to as the "topological projective
limit" of the "topological projective system" consisting of the net of
topological spaces (X(B)'1)Bf® and the continuous restrictions maps Rgg i
defined for each pair (B,B') such that В < B1.
More generally, a "topological projective system" (Definition 4.6-1)
is a set (Z ) of topological spaces indexed by a preordered set M and a
m mfM
collection of continuous maps R :Z. -> Zm, one defined for each pair (m, t)
mt t m
336
6. LB-ALGEBRAS
for which m < t, such that R is the identity map for each mCM and R oR =
—	mm	ms s t
R for all m < s < t. The topological projective limit Z of this system
is defined to be the topological subspace of all elements (z	suc'1
that R (z )=z for all m < t.*
mt t m	—
Suppose that each is a non-empty compact Hausdorff space; we claim
that Z is also a non-empty compact Hausdorff space. That Z is a compact
Hausdorff space will follow from having shown it to be closed in the com-
pact Hausdorff product m^Zm. Indeed, if (w^)^]. is a net °f elements w^=
(z . )fZ converging to the element z=(z^)f m^4^m t'len’ denoting the continu-
ous projection of the product onto Z by pr , we see that (z .).	=
" J	mm	mi ifl
(pr (w.)) converges to z in Z . Since (z .)€Z, then for each id we
m 1 ifl	mm	mi
have R (z . )=z . whenever m < t. Since each such R is continuous it fol-
mt ti mi	—	mt
lows that R (z )=z for each pair (m,t) for which m < t. Hence (z )CZ and
mt t m	—	m
Z is closed.
To see that Z is non-empty it is enough to show that Z is the intersec-
tion of a family of closed subsets of the compact spaces	having the
finite intersection property. Fix tfM and let Y be the set of all
(z )C U.Z with the property that R (z )=z for all m < t. That Y is
m mCM m	mt t m	—	t
closed follows in the same way as the closedness of Z. Now if we choose
some z from the non-empty space Z and set z =R (z ) for each m < t then,
t	t	m mt t	—
by arbitrarily assigning values to all other z^'s, we obtain a member of Y .
Hence each Y ^0. Suppose that m , . . . ,m fM; then by choosing t > m , . . . ,m
t	n	1 n	In
it follows that Y CL /\Y .. Thus the collection (Y Itcn} has the finite
t j=l mj	1 tI
intersection property and Z= Q,Y ^0.
tCM t
Returning to the situation at hand we recall that the correspondence
T:h <—> (h ) _ is a 1-1 mapping of H(X) onto the topological projective lim-
В В	h
it Z of the compact Hausdorff spaces X(B) . Next we claim that this corre-
spondence establishes a homeomorphism when H(X) carries the Gelfand topology.
A typical subbasic neighborhood of the point (h^)in Z is a set of the
form
* If one takes the
intuitive viewpoint that the mappings R act to "reduce
mt
the size" of the Zt's then the projective limit may
creasing" limit of the net (Z ) This viewpoint
in the case when the projective limit of the spaces
be considered as a "de-
ls especially appealing
of homomorphisms X(B)^
is formed, for then В < B' implies that X(B)CX(B') and, therefore,
X(B’)h "C" X(B)h.
6.2 PROPERTIES OF LB-ALGEBRAS
337
< V(hB ’ Xo> e) > = {(fB)eZ lfB eV(hB ’ V
О	о о
where В is fixed in® and V(h , x ,f)={hfX(B )h I |h(x )-h (x ) I < €},
o	BQ о	о I 1 о В о I J
with x €X(B ), and € >0, is a typical subbasic neighborhood of h in
h° °	°°
X(B ) . Now it is clear that T(V(h,x ,f))C2 < V(h ,x , c) >, from which
о	О	Bq o
continuity follows. T is bicontinuous since it is a 1-1 continuous mapping
of a compact Hausdorff space into a Hausdorff space. V
As an immediate consequence of the fact that Н(Х)^0 we obtain a result
for LB-algebras similar to the Gelfand-Mazur theorem [Theorem 4.9-1].
(6.2-2) IF THE LB-ALGEBRA X IS A FIELD, X"="Q. If X is an LB-algebra and
a field then X is algebraically isomorphic to C.
Proof By Theorem 6.2-2 (b) there exists a non-trivial homomorphism h:X-,_C.
If we assume that h is not 1-1 then there exists a non-zero x in the field
X, such that h(x)=0. Thus, for each yCX,
h(y) = h(ye) = h(yx • x) = h(yx h(x) = 0
which contradicts the non-triviality of h. Thus h is 1-1 and X is algebra-
ically isomorphic to С. V
Recall that in any Banach algebra the maximal ideals are in 1-1 corre-
spondence with the non-trivial complex homomorphisr;. It is always the
case that the kernel of a non-trivial (complex) homomorphism of an algebra
is a maximal ideal. On the other hand if M is a maximal ideal in an alge-
bra X then X/M is certainly a field. Hence if we can show that X/M is an
LB-algebra whenever X is an LB-algebra and M is a maximal ideal, it will
follow by (6.2-2) that M is the kernel of the complex homomorphism
h:X—ь X/M + C
x —> x+M-+ X
where x+M — X is an isomorphism between X/м and C.
To do this we prove a more general result:
(6.2-3) QUOTIENTS OF LB-ALGEBRAS Let X be an LB-algebra with respect to
the bound structure® and I be an ideal of X with the property that 1ЛХ(В)
is a closed ideal in X(B) for each Be®. Then the algebra X/I is an LB-
algebra with respect to the bound structure 6B/l={B+I jBffl]. In particular
X/M is a LB-algebra for any maximal ideal M of X.
Proof First we note that by some routine considerations ®/l is a bound
structure on X/l. It is also clear that for each B€0 the subalgebra of X/I
generated by B+I is just X(B)+I=[x+I |xfX(B)} and Х/1=в*^^(В)+1. Thus it
338
6. LB-ALGEBRAS
remains to show that each algebra X(B)+I is a Banach algebra with respect
to the gauge of B+I. To this end suppose that xfX(B) and consider
pn,T(x+l) = inf{\ >0 lx + IfX(B + I)}
= lnf zhnX(B) u >° Iх +
= inf inf{X > o |x + zfXB]
zei nx(B)
= inf	p (x + z) = p (x + iHX(B)).
zeinx(B)
Thus the mapping
X(B) + I----------------------»- X(B)/If) X(B)
x + I --->- x + 1П X(B)
is seen to be an onto "isometric" isomorphism when X(B)+I carries the semi-
norm p T and X(B)/I(1X(B) the semi-norm p induced by p . Since lOX(B)
B+I	t	в	в
is closed in X(B), p is a norm which, because X(B) is complete in the norm
В
p , renders X(B)/l/lX(B) a Banach algebra. Hence each algebra X(B)+I is a
rB
Banach algebra and X/l is a LB-algebra.
Now if M is a maximal ideal in X then we claim that each ideal Mf)X(B)
is closed, in the Banach algebra X(B). Suppose that this isn't the case,
i.e. there exists a Bf® such that MflX(B) is not closed in X(B). Then
there exists a sequence (ii^)CMOX(B) convergent in X(B) to some x&l.
Since M is maximal in X elements yfX and mfM exist such that e=yx+m. Choose
B’f® such that X(B')^>X(B) and yfX(B'). As the injection mapping I ,
В в
taking X(B) into X(B') is continuous by Theorem 6.2-1 it follows that
ym^+m -> yx+m=e in X(B'). Since a Banach algebra is a Q-algebra [(4.8-2)]
we may choose к so large that ym^+mfM belongs to the open set of units.
However у m^+rnCM and M, being proper, can contain no invertible elements.
This contradiction implies that each MflX(B) is closed in X(B) thereby con-
cluding the proof.V
As an immediate consequence of this result and the remarks preceding
it we have
(6.2-5) IN AN LB-ALGEBRA H(X)	Each maximal ideal of a LB-algebra is
the kernel of some hfH(X).
In light of the last result it is evident that whenever M is a maximal
ideal in an LB-algebra X each element xfX can be written uniquely in the
form x=\e+tn where XfC and mfM. Thus M is a subspace of X of codimension
one consisting of singular elements. Gleason 1967 (cf. Beckenstein, Narici,
6.2 PROPERTIES OF LB-ALGEBRAS
339
and Bachman 1971) proved the converse for Banach algebras, i.e. any subspace
of a Banach algebra consisting solely of singular elements and having co-
dimension one is a maximal ideal. We conclude this section with a presen-
tation of Gleason's result followed by an extension of it to LB-algebras.
(6.2-b) IN AN LB-ALGEBRA. МЛ 0 = 0 AND COD(M) ° 1 IMPLIES If M is a
subspace of the LB-algebra X of codimension one, consisting of singular ele-
ments then M is a maximal ideal.
Proof First suppose that X is a complex Banach algebra. The subspace M,
being of codimension one, must be either dense or closed in X. Since X is
Q-algebra (4.8-2), Q, the set of units in X, is non-empty and open; so M,
containing only singular elements, must be closed. If we let h be that
linear functional on X having M as its null space and mapping e into 1, then
since M is closed, h is continuous. Now it suffices to show that h is a
homomorphism. Furthermore, the equation
.	.2	2	2
xv = ,(*+x)___~ x-_У
y	2
2
together with the linearity of h reduces the problem to showing that h(x ) =
2
(h(x)) for each xfX.
To this end consider the function
exp(Xx)
<» n n
n=0 n.
Since ||x || < ||x|| in a normed algebra it follows that the defining series
is absolutely convergent and, therefore, convergent in the Banach algebra X
for each Xf£. Thus, by the continuity of h,
co •>	, П к
h(exp(\x)) = S ------ ,	XeC,
n=0 n-	~
and h(exp(\x)) is an entire function with no zeros. Furthermore, setting
M(r)=sup lh(exp(Xx))) we obtain
, zu ,	Inin M(r)
order (h(exp(Xx))) = lim----------
, Хпг
< lim Inin ||h|| JQ №1^
* The notion of the order of an entire function may be found in Markushevich
(1965), Vol. 2, p. 251.
340
6. LB-ALGEBRAS
lim
< lim
ln(,l,n||h|| + r||x||)
Inr
lnln||h|| + Lnr + lji||x||
Lnr
= 1.
Thus, by a weak version of Hadamard's factorization theorem', the fact
that h(exp(Xx)) never assumes the value zero, and, h(exp().x)) =1, it follows
tha t
co n n
h(exp(Xx)) =	= S “-7-
n=0 n-
for some	Now by the identity theorem tor power series , h(x )ea
for each n > 0; so
2	2	2
h(x ) = a = (h(x))
for each x€X.
Next suppose that X is an LB-algebra with respect to the bound Struc-
ture'S and M is linear subspace of X as in the hypothesis. Since the co-
dimension of M is one, we may write each xfX uniquely in the form х=\е+т
where XcC, and mfM. Now if xfX(B), (Be®), then, since efB, mfX(B) , and
it follows that the codimension of МПХ(В) is one in X(B). As the elements
of M are all singular so are the elements of MflX(B) singular in X(B) .
Hence by the result established above we conclude that МПХ(В) is a maximal
ideal in X(B). It remains to show that M is an ideal in X. To this end
let xfM and ycX and choose B£$ such that x, y€X(B). Then xfMf)X(B) and
ху£МПХ(В)СМ. V
Section b.3-1. Complete LMC LB-Algebras The concept of an LB-algebra is
primarily algebraic - it need not be a topological algebra to begin with,
nor do we have to add topological structure to it for its salient proper-
ties (see Sec. 6.2). However, it is the case that we can completely char-
acterize complete LMC LB-algebras, i.e. those complete-U4CH-a Igebras that
* (Markushevich (1965), Vol 2, p. 266) if the entire function f(X) of
order p never assumes the value wcc then p is an integer and f(X) is of the
form f(X)=w+eP(^ where p(X) is a polynomial of degree p.
**(Markushevich (1965), Vol. 1, p. 352). If the complex power series
as
nS0 an(z-ZQ)	agree on a bounded infinite set of complex numbers
then a =b for each n >0.
n n	—
6.3 COMPLETE LMC LB-ALGEBRAS
341
are also LB-algebras. A bound structure of a Banach algebra consists of
the unit ball, a topologically bounded closed absolutely m-convex set con-
taining the identity; a natural bound structure to consider in an IMC al-
gebra is the collection of all absolutely m-convex closed bounded sets
which contain the identity. We shall show in the main theorem of this sec-
tion (Theorem 6.3-1) that a complete IMC algebra is an LB-algebra with re-
spect to this bound structure whenever the space of non-trivial complex-
valued homomorphism is compact in its Gelfand topology. It follows from
this and the fact that the space of non-trivial complex-valued homeomor-
phisms is always compact for an LB-algebra (Theorem 6.2-2) that if a com-
plete IMC algebra is an LB-algebra with a bound structure® then it must be
an LB-algebra with respect to (3^. It is also established that the Frechet
LB-algebras are precisely the Frechet Q-algebras (Theorem 6.3-2).
Definition 6.3-1. THE NATURAL BOUND STRUCTURE Let X be a complex topolog-
ical algebra. Then the associated na tura 1 bound s tructure, denoted by ®n >
is the collection of all subsets BCX such that
(a) В is absolutely m-convex and efB ,
(b) В is closed and bounded.
The collection X°= X(B) where X(B) = {ay |ae£> у€В} is referred to as the
bounded elements of X and is a subalgebra of X. If xfX while x^Xq then x
is an unbounded element.
Prior to presenting examples we characterize the elements of X .
о
(6.3-1) A CHARACTERIZATION OF X The element of xfX is a bounded element
---------------------------------о
(i.e. xfXj iff there exists a scalar x such that {(\x) |n€N} is a bounded
set.
Proof If xfXQ then x€X(B) for some	Since X(B) = {ay |a€£, yfB} , x=oy
for some afC and yfB. If x=0 then {(Xx)n|nfN} is certainly a bounded sub-
""1	n
set of X. If x^O then y=a xfB, and, since В is multiplicative, у fB for
any n. Thus {(a ^x)n|n€N}c.B and is therefore bounded.
Conversely, suppose that {(Xx)n|neN} is bounded. It is clear that S=
(Xx)n |n€N}(J [e} is multiplicative and bounded. Since multiplicativity and
boundedness are preserved when forming the closed absolute convex hull of a
set, B=cl(Sbc)€<an. Thus xfX(B)CXQ. V
Certainly all elements of a normed algebra are bounded as can be seen
directly from the definition of Xq or from the characterization given in
(6.3-1). In our next example we characterize the elements of C(T,£,c) that
are bounded. As might be expected they are just the set of uniformly bound-
ed complex-valued functions on T.
342
6. LB-ALGEBRAS
Example b.3-1. C(T,C,c) "=" UNIFORMLY BOUNDED FUNCTIONS Let xeC(T,C, c);
we claim that s is a bounded element iff it is uniformly bounded on T. In-
deed if x is uniformly bounded on T by the number M > 0, i.e. sup|x(T) | < M,
then the function (1/m)x and all of its positive powers are uniformly bound-
ed by 1. Thus it follows that each function ((1/Мх)П, n=l,2,..., is bounded
by 1 on each compact set K. Since the seminorms PK(y)=sup |y(K) | ,yeC(T,£, c),
generate the compact-open topology, we see that {((1/м)х)П |nfN] is a bound-
ed set in C(T,C,c). Conversely, suppose that x is not uniformly bounded on
T; then neither is the function Xx, for any X^O. Thus there exists a com-
pact set К on which Xx assumes a modulus larger than 1. It follows that the
set of numbers {p^((Xx)n) |nfN} is unbounded and [(Xx)n|n€N} is not a bound-
ed set. Since X was arbitrary, x^C(TJ(C,c)o by (b.3-1) and the proof is com-
plete. v
As a result of the preceding example we see that C(T,C,c) possesses
only bounded elements iff T is pseudo-compact (Sec. 1.5). We shall make fur-
ther use of this example at the end of this section when we prove that a
completely regular Hausdorff hemi-compact k-space (Secs. 2.1 and 2.2) is
pseudo-compact iff it is compact.
In order for X to be an LB-algebra with respect to it is necessary
that each X(B) be a Banach algebra with respect to the gauge p of B. In
В
our next result we present a sufficient condition on X for this to be the
case.
(b.3-2) EACH X(B), Bc<8 , IS COMPLETE IF X IS SEQUENTIALLY COMPLETE If X
is a sequentially complete locally convex Hausdorff algebra then each X(B),
BfSn, is a Banach algebra with respect to the gauge p^, i.e. (X,®n) is a
complete bound algebra.
Proof Since each Bf0 is a topologically bounded set in X, no such В can
contain a non-trivial subspace;
hence
each p
В
is a norm.
Next we contend
that the norm topology defined by p on X(B) is stronger than the topology
В
induced by X. To see that this is so let UeV(O). Then there is a scalar
a > 0 such that a BCZ.U. Now suppose that xfaVp^, i.e. Pg(x)=inf[a > 0 |x€oB)
< a. Since В is balanced, x^aBCU. Thus a Vp.C L'CX(B) .
В
Consequently we see that any sequence (x^)cX(B) which is Cauchy in the
norm p is also Cauchy in the induced topology on X(B). By the sequential
В
completeness of X, (xn) has a limit xfX. It remains to show that p^(x^-x)-«0
and xfX(B) . Since (x ) is Cauchy in the norm p for each 6 > 0 there is an
n	В
index N > 0 such that p„(x -x ) < A whenever n3 m > N. Thus x -x c6B for n.
В n m	—	n m
m > N and, noting that В is closed in X, it follows, that x -x=lim(x -x IffiB
n m n m
6.3 COMPLETE LMC LB-ALGEBRAS
343
for n > N. Hence хех^+6ВС-Х(В) and Pg(xn'x) < ® tor n > N. V
The next result, the central theorem of the section, provides a charac-
terization of complete INCH LB-algebras.
Theorem b.3-1. THE COMPLETE INCH ALGEBRA X IS AN LB-ALGEBRA IFF H(X) IS
COMPACT Let X be a complete INCH complex algebra. If either X*1 or H(X) is
compact in its Gelfand topology then X is an LB-algebra with respect to
Remark Theorem 6.2-2 shows H(X) to be Gelfand-compact if X is an LB-algebra.
Thus the compactness condition on H(X) is necessary as well as sufficient.
Proof We already know by the previous result that (X, ®n) is a complete
bound algebra. Thus it only remains to show that X=X =U „ X(B) . Further-
o B€
more, as a result of the characterization of Xq given in (6.3-1) and the
fact that a subset of a IMCH algebra is bounded iff it is weakly
bounded, it suffices to show that for each xfX a scalar XfC, exists such that
{x‘ (Qx)n)nQj} is a bounded set of complex numbers for each x'cX'. In order
to do this we make use of the analyticity of the resolvent function of a
П"" 1
element in an LMCH algebra to construct a power series having -x'(x	) as
its n-th coefficient. We then apply the Cauchy-Hadamard formula for the
radius of convergence to prove that the set {x'((Xx)n) |n€N} is bounded.
We begin the proof by observing that in the complete IMCH algebra X,
the spectrum of each element xfX is obtained as either фх(Н(Х)) or ^(X*1)
((4.10-8)), and is therefore compact. Hence p(x), the resolvent set, is
open, and by (4.8-5) and (4.8-6), the resolvent function r (X)“(x-Xe) is
analytic on p(x). Furthermore, again by the compactness of a(x), there is
an r > 0 such that {Xec | |X | > r }c. p(x) . Thus, setting f=l/r, we can define
the function s at X by
( r (1/X) for 0 < IX I < c
, \ \	_ I x	11
s(a) = j
( 0 for X ~ 0
which is clearly analytic for 0 < |X | < C. Moreover we claim that s is ana-
lytic S (0) = {X | |X I < €} and the n-th derivative of s at X is given by
/.л	(п)оч , on i П'1/.	.-(n+1)
(*)	s4 (X) = (-1) n! x (Xx-e)
for XcS (0) and n=l,2,... . First consider the case n=l. Since s(x)=
C _ i	_ j
(x-(l/x)e) =X(Xx~e) for 0 < |x | < € and inversion is continuous in any
LMC algebra [(4.8-6)]
* The Cauchy-Hadamard formula for. the radius of convergence of Еа^ХП is
1/lim supn/ |an |. (Markushevich ,(1965) , Vol. 1, p. 344.
344
6. LB-ALGEBRAS
s (1) (0) = lim	~ "n= li™ (Xx-e) = -e
X - 0 X	X - o
which agrees with (*) for X=0 and n=l. Next suppose that 0 < |p|, |X | < €
and consider
s (1) (p) = lim S<\>- - S-^
X - p X "
rx(l/X) - rx(l/|i)
(1/X - l/ц) rx(l/x) rx(l/p.)
2	?	-2
= (-l/ц ) (rx(l/n))	= -(px - e)
Hence we have established (*) for n=l. Proceeding by induction we assume
that (*) is valid for the integers l,...,n and then for X, p€S (0), we con-
sider
(n).,.	(n) , .	, ..n	,	n-1 r/,	.-(n+1) .	.-(n+l)i
sv (X)-s	(n) = (-1) n. x [(Xx-e) v -(px-e) v J
_ . ..n	,	n-1	.-(n+1)	.-(n+1) r. . n+1	.n+l-|
= (-1) n. x (Xx-e) v (px-e) v [(px-e)	- (Xx-e) J
- (-1) n. X (Xx-e) (px-e) v [(p*X)x][ + (px-e) (Xx-e) J
k=0
Therefore,
fn) fnl	n
s4	(X) -s4	(ц)	. ..n+1 , n. .-(n+1).	.-(n+1). 7 .	.k,,	. n-lq
\ _ p,--= (-1) n.x (xx-e) (ux-e) 4	’ [^(цх-е) (Xx-e) J
and, by the continuity of inversion, addition and multiplication, it follows
that by taking the limit as X that
(n+1). .	. ..n+1.	... n, .-(n+2)
sv '(p) = (-1)	(n+l)!x (px-e) 4	'
for each p€S (0) .
Thus having established (*) , it follows that for x'fX', the complex-
valued function x'-s is analytic on S^(0), and, by induction together with
a routine computation,
, I \ (П)	1,4-1.	. * (П+1),
(x -s)v (p) - (-1) n.x (x (px-e) 4	')
for each peS^(O) and n=l,2,... . By the analyticity of x'-s on 3^(0) the
Taylor series of x'-s centered at u=0 converges in S (0). Since
6.3 COMPLETE LMC LB-ALGEBRAS
345
(x'.s)	(0)/n]=-x ' (x11 1) for n > 1, we may apply the Cauchy-Hadamard for-
mula to obtain
, .	I l / tl 1 \ |1/п i /	__
lim sup x (x )	< 1/e = r
which implies
, . if  / / f ..nl. .1/n l
lim sup x ((*• x) ) |	< -
Thus for all but at most a finite number of values of n,
If X X) > I 1
and the final conclusion that the set [x'((^ x)n)njN} is bounded follows. 7
Suppose now that X is not an LB-algebra with respect to	Is it
possible for X to be an LB-algebra with respect to some other bound struc-
ture? The answer is no, in view of our previous result and the fact that
an LB-algebra has a Gelfand-compact set of non-trivial homomorphisms.
Any complex homomorphism of a Banach algebra is bounded or equivalent-
ly continuous - as mentioned prior to (Theorem 6.2-2). Boundedness cannot
possibly be equivalent to continuity for LB-algebras since they don't carry
a topology. Suppose however that (X,®) is an LB-algebra and that a compat-
ible topology-^ for X exists such that each Bf<B is -^-bounded. By (6.2-1)
any complex homomorphism of X maps elements of ® into bounded subsets of
the complex plane. Since, algebraically, X is structurally related to
Banach algebras (as an inductive limit) and, since a compatible topology
for X is linked to this algebraic structure it is natural to ask if all ho-
momorphisms of X onto C are ^/'-continuous. As shown below, this needn't
happen.
Example 6.3-2. A DISCONTINUOUS HOMOMORPHISM Let T be a compact Hausdorff
topological space with a non-isolated point t° (e.g. T=[0,l] and t =0).
Since C(T,£,c) is a Banach algebra each element of Н(С(Т,С!,с)) is continuous
and, moreover, by Example 4.10-2 H(C(T,£,c))=T* (the evaluation maps). De-
noting by J' the weakest topology of С(Т,_С) for which the homomorphisms
T*-[tQ*} are continuous, it follows tha'\T is weaker than the compact-open
topology. Thus the elements of the natural bound structure are all
-bounded while, by Example 4.10-1, t * is not continuous with respect to
As the above example illustrates not all homomorphisms of an LB-algebra
(X,® ) are ^Г-continuous even tho -J is a compatible topology in which
the elements of® are cT-bounded. Reversing direction somewhat, starting
346
6. LB-ALGEBRAS
with a Frechet algebra (X/sT ) , if S is the natural bound structure deter-
mined by^f and (X,®) is an LB-algebra then all complex homomorphisms of X
must be continuous as shown by Theorem 6.3-2: The main thrust of Theorem
6.3-2 is that in the class of Frechet algebras, being an LB-algebra is
equivalent to being a Q-algebra.
Theorem 6.3-2. FOR FRECHET ALGEBRAS, LBQ If X is a Frechet algebra
then the following are equivalent:
(a)	X is an LB-algebra.
(b)	X is a Q-algebra - hence all homomorphisms are continuous.
(c)	Xh is compact in its Gelfand topology.
(d)	r (x) < <= for each x€X.
a
Proof (a) — (d) : By Theorem 6.2-2, H(X) is compact in its Gelfand topolo-
gy. Since each singular element lies in a maximal ideal and each maximal
ideal is the kernel of some complex homomorphism [(6.2-5)], it follows that
a(x)={h(x) |hfH(X) } = (фх) (H(X) ) - where ф is as in Def. 4.12-1 - and is there-
fore a bounded set of complex numbers for each xfX.
(b) — (c) : We know that a Frechet algebra, being nonmeager, is bar-
reled. In (4.12-3) we established the equivalence of (b) and (c) for bar-
reled complete complex LMCH algebras.
(d) -> (b) : Recall that X is a Q-algebra iff the set U(a)=fxeX|r (x)<rl}
1 a ~
has non-empty interior (4.8-3). In a complete complex LMCH algebra U(d)=
(Xh)° (4.12-2). Since the polar of a set is weakly closed and absolutely
convex and a a(X,X') - closed absolutely convex set is closed in any topol-
ogy of the dual pair, U(a) is a closed subset of X. Now by (d), we have
that X=U„nU(a), a countable union of closed sets, and, so by the Baire
nCN	7
category theorem, U(a) has non-empty interior.
(c) -• (a): By Theorem 6.3-1. v
In Chap. 1 it was shown that for replete spaces, compactness and pseu-
do-compactness are equivalent ((1.5-3)) As any completely regular Hausdorff
hemicompact k-space is replete (Th. 1.5-3 ) such a space is compact when-
ever it is pseudo-compact. An alternate proof of this result can be given
using the ideas developed in this chapter: More precisely, the previous
theorem and the characterization of the bounded elements of C(T,C,c) given
in Example 6.3-1. Suppose that T is a. completely regular Hausdorff hemi-
compact k-space which is not compact. We know from Example 4.10-2 that
because T is completely regular C(T,C,c)h=T*. Since the Gelfand topology
of C(T,C,c) is the weakest with respect to which each function фх, where
6.3 COMPLETE INC LB-ALGEBRAS
347
(фх ) (t*)=x(t) for xfC(T,C) and t€T is continuous, the original topology of T
is stronger than the Gelfand topology. To see that the two topologies are
in fact the same, let U be a proper open subset of T in the original topol-
ogy. In that T is completely regular, it follows that corresponding to each
tfU there exists a continuous real-valued function x such that x (t)=0 and
х^(си) = {1}. Hence the Gelfand neighborhood of t, x^ (S^(0)) , is contained
in U and U is open in the Gelfand topology. Having established that the
Gelfand topology of T coincides with the original non-compact topology, it
follows by the previous theorem ((a) and(c))that the Frechet algebra C(T,,C,c)
is not an LB-algebra. Now by virtue of the fact that X=C(T,C,c) is complete,
each X(B) , for	is a Banach algebra by (6.3-2). Consequently it must
be that C(T,£, c)?tC(T,(C, c)q. Thus by Example 6.3-1 there are continuous un-
bounded complex-valued functions defined on T which precludes the possibility
of T being pseudo-compact.
We are now in a position to give specific examples of complete INCH
algebras that are not LB-algebras. If T is any non-compact completely reg-
ular Hausdorff hemicompact k-space, e.g. T=R, then C(T,,C,c) is a complete
INCH algebra that is not an LB-algebra.

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INDEX OF SYMBOLS
A', 227
A°, 227
ac> ™
A or (X , algebra generated by zerosets, 33
z z
CS, 76
76
₽ > 9
T’
P(X,X'), 4
or C , 6
G, complex numbers
CA, complement of A
4 or 6>b’ 6
C (S,T), 2
£b(T,R) or£b(T,R), 6
&G' 310
£bk’ 310
C (0), closed unit disc about 0 in any normed space
clgA, closure of A in S
C(S,T), 2
C(T,F,c), 3
£(T,R) or (T,R), 6
JQ, 187
F, real or complex numbers, 2
H, H(T), characters of C(T,R), 29
H(I), 300
h(I), 300
H(X), non-trivial scalar homomorphisms of X
ker A, (kernel of A), 177
K(S), 300
k(S), 300
M, 14
M , 223
c
, 0-1 measures, 223
Д5, positive integers
4>, 243
rational numbers
К real numbers
363
364
INDEX OF SYMBOLS
Rad X, 236
p(x), 201
ra(x), 201
S , 4
Sf(0), open unit disc of radius f > 0 in any normed space
a (x), 201
a(X,X'), 4
supp (support), 91, 92
Л’302
^hk’ 300
''hk’
J" > 255
r
T* = [t |t€T, t :C(T,F) - F, x - x(t)}
V J , 190
p*
# , 248
wc
т(X.X1), 4
U(a), 201
uT, 21
uxT> 311
V , 183
_P .
Vp = {x |p(x) < 1}
V(x), neighborhood filter at x in any topological space
wL, 143
w(T,£), 143
(x), 220
X(B), 329
X11, 223
X1, continuous dual of TVS X, 227
Й, Gelfand map, 223
X , algebraic dual of vector space, X
Z, zero sets
Z, integers
z (M), 10
z (x), 10
INDEX
absolutely m-convex, 181
absolutely K-convex, 134
adjoint, 227
Allan, 329
Alexandrov, 84
algebra, 176
algebra homomorphism, 177
algebra with involution, 259
symmetric, 259
almost open, 257
Alo, 170
ap-lattice, 148
a-lattice, 147
analytic, 211
analytic functions on a disc, 195, 305
Arens, 199, 210, 296
Ascoli theorems, 120
atom, 165
atomic, 166
Bachman, 51, 55
Bade, 317, 318, 320, 321
Bagley, 118, 120
Baire measure, 124
Banach algebra, 179
Banaschewski compactification, 53
Barreled, 93
Barreledness of C(T,F,c), 94
Beckenstein, 51, 55,~296, 339
^-lattice, 148
Boolean lattice, 137
bound algebra, 329
complete, 329
bounded in a topological ring, 49, 211
bounded element, 341
bounding set, 132
bound structure, 329
bornological, 99
bornologicity of C(TJ(F,c), 99
bornology, 110
Brooks, 170, 172
Browder, 254
C-extension, 50
C -extension, 50
C-embedded, 21
C, -embedded, 21
Chandler, 50
character, 28, 52
clopen, 1
Comfort, 58
compactification, 8
compact-open topology, 3, 186
compatible topology, 177
compatible uniformity, 5
complement, 137
365
366
INDEX
complemented lattice, 137
complete Boolean lattice, 166
completely regular, 1
completely separated, 12
completeness of C(T,F,c), 64
completion, 6
condition hH, 306
continuous inverse, 208
Correl, 49
Curtis, 241, 317, 318, 320, 321, 323
Dales, 329
d-discrete, 44
6-z-ultrafliter, 42
derivation, 237
De Wilde, 115
Dieudonne, 99
differentiable functions, space of, 187
discontinuous homomorphisms, 276, 345
dual of C, (T,R), 77
dual of CtTjF^c), 88
E-closed, 171
E-compact, 54
E-compactification, 54, 171
E-completely regular, 53
Engelking, 53, 54
Equivalent Wallman spaces, 161
ES-algebra, 293
factor algebra, 193
of C(T,F,c), 194
F-algebra, 290
filter, 140
final topology, 3, 180, 190
fixed ideal 17, 51
formally real, 218
free ideal, 17, 51
free union, 117, 252
Frolik, 58
full algebra, 242
fully complete 121, 258
functionally continuous, 275
Gelfand, 175
Gelfand-Kolmogorov theorem, 18
Gelfand map, 223
Gelfand-Mazur theorem, 212
Gelfand topology, 223
generic point, 168
Gleason, 338
Glicksberg, 58
G-normal, 304
Goldhaber, 49
Gulick, 241
INDEX
367
hemicompact, 62
Henriksen, 32, 49, 59
HK, 300
hk, 300
hk-normal, 304
homomorphism, 28, 177, 222
homomorphism topology (7(X,X ), 269
HTVS, 4
hull, 300
hull-kernel normal, 304
hull-kernel topology, 300
ideal, 189
inductive limit, 332
inductive system, 332
inf norm, 222
infrabarreled, 98
infrabarreledness of C(T,F,c), 98
initial topology, 2, 180,~188
initial uniformity, 5
irreducible closed set, 167
Isbell, 59
Jacobson complete, 168
Jacobson filter, 169
Johnson, 241
Kaplansky, 32, 49, 175, 214
K-barreled, 134
K-bornological, 134
kernel, 300
к-extension topology, 70
Kowalsky, 49, 320
K-pseudocompact, 134
kr-space, 65
k-space, 65, 116
Kuczma, 296
k^-space, 116
lattice, 32, 136
lattice filter, 140
lattice ultrafilter, 140
LB-algebra, 329
LCHS, 4
LCS, 4
Lindeldf, 1
Lionville's theorem, 211
LMC algebra, 184
LMCH algebra, 184
LMC topology, 184
initial, 188
final, 190
locally A-convex, 281
locally compact, 1
locally constant, 158
locally m-convex algebra, 184
368
INDEX
locally p-convex, 286
J-uniform continuity, 153
Mackey topology, 4
maximal filter subbase, 142
McClure, 329
m-convex, 181
measurable cardinal, 43
measurable functions, space of, 214
measure
finitely additive, 34
regular, 35
signed, 34
0-1, 34
metrizability of C(T,F,c), 62
Miller, 241
Michael, 176, 242, 257, 282
multiplicative convexity, 181
multiplicative seminorm, 182
Mrowka, 53, 54
Nachbin, 22, 99
Nanzetta, 57
Narici, 51, 55, 296
natural bound structure, 341
Nielsen, 51
Noble, 119
nonarchimedean, 2
nonarchimedean IMCH algebra, 296
nonarchimedean normed algebra, 296
normal algebra, 304
normal lattice, 145
normed algebra, 179
order-bounded linear functional, 128
partition of unity, 82, 265
p-Banach algebra, 330
permanently singular, 296
Piacun, 172
Pierce, 48, 50, 53
Plank, 57
p-norm, 330
p-normed space, 285
point derivation, 323
point-open topology, 3
polar, 227
positive linear functional, 77
prime filter, 169
principal ideal, 220
projective limit, 198
topological, 198
projective system, 198
topological, 198
pseudo-Banach algebra, 329
pseudo-compact, 27, 57
INDEX
369
relatively, 28
Ptak, 258
Q-algebra, 204
quotient topology, 189
radical, 236, 289
real maximal ideal, 29
reflexivity, 111
of C(T,X,c), 111
regular, 201
regular algebra, 241, 303, 318
replete, 21
example of non-replete space, 27
repletion uT of T, 21, 54, 55
resolvent map, 208
Rosenfeld, 241, 325
saturated, 183
Schmets, 115
semisimple, 236
strongly, 236
separability of C(T,F,c), 107
separated
by open sets, 1
by a continuous function, 1
separating family, 172
Shapiro, 170
Shepherdson, 34
Shirota, 32, 99
Shirota's theorem, 44
a -compact, 1
Siriblair, 322
Singer, 241, 323
singular, 201
Sloyer, 51
spectrally complete, 168
spectral norm, 287
spectral radius, 201
spectrum, 201
spherically complete, 134
square algebra, 250
square-preserving, 250
star algebra, 259
star homomorphism, 263
Steiner, 172
Stone, 32
Stone-Cech compactification, 9, 48, 51, 58
Stone Representation theorem, 165
Stone's theorem, 49
Su, 172
Suffel, 296
support
of a continuous linear functional, 92, 132
of a set function, 90, 132
supremum topology, 189
370
INDEX
topological algebra, 177
topological divisor of zero, 296
topological isomorphism, 177
topological projective limit, 335
TVS, 4
Ulam cardinal, 34, 43
Ulam measure, 33, 43
uniform algebra, 254
uniformizable, 5
unit, 201
ultrabornological, 115
ultrafilter, 140
ultranormal, 53
ultraregular, 52
vague topology, 35
valuation, 2
valued field, 1
vanishing set, 132
Varadarajan, 33
very dense, 150, 167
Wallman compactification, 143, 171, 308
Wallman space, 143
Warner, 51, 55
weakened compact-open topology, 248
weakened topology, 4
weak-*normal, 304
weak topology, 4
Wermer, 241, 257, 323
Williamson, 214
Wolk, 49
X-holomorphic, 331
X-repletion, 311
Young, 118, 120
Zelazko, 285, 296
zero-dimensional, 1
zero-one (0-1) measure, 33, 34, 132
zero set, 10
z-filter, 13
z-ultrafilter, 13