Fields Institute
Monographs
The Fields Institute for Research in Mathematical Sciences
C*-Algebras by Example
Kenneth R. Davidson
American Mathematical Society

Fields Institute
Monographs
The Fields Institute for Research in Mathematical Sciences
C*-Algebras by Example
Kenneth R. Davidson
American Mathematical Society
Providence. Rhode Island

The Fields Institute
for Research in Mathematical Sciences
The Fields Institute is named in honour of tue Canadian mathematician John Charles
Fields (1863-1932). Fields was a remarkable man who received many honours for his
scientific work, including election to the Royal Society of Canada in 1909 and to the
Royal Society of London in 1913. Among other accomplishments in the service of the
international mathematics community, Fields was responsible for establishing the world's
most prestigious prize for mathematics research—the Fields Medal.
The Fields Institute for Research in Mathematical Sciences is supported by grants from
the Ontario Ministry of Education and Training and the Natural Sciences and Engineering
Research Council of Canada. The Institute is sponsored by McMaster University, the
University of Toronto, and the University of Waterloo and has affiliated universities in
Ontario and across Canada.
1991 Mathematics Subject Classification. Primary 46L05.
Library of Congress Cataloging-in-Publication Data
Davidson, Kenneth R.
C*-algebras by example/Kenneth R. Davidson.
p. cm. —(Fields Institute monographs, ISSN 1069-5273; 6)
Includes bibliographical references (p. - ) and index.
ISBN 0-8218-0599-1 (alk. paper)
1. C*-algebras. I. Title. II. Series.
QA326.D375 1996
512/.55-dc20 96-20184
CIP
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10 9 8 7 6 5 4 3 2 01 00 99 98 97

To my sons, Colin and Stuart
The joy of doing any scientific research is in the quest for new
ideas and deeper understanding. Finding something new yourself is
exhilarating. Second to that, but still very important, is the pleasure
of explaining it to someone else who cares.
In the pursuit of the unknown, you are never sure what you need
to know. You can never know too much. Yet when approaching
a new problem, you always know both too much (about the wrong
things) and too little (about the question at hand). A big step towards
the solution is figuring out exactly what it is that you really need to
know.

Contents
Preface
xin
I. The Basics of C*-algebras 1
1.1 Definitions 1
1.2 Banach Algebra Basics 3
1.3 Commutative C*-algebras 7
1.4 Positive Elements 9
1.5 Ideals, Quotients and Homomorphisms 12
1.6 Weak Topologies 15
1.7 The Density Theorems 19
1.8 Some Operator Theory 23
1.9 Representations of C*-algebras 26
1.10 C*-algebras of Compact Operators 36
Exercises 40
II. Normal Operators and Abelian C*-algebras 46
II. 1 Spectral Theory 46
H.2 The L°° Functional Calculus 48
II.3 Multiplicity Theory 53
H.4 The Weyl-von Neumann-Berg Theorem 57
H.5 Voiculescu's Theorem 64
Exercises 72
III. AF C*-algebras 74
HI. 1 Finite Dimensional C*-algebras 74
m.2 AF Algebras 75
m.3 Perturbations 79
111.4 Ideals and Quotients 84
111.5 Examples 86
111.6 Extensions 91
Exercises 95

Contents
IV. K-theory for AF C*-algebras 97
IV. 1 Idempotents 97
IV.2 K0 100
IV.3 Dimension Groups 102
IV.4 Elliott's Theorem 109
IV.5 Applications 112
IV.6 Riesz groups 118
IV.7 The Effros-Handelman-Shen Theorem 120
IV.8 Blackadar's Simple Unital Projectionless C*-algebra 124
Exercises 129
V. C*-AlgebrasofIsometries 132
V. 1 Toeplitz Operators 132
V.2 Isometries 136
V.3 Bunce-Deddens Algebras 137
V.4 Cuntz Algebras 144
V.5 Simple Infinite C*-algebras 147
V.6 Classification of Cuntz Algebras 150
V.7 Real Rank Zero 156
Exercises 162
VI. Irrational Rotation Algebras 166
VI. 1 The algebras Ae 166
VI.2 Projections in Ae 170
VI.3 An AF algebra 172
VI.4 Berg's technique 174
VI.5 Imbedding Ae into 21* 177
Exercises 180
VII. Group C*-Algebras 182
Vn. 1 Group Representations 182
Vn.2 Amenability 185
Vn.3 Primitive Ideals 190
Vn.4 A Crystallographic Group 193
Vn.5 The Discrete Heisenberg Group 200
Vn.6 The Free Group 203
Vn.7 The Reduced C*-algebra of the Free Group 206
Vn.8 CJ(F2) is Projectionless 210
Exercises 214

Contents
XI
VIII. Discrete Crossed Products 216
Vm.l Crossed Products 216
Vm.2 Crossed Products by Z 222
Vm.3 Minimal Dynamical Systems 223
Vm.4 Odometers 230
Vffl.5 K-theory of Crossed Products 232
Vm.6 AF Subalgebras of Crossed Products 235
Vm.7 Crossed Product subalgebras of AF Algebras 238
Vm.8 Topological Stable Rank 244
Vm.9 An Order 2 Automorphism 247
Exercises 250
IX. Brown-Douglas-Fillmore Theory 252
DC.l Extensions 252
DC.2 An Addition and Zero Element for Ext(X) 254
DC.3 Some Special Cases 258
IX.4 Positive maps 259
DC.5 Ext(X) is a group 266
DC.6 First Topological Properties 268
DC.7 Ext for planar sets 273
DC.8 Quasidiagonality 281
DC.9 Homotopy Invariance 286
DC. 10 The Mayer-Vietoris Sequence 289
DC 11 Examples - 294
Exercises 299
References 303
Index 307

Preface
These notes were developed in the fall of 1993 for a graduate course on C*-
algebras. The subject of C*-algebras received a dramatic revitalization in the 1970s
by the introduction of topological methods due the deep work of Brown, Douglas
and Fillmore on extensions of C*-algebras and Elliott's use of K-theory to
provide a useful classification of AF algebras and Kasparov's melding of the two into
KK-theory. These results were the beginning of a marvelous new set of tools for
analyzing concrete C*-algebras. The subject flourished by virtue of a rich and
varied group of examples which were the perfect fodder for these new methods.
Moreover, these examples served as a beacon for the kinds of general tools needed
to study C*-algebras. Today, a student cannot get very far in the C*-algebra
literature without being somewhat familiar with the lexicon of examples that now dot
the landscape.
These notes are not intended as a systematic study of the general theory of
C*-algebras, nor of K-theory. There are several excellent books on both of these
aspects. Rather, I develop a modicum of the general theory for the sake of self-
containment and then launch into a study of various important classes of examples.
A number of choices had to be made. I have developed the theory of AF
algebras to quite an extent. However, in some of the other topics, such as irrational
rotation algebras and Cuntz algebras, I have limited myself to the more modest goal
of obtaining enough information to classify the algebras within their narrow
context. Because this is not a course in K-theory, certain stronger results that require a
more systematic study have been omitted. I have also given a detailed treatment of
the Brown-Douglas-Fillmore theory. The discussion is limited to the commutative
case as in their original work because of the technical simplifications. However the
informed reader will notice that a number of the proofs have been influenced by
the more general theory.
The hope is that these notes will open a student's eyes to some of the power,
beauty and variety of these amazing algebras. This is just a glimpse into an exciting
area of current research interest.
These notes were compiled during the author's participation in the special year
on C*-algebras at the Fields Institute of Mathematics held in Waterloo, Canada
during the 1994-95 academic year. The author wishes to thank the Fields Institute
and the University of Waterloo for providing the release time that made this project
Xlll

XIV
Preface
possible. I also wish to thank everyone who provided feedback on the early drafts
of these notes. I am especially indebted to Keith Taylor, who provided me with
some wonderful material on the C*-algebras of crystallographic groups and Alan
Paterson for advice on amenability. I am also grateful to Florin Boca, who provided
me with detailed notes on showing that the irrational rotation algebras are limit
circle algebras even though, in the end, I decided not to include them. I must
also mention Ileana Ionascu and Raul Curto, who read the whole manuscript and
made many detailed comments. In addition, I thank Ola Bratteli, Larry Brown,
John Conway, Don Hadwin, Dick Kadison, Ian Putnam, Norberto Salinas, Jack
Spielberg and Hong Sheng Yin who put me right at various points of the book. In
spite of all their help, I know that I have been unable to get all the minor blemishes
out. For this, I must bear the burden myself.
The text was prepared using X^£-I£IeX, and typeset in llpt Times roman.
The commutative diagrams and figures were produced using the macro package
X$f-pic version 3.2 written by Kris Rose and Ross Moore.
Kenneth R. Davidson
Waterloo, February, 1996

CHAPTER I
The Basics of C*-algebras
A Banach algebra is a complex normed algebra 21 which is complete (as a
topological space) and satisfies
\\AB\\ < \\A\\ \\B\\ for all A,B(E2l.
A Banach *-algebra is a complex Banach algebra 21 with a conjugate linear
involution * (called the adjoint) which is an anti-isomorphism. That is, for all A, B in
21 and A in C,
(A + B)* = A* + B*
(XA)* = XA*
A** = A
(AB)* = B*A*.
A C*-algebra is a Banach *-algebra with the additional norm condition
\\A*A\\ = \\A\\2 for all A € 21.
Certain properties follow easily from these definitions. For example, the
adjoint map is isometric. Indeed,
||A||2 = \\A*A\\ < ||A*|| ||4|;
whence || A\\ < ||A*|| < ||A**|| = ||A||. Other facts, such as some of the properties
of positive elements, are surprisingly subtle.
Example 1.1.1 The algebra of all bounded operators B(H) on a Hilbert space %
is a C*-algebra with the usual adjoint operation. This follows from the well known
identity
\\A*A\\ = sup \{A*A*,y)\ = sup \(A*,Ay)\ = \\A\\\
IMI=llvll=i IMI=llvll=i
It follows easily that a norm-closed subalgebra of a C*-algebra which is closed
under adjoints is also a C*-algebra. A norm closed self-adjoint subalgebra of B(H)
will be called a concrete C*-algebra.
The simplest example of a concrete C*-algebra is the algebra .£ of all compact
operators on a separable Hilbert space. Another important example is C*(T), the
unital C*-subalgebra of B(H) generated by an operator T in B(H), which is the
closure of all polynomials in T, T* and J.
1

2
I. The Basics of C*-algebras
Example 1.1.2 Let X be a locally compact Hausdorff space. Then Co(X), the
space of all continuous functions on X vanishing at infinity, forms a C*-algebra
with complex conjugation as the adjoint operation. Then
117/iu = sup \mm\ = ^p i/(*oi2=ii/iii.
Following the definitions for operators, we say that an element A of a C*-
algebra 21 is self-adjoint if A* = A; N is normal if N*N = iViV*; and U is
unitary if U*U = I = UU*. We also define A to be positive if A = A* and the
spectrum (see section 1.2) cr(A) is contained in the non-negative real line [0, oo).
It is often convenient to have a unit around, even when the algebra is not unital.
So we show how to adjoin one while maintaining the C*-algebra structure. In fact,
it is unique (see the exercises).
When a C*-algebra 21 has an identity element J, compute
/*A = (A*I)* = A** = A for all A e 21.
So I* = J* J = J. Hence ||/||2 = || J*J|| = ||/||. Since J ^ 0, this shows that
IMI = i.
Proposition 1.1.3 Every non-unital C*-algebra 21 is contained in a unital C*
algebra 21" as a maximal ideal of codimension one.
Proof. Form 21" := 210 C and define
(A,A)(flf/0 := (AB + XB + //A,A/x)
(A,A)*:=(A*,A)
||(A,A)||:= sup ||AB + AB||
This makes 2l~ into a *-algebra. The norm is a Banach algebra norm because it
is the norm induced from the space Z?(2l) of bounded operators on 21 given by the
*-algebra of operators {La + A/ : A e 21, A e C}, where La{B) = AB is a
left multiplication operator. Thus this is a Banach *-algebra with unit (0,1). By
design, 21 is a maximal ideal of co-dimension one. The imbedding of 21 into 21" is
isometric because
||A|| = ||A(A7||A||)||<||(A,0)||= sup ||il£|| < HAH.
I|B||<1

1.2. Banach Algebras Basics 3
It remains to verify the C*-algebra norm condition.
||(A,A)||2 = sup ||AB + AB||2
Pll<i
= sup \\B*A*AB + \B*A*B + \B*AB + \\\2B*B\\
\\B\\<1
< sup \\A*AB + XA*B + XAB + \X\2B\\
\\B\\<1
= \\(A*A + \A* + \A,\\\2)\\
= \\(A,\y{A,\)\\<\\(A1\y\\\\(A,\)\\
Thus||(A,A)|| < \\(A, A)*||. By symmetry, we have \\(A, A)|| = ||(A,A)*||. Hence
the inequality above is an equality, and so
\\(A,\)\\2=\\(A,\y(A,\)\\
as claimed. ■
1.2 Banach Algebras Basics
For the convenience of the reader, we review the necessary background from
Banach algebras that we need.
The spectrum of an element A of a unital Banach algebra 21 is the set
<r(A) := {A e C : XI - A is not invertible}.
The complement of the spectrum is called the resolvent, and Ra{X) = (A/- A)"1
is the resolvent function.
Theorem 1.2.1 In any unital Banach algebra 21, the spectrum of each A in 21 is a
non-empty compact set; and the resolvent function is analytic onC\ cr(A).
Proof. If |A| > ||A||,then||A-Mn|| < (|A|-1||A||)n decreases geometrically fast;
so the series
J^A-n"1An
n>0
is norm convergent. The limit is (XI — A)"1 since
k
{XI - A) J] X~n-lAn = I - X-k-2Ak+x
n=0
which converges to J. Moreover, this shows that Ra{X) is analytic, and has a
Laurent expansion about the point at infinity. Furthermore,
Urn ||*a(A)||< Hm lAl-^l-IAniiiH)-1^.
|A|-*oo |A|-*oo

4
I. The Basics of C*-algebras
Similarly, if X0I - A is invertible and |A - A0| < ||(A0J - A)"1!!"1, then
(XI - A)'1 = ]T(A - Ao)n(A0/ - A)—1
n>0
is the Taylor expansion of Ra in a neighbourhood of Ao. So the resolvent function
is analytic on the complement of the spectrum, known as the resolvent of A. In
particular, /(Ra(X)) is an analytic scalar valued function for every continuous
linear functional / on 21. The resolvent is therefore an open set containing all A in
C with |A| > ||A||. So cr(A) is a compact subset of {A e C : |A| < ||A||}.
If the spectrum of A were empty, then Ra(X) would be a bounded entire
function. By Liouville's Theorem, this leads to the absurd conclusion that Ra is the
constant zero function. Indeed, for every functional /, /(Ra(X)) is a scalar entire
function vanishing at infinity; whence it is zero. So Ra = 0 by the Hahn-Banach
Theorem. Hence the spectrum must be non-empty. ■
Note that the power series technique of the proof provides the useful fact that if
|| A|| < 1, then I-A is invertible with inverse J2n>o ^n- The following elementary
functional property of the spectrum is known as tHe spectral mapping property.
Lemma 1.2.2 If p is a polynomial and A € 21 is an element of a unital Banach
algebra, then
<r(p(A)) = p(<r(A)).
Proof. For a in C, factory - a as
n
p(z)-a = cfJ(z-A).
i=i
Then it follows that
n
i=i
Because all the terms commute, p(A) - al is invertible if and only if A - fcl is
invertible for every i. Thus a lies in cr(p(A)) if and only if there is a fy in cr(A)
such that p(/3i) = a. ■
The spectral radius of A is defined to be
spr(A) = sup |A|.
\e<r(A)
Proposition 1.2.3 For each A in a Banach algebra 21, the spectral radius is
determined by spr(A) = Jirr^ \\An\\^n.
Proof. Look more carefully at the Laurent series
{XI-A)-1 = J2x"n'lAn
n>0

1.2. Banach Algebras Basics
5
for Ra at infinity. Since Ra is analytic on {A : |A| > spr(A)}, this series
converges absolutely and uniformly for |A| > r > spr(A). In particular, the Taylor
series coefficients r"*1""1!!^!! converge to 0 for r > spr(A) which implies that
limsupll^H^^spr^).
n-+oo
On the other hand, there is a a in <r(A) with \a\ = spr(A). By the spectral
mapping property, an lies in the spectrum of An. Hence
spr(A) = la"!1/" < ||An||1/n for all n > 1.
Thus
limsupHAl1/" < spr(A) < inf \\An\\^n,
n-*oo ™>1
which shows that the limit exists. ■
The rest of this section is devoted to the Gelfand theory of abelian Banach
algebras.
Theorem 1.2.4 The only simple abelian unital Banach algebra is C
Proof. Suppose that 21 is a unital abelian Banach algebra containing an element A
which is not scalar. Let a be an element of cr(A), and consider the closed ideal
3 = (A - al)21. No element of the form (A - aI)B is invertible, and thus
\\(A-aI)B-I\\>l.
Therefore J is not in the closure of such elements; whence J is a proper ideal.
Thus when 21 is simple, every element of 21 must be scalar; and therefore
21 = C ■
A multiplicative linear functional on a commutative Banach algebra 21 is a
non-zero homomorphism of 21 into C The set M% of all multiplicative linear
functionals on 21 is called the maximal ideal space of 21. We shall show that
multiplicative linear functionals have norm one. So M% may be endowed with the
topology induced by the weak-* topology on the Banach space dual of 21.
Theorem 1.2.5 The multiplicative linear functionals on a unital abelian Banach
algebra are continuous of norm 1. The map taking each multiplicative linear
functional to its kernel is a bisection onto the set of maximal ideals of%.
Proof. Suppose that (p is a multiplicative linear functional and A € 21 such that
||A|| < 1 = <p(A). Let B = £n>1 An. Then since A + AB = J5,
<p(B) = <p(A) + <p(A)<p(B) = 1 + <p(B)
which is absurd. So ||y>|| < 1. Since (p(I) must equal 1, this is an equality.
It follows that M = ker (p is a closed ideal of codimension 1 in 21, and thus is
maximal. Since cp is determined by M and the fact that (p(I) = 1, this
correspondence is one to one. Conversely, if M is a maximal ideal of 21, then dist(J, M) = 1

6
I. The Basics of C*-algebras
because the unit ball about J consists of invertible elements. It follows that the
closure of M still does not contain J. As this is a larger proper ideal, we deduce that
M is closed. So the quotient algebra 21/M is a simple abelian unital Banach
algebra. By Theorem 1.2.4, this quotient is isomorphic to C. So the quotient map cp is
a continuous homomorphism of 21 onto C with kernel M. ■
Corollary 1.2.6 The maximal ideal space of a unital abelian Banach algebra is
a compact Hausdorff space. lf% is abelian but not unital, then M%, is locally
compact
Proof. It is clear that a weak-* limit of multiplicative linear functionals is again
multiplicative. So M.% is a weak-* closed subset of the unit ball of the dual space
of 21. By the Banach-Alaoglu Theorem, it is weak-* compact and Hausdorff.
In the non-unital case, form the algebra 21" as in Proposition 1.1.3 (which will
be a Banach algebra but not a C*-algebra in this case). Then 21 itself is a maximal
ideal of 21". This corresponds to the functional <po(A + XI) = A. Thus every other
maximal ideal M of 2l~ must have proper intersection with 21; and therefore yields
an ideal of codimension one in 21. This then determines a non-zero homomorphism
of 21 into C. Conversely, if cp is a non-zero multiplicative linear functional on 21, it
is clear that <p(A + XI) = cp(A) + A is the unique extension of cp to a multiplicative
linear functional on 2l~. (VERIFY!) So there is a bijective correspondence
between M.% and M<%~ \ {^o}- In particular, this is locally compact because M%~
is the one-point compactification of M% and it is Hausdorff. ■
Now we define the Gelfand transform T of a commutative Banach algebra
into C0(.M2i) by T(A) = A where
Mf) = <P(A)-
Theorem 1.2.7 The Gelfand transform is a contractive algebra homomorphism of
an abelian Banach algebra 21 into Co(M%). The image algebra separates the
points of M%.
Proof. The functions A are continuous because of the definition of the weak-*
topology. In the non-unital case, M% = M%~ \{y>o}- Since A(<p0) = <fo(A) = 0,
it follows that A lies in Co(.Ma)• The map is contractive because each (p is
contractive and the norm on C0(My) is the sup norm. Finally, T(2l) separates points
of M% because the points correspond to distinct multiplicative linear functionals
of 21. ■
Corollary 1.2.8 In a unital commutative Banach algebra 21, A is invertible if and
only if A is invertible, which is precisely when A does not vanish on M%. Thus
<r(A) = a(A) = {<p(A) : <p € M*}
and ||i||oo = spr(A).

13. Commutative C*-algebras
7
Proof. If A has an inverse, then since T is a homomorphism, T(A"1) = T(A)"1.
Conversely, if A is not invertible, then as in the proof of Theorem 1.2.4, the ideal
3 = A21 is proper and thus is contained in a maximal ideal M. Let (p be the
associated multiplicative linear functional. It follows that A(cp) = 0 and thus A
is not invertible in C(M%). It is now immediate that A has the same spectrum
as A, and that this coincides with the range of A. As the norm in C(My) is the
supremum norm and the range of A is <r(A), we conclude that || A||oo = spr(A).H
1.3 Commutative C*-algebras
We will show that every commutative C*-algebra has the form Cq(X) as in
Example 1.1.2. This will be applied to the structure of normal operators.
Theorem 1.3.1 Let 21 be an abelian C*-algebra. Then the Gelfand transform is an
isometric ^-isomorphism of$l onto Co(My).
Proof. First assume that 21 is unital. Let (p be a multiplicative linear functional on
21. We will show that ff{A*) = ff(A). Suppose first that A = A* is self-adjoint.
Then form a family of unitary elements
^ n!
n>0
Indeed,
u: = y^ = y(j±Ar = e-itA = ui
n>0 n>0
So each Ut is unitary. Hence ||tft||2 = \\U?Ut\\ = \\I\\ = 1.
Therefore
(iMA)r\
n>0
As this holds for all real t, we deduce that (p(A) is real. Now if X is an arbitrary
element of 21, write it as X = A + iB where
A := (X + X*)/2 and B := (X - X*)/2i
are the self-adjoint elements known as the real and imaginary parts of X. We
know that <p(A) and <p{B) are real, and thus
<p(X*) = <p{A - iB) = <p(A) - icp(B) = <p(A) + i<p(B) = <p{X).
So the Gelfand transform V satisfies A* = A*. Therefore T is a *-homomorphism.
For A = A\ we have ||A||2 = ||AM|| = ||A2||. Thus
HiiHoo = spr(A) = Km \\A*y/*n = Km (||A|f J1/2" = \\A\\.

8
I. The Basics of C*-algebras
So Halloo = || A||. For a general element T, we obtain
||T||2 = \\tt\\ = Hf^iico = ||***IU = ||.f||»..
Consequently, the Gelfand transform is isometric
Thus the image of 21 under the Gelfand map is a unital norm closed self-adjoint
subalgebra of C(X) which separates points. By the Stone-Weierstrass Theorem,
T is surjective and hence is a *-isomorphism.
Now consider the non-unital case. By Proposition 1.1.3, 21 is contained in a
unital C*-algebra 2l~ as a maximal ideal of co-dimension one. Clearly 2l~ is also
abelian. Moreover, from Corollary 1.2.6, M%, is canonically homeomorphic to
M%~ \ {y>o}> where y>0 is the multiplicative linear functional on 2l~ with kernel
21. Using the argument above, we see that the Gelfand transform is an isometric
♦-isomorphism of 21" onto C(M%~). This takes 21 onto the ideal of functions
vanishing at y>o> which is naturally identified with Co(M%). ■
In particular, this applies to normal elements in C*-algebras.
Corollary 1.3.2 If N is a normal element of a unital C^-algebra 21, then C*(N) is
isometrically ^-isomorphic to C(cr(N)), the continuous functions on the spectrum
ofN, via a map that takes N to the identity function z(t) = t. The (not necessarily
unital) C*-algebra generated by N and N* is carried onto Co{cr(N) \ {0}).
Proof. Since N is normal, C*(N) is abelian. It suffices to determine the maximal
ideal space X. Note that a multiplicative linear functional cp in X is determined
by ff(N) = A for then <p(p(N, N*)) = p(\, A) for every polynomial p. Thus the
map from X into C taking cp to <p{N) is a homeomorphism onto N(X). From the
Gelfand theory, N(X) = cr(N). This map identifies N with the identity function
z as desired. By Theorem 1.3.1, this map is an isometric *-isomorphism.
When N is not invertible, the subalgebra generated by N and N* (but not J)
corresponds to the ideal of functions vanishing at 0. ■
We obtain some useful consequences from this basic result. The first is an
important fact known as the continuous functional calculus for a normal element.
Many useful facts follow readily from this.
Corollary 1.3.3 IfN is a normal element of a unital C*-algebra and f is a
continuous function on <r(N), the operator f(N) is defined as the inverse off under the
Gelfand transform ofC* (N). This map is an isometric ^-isomorphism ofC(cr(N))
onto C*(N). 7/0 e <r(N) and /(0) = 0, then f(N) lies in the non-unital algebra
generated by N and N*.
Moreover cr(f(N)) = f(cr(N)). Also ifg is continuous on f(<r(N)), then
9(f(N)) = (gof)(N).
Proof. The first part is immediate from Corollary 1.3.2. We have
a(f(N)) = <r(f(N)) = a{f) = f(<r(N)).

1.4. Positive Elements 9
When p is a polynomial in z and z, it is immediate from the fact that the functional
calculus is a homomorphism that p(f(N)) = (pof)(N). The general case follows
by approximating the continuous function g by polynomials. ■
Corollary 1.3.4 Let 21 be a C*-algebra.
(i) IfN in 21 is normal, then \\N\\ = spv(N).
(ii) IfA = A*, then cr(A) is real.
(iii) IfUinVl is unitary, then <r(U) is contained in the unit circle.
Proof. By Corollary 1.3.2, we have
ll^ll = II^IU(-AT) = IklU(JST) = spr(iNT).
If A is self-adjoint, then A = A* = A is a real valued function. Since <r(A) is the
range of A, part (ii) follows. Similarly, if U is unitary, then
\U\2 = U*U = I = 1. ■
1.4 Positive Elements
Positive elements play an important role in C*-algebras. They determine an
order on the self-adjoint elements 2l,a of 21 by setting A < B if B - A is
positive. Some of the properties of positive elements require non-trivial finesse. For
convenience, we will work in unital algebras. This is no serious restriction because
of Proposition 1.1.3. We begin with another easy consequence of the continuous
functional calculus.
Corollary 1.4.1 Each positive element A of a C*-algebra has a unique positive
square root.
Proof. The square root function f(x) = yfx is a continuous function on [0, ||A||]
which contains <r(A). Thus B := f(A) is self-adjoint because / is real valued; and
a{B) = f(cr(A)) is contained in the positive real line. So B is positive. Moreover
B2 = f2{A) = z{A) = A.
Suppose that C is another positive square root of A. Then by the functional
calculus,
C = f(C2) = f(A) = B. M
Another useful application is the following analogue of the Hahn
decomposition in measure theory.
Corollary 1.4.2 If A in 21 is self adjoint, then there are positive elements A+ and
A- in 21 such that A = A+ - A_ and A+ A_ = 0.
Proof. Let f(x) = (x + \x\)/2 and g(x) = f(-x)\ so that / and g are positive,
fg = 0 and f{x) - g(x) = x. Set A+ = f(A) and A_ = A+ - A = g(A). Then
A+A- = f(A)g(A) = (fg)(A) = 0. ■

10 I. The Basics of C*-algebras
The following lemma contains some useful characterizations of positivity.
However the main result about positivity is Theorem 1.4.5 below.
Lemma 1.4.3 For A = A* in a C*-algebra 21, the following are equivalent:
(i) A > 0;
(ii) A = B2 for some B = B*;
(hi) \\cl - A\\ < cforallc> \\A\\;
(iv) \\cl - A|| < c for some c > \\A\\.
Proof, (i) implies (ii) is Corollary 1.4.1. Assuming (ii), there is a self-adjoint
element B such that A = f{B), where / € C{a(B)) is the function /(a?) = x2.
Consequently H/H^B) = ||.A||; and thus 0 < / < ||A|| < c. Therefore 0 < c - / < c.
So
\\cI-A\\ = \\(c-f)(B)\\ = \\c-f\\<r(B)<c.
This establishes (iii), which clearly implies (iv).
Assuming that (iv) holds for a particular value of c, we conclude that
c>||c/-A|| = ||(c-z)(A)|| = ||c-z|U(jl).
Thus the identity function z is non-negative on <r(A). That is, <r(A) is contained in
R+; and thus A is positive. ■
Corollary 1.4.4 If A and B are positive elements of% then A+Bis also positive.
Proof. Choose R > ||A||andS> ||£||. Then.R + S > ||A +J3|| and
\\(R + S)I-(A + B)\\<\\RI-A\\ + \\SI-B\\<R + S.
So by part (iv) above, we see that A + B is positive. ■
We have accumulated enough tricks to tackle the main result. Notice that the
proof is quite devious considering how straightforward this fact is for operators.
Theorem 1.4.5 If A belongs to a C*-algebra 21, then A* A is positive.
Proof. Let B = A* A. Since this is self-adjoint, Corollary 1.4.2 allows us to write
B = J3+ - J5_ where J3+ and J3_ are positive and J3+J3_ = 0. Let C be the
positive square root of J3_, and set T = AC. Note that since C is a limit of
polynomials in J3„ with zero constant coefficient, we have CJ3+ = 0. Then
-T*T = -CA*AC = -C(B+ - BJ)C = CB_C = B2_.
In particular, -T*T is positive, and thus has spectrum contained in R+.
Now write T = X + iY where X and Y are the real and imaginary parts of T.
Then
T*T + TT* = (X + iY)*(X + iY) + (X + iY)(X + iY)* = 2(X2 + Y2).

1.4. Positive Elements 11
This is a sum of positive elements, and hence is positive by the previous corollary.
Consequently,
is a sum of positive elements, and hence is positive.
It is a well known ring theoretic fact (see Exercise 1.4) that
<r(AB) U {0} = <r(BA) U {0}.
Since TT* has non-negative spectrum, so does T*T. Hence ±T*T are both
positive, and therefore cr(T*T) = {0}. Since T*T is self-adjoint, its norm equals its
spectral radius; so J5i = -T*T = 0. As J3_ is self-adjoint, we obtain J3_ = 0;
and so B is positive as required. ■
We collect a couple of other results about the order structure that will be used
frequently in the future.
Corollary 1.4.6 If A < B in 2l,a and X belongs to % then X*AX < X*BX.
Proof. Let C be the positive square root of B - A. Then
X*BX - X*AX = X*(B - A)X = (CX)\CX) > 0. ■
Lemma 1.4.7 If0<A<Bare invertible in 2l*a, then B"1 < A"1.
Proof. By the previous corollary,
/ - B'^AB"1'2 = B-^2{B - A)B-1'2 > 0.
Thus (A^B-^yiA^B-1'2) < J; whence WA^B"1^ < 1. The adjoint
has the same norm, so
/ > (A^B-^iA^B-1'2)* = A^B^A1'2.
Multiplying on both sides by A"1/2 and applying the previous corollary yields
A'1 = A-^IA-1'2 > A-^A^B^A1'2^1'2 = B'1. ■
An approximate identity for a Banach algebra 21 is a net Ex for A G A which
is bounded and satisfies
lim Ex A = lim AE\ = A for all A G 21.
aga aga
In a C*-algebra, we further stipulate that 0 < Ex, \\EX\\ < 1, and that Ex < E^
when A < //. Since A is directed, for each A and /z in A, there is an index v G A
such that Ev > Ex and Ev > E^.
Theorem 1.4.8 Every C*-algebra has an approximate identity.

12
I. The Basics of C*-algebras
Proof. Let A = {A e 21: A > 0 and \\A\\ < 1} ordered by the order on 2l,a. First
let us show that A is directed. Suppose that A, B e A. Let f(t) = t(l - t)"1 for
0 < t < 1 and g{t) = til + t)"1 = 1 - (1 + *)"1 for* > 0. Note thatg(f(t)) = t.
Set Y = f{A) + /(B) and C = </(Y) (which lie in 21 because /(0) = </(0) = 0).
Since g is positive and HtfH^y) < 1, we see that C belongs to A.
Since X := f(A) < Y, we have J + X < I + Y; and hence by Lemma 1.4.7,
(J + X)-1^ (J + Yj^.Thus
A = /-(/ + X)"1 </-(/+ Y)-1 = C.
Similarly J3 < C. So A is directed.
Next we need to establish that approximation on a cofinal subset suffices. Care
must be taken because 0 < A < B does not imply that A2 < B2 when A and B
don't commute (see Exercise 1.8). Now if 0 < A < B belong to A, and X belongs
to 21, compute
\\X - BX\\2 = \\X*(I - B)2X\\ < \\X*(I - B)X\\ < \\X*{I - A)X\\.
Similarly, \\X - XB\\2 < ||X(/- A)X*\\.
For any X > 0, let An = g(nX) which lies in A by the argument in the first
paragraph. Set h(t) = t2(l - g(nt)) = t2(l + nt)"1 < t/n. Then
\\X(I - An)X\\ = \\h(X)\\ < ||fc|U(jr) < ||X||/n.
Hence
Um \\X - BX\\2 < lim sup ||X - BX\\2
BGA n-+oo 5gA
B>An
< lim ||X(/-An)X|| = 0.
Similarly, limBGA ||* - XB\\2 = 0.
For arbitrary X,
\\X - XS||2 = ||(/ - B)X*X(I - B)\\ < \\X*X - X*XS||.
This tends to 0, as does \\X - J5X||2. So A is an approximate identity. ■
In the separable case, we can replace nets by sequences. The following
corollary is left as an exercise.
Corollary 1.4.9 If 21 is a separable C*-algebra, then there is an increasing
sequence 0 < Ei < E2 < ... of positive norm-one elements which form an
approximate identity for 21.
1.5 Ideals, Quotients and Homomorphisms
By an ideal in a C*-algebra, we will mean a norm-closed two-sided ideal.
One-sided ideals will be specified as left or right ideals, and are also assumed to be
closed unless otherwise specified.
Lemma 1.5.1 Every ideal of a C*-algebra is self-adjoint.

1.5. Ideals, Quotients and Homomorphisms 13
Proof. Let J be an ideal of 21. Then 25 := 3 n T is a C*-subalgebra of 21. Notice
that 25 contains 33*. By Theorem 1.4.8, 25 contains an approximate identity E\.
Then for any J in 3,
lim || J* - J*EX\\2 = lim ||(JJ* - JJ*EX) - J57A(JJ* - JJ*£A)|| = 0.
AgA AgA
As E\ belongs to the ideal 3, it follows that J* also belongs to 3. Therefore 3 = 3*
is self-adjoint. ■
A subalgebra 25 of a C*-algebra 21 is called hereditary if B € 25 is positive,
A e 21 and 0 < A < B implies that A belongs to 25. It is an important fact that
ideals are hereditary. The proof requires a factorization trick.
Lemma 1.5.2 IfX*X < A in a C*-algebra % then there is an element B in 21
with \\B\\ < \\A\\XIA such that X = BA1/4.
Proof. Define Bn := X(A + £ J)-1/2^1/4. (This lies in 21 even if it is not unital,
but the computation is done in 21" for convenience.) Set
Snm^A+i/)-1/2-^^/)-1/2 and /n(«) := <3/4(< + iri/2
Notice that fn converges uniformly to i1/4 on K = [0, ||A||]. Then
\\Bn - Bm\\2 = \\XDnmAll*tf = U^D^X'XD^A^W
< \\A^4DnmADnmA^4\\
= \\DnmA^f = \\fn(A) - /m(A)||2 < ||/n - fmfK.
Since fn is Cauchy, so is Bn.
Let B = limn^oo Bn. Then
BA1'* = lim BnA^4 = lim X(A+ ^iy^A1'2 = X. ■
n-*oo n-*oo n
Theorem 1.5.3 Suppose that 3 is an ideal of a C*-algebra 21. If J in 3 is positive
and A* A < J, then A belongs to 3. In particular, ideals are hereditary.
Proof. Factor A = BJ1/4 using the previous lemma. Then J1/4 belongs to the
non-unital algebra generated by J and hence belongs to X Thus so does A. ■
Theorem 1.5.4 If 3 is an ideal of a C*-algebra 21, then the quotient algebra 21/J
is a C*-algebra.
Proof. Let us write A for the coset A + X As usual, the adjoint is defined as
(A + 3)* = A* + 3; and the norm is defined as ||A|| := infjG3 \\A- J\\. Since J is
self-adjoint, || A* || = ||A||. The only thing that needs to be checked is the C*-norm
condition.
Let E\ be an approximate identity for X We claim that
||i|| = lim||A-AE?A||.

14
I. The Basics of C*-algebras
Indeed, as AE\ belongs to 3, ||A - AE\\\ > \\A\\. On the other hand, for e > 0
there is an element J in J so that \\A - J|| < \\A\\ + e. So
Urn \\A - AEX\\ < lim ||(A - J)(I - Ex)\\ + \\J - JEX\\
<Hii-j||<l|ii||+«.
Letting e decrease to 0 establishes the claim.
Now compute
||A*A|| = lim \\A*A(I - Ex)\\ > Hm \\(I - EX)A*A(I - Ex)\\
AfcA AfcA
= lim ||A(/ - Ex)f = ||A||2 = ||A*|| ||i|| > \\A*A\\
So this is an equality. Hence the quotient norm is a C*-norm. ■
Now we may deduce a fundamental fact about homomorphisms between C*
algebras.
Theorem 1.5.5 Let n be a non-zero *-homomorphism of a C*-algebra 21 into an-
other C*-algebra 25. Then \\k\\ = 1 and 7r(2l) is a C*-subalgebra of*B. Ifn is
injective, then it is isometric. So in general, n factors as nq where q is the
quotient map of% onto 21/ ker n and n is the induced isometric ^-isomorphism of the
quotient onto 7r(2l).
Proof. If A = A* in 2l*a, then the spectrum a^(n(A)) is contained in cr^{A)\ and
hence
||7r(A)|| = sprWA))<spr(A)-||A||.
For general A in 21,
||7r(A)||2 = |HA*A)|| < ||A*A|| = ||A||2.
Thus ||7r|| < 1. In particular, n is continuous. So ker7r is closed and n factors
through the quotient 21/ ker7r. Write n = irq where q is the quotient map and 7r is
the induced map on 21/ ker7r.
Now 7T is injective. If ir were not isometric, there would be an element A such
that ||7r(A)|| < ||A||. Hence r := ||7r(A*A)|| < \\A*A\\ := s. Let / in C([0, s])
be defined so that f(i) = 0 for 0 < t < r and f(s) = 1. Then by the continuous
functional calculus Corollary 1.3.3,
0 = /(;r(A*A)) = rr(/(A*A)).
But as / does not vanish on cr(A*A), Corollary 1.3.2 shows that /(A* A) ^ 0. This
contradicts the injectivity of 7r. Therefore ic must be isometric. It follows that 7r(2l)
is closed, and hence is a C*-subalgebra of 25. Finally, since ||7r|| = 1 and q is a
quotient map, it follows that ||7r|| = 1 as well. ■

1.6. Weak Topologies
15
As a consequence, we recover a basic isomorphism theorem for rings in the
C*-algebra context.
Corollary 1.5.6 Suppose that Z is an ideal of a C*-algebra 21, and that 25 is a
C*-subalgebra of '21. Then 25 + 3 is a C*-algebra, and
25/(25 nor)-(25+ 30/3
is a ^isomorphism.
Proof. Consider the map n : 25 -» 21/J given by n(B) = B + J. Clearly, ker7r =
25 H X So we may factor n = nq through the quotient map q : 25 -■> 25/(25 fl J).
However, we also have the factorization n = Qj where j is the natural injection of
25 into 21 and Q is the quotient map of 21 onto 21/J.
Since 7r(25) is closed, so is Q-1(7r(25)) = 25 + X As 25 + J is evidently a
*-algebra, it is a C*-algebra. Applying Theorem 1.5.5 to the restriction of Q to
25 + J shows that there are isometric *-isomorphisms
(25 + 3) ft ~ tt(25) ~ 25/(25 n 3). ■
Recall that an algebra 21 is called inverse closed if, whenever 21 is contained
in a larger algebra 25 (in the appropriate category of algebras), an element A in 21
is invertible in 25 only if it is already invertible in 21.
Corollary 1.5.7 IfVlisa unital C*-subalgebra of*B and A G 21, then
cr%(A) = ct<b(A).
So C*-algebras are inverse closed.
Proof. Since <*b(A) C <r&(A), it suffices to show that if A in 21 is invertible in 25,
then the inverse lies in 21.
First suppose that A = A*. Then <£ := C*({A, A"1}) is a subalgebra of 25
isomorphic to some C(X) by Theorem 1.3.1. Let A denote the image of A under
this isomorphism. Then A ^ 0 on X. So 0 ^ <rc{A) which is a subset of the
real line. Choose polynomials pn such that pn{z) converges uniformly to x'1 on
crc(A). Then since A"1 = limn_>00pn(A), we see that A"1 = limn_>00pn(A)
belongs to C*(A), which is contained in 21.
For general A, if A"1 belongs to 25, then (A*A)'1 = A'1 A'1* lies in 25 and
therefore also in 21. Thus (A*A)'1 A" = A"1 belongs to 21. ■
1.6 Weak Topologies
There are several important topologies on B(H) that are weaker than the norm
topology. We will introduce two here, and develop some others in the exercises.
These topplogies will be applied to relate algebraic and analytic information about
C*-algebras acting on Hilbert space.

16 I. The Basics of C*-aIgebras
The weak operator topology (wot) on B(H) is defined as the weakest
topology such that the sets
W(T}x,y):={Ae B{U) : |((T - A)z,y)\ < 1}
are open. The sets
n
W(Tit xit K; 1 < i < n) := f| W{Ti, xit Vi)
»=i
form a base for the wot topology. A net Ta converges wot to an operator T (write
Ta-^T) if and only if
lim(Taa;,y)-(T.?,3/) for all x,yeU.
Analogously, the strong operator topology (sot) is defined by the open sets
S{T,z):={AeB{H):\\{T-A)z\\<l}.
A net Ta converges SOT to T (write Ta --4 T) if and only if
\imTax = Tx for all xeH.
a
Clearly, the strong operator topology is stronger than the weak operator topology
but weaker than the norm topology.
It is routine to verify that the adjoint is WOT-continuous. However, it is not
SOT-continuous. Indeed, consider the unilateral shift S on £2 given by Sen = en+i
on the orthonormal basis {en : n > 0}. The reader may readily verify that 5*n
converges SOT (and thus also WOT) to 0. But Sn is isometric for all n > 1 and
does not converge at all in the strong operator topology; while it converges WOT to
0.
Left and right multiplication by a fixed operator is continuous in both the WOT
SOT
and SOT topologies. In other words, if Ta —> T, then
ATa ---> AT and TaA -^ TA]
andifTa -^-4 T, then
ATa -^> AT and TaA -^> TA.
However multiplication is not jointly continuous in either topology. It is not even
WOT-continuous when restricted to the unit ball. For example, using the unilateral
shift again, we have
WOT-limS*n = WOT-limSn = 0; but WOT-limS*nSn = J.
n->oo n->oo n->oo
An example showing that multiplication is not jointly SOT-continuous is more
delicate, and will be left to an exercise. However, multiplication is SOT-continuous

1.6. Weak Topologies 17
on the unit ball. Indeed, suppose that
Sa^S and Ta-^T,
and that ||5a|| < 1 for all a. Then
\\(ST - SaTa)x\\ < ||(5 - Sa)Tx\\ + \\Sa\\ \\(T - Ta)x\\.
The right hand side tends to 0, and thus SOT-lirria SaTa = ST.
It is an easy application of the uniform boundedness principle to show that a
sequence which is converging WOT or SOT is necessarily bounded. The Banach-
Alaoglu Theorem can be adapted to show that the unit ball of B(H) is compact in
the weak operator topology. It is not compact in the strong operator topology, as a
consideration of Sn again shows.
Proposition 1.6.1 The v/OT-continuous linear junctionals on B(H) and the SOT-
continuous linear junctionals coincide, and each junctional has the form
f(T) = J2(Txi,yi)
t=l
for a finite set of vectors x\,..., xn, y\,..., yn in W.
Proof. The functional described are clearly WOT-continuous, and hence are also
SOT-continuous. Conversely, suppose that / is a SOT-continuous linear functional.
Then /_1 (D) (where ED is the unit disk) is SOT-open, and hence contains a basic
neighbourhood of the origin
rx(D) D{Te B(U) : ||T^|| < 1, 1 < i < n}
D{TeB(H):J2\\T*i\\2<l}
for certain vectors x\,..., xn. Therefore
1/(201 < (Di^ii2)172.
t=i
Define a map $ of B(H) into vSn^, the direct sum of n copies of H, by the formula
$(T) = (Ts1,Ts2,...,Tsn).
We may define a linear map F on the range of $ by F($(T)) = f{T). Since
||F|| < 1, we may extend it by the Hahn-Banach Theorem to a norm one linear
functional F on VSn\ By the Riesz representation theorem for continuous linear
functionals on Hilbert space, there are vectors y\,..., yn such that
F((vu.. .,vn)) = ^Kyt)-
t=i
In particular,

18
I. The Basics of C*-algebras
n
/(r) = F(*(r)) = 5^(r^lW)- ■
t=i
Since closed convex sets are the intersection of closed half spaces
(corresponding to continuous linear functionals), we obtain an immediate corollary.
Corollary 1.6.2 B(H) has the same closed convex sets in the weak operator and
strong operator topologies.
While B(%) is not metrizable in either of these two topologies, it is a useful
fact that the unit ball is metrizable in both.
Proposition 1.6.3 If % is a separable Hilbert space, then the unit ball B(H)X of
B(H) is metrizable in both the weak and strong operator topologies.
Proof. Choose a countable sequence 21,22,... which is dense in the unit ball of
U. Define
dw(S,T):= ]T 2-i-S\((S-T)zhzj)\
and
ds(S,T):=Yl2~iUS-T)xiW
It is left as an exercise to verify that these are indeed metrics equivalent to the weak
operator and strong operator topologies respectively on the unit ball. ■
The order structure is intimately related to the strong operator topology. Here
is an easy lemma that will prove useful.
Lemma 1.6.4 Suppose that Aa, a G A, is an increasing net of self-adjoint
operators which is bounded above by an operator M. Then A = SOT-lirria Aa exists,
and is the least upper bound for the net Aa. IfAa are all projections, then the limit
is the projection onto the closure of the union of the ranges ofAa.
Proof. For each x in %y the net (Aax, x) is an increasing sequence of real numbers
bounded above by (Mx, x). Thus we may define a quadratic form
A(x) = lim(Aaa;,a;).
Then a linear operator is defined by the polarization identity
(Ax, y) := \ (A(x + y) - A(x - y) + iA(x + iy) - iA(x - iy)).
As the corresponding identity is valid for each Aa> it follows that
\im(Aax1y) = (Ax,y)
for every 2, y in H. So A = WOT-Iirria Aa. Since
(Ax, x) = lim(Aaa;, x) = sup(Aa;c, x)y
a a

1.7. The Density Theorems
19
it is apparent that A > Aa for all a and that no smaller operator has this property.
So A = supa Aa.
To obtain strong convergence, notice that if B > 0, the Cauchy-Schwarz
inequality for the form («, y) := (Bx, y) yields
\\Bx\\2 = (x,Bx) < {x,x)1l2(Bx,Bx)1l2={Bx,x)1l\Bzx,x)1l2.
Thus for x in %, a0 € A and any a > c*o,
\\(A- Aa)x\\2 < ((A- Aa)x,x)^2((A- Aa)3x,x)^
<\\A-Aaof\\xf((A-Aa)x,x)^
The right hand side converges to 0, and hence Aa converges to A strongly.
When Aa are all projections, the limit A is also a projection because the set
of projections is SOT-closed. Indeed, P is a projection if and only if 2P - J is a
self-adjoint isometry (a symmetry). Clearly both the set of self-adjoint operators
and the set of isometries are SOT-closed. Thus the same is true for symmetries,
and thus for projections. The limit projection A satisfies Ax — x for all x in
Ua Ran(Aa) and Ax = 0 for all x in fla ker(Aa). This is evidently the projection
onto Ua Ran (Aa). ■
1.7 The Density Theorems
A C*-subalgebra of B(H) which contains the identity operator and is closed in
the weak operator topology is called a von Neumann algebra. If S is any subset
of B(H), let the commutant of S be
Sf := {T e B(U) :ST = TS for all S e S}.
It is easy to verify that if S is self-adjoint, then S' is a self-adjoint unital algebra.
WOT
Moreover, it is woT-closed. Indeed, if Ta € S and Ta > T, then for every 5
in S
ST = WOT-lim STa = WOT-Iim TaS = TS.
a a
In particular, the commutant of a C*-algebra is always a von Neumann algebra.
The von Neumann algebra generated by an operator T is denoted by W* (T).
The following important result relating the double commutant 21" := (217)7 to
the wOT-closure is called the von Neumann Double Commutant Theorem.
Theorem 1.7.1 Suppose that 21 is a C*-subalgebra ofB(H) with trivial null space.
Then
a// = 2tW0T = 2tS0T.
—"S OT —W OT
Proof. Clearly 21 C 21 C 21" since the SOT is stronger than the WOT; and
since 21" is WOT-closed and contains 21. So fix an operator T in 21" and vectors
si,..., xn. It suffices to find A in 21 such that ^=1 \\(T - A)xi\\2 < 1 as this
represents a basic SOT open neighbourhood of T.

20
I. The Basics of C*-algebras
First consider n = 1. Let P be the (orthogonal) projection onto the subspace
21x7. Then P belongs to 21'. Indeed, %PU C PU and thus PAP = AP for every
A in 21. Therefore
PA = (A*P)* = (PA*P)* = PAP = AP.
If y = PLxXy then 2ly =_2lP1;ci = P1^ = 0. As 21 has trivial null space,
y = 0; that is, xx lies in 91*i. So PT = TP, and Txx belongs to 2l*i. Therefore
there is an operator A in 21 such that || (T - A)*i || < 1.
Now consider n > 2. Form the Hilbert space H^ which is a direct sum of n
copies of %. Let A^ denote the operator on H^ given by
AW(vu...,vn) = (Avu...yAvn).
This is the direct sum of n copies of A. Then let 2l<n) := {A<n) : A e 21}. An
operator X in 5(7^) can be represented as an n x n matrix with coefficients Xy
in B(U).
We wish to compute 2l(n)". It is easy to see that an operator X = [Xy] lies
in 21^' if and only if each matrix entry Xij belongs to 2l;. Hence an operator
T = [Tij] in 2l(n)" must commute with each matrix unit Ey, the operator with (i, j)
coefficient equal to J and all other coefficients equal to 0. A routine calculation
shows that this forces T to be diagonal with T# = 2\i for 1 < i < n. That is,
T = T^. In addition, T commutes with X^ for each X in 2l;, which means Tn
belongs to 21". So 2l<n)" = (2l")H
Now apply the n = 1 case to T<n) in (2l")(n> and x = (*i,..., xn) to obtain
an operator A in 21 such that
1 > ||(T<n> - AW)x||2 = f^ ||(T - A)Xi\\2.
t=l
Thus A lies in the given SOT neighbourhood of T. ■
There is a minor deficiency in the Double Commutant Theorem because of
the fact that convergent nets need not be bounded. In order to obtain a limit A,
it may be necessary to use operators of arbitrarily large norm. This is remedied
in Kaplansky's Density Theorem. First we need a technical result. Say that a
function / on R is strongly continuous on B(H)8a provided that whenever Aa is
a net of self-adjoint operators with SOT-liiria Aa = A, then
SOT-lim/(Aa) = /(A).
Lemma 1.7.2 Every continuous function f onR such that
limsup---|-T-^ < oo
|t|-+oo 1*1
is strongly continuous on B(H).

1.7. The Density Theorems
21
Proof. Let S denote the family of strongly continuous functions on R; and let Sb
denote the set of bounded functions in S. Since ||/(A)|| < ||/||oo» it is easily
verified that S and Sb are closed under uniform limits.
First we show that SbS is contained in S. Indeed, suppose that / is in Sb, g is
in S and SOT-liiria Aa = A. Then for x in %y
W(f9(Aa) - fg(A))x\\ < 11/(4011 \\(g(Aa) - g(A))x\\
+ \\(f(Aa)-f(A))g(A)x\\.
Since ||/(Aa)|| < ||/||oo» both terms on the right tend to 0. Thus fg is strongly
continuous. In particular, Sb is an algebra.
Next we show that h(t) = (1 + t2)"1 belongs to Sb. Indeed,
||(h(Aa) - h(A))x\\ = ||((/ + Al)"1 (A2 - Al)(I + A*)-i)x\\
= ll((/ + ^)--1(^«(A--Aa) + (A~Acx)A)(/ + A2)-1)x||
<\\(I + Al)-iAa\\\\(A-Aa)(I + A*)-ix\\
+ \\(I + A2^\\\\(A^Aa)A(I+A2)-'x\\
< ||t(l + ^J^HoolKA - i*a)y|| + 11(1 + ^-'lloolKA - Aa)z\\
= l\\(A-Aa)y\\ + \\(A-Aa)z\\
where y = (I + A2)~lx and z = A(I + A2)'xx. When SOT-lirria Aa = A, the
right hand side converges to 0. Thus SOT-liiria h(Aa) = h(A).
Clearly, S is invariant under translation and dilation by R. Therefore the
functions h8(t) = (1 + sH2)'1 belong to Sb for all real s. Since z(t) = t belongs to <S,
the multiplicative property shows that k8(i) = st h8(i) belongs to Sb for s in R. In
fact, these functions belong to Co(R). Moreover they are read valued and separate
points in R. Thus by the Stone-Weierstrass Theorem, they generate Co(R) as a
uniformly closed algebra.
Now suppose that / is a continuous function on R such that |/(*)| < C|*| for
\t\ > 1. Then g(t) = (1 + t2)"1/^) belongs to C0(R) and hence lies in Sb. Thus
zg(t) = t(l + t2r1f(t)
belongs to S by the multiplicative property. However, this function is also bounded
(by C)\ and so lies in Sb. Thus multiplying by z again, we deduce that
z*g(t) = t2(l + t2)-1f(t)
belongs to S. Hence g + z2g = f belongs to S. In other words, / is strongly
continuous. ■
We are now ready to prove Kaplansky's density theorem.
Theorem 1.7.3 If$l is a C*-subalgebra ofB(H) with trivial null space, then the
unit ball of%8a is SOT-dense in the unit ball of^a and the unit ball of% is SOT-
dense in the unit ball of$l". Likewise, the positive operators in the unit ball of%

22
I. The Basics of C*-algebras
are SOT-dense in the positive contractions in 21". In addition, if% is unital, the
unitary group of$l is SOT-dense in the unitary group o/2l".
Proof. Suppose that T belongs to 2l"a and ||T|| < 1. By the Double Commutant
Theorem 1.7.1, there is a net Aa in 21 such that T = WOT-Iirria Aa. Since the
adjoint is weakly continuous,
T = WOT-lim(Aa + A* )/2.
g OT *——W OT
Since 2l*a is convex, Corollary 1.6.2 shows that 2l,a = 2l,a . Hence there is
a net Bp in 2l,a such that T = SOT-lim^ Bp.
Let
f(t) = (t A 1)V (-!)={
1 for t> 1
t for -1<*< 1
-1 for t<-l
By Lemma 1.7.2, / is strongly continuous. Hence
T = f(T) = SOT-lim f(Bp).
Since ||/(.B/3)|| < ||/||oo = 1» this is the desired net in the unit ball of 2l,a.
The same argument works for the positive part by using the function
</(*) = (* A 1)V0.
Similarly, when 21 is unital, suppose that U is a unitary operator in 21". By the
spectral theorem (see the next section), we can write U = etT for some T in 2l"a.
Let T = SOT-lim^ Bp be a limit of self-adjoint elements. Since h(t) = eil is
strongly continuous by Lemma 1.7.2, we obtain
U=h(T) = SOT-lim h(Bp).
Since the h(Bp) are unitary in 21, the result follows.
Finally consider the case of the whole unit ball. Let X be an element of 21"
with ||X|| < 1. Form the C*-algebra ^2(21) of 2 x 2 matrices with coefficients in
21 acting on % © H. It is readily apparent that
M2{%)" = X2(2l)W°T = M2(2lWOT) = M2(2l,;).
In particular, T = [ £* -J ] belongs to the unit ball of ^2(21")^- Hence it is the
SOT-limit of a net in the unit ball of ^2(21)^. For such a net to converge, each
matrix coefficient must converge SOT. Thus the (1,2) entries form a net in the unit
ball of 21 converging SOT to X. ■
Remark 1.7.4 When % is separable, the unit ball of B(H) in the strong operator
topology is metrizable by Proposition 1.6.3. Thus the nets in the Kaplansky Density
Theorem may be replaced by sequences.

1.8. Some Operator Theory
23
1.8 Some Operator Theory
Recall that a partial isometry is an operator U such that U = UU*U.
Associated to a partial isometry U is its initial projection P = U*U and it range
projection Q = UU*. Then U maps PH isometrically onto QH, and vanishes
on PL%. The polar decomposition of an operator T in B(H) is a factorization
T = UA where A is positive and U is a partial isometry with initial space Ran (A)
and range space Ran(T).
Theorem 1.8.1 Every operator T on a Hilbert space % has a unique polar decom-
positionT = UA. The positive operator A = \T\ := (T*T)X/2 UesinC*(T); and
the partial isometry U belongs to W*(T). IfT is invertible, then U is a unitary
element of C*(T).
Proof. If T = UA, then T*T = A*U*UA = A*A = A2. By Corollary L4.1, A is
the unique positive square root of T*T, known as \T\. If x is in H,
\\Ax\\2 = (Ax, Ax) = (A*Ax,x) = {T*Tx,x) = {Tx,Tx) = \\Tx\\2.
Thus we may define an isometric operator U on Ran(A) by U(Ax) = Tx. Clearly,
the range is precisely Ran(T). Extend U by continuity to the closure Ran(A).
Then define U to be 0 on Ran (A)1 = ker(A) = ker(T) and extend by linearity
to all of H. By construction, U is a partial isometry with initial space equal to
Ran(A) and range Ran(T). Moreover, the choice of U is uniquely determined
since we require that U(Ax) = Tx and that the range of U*U equals Ran (A).
To verify that U belongs to W*(T), the double commutant theorem states that
it suffices to show that U belongs to C*(T)". For X in C*(T); and x in ker(T),
it follows that TXx = XTx = 0; so Xx belongs to ker(T) as well. Therefore
UXx = 0 = XUx for every x in ker(T). Also if x = Ay lies in Ran(A),
UXx = UXAy = (UA)Xy = TXy = XTy = XUAy = XUx.
By continuity and linearity, it follows that X and U commute.
When T is invertible, we have the identity U = T(T*T)~1/2 which lies in
C*(T).Also
jj*jj — /y*y\-l/2y*y/y*jn-l/2 _ j
Since U is invertible, this shows that U is unitary. ■
Corollary 1.8.2 If T is an operator on a Hilbert space H, then the projection
[RanT] onto the closure of the range ofT and the projection [ker(T)] onto the
kernel ofT belong to Wr*(T).
Proof. If T = U\T\ is the polar decomposition, then
[RanT] = tftT and [ker(T)] = / - U*U ■

24
I. The Basics of C*-algebras
In fact, von Neumann algebras contain a plentiful collection of projections.
Recall that the spectral theorem for self-adjoint operators expresses an operator
A = A* as an integral A = $atA\ A EA(dX) where EA is the spectral measure
for A. We will prove the spectral theorem in Chapter II. For the moment, we just
recover a few of its components that we need at this stage.
Let O be an open subset of R. We will show that the spectral projection EA(0)
belongs to W*(A). Indeed, consider the collection To of all non-negative
continuous functions / of compact support in R such that / < Xo. This set is
upwards directed in the usual order. Let P = SOT-lim/e^ /(A), which exists by
Lemma 1.6.4. We will show that P = EA(0).
First note that P is a projection. Since f(A) < I for all / in To, we have
P < I. Since P lies in Wr*(A) by construction, it commutes with each f(A). For
each / in To. one also has f1'2 in To- Hence P2 > fll2(A)2 = f(A). (Careful,
this uses commutativity!) As this is true for all / G To, we have J > P2 > P.
Thus cr(P) is contained in {0,1}; and so P is a projection.
If g in C(R) has compact support in O, then there is a function / in To such
that g = fg. Hence (again using commutativity)
g(A)>Pg(A)>f(A)g(A)=g(A).
Taking strong limits shows that Pg{A) = g(A) whenever g = gXo- On the other
hand, if g\o = 0, then f(A)g(A) = 0 for every / in To- Thus Pg{A) = 0.
Consequently, P is the projection onto the closed union of the ranges of all g(A)
with support contained in O. This is the spectral projection EA(0).
Now if <Y is a Gs subset of R, then it is the intersection of all open sets
containing it. Thus we may similarly show that EA(X) = info?* EA(0). Again this
spectral projection lies in W*(A). Since every Borel measurable set agrees with
a Gs set up to a null set, this shows that every spectral projection of A belongs to
W*(A).
It is not difficult to recover the integral formula for A. For convenience assume
that || A|| < 1. It is an exercise to show that the Riemann sums satisfy
E Ssa([£, **))<*< E *^A((S,*±i])
and that both sides converge to A.
We will use these spectral projections to show that there are abundant
projections in every von Neumann algebra.
Theorem 1.8.3 If 21 is a von Neumann algebra, then the closed convex hull of
the set of projections in 21 equals the set 2l+i of positive contractions in 21; and
the closed convex hull of the set of symmetries equals the set 2l*ai of self-adjoint
contractions in 21.

1.8. Some Operator Theory
25
Proof. Let 0 < A < I be a positive contraction in 21. Then
t=i
Analogously, if -J < B < J, then
B=n1i-^E2^((f-i,i])-/. ■
t=i
Since a C*-algebra may not have any projections at all, this result does not
generalize to arbitrary C*-algebras. Surprisingly, we are able to show that the unit
ball of a unital C*-algebra is always the closed convex hull of the unitary elements.
This is true even though they are not the full set of extreme points, nor is there
necessarily any compactness of the unit ball in some weak topology. This result
is known as the Russo-Dye Theorem; however, the proof given here is a stronger
variant of the original.
Theorem 1.8.4 In any unital C*-algebra 21, the closed convex hull of the set of
unitary elements is the whole unit ball
Proof. It suffices to show that if ||A|| < 1, then A is in the convex hull of a set
of unitary operators. When A is invertible, this is easy. For in this case, the polar
decomposition A = U\A\ lies in 21 by Theorem 1.8.1. It is a simple computation
to verify that V± := \A\ ± i(I - A*A)1!2 are both unitary. Then
A = {UV+ + UV-)/2.
Next we show that if ||A|| < 1 and U is unitary, then U + A is the sum of two
unitaries. Indeed, (U + A)/2 = U(I + U*A)/2 is an invertible contraction since
||?7*A|| < 1. So this is the average of two unitaries U\ and V\ by the previous
paragraph. That is
U + A = U1 + V1.
By induction, we will show that there are unitaries such that
U + nA = U1 + .-- + Un + Vn.
Indeed, assuming this holds, find unitaries Un+i and V^+i such that
^ + ^ = ^+1+1^+1.
Then
U+(n+l)A = U1 + ---+Un+(Vn + A)
= U1 + ---+Un + Un+1+Vn+1.
Now suppose that ||T|| < 1 - £. Then let A = -^-T - ^ J and U = I.
Since
U\\< ^t2^ + ^r = i.

26
I. The Basics of C*-algebras
we obtain unitaries U\,..., Un-i and Vn-\ such that
nT = I + (n - I)A = Ux + • • • + tfn_i + Vn-L
Hence T is the average of n unitaries.
It follows that the convex hull of the unitary group includes the whole open
ball. So the norm closure is equal to the closed unit ball. ■
1.9 Representations of C*-algebras
In this section, we will develop the basic properties of representations of C*-
algebras. This will culminate in a proof of the fundamental theorem of Gelfand and
Naimark that every C*-algebra is isomorphic to a concrete C*-algebra of operators.
A representation n of a C*-algebra 21 on a Hilbert space % is a *-homomor-
phism of 21 into B(H). We say that n is topologically irreducible if 7r(2l) has no
proper closed invariant subspaces. It is called algebraically irreducible if it has
no proper invariant manifolds (subspaces that are not necessarily closed). Our first
theorem shows that these two notions coincide for C*-algebras. So we will call n
irreducible when this condition holds.
Lemma 1.9.1 Let n be a representation of a C*-algebra %ona Hilbert space H.
Then 7r(2l) is topologically irreducible if and only (/,tt(21)/ = CI.
Proof. If ^(21)' is larger than the scalars, then it contains a non-scalar positive
operator; and thus by Theorem 1.8.3, it contains a proper projection P. Thus PH
is an invariant subspace for 7r(2l).
Conversely, suppose that M is a proper invariant subspace for 7r(2l); and let
P be the orthogonal projection onto M. Invariance is expressed algebraically as
n(A)P = Pn(A)P for every A in 21. However, it then follows that
Ptt(A) = (7r(A*)P)* = (Ptt(A*)P)* = Pn(A)P = n(A)P
for every A in 21. Thus P is a non-scalar operator in 7r(2l)/. ■
Lemma 1.9.2 Let n be a topologically irreducible representation of a C*-algebra
%ona Hilbert space H. Suppose that T in B(H), a finite dimensional subspace
K C % and e > 0 are given. Then there is an element A in 21 such that
P|| < HrWI and \\(w(A)-T)\K\\<e.
Proof. Since 7r (21) is irreducible, Lemma 1.9.1 shows that the commutant of n(21) is
just the scalars. Hence 7r(2l)" = B(U). By the Double Commutant Theorem 1.7.1,
^)SOT = B(U).
Assume that \\T\K\\ = 1 and let 5 = TPK. By Kaplansky's Density
Theorem 1.7.3, there is an element A\ in 21 with ||7r(Ai)|| < 1 such that
\\{*{Ad-S)PK\\<el2.

1.9. Representations of C*-aIgebras 27
By Theorem 1.5.5, there is an element A2 in 21 such that ||A2|| < (1 - e/2)-1 such
that 7r(A2) = n(Ai). Let A = (1 - e/2)A2. Then ||A|| < 1 and
||(7r(A) - T)k|| < HWiij) - S)PK\\+e/2\\n(A1)\\ < s. ■
Theorem 1.9.3 Every topologically irreducible representation nofa C*-algebra
21 is algebraically irreducible.
To prove this theorem, it suffices to show that if x and y are unit vectors in H,
then there is an element A in 21 such that n(A)x = y. So this result clearly follows
from the following stronger variant known as Kadison's Transitivity Theorem,
which is the C*-algebra version ofJacobson's density theorem for rings.
Theorem 1.9.4 Let nbea topologically irreducible representation of a C*-algebra
%ona Hilbert space %. Suppose that T in B(H), a finite dimensional sub space
K C % and e > 0 are given. Then there is an element A in 21 such that
tt(A)\k = T\k and \\A\\ <\\T\\ + e.
Proof. Use Lemma 1.9.2 to find A0 in 21 such that
||A0||<||r|| and ||(*(i4o)-T)flc||<e/2.
Recursively find An in 21 such that || An|| < 2~ne such that
\\(jr«(Ak)-T)pK\\<2-n-1e.
fc=o
Indeed, suppose that this holds for n. Apply Lemma 1.9.2 to the operator
n
5 = J>(Afc)-T,
k=0
the subspace K and 2~n~2e to obtain an approximant An+i in 21 with
n+l
||An+i|| < 2-"~1e and ||Q>(Afc) - T)PK\\ < 2~n~2e.
fc=0
LetA = EfcLo Ak- & is clear that ir{A)\K = T\K and
||A||<||T|| + f;2-»e = ||T|| + e. ■
n=l
A positive linear functional on a C*-algebra is a linear functional such that
f(A) > 0 whenever A > 0. A state is a positive linear functional of norm 1.
For example, if n is a representation of 21 on a Hilbert space % and a; is a vector
in H, then

28
I. The Basics of C*-algebras
is positive. Indeed, if A > 0, then
f{A) = W2Yx,x) = ||»(Al/«)«||* > 0.
In the unital case, this is a state when ||*|| = 1.
Associated to any positive linear functional is a positive semidefinite sesquilin-
ear form on 21 given by
[A,B]:=f{B*A).
That is, [•, •] is linear in the first variable, conjugate linear in the second, and
[A, A] > 0 for all A in 21. Hence it satisfies the Cauchy-Schwarz inequality
\[A,B]\2<[A,A)[B,B]
or equivalently
\f(B*A)\> < f{A*A)f{B*B).
Lemma 1.9.5 Positive linear junctionals are continuous. IfE\ is an approximate
unit for % then \\f\\ = lirn\ f(E\). In particular, when 21 is unital, \\f\\ = f(I).
Proof. As the unital case is more straightforward, we prove it first. Positivity
implies that if A < B, then f(A) < f(B). So for 0 < A < I, we have
0 < f(A) < /(/).
Now if X is in 21 with ||X|| < 1, then 0 < X*X < I and so
|/(X)|2 = |/(/*X)|2 < f(X*X)f(I) < /(/)*.
In general, suppose that / is not bounded on 2l+1. Then there are Ak in 2l+i
such that f{Ak) > 2k for k > 1. Let A = Y,k>i 2~fc4fe. Then
oo > f(A) > /(^2"fcAfc) =J2*-kf(Ak) > n.
fc=i fc=i
This holds for all n which is absurd; thus / is bounded on 2l+1. However, an
arbitrary element X in 21 may be written as
X = {X + X*)/2 + i{X - X*)/2i = Ax - A2 + iAz - iA4
where (X + X*)/2 = A1 - A2 and (X - X*)/2i = A3 - A4 are the Hahn
decompositions of Corollary 1.4.2. Thus \\Ai\\ < \\X\\ for 1 < i < 4. Consequently,
we deduce that / is bounded on the whole unit ball of 21.
Let M = HrriA f{E\)> which exists because this is an increasing net bounded
above by ||/||. Then for any X in 21 with ||X|| < 1,
\f(X)\2 = lim|/(JE?AX)|2 < \imf(El)f(X*X) < M\\f\\.
Choosing X so that |/(-X")| approximates the norm of / yields ||/||2 < M||/||, or
equivalently, ||/|| < M. ■

1.9. Representations of C*-algebras 29
The key to representing a C*-algebra on a Hilbert space is to build
representations from states. This important procedure is called the GNS construction named
after Gelfand, Naimark and Segal.
Theorem 1.9.6 Let f be a positive linear functional on 21. Then there is a
representation nf of 21 on a Hilbert space H and a vector xj inH which is a cyclic
vector for 7r(2l) such that \\xf\\2 = \\f\\ and
f(A) = {nf{A)xflxf) for all A G 21.
Proof. Let^ = {A G 21: /(A*A) = 0}. Then
M = {A G 21: f(B*A) = 0 for all B G 21}
because
\f(B*A)\2 < f{B*B)f(A*A) = 0 for all A erf, Be 21.
Hence Af is a closed subspace. Moreover, M is a left ideal since if N lies in M and
A, J5 belong to 21, then
f{B*{AN)) = f((A*B)*N) = 0 for all A,BG2l.
Whence, AiV belongs to Af.
Define a positive definite inner product on %jM by
(X,y):=/(Y*X)
where -X" denotes X +Af. This is well defined because if Ni, iV2 are in Af, then
/((Y + N2)*(X + Nx)) = /(Y*X) + f((Y + N2rNx) + f(X*N2)
= /(Y*X).
Let H denote the Hilbert space obtained by completing %l/Af in the inner product
norm.
Let 7T0 denote the left regular representation of 21 on 21/^/*:
n0{A)X := AX
which is well defined because Af is a left ideal. This is a ^representation because
(MA)X,Y) = f(Y*AX) = f{{A*Y)*X)
= {XMA*)Y) = {ko{A*)*X,Y).
Henceir0(A*) = n0(A)*. Moreover, ||7r0|| < 1 because
||7r0(A)||2 = sup ||7r0(A)X||2 = sup f(X*A*AX)
imi<i imi<i
< sup \\A*A\\f(X'X) = \\Atf.
Thus 7T0 extends by continuity to a *-representation irf of 21 on H.

30 I. The Basics of C*-algebras
In the unital case, let xj = J. Then
(nf(A)xf,xf) = f(I*A) = f(A).
The vector Xf is cyclic because ny (2l)a:/ = 21/A/" which is dense in H. And
ii/ii=m = imp-
Otherwise, let Ex be an approximate identity. The net E\ is Cauchy because
limA f(E\) = II/II and thus there are indices a < 0 such that
f(Ea)>\\f\\-e and \\ExEa - Ea\\ < e for all X>/3.
Hence if A > 0,
Re f{ExEa) = f{Ea) + Re f(ExEa - Ea) > \\f\\ - 2$.
Therefore
||4 - 4||2 = f((Ex - Ea)2) = f(E2x) + f{El) -2Re f(ExEa)
<f{Ex) + f{Ea)- 2(||/|| -2e)<4e.
Thus for A, /z > /?,
\\ex - 411 < 114 - 4|| +1|4 - 411 < 4*1/2-
Let Xf — lirriA 4- Then
(nf(A)xf}Xf) = lim(ir/(il)4,4) = Hm/(J57AAJ57A) = /(A).
This vector is cyclic because the vectors nf(A)xf = A form a dense subset of W.
And since
\imnf(Ex)A = \imnf(Ex)7Tf(A)xf = nf(A)xf = A
for this dense set of vectors, it follows that SOT-lirriA itf{Ex) = I- Hence
H/ll = lim|/(£7A)| = lim(nf(Ex)xf,xf) = \\xf\\\ ■
Corollary 1.9.7 Every state on 21 has a unique extension to a state on 2l~.
Proof. Existence follows since the representation -kj constructed above extends to
2l~ by setting tt/(/) = J. Thus we define f(A) = (7Tf(A)xf, Xf) for all A in 2l~.
Clearly this is a positive linear functional of norm \\xf\\2 = 1. Uniqueness follows
because /(J) = ||/|| = 1 by Lemma 1.9.5. ■
Let 5(21) denote the set of all states on 21, known as the state space of 21. A
state is called pure if it is an extreme point of 5(21).
Theorem 1.9.8 Let n be a representation of a C*-algebra 21 with a cyclic unit
vector x. Then the state f(A) := (7r(A)x, x) is pure if and only ifn is irreducible.

1.9. Representations of C*-algebras 31
Proof. Assume that n (21) is not irreducible. Then by Lemma 1.9.1, there is a proper
projection P in 7r(2l)'. Let t = \\Px\\2. If t = 1, this means Px = x and thus
Pn(A)x = n(A)Px = 7r(A)x for all A in 21. As x is cyclic, these vectors are
dense in H and hence P = J, contrary to fact. Likewise if t = 0, we conclude that
P1 = J. So0<*< 1.
Let </i(A) = rHTrUJPaj^aj) and (/2(A) = (1 - ^-^(AJP^P1*).
These are states because
*(/) = r^lPsll2 = l = (l -1)-1!^1*!!2 = *(/).
And since PLn(A)P = 0 for all A in 21, we obtain
(tgi + (1 -1)*2) (A) = (V(A) P*, P*) + (t(A)Px*, P1*)
-(ir(A)*l*) = /(A).
So / is a convex combination of 51 and 52- If / were pure, then </i = </2- But this
leads to an absurdity. Indeed, suppose that
0 = 9i(A) - g2(A) = t'1(n(A)Px} Px) - (1 - ^(ir^F1*, P1*)
= (tt^m^p* - (1 -*rljDl*)-
Then y = ^~1P« - (1 - t)~1PLx is orthogonal to 7r(2l)a; which is dense in %\
whence y = 0. This is false, and thus / is not pure.
Conversely, suppose that / = tg\ + (1 - t)g2 for 51 ^ g2 and some 0 < t < 1.
Define a sesquilinear form on 7r(2l)a; by
[*(A)z,K(B)z]:=tgi(B*A).
Then
0 < [n{A)x,7r{A)x] = t9l{A*A) < f{A*A) = \\n{A)x\\2.
In particular, this form is well defined because n(A)x = 0 implies that
0=gi(A*A) = f(A*A).
So it is a positive semi-definite sesquilinear form of norm at most 1. Thus there
is an operator H with 0 < H < I such that [u, v] = (tz, Hv). Now calculate for
A, X, Yin 21,
({n{A)H - Hn{A))n{X)x,ir{Y)x)
= (n(X)x,H7r(A*Y)x) - (7r(AX)xyH7r(Y)x)
= t9l{Y*AX) - tgx{Y*AX) = 0.
Hence H commutes with 7r(2l).
If 7r were irreducible, then H would be scalar, say H = cl. Then
tgi(B*A) = c(n(A)x,7r(B)x) = cf(B*A).
Both / and 51 are states; thus c = t and </i = /, contrary to fact. Therefore the
representation n is reducible. ■

32
I. The Basics of C*-algebras
Now we return to the proof of the Gelfand-Naimark Theorem that states
that every C*-algebra can be isometrically represented as a concrete C*-algebra of
operators. In view of the GNS construction, it is sufficient to show that there are
enough states to determine the norm. We need some more results about states.
Lemma 1.9.9 Let f be a continuous linear functional on 21 such that for some
approximate identity E\,
ll/H = l = lim/(J5A).
Then f is a state.
Proof. First let us reduce to the unital case. Let / be any Hahn-Banach extension
of / to 2l~. Let /(/) = a. Since ||/|| = ||/|| = 1, we have \a\ < 1. Also
||2J5A - J|| < ! and hence
l>lim|7(2J57A-/)| = |2-a|.
Together, these inequalities imply that /(/) = 1. So we may assume that 21 is
unital and/(J) = 1.
Next we show that / is self-adjoint meaning that if A = A*, then f(A) is real.
Let A be a self-adjoint element of norm one. Notice that \\A ± inl\\2 = n2 + 1.
Hence
\f(A) ± ni\ < \/n2 + l for all n G N.
That is, f(A) lies in the intersection of all disks centred at ±ni of radius y/n2 + 1.
This intersection is the interval [-1,1].
Now if 0 < A < J, then \\2A - I\\ < 1. Hence -1 < 2f(A) - 1 < 1, and
thus 0 < f(A) < 1. So / is positive. ■
Now the key step in constructing sufficiently many states uses the Hahn-
Banach Theorem again.
Lemma 1.9.10 Let A be a self-adjoint element ofVL Then there exists a pure state
f on% such that \f(A)\ = \\A\\.
Proof. We will work in 2l~ if 21 is not unital. Then C*(A) is a subalgebra of
2l~. Since ||A|| = spr(A), there is a multiplicative linear functional y> in the
maximal ideal space of C*(A) such that \<p{A)\ = \\A\\. It is always the case that
\\<p\\ = 1 = ¥>(/). Let $ be any Hahn-Banach extension of cp to a functional on
2l~. Then since ||$|| = 1 = $(/), the previous lemma shows that $ is a state on
21" and thus restricts to a state on 21.
Let T be the set consisting of all states / on 21 such that f(A) = $(A). This
is a weak-* closed bounded convex subset of the dual space of 21. By the Banach-
Alaoglu Theorem, T is weak-* compact. So by the Krein-Milman Theorem, it has
an extreme point /0. To see that /o is an extreme point of 5(21) and hence is pure,

1.9. Representations of C*-algebras
33
it suffices to show that T is a face of 5(21); for this implies that extreme points of
T are extreme points of 5(21).
But this follows from the fact that f(A) = $(A) = ±\\A\\ for / in T. So if
f = {9i+ 92)/2 for / in T and # in 5(21), then
Pll = l/M)l<(l5iM)l + l52M)|)/2<||^||.
This can only occur if gx(A) = g2(A) = f(A) = $(A). Hence gi belongs to T\
which shows that T is a face. ■
Corollary 1.9.11 If A is an element of% then there exists an irreducible
representation n o/2l and a unit vector x such that ||7r(A)a;|| = || A||.
Proof. Apply Lemma 1.9.10 to A*A to obtain a pure state / with /(A*A) = \\A\\2.
Then let tsj and xj be obtained from the GNS construction applied to /. By
Theorem 1.9.8,7T/ is irreducible. Finally,
\MA)z,f = (n(A*A)zf} *,) = f{A*A) = \\A\\>. ■
Finally we have all the pieces to complete the proof of the Gelfand-Naimark
Theorem.
Theorem 1.9.12 Every abstract C*-algebra 21 is isometrically ^-isomorphic to a
concrete C*-algebra of operators. lf% is separable, then one may take the Hilbert
space to be separable.
Proof. Take n := X) ®/€5(a) nf- Then 11^(^)11 = WA\\ for every ^ in 21 by the
previous corollary. When 21 is separable, it suffices to choose one representation nn
for each element An of a countable dense subset of 21 such that ||7rn(An)|| = ||An||.
Then X) ©n ^n works and is separably acting. ■
The representation used in this proof is called the universal representation of
21. Note that every state on 7r(2l) is a vector state in this representation; meaning
that for each state /, there is a unit vector x such that f(7r(A)) = (7r(A)x, x). As
every linear functional on 21 is a linear combination of states (see Exercise 1.32),
it follows that every functional / on 21 may be represented by f(A) = (n(A)x, y)
for certain vectors x and y. Thus they extend to wOT-continuous linear functionals
on tt(21)".
The Jacobson radical rad(>t) of a Banach algebra A is the intersection of
the kernels of all algebraically irreducible representations. An algebra is called
semi-simple if the Jacobson radical is {0}.
Corollary 1.9.13 C*-algebras are semi-simple.
Proof. By Theorem 1.9.3, topologically irreducible representations of C*-algebras
are algebraically irreducible. So this applies to tsj when / is a pure state. Since the

34 I. The Basics of C*-aIgebras
pure states separate points in 21 by Corollary 1.9.11, the representation
/pure
is isometric. Hence
rad(2l)= f] kerrr, = {0}.
/pure
Therefore 21 is semi-simple. ■
We conclude this section with a couple of results about the relationship
between representations of a C*-algebra and its ideals. A representation n is non-
degenerate if 7r(2l)K is dense in H.
Lemma 1.9.14 Suppose that 3 is an ideal of a C*-algebra 21, and that n is a non-
degenerate representation of 3 on a Hilbert space %. Then there is a unique
representation if of VI on H extending n. Moreover, if is irreducible if and only if it is
irreducible.
Proof. For A in 21, J in 3 and x in H, define
if(A)(ir(J)x):=n(AJ)x. (1)
To verify that this is well defined, suppose that n( Ji)xi = n(J2)x2. Let E\ be an
approximate unit for 3. Then
7r(A7i);ci = Um7r(AE\Jx)xi = limir(AE\)ir(Ji)xi
= limn (AE\)n(J2)x2 = limn (AE\J2)x2 = n(AJ2)x2.
Hence if (A) is well defined on a dense subset of H. Moreover
||7r(A)(7r(J)aO|| < limHA2?A)|| ||ir(J)«|| < ||A|| \\n(J)x\\,
and thus if (A) extends uniquely to a bounded operator in B(H).
To see that this is a ^representation, one should verify that if is multiplicative
and self-adjoint. We do the latter only. For A in 21 and J\, J2 in 3,
(ifiA^iriJJxuiriJ^) = {ir{A^{)xuiT{J2)x2) = (ir(j;A*J1)xlyx2)
= {n{Ji)xi,ir{AJ2)x2) = {n{Ji)xuif(A)ir(J2)x2)
= {if(Ayir(Ji)xuir(J2)x2).
Hence if (A*) = if(A)*. The uniqueness follows from the fact that any extension
of it must satisfy (1) which determines n on a dense set.
Suppose that n is not irreducible, and thus there is a proper invariant subspace
M for 7r(3). Then since n is non-degenerate
U = ir(3)(M + ML) C n(3)M + n(3)ML.

1.9. Representations of C*-aIgebras
35
Since irfflM1 C ML% we must have M = 7r(3)M. Therefore
*{Vk)M = *{Vk)*Q)M = n(d)M = M.
So n is not irreducible either. The other direction is trivial. ■
Our other result in this direction turns the tables and considers the restriction
of a representations of 21 to an ideal.
Lemma 1.9.15 Suppose that n is a representation of% on a Hilbert space H. Let
Z be an ideal of%. Then the projection P onto n(3)H lies in the centre 0/tt(21)".
Ifn is irreducible and n(d) 7- 0, then the restriction n\ 3 is also irreducible.
Proof. Since n(H)ic(3)H = n(3)H, it follows that n(d)U is invariant for tt(21)
and thus P lies in tt(21)'. If X lies in 7r(3)', then Xn(J)x = n(J)(Xx) belongs
to 7r(3)H for every J in 3 and x in H. Thus P% is invariant for n( 3)'; whence
P e Trp)" n tt(21)' c tt(21)" n *(»)'
which is the centre of 7r(2l)".
If n is irreducible, then P is scalar. As n(3) 7- 0, we have P = I. Thus 7r| J is
non-degenerate. So it is irreducible by the previous lemma. ■
Using these results, we will derive a useful extension of Theorem 1.4.8. A
quasicentral approximate unit for an ideal 3 of a C*-algebra 21 is an approximate
unit E\ such that
lim \\EXA - AEX\\ = 0 for all A G 21.
Theorem 1.9.16 Every ideal Z of a C*-algebra 21 has a quasicentral approximate
unit
Proof. Let E\, A G A, be an approximate unit for 3 given by Theorem 1.4.8. Recall
from the proof of that theorem that ifE\<E<I, then
\\X-XE\\<\\Xm{I"Ex)X\\1^ for all I6X
Thus the convex hull £ of {E\ : A G A} is directed and forms an approximate unit
for 3, again as in the proof of Theorem 1.4.8. Therefore, it suffices to show that for
A G A and A\,..., An in 21, then there is an element F in £ such that F > E\ and
\\AiF - FAi\\ < 1/n for 1 < i < n.
To construct F, form the C*-algebra .Mn(2l). Notice that J5^n), the direct sum
of n copies of E\y forms an approximate unit for the ideal Mn{3)- Let
be the diagonal operator with diagonal entries Ai. First we show that 0 is in the
closure of S := {AF^ - F^A : F eJF} where T = convfJ^ : \i > A}.
Indeed, S is a convex set. So if 0 is not in <S, the separation theorem version of the
Hahn-Banach Theorem shows that there is a linear functional / so that
Re f(AE^ - E^A) > 1 for all \i > A.

36
I. The Basics of C*-algebras
Let 7r be the universal representation of 21, and let vectors x and y be chosen
so that f(A) = (n(A)x, y). Let P be the projection onto n(3)H. Then it is easy
to verify that
SOT-lim7r(J57A) = P.
But P belongs to 7r(2l)' by Lemma 1.9.15. Hence
lim f(AEx - EXA) = lim{n{AEx - ExA)x, y)
AGA AGA
= \\m(ir{A)P - Pn(A)x, y) = 0.
AGA
This contradicts the assumption that 0 is not in the closure of S. Thus there is an
element F in T so that
\\AF^ - F^A\\ = max \\A{F - FA{\\ < 1/n.
l<t<n
Now define F = F^x indexed by finite subsets A of 21 and A G A. To see that
it is upward directed, suppose that F^\ and Fb^ are two such elements. Since
they lie in 5, we have
k l
FAA = E s{EXi and FB^ = £J tjE^.
t=l i=j
Let C = A U B and choose i/ in A such that v > A; for 1 < i < k and v > hj
for 1 < j < L Then it is easily seen that Fc,u dominates both F^\ and Fa)Ai. It is
routine to verify that it is a quasicentral approximate unit. ■
1.10 C*-algebras of Compact Operators
In this section, we develop the complete structure theory for C*-subalgebras
of the compact operators. This applies, in particular, to finite dimensional C*-
algebras which are the basic building blocks used in Chapter HI. The C*-algebra
algebra £ of compact operators is one of the simplest and most basic C*-algebras.
It will play a central role in the theory. As for matrix algebras, the existence of
minimal projections in £ is crucial to the analysis.
Lemma 1.10.1 If % is a non-zero C*-algebra of compact operators, then it con-
tains a minimal projection E. Moreover, J5721JS7 = CE.
Furthermore, ifn is a non-zero representation of% then there is a minimal
projection E of% such that rr(E) ^ 0. When n is irreducible, n(E) has rank one.
Proof. As 21 is non-zero, it contains a non-zero positive operator A. Because A is
compact, its spectrum is a countable set with 0 as the only possible cluster point.
From the continuous functional calculus Theorem 1.3.3, the spectral projection P
corresponding to a non-zero eigenvalue of A lies in 21. Since P is compact, it
has finite rank. Clearly, P dominates a projection E in 21 with minimal positive
rank. This projection is therefore minimal in 21. If J5721JS7 contained a self-adjoint

1.10. C*-aIgebras of Compact Operators
37
element T which is not a scalar multiple of E, then the same argument produces
a spectral projection of T which is dominated by E, and so has smaller positive
rank, contrary to fact. Hence J5721JS7 = CE.
Suppose that n is a non-zero representation. Then there is a positive element A
such that 7r(A) ^ 0. From the functional calculus, we deduce that there is a finite
rank spectral projection P of A such that 7r(P) ^ 0. Now P is the sum of minimal
projections of 21 because it is finite rank. So there is a minimal projection E such
that tt(J57) 7-0.
If n(E) has rank greater than 1, choose orthogonal unit vectors x and y in the
range ofn(E). For any A in 21, there is a scalar A so that EAE = XE. Hence
{n{A)z,y) = {ir{EAE)z,y) = (\z,y) = 0.
Thus 7r(2l)a; is a proper invariant subspace, and so n is not irreducible. ■
We write £(H) to denote the C*-algebra of compact operators on H, which
is isomorphic to Mn if % is n-dimensional, and to .£ when % is separable and
infinite dimensional.
Lemma 1.10.2 Let Hbea Hilbert space. Then the only irreducible C*-subalgebra
of&(U) is itself.
Proof. Suppose that 21 is a non-zero irreducible subalgebra of £{%) acting on a
Hilbert space % of the appropriate dimension. Let E be a minimal projection
in 21 provided by Lemma 1.10.1 such that E is rank one. Then there is a unit
vector e so that E = ee*. Since 21 is irreducible, it is algebraically irreducible by
Theorem 1.9.3. So if z and y are vectors in H, one may choose elements A and B
in 21 so that Ae — z and Be = y. Then 21 contains
AEB* = Aee*B* = (Ae)(Se)* = zy\
Thus 21 contains every rank one operator. As these operators span £, we obtain
2l = £. ■
Before going on, we collect two useful corollaries.
Corollary 1.10.3 The C*-algebra of compact operators is simple.
Proof. As .£ is irreducible, any non-zero ideal of .£ is irreducible by Lemma 1.9.15.
Hence the only non-zero closed ideal is all of .£ by Lemma 1.10.2. ■
Corollary 1.10.4 If 25 is an irreducible C*-subalgebra ofB(H) which contains a
non-zero compact operator, then 25 contains &
Proof. Since 25 fl ^ is a non-zero ideal of 25 by hypothesis, it is irreducible by
Lemma 1.9.15. Hence it is all of .£ by Lemma 1.10.2. ■
This next lemma contains the key idea. A minimal idempotent which is not in
the kernel of a representation a generates a cyclic irreducible representation both
for 21 and for <r(2l); and they are unitarily equivalent.

38
I. The Basics of C*-algebras
Lemma 1.10.5 Every non-degenerate representation of a C*-algebra 21 of
compact operators has an irreducible subrepresentation which is unitarily equivalent
to the restriction of% to a (minimal) reducing subspace.
Proof. Let a be a non-degenerate representation of 21. By Lemma 1.10.1, there is
a minimal projection E such that P = <r(E) is non-zero. Let / be a unit vector
in the range of P; and let HZ = cr(2l)/. Also let e be a unit vector in the range
of E\ and let Ue = 2le. The subspaces Ue and Wf are invariant for 21 and cr(2l)
respectively. Since JE721JS7 = CE, the state on 21 given by cp(A) = (Ae, e) satisfies
EAE = <p(A)E for all A G 21.
Define a linear map U from We to Wj by the formula
U(Ae) := <r(A)f = <r(AE)f.
A simple computation shows that for A, B in 21
(UAe, UBe) = [a(AE)f, a(BE)f) = (<r(EB*AE)f, f)
= y>(S*A)(P/,/) = (S*Ae,e) = (Ae,Be).
Consequently, U is an isometry from Ue onto Wj. In particular, J7 is well defined.
For A, B in 21, we have
a(A)(UBe) = a(A)a(B)f = <r(AB)f = U(ABe) = UAU*(UBe).
Thus a(A)\ii<r = UAImcU* for every A in 2l. So cr|#y is unitarily equivalent to
the restriction of 2l to He.
Let 7re denote the restriction map of 2l to Ke- Notice that ne(E) = ee* is rank
one. Indeed, J5?Ae = EAEe belongs to Ce for every A in 2l, and thus Ce is the
range of ne(E). Suppose that P is a projection in B(Ke) which commutes with
7re(2l). Then
Pe = P7re(E)e = 7re(E)Pe
is a multiple of e. As Pe = P2e, this multiple is 0 or 1. By considering P1 if
necessary, we may suppose that Pe = 0. But then
PAe = Pire(A)e = ire(A)Pe = 0 for all A G 21;
whence P = 0. Therefore ^e^y = CI; and so 7re is irreducible. Hence the
subspace He is minimal. ■
Corollary 1.10.6 Every irreducible representation ofMn or £ is unitarily
equivalent to the identity representation.
Proof. The only invariant subspace for Mn or £ is the whole space. ■

1.10. C*-algebras of Compact Operators
39
It remains to show that representations of these algebras are direct sums of
irreducible ones, not something more complicated. As we shall see in the next
chapter, the situation is not as simple even for most abelian C*-algebras.
Theorem 1.10.7 Let %bea non-degenerate C*-subalgebra of the compact
operators. Then every representation vof% is the direct sum of irreducible
representations which are unitarily equivalent to subrepresentations of the identity
representation.
Proof. By Lemma 1.10.5, H* contains a subspace H^ such that the restriction of
7r(2l) to HZ is an irreducible representation which is unitarily equivalent to a sub-
representation 7re of the identity representation. Choose a maximal family {H„}
of pairwise orthogonal reducing subspaces with this property. Then Ha is spanned
by the H%'s. For otherwise, the complement (]T)n H^)1 would dominate another
such subspace H^ by Lemma 1.10.5 again. This would contradict the maximality
of the original family. Let an be the restriction of a to H„. Then we obtain the
decomposition a = £n ®anr and each <rn is equivalent to an irreducible subrep-
resentation of the identity. ■
Finally, we apply this to the identity representation itself. Recall that H^
denotes the Hilbert space direct sum of k copies of H. And for each A in B(H),
A^ denotes the operator on H^ given by A^fei,..., x^) = {Azi,..., Axn).
Theorem 1.10.8 Let %bea C*-subalgebra of the compact operators & Then there
are Hilbert spaces Hi of dimension n» for i > 0 and non-negative integers ki so
that
ft~ft0e]Te?4fc,') and 2i-oe^©£(^)(fc,,)-
t>i i>i
Proof. Let H0 = ker2l. By Theorem 1.10.7, the Hilbert space Hfr may be
decomposed as a direct sum of reducing subspaces Kj such that the restriction <tj of
21 to Kj is irreducible. By Lemma 1.10.2, (7,(21) = R(Kj). Collect together all
equivalent representations into classes {ctj : j G Si}. Let Hi be a Hilbert space of
dimension n» = dim Kj for j in <%; let ki be the cardinality of <%; and let tt; denote
the identity representation of £(Hi). Then it is evident that
H ~ Ho © J] ®Hfi] and id ~ 0 0 ][] ©^•
t>l i
Finally, notice that a rank one projection Ei in R(Hi) is mapped onto a projection
of rank ki. As this projection is compact, it follows that ki is finite for all i. ■

40
I. The Basics of C*-algebras
Exercises
LI Show that if S and T are two normal operators, then there is a
♦-isomorphism 7T from C*(S) onto C*(T) such that tt(5) = T if and only if 5 and
T have the same spectrum.
1.2 Show that Co(X, 21), the space of continuous functions vanishing at
infinity on a compact Hausdorff space X with values in a C*-algebra 21, is a C*-
algebra with the operations f*(x) = f(z)* and ||/|| = sup^jr ||/(«)||.
1.3 If 2ln is a sequence of C*-algebras for n > 1, form the infinite product
9JT = IIn>i %n of all bounded sequences (An) with An in 2ln and the
infinite sum Z — Y^ ©n>i ^n consisting of such sequences which converge
to 0. With the supremum norm and point-wise multiplication and
conjugation, these become Banach *-algebras. Show that they are C*-algebras,
and that J is an ideal of 9JT.
1.4 Show that in any algebra,
(A{BA - XI)'XB - /) {AB - XI) = XL
Hence show that <r(AB) U {0} = <r(BA) U {0}.
1.5 Prove Corollary 1.4.9.
HINT: List a dense subset An of 21. Then choose an increasing sequence
of elements En in A (from Theorem 1.4.8) such that || AkEn - Ak\\ < 1/n
and \\EnAk - Ak\\ < 1/n for all 1 < k < n.
1.6 Use the identity ||A||2 = \\A*A\\ = spr(A*A) to prove that there is a
unique C*-algebra norm on a given *-algebra. In particular, show that the
unitization 2l~ of a non-unital C*-algebra 21 is unique.
1.7 Show that there is a C*-algebra norm on Mn{%).
HINT: Prove it first for concrete C*-algebras.
1.8 Show that there are positive 2x2 matrices A and B such that A < B but
A2 ^ J52. Prove that the inequality is valid in every C*-algebra when A
and B commute.
1.9 Show that if 0 < A < S, then 0 < A1'2 < B1'2.
HINT: Use Lemma 1.4.7 to show that ||A1I2B-1I2\\ < 1. Then use
Exercise 1.4 to show that spr(JB-1/4A1/2S-1/4) < 1.
1.10 A function on R+ is called operator monotone if 0 < A < B implies
that/(A) </(£).
(a) Use Lemma 1.4.7 to show that f3(t) = st(l + st)~x is operator
monotone for s > 0.
(b) Show that an integral of f3 by a positive measure is also operator mono-

Exercises
41
tone. Apply this to /0°° /,(05_/3 ds to show that f(t) = t& is operator
monotone for 0 < /? < 1.
(c) Show that f(t) = min{£, 1} is not operator monotone on Mi.
1.11 If A is a positive element in 21, show that A&A contains A. Hence this
is the hereditary C*-subalgebra of 21 generated by A. Show that every
separable hereditary subalgebra has this form.
1.12 For a separable Hilbert space H, show that the ideal .£ of compact
operators is the unique proper ideal of B(H).
1.13 Letu; = {n_1 : n > 1}U{0}. Find all ideals of the C*-algebraC(u;,.M2).
1.14 Show that every closed ideal of Co(-X") has the form
Vb = {/ € Co(X): f\E = 0}
for a closed subset E of X.
HINT: Use the Stone-Weierstrass theorem.
1.15 Verify the integral formula of the spectral theorem following the outline in
section 1.8.
1.16 For a von Neumann algebra 21, show that the extreme points of VHaa\ are
precisely the set of symmetries; and the extreme points of 2l+i are the
projections. Show that for B(H), the set of extreme points of the unit ball
is strictly larger than the set of unitaries.
1.17 Use the polar decomposition of a compact operator K to show that it may
be written as K = X)n>1 snenf£ where sn is a sequence of real numbers
decreasing to 0, and {e^} and {/n} are orthonormal sequences, where ef*
denotes the rank one operator ef*(x) = (x} f)e. The sequence sn(K) are
called the singular values of K.
1.18 A compact operator K in .£ is trace class if
PTIIx:- Y%n(tf)<oo.
n>l
The collection of all trace class operators is denoted by C\.
(a) Show that if xn and yn are orthonormal sequences, then
El(^n.»n)l<ll^lll-
n>l
(b) Show that C\ is complete in the trace norm, and that the ideal of finite
rank operators is dense in C\.
(c) Show that ||AlfB||i < ||A|| ||tf ||i ||B||. Hence deduce that d is an
ideal of B{H).

42
I. The Basics of C*-algebras
(d) Fix an orthonormal basis {en} and define the trace by
Tr(K) = J2(Kenien).
n>l
Show that Tr(KT) = Tr(TK) for all K in d and T in B(H). Hence
deduce that Tr is independent of the choice of basis.
(e) Each T in B(H) defines a linear functional ifT on C\ by <pt{K) =
Tr{TK). Show that ||y>T|| = ||T||.
(f) Show that if (p is a linear functional on Ci, then the sesquilinear form
(x, y) := (p(xy*) determines a bounded linear operator T such that y> =
(g) Deduce that B(%) is the dual space of C\. This defines the weak-*
topology on B(H) given by the functional (fK on B(H) for if in Ci.
L19 (a) Show that the weak operator topology corresponds to the functionals
<pp for finite rank F. Hence deduce that the weak-* topology is stronger
than the wot.
(b) Show that every von Neumann algebra is a dual space.
1.20 Show that the dual of the compact operators is C\.
1.21 The strong-* topology on B(H) is the weakest topology such that T -» Tx
for x in H and T -» T* are continuous maps. Show that the functionals
which are strong-* continuous on B(H) coincide with the WOT-continuous
functionals.
1.22 Show that multiplication is not jointly continuous in the strong operator
topology.
HINT: Let A consist of ordered pairs (M, x) where M is a finite
dimensional subspace and a; is a unit vector orthogonal to M ordered by the
relation (M} x) < (N, y) if M and x are both contained in N. Let e be a fixed
unit vector. Set A(m,x) = (dimM)ex* and J3(Af,a>) = (dimM)~1a;e*.
1.23 Complete the proof of Proposition 1.6.3.
1.24 Prove the converse of Lemma 1.7.2.
HINT: Let A be a net consisting of ordered pairs (M,«), where M is a
finite dimensional subspace and a; is a unit vector which is not orthogonal
to M, and ordered by the relation (M, x) < (iV, y) if M and x are both
contained in N. Set A(MtX) = (dim M II-Pm^I!)""1^*- Show that this net
converges strongly to 0, but that limsup(M ^ ||/(-A(Af,«)y|| = °° for every
y^O and / in C(R) such that lim sup^j^^ | /(t)/t | = oo.
1.25 Suppose that A belongs to a concrete C*-algebra 21 C B(H)> and has a
polar decomposition A = U\A\. Show that Uf(\A\) belongs to 21 provided
that /(0) = 0.

Exercises
43
1.26 Show that if J is an ideal of a C*-algebra 21 and A lies in 21, then there is
an element J in J such that \\A - J|| = dist(A, 3).
HINT: Use Corollary 1.4.2 on \A\ - \\A + 3|| J.
1.27 If a C*-algebra 21 contains a non-unitary isometry 5, show that
\\S-A\\>±
for every A = ^£=1 A^C/i which is the convex combination of n unitaries.
HINT: You may suppose that Ax > 1/n. Estimate \\U^S - Xil\\ and
compare it with the fact that cr(UiS) = 5.
1.28 Show that in C(D), the function f(z) = (1 - -)z is not a convex
combination of fewer than p unitaries.
HINT: If / = X)r=i ^*ui1S a convex combination of unitaries, show that
/ - XiUi maps D into itself, and thus has a fixed point. Hence deduce that
X{ < 1/p.
1.29 Show that states on Co(-^) correspond to regular Borel probability
measures on X. Which are the pure states?
1.30 Show that irreducible representations of abelian C*-algebras are one
dimensional.
HINT: Use the spectral theorem.
1.31 Show that every linear functional on a C*-algebra decomposes as a sum of
a self-adjoint functional and a skew-adjoint functional.
1.32 Show that every self-adjoint functional / decomposes as / = </i - g2
where gi are positive linear functional such that
ii*iii+M-ii/ii-
HINT: Mimic the proof that a real measure decomposes as the difference
of two positive measures supported on disjoint sets. This is known as the
Jordan decomposition.
1.33 Show that every irreducible representation comes from a (pure) state.
HINT: Use a state of the form f(A) = (n(A)x, x).
1.34 Show that if J is an ideal of a separable C*-algebra 21, then there is a
sequential quasi-central approximate unit for J relative to 21. Then show
that this is still true even if J is not closed (but is still self-adjoint).
1.35 Suppose that p is a representation of a C*-algebra 21 on a Hilbert space %
with a unit cyclic vector x. Define the state f(A) = (p(A)x, x). Show that
p is unitarily equivalent to the GNS construct -kj via a unitary operator U
such that Uxf = x. This shows that the GNS construction is unique.
HINT: Set UX = p(X)x.

44
I. The Basics of C*-aIgebras
1.36 Show that every finite dimensional C*-algebra may be faithfully
represented on a finite dimensional Hilbert space.
HINT: Show that finitely many states suffice for the Gelfand-Naimark
Theorem in this context.
1.37 Suppose that 21 is a C*-algebra containing the compact operators. Show
that every faithful irreducible representation is unitarily equivalent to the
identity representation.
1.38 A representation a is a subrepresentation of a representation p if there is
a central projection P in />(2l)' such that a = Pp\PH. Hence p splits as
a direct sum p ~ <r (B p'- Show that the Schroeder-Bernstein Theorem is
valid for representations. That is, if cr and p are each unitarily equivalent
to subrepresentations of the other, then they are unitarily equivalent.
HINT: (a) Show that p~p@(p'@ a').
(b) By iterating (a), show that there are countably many pairwise
orthogonal projections Pn in />(2l)' such that Pnp\Pn% are each unitarily
equivalent to p1 © a'.
(c) Hence deduce that
p ~ p" © (p' © (/)(oo) ~ p © (/>' © </)(oo).
(d) Hence show that
/> ~ (/> © *') © (/>' © (/)(oo) ~ a © (/>' © (/)(oo).
1.39 (a) Suppose that p and a are representations of a C*-algebra 21 on Hilbert
spaces Hp and Ha. Suppose that T in B(/Haj/HP) is a non-zero operator
such that
p(A)T = T<r(A) for all A G 21.
Show that the partial isometry {/ in the polar decomposition of T also
intertwines p and cr. Moreover, the range and domain projections are
reducing subspaces for cr(2l) and />(2l) respectively. The restrictions to these
subspaces are unitarily equivalent.
(b) If a is irreducible in the above situation, then a is unitarily equivalent
to a subrepresentation of p. If both p and a are irreducible, then they are
unitarily equivalent.

Exercises
45
Notes and Remarks.
Commutative Banach algebras and the Gelfand transform were introduced
by Gelfand [1941]. The notion of an abstract C*-algebra is due to Gelfand and
Naimark [1943], who proved Theorem 1.9.12 that C*-algebras can be isometrically
imbedded into B{%). Their definition of a C*-algebra was somewhat stronger than
the present-day definition. This gap (Theorem 1.4.5) was filled by Fukamiya [1952]
and Kaplansky (unpublished). Segal [1947] proved the existence of an approximate
identity, and further refined the GNS construction proving Theorem 1.9.8.
Theorem 1.5.4 on ideals is also due to Segal [1949] and to Kaplansky [1949]. Weak
topologies and the double commutant Theorem 1.7.1 are due to von Neumann
[1929]. Kaplansky [1951] extended this to his density Theorem 1.7.3. Kadison
[1955] proved the transitivity Theorem 1.9.4. Theorem 1.8.4 is due to Russo and
Dye [1966], but the proof presented here is from Kadison and Pedersen [1985].
Quasicentral approximate units were introduced by Arveson [1977] and Ackemann
and Pedersen [1977].

CHAPTER H
Normal Operators and Abelian C*-algebras
ILl Spectral Theory
Early in Chapter I, we learned that if N is a normal operator with spectrum X,
then the continuous functional calculus Corollary 1.3.3 provides a *-isomorphism
of C(X) onto C*(N). So to classify normal operators, we need to classify the
♦-representations of C(X) for a compact metric spaces X.
Two normal operators M and N are unitarily equivalent if there is a unitary
operator U such that N = UMU*. This is the most rigid notion of equivalence,
as a unitary operator preserves the complete spatial structure of a Hilbert space.
Likewise, two representations p and cr of C(X) are unitarily equivalent if there is
a unitary operator U such that cr = Ad U />, where Ad U(X) = UXU*.
Recall that a representation p of a C*-algebra 21 on % is cyclic if there is a
cyclic vector x such that />(2l)a; is dense in %. Associated to a unit cyclic vector is
the state
<p(A):=(p(A)x,x).
By Exercise 1.35, /> is unitarily equivalent to n^, the representation obtained from (p
by the GNS construction. In the commutative case, this is made even more explicit.
Theorem IL1.1 Let X be a compact metric space. Every cyclic representation
p ofC(X) is unitarily equivalent to a representation given by cr^(f) = M?, the
operator of multiplication by f on L2 (X, //), where fi is a regular Borel probability
measure on X.
Proof. Let a; be a unit cyclic vector for />, and consider the state on C(X) given
by ¥>(</) := (p(g)xix)- By the Riesz Representation Theorem for positive linear
functionals on C(X), there is a regular Borel probability measure // on X such that
<p{g)= / gdfi.
Jx
Define U from C(X) into U by Ug = p{g)x. Then calculate
\\Ug\\2 = (p(g)z,p(g)x)=(p(gyp(g)x)x)
= <p(\g\2)= f \g\2^ = \\g\\lM
j x
46

II.l. Spectral Theory
47
As C(X) is dense in L2(fij and p(C(X))x is dense in H, this operator extends by
continuity to a unitary operator of L2(/x) onto %.
For /, g in C(X), compute
/>(/)ff* = />(/)/>(</)* = />(/</)* = U(fg) = UMfg.
Again by the density of C(X) in L2(/x), we deduce that p(f) = UMfU*, and
hence that p = AdUcr^. ■
Of course, not every representation is cyclic. However, we have the next best
thing.
Proposition II.1.2 Every non-degenerate representation crofa C*-algebra 21 is
the direct sum of cyclic representations.
Proof. If x in % is a unit vector, let Hx = cr(2l)sc. Since cr(2l) has non-trivial
null space, it follows that x belongs to Hx. (See the proof of Theorem 1.7.1.) It is
evident that this subspace is invariant for cr(2l) and the restriction ax of a to Hx
is cyclic with cyclic vector x. Then we apply Zorn's Lemma to obtain a maximal
family {xa : a G A} of unit vectors with the property that %Xa are pairwise
orthogonal. It suffices to show that % = Z)©aG^^ajQ- Indeed, the right hand
side is a direct sum which forms a subspace of %. If it is proper, choose a unit
vector y orthogonal to it. For any a, and any A} B in 21,
(<r(A)y,*(B)za) = (y,*(A*B)xa) = 0.
Thus %y is orthogonal to each %Xa. This contradicts the maximality of the
family constructed above. Therefore % must be the direct sum of a family of cyclic
subspaces. ■
Applying this to the abelian case yields one version of the Spectral Theorem.
Theorem II.1.3 Every normal operator on a separable Hilbert space is unitarily
equivalent to a multiplication operator Mj on some L2(fi) space.
Proof. Let N be a normal operator with spectrum X\ and let p(f) = f{N) be
the associated representation of C(X). By the preceding proposition, we may
decompose p as a direct sum of cyclic representations. As % is separable, this is a
countable direct sum. So we may write p=Yl ©n>i Pn> Then by Theorem HA A,
there are regular Borel probability measures /xn such that pn ~ a^ and
N\<Hn = Pn(z)~M?».
Form the space Xqo = X x N of countably many disjoint copies of X, and
define a measure // on A"^ by putting the measure /xn on X x {n}. Then it is evident
that L2(fi) :_ X) ©n>i -/2(//n)- If % denotes the coordinate ftmction Z{x% n) = x,
then
Mz~J2®M?n-N- ■
n>l

48
II. Normal Operators and Abelian C*-algebras
II.2 The L°° Functional Calculus
Most formulations of the spectral theorem include more information,
specifically about a more powerful measurable functional calculus. Part of this can be
deduced immediately, but first we make some additional preparations.
A vector x is called separating for a C*-algebra 21 of operators if A in 21 and
Ax = 0 implies that A = 0. It is an elementary fact that ifx is a cyclic vector for
a C*-subalgebra 21 ofB(H), then x is a separating vector forty'. Indeed, suppose
that T is in 21' and Tx = 0. Then for every A in 21, one has TAx = ATx = 0.
But 21a; is dense in H, whence T = 0.
Proposition II.2.1 Let 21 be an abelian subalgebra ofB(H) for a separable Hil-
bert space %. Then 21' has a cyclic vector; whence 21" has a separating vector.
Proof. By Proposition n.1.2, H = Z)©n>1 HXn is the sum of cyclic subspaces
for 21'. The subspace HXn is invariant for 21'. Thus the projection Pn onto HXn
commutes with 21', and so lies in 21". By the Double Commutant Theorem 1.7.1,
21" is the weak operator topology closure of 21; and thus is abelian. Hence 21' = 21'"
contains 21". Let x = J2n>i 2"najn. Then 21's contains W2nPnx = Wxn = UXn
for all n > 1. So x is cyclic for 2l;. ■
Say that a representation p of C(X) is multiplicity free if p(C(X))f is abelian.
An abelian subalgebra of B{%) is called maximal abelian if it is not contained in
any larger abelian subalgebra. A maximal abelian von Neumann algebra is often
abbreviated as a masa, which stands for maximal abelian self-adjoint algebra.
Theorem II.2.2 Let p be a representation ofC(X)ona separable Hilbert space
%. Then the following are equivalent.
(i) p(C(X)) has a cyclic vector.
(ii) p is multiplicity free.
(iii) p(C(X)Y is maximal abelian.
(iv) p(C(X))ff is unitarily equivalent to the algebra L°°(fi) acting by
multiplication on L2 (//) for some regular Borel probability measure fionX.
Proof. Assume that (i) holds, and apply Theorem n. 1.1 to show that p = k&U a^
for a unitary operator U and a Borel probability measure // on X. So for
convenience, we may suppose that p = cr^. Suppose that T commutes with p(C(X)).
Let f = Tl where 1 represents the constant function 1 in L2(p). Then for g in
C(X)9
Tg = TM£l = M£Tl = M»f = fg.
Moreover, since ||/flf||2 = \\Tg\\2 < \\T\\ \\g\\2 for all g G C{X) which is dense in
L2(/x), it follows that ||/H*, < ||T||. Consequently, T = Mf is a multiplication
operator by a bounded measurable function / in C°°(p). Conversely, every such
multiplication operator commutes with p(C(X)). Thus p(C(X)Y = L°°(fi) is
abelian; whence (ii) p is multiplicity free and (iv) holds.

n.2. The L°° Functional Calculus 49
When p(C(X)) is abelian, we have
rWOT
p(C(X)Y = />(C(X)) C P(C(X))'.
But (ii) implies that p{C(X))f C p(C{X))". Thus p(C(X))" is equal to its own
commutant, so it is maximal abelian. Hence (ii) implies (iii).
Now (iii) together with Proposition n.2.1 shows that p(C(X)Y has a cyclic
vector x. But
p(C(X))x = p(C(X)y>x = H,
so p is cyclic. Hence (iii) implies (i).
Finally if (iv) holds, then by Theorem n.1.1, p is cyclic; and therefore (iv)
implies (i). ■
Of course, most representations of C(X) are not cyclic, but the general
structure can be obtained by applying Proposition n. 1.2. In order to get the full power of
the result, we need a couple of preparatory lemmas. Recall that Lx(p)* = L°°(/z),
and that this pairing determines the weak-* topology on L°°(/z).
Lemma II.2.3 Let pbea Borelprobability measure on a compact metric space X.
Suppose that fa is a net of functions in L°°(p) converging in the weak-* topology
to /. Then M? converges in the weak operator topology to M£\
Proof. Fix vectors x, y in L2(p). Then h= xy belongs to L1(/x). Hence
lim(Mil x,y) = lim / fahdp = lim / fhdp = lim(Mjla:,y). ■
a Ja a. J a J a *
Lemma II.2.4 Let p and v be regular Borel probability measures on a compact
metric space X. Suppose that a is a ^-isomorphism of L°°{p) onto L°°(u) which
is the identity onC(X). Then p and v are equivalent measures (so that L°°(p) =
L°°(v)), and a is the identity map.
Proof. Since a is a *-isomorphism, it preserves order; and therefore it preserves
suprema. Let O be an open subset of X. Then Xo is the supremum of an increasing
sequence of continuous functions. For example, let
-J*
[2*
f(c) = r if dist(^c) > 2~*
nK<i) |2fcdist(£,0c) if dist(£,0c)<2-fc
Hence
<r(Xo) = sup a(fk) = sup fk = Xo-
k>l fc>l
If <r(XE) = XE and <t(Xf) = Xf, then
<t(Xe') = o-(l - Xe) = 1 - ct(Xe) = 1-XE = Xe',

50
II. Normal Operators and Abelian C*-algebras
and
<r{XEnF) = <r{XEXF) = <r(XE)<r(XF) = XEXF = XEnF.
The result for unions follows from these two relations. Finally, if <r(XEk) = XEk
for k > 1, then
a(xVk>iBk) = (7(suPxU2=i^) = suP*(xU2=1is*)
- n>l n>l
= 8UpXU»_lH»=XU J,.
n>l -
This shows that the set {J57: <r(XE) = X#} is a sigma algebra containing the open
sets. So it contains all Borel sets, which yields all measurable sets up to subsets
which are measure zero relative to // + v. In particular, fi(E) = 0 if and only if
XE = 0 in L°°(/x); which is therefore if and only if XE = cr(XE) = 0 in L°°{y)\
which is if and only if u(E) = 0. So \i and v are equivalent measures. Finally, a is
the identity on the span of the characteristic functions. As this space is norm-dense
in L°° (//), it follows that a is the identity map. ■
With these two measure theoretic lemmas in hand, we can prove the main result
of this section.
Theorem II.2.5 Let p be a representation ofC(X)ona separable Hilbert space
%; and let 9Jt = p{C(X))". Then there is a regular Borel measure fionX so that
9JI is ^-isomorphic to L°°(/x). Moreover, this map is a homeomorphismfrom the
weak operator topology on QJt to the weak-* topology on L°°(/x).
Proof. By Proposition H.2.1, QJt has a separating vector x. Let K be the cyclic
subspace generated by x. Consider the restriction map n(A) = A|/c for A in
9JI. Clearly, this is a *-homomorphism. Since a; is a separating vector, n is an
isomorphism. Moreover, this map is obviously continuous in the weak operator
topology.
Now 7r(9Jl) has a cyclic vector x, whence by Theorem n.2.2, there is a regular
Borel probability measure v on X so that QJt is unitarily equivalent to L°° (u) acting
on L2(v). Note that the weak operator topology on L°°(u) coincides with the
weak-* topology it inherits as the dual of L1 (u). Indeed, if hi and h2 are functions
in L2(v), then the WOT-continuous linear functional
<p(f) = (M?huh2) = f f(hih)du
is integration against the L1 (u) function h = hih2, and therefore is weak-*
continuous. Conversely, if ft is in L1^), then we set ft2 = |ft|ly^2 and fti = ft/ft2. Clearly
11ft;11| = ||ft||i; so they lie in L2(v). Thus the weak-* continuous functional
^(/)= f fhdp= f f{hjr2)dv = (Mufhuh2)
J X J X
is WOT-continuous.

II.2. The L°° Functional Calculus
51
Now turn the tables and work from the other direction. Apply Proposition II. 1.2
to p to decompose it as a direct sum p = ^ ®pn of at most countably many cyclic
representations pn. By Theorem n.1.1, there are Borel probability measures fin on
X so that pn is unitarily equivalent to a^. Let // = ]T)n 2~nfinj which is a measure
equivalent to the family {/xn}. For each bounded Borel function / on X, choose a
bounded sequence /*. in C(X) converging to / a.e. dfi (which is possible by Ego-
rofFs Theorem). Then /*. converges to / in the weak-* topology on L°°(/x) by the
Lebesgue Dominated Convergence Theorem; and thus also converges weak-* to /
in each L°°(/xn). So by Lemma n.2.3,
^2®Mfn = WOT-limJ]eM£n
n n
belongs to 9JT. Moreover, the same lemma now shows that the *-homomorphism
/> from L°°(n) to 9JT given by p(f) = ]T)n ©M^n is continuous from the weak-*
topology on L°°(n) to the weak operator topology on 9JT.
The map p is injective. Indeed, if p(f) = 0 for a non-zero function / in
L°°(/x), then there is a function g in L°°(n) such that fg = Xe for a set E of
positive measure. But then
J2®MpB = HxE) = p(f)p(9) = o.
n
This implies that iin(E) = 0 for all n; whence n(E) = 0, contrary to hypothesis.
Consequently, p is a *-isomorphism. It carries C(X) onto p(C(X)). Since it is
weakly continuous, it carries L°°(/x) onto 9JT.
Finally, put the two pieces together. The map npis a *-isomorphism of L°° (//)
onto L°°(v) which is the identity on C(X). Hence by Lemma II.2.4, this is the
identity map. Consequently, both p and n = p"1 are continuous between L°°(/x)
endowed with the weak-* topology and DJl with the WOT topology. Therefore p is
a homeomorphism. ■
Applying this to a single normal operator yields the so called L°° functional
calculus.
Corollary II.2.6 If N is a normal operator on a separable Hilbert space, then
there is a regular Borel measure fi on cr(N) and a ^-isomorphism <pn of L°°(n)
onto W*(iV) which is a homeomorphism from the weak-* topology on L°°(fi) to
the weak operator topology on W*(iV).
This yields the spectral measure of a normal operator as follows. Define a
projection valued measure En on Borel subsets X of C by En(X) := (Pn(Xx)-
This assigns a projection to each Borel subset of the plane. Since y># is a homo-
morphism, we obtain the identities
EN(X DY) = EN(X)EX(Y)

52
II. Normal Operators and Abelian C*-algebras
and
EN(X U Y) = EN(X) + EN{Y) - EN{X n Y).
Countable additivity is a consequence of the fact that y># is a homeomorphism
from the weak-* topology on L°°(/x) to the weak operator topology on W*(N).
Indeed, suppose that Xn are pairwise disjoint Borel sets; and let X = Un>i -^n-
Then Z)fc=i *** converges weak-* to Xx and thus X)fc=i -^N(-^fc) converges in
the weak operator topology to En(X). As the limit is a projection, it follows that
this converges in the strong operator topology as well. Hence we obtain that
SOT-J2EN(Xk) = EN(X).
We also observe that the measure fi is determined only up to its measure class
of equivalent measures. Indeed, two measures have the same non-zero
characteristic functions if and only if they have the same null sets which occurs if and
only if they are equivalent measures. On the other hand, if fi and v are
equivalent measures, then the Radon-Nikodym Theorem says that dv = hdfi where
h > 0 a.e. dp. Hence we may define a unitary operator from L2(v) onto L2(/x) by
Uf = h>l2f. Indeed,
\W\\hM = J \f\2h d„ = J |/|2 dv = \\f\\l^y
Moreover,
M2Uf = gfh1'2 = UMZf;
whence Ad U is a unitary equivalence between the two cyclic representations.
Therefore we have proven:
Corollary IL2.7 Two cyclic representations ofC(X) are unitarily equivalent if
and only if the corresponding measures are equivalent
A simple device allows us to obtain the same result for arbitrary separably
acting abelian von Neumann algebras by showing that they are singly generated,
and thus arise as a representation of some C(-X").
Lemma II.2.8 Let QJt be an abelian von Neumann algebra on a separable Hilbert
space. Then there is a self-adjoint operator A such that 9JT = W*(A). Moreover,
if 21 is a separable C*-subalgebra of9Jl, it can be arranged that 21 C C*(A).
Proof. The unit ball of B(H) in the weak operator topology is metrizable by
Proposition 1.6.3 and compact. Hence there is a countable family of self-adjoint operators
which are WOT-dense in the unit ball of 9Jl,a. Include in this family a countable
dense subset of 2l,a. Each self-adjoint operator is in the norm-closed span of its
spectral projections corresponding to diadic intervals. Consequently, there is a
countable collection {En} of projections in QJt that spans a C*-algebra <£ which
contains 21 and is WOT-dense in 9JT.

113. Multiplicity Theory 53
Let A = J2n>i 3~n£!n. We claim that C*(A) = <£. Since A belongs to <£, it
suffices to show that each En belongs to C*(A). To this end, note that
0 < EtA = Et Y, 3""nj5?n < l^i1
n>2
and similarly
|J57i < Ex A < \EX.
Thus it follows that the spectrum <r(A) is contained in [0, |] U [|, |]. So C*(i4)
contains the projection £U([§> |]) = E\. After subtracting off \E\ from A, we
may likewise show that E2 belongs to C*(A); and recursively we can establish that
C*(A) = <£. Thus W*(A) = £W0T = 9Jt. ■
Corollary II.2.9 Lef 9JI &e an abelian von Neumann algebra on a separable
Hubert space. Then there is a compact subset X ofK and a regular Borel probability
measure on X such that 9JI is ^-isomorphic and WOT-homeomorphic to L°°(/z).
Proof. By Lemma II.2.8, there is a self-adjoint operator A such that QJt = W*(A).
Let X = <r(A). Then the L°° functional calculus for A of Corollary n.2.6 yields
the measure // on X and the desired *-isomorphism. ■
II.3 Multiplicity Theory
In the previous section, we identified representations which are multiplicity
free, or multiplicity one; and showed that every representation always has a cyclic
subspace such that the restriction to that subspace is a *-isomorphism. The question
remains about how a general abelian von Neumann algebra is structured. If 9JI acts
as L°°(n) on U = L2(/x), form the algebra 9Jt<n) acting on the direct sum U^ of
n copies of % by
M}n)(fti e • • • e hn) = fhx © • • • e fK\
where My is the direct sum M/ © • • • © M/ of n copies of M/. We justify saying
that this algebra has uniform multiplicity n by the following lemma.
Lemma II.3.1 Let 9JI act as L°°(fj) onU = L2(fj). The commutant of VJl^ is
Mn(%ft)> the algebra ofn x n matrices with coefficients in L°°(n) acting on VSn\
Proof. It is evident that every n x n matrix with coefficients in L°° (//) commutes
with 9#(n). Conversely, if A commutes with 93t(n), write it as an n x n matrix
[Aij] where A^ = P^Alty maps the j-th copy of U into the i-th copy. A routine
matrix calculation shows that each A^ must commute with 9JT; and hence belongs
to QJt because it is maximal abelian. ■

54
II. Normal Operators and Abelian C*-algebras
In a more coordinate free way, we say that an abelian von Neumann algebra
91 has uniform multiplicity n provided that 91' is unitarily equivalent to .Mn(9Jt)
for some maximal abelian algebra 9JI. This means that 91' contains a system of
matrix units {Ey : 1 < i}j < n} such that EiiW\EiiH is maximal abelian.
This makes sense even when n = N0 if we replace Mn{^l) by the algebra of all
bounded operators on H(°°) such that the matrix coefficients A# lie in 9JL By
Theorem IL2.2, this general situation reduces to the case we have just considered.
However, there is one important subtlety:
Lemma II.3.2 If 91 has uniform multiplicity, then the multiplicity is well defined.
Proof. Suppose that there are cardinal numbers m <n such that 91 has both
uniform multiplicity m and n. Hence 91' is unitarily equivalent to Mm(Wl) and to
Mn(9tt) for two masas 9JI and 9JI. Thus there is a unital homomorphism of Mn
into Mm defined as follows: take the unital imbedding j of Mn into Mn(%R)
by sending a matrix (or operator when n = oo) to the corresponding matrix with
scalar entries. Follow that with the identification of Mn(3)t) with 91' and hence
with .Mm (9JT). Then follow this with evaluation at any multiplicative linear
functional cp of 9JT. This determines a unital *-homomorphism y>(m) of Mm{%K) onto
Mm- The composition is a unital *-homomorphism of Mn into Mm- This is only
possible when n = m. (Recall that Mn is a simple algebra, and B(H) has only the
Calkin algebra as a possible quotient. So by dimension alone we may distinguish
these algebras.) ■
So now we may try to decompose an arbitrary abelian von Neumann algebra
into parts of different multiplicities. To this end, say that a projection P in an
abelian von Neumann algebra 91 has uniform multiplicity n if y\\P% has uniform
multiplicity n.
Lemma II.3.3 Let 91 be an abelian von Neumann algebra acting on a separable
Hilbert space. The supremum of all projections in 91 of uniform multiplicity n also
has uniform multiplicity n.
Proof. By Corollary II.2.9, 91 is *-isomorphic to L°°(/x). Thus every projection
P in 91 corresponds to multiplication by a characteristic function Xx under this
isomorphism. Thus if P has uniform multiplicity n, then {^fl\PH)f is isomorphic
toMn(L~(ti\x)).
Now notice that if P has uniform multiplicity n, then so does every subprojec-
tion of P in 91. For any P' < P corresponds to Xx1 for some measurable subset
X1 C X; and thus (%P"H)' is isomorphic to M^L00^*))-
Suppose that Pa are projections of uniform multiplicity n corresponding to
subsets Xa. Then the supremum of the Pa is the supremum of a countable subset.
(Equivalently, there is a countable collection of the measurable sets Xa whose
union contains all the others modulo null sets.) Denote this set by {P*. : k > 1}.
Replace P*. by P'k = Pfc(Vi<j<fc Pj)1 so that they are pairwise orthogonal with

113. Multiplicity Theory
55
the same supremum. By the previous paragraph, each Pk has multiplicity n and
corresponds to a characteristic function Xx> - The sets Xk are essentially disjoint;
so we may assume that they are actually disjoint.
Then let P = J2k>i Pk- Th*s *s ^e supremum of the Pa's, and corresponds to
the characteristic function Xx where X = \Jk>x Xk. Because 9t' commutes with
each P'k> the restriction 9t'|PH is unitarily equivalent to the direct sum
Hence P has uniform multiplicity n. ■
Lemma II.3.4 Let 9t be an abelian von Neumann algebra acting on a separable
Hilbert space. Then there is a non-zero projection P in 9t of uniform multiplicity.
Proof. Recursively apply Proposition EL2.1 to choose unit vectors xn and
projections Qn in 9t' onto the cyclic subspace span{A«n : A € 91} such that xn is a
separating vector for OTlCCSTi1 Qij^W- Let Qn denote the smallest projection in
9t such that Qn < Qn. This is known as the central cover of Qn. Since xn is
always chosen to be a separating vector for the compression algebra, it follows that
Qi = I and Qn > Qn+i for all n > 1.
There are two cases to deal with. Either (i) there is an integer no such that
Q^= I > 5^,or(ii)Q^=/foralln> 1.
In the first case, let P = Qno+i » and consider the algebra 9t| PH. Then
Pzno+i = 0. Since x^+i is a separating vector for 9t|(]Cr=i Qi)L^ ** follows
that P(E?=i Qi)"1 = 0;_whence P^< £?=i Q*- Let ^« = pQi for 1 < i < n0.
Then it is easy to see that En = PQi = P. Moreover Pxi is a cyclic and separating
vector for yi\EiiH. If P corresponds to the characteristic function Xx, then we
see that ^l\EuH is unitarily equivalent to 9JT = L°°{p\x) acting on L2(p\x).
Hence Vl\PH is unitarily equivalent to Qjtf"0). So by Lemma II.3.1, P has uniform
multiplicity n0.
The second case is similar, though a bit more complicated. We start over,
and use Zorn's Lemma to construct a maximal family of separating vectors xn
for 9t such that the cyclic subspaces Qn onto span{Asn : A € 0T} are pairwise
orthogonal. The central cover R of R = (Sn>i Qn)1 must have R < I. For
otherwise by Proposition II.2.1, there is a separating vector y for yi\RH which
would be separating for 9t contradicting the maximaJity of the family {xn}. Let
P = RL.
As before, let 2?tt- = PQ{. The restriction 9112^% has a cyclic and
separating vector Pxi> and thus is maximal abelian. As above, each is isomorphic to
9JT = L°°(p\x) acting on L2(p\x) for i > 1. Consequently, Vl\PH is unitarily
equivalent to 3Jt(°°). Hence by Lemma II.3.1, P has uniform multiplicity N0- ■

56
II. Normal Operators and Abelian C*-algebras
We now have all the ingredients to describe the structure of a general abelian
von Neumann algebra.
Theorem II.3.5 Let VI be an abelian von Neumann algebra acting on a separable
Hilbert space. Then there is a uniquely defined family {Pn : 1 < n < N0} of
projections in VI which are pairwise orthogonal, partition the identity (that is,
J2n P* = I) and such that Pn has uniform multiplicity nfor each n.
If we identify VI with L°°(p) via the *-isomorphism of Corollary II.2.9, then
there are mutually singular measures p,n « p, such that VI is unitarily equivalent
to
oo
n=l
acting on
oo
w = £ei;Wn)ei2(MK.)(Ho)-
n=l
Proof. For each cardinal n, 1 < n < No> let Pn be the supremum of all projections
in VI of uniform multiplicity n. By Lemma II.3.3, each Pn has uniform multiplicity
n and is the largest projection with this property.
For m/n,we must have Pm orthogonal to Pm because the projection PmPn
has uniform multiplicity both m and n, which would contradict Lemma II.3.2
unless PmPn = 0. Let Q = (X)n Pn)*-; and set 9t0 = Vl\QU. If Q ^ 0, we may
find a non-zero subprojection Qf < Q of uniform multiplicity by Lemma II.3.4.
This contradicts the maximality of one of the Pn. Hence Q = 0; and therefore
{Pn : 1 < n < No} forms a partition of the identity.
Each Pn is identified with a characteristic function Xxn in L°°(p). The
orthogonality implies that Xn are essentially disjoint. Hence the measures \in — p\xn
are mutually singular. And the fact that Pn partition the identity shows that the
union UnXn = X is the whole space up to a null set. This yields the desired
decomposition of VI. ■
Applying this to the von Neumann algebra generated by a single normal
operator yields a much stronger version of the Spectral Theorem II.L3. The spectral
measure E^ given by the L°° functional calculus is now seen to be determined by
the algebra L°°(p) and the multiplicity function mjy defined by m^{x) = k for
x G Xk, 1 < k < No- This function is determined almost everywhere dp, because
of the uniqueness of the decomposition. The measure p. is determined up to its
measure class of all equivalent measures by Corollary II.2.7.
This immediately yields complete unitary invariants for normal operators.
Corollary II.3.6 Two normal operators M and N acting on separable Hilbert
spaces are unitarily equivalent if and only if they have equivalent spectral measures
and multiplicity functions.

II.4. The Weyl-von Neumann-Berg Theorem
57
II.4 The Weyl-von Neumann-Berg Theorem
There are other weaker notions of equivalence that are important. Say that two
operators A and B are approximately unitarily equivalent (write A ~a B) if
there is a sequence of unitary operators Un such that B = limn-^ UnAU*. It is
evident that two operators are approximately unitarily equivalent if and only if they
have the same norm-closed unitary orbit
U{A) = {UAU*:U unitary}.
Two operators with the same closed unitary orbit have the same observable data in
the sense that no finite set of measurements determined by vectors can distinguish
the two operators. It will be convenient to simultaneously consider several related
notions. Two operators A and B are approximately unitarily equivalent relative
to £ (write A ~& B) if in addition to having B = lirrin^oo UnAU*> one also has
that B - UnAU* belongs to the ideal £ of compact operators for all n. Evidently,
this second notion is stronger. However, we shall see that the two relations are, in
fact, equivalent.
Analogously, we say that two representations a and p of a separable C*-
algebra 21 are approximately unitarily equivalent (relative to £) if there is a
sequence of unitary operators such that
p(A) = lim Ad U alA) for all A e 21
n-j>oo
(and in addition the range of p - Ad U a is contained in the ideal £ for all n > 1).
Again we write p ~a & and p ~# a respectively.
There is another apparently much weaker relation in the same vein, we will say
that two representations p and a are weak approximately unitarily equivalent
(p ~wa o) if there there are sequences Un and Vn of unitary operators such that
a{A) = WOT-lim Unp(A)U* and p(A) = WOT-lim Vna(A)V*
n—j>oo n—j>oo
for all A in 21. Both directions are needed to obtain an equivalence relation. This
notion readily implies a stronger version of itself with convergence in the strong-*
topology. Indeed,
\\{a(A) - UnP{A)UZ)m\\* = ((<r(A) - Unp(A)UX)x,a(A)x)
+ ((<r(A*) - Unp(A*)K)<r(A)z,z)
-({<r(A*A)-UnP(A*A)U;)x,x).
This demonstrates convergence in the strong operator topology for both A and A*,
which is strong-* convergence.
The corresponding notion for a single operator using the weak operator
topology does not yield a satisfactory relation (Exercise 11.10). However, the strong-*
topology does yield a useful notion. So to have parallel notation, we say that two

58
II. Normal Operators and Abelian C*-aIgebras
operators A and B are weak approximately unitarily equivalent if there there
are sequences Un and Vn of unitary operators such that
B = sOT*-limtfnACf* and A = scrr*-]imVnBVZ.
This relation is sufficient to guarantee that there is a *-isomorphism p from C*(A)
onto C*(JB) such that p(A) = B and p ~wa id. To see this, first notice that if
UnXiU* converges SOT to Y{ for i = 1,2, then
(UnX1X2U* - YxY2)x = {UnXiK " Y!)Y2z + Y1(UnX2UZ - Y2)x
+ (UnXxUZ - Y1)(UnX2UX - Y2)x.
Whence UnXiX2U^ converges SOT to Y{Y2. Consequently, if p is any polynomial
in non-commuting variables, it follows that
SOT-limUnP(A, A*)UZ = p(B, B*)
and
S0T-limynp(5, B*)V* = p(A, A*).
Thus the map p(p(A, A*)) = p(B, JB*) is an isometric *-isomorphism, and extends
to their C*-algebras. Evidently, we have shown that p ~wa idc*(A)-
The following theorem shows that normal operators are diagonal plus compact.
This will enable us to show that every normal operator is approximately unitarily
equivalent to a diagonal operator, which will facilitate their classification.
Theorem II.4.1 Suppose that 21 is a separable abelian C*-subalgebra ofB(H).
Then there is an orthonormal basis {e* : k > 1} for H so that 21 is contained in
V + &, where V is the algebra of diagonal operators with respect to this basis.
In addition, one may arrange that any given finite collection of projections in 21"
belongs to D.
Proof. By Lemma II.2.8, 21 is contained in a C*-algebra <£ = span(£) generated
by a countable, commuting family £ = {En, n > 1} of projections. If there are
finitely many projections given, we may assume that they are the first N in the list.
Construct an approximate unit of projections for A which is quasi-central for
<£ as follows. Fix an orthonormal basis »i, x2,... for H. For each projection 2%,
let us denote e\"1) := I - Ei and E^ := E{. Then for k > i\T, set
k
Ck = span{JJj57^)»i : 1 < j < fe, e{ = ±1}.
*=i
Then Ck is an increasing sequence of finite dimensional subspaces with dense
union. So the orthogonal projections Fk onto Ck increase strongly to J; whence it
is an approximate unit for A. Also notice that Ck is invariant for En when n < k9
and thus Fk commutes with these projections. Consequently,
Km FkEn - EnFk = 0 for all n > 1.
ifc-j>0O

II.4. The Weyl-von Neumann-Berg Theorem
59
This conclusion extends to the closed linear span.
Let Dn = En for 1 < n < N and Dn = En(I - Fn) for n> N. Then the
projection Dn exactly commutes with every Fk fork>N and with all the other
Z?m's. Let V be the C*-algebra generated by {Dn : n > 1}. It is immediate that
this is an abelian C*-algebra such that
21 C € C D + £.
It remains to show that V is diagonalizable. As V commutes with each Fk for
k > N, the finite dimensional subspaces Hn = £n and Hk = £>k 0 £fc-i for
k > N are all invariant for V. The restriction of *D to Hk is a commuting family
of normal matrices, which is diagonalizable by the finite dimensional spectral
theorem. Choose such a basis for each Hk- The basis for % obtained by combining
these bases for each k diagonalizes V. ■
The following corollary asserts that every normal operator is a small compact
perturbation of a diagonalizable operator. It is known as the Weyl-von Neumann-
Berg Theorem.
Corollary II.4.2 Every normal operator N on a separable Hilbert space can be
expressed as a sum N = D + K of a diagonal normal operator D and a compact
operator K. Moreover for any e > 0 and any n commuting Hermitian
operators A\,..., An9 there are simultaneously diagonal Hermitian operators D{ and
compact operators K{ such that A{ = D{ + K{ and ||ift|| < £•
Proof. We require an efficient way to construct each A{ from projections. For
convenience, let us translate each Ai by a multiple of the identity and scale them
so that 0 < Ai < I. This may require redefining e.
For each A», consider the spectral projections
fork > 1. Then
fc>l
Choose N large enough that 2""^ < e. Then apply the previous proof to the family
£ = {e£' : k > 1, 1 < i < n}. We obtain a diagonal algebra V which contains
DJp "= E^ for 1 < k < N and 1 < i < n, and for each other E^\ V contains
Djp = E%' - Rik where Rik is a finite rank projection.
Then let
fc>l

60 II. Normal Operators and Abelian C*-aIgebras
These are positive diagonal contractions in V. Moreover,
Ki := Ai -B{=J2 *~kRik-
k>N
It follows that K{ are compact and \\Ki\\ < 2"N < e. To obtain the result for a
normal operator N9 apply this result to its real and imaginary parts. ■
Pushing this a bit harder, we obtain our classification theorem of normal
operators up to approximate unitary equivalence. For convenience, we isolate a
combinatorial part of the argument for later use.
Lemma II.4.3 Suppose that X is a compact metric space. Let {& : k > 1} and
{Ck : k > 1} be two countable dense subsets ofX such that each isolated point of
X is repeated the same number of times in each sequence. Then given e > 0, there
is a permutation irofNso that dist(ffc, Cx(fc)) < efor allk>l and
lim dist(0k> Cr(ib)) = 0.
Proof. Let Xe denote the cluster set of X together with all isolated points which are
repeated infinitely often in the sequences. First pair off the (finitely many) terms in
each sequence which are isolated points that are at a distance at least e/2 from Xe.
These pairings are at zero distance from each other, and so do not affect the final
result. Then restricting our attention to the remainder, we may suppose that every
point in X is within distance e/2 of Xe.
We construct n recursively. At the fc-th stage, we will arrange that ir(j) and
n^U) are defined for 1 < j < k so that when 7r(i) is defined,
|6 - Crwl < max{dist(£,Xe) + 2~*et dist(C(i), Xe) + 2~^e}.
Indeed, if 7r(fc) is as yet undefined, choose I not yet in the range of the partially
defined function w so that
|a-C/|<dist(6,Xe) + 2-^.
This is possible since {&} are dense in Xe and isolated points of Xe are repeated
infinitely often. Then we set n(k) = £. Likewise, if k is not in the range of the
partially defined function 7r, choose an integer I which is not in the domain of n
yet so that
|6-ai<dist(a,Xe) + 2^.
Then we set n(£) = k. Proceeding in this way, the function n is eventually defined
on all of N so that it is a bijection satisfying the desired estimates. ■
Theorem II.4.4 Suppose that M and N are normal operators on separable Hil-
bert spaces. Then M and N are approximately unitarily equivalent (relative to £)
if and only if
(i) *e(M) = *e(N), and
(ii) null(M - XI) = null(i\T - XI) for all XinC\ ae(M)

II.4. The Weyl-von Neumann-Berg Theorem
61
Proof. Suppose that M and N are approximately unitarily equivalent normal
operators. If Un are unitary operators such that N = lim^oo UnMU*9 then it is
evident that X := a(N) = a(M) and that
f(N)=VbtooUnf(M)UZ for all feC(X).
In particular, if / = X{xy is the characteristic function of an isolated point of X,
then
EN({x}) = X{x}(N)= lim UnX{x}(M)U:= lim UnEM({x})K.
Hence it follows that these two projections have the same rank. This establishes
(ii), and shows that M and N have the same isolated points in their essential
spectra. Since every cluster point of a(M) = cr(N) is in the essential spectrum, (i)
follows.
For the converse, we first show that the result is true when both M and N are
diagonalizable. In this case, we may write M = diag(^) with respect to a basis
{ek} and N = diag(^) with respect to a basis {/*} where {^} and {uk} are
dense subsets of X = a(M) = cr(N). Moreover, any isolated point of X must be
repeated according to the multiplicity of the eigenvalue, which is the same for M
and N by (i) and (ii). Let Xe = ae(M).
Given any e > 0, Lemma II.4.3 produces a permutation it of positive integers
so that \pk - *V(fc) \ < sforallk and
£m l«k-**(*)! = 0.
Thus the unitary operator given by Uek = U(k) satisfies
M-irNU = du*(ph-vw(k))t
which is compact and has norm \\M - U*NU\\ < e. As e was arbitrary, it follows
that M and N are approximately unitarily equivalent relative to £.
Now turn to the general case. It suffices to show that every normal
operator M is approximately unitarily equivalent to a diagonal operator relative to £.
Now M may be decomposed as M = M0 ® M7 where M0 is the summand of M
corresponding to all isolated eigenvalues of finite multiplicity. The remaining
summand M' is normal and a(M') is contained (possibly properly) in Xe = ae(M).
By Corollary n.4.2, for each positive integer fc, there is a diagonalizable operator
Dk such that M' - Dk is compact and \\M' - Dk\\ < l/2k. In particular, the
eigenvalues of Dk are at a distance of at most l/2k from Xe and asymptotically,
they approach Xe. So there is a diagonal compact perturbation Dfk of Dk so that
a(D'k) c Xe and \\M' - D'k\\ < 1/Jfe. Let Ek = M0 0 D'k. Each Ek is
diagonalizable with the same isolated eigenvalues with the same multiplicities as M and
essential spectrum Xe. So by the diagonal case, Ek ~& Ei for all k > 1. But
M - Ek is compact and lim^oo \\M - Ek\\ = 0. Hence it follows that M ~# 2?i.

62
II. Normal Operators and Abelian C*-aIgebras
The same holds for iV, and thus M and N are approximately unitarily
equivalent relative to £. ■
Looking at the Weyl-von Neumann-Berg Theorem in a slightly different way,
we obtain the corresponding result for representations of C(X). Say that a
representation of C(X) is a diagonal representation if the range is diagonalizable.
Corollary II.4.5 Let X be a compact metric space. Then every representation
ofC(X)ona separable Hilbert space is approximately unitarily equivalent to a
diagonal representation relative to £.
Proof. Let p be a representation of C(X). Let Pn be an enumeration of all minimal
finite rank projections in p{C(X))"\ and let P be the sum of these projections.
Then p0 = Pp\P% is easily seen to be diagonalizable. Thus we may restrict our
attention to p = PLp\PL%. This has the property that p(f) is never a non-zero
compact operator. Indeed, if p(f) = K ^ 0 is compact, then p(C(X)) contains
C* (K) which contains a finite rank projection E of minimal rank. Since P belongs
to p(C(X))", it follows that K1 = PLp(f)PL is a non-zero compact operator in
p(C(X)Y unitarily equivalent to K © 0. Hence there is a finite rank projection E'
in p(C(X)Y orthogonal to P, contrary to the definition of P.
By Lemma n.2.8, there is a positive operator A such that
p(C(X)) C V(A) C p(CpO)" = W*(A).
Note that ae{A) = cr(A). Indeed, if A were an isolated eigenvalue of A, then
the corresponding spectral projection would be a finite rank projection in C*(-A),
which is contained in p(C(X))" contradicting the construction of P. By the Weyl-
von Neumann-Berg Theorem, A is approximately unitarily equivalent relative to
A to a diagonalizable operator B with
a(B) = ae(B) = ae(A) = a(A).
Thus by Theorem II.4.4, the *-representation of C*(A) given by
r(f(A)) = f(B) for feC(<r(A))
is approximately unitarily equivalent to the identity representation relative to £.
Therefore rp is approximately unitarily equivalent to p relative to £. So po © rp
is a diagonal representation of C(X) which is approximately unitarily equivalent
relative to £ to p. ■
Now we can complete the picture for approximate unitary equivalence of
representations of C(X). The last condition (iv) should be seen as measuring the
support of the representation and the rank of (characteristic functions of) isolated
points in this support.

II.4. The Weyt-von Neumann-Berg Theorem
63
Theorem II.4.6 Let X be a compact metric space. Suppose that p and a are
separable representations ofC(X). Then the following are equivalent:
(i) p ~& a;
(ii) p ~a a;
(iii) p ~wa a;
(iv) rankp(/) = rank(j(/)/ora/// G C(X).
Proof. The implications (i) implies (ii) implies (iii) are trivial, and (iii) implies
(iv) follows because rank is lower semi-continuous in the WOT topology. (To see
this, note that the set of operators of rank at most k is WOT-closed.) So assume
that (iv) holds. By Corollary n.4.5, we may replace p and a by diagonalizable
representations which are approximately equivalent relative to £ to p and a. Thus
it suffices to prove (iv) implies (i) in this special case. Clearly condition (iv) implies
that ker p = ker a. By Exercise 1.14, there is a closed subset E of X so that ideal
ker p has the form
V(E) = {/ € C(X): f\E = 0}.
Thus both p and a factor through the restriction to C(E). Without loss of
generality, we may suppose that E = X and that p and a are injective.
There are sequences {fn : n > 1} and {£n : n > 1} in X such that
p(f) = diag(/(£n)) and <r(/) = diag(/«n))
with respect to orthonormal bases {en} for %p and {/n} for %a. The condition
ker p = ker a = {0} is equivalent to the condition
{£n : n > 1} = {(n : n > 1} = X.
For each isolated point {tj} of X, consider the characteristic function / = X^.
Condition (iv) guarantees that the spectral projections p(f) and a(f) have the same
rank. Hence the isolated points of X are repeated the same number of times in the
two sequences {fn} and {Cn}- The remainder of the proof follows the proof of
Theorem n.4.4. By Lemma II.4.3, given any positive integer n, there is a
permutation 7rn of N such that
maxdist(&, („n{k)) < 1/n and Jur^dist(&, („n{k)) = 0.
Let Un be the unitary operator that implements this permutation between the bases
for Up and «,. For any / in C(X),
P(f) - Un*(f)K = diag(/(6) - f{C*nW)).
The uniform continuity of / shows that this is a compact operator and that it has
small norm for sufficiently large n. Thus Un implement an approximate unitary
equivalence relative to £. ■

64
II. Normal Operators and Abelian C*-aIgebras
II.5 Voiculescu's Theorem
In this section, we will prove a generalization of the Weyl-von Neumann
Theorem valid for arbitrary separable C*-algebras. While this has nothing directly to
do with the title of this chapter, it is an important direct generalization of the results
in the previous two sections on the Weyl-von Neumann-Berg Theorem.
The starting point is an approximation theorem for states known as Glimm's
Lemma. Using nets rather than sequences, this lemma is valid for arbitrary C*-
subalgebras of B(H).
Lemma II.5.1 Let (pbea state on a separable C*-subalgebra 21 ofB(H) with the
property that y>(2ln£) = 0. Then there is a sequence of unit vectors xn converging
weakly to 0 such that the corresponding vector states ipn(A) = (Axn, xn) on 21
converge weak-* (pointwise) to (p.
Proof. Without loss of generality, we may assume that 21 contains £ by the simple
expedient of replacing it with 21 + £ and defining (p(A + K) = <p(A), which is
well defined by Corollary 1.5.6.
Consider the set Se of all states tp on 21 for which there is an sequence of unit
vectors xn converging weakly to 0 in H such that ip(A) = lirrin^oo^AaJn, xn) for
all A in 21. First note that Se is non-empty. Indeed, take a countable dense subset
{Ai} of 2l,a and any orthonormal sequence yk. Then a standard diagonalization
argument yields a subsequence so that ^(-^t) = H*nk-*oo(Aiynk, ynk) exists for all
i > 1. By uniform continuity, it converges on all of 21. Clearly every state tp in Se
annihilates the compact operators.
Secondly, we claim that Se is convex. For suppose that tp and (p are two states
of this form corresponding to sequences Xk and yn converging weakly to 0.
Recursively choose unit vectors y'k as follows. For each k> let Pk be the projection
onto
span{AiSfc : 1 < i < k} U {y'j : 1 < j < k}.
Since lirrin^oo ll-Pfcynll = 0, there is an integer n*. so large that ||jPfcynJ| < 2~"fc.
Now a small perturbation will yield a unit vector y'k near ynk orthogonal to the
range of P&. Then {yfk} is orthonormal by construction. Since it is asymptotic to
ynu, it determines the same state (p. Let z* = [xk + y'k)/\/2. This is a sequence of
unit vectors converging weakly to 0. Moreover, when k > i,
(AiZfc, zk) = \{{AiXk, xk) + (Aixk, y'k) + (y'k, AiXk) + (Aiy'k, y'k))
= ±(AiXk,xk) + ±(Aiy'k,y'k)
which converges to \ (^(^t) + ¥>(^t))- Hence |(V> + <P) belongs to Se.
Another diagonalization argument shows that Se is closed in the weak-*
topology. So suppose that (p is not in Se. Then by the Hahn-Banach Theorem, there is

II.5. Voiculescu's Theorem
65
a self-adjoint operator A in 21 such that
<p(A)tWe(A):=W(A):ii>eSe}. (1)
Because We(A) is the continuous image of a compact convex set by a linear map,
it is a closed convex set. Since A is Hermitian, We(A) is a subset of R.
Let the convex hull of <re(A) be the interval [a, 6]. It will be shown that We(A)
equals [a, 6] also. Replace A by f(A) where / is the function a V x A 6. Since
irf(A) = f(nA) = n(A), it follows that f(A) - A is compact. Therefore
<p(f(A)) = <p(A) and We(f{A)) = We(A). We have arranged that al < A < bl,
so that a < <p(A) < b and a < (Ax, x) < b for every unit vector x. Hence We(^)
is contained in [a, 6].
The spectral projection J5?^(6 - £, 6] is infinite rank for all n > 0. If xn
are orthogonal unit vectors in J5?^(6 - £, b]H, then (i4ajn> »n) > b - £. Use a
diagonalization argument to obtain a subsequence which determines an element V>
in Se. Then V>(^) > b. Similarly, there is state ip' in Se so that ip'(A) < a. Hence
We(.A) contains [a, 6]. This establishes the claim. However, (p(A) belongs to [a, 6]
as well, contradicting (1). ■
This lemma can be strengthened to apply to certain finite dimensional maps.
A positive linear map between C*-algebras is just a linear map which takes
positive operators to positive operators. Hence it is a map which preserves the order
structure. If (p is a map between C*-algebras 21 and 25, it induces a map (p^ from
A*n(2l)intoA*n(»)by
<P{nH[Aij]) = MAij)].
Say that (p is n-positive if <p(n) is positive and completely positive is it is n-positive
for all n > 1. These notions will be explored in more detail in Chapter IX. For
the moment, we notice that if p is a *-representation of 21 into B(H) and X is a
bounded operator from K into W> then (p(A) = X*p(A)X is completely positive.
Indeed, p^ is a ^representation, and therefore is positive. Then a calculation
shows that
^(n)(M) = ^(n)V(n)(M)^(n)
which is positive when [Aij] is positive.
Lemma II.5.2 Let (p be a unital completely positive map of a separable C*-sub-
algebra 21 ofB(H) into Mn such that y>(2l fl R) = 0. Then there is a sequence V*.
ofisometries ofC1 into H such that WOT-limfc._j.oo V*. = 0 and
lim \\<p(A) - V£AVk\\ = 0 for all A e A.
k-^oo

66 II. Normal Operators and Abelian C*-aIgebras
Proof. Notice that y>(n) is a positive map of Mn($l) into Mn(Mn). Fix an or-
thonormal basis y», 1 < i < n, for C1 and consider the state
*([4il) = i X>(4ite.w) = (y(n)(M)y,y)
where y = (yi, y2,..., VnY/y/n. By the previous lemma, there are unit vectors
xk = (a?*,..., a;*) in WSn) converging weakly to zero such that
*(M)= lim([^]xfc,x*).
fc-*oo
Define £/fc2ft = ^/ns*. Then
&ij = (yj,Vi) = n^Eij) = lim n(Syx*,x*) = lim n(*J,*J).
fc-j>oo fc-j>oo
It follows that the set v^2^ is almost orthonormal. A routine estimate shows
that U£Uk - I has small norm. Therefore the partial isometry V*. in the polar
decomposition of Uk is an isometry; and lim^oo ||Vjk - Uk\\ = 0. Also
£m| ((<p(A) - U*kAUk)yhy) | = limJu^AEij) - n{Ax),z*)\ = 0.
Hence
lim \\<p(A) - V£AVk\\ = lim \\<p(A) - U^AUk\\ = 0.
fc-i>00 K-HX>
If P is any finite rank projection, then
0 = <p(P) = lim \\VJ!PVh\\ = lim ||P^||2.
k-¥oo k-¥oo
Thus Vk converges to 0 in the weak operator topology. ■
We wish to note a minor variant of this lemma which will be useful. Suppose
that A is a finite subset of 21, e > 0 and Af is a finite dimensional subspace of %.
Then there is an isometry V from C1 into .Af1 such that \\<p(A) - VMV'H < e for
each A in A. Indeed, for k sufficiently large, one obtains
\\<p{A)-V£AVh\\<e/2 for AeA and \\PMVk\\<e/A.
Let V be the isometry from the polar decomposition of PjjVk. A simple estimate
yields the desired inequality.
This brings us to the key technical result of this section which is a
generalization of the Weyl-von Neumann Theorem to arbitrary C*-algebras.
Theorem II.5.3 Suppose that <p is a completely positive map from a separable C*-
subalgebra 21 ofB(H) into B(K) such that y>(2lfl.£) = 0. Then there is a sequence
Vk ofisometries ofK into % such that <p(A) — Vj*AVk is compact for all k > 1
and all A in 21, and
Km\\<p(A)-V£AVk\\ = 0 for all A e 21.
k-^-oo

II.5. Voiculescu's Theorem 67
Proof. By Theorem 1.9.16 and Exercise 1.34, there is a sequential quasicentral
approximate unit for the finite rank operators on K relative to 25 = C*(y>(2l)) + &.
Thus there is an increasing sequence of finite rank positive contractions which
tends strongly to the identity and is quasicentral for 2$.
By Exercise II.8, there is a sequence 8n > 0 with the property that for all
E > 0 and ||A|| < 1 in any C*-algebra,
\\EA - AE\\ < Sn implies \\El'2A - AEl'2\\ < 2~n.
(In fact, 8n = 2""2n""1 will suffice. See Exercise II.9.) Let {A;, i > 1} be a dense
subset of the unit ball of 2l,a. By dropping to a subsequence En of the quasicentral
approximate unit, one may arrange that
MAi)En-Enip{Ai)\\<\Sn^ for l<t<n + l.
Define Fn = (En - En-i)1/2 (using E0 = 0). Then
||^t0^n-Fn^(At0||<2-"n for l<i<n.
Thus it follows that
Y^\\<p(Ai)Fn-Fn<p(Ai)\\<oo for all i > 1.
n>l
Let Pn be the (increasing) sequence of finite rank projections onto the ranges
of En. By the previous lemma and the subsequent remark, we may construct a
sequence Un of isometries from PnK into H such that Ran(C/n) is orthogonal to
Mn = span{Ran(C/i), Aj Ran(CTi), 1 < i < n - 1, 1 < j < n}
and
\\Pn<p(Ai)Pn-UZAiUn\\<2-n for 1 < i < n.
Hence
^||Pn^(^)Pn-l^ilil7n||<oo forall i > 1.
n>l
Define V = SOT-£n>1 UnFn. This is an isometry because the Un's have
orthogonal ranges, whence
oo
n=l
Notice that the orthogonality of Nn and Ran Un shows that
U*AiUm = 0 for 1 < i < max{m, n}.

68 II. Normal Operators and Abelian C*-aIgebras
Hence
<p(Ai) - V*A{V = Y,f{M)Fl - J2J2F™U^AiUnFn
n>l ro>ln>l
= J2(v(Ai)F» - FMM))Fn + Y, FmKAiUnFn
n>l mjzn
+ Y<Fn{Pn<p{Ai)Pn " KAiUn)Fn
The first and third sums are norm convergent sums of finite rank operators, and
thus are compact for all i > 1; and the second sum is finite for each At\ Thus
this sum is compact. An arbitrary element A of 21 can be approximated by a linear
combination of the iVs, from which we deduce that <p(A) - V*AV is compact for
every A in 21.
To get a sequence with the asymptotic norm property, we employ a simple
trick. Apply the result just demonstrated to the map <p(°°)(A) = <p(A)(°°) acting
on K(°°\ The resulting isometry V may be written as [Vi V2 V3 • • •]. The
compactness of (p(A)(°°) - V*AV implies that
lim\\<p(A)-Vk*AVk\\ = 0. ■
K-tCO
Corollary II.5.4 Suppose that p is a non-degenerate representation of a separable
C*-subalgebra 21 ofB(U) into B(K) such that />(2l n £) = 0. Then there is a
sequence V*. ofisometries ofK into H such that Vkp(A) — AVk is compact for all
k > 1 and all A in 21, and
lim \\Vkp(A) - AVk\\ = 0 far all A e 21.
fc-foo
Proof. Find isometries Vk using the theorem above. Then
(Vkp(A) - AVky(VkP(A) - AVk) =
p{A*){p{A) - Vk*AVk) + {p(A*) - VZA*Vk)p{A) - {p{A*A) - Vk*A*AVk).
Hence this is compact, and the norms tend to 0 as k increases. ■
We now obtain the most important corollary known as Voiculescu's Theorem.
Corollary II.5.5 Suppose that p is a non-degenerate representation of a separable
C*-subalgebra 21 ofB(U) into B(K) such that />(2l n £) = 0. Then id ~tf id 0/9.
Proof. Apply the previous corollary to p(°°) acting on K^°°\ Let V be an isometry
such that AV - Vp(A)(°°) is compact for all A in 21. Then let Jn be the canonical
injection of K onto the nth summand of KS°°}\ and let Sn be the infinite shift
operator
n-l
Sn = Yl JiJi + SOT-]T Ji+iJ?.
*=1 i>n

IL5. Voiculescu's Theorem 69
Then define unitary operators Wn from H(&K onto H by
Wn=[l- VV* + VSnV* VJn] .
A simple calculation shows that this is unitary. It has the effect of imbedding p(A)
into the range of Vn, which is made available by shifting the ranges of V(ti onto
Vi+iH for i > n.
One computes that AWn - Wn(A © p(A)) equals the 1 x 2 matrix
[a(J - VV*) -(I- VV*)A + AVSnV* - VSnV*A AVJn - V Jnp(A)].
Now
AW* - VV*A = (AV - Vp(A)W)V* + V(A*V - Vp(A*)W)*;
and because p(°°) commutes with 5n,
AVSnV* - VSnV*A =
{AV - Vp{A)W)SnV* + VSn(A*V - Vp(A*)W)*.
Let Pn denote the projection SOT-£i>n JiJ?. Since Sn-I = Pn(Sn - J), we see
that the 1,1 entry of AWn - Wn(A © p{A)) is
(AV - Vp(A)^)Pn(Sn - I)V* + V(Sn - I)Pn(A*V - Vp(A*)W)*
which is compact for all A in 21. Similarly, the 1,2 entry is
AVJn - VJnp(A) = (AV - Vp(A)W)Jn.
Moreover, since Pn and Jn tend strongly to 0, the norm of these terms converges
to 0. Hence id ~# id ©/>. ■
Let 7r denote the quotient map of B(H) onto the Calkin algebra.
Corollary II.5.6 Suppose that p\ and p2 are representations of a separable C*-
algebra 21 on separable spaces such that
ker p\ = kempi = kei p2 = ker7T/92.
Then p\ ~& P2-
Proof. By hypothesis, both 21* = />t(2l) are separable C*-algebras containing no
compact operators. There is a unique *-isomorphism a of 2li onto 2I2 such that
p2 = ap\. Let idt- denote the identity representation of 21*. By the previous
corollary,
Pi = idi p\ ~& (idi ®o)pi = p\ © p2.
Likewise, p\ © p2 ~& P2- ■

70 II. Normal Operators and Abelian C*-aIgebras
We are now ready to classify all representations up to approximate unitary
equivalence analogous to Theorem II.4.6. The remaining technicality is an easy
result about C*-algebras of compact operators.
Lemma II.5.7 Let %bea C*-subalgebra of the compact operators, and let a and p
be two non-degenerate separable representations o/2l. Then a and p are unitarily
equivalent if and only if
rank((j(A)) = rank(/9(A)) for all A G 21.
Proof. The necessity of the condition is clear. For the converse, let 7rt- denote
the inequivalent irreducible representations of 21. By Theorem 1.10.7, there are
cardinals nt- and m* in {0,1,..., N0} so that
i i
Let Ei denote a rank one projection in R(Hi) = 7r;(2l). Then
Hi = rank((j(2?t-)) = rank(/9(2?j)) = mt-
for all i. Hence a and p are unitarily equivalent. ■
Theorem II.5.8 Let %bea separable C*-algebra, and let a and p be
non-degenerate representations of 21 on separable Hilbert spaces. Then the following are
equivalent:
(i) a ~si P>
(ii) a ~a p,
(iii) a ~wa pt
(iv) rank((j(A)) = ia,nk(p(A))forall A G 21.
Proof. Clearly (i) to (iv) are successively weaker notions; where the implication
(iii) implies (iv) follows from the fact that the set of operators of rank at most k is
WOT-closed for every k > 0.
Condition (iv) applied to rank zero yields that ker a = ker p. The spectral
theory of compact Hermitian operators shows that every compact operator in any
C*-algebra is the norm limit of finite rank operators in the C*-algebra. Thus if
# denotes the space of finite rank operators and n is the quotient of B(H) by the
compact operators, we obtain
ker7T(7 = (7-"1(^) = (7~1(y)
= 7TtfJ = p-1(£) = te™p-
Let 25 = <r(2l); and define a representation of 25 by
t(B) = p{<r-\B)).

II.5. Voiculescu's Theorem
71
This is well defined since a and p have the same kernel; and condition (iv) shows
that r preserves rank. Let
«o = Ran(25 n £) «i = ?{, 6 «o
/Co = Ran(/>(21) n £) and Ki = Upe JCo-
Define 7 to be the identity representation of 25. Let
li = l\-Hi and Ti = r\Ki for i = 0,1.
We have observed that r maps 25fl£ onto r(25)fl.£, and preserves rank. Hence
by Lemma II.5.7, the restriction of r0 to this subalgebra is unitarily equivalent to
the identity representation 70 on this subalgebra. By Lemma 1.9.14, it follows
that 70 and r0 are unitarily equivalent. Since n is a representation of 25 which
annihilates 25 fl £, Corollary II.5.5 shows that
7 ~* 7 © n = To © 71 © n - To © n © 71 = T © 71 •
By symmetry, consideration of />(2l) shows that r ~# r © 71 as well. Hence
7 ~£ r, which is equivalent to the statement a ~& P- ■
Finally we will obtain a corollary that generalizes the Weyl-von Neumann
Theorem directly. Recall that the irreducible representations of C(-X") are one
dimensional. Thus a representation of C(-X") is diagonalizable if and only if it is the direct
sum of irreducible representations.
Corollary II.5.9 Every representation of a separable C*-algebra 21 on a
separable Hilbert space is approximately unitarily equivalent to a direct sum of
irreducible representations.
Proof. As in the previous section, a representation of 21 decomposes as a direct
sum p = pe © p where pe is the compression to the range K of (/>(2l) fl $)%.
Let ae be the compression representation of 25 := /?(2l) onto /C, and let a' be
the compression map to KL. By construction, p(A) factors as <r'p. Clearly, <r*
annihilates the compact operators in 25.
By Theorem 1.10.7, the identity representation of 25 fl £ is the sum of
irreducible representations. By Lemmas 1.9.15 and 1.9.14, it follows that ae is the
direct sum of irreducible representations.
From the GNS construction Theorem 1.9.12, there is a direct sum a of
irreducible representations of 25 such that ker a = ker a1. Now the range of r = <r(°°)
contains no compact operators either. So by the Corollary II.5.5,
id© = ae © <r' ~fi ae © <r' © r ~R ae © T.
Hence
p = id© p ~si (<re © r)p = pe®rp
which is the direct sum of irreducible representations. ■

72
II. Normal Operators and Abelian C*-aIgebras
Exercises
II. 1 (Fuglede 's Theorem) If X commutes with a normal operator N> show that
it also commutes with iV*.
HINT: Consider the bounded entire function
f(z) = e*N'Xe-*N' = e*N*e-*NXe*Ne~*N\
n.2 Show that if M and N are normal and MX = XN, then M*X = XN*.
HINT: [*f £] commutes with [g *].
11.3 Show that two similar normal operators are unitarily equivalent.
HINT: Take the unitary part of the polar decomposition of the similarity
and apply Fuglede's Theorem.
11.4 (a) Suppose that N is normal. If r > 0, show that
{x e U : lim r~n\\(N - XI)nx\\ = 0}
n-i>oo
is the spectral subspace of N corresponding to Dr (A) = {z : \z - A| < r}.
(b) Show that if X commutes with iV, it commutes with the spectral
projections for every open disk. Hence deduce that X lies in C*(N)f. (This is
another proof of Fuglede's Theorem.)
11.5 (a) If A = A* has a cyclic vector x> show that the subspaces
k
Mk = svw{]\E\ei)x for €i = ±l}.
t=i
have dense union. With this choice, show that the perturbations Rk of
Corollary n.4.2 have rank(.Rfc) < 2k. Hence show that the compact
operator K constructed lies in the Schatten Cp class, and has small Cp norm for
any p > 1.
(b) Remove the condition on cyclicity by using the fact that A is the direct
sum of cyclic operators.
(c) Generalize this to show that n commuting Hermitian operators may be
diagonalized modulo the Cp class for any p> n.
11.6 Say that two operators A and B are compalent if there is a unitary operator
U such that B - UAU* is compact. Prove that two normal operators are
compalent if and only if they have the same essential spectrum.
11.7 Two operators A and B are quasi-similar if there are injective operators
X and Y with dense range such that AX = XB and YA = BY. Show
that quasi-similar normal operators are unitarily equivalent.
NOTE: in general, quasi-similar operators need not even have the same
spectrum.

Exercises
73
11.8 Show that if / is continuous on a compact subset IcC, then for each
e > 0, there is a S > 0 so that whenever A and B are normal operators
such that a(A) U a(B) c X and ||A - B\\ < 8, then ||/(A) - f(B)\\ < e.
HINT: Prove it first for polynomials in z and z; then approximate /.
11.9 If A is positive, show that \\All2T - TAll2\\ < 2||r||1/2||AT - TA\\ll2.
HINT: Show that when T = U is unitary, this is valid with constant 1
instead of 2. Then prove it for T = T* by applying the result to the
unitary U8 = (sT + U)(sT - il)~l for s > 0.
ILK) Let A be a positive contraction such that both 0,1 G <re(A); and let P be
a projection such that rank(P) = rank(P1) = oo. Show that there are
unitary operators Un and Vn such that
P = WOT-Iim UnAU* and A = WOT-lim VnPV*.
This shows that the weak operator topology yields a very weak relation for
single operators.
II. 11 For nonseparable C*-algebras, the definition of approximate unitary
equivalence requires nets of unitaries. Show that two representations of a C*-
algebra 21 are (weak) approximately unitarily equivalent (relative to £) if
and only if this holds for the restriction to every separable subalgebra.
11.12 Let V denote the diagonal algebra of £°° acting on £2. Let r denote a
multiplicative linear functional on £°° which annihilates Co. Show that
id and a = id ©r(°°) are approximately unitarily equivalent relative to
A. However, show that there is no sequence of unitaries Un such that
WOT-lirrin^oo UnDU* = a(D) for all D in V. Nor is there a unitary U
such that UDU* - <r(D) is compact for all D in V.
Notes and Remarks.
Spectral theory for self-adjoint operators goes back to Hilbert at the turn of the
century. For normal operators, it is due to von Neumann[1929]. The L°°
functional calculus is developed later in von Neumann [1931]. The multiplicity theory
is due to Hellinger and Hahn. Wey 1 and von Neumann showed that Hermitian
operators are diagonal plus compact. The normal case is due to Berg [1971]. Glimm's
lemma was proven for a different context in Glimm [1960b]; and the generalization
to finite dimensional maps is due to Bunce and Salinas [1976]. Voiculescu [1976]
proved his non-commutative extension. Arveson [1977] put this all in an
appropriate abstract framework using quasicentral approximate units. It is this approach
that is used here even in the commutative case. His paper also shows the
equivalence of weak approximate equivalence with the other stronger notions. Hadwin
[1977] proves the even weaker version in terms of rank. In Hadwin [1980], he
develops the appropriate analogue of these results for the non-separable case.

CHAPTER m
Approximately Finite Dimensional (AF) C*-algebras
The purpose of this chapter is to make a detailed study of a special class of
C*-algebras which are built up in a natural way from finite dimensional algebras.
This class has an interesting variety of examples but at the same time it admits a
complete classification which will be dealt with in the next chapter. This
classification is described in terms of a group called K0 that reflects the structure of the
projections in the algebra. It is significant that one can also give a complete
description of the allowable groups. This will permit the construction of interesting
algebras.
We begin with the building blocks, the finite dimensional C*-algebras. We
may apply the structure theory for algebras of compact operators, Theorem 1.10.8,
to finite dimensional C*-algebras since every finite dimensional C*-algebra can be
represented *-isomorphically as a C*-algebra acting on a finite dimensional Hilbert
space. (See Exercise 1.36.)
Theorem III.l.l Every finite dimensional C*-algebra 21 is ^-isomorphic to the
direct sum of full matrix algebras
In particular, every finite dimensional C*-algebra is unital.
Proof. This is an immediate application of Theorem 1.10.8. The number of sum-
mands must be finite and the dimensions n» must also be finite for every i in order
that the direct sum be finite dimensional. The direct sum of the unit on each sum-
mand is the identity element. ■
We will make frequent use of the fact that Mn has a nice basis consisting of
the matrix units E{j := et-e*> for 1 < i, j < n associated to any orthonormal
basis ei,..., en for the underlying Hilbert space. Thus any finite dimensional C*-
algebra 21 ~ Mnx © • • • © Mnje has a system of matrix units which we will denote
by
{E$:l<s<k, l<ij<n8}.
The representation theory of finite dimensional algebras is now a direct
consequence of Theorem 1.10.7.
74

III.2. AF algebras
75
Corollary III.1.2 Ifn is a non-degenerate ^-representation of a finite dimensional
C*-algebra 21 = Mm © • • • © Mnk, then there are cardinal numbers ol\ ,..., a* so
that it is unitarily equivalent to id} *' © • • • © idj^ where idj denotes the identity
representation ofMUj-
III.2 AF algebras
The next simplest C*-algebras may well be those algebras that can be built
up from finite dimensional ones. While one might come up with several possible
definitions of what this might mean, we shall see that several of the most natural
choices turn out to define the same class.
A C*-algebra 21 is approximately finite dimensional or AF if it is the closure
of an increasing union of finite dimensional subalgebras 21*. When 21 is unital, we
further stipulate that 2l0 consist of the scalar multiples of the identity element 1.
To understand how such a union might develop, we need to understand how
one finite dimensional C*-algebra may fit into another.
Lemma III.2.1 Suppose that cp is a unital homomorphism of a finite dimensional
C*-algebra 2li ~ Mni © • • • © Mnk into another finite dimensional C*-algebra
2I2 ~ Mmi © • • • © Mmr Then <p is determined up to unitary equivalence (in 2I2)
by an £ X k matrix A = [ay] in Mik(^o) with non-negative integer entries such
that
*(I)-G)-
Proof. Let £t-, for 1 < i < £, denote the canonical surjection of 2t2 onto the
summand Mmr Then <pi := sup is a unital homomorphism of 2li into Mmi. By
Corollary HI.L2, (pi is determined up to conjugation by a unitary in Mmi by the
multiplicities ay in No, 1 < j < k, so that
^-id^)©---©id^).
Since (pi is unital, a comparison of dimensions yields
k
i=i
Since cp 2^ (p\ © • • • © (pi, we see that (p is determined up to unitary equivalence
by the non-negative integer valued £ x k matrix A = [ay] of partial multiplicities.
The integer ay is the multiplicity of the imbedding of the summand Mnj of 2li
into the summand Mm. of 2l2. ■
Consider imbeddings which need not be unital. Then <p(I) is a projection in
2I2. By restricting to the range of this projection, we reclaim the unital case at the
expense of reducing the size of the blocks in 52l2. Hence we obtain:

76
III. Approximately Finite Dimensional (AF) C*-aIgebras
Corollary III.2.2 Suppose that (p is a homomorphism of a finite dimensional C*-
algebra 2li ~ Mni © • • • © Mnje into another finite dimensional C*-algebra
2I2 — Mmi © • • • © Mmr Then (p is determined up to unitary equivalence (in 2I2)
by ant X k matrix A = [ay] in Mik{^o) of partial multiplicities such that
k
\_] Q>ijnj < ™>i for 1 < i < £.
t=i
This allows us to describe the imbedding of 2li into 2t2 in a simple graphical
way. Represent 2li by the fc-tuple {(1,1) = ni,..., (1, k) = n*.} and 2I2 by
the ^-tuple {(2,1) = mi,..., (2, £) = mi}. Denote the imbedding of 2li into
2I2 by drawing ay arrows from (l,i) to (2, i) to indicate the partial multiplicity
of the imbedding. The sequence of these pictures for a sequence of imbeddings
ap of 21^ into 2lp+i is called the Bratteli diagram of the sequence. This is an
infinite graph consisting of nodes (p, i) = n representing the fact that the i-th
summand of 2lp is isomorphic to Mn> There will be af^ arrows from (p, i) to
(p + 1, j) corresponding to the partial multiplicity of the mapping. This is also
often called the Bratteli diagram of the algebra, but this is an abuse of terminology
as the sequence is not unique. We will see though how to compare two diagrams.
Example III.2.3 Consider the C*-algebra C/+£. Choose an increasing sequence
Pn of projections with rank(Pn) = n converging strongly to the identity. Let
2ln := CP£ + Pn£Pn ~ C © Mn. It is easy to see that Un2ln is dense in
CI + £. The imbedding of 2ln into 2ln+i imbeds Mn once into Mn+i. Since
P„ = P„+i + 2?n+i where 2?n+i is a rank 1 projection less than Pn+i> the scalars
are imbedded once into each summand of 2ln+i. We obtain the Bratteli diagram:
The ideal £ corresponds to the sequence
<Bn := 2l» fl £ = Pn£Pn ~ Mn.
So we obtain the (directed) subset of the diagram above to represent £:
1—> 2 —► 3 —► 4 —>
Example III.2.4 The CAR algebra 21 is obtained as the union of subalgebras
21*. ~ M2k where the imbedding (pk of M2k into M2*+i is given by the
multiplicity 2 imbedding
<pk(A) =
A 0
0 A

IH.2. AF algebras
77
This has the Bratteli diagram
-X.T.
:8
16
32:
The image of <pu sits inside the smaller subalgebra 95* ~ M2k © M2k of
2lik+i. So 21 is also the increasing union of the 2$*. An inspection of the action of
<Pk+i on <8fc shows that
<Pk+i
Si
0
0
S2
])■
" Si 0
0 B2
0 0
0 0
0 0 1
0 0
Si 0
0 S2 J
Hence the partial multiplicities of the imbedding of 2$* into 2Jfc+i is given by [ \ \ ].
So we obtain an alternative diagram
These two diagrams can be intertwined because of the relationship
2li C ©i C 2l2 C ©2 C 2l3 C ©3 C 2l4 C ©4 C ...
Except for dropping to subsequences and making small perturbations, this kind of
intertwining must occur for two diagrams to represent the same algebra.
Example III.2.5 A finite dimensional commutative C*-algebra is spanned by
finitely many pairwise orthogonal projections. Thus a separable commutative C*-alg-
ebra is AF precisely when it is spanned by its projections. This is equivalent to the
spectrum being totally disconnected. A typical case of this phenomenon occurs for
C(-X") where X is the Cantor set. Using the traditional 'middle thirds' construction,
we obtain X as the intersection of a decreasing family Xn of closed sets where Xn
consists of 2n disjoint intervals, and each interval contains exactly two intervals of
Xn+i. Let 2ln be the subalgebra of functions in C(X) which are constant on the
intervals of Xn.
Notice that 2ln is isomorphic to C2". The imbedding of 2ln into 2ln+i splits
each minimal projection into the sum of two minimal projections. Thus we obtain
the diagram

78
III. Approximately Finite Dimensional (AF) C*-aIgebras
Example III.2.6 Finally we give an example which we won't be able to identify
at the moment, but will come up later in our discussion. Start with a sequence
21*. = Mmk © Mnk given recursively by mi = n\ = 1 and a repeated use of the
partial imbedding matrix [ \ J]. This yields the recursion relation
mfc+i = rrtk+rik and rik+i = mk
which is easily recognized as the Fibonacci sequence. We obtain the diagram
1 >2 ^3 ^5 ^8 >- •••
<xxv
112 3 5
While it is premature to discuss when different diagrams yield the same
algebra, let us at least observe that a diagram yields a unique algebra.
Proposition III.2.7 If 21 = Un>i2ln and*8 = Un>i®n have the same Bratteli
diagram, then they are isomorphic.
Proof. Let the injection of 2ln into 2ln+i be denoted by an, and similarly let /3n
denote the imbedding of 2Jn into 2Jn+i- At each stage, there is a natural
isomorphism <pn of 2ln onto 2Jn. Thus v?n+i<*n and f5n<Pn are injections of 2ln into 2Jn+i
with the same partial multiplicities (because the partial multiplicities of an and j3n
agree). Hence by Corollary m.1.2 there is a unitary Un+i in 2Jn+i so that
/3n<pn = Ad(Un+i)<pn+ian.
Let V>i = ¥>i and set Vi = I. Recursively define
Vn+i = PniVrjUn+x and Vn+i = Ad(Fn+i)<pn+i for n > 1.
Then a calculation shows that
Mn = Pn Ad(Vn)<pn = Ad(/3n(Fn))/3n¥>n
= Ad(/3n(V„)) Ad(C/n+1)^n+1an
= Ad(/3n(Vn)Un+i)<pn+ian = V>n+i<**-
Hence the following diagram commutes.
a2
2li
»i
ai
*2l2
01
*<8
A
2l3-
■®3
a3
■su
ft
a<
/34
From this it is evident that ^ '•= U„^n is a *-isomorphism of UnM^ onto
Un>i35n- As t{> is isometric, it extends by continuity to a *-isomorphism of 21
onto <8. ■

III3. Perturbations
79
III.3 Perturbations
In making approximations in AF algebras, any finite dimensional subalgebra
25 of 21 = Un>i2ln will be 'almost contained' in 2ln for n sufficiently large. We
will show that by twisting 25 slightly by a unitary close to the identity, it can be
moved inside 2ln. This first lemma shows how to move projections.
Lemma III.3.1 Given any e > 0 and n € N, there exists a positive real number
8 = 8(e, n) so that if Pi,..., Pn arepairwise orthogonal projections in a C*-alg-
ebra V and 25 is a C*-subalgebra of® such that dist(P$, 25) < 8 for 1 < i < n,
then there arepairwise orthogonal projections Qi in 25 such that \\Pi — Q%\\ < e
for 1 < i < n. IfY%=i P% = 1» then we may arrange that E3Li Qi = 1 a^so-
Proof. First let n = 1 and set 8 = min{e/2,1/3}; and suppose that Pi is a
projection in V such that dist(Pi, 25) < 8. Pick X in 25 such that ||Pi - X\\ < 8.
Since Pi = Px*, we may let B = (X + X*)/2 and obtain ||Pi -B\\<6. Because
IKP-A/^l^distCA^O,!})-1,
it follows that B - XI is invertible if |A| > 8 and | A - 1| > 8. Hence the spectrum
<t(B) is contained in [-8,8] U [1 - 8,1 + 8]. These intervals are disjoint because
8 < 1/3. Using the continuous functional calculus for normal operators, we obtain
a projection Qi := X[is,i+8]{B) in <8. Moreover,
\\Pi -Qi\\< \\Pi - B\\ + \\z - X^^m*) < 8 + 8 < e.
For the general case, proceed by induction. Suppose that we have found
projections Qi in <B for 1 < i < n - 1 such that ||P; - Qi\\ < erf. Set P = YJ^1 Pi
and Q = Y%=i Qi- Choose B = B* in <B as above with ||P„ - B|| < 8. Then
\\Pn -(I- Q)B(I -Q)\\ = ||(/ - P)Pn(I - P) - (/ - Q)B(I - Q)\\
<\\(Q-P)Pn{I-P)\\ + \\(I-Q)(Pn-B)(I-P)\\
+ \\(I-Q)B(Q-P)\\
<(l + \\B\\)\\Q-P\\ + \\Pn-B\\
n-1
< (2 + 8)J2ci8 + 8 <c'J
i=l
where c'n = l + 3YZ=? *•
Proceeding as above, we obtain a projection Qn in (J - Q)25( J - Q) such that
||(/ - Q)B(I -Q)- Qn\\ < c'J; whence \\Pn - Qn|| < 2c'J. It remains to
choose 8 sufficiently small to obtain the desired estimate. When S£=i P% = I>we
see that
\\i - E Q*w * E h * - Q*w * (£ «)' <1

80
III. Approximately Finite Dimensional (AF) C* -algebras
provided that S is sufficiently small. But the only projection within distance 1 of
the identity is J itself. Thus Y^i Qi = L ■
Now we are able to show that when a finite dimensional C*-algebra is almost
contained in another C*-algebra, it can be twisted inside by conjugating by a
unitary near the identity. We make no effort to make our estimates independent of the
dimension, but this can be done with a lot of work. A natural measure of how close
21 is to being included in 25 is
sup dist(A, 25).
Aea,||A||<i
However, as we will be working with matrix units, it will be convenient to take
this supremum only over a standard basis. These two measures will be comparable
within a constant factor that depends on the dimension.
Lemma III.3.2 Given any e > 0 and n € N, there exists a positive real number
5 = 5(e,n) so that whenever 21 and 25 are C*-subalgebras of a unital C*-algebra
V with dim 21 < n and such that 21 has a system of matrix units {E^'} satisfying
dist(J57^j), 25) < 8, then there is a unitary U in C*(2l, 25) with \\U - I\\ < e so
that UVIU* is contained in 25.
Proof. We may suppose that 21 and 25 are unital by adjoining the identity if
necessary. It is easy to see that if J? is a unit for 21, then 2l~ = 21 © CEL. So the set
of matrix units for 21 may be expanded to include EL. An easy estimate which we
omit shows that dist(J5?-L, 25") < n&.
First let us prove this theorem for 21 :_ ££. The matrix units for 21 are the n
pairwise orthogonal minimal projections Pi,..., Pn. We may assume that e < 1;
set 77 = (n+l)"1^. Using S = 8(r], n) of the preceding lemma, we obtain pairwise
orthogonal projections Qi in 25 such that ||P» -Q%\\ < rj. Define X = YX=i QiPi-
Then a calculation shows that
n n
i=l i=l
Similarly XX* > (1 - rj)I. So X is invertible. Moreover, we have the identities
QiX = XPi for 1 < i < n.
Let U be the unitary in the polar decomposition of X = U\X\. As X is
invertible, Theorem 1.8.1 shows that U belongs to C*(X) which is contained in
C*(2l, 25). Clearly, each Pi commutes with \X\. So
UPi = Xpq-1^ = XPilXl"1 = QiX\X\~l = QiU.

IIU. Perturbations 81
Hence UP{U* = Qi for 1 < i < n. It remains to make the norm estimate:
Hi7-j||<||.7-*|| + pr-j||
<||Jr|||||Jr|-1-/|| + £||(Q*-fl)fl||
i=l
< ((1 - ?7)""1/2 - l) + nrj < (n + 1)77 = e.
For general 21 with matrix units {E;y : 1 < s < k, 1 < i, j < n,}, let 9JT
be the masa spanned by the diagonal matrix units {E;*'}. Let <S = 8(7]', n) where
t;' = (6n + 6)_1£. Using the unitary obtained above for QJt and this choice of 8,
we obtain a unitary U so that UfflU* is contained in 25 and ||J7 - I\\ < e/6. The
matrix units for {721*7*, say l^y' := UE^yU*9 satisfy the norm estimate
dist(F^\<B)<S + \\E^-F^
<S + 2\\U-I\\<3e/6 = e/2.
Choose Xj^ in <B so that HF^ - x[f\\ < e/2. As F$ belong to 25, we may
replace X\y with i<\y x\jFjj without increasing this norm estimate. Assume
that this has been done.
Notice that x[y is bounded below by 1 - e/2 on the range of FJy and has
range equal to the range of F^ . Thus the polar decomposition for X\*j yields a
partial isometry G[8J in 2$ with the same co-domain and range. As in the
commutative case, we find that
ii^i(?-^?ii<a-^/2)-1/2-i<^/3.
Extend this to a full system by defining G^y = G\^G^. Then define a unitary
operator
8=1 j=l
An easy computation confirms that G^W = WF>p, whence
WFWW* = G\? for all 1 < s < jfe, l<i,j<n9.
Finally we obtain the norm estimate (using the fact that a sum of terms with
orthogonal domains and ranges has norm equal to the maximum of the norm of
each term)
||W-/|| = max||G(f^-^f^||
= mzx\\G[')-F$\\<5e/6.

82
III. Approximately Finite Dimensional (AF) C*-aIgebras
The desired unitary WU thus satisfies
\\WU - I\\ < \\W - I\\ \\U\\ + \\U - I\\ < 5e/6 + e/6 = e. U
For ease of our constructions, we improve this result in the case that 21 and 25
contain a common subalgebra.
Corollary III.3.3 If in addition to the hypotheses of the preceding lemma, we have
an algebra 211 contained in 21 fl 25, then we may choose the unitary U to lie in the
commutant of%\.
Proof. This can be accomplished with superior constants by doing the
construction over again with 2li in mind. However, we choose to give a more transparent
argument that has the minor flaw of requiring a smaller choice of 5.
Let S = 8(e/3, dim2l) from the last lemma. Obtain a unitary operator U in
C*(2l, 25) such that OW* C © and \\U - I\\ < e/3. Fix a set {F^} of matrix
units for 2li; and let G$ = UF^U* be the corresponding system for J72liCT\
Define
*=1 J=l
as in the proof above. This lies in 25 and satisfies WF;yW* = Gy for all i, j, s
and
\\W-I\\<mzx\\G$-F$\\
= maxIKtf - J)jsg> - F$(U - I)\\ < 2\\U - I\\.
Thus the unitary V = W*U commutes with 2li, satisfies VW* C 25 and
\\V - I\\ < \\W - I\\\\U\\ + \\U - 1|| < W - I\\ <e. U
Now we can obtain a characterization of AF algebras which is not dependent
on choosing a sequence of subalgebras.
Theorem III.3.4 A C*-algebra 21 is AF if and only if it is separable and:
(*) for all e > 0 and Ai,..., An in 21, there exists a finite dimensional C*-
subalgebra 25 o/2l such thatdv&t{A^ 25) < efor 1 < i < n.
Moreover, (f2li is a finite dimensional subalgebra of% then we may choose 25 so
that it contains 2li.
Proof. If 21 is AF, the other properties are easily established. So assume that 21 is
separable and (*) holds. Fix a countable dense subset {A{ : i > 1} of the unit
ball of 21 with Ai = 0 and a sequence €{ decreasing monotonely to 0. Proceed
recursively starting with 2li if it is given, and with the identity adjoined if 21 is
unital. (Otherwise start with the subalgebra of scalars.)
For the purpose of proceeding by induction, assume that a finite dimensional
subalgebra 21^ has been found so that dist(Ai,2lfc) < Sk for all 1 < i < k.

III3. Perturbations
83
Let S = <S(£fc+i/3,dim2lfc) as in Lemma m.3.2; and fix a set of matrix units
{E>y} for 2lfc. Using property (*), find a finite dimensional subalgebra 23 of
21 so that diat(E^\ 23) < S for all the matrix units and dist(.Ai, 23) < ek+i/3
for 1 < i < k + 1. Then by Lemma HI.3.2, there is a unitary U in 21 so that
UWkU* C 23 and || tf - 7|| < ^+i/3. Let 2l*+i := U*<BU. Clearly this is a finite
dimensional algebra containing 21*. Moreover,
dist(Ai, 2lfc+1) = dist(C7AiU\ 23) < 2||tf - I\\ + efc+1/3 < efc+1.
Proceeding recursively, we construct an increasing sequence of finite
dimensional algebras with dense union. So 21 is AF. ■
It turns out that there is quite a strong uniqueness of the chain of subalgebras
of an AF algebra. Of course, one may always drop to a subsequence, but this will
not change the union. The union is unique up to conjugation by a unitary. This
allows us to (almost) intertwine two chains determining the algebra.
Theorem III.3.5 Suppose that 21 is an AF algebra which is the closure of the
increasing union of two chains 21 = Un>i2ln = Un>i23n. Then for any e > 0, there
is a unitary operator W in 2l~ with \\W - I\\ < e so that
Un>12ln = ^Un>i©n^*.
In particular, there are subsequences m» and n» ofNso that
2U, C WXmW* C %n^ for alii > 1.
Proof. Choose positive numbers Si so that ]~[i>i(l + ei)2 < 1 + e- Let mi = 1
and Si := S(ei, dim2li) as in Lemma HI.3.2. As the matrix units of 2li are almost
contained in the dense union of the 23n's, there is an integer n\ such that 2li is
within &\ of being contained in 23ni. Hence there is a unitary U\ so that
tfi2lxtf* C <Bni and ||tfi-/||<£i.
Now let 771 := 8(ei, dim23ni). Proceeding in the same way, find an integer
ra2 > 1 so that U^23ni U\ is 771-contained in %n2. By Corollary ni.3.3, there is a
unitary V\ in 21 commuting with 2li so that
VxU&^UxVtc*^ and ||Vi-/||<ei.
We will proceed recursively. Suppose that at the fc-th stage, we have found
integers m\ < • • • < rrik+i and n\ < • • • < nk and unitary operators U{ and V{ for
1 < i < k satisfying \\U{ - I\\ < e; and \\Vi - I\\ < Si such that
SU, C ©n, := ViU? ... ViC/rSn^iV; ... UiV? C SUi+1
and so that V{ commutes with 21* and Ui+i commutes with 23n,for 1 < i < k. To
obtain the next stage, repeat the argument of the second paragraph.
The condition on 6{ guarantees that the infinite product
W := HmViU?...ViU;
i-¥oo

84
III. Approximately Finite Dimensional (AF) C*-aIgebras
exists and satisfies || W - I\\ < e. Moreover, if we set
Wk:=VhU£...V1U; and W™ = lim 1*0? .. .T^+i^i,
t-j>oo
we see that W^ commutes with 2Jnfc; whence
W<Bnk W* = WW&nk WW* = %nk . U
We obtain the following immediate consequence.
Corollary III.3.6 7f2l = Un>i2ln and*8 = Un>i2Jn are ^isomorphic AF
algebras, then Un>i2ln an<iUn>i2$n are ^-isomorphic.
Example III.3.7 The purpose of this example is to show that even if 2ln is
isomorphic to 2Jn for all n > 1, the algebras need not be isomorphic. The nature of the
imbeddings is critical.
Consider the algebra 21 = C(X) where X is the Cantor set with a chain of
subalgebras 2ln isomorphic to £^ as in Example HI.2.5. For the second algebra,
let <8 = C(Y) where Y = {0} U {n"1 : n > 1}. Let <8n be the subalgebra of
functions in C(Y) which are constant on [0,2~n]. It is readily apparent that <Bn
is also isomorphic to £§S. But as the spectrum of isomorphic C(K) spaces are
homeomorphic, we deduce that 21 and 2$ are not isomorphic.
The Bratteli diagram of 2$ is quite different from that of 21:
III.4 Ideals and Quotients
We will now see that the ideal structure of an AF algebra can be read off of
its Bratteli diagram. First we show that ideals are inductive, meaning that they
come from ideals of the defining sequence. Notice that the finite dimensionality of
the subalgebras 2ln is not used. So this result may be used later for more general
inductive limits.
Lemma III.4.1 Let 3 be an ideal of a C*-algebra 21 = Un>i2ln, where 2U is an
increasing sequence ofC*-subalgebras of$l. Then
J = Un>ipnan) = 3rnUn>iSin.
Consequently, when 21 is AF, so is 3.

III.4. Ideals and Quotients
85
Proof. Consider the commutative diagram
3—— ^ + J—^a» + J/J
Here t denotes the canonical injections, and ir denotes the canonical quotient maps.
The map a is an isomorphism by Corollary 1.5.6. In particular, for A in 2ln,
dist(i4,2tnfiJ)=dist(i4,J).
Hence if J belongs to 3 and e > 0, there is an element A in 2ln (for n large
enough) so that || J - -A|| < e. Thus there is an element J' in 3 0 2ln so that
||A - J'|| < e. Hence || J - J'|| < 2e. It follows that 3 = Un>i(Jn2ln).
If 2ln are all finite dimensional, then so are 3 0 2U; and hence 3 is AF. ■
Now we can describe all the ideals. A subset S of a Bratteli diagram V is
directed if whenever some (p, i) belongs to S and (p, i) ->- (p + 1, j) in V> then
(p + 1, j) also belongs to 5. The subset S is hereditary if whenever every image
of a vertex at the next level belongs to <S, then the vertex also belongs to S. That
is, whenever for some (p, i) in V all the (p + 1, j) belong to S for which there is
an arrow (p, i) -)- (p + 1, i) in P, then (p, i) also belongs to S.
Theorem III.4.2 Let 21 be an AF algebra with a Bratteli diagram V. Then the
ideals of% are in a one to one correspondence with directed hereditary subsets S
ofV.
Proof. An ideal 3P of Qlp c^ M(Pti) © • • • © M(Ptk) corresponds to the sum of a
subset of these summands because Mn is simple. So given an ideal 3 of 21, let S
denote the subset of V corresponding to the union of the summands corresponding
to each 3P := 3 D 2lp. Since Lemma m.4.1 shows that the sequence 3P uniquely
determines 3, one sees that 3 can be recovered from knowledge of S.
Let ap denote the injection of 2lp into 2lp+i. To see that S is directed, note that
if M{pyi) belongs to 3P> then whenever (p, i) ->- (p + 1, j)9 there is a non-empty
intersection
3n -Moh-ij) 3 OLp{M{Pyi)) n -Mfp+ij) ^ {0}.
Hence J contains .Mp+ij and so (p + 1, j) belongs to S.
To see that S is hereditary, suppose that (p + 1, j) belongs to S for all j in
J:={i:(p,i)-^(p + l,i)inP}.
Then since
<*p(MM) C X>A*(p+U) c J>
it follows that (p, i) belongs to S as well.

86
III. Approximately Finite Dimensional (AF) C*-aIgebras
Conversely, if S is a subset of V which is directed and hereditary, let 3P denote
the ideal of 2lp corresponding to the set of vertices (p, i) in S. Since S is directed,
this is an increasing net of ideals. The closure of the union is an ideal 3 of 21.
The hereditary property ensures that 3P = 2lp fl 3p+i from which we deduce that
3P = 2lpfl 3. Hence this is the sequence we associated to 3 as in the first paragraph.
So the subset S is the canonical choice associated to 3. ■
Corollary III.4.3 A unitalAF algebra 21 with Bratteli diagram V is simple if and
only if for each (p, i) in V, there is an integer q > p so that (p, i) —> (g, j)for all
J-
Proof. If V has this property, then as every non-zero ideal 3 will have non-zero
intersection with some 2lp, we find a (p, i) in the associated subset S of V. Hence
by the directed property and the hypothesis of the corollary, S contains all (g, j)
at some higher lever. But this means that 3 contains 2lq, and so it contains the
identity. Therefore 21 is simple.
Conversely, suppose that the condition fails for a certain (p, i). Then the
directed subset Sf of V generated by (p, i) is a proper subset at every level. Increase
this to the smallest directed and hereditary subset S containing (p, i) by adding in
any vertex that maps completely into S' at some higher level. This is still proper
at every level because each 2ln contains the identity of 21. Consequently, it
corresponds to a proper ideal and 21 is not simple. ■
Now we can proceed to compute the quotient algebra.
Theorem III.4.4 If 3 is an ideal of an AF algebra 21 corresponding to a subset S
of the Bratteli diagram V, then %/3 is AF and corresponds to the diagram V\S.
Proof. First notice that 21/3 is the increasing union of subalgebras Qln+3/3, which
are finite dimensional, and thus the quotient is AF. By the proof of Theorem III.4.2,
we have that 2ln + 3/3 c^ 2ln/3n. This algebra is isomorphic to
which corresponds to the n-th level of V \ S. Moreover the partial imbeddings
of (n, i) into (n + 1, j) are unchanged if neither belongs to S. Hence the Bratteli
diagram of 21/3isV\S. ■
III.5 Examples
Example III.5.1 UHF Algebras. A C*-algebra is called uniformly hyperflnite
or UHF if it is the increasing union of unital subalgebras isomorphic to full matrix
algebras Mkn> Since a unital imbedding of Mm into Mn requires m|n, we have
an increasing sequence fci|fc2|fc3| Thus for each prime integer p, there is a
unique ep in Nu{oo} which is the supremum of the exponents of powers of p which
divide knasn tends to infinity. We define the supernatural number associated

III.5. Examples
87
to the sequence 2ln to be the formal product <S(2l) := lip primed- ^e following
theorem of Glimm classifies UHF algebras.
Theorem III.5.2 Two UHF algebras 21 and 25 are isomorphic if and only if
<J(2l) = *(»).
Proof. Let 21 and 25 correspond to sequences km and tn respectively. If 21 and
25 are isomorphic, Corollary HI.3.6 shows that Um>i2lm and Un>i25n are
isomorphic Thus as in Theorem HI.3.5, there are subsequences mt- and n»- so that
21m,. ~ Mkm. is isomorphic to a subalgebraof 25n>. ~ Min. and thus kmi\tni\ and
25n. is isomorphic to a subalgebra of 21™^, whence £ni |fcmi+1. Hence 21 and 25
determine the same supernatural number.
Conversely, suppose that <S(2l) = <S(25). Then there are subsequences mt- and
nt- such that kmi Kn, |^n,+i - For convenience, let us renumber so that rrii = ni = i\
and let an denote the imbedding of 2U into 2ln+i; and similarly let j3n denote
the imbedding of 25n into 25n+i- Define a *-isomorphism from Un>i2ln onto
Un>i25n as follows. There is a unique imbedding (up to unitary equivalence) <p\
of 2li into 25i by Lemma HI.2.1. Likewise there is an imbedding xf>[ of 25i into
2I2. The composition tp[(pi is equivalent to the imbedding ol\ of 2li into 2I2. Thus
by Lemma HI.2.1, there is a unitary operator W\ in 52l2 so that Ad(Wi)^iv?i = <*i.
Let^i = Ad(T^i)^i.
Continue in this way to construct injections <pn of 2ln into 25n and if>n of 25n
into 2ln+i so that the diagram
commutes. Then by construction, cp = Un>iv?n is the desired isomorphism; and
xff = Un>i^n is its inverse. ■
Corollary III.5.3 UHF algebras are simple.
Proof. Since the Bratteli diagram consists of a single node at each level, it is evident
that the only directed hereditary and non-empty subset is the whole diagram. Thus
UHF algebras are simple by Corollary HI.4.3. ■
Example III.5.4 The CAR algebra. The acronym CAR abbreviates canonical
anticommutation relations, a notion which comes from quantum mechanics.
Consider a linear map a of a Hilbert space % into 8(H) such that for all /, g in H
<*(f)a(9) + *(9)a(f) = 0
(1)

88 III. Approximately Finite Dimensional (AF) C*-algebras
and
<*(fra(g) + a(g)a(fy = (g,f)I. (2)
Let 21 denote the C*-algebra generated by {<*(/) : / G H}. We will establish that
this algebra is independent of the choice of the function a, and that it is in fact the
2°° UHF algebra.
First, notice that if / is a unit vector, then using g = f yields
a(/)2 = 0 and <*(/)*<*(/) + <*(/)<*(/)* = J.
Multiplying the second equation by a(/)*a(/) yields
(a(/)*a(/))2 = a(/)*a(/) =: E{f)
is a projection. Hence a(f)a(f)* = E(f)L. The operator a(f) is a partial isom-
etry with domain E(f) and range E(f)L. Hence
C*(a(/)) = span{a(/), <*(/)*, E(f), E(f)L} ~ A<2.
Let £<!> = «(/), sg> = <*(/)*, Eg> = £?(/), and ^ = U(/) J-.
Next note that if / and g are orthogonal unit vectors, then
[«((,), E(f)) = a(g)a(f)*a(f) - a(f)*a(f)a(g)
= a(g)a(f)*a(f) + <*(/)*«($)<*(/)
= (5,/H/) = 0.
Thus a(flr) commutes with E{f), and consequently so does E(g).
Let Vi := J - 2E(f) = E(f)x - E(f). Then a calculation using (1) shows
that
Via(g)a(f) = -Via(/)a(» = a(/)Via(5)
and
VMg)a(fy = -Vio(/)*o(» = a(/)*Viafo).
Hence ^ia(^) commutes with C*(cc(/)).
Note that C*(Via(5)) = span{Via(g),Via{g)*,E(g),E(g)L} ~ M2 and
commutes with C*(a(f)). Set
^ff-Viafo). sff-Vkafo)*, *g> - £(,,), and «g> - ^)A.
It is now evident that C*(a(/), a(flf)) is isomorphic to Ma with a standard basis
E^E™ fori <i,j,k,£< 2.
Now fix an orthonormal basis {/n : n > 1} for H. Set
n-1
V0 = I and Fn = fj (J - 2£(/t)) for n > 2.

in.5. Examples 89
Then we may define matrix units for a copy of M 2 by
E{$ = E{fn), Eg> = Vna(fn), E{$ = Vna{fny, and E%> = E{fn)"-.
It follows by extending the analysis above that these copies of M2 commute with
each other, and hence 2ln := C*({a(/t) : 1 < i < n}) is isomorphic to M2"
with a standard basis consisting of the matrix units E^, := n£=i E (kU(k) ^or ^
functions <p and xff of {1,2,..., n} into {1,2}.
Thus 21 = U^i is the 2°° UHF algebra of Example ffl.2.4.
Example III.5.5 The GICAR algebra. GICAR stands for gauge invariant CAR
and is also called thecurrent algebra. If U is a unitary operator on %, define an
automorphism of the CAR algebra 21 by
<pu(a(f)) = a(Uf) for all / G U.
It is easy to see that this map preserves the relations (1) and (2) above, and thus
(pu extends to *-automorphism of 21. In particular, consider the scalar unitaries XI
for A in the unit circle T. These give rise to the so called gauge automorphisms
Xa := <p\i for A in T determined by Jhe identities
Xa(<>(/)) = a(Xf) = Xa(f) and XA (<>(/)*) = Xa(f)
for all/ in U.
Define the subalgebra 21° of the CAR algebra which is invariant for all the
gauge automorphisms:
21° := {A e 21: XX(A) = Aforall A € T}.
Note that if /1,..., /n, 51,..., gm are vectors in H, then
A:=a(/1r...a(/nra(5i)...a(i7m)
satisfies X\(A) = Am"~nA In particular, A belongs to 21° when m = n. It
will turn out that 21° is spanned by the terms of this type. In particular, note that
X\(E(f)) = E(f) for all / in U\ and hence we also have X\(Vn) = Vn for the
symmetries Vn described in the previous section. Therefore the matrix unit E\"' is
transformed by the rule
Xx{E\f) = Xi-*E\f for l<i,i<2.
n
Define 2l£ := 21° fl 2^. Let us show that 2l£ is isomorphic to Y]®Mfny
fc=0
Consider the action of Xa on a standard basis element of 2ln
U=l / k=l
So with respect to this basis, Xa acts as a diagonal matrix on 2ln. So we see
immediately that the fixed point algebra is spanned by those matrix units satisfying

90
III, Approximately Finite Dimensional (AF) C*-algebras
££=i *k = ££=i h- As these are sums of l's and 2's, the possible sums are
n, n + 1,..., 2n.
For each integer 0 < s < n, the span
n n
^ = span{£7^ : ]T y>(fc) = £ ^(fc) = n + s}
k=i fc=i
is an algebra. Indeed, a product E^E^i = S^E^t is non-zero if and only
if ^ = xf)y which implies that ££=i ¥>(&) = n + s = ££=1 ^'(*)- -^ fact *is
formula shows that this basis is a set of matrix units for a full matrix algebra. The
dimension is computed from the fact that there must be exactly s 2's and n- sVs
in each term. This means that there are (") possible functions (p of {1,2,..., n}
into {1,2} with this sum. So T? ~ M/n\. Moreover our product identity shows
that T^Tt1 — 0 for s ^ t. So this decomposition is a direct sum as claimed.
Next consider the imbedding of 21° into 21°+1. For j = 1,2, let <pj denote the
function of {1,2,..., n + 1} into {1,2} given by (pj(k) = (p(k) for 1 < k < n
and <pj(n +1) = j. In the imbedding of 2ln into 2ln+i, the matrix unit E^ is sent
to E^i + 2^2V>2. If -EW' ^es *n ^T> then
n+l n+l
J] pl(ft) = ]T ^l(ft) = (n + 1) + s
fc=i fc=i
and
n+l n+l
J>2(fc) = J2l>2(k) = (n+ 1) + s + 1.
fc=i fc=i
Hence .E^i^i lies in ^J1"*"1 and E^w lies i» ^T+i.1- This yields a Bratteli diagram
for the sequence 21° which looks like Pascal's triangle:
Next we show that the map A -+ X\(A) is continuous for each A in 21. This
is evident for the matrix units E^ from the explicit formula established above.
Since each 2tn is finite dimensional, we see that A -+ X\(A) is continuous for A

III.6. Extensions
91
in 2ln. Given A in 21 and e > 0, choose an approximant B in 2ln for some n
so that ||A - JB|| < e. Then choose a J > 0 so that |A - A7| < S implies that
||XA(B) - XX'(B)\\ < e. Then for |A - A'| < 8,
\\Xx(A)-Xx>(A)\\
< \\XX(A) - Xx(B)\\ + \\Xx(B) - Xx>(B)\\ + \\XX>(B) - xx*(A)\\
< e + e + e = 3e.
This establishes continuity at A.
We wish to establish that 21° = Un>i2l£. Let A belong to 21° and let e > 0.
Choose an element B in 2ln for some n so that ||A - JB|| < e. Then define
B0:= fxx(B)d\
Jt
where dX is normalized Lebesgue measure on the circle. This integral makes sense
in the Riemann sense because the integrand is norm continuous. I claim that JBo
lies in 2l£. It lies in 2ln because XA(2ln) = 2tn. But it is also gauge invariant
because
Xa'(#o) = f Xx>Xx(B) d\= [ XA+A'(£) d\
Jt Jt
= [ X„{B) dfi = B0
Jt
Moreover this is a good approximant to A because
\\A-B0\\ = ^Xx(A-B)dxj
< [\\Xx(A-B)\\d\=\\A-B\\<e
Jt
Hence Un>i2l£ is dense in 21°.
This algebra has a rich ideal structure. The reader can verify that the ideals
correspond to the directed sets generated by any single vertex of the diagram.
III.6 Extensions
A C*-algebra <£ is an extension of a C*-algebra 21 by a C*-algebra 25 if there
is a monomorphism j of 21 onto an ideal of £ such that <£/i(2l) c^ 25. In other
words, there is a short exact sequence
0 *• 21 -— £ —^ » *• 0.
This is the inverse operation of taking quotients. However, there is in general
more than one extension of 21 by 25. Later, we will see some examples of how
such extensions are classified. In this section, we will show that extensions of AF
algebras by AF algebras are AF.

92 III. Approximately Finite Dimensional (AF) C*-aIgebras
The key device is to lift a projection in the quotient algebra to a projection in
21. After that, the result follows using straightforward arguments.
Lemma III.6.1 Suppose that 3 is an AF ideal of a C*-algebra 21. Then for each
projection p in 21/3, there is a projection P in 21 such that P + 3 = p.
Proof. Let the quotient map be denoted by r. First lift p to a positive contraction
in 21. Indeed, if r(X) = p, then H = f(ReX) suffices, where f(x) = x A 1 V 0.
So
t(H) = f(r(ReX)) = f(Rer(X)) = f(p)=p.
Then r(e2*iH) = e2*ip = I because a(p) = {0,1}; and hence W = e2iriH is a
unitary element in 3 + CI.
Since 3 is AF, we may write it as an increasing union of finite dimensional
subalgebras 3n. For any 8 > 0, there is an integer n and a unitary U in 3n + CI
such that || W — U\\ < S. As r(U) = al for some scalar a, we have
|a-l| = ||r(l7-W0||<*.
Replace U by V = a~lU. Then r(V) = I and \\W - F|| < 28.
Since V lies in a finite dimensional algebra, it has finite spectrum. Thus there
is a positive operator A in 3n + CJ such that F = e2xti4 and 0 < A < I. Because
/ = T(V) = e2"iT(A\
it follows that r(A) is a projection. And as ||A|| < 1, this means that t(A) = 0; so
that A belongs to 3n.
Since the spectrum of A is finite, E := X \ 2 (A) is a projection in 3n. Define
K = (I - £)#(/ - S). We claim that for 8 sufficiently small, \ is not in a(K).
Once this claim is established, we may set P = X 1 .AK)- This is a projection in
21. Moreover, since E belongs to 3,
r(P) = \^(r(K)) = X(i A]ME)) = P-
Consider the functions f(e2vit) = < -12 and sCe2™'*) = t2 -13 for 0 < t < 1,
which are defined and continuous on the unit circle T. By continuity, for each
e > 0, there is a S > 0 so that if U and V are unitary such that \\U - V\\ < S, then
\\f(U) - f(V)\\ < e and \\g(U) - g(V)\\ < e (see Exercise E.8). Thus with this
choice of 8 for any given e, we obtain
\\(H - H2) -(A- A2)\\ = ||/(W0 - f(V)\\ < e
and
\\(H2 - H3) - (A2 - A3)\\ = \\g(W) - g(V)\\ < e.

III.6. Extensions 93
Notice that
sup (a -
0<a<|
- a2) = sup (a - a2) = |
§<"<i
= inf (a — a2) < sup
§<"<§ §<a<§
,2\_ 1
Hence
\E<A-A2 <\E+\EL.
Therefore
ELH2EL - K2 = ELH2EL - ELHELHEL
= ELHEHEL = EL(H - A)E(H - A)EL
< \EL{H - A)(A - A2){H - A)EL.
Thus
\\E^H2E^ - K2\\ < §(||i?(A - A2) - (A2 - A3)||)||i? - A\\
< § (||iT||||(A - A2) - (H - H2)\\ + \\(H2 - i?3) - (A2 - A3)||) \\H - A\\
< §(2e) = 9e.
We used the fact that \\H - A\\ < 1 because both are positive contractions. Finally
we compute
K-K2 = ELHEL -K2 = EL(H - H2)EL + ELH2EL - K2.
Thus
||jKT - K2\\ < \\EL ({H - H2) - (A - A2)) EL\\
+ \\EL(A - A2)EL\\ + \\ELH2EL - K2\\
<e + l + 9e<\
provided that e < ^. Since j — (j) = |> this implies that \ does not belong to
<t{K), completing the proof. ■
The next lemma manipulates matrix units much like we did earlier in this
chapter.
Lemma III.6.2 Suppose that 3 is an AF ideal of a C*-algebra 21 and that 2$ is
a finite dimensional subalgebra of $1/3. Then there is a (not necessarily unital)
*-monomorphism p of ^8 into 21 such that rp = id©.
Proof. Use Lemma HI.6.1 to lift the minimal central projections of 2$ to pairwise
orthogonal projections in 21. To do this, suppose that the minimal central
projections are pi,... ,pfc. First lift p\ to a projection Pi. Then consider the ideal
Pf- XPj1 in P1-L21P1-L. This is also an AF ideal, so the procedure may be repeated.
This reduces the problem to lifting Mn from pi*Bpi to i^2lP».

94
III. Approximately Finite Dimensional (AF) C*-aIgebras
Let eij for 1 < i, j < n be a set of matrix units for «Mn- As above, find
pairwise orthogonal projections 2?i,..., Fn such that r(Ej) = ejj for 1 < j < n.
For each j9 let Vj be chosen so that r(Vj) = eji. Then define Qj = EiVjEjVjEi.
Since r(QJ) = en, we see that Qj = E\ - Jj where Jj is a positive contraction
in 3.
By Exercise HI.l, there is an increasing sequence of projections which form an
approximate identity for Ei3Ei. Choose a projection F in E{3Ei such that
\\(Ei-F)Jj{Ei-F)\\<\ for l<j<n.
By replacing Ei with E[ = E\ - F in the definition of Q^ above, we obtain a
perturbation Qj = E[ - Jj with ||Jj|| < |. So the polar decomposition yields
the factorizations EjVjE[ = EjiQj where Eji is a partial isometry with initial
space E[ and r(J57Ji) = eji. Thus the range space is Fj := EjiE^ < Ej still
satisfies r(J5?j) = ejj.
Set Fy := EnEjV It is readily verified that this forms a set of matrix units in
21 which maps onto ey in the quotient. ■
Now we have the necessary ingredients to complete the proof of the main
result.
Theorem IH.6.3 Suppose that
0 *3 —^21 —^<B ^0
is an exact sequence of C*-algebras and that 3 and 2* are AF. Then 21 is AF.
Proof. Using Theorem HI.3.4, we take Ai,..., An in 21 and e > 0 and try to find a
finite dimensional subalgebra that almost contains them. Since t(.Ai), ..., r(An)
lie in the AF algebra *B, there is a finite dimensional subalgebra 5S0 of 25 and
elements 6» in <B0 such that \\r(Ai) - 6;|| < s/3 for 1 < i < n.
Lemma IH.6.2 provides a *-monomorphism /? of 2$o onto a subalgebra 2lo of
21 such that r/> = id<B0. Choose J* in J so that ||i4; - p(bi) - J;|| < e/3. Then
it suffices to approximate Bi := p(bi) + J{ within 2e/Z by elements of a finite
dimensional subalgebra. Note that the JB; lie in 2lo + 3.
The proof is completed by constructing an approximate identity of projections
for 3 which commutes with 2lo. To this end, let Ely for 1 < s < fc, 1 < i, j < n8
be a set of matrix units for 2lo; and let P be the unit of 2l0. For each s,2?}^3E[{ is
AF, and thus has a sequential approximate identity of projections, say FA \
Similarly, PL3PL has an approximate identity FA°' of projections. Define
5=1 t=l

Exercises 95
This is readily seen to be an approximate identity for 3. In addition,
E$Fn = E$E$fMe[?
= eI?f(»eI?
= El?F(')Ei?Ei? = FnE$
Hence each Fn commutes with 2lo.
For each A in 2lo and J in 3, we have
lim Fn(A + J)Fn + F^AF^ = A + lim FnJFn = A + J.
n-*oo n-*oo
So there is an integer n sufficiently large so that
\\Bi - (FnBiFn + F^p^F^W < e/8.
These approximants all lie in the algebra Fn3Fn + F^-2loF^. Since this is a direct
sum, and Fn3Fn is AF and F^^i0F^ is finite dimensional, this algebra is AF. So
there are elements C% lying in a finite dimensional subalgebra such that
\\FnBiFn + F£p(bi)F£ - Ci|| < 6/3.
This is the desired approximating set. So 21 is AF. ■
Exercises
III. 1 If 21 is AF, show that it has an approximate identity consisting of an
increasing sequence of projections.
HINT: Use the units of the subalgebras 2U.
III.2 Show that if P is a projection in an AF algebra 21, then P21P is AF.
NOTE: This result was used several times in the last section.
HI.3 Show that a direct limit of finite dimensional C*-algebras can be replaced
by a direct limit with injective connecting maps with the same limit.
HI.4 Suppose that 21 is a C*-subalgebra of a finite dimensional C*-algebra 25.
If a system of matrix units for 21 is given, show that there is a system of
matrix units for 25 such that each matrix unit of 21 is a sum of matrix units
of©.
III.5 Let 21 be a UHF algebra with unique trace r, and let nT be the
representation of 21 associated to r by the GNS construction. Prove that r extends to
a faithful WOT-continuous trace on 7rr(2l)".
IH.6 Show that a C*-subalgebra of an AF algebra need not be AF.
HINT: Show that C[0,1] can be imbedded as a subalgebra of C(X) where
X is the Cantor set.

96
III. Approximately Finite Dimensional (AF) C*-aIgebras
HI.7 Show that $(A) = fTX\(A) dX is a contractive projection of the CAR
algebra onto the GICAR algebra. Moreover, show that $ is positive and
faithful.
HI.8 Find all the ideals of the GICAR algebra.
m.9 A derivation of a C*-algebra 21 is a (bounded) linear map 8 of 21 into itself
such that
8(AB) = 8(A)B + A8(B).
A derivation 8 is called inner if there is an element X in 21 such that
8(A) = Sx{A) = XA - AX. It is approximately inner if it is the point-
wise limit of inner derivations.
(a) Show that every derivation of a finite dimensional C*-algebra is inner,
hint: Choose a finite group G of unitaries which span 21. Define
x = \g\~1Y,s(u)u-
ueg
(b) Show that every derivation of an AF algebra is approximately inner.
HINT: Show that \\X\\ < \\8\\ in part (a).
(c) Show that every derivation of the compact operators has the form 8x
forsomeXinB(W).
HI. 10 A derivation is self-adjoint or a *-derivation if 8(A*) = 8(A)*. In this
case, show that at = etS := Sn>o h^)n *s a *-automorphism for all
HINT: Show that Sn(AB) = ££=o (Z)8k(A)Sn-k(B).
Notes and Remarks.
Almost all of the material on AF algebras in this chapter is taken from the
seminal paper of Bratteli [1972]. The exceptions are the classification of UHF
algebras which is taken from Glimm [I960]; and the last section. The original
proof of the result on extensions due to Brown [1981] used K-theory. The proof
given here is due to Choi [1983].

CHAPTER IV
K-theory for AF C*-algebras
Ko is a functor that assigns an ordered abelian group to each ring based on
the structure of idempotents in the matrix algebras over the ring. It turns out to be
a useful and frequently computable algebraic invariant. For C*-algebras, it often
carries a lot of important information. In particular, it turns out to be a complete
invariant for AF algebras. Moreover, there is a complete description of all possible
Ko groups of AF algebras. This has proven to be a useful tool for constructing
C*-algebras with particular properties.
In a ring 11, say that two idempotents P and Q are (von Neumann) equivalent
if there are elements X, Y in 1Z such that
P = XY and Q = YX.
In a C*-algebra 21, two projections P and Q are *-equivalent if there is an element
X in 21 (necessarily a partial isometry) such that
P = X*X and Q = XX\
It is evident that these are equivalence relations. In a C*-algebra, it is more
natural and more convenient to consider only projections rather than arbitrary
idempotents. The first easy proposition shows that this choice does not affect things.
Moreover, since equivalence and *-equivalence are the same relation on
projections, we will use the term equivalence only after this proposition.
Proposition IV.1.1 In a C*-algebra, every idempotent is equivalent to a
projection; and equivalent projections are ^-equivalent
Proof. Suppose that J57 = J572 is an idempotent in a C*-algebra 21. Set
X = I + {E - E*)*(E-E*) = I-E-E* + EE* + E*E.
This is a strictly positive (hence invertible) element of 21". A calculation yields
EX = EE*E = XE. Thus E and J57* both commute with X"1. Therefore, we
may define a self-adjoint element P = EE*X~X. Furthermore,
P2 = X-1(EE*E)E*X~1 = X~lXEE*X~l = P.
So P is a projection; and it satisfies EP = P and PE = X~XEE*E = E. Thus
E and P are equivalent.
97

98
IV. K-theory for AF C*-aIgebras
Now suppose that P and Q are equivalent projections, and let X and Y be
given so that P = XY and Q = YX. Let X0 = PXQ and Y0 = QYP. Then
X0Y0 = PXQYP = P(XY)2P = P
and
Y0X0 = QYPXQ = Q(YX)2Q = Q.
Notice that since the range of X0 equals the range of P and the range of X£ equals
the range of Q, the partial isometry U in the polar decomposition X0 = U\X0\
satisfies UU* = P and U*U = Q. It must be shown that U belongs to 21. Now
P = P*P = XSY0*Y0Xo < \\Yo\\2XZX0 < ||y0||2||Xo||2P
Thus ||yoH"2P < XSX0 < \\X0\\2P. So a(X*X0) is contained in the interval
{0} U [||lo|r2, ll^oll2]; and X{X£X0) = P, where X is the function such that
X(0) = 0 and X(z) = 1 on a(X^X0) \ {0}. Therefore U = X0g(X^Xo) where
g is the continuous function on <t(XqXq) such that g(0) = 0 and g(x) = x~ll2
for x > ||lo||""2. In particular, U belongs to 21 and thus P and Q are *-equivalent
projections in 21. ■
The first clue that K0 has a topological nature comes from the fact that it is
a homotopy invariant. Equivalence satisfies homotopy invariance if whenever
p[t) = Pu for 0 < t < 1, is a continuous path of projections in 21, then P0 and Pi
are equivalent. This will then imply the homotopy invariance of jK"o-
Proposition IV.1.2 IfP and Q are projections such that \\P - Q\\ < 1, then they
are equivalent Thus equivalence is a homotopy invariant
Proof. For the first part, suppose that ||P-Q|| = 1 - e < 1 and consider the
operator X = PQ. Then
P > XX* = PQP = P- P(P - Q)P > eP.
Similarly, eQ < X*X < Q. As in the previous proposition, the partial isometry U
in the polar decomposition of X belongs to 21 and yields UU* = P and U*U = Q.
If p(t) = Pt is a continuous path of projections, continuity implies that there
are real numbers 0 = t0 < tx < • • • < tn = 1 so that ||Pt|. - PtM \\ < 1/2 for
1 < i < n. Hence by the first part of this proposition, each Ptimml is equivalent to
Ptr Thus Po and Pi are equivalent. ■
Corollary IV.1.3 If 21 = Un>i2ln is an AF algebra, then every projection in 21
is equivalent to a projection in Un>i2ln. Moreover, if P and Q in Un>i2ln are
equivalent in 21, they are equivalent in Un>i2ln.
Proof. By the density of Un>i2ln in 21, each projection P in 21 is close to 2ln for n
sufficiently large. So Lemma III.3.1 shows that there is a projection Q in 2ln with
||P - Q|| < 1/2. Therefore P and Q are equivalent by Proposition IV. 1.2.

IV. K-theory for AF C*-aIgebras
99
If P and Q in Un>i2ln are equivalent in 21 by a partial isometry U, choose an
element X = PXQ in 2ln for some n sufficiently large so that \\U - X|| < 1/2.
As above, the partial isometry V in the polar decomposition of X belongs to 2ln
and implements the equivalence. ■
Now we extend our definitions to deal with matrix algebras.
Let P(2l) denote the collection of all projections in Un>i Mn(%). This is a
semigroup under the operation of direct sum. Say that two projections in P(2l),
say P in Mm(^i) and Q in Mn(%) with m < n, are equivalent (write P ~ Q) if
P © 0n_m is equivalent to Q in A4n(2l). Say that P and Q are stably equivalent
(write P « Q) if there is a projection R in 7>(2l) so that P © J? ~ Q © i2. Stable
equivalence is evidently an equivalence relation. Let #o"(2l) denote the collection
of stable equivalence classes, denoted by [P] for P in P(2l).
We need the following routine results about jFf0(2l)+. In particular, part (a)
shows that direct sum induces a well defined operation on jKo(21)+.
Lemma IV.1.4 Let P, P', Q, Q' and R be projections in P(2l).
(a) IfPttP'andQ&Q',thenP®Q&P'®Q'.
(b) P®Q~Q®P.
(c) IfPQ = 0 in A*n(2l), thenP + Q~P®Q.
(d) IfP®R&Q®R, then P « Q.
Proof. All parts are easy. We will prove (c) as an example. Suppose that P, Q are
mutually orthogonal projections in Mn($l). Let X = [q g]. Then
X*X =
P + Q 0
0 0
and XX* =
P 0
0 Q
Corollary IV.1.5 Kq (21) is an abelian cancellation semigroup with the operation
[P] + [Q] := [P © Q] and with zero element [0].
Proof. By Lemma IV.1.4 (a), the sum operation on 1^(21) is we^ defined. By part
(b), it is commutative. By (d), [P] + [R] = [Q] + [R] implies that [P] = [Q]9 so
jK"o"(21) has the cancellation property. Associativity follows because direct sum is
an associative operation. Finally, it is clear that [0] is a zero element. ■
Say that a C*-algebra 21 satisfies cancellation if ~ and « are the same
equivalence relation on P(2l). This isn't always the case, but it simplifies things when it
is true.
Theorem IV.1.6 7f2l is an AF algebra, then 21 satisfies cancellation, and 1^(21)
is generated by the projections in 21.
Proof. Suppose that 21 = Un>i2ln. By Corollary IV.1.3, every projection in
tMp(2l) is equivalent to a projection P in Mp(Vln) for some n sufficiently large.
Suppose that 2U ~ Mkl © • • • © Mkt- Then Mp(%n) ^ Mpkx © • • • © Mpk,-
So we may write P ~ Pi © • • • © P,, and set 7r» = rank(P»).

100
IV. K-theory for AF C*-aIgebras
Since two projections in Mn are equivalent if and only if they have the same
rank, two projections P ~ Pi © • • • © P8 and Q ~ Qi © • • • © Q8 in 2ln are
equivalent in 2U exactly when rank(Pi) = rank(Q») for 1 < i < s.
Suppose that P « Q in 21, and let R be chosen so that P © R ~ Q © R. By
Corollary IV. 1.3, P © R and Q © R are equivalent in 21m for some sufficiently large
m. We may suppose without loss of generality that P and Q lie in Mp(%rn) and
R lies in Mq(%n). Using the notation above, write
P~Pi©...©P„ Q~Qi©- -©Q„ and i? - Rx © •••©£,.
Then in 21™, P®R~Q®R if and only if rank(P; © #;) = rank(Q; © #;),
which occurs if and only if rank(Pi) = rank(Q») which is equivalent to P ~ Q.
Let 2?}^ denote the rank 1 projection in the ith summand of 21m. Then
[j>] = !>[£!?]•
So K£ (21) is generated by the projections in 21. ■
IV.2 K0
An abelian cancellation semigroup S generates an abelian group G, called the
Grothendieck group of 5, consisting of all 'differences' of elements of 5 modulo
the natural equivalence. That is, consider the collection of all formal differences
s - t for 5, t in S and identify si - ti with s2 - *2 if and only if si+t2 = s2 + ti
in S. The cancellation property is exactly what is needed to show that this is an
equivalence relation. Indeed, suppose that si — ti = $2 - *2 and 52 —t2 = 53 - £3.
By definition this means that
s\+t2 = s2 + ti and 52 + £3 = 53 + *2-
Adding these two equations and using commutativity, we obtain
$1 + *3 + ($2 + t2) = 53 + *1 + ($2 + t2).
So by cancellation, si + £3 = 53 + ti and thus si - *i = 53 - £3.
Define addition by the rule (si - £1) + (s2 - t2) = (si + 52) - (*i + h)- It
is routine to verify that addition is well defined, commutative and associative. The
inverse of s — t is t - s, and the zero is 0 — 0. So G is an abelian group.
There is a natural imbedding of S into G given by the map taking s to s - 0.
This is injective because of the cancellation argument above. So with this
identification, we have G = S - S. Under good conditions, S determines a cone in G
which yields a useful partial order.
If 21 is a C*-algebra, define jFT0(21) to be the Grothendieck group of K$(21).
Example IV.2.1 For the moment, we content ourselves with a very elementary
example. Let 21 be a finite dimensional C*-algebra, say that 21 ~ Mni ©• • -®Mnk.
As we have seen, two projections P = Pi © • • • © P*. and Q = Q\ © • • • © Qk

IV.2. K0
101
in matrix algebras over 21 are equivalent if and only if rank(P») = rank(Qt) for
all 1 < i < k. As all ranks are possible, K£(9L) equals N§, the semigroup of
non-negative integer fc-tuples. Hence that the enveloping group is jK"o(21) = Z*.
Define a relation on jK"o(21) by setting x <yify - x € K£(Vl). Say that u is
an order unit of lfo(2l) if for every x in jFTo(21) there is a positive integer n such
that — nu < x < nu.
The following easy proposition establishes that < is a preorder (a reflexive and
transitive relation that is not necessarily symmetric). Then we establish a sufficient
condition to imply that it is a partial order.
Proposition IV.2.2 Let 21 be a C*-algebra. Then
(i) < is a preorder on jK"o(21).
(ii) ifx < y, then x + z <y + zfor all z € K0(Vl).
(iii) i/9l is unital, then [I] is an order unit for Ko($l).
Proof. Since 0 belongs to 1^(21), x < x for all x in jFT0(21). Also since 1^(21)
contains K£ (21) + 1^(21), we see that x < y and y < z implies that x < z. So <
is a preorder.
Part (ii) is immediate from the definition.
When 21 is unital, any projection P in Mn(Vl) satisfies 0 < P < In. Hence
0 ^ [P] < **[/]. So if Q is a projection in .Mm(2l), we obtain
-m[/] < [P] - [Q] < n[I]. M
A subset 5 of an abelian group G is a cone if S n (-5) = {0} and G = S-S.
In this case, the preorder < defined on G by S is a partial order. For if x < y < x,
then x - y belongs to both S and -5, whence x = y. The property G = S - S
implies, among other things, that given x and y in G, there is an element larger
than both of them. To see this, write x = a — b and y = c — d for elements a, 6, c, d
in 5. Then both » and y are dominated by a + c. Note that this does not mean that
there is necessarily a least element greater than x and y.
A C*-algebra 21 is called finite if whenever P and Q are equivalent projections
in 21 such that P < Q, then P = Q. The algebra is stably finite if A^n(2l) is finite
for all n > 1. Since equivalence in a matrix algebra Mn is determined by rank, it
is evident that finite dimensional C*-algebras are stably finite.
When 21 is unital, finiteness is equivalent to the statement that 21 contains no
proper isometries. Indeed, if 5 is an isometry in 21, then SS* < S*S = J. So
finiteness implies that SS* = J; whence S is unitary. Conversely, suppose that
21 is a unital C*-algebra with no proper isometries, and let P and Q be equivalent
projections in 21 such that P < Q. If X were a partial isometry in 21 such that
P = XX* and Q = X*Xy then S = X + QL would be an isometry in 21 with
range projection P + QL. Since X must be unitary, it follows that P = Q.
In fact, the term finite comes from the well known property of matrix algebras
that left invertibility implies invertibility (see Exercise IV.6).

102
IV. K-theory for AF C*-aIgebras
Theorem IV.2.3 AF algebras are stably finite.
Proof. Let 21 = Un>i2ln. Since Mk{W) = Un>iMk(%i)9 it follows that Mk{%)
is also AF. So it suffices to show that AF algebras are finite.
Suppose that P and Q are equivalent projections in 21 with P < Q. Use
Lemma 111.3.1 to obtain projections P' and Q' in some 2ln (for n sufficiently large)
so that P' < Q'and ||P'-P|| < 1/2 and ||Q'-Q|| < 1/2. By Proposition IV. 1.2,
[P>] = [P] = [Q] = [Ql
By the finiteness of 2ln, P' = Q'. Hence the projection Q - P has norm less than
1, which implies that P = Q. ■
Theorem IV.2.4 7/21 is stably finite, then K}(%) is a cone.
Proof. Suppose that P in Mk(%) and Q in Mi(%) are projections in matrix
algebras over 21 such that [P] = -[Q] is an element of iiC^(2l) D (-iiC^(2l)). Then
[0] = [P] + [Q] = [P®Q].
Since 0 < P © Q in .Mfc+/(2l), finiteness shows that P © Q = 0; and therefore
[P] = [Q] = [0]. The requirement that tf0(2l) = #+(21) - ^(21) follows from
the definition of K0 (21). ■
An immediate consequence of our results is the following.
Corollary IV.2.5 7f2l is stably finite, then ifo(2l) is a partially ordered group.
When 21 is unital, it also has an order unit.
IV.3 Dimension Groups
For the rest of this chapter, we restrict our attention to AF algebras. Call the
ordered group (jK"o(21), jK"o"(21)) associated to an AF algebra a dimension group.
Notice that {K0(Mn), K£(Mn)) = (Z, Z+) is the same for all the full matrix
algebras Mn, n > 1. To distinguish between dimensions, we introduce another
notion. A scale on a dimension group (G, G+) is a subset S of G+ which is
• hereditary: if 0 < g < s and s € 5, then g € S.
• directed: if si, s2 G S then there is an s £ 5 so that si < s and s2 < s-
• generating: every 5 € G+ is the sum of finitely many elements of 5.
For an AF algebra, we define the scale
T(2l) := {[P] : P a projection in 21}.
This is a generating set by Theorem IV. 1,6. Suppose that Q is a projection in 21 and
P is a projection in -Mn(2l) such that [P] < [Q]. Then there is a partial isometry
X in A^n(2l) such that XX* = P and X*X < Q © On-i. Therefore there is a
projection P' in 21 such that X*X = P' © On_i. Hence [P] = [P^ belongs to
T(2l). This shows that T(2l) is hereditary. To verify that T(2l) is directed, let P and
Q be projections in 21. By Corollary IV. 1.3, there is an integer n and projections

TV3. Dimension Groups
103
P' and Q' in 2U equivalent to P and Q respectively. Write 2U = Ya=i ®Mn.9
P = ELi ®pi and Q = £?=i ®Q{. By Example IV.2.1, the order in jFT0(2ln) is
determined by the rank of each summand. Choose a projection R{ in Mni with
rank(iii) = max{rank(Pi),rank(Qi)}.
Then R = £*=1 eii; is a projection in 2U with [P] < [fl] and [Q] < [R]. So
T(2l) is directed. This shows that T(2l) is a scale.
In particular, if 21 is unital, [J] is an order unit for jK"o(21); so
W) = {geK+(*):0<g<[T]}.
In the non-unital case, there need not be an order unit. For example, the abelian
C*-algebra Co of all sequences converging to 0 has countably many pairwise
orthogonal minimal projections E{. Together they generate K£(c0) ~ ]C£i ^+
which is a cone in K0(co) ~ 2Si ^- The scale is
r(c0) = {(ni) : Hi £ {0,1}, U{ = 0 except finitely often}.
It is easy to see that there is no order unit in this case.
A positive homomorphism of a dimension group (G, G+, T(G)) into another
dimension group (if, if+, T(H)) is a homomorphism tp of G into if such that
V>(<3+) C J5T+. The map tp is called contractive if rp(T(G)) C r(if). Also rp is
said to be unital if G and if have order units uq and uh determining T{G) and
r(iT) respectively, and iP{ug) = uh- A positive unital homomorphism is always
contractive.
If (p is a *-homomorphism of a C*-algebra 21 into another C*-algebra 25, then
there are *-homomorphisms <p(n) of Mn($l) into Mn(i8) obtained by applying (p
to each matrix entry. So we may define a natural map cp* from jFTo(21) into .K"o(25)
by
^([P]):=[^H(P)] for all P € 7>(2l).
In order to show that y>* is well defined, one must verify that P ~ Q implies
(p(P) ~ (p(Q), and that P « Q implies v?(P) « ¥>(Q). Both of these facts follow
trivially from the definitions. Also since
y>(m+n)(P 0 Q) = p<m>(P) 0 y>(n)(Q),
it follows readily that (p* is a homomorphism of K% (21) into iif^" (25). This extends
uniquely to a positive homomorphism of ifo(2l) into .Ko(2$). Moreover, since
projections in 21 are taken to projections in 25, 9?* is contractive. When (p is a
unital map of unital algebras, y>* is also unital.
Example IV.3.1 Consider the case of a homomorphism (p : 21 -> 25 between finite
dimensionalC*-algebras. Let21 ~ A4ni©' * -®^nfc and 25 ~ Mmi@- • -©.Mmr

104
IV. K-theory for AF C*-algebras
By Corollary in.2.2, (p is determined up to unitary equivalence in 2$ by an £ x k
matrix A = [ay] in Mik(^o) such that
k
y] dijUj < mi for 1 < i < t.
%=i
In the unital case, these are all equalities.
lifo(2l) is the scaled dimension group (Zfc, Z+, [0, [Jg]]), and jK"o(2$) equals
(Zl, Z{, [0, [/©]]), where [Ja] = (m,..., nky and [/*] = (mu ..., mt)K Let £,•
denote a minimal projection in the j-th summand of 21. Then [Ej] = ej, the vector
with j-th coordinate 1 and all others equal to 0. Now (p(Ej) = Fi © • • • © Ft,
where Fi is a projection of rank ay in Mni- Consequently,
[iP{Ej)] = (alh...,anj)t = Aei.
Hence y>* = A.
We need to recall some properties of direct limits of groups, suitably enhanced
for dimension groups. Suppose that (Gm, G+, T(Gm)) are dimension groups and
Tmn • Gm -> Gn are contractive homomorphisms for all m < n. The direct limit
G = lirnGn is the group (G, G+, T(G)) with the following properties:
(i) there are homomorphisms ym : Gm —Y G so that ynymn = 7m
for all
m < n.
(ii) G+ := Um>i7m(G+) and T(G) := Um>i7m(r(Gm)).
(iii) if H is any (ordered) group and pn : Gn ^ H are homomorphisms such
that pm = Pnjmn for all m <n, then there is a homomorphism p :G -t H
so that /?m = /?7m for all m > 1. If pm are all positive or contractive, then
so is p.
Condition (iii) is the universal property that distinguishes the direct limit. For
detailed information about direct limits, we recommend consulting a book on
algebra. However, we will prove one basic result that will be used repeatedly.
Lemma IV.3.2 Let G = lirnGn with connecting maps ymn. Suppose that <p is a
homomorphism of a finitely generated group H into Gm such that ym(p = 0. Then
there is an integer n> m such that ymn<p = 0.
Proof. The key is the fact that if g is in Gm and ymn{g) ^ 0 for all n > ?n,
then ym(g) ^ 0. To see this, consider the group K := Iln>i Gn/ J2n>i Gn> the
group of all sequences (gn) of elements gn in Gn modulo the relation thai identifies
sequences which eventually agree. Define homomorphisms Tn of Gn into K by
rn(flO = [{9k)] where
{0 k < n
7nk{g) k>n'

IV3. Dimension Groups
105
It is readily verified that Tn7mn = Tm for all m < n. Therefore by the universal
property (iii) of the direct limit, there is a homomorphism r of G into K so that
Tym = Tm for all m > 1. If g belongs to Gm and ymn(g) ^ 0 for all n > m, then
r7m(<7) = Tm{g) ± 0. Henceym{g) ± 0.
Suppose that <p : H -* Gm satisfies the hypotheses of the lemma. Applying
the key fact established above to each generator hi of if, we find integers nt- > m
so that ymni<p{hi) = 0. Thus with n = max{nt}, we obtain 7mn¥> = 0. ■
The following result will allow us to compute the K0 group of an AF algebra.
Theorem IV.3.3 7/21 = Un>i2ln is an AF algebra, then
as a scaled dimension group. If all imbeddings are unital, then the order unit is the
direct limit of the order units.
Proof. We are given *-homomorphisms (pmn : 21™ -> 2tn and (pn : 2ln -> 21
so that (pm = (pn(pmn for all m < n. Therefore we obtain contractive homo-
morphisms <pmn* : jFTo(2lm) -» ifo(2ln) and <pn* : K0($ln) -■> jFT0(21) so that
¥>m* = <Pn*<Pmn* for all TR < 71.
Let (G,G+,r(G)) := V^K0(fHn). By the definition of direct limits, there
are homomorphisms V>n • ^o(2ln) -► G so that tpm = rpn<Pmn* for all m < n.
Applying the universal property (iii) of direct limits of scaled dimension groups,
we obtain a contractive homomorphism ip : G —> jKo(21) so that the following
diagram commutes:
G-^KoW
We will complete the proof by showing that tp is an isomorphism.
Let [P] be an element of r(jK"0(2l)). By Corollary IV.1.3, there is an integer n
and a projection Q in 2tn so that <pn*([Q]) = [P]- Let g = VVi([Q])- This l*es in
r(G) by definition. Now ^(g) = [P]. Hence V> maps T(G) onto r(jFf0(2l)). Since
r(G) generates G+ and G, and similarly r(jFf0(2l)) generates i^(2l) and jFC0(21),
it follows that ip is surjective.
Suppose that tp(g) = 0. Choose an element x in ifo(2lm)> for m sufficiently
large, such that tpm(x) = g. Then choose projections P and Q in P(2lm) so that
x = [P]m - [Q]m (where the notation [P]m denotes the equivalence class of P in
K0(%n)l Since
0 = *(1>m(x)) = <pfn*(z) = [P\-[Q],

106
IV. K-theory for AF C*-algebras
we see that P « Q in 21. By Corollary IV. 1.3, there is an integer n > m so that
[P]n = [Q]n- Thus (pmn*(x) = 0. So g = lt>n(<Pmn*(*)) = 0. It follows that ^ is
injective. ■
Example IV.3.4 Consider the CAR algebra 21 (see Examples m.2.4 and III.5.4)
which is a direct limit of the full matrix algebras 2tn = My* with connecting
maps (pmn{A) = A^2n"m\ where A^ denotes the direct sum of k copies of A.
We have seen that K0(M2") = (Z, Z+). Since these algebras and the imbeddings
are all unital, we need to keep track of the order unit. The map of Kq(M2»)
into Z+ is given by rank. So the order unit is un = 2n. Thus the dimension
group is K0(M2n) = (Z, Z+, [0,2n]). By computing rank again, one sees that the
intertwining maps are given by <pmn*(k) = 2n~mk. It is convenient to normalize
so that the order unit is always 1. Then jFT0(2ln) = (2"nZ, 2-nZ+, [0,1]) and the
maps (pmn are the natural inclusions. It follows that
Ko(*) = (Z
.ZH
,[0,1])
where Z [|] is the group of diadic rationals.
Example IV.3.5 A much easier example to analyze is the algebra of compact
operators £. Think of A as the direct limit of matrix algebras Mn with imbeddings
(pmn : Mm -> Mn given by (pmn(A) = A © On-m. Then the corresponding
imbeddings yw of K0(Mm) = (Z, Z+l [0, m]) into if0(A<n) = (Z, Z+l [0, n])
is the identity map. Thus jFT0(^) = (Z, Z+> Z+).
Example IV.3.6 Consider the Fibonacci algebra of Example in.2.6. Recall that
21*. = Mmk © Mnjt, where m*. and n^ are given recursively by mi = ni = 1
and a repeated use of the partial imbedding matrix A = [J J], which yields the
recursion relation
rafc+i = ruk + r^ and nfc+i = m*.
It is easy to see that
^o(2lfc) = (Z2,Z|,[(8))(^)]).
Since a projection P©Q is sent to (P©Q)©P, the map from K0($lk) to jFTo(2lfc+i)
is given by the matrix A:
*^i((S)) = ii (5) = ("£")•

rV«3. Dimension Groups
107
Since v?fc,fc+i is an isomorphism for each k > 1, the limit group -Ko(2l) is
isomorphic to Z2. Indeed, map jK"o(2ln) onto jK"o(21) by the map A~n. So
Ko(Vlm) = Z2 ^ if0(2ln) = Z2
#o(2l) = Z2
is a commutative diagram for all m < n.
The order unit for K0^ik) is (£*) = A* (J). Thus the order unit for jFT0(21)
is (J). The positive cone is obtained as
K+(*)=\jA-nZl.
n>l
Setting no = 0, we obtain the Fibonacci sequence nk = rik-2 +n>k-i for all k > 2.
An easy induction argument shows that
A~k =
'(-l)*n*_i (-l)*-1^'
(-l)*"1^ (-l)*n*+i.
These are unimodular matrices, so they map Z2 onto itself. The image of li\ is the
cone generated by
\fc-i.
= /(-l)-*nfc
U-l)*n*+i
)•
From the well known formula
n>k =
r*+i + (_T)i-fe
T2 + l
where r := ^^, we obtain
hm —— = r+
fc-i>oo ri2fc
and
^2ib
lim
fc-i>oo Tl2fc-l
= T_.
Thus a2k = fck-i is approaching the line segment y = tz, x > 0; and similarly,
a2fc+i = /?2fc is approaching the line segment y = tz, x < 0; and r» + y > 0 for
all of these vectors. Thus the union of these cones is
*o+(a) = {(™):rm + n>0}.
The map from Z2 into (R, R+, 1) given by / ((™)) = m + r^n is a
positive unital homomorphism. Thus we see that the order on K0(9l) is a total order.
Indeed, we obtain the isomorphism
(ffo(Sl). K+(K), (J)) ~ (Z + t-'Z, (Z + r-^n R+l 1).

108 IV. K-theory for AF C*-aIgebras
Example IV.3.7 In this last example, we will compute the K0 group of the GICAR
algebra of Example HI.5.5. From the Bratteli diagram, which is like Pascal's
triangle, we obtain that jFf0(2ln) ^ (Zn+1, Z£+1). The map an* from K0{%,) into
^o(2ln+i)isgivenby
<*n*(ao. Oli • • -,an) = (ao,a0 + ai,ai + ^2. •• -.an-l +a>n,a>n)-
It is helpful to represent the group jK"o(2ln) as Vn = {p e Z[x] : degp < n} and
P+ = Z+[»] fi 7\i- Then the map an* becomes multiplication by » + 1. As we
have seen before, the natural choice of maps from Vn into the limit group (G, G+)
is given by multiplication by (x + l)~"n. The limit group then consists of
G = {(x + l)~np(x) : p e Z[z], degp <n,n> 1}.
The positive cone consists of those elements for which p has all positive
coefficients. Thus in order to determine if [x + l)~np(x) belongs to G+, one must
check the other representatives (x + l)~n~N((x + l)Np(x)). So this element
belongs to G+ if there is an integer N for which (x + l)Np(x) has all positive
coefficients.
Lemma IV.3.8 For each pair (a, 6) in R2 with 6^0, there is an integer N > 0
such that (x + l)N(x2 - 2ax + a2 + b2) has all positive coefficients.
Proof. Rewrite the quadratic as x2 - ax + /?; and notice that 4/3 > a2. We may
assume that a > 0, for otherwise N = 0 suffices. Compute
N
x
JV+2
h\(N + Z-bMaN>kX
m n 9u
where
ajv.fc = P(N + 2 - *)(# + 1 - *) - ak(N + 2 - Jfe) + k(k - 1)
= (1 + a + /?)Jfe2 - (2/3 + a) WJfe - (3/3 + 2a + l)k + p{N2 + 3N + 2)
= {l + a + fi)(k- f + a/2aN)2+/3(3N-M + 2)-(2a+l)k
1 + a + p
+ (1 + a + P)~1N2 ((1 + a + fi)fi -(13 + a/2)2)
> (1 + a + p)-xN2tf - £L) - 40 - (2a + 1)(JV + 2)
This is positive for large N independent oik. I

IV.4. Elliott's Theorem
109
Proposition IV.3.9 Given a non-zero polynomialp in R[x], there is an integer N
so that (x + l)Np(x) has all non-negative coefficients if and only ifp(x) > 0 on
(0,oo).
Proof. Clearly every non-zero polynomial with non-negative coefficients is strictly
positive on (0, oo). So this condition is necessary. Conversely, a polynomial is
strictly positive on the positive real line precisely when it has no roots there and its
leading coefficient is positive. So it factors as a product
p(x) = c]l(x + Xi) IT*2 - 2ajx + a) + b])
* 5
where A; > 0 and bj > 0. By the previous lemma, there are integers Nj > 0 so
that (x + l)^;*?2 - 2ajx + a2- + 6?) each has non-negative coefficients. Thus
N = J2jNj will suffice. ■
Hence G+ consists of those elements (x + l)~np(x) such that degp < n and
p(x) > 0 on (0, oo). We can obtain a less unwieldy formulation by making a
change of variables. Substitute y = (x + l)"1. The function [x + l)~np(x) for a
polynomial Sj=o a5x* *s transformed into
ynp(l-i) = fl"j(i-y)jyn-j-
i=o
Thus this map carries G onto Z[y], This transformation is a conformal map that
carries the positive real line onto (0,1). Thus a function is positive on the positive
real line if and only if the transformed function is positive on (0,1). So G+ consists
of those polynomials in Z[y] which are positive on (0,1). The order unit is the
constant polynomial 1.
IV.4 Elliott's Theorem
In this section, we will prove that the dimension group of a C*-algebra is a
complete invariant for AF algebras. As well as classifying AF algebras in a useful
algebraic way, this result has proven to be the prototype for a much larger
classification scheme for more general C*-algebras.
In order to recover an AF algebra from its K-theory, we need to be able to
recover maps from their induced homomorphisms. Example IV.3.1 essentially
indicates how this can be done in the finite dimensional case.
Lemma IV.4.1 Suppose that 21 and 25 are finite dimensional C*-algebras and that
ip is a contractive homomorphism of Ko($l) into jFCo(2$). Then there is a *-ho-
momorphism (pof$l into 25 such that <p* = tp, and (p is unique up to unitary
equivalence in 25. Moreover, iftp is unital, then so is (p.
Proof. Let 21 c=: Mni 0 • • • 0 Mnje and 25 ~ Mmi © • • • © Mmr #o(2l) is the
scaled dimension group (Zfc, Z£, [0, [/a]]) where [I%\ = (ni,..., n*.)*. Similarly,

110
IV. K-theory for AF C*-aIgebras
K0(<B) equals (Z*, Z|, [0, [J©]]) where [/»] = (mi,..., m/)'. The homomor-
phism if) of Z* into Z/ is given by an I x ft integer matrix A = [ay] so that
tp(z) = Az for each vector z in Z*. Since V> is contractive, it follows that ay > 0
and ip([I%]) < [/<b]- This latter condition translates into the inequalities
k
y^ aijnj < m» for 1 < i < £.
i=i
When ip is unital, these are equalities.
From Corollary m.2.2, we see that these are precisely the conditions required
on a set of partial multiplicities of an imbedding of 21 into 25. By that corollary,
there is a *-homomorphism <p of 21 into 25 with these partial multiplicities. It is
unique up to unitary equivalence in 25, and is unital when all the inequalities are
equalities. By Example IV.3.1, it follows that (p* = t{). ■
We next improve this result by replacing the image algebra 25 by an AF
algebra.
Lemma IV.4.2 Let 21 be a finite dimensional C*-algebra; and let 25 = Un>i25n
be an AF algebra with the imbeddings 6>/25m into 25 be denoted by (3m. Suppose
that ip is a contractive homomorphism ofKofil) into jFCo(25). Then there is an
integer m and a ^-homomorphism cp of$l into 25m such that (3m*(p* = tp. Moreover,
cp is unique up to unitary equivalence in 25; and iftp is unital, then so is (p.
Proof. Let 21 be isomorphic to Mni © • • • © Mnk- Let 1% be the unit for 21, and
let Ejy for 1 < j < ft, be the minimal projections of 21. Since ip maps projections
of 21 into T(25) = Un>i/3n*r(25n), there is an integer n sufficiently large so that
ip([I%\) and tp{[Ej])y fori < j < ft, all belong to /3n*r(25n). Choose projections J
and Fj in 25n so that pn*([J]) = Wa]) and jMM = V>([#j]) for 1 < j < ft.
In the unital case, take J = I<sn. Since
k
/MM-£";[^]) = o,
Lemma IV.3.2 shows that there is an integer m > n so that
k
i=i
Let Fj := Pnm(Fj) and J' = /3nm(J). Define a map p from jFT0(21) into
#o(25m) by setting p{[Ej]) = [Fj] and extending it to a homomorphism. By
construction, p is positive. Since
^) = £«;[^] = [^]er(<8m),

IV.4. Elliott's Theorem
111
it follows that p is contractive. By construction, we also have /3m*/> = rp.
Apply the previous lemma to p to obtain a homomorphism (p of 21 into 2Jm
such that <p* = p (and is unital in the unital case). This is the desired map.
Suppose that iff : 21 —■> 2Jm/ is another map with these properties. Choose an
integer £ > max{ra, m'} and replace (p and iff by Pmi<p and /3m'/¥>' respectively,
so that they map into the same subalgebra. Now /?z*(y>* - (p+) = 0. Again by
Lemma IV.3.2 there is an integer p > £ so that /?/p*(y>* - y>*) = 0. Hence by
Corollary HI.2.2, (3ip<p and Pip<p' are unitarily equivalent in 2JP. ■
We are now able to prove the main classification theorem for AF algebras.
Theorem IV.4.3 Two AF algebras 21 and 2$ are ^-isomorphic if and only if their
scaled dimension groups are isomorphic. Moreover, given a scaled dimension
group isomorphism p : jK"o(21) -> Ko(*B), there is a ^-isomorphism if of$l onto
2$ such that <p* = p.
Proof. As the second statement implies the non-trivial part of the first, we will only
prove the second statement. Let 21 = Um>i2lm and 2$ = Un>i 2Jn with connecting
maps am and j3n respectively. By Theorem IV.3.3,
#0(91) = ^K0(%n) and K0(<B) = KyKoVBn)-
The plan is to construct maps to make the following diagram commutative:
0£mirT»2
0£rT»2n»3
&m$m4
fir%ir%2
Pn2n3
Pn3n4
and so that <p* = p and ^* = p~l.
Choose two increasing sequences of positive integers mi < m^ < ... and
n\ < 712 < ... as follows. Let mi = 1. Apply Lemma IV.4.2 to the map pami*
of <Ko(2lmi) into KotfB) to obtain an integer rti > 1 and a *-homomorphism (pi
of 21m! into<8ni such that pni*<pu = P<Xmx*.
Now apply Lemma IV.4.2 to the map p^fim* of K0(*Bni) into K0($l) to
obtain a positive integer m and a *-homomorphism V> of 2Jni into 21m such that
/3m*V>* = P~lPmx*- Then notice that
OLn
*{ll>*<Pl* - Onum*) = 0.
Thus by Lemma IV.3.2 there is an integer m^ > m such that
«mm2*(V,*V?l* - «mim*) = 0.
Hence by Corollary HI.2.2 there is a unitary U in 2lm2 so that
Ad(tf)amm2V>¥>i = otmim2.
Set V>i := Ad(J7)amm2^.

112
IV. K-theory for AF C*-aIgebras
Now proceed recursively. If mi,..., m^, ni,..., n^-i, y>i,..., (fk-i and
V>i,..., V>fc-i are all defined so that the diagram above commutes, follow the
procedure of the previous paragraph to obtain n^ > n^-i and a map <pk of %mh into
2Jnfc so that
Pnk*<Pk* = /><*1
mfc*
and
<Pkfa-l=Pnk-Xnk-
Then choose m^+i > m* and a homomorphism V>fc from 2Jnfc into 2lmfc+1 so that
<Xmk+x*fa* = P~lPnk* and Vfc¥>fc = amfcmfc+1 •
Let (p : 21 -> 25 and V> : ® -> 21 be the maps determined by the maps (pk, fa
and the properties of direct limits so that the diagrams
21m,
-^21
2W+Ia—21
<Pk
®,
n*+i
and
**
Ai,
*»
fa
®.
tA
**
/3„fc
-^<B
commute for all fc > 1. As the restriction of ¥>lam*(2lm*) = <Pk is injective for
all k > 1 and V^I^W = <*mfc, it follows that xfxp = id^. Similarly, (pxf) = id©.
So 21 and <B are isomorphic. Moreover, at the level of K-theory, we have the
commutative diagrams
K0(%mh)a-2^K0(?L)
#o(2lmt)--^iro(20
M*nk)
Pnk*
\<P*
T
■*o(»)
and
K0(%nk)
Pnk*
\P
■*o(»)
By the uniqueness of the direct limit of the maps (pk*> we obtain (p* = p. Similarly,
fa = p~x. ■
IV.5 Applications
Since jK"o(21) is a complete invariant, it must be possible to read off various
algebraic information about 21 from jK"o(21). In this section, we will see some
examples of this.
Ideals. First consider the ideals of 21, which were classified in terms of subsets
of the Bratteli diagram in Theorem HI.4.2. A subgroup H of a partially ordered
group (G, G+) is called an order ideal provided that if+ := Hf\G+ is hereditary
(meaning that if 0 < g < h for some g in G and h in H+, then g belongs to H)
and H = H+ -H+.

IV.5. Applications
113
Proposition IV.5.1 The ideals of an AF algebra 21 are in one-to-one
correspondence with the order ideals of Ko($l) via the map which takes each ideal 3 to
K0(3).
Proof. Suppose that 3 is an ideal of 21. Then K0(3) is the subgroup of K0(%1)
generated by T(3) = {[P] : P = P2 G 3}. In particular,
K+(3) = K+(*)nK0(3).
To see that K$(3) is hereditary, suppose that 0 < [Q] < [P], where [Q] is in
jK"o~(21) and [P] belongs to K$(3). We may suppose that there is a common
integer n so that Q is a projection in Mn($l) and P is a projection in Mn(3).
By Theorem IV. 1.6, Q is unitarily equivalent to a projection Q' < P. Thus
Q' belongs to Mn{3) by Theorem 1.5.3, and so Q also lies in Mn{3). Since
K0(3) = K+(3) - K£(3)9 we see that K0(3) is an order ideal of tf0(2l).
Conversely, suppose that H is an order ideal of jK"o(21). Let 3 be the ideal
generated by the set S consisting of those projections P in 21 such that [P] belongs
to T(H) := H n r(tf0(2l)). We claim that
3 = {X e 21: there exist P» G S sothatX*X < ^2Pi}.
Indeed, the right hand side, say X, is contained in the ideal 3 by Theorem 1.5.3.
One shows that X is closed under sums by using a clever identity. Suppose that
m n
X*X<J2pi and Y*Y < ^Qj
for projections Pi and Qj in 5. Then
(x ± r)*(x ± y) < (x + y)*pr + r) + (* - ^)*(x - r)
m n
= 2X*x + 2y*y <2^pi + 2^gi
*=i j=i
and so X ± Y belong to X. To show that X is an ideal, it suffices by Theorem 1.8.4
to show that UX and XU belong to X for every X in X and every unitary U in
2T. Now UX belongs to X because
m
{uxyux = x*x < J] Pi.
t=i
Also
m
(xuyxu = u*x*xu < ]T crPitf.
t=i
Since [J7*Pit7] = [Pi] belongs to T(H), we see that U*PiU belongs to S and so
UX is in X. This establishes our claim.

114
IV. K-theory for AF C*-aIgebras
If Q is a projection in 3, the claim shows that there are projections Pi in S so
that Q < YT=i pi and therefore [Q] < E^P;]. As if+ is hereditary, Q also
belongs to S. Therefore T(K0(3)) = r(if); and hence K0(3) = H. This shows
that the map from ideals 3 to order ideals Ko(3) is surjective. On the other hand,
suppose that K0(3i) = Ko(32) = H for two ideals 3i and 32. As 3i and 32 are
AF, they are generated by their projections. If 32 contains a projection P, then [P]
belongs to T(K0(3i)) and thus P also belongs to 3\. Consequently 3i and 32 have
the same projections and are therefore equal. ■
A dimension group is called simple if it has no proper order ideals. We obtain
the following immediate corollary.
Corollary IV.5.2 An AF algebra 21 is simple if and only if (<Ko(2l), #<j"(2l)) is a
simple dimension group.
Traces. Generalizing the notion of the trace of a matrix, a (normalized) trace
on a C*-algebra 21 is a state r such that r(XY) = r(YX) for all X and Y in 21.
Using Theorem 1.8.4, we can easily see that to verify that r is a trace, it suffices
to show that t(UXU*) = r(X) for every X in 21 and every unitary element U
in 2l~. Also, because the projections span an AF algebra, it suffices to verify that
t(UPU*) = t(P) for every projection P and unitary U in 2l~. In particular, r
induces a well defined map r* on jK"o(21) by ^([P]) = r^(P) for projections P
in A/tn(2l). Hence we can reasonably expect to find a K-theoretic characterization
of traces.
The key observation is that r* is a contractive homomorphism of .Ko(2l) into
(R, R+, [0,1]). Indeed, for every projection P in 21, r{P) belongs to [0,1] because
r is positive and contractive. If P and Q are projections in .Mm(2l) and -Mn(2l)
respectively, then
T(m+n)(p 0 Q) = T(m)(p) + TW(Q);
from which we deduce that r* is a homomorphism. So we define a state (p on a
dimension group (G, G+, T(G)) to be a positive homomorphism into (R, R+, [0,1])
such that (p(T(G)) = v?(G+) fl [0,1]. In the unital case, this latter condition is
merely that (p([I]) = l.
Theorem IV.5.3 The traces on an AF algebra 21 are in a natural one-to-one
correspondence with states on jFTo(21) via the map which takes a trace r to r*.
Proof. Any trace on 21 extends to a trace on the unitization 21" so that r(I) = 1. So
for convenience, we suppose that our algebras are unital. We have already observed
that if r is a trace, then r* is a state on K0($l).
Conversely, suppose that p is a state on -Ko(2l). Define a map r on the rational
span of the projections in 21 by
m m
i=l t=l

IV.5. Applications 115
To see that this is well defined and contractive, suppose that
m n
||E'.p.-£^;||<l/*.
i=l J=l
Choose an integer N such that Nri, Nsj and N/k are all integers for 1 < i < m
and 1 < j < n. Then
AT rn n AT
»=1 j=l
It follows (see Exercise IV. 12) that
N
k
AT AT
i=l j=l
Hence applying p and dividing by N yields
1 m n 1
*=1 J=l
Thus r extends to a well defined state on 2l5a by continuity and hence by complex-
ification to all of 21. Moreover, since
t(UPU*) = p{[UPU*}) = p([P]) = r(P)
for every projection P and unitary U in 2l~, the remarks preceding the proof imply
that r is a trace. As this definition of r is the only possible one with r* = /?, there
is a unique trace for each state on Ko (21). ■
Example IV.5.4 Consider the algebra £ of compact operators. Then
tfo(£) = (Z,Z+,Z+).
The only homomorphism taking Z+ into [0,1] is the zero map. So there are no
(finite) traces.
On a finite dimensional C*-algebra 21 = Mnx © • • • © Mnu we have
tfo(2l) = (Z\ Z{, [0, [/]]) where [/] = ( i' ) •
Let Ej denote a rank one projection in the j-th summand, and let Ij denote the unit
of the j-th summand. If r is a trace on 21, let tj := t(Ij). Since [Ij] = rtj[Ej] and
[i] = EjU VA we must have
k
J]ti = l and r{Ej) = njltj.
i=i
Thus r is a convex combination of the unique normalized trace on each summand.

116
IV. K-theory for AF C*-aIgebras
Example IV.5.5 Consider the Fibonacci algebra 21 from Example IV.3.6. Recall
that
ifo(2l) = (Z2,{(-):rm + n>0},(J))
which is isomorphic to (Z + r~1Z> (Z + t_1Z) n R+, 1). Since this is a total
order, there is a unique unital homomorphism of K0($l) into (R, R+, 1), namely
r* ((™ )) = m + r_1n. So there is a unique trace on this algebra.
Automorphisms. The automorphism group of a C*-algebra is often difficult
to deal with. In the case of AF algebras, certain simplifications occur.
An automorphism a of a C*-algebra 21 is called inner if there is a unitary
operator U in 21" such that a = Ad U, which means that a(A) = UAU*. The set
of all inner automorphisms forms a normal subgroup Inn (21). An automorphism a
is approximately inner if it is the pointwise limit of inner automorphisms. That is,
there is a net a* of inner automorphisms such that a(A) = lim* <*k{A) for every A
in 21. When 21 is separable, a sequence will suffice. The subgroup of approximately
inner automorphisms is denoted by Inn (21).
We also define several other normal subgroups of Aut(2l). Let Aut0(2l)
denote the connected component of the identity map id in the topology of pointwise
convergence; and let Autp(2l) denote the path connected component of the
identity. Finally, let Id (21) denote the set of automorphisms a such that a* = id* on
#o(2l).
Remark IV.5.6 (a) Autp(2l) C Aut0(2l) C Id(2l) for any C*-algebra. The first
inclusion is true for all topological spaces. For each projection P in .Mn(2l), let
UP = {ae Aut(2l) : a^(P) ~ P}
= {aGAut(2l):a,([P]) = [P]}.
Suppose that P and Q are projections in 21 such that \\P - Q\\ < 1. Then by
Proposition IV. 1.2, P is equivalent to Q\ whence [P] = [Q]. Thus Up is both open
and closed, and so contains Auto(21). Thus
Aut0(2l)c f| WP = Id(2l).
Pe?(a)
(b) IrIrT(2t) C Id(2l). If a = Ad tf, then a*([P]) = [U^PU^*] = [P],
Thus Inn(2l) C Id(2l). By part (a), Id(2l) is closed. So it contains Inn(2l) too.
Theorem IV.5.7 If 21 is an AF algebra, then
irlri(2i) = Autp(2l) = Aut0(2l) = Id(2l).
Proof. Using the remarks above, it suffices to show that
Id(2l)cInn(2l)flAutp(2l).

IV.5. Applications
117
Write 21 = Un>i2ln as an increasing union of finite dimensional subalgebras.
Fix an a in Id(2l). Since 21 = Un>ia(2ln), Theorem HI.3.5 shows that there is a
unitary W in 2l~ such that
WUn>1a(%l)W* = Un>1%l.
Furthermore, we may assume that \\W - I\\ < 1/2 so that W is in the connected
component of J, and thus Ad W belongs to Autp(2l) fl Inn(2l). Hence we may
replace a by (Ad W)a.
So for every integer n, there is an integer m such that a(2ln) is contained in
21m. Let E{y 1 < i < fc, be a set of minimal projections in 2ln. Since a* = id*,
we have a(E{) ~ E{ in 21. By Corollary IV. 1.3, these projections are equivalent
in some 2lm.. Taking m' = max{ra, mt}, Corollary in.2.2 provides a unitary
Un in 2lm/ such that a\%n = Ad C/n|2ln- By dropping to a subsequence, we may
suppose that a(2ln) is contained in 2U+1 and that Un belongs to 2ln+i. Clearly,
a(A) = lirrin^oo Ad Un(A) for all A in Un>i2ln. Since all automorphisms are
contractive, this extends to the closure; whence a belongs to Inn (21).
Next, notice that since Ad Un+\ |2ln = Ad Un |2ln, the unitary element Un+\ U*
belongs to the finite dimensional C*-algebra 21^ fl 2ln+2. The unitary group is
connected in Mn and thus in any finite dimensional C*-algebra. Therefore, we may
choose a continuous path of unitaries Wjn* in 21^ fl 2ln+2 for 0 < s < 1 so that
Wq = I and W{' = Un+iU*. Then define a continuous path of unitaries in 21
by
Ut = W%t)U» for l"£<*<l-;iT,n>l
where an(t) = (n2 + n)t - n2 + 1 maps [1-^,1- ^~j] linearly onto [0,1].
By construction, AdC/t|2ln = a\%n for all 1 - £ < £ < 1. Consequently,
limt_n Ad Z7t(A) = a(A) for all A in 21. Thus a is the endpoint of the path
Ad Uu and therefore belongs to Autp(2l) fl lnn(2l). ■
Corollary IV.5.8 If '21 is a unital AF algebra such that jFTo(21) is totally ordered
and Archimedean, then Aut(2l) = lnn(2l). In particular, this is the case for UHF
algebras.
Proof. Since jK"o(21) is totally ordered and Archimedean, there is a unique unital
homomorphism into (R, R+, 1). Hence there is a unique unital automorphism of
jFT0(21), namely id*. But for any automorphism a, the map a* is a unital
automorphism of jFT0(21). Hence Aut(2l) = ld(2l) which, by Theorem IV.5.7, equals
Irlrifa).
A UHF algebra 21 = liir^A^nfc has the K0 group jFT0(21) = Ufc>i nl^ with
order structure inherited from the reals and order unit 1. Thus the hypotheses hold
in this case. ■

118
IV. K-theory for AF C* -algebras
IV.6 Riesz Groups
In this section, we develop some properties of a class of partially ordered
groups that will turn out to determine the possible K0 groups of an AF algebra.
The proof of this important fact will come in the following section.
A partially ordered group (G, G+) is called unperforated if whenever g G G,
k e N and kg G G+, then g belongs to G+.
Proposition IV.6.1 If 21 is an AF algebra, the group K0($l) is unperforated.
Proof. When 21 is finite dimensional, this is clear since G is isomorphic to (Zn, Z+)
in this case. Suppose that 21 = Un>i2ln with connecting maps amn. Then
tf0(2l) = lijri tf0(2ln) by Theorem IV.3.3.
Suppose that g is in K0($l) and k > 1 such that kg belongs to jFT^(21). Then
there is an integer m sufficiently large so that there are elements x in 1^(21™) and
y in K^(%n) for which am*(x) = g and am*(y) = kg. Then am*(y - kx) = 0.
Hence there is an integer n > m so that amn*(y - kx) = 0. Let x1 = amn*(»)
and y' = amn*(y). Then y' = kx' is positive in lfo(2ln)> which is unperforated.
Therefore x1 is positive. So 5 = an*(a;') is also positive. ■
The following proposition establishes the equivalence of several
decomposition properties that are somewhat weaker than a lattice property. This property will
turn out to be the key notion. A partially ordered group (G, G+) which is
unperforated and satisfies the Riesz interpolation property of the following theorem will
be called a Riesz group.
Theorem IV.6.2 For a partially ordered group (G, G+) the following are
equivalent:
(i) Riesz interpolation property: For every x\,xi,y\,yiinG such that X{ < yj
for i, j € {1,2}, there is an element z in G such that X{ < z < yj for
i,ie{i,2}.
(ii) Riesz decomposition property: For every x, 2/1,2/2 belonging to G+ such
that x < 2/1 + 2te> there are elements X{ in G+ such that x = x\ + x^ and
**' < y%fori = 1,2.
(iii) another Riesz decomposition property: For every xi)x2} 2/1.2/2 in G+ such
that X1 + X2 = yi+y2, there are elements Z{j in G+ such that X{ = zn + zt*2
and yj = zxj + z2jfor i, j G {1,2} .
Proof. (i)=Kii). Given the hypotheses of (ii), notice that 0 and x - 2/1 are both less
than x and 2/2- Hence by (i), there is an interpolating element z so that
{0,s-2/i} <*< {s.2/2}.
Then let xi = x-z and x2 = z. Clearly x = xi+x2. We already have 0 < z < y2.
Since x - y\ < z < x, we have 0 < x - z < yi.
(ii)=>(iii). Given the hypotheses of (iii), apply (ii) to «i < yi + 2/2 to obtain
positive elements zn < 2/1 and Z12 < 2/2 such that x\ = z\\ + z\2. Then define

IV.6. Riesz Groups
119
^21 = 3/i - *n and z22 = V2 - *i2• These are positive and are designed to satisfy
the necessary identities for y\ and 3/2. It remains to verify
*21 + *22 = Vl — -Zii + 3/2 - ^12
= (S/l + V2) - (*11 + 2i2) = 3l + «2 - 31 = 32-
(iii)=^(i). Given the hypotheses of (i), apply (iii) to the identity
(3/1 - xi) + (3/2 - 32) = (3/1 - x2) + (1/2 - si).
This yields elements z# in G+ such that
3/1 - xi = zn + Z12 3/2 - «2 = 221 + 222
J/l - «2 = -Zn + ^21 J/2 - 2l = *12 + z22
Set z = x\ + Z12 = 3/1 - zn = «2 + ^21 = 3/2 - ^22- From this, it is evident that
x% < z < Vj for i, j e {1,2}. ■
An ordered group G is lattice ordered if for each 2, y in G, there is a /e0.sf
upper bound z = 2 V 3/ in G; that is, 2 < z, 3/ < z, and if x < a and y < a,
then z < a. The greatest lower bound of x and 3/ is x A 3/ = -((-«) V (-3/)).
Lattice ordered groups satisfy the Riesz interpolation property; for if X{ < yj for
i, j G {1,2}, then z = «i V x2 is an interpolant.
A routine induction argument yields the following generalizations.
Corollary IV.6.3 IfG is a Riesz group, then
(i) For all X{, yj in G such that X{ < yj for 1 < i < k and 1 < j < £, there is
an element z in G such that X{ < z < yjfor 1 < i < k and 1 < j < £.
(ii) For every x, yj in G+ such that x < Y^j=i Vi> there are elements Xj in G+
such that x = ]Cj=i xi an<^ x5 ^ Vi for - ^ J -: '•
(iii) For all X{, yj in G+ such that ]Ct=ix* = 2j=i 30> ^r^ are elements Z{j
in G+ for 1 <i < k and 1 < j < £ such that
t k
i=i »=i
To bring this back to C*-algebras, we need the following easy result.
Proposition IV.6.4 If% is an AF algebra, then iCo(2l) is a Riesz group.
Proof. If 2ln is a finite dimensional C*-algebra, K0(%n) ~ (Z*, Z%) which is a
lattice ordered group. Hence it satisfies the Riesz interpolation property.
In the general case, K0(9i) = Ivv^Koi^). Suppose that X{ < yj in K0($l) for
hi £ {1)2}. Proceeding as in the proof of Proposition IV.6.1, we can find a; < bj
in Ko(%i) such that an*(ai) = X{ and an*(6J) = yj. Let c be an interpolating
point in Ko(%i) for a* and bj. Then z = an*(c) is the desired interpolant for Xi's
andy/s. ■

120
IV. K-theory for AF C*-algebras
IV.7 The Effros-Handelman-Shen Theorem
Our goal is to show that every Riesz group is the Ko group of an AF algebra.
Since these K0 groups are all direct limits of copies of (Zfc, Z£), we first obtain a
technical result about homomorphisms from (Zfc, Z+) into a Riesz group that will
be the key building block in the proof of the main result.
Lemma IV.7.1 Let (G, G+) be a Riesz group. Suppose that <p is a positive homo-
morphism of(lik, Z+) into (G, G+). Then there is an integer n and positive
homomorphisms a :Zk —J> Zn and if>: Zn -» G such that (p = tpa and ker a = ker (p.
Proof. As every subgroup of Zk is a finitely generated free group, there is a basis
ai» 1 < i < m, for ker (p so that ker (p = ]Ci=i ^-ai- It suffices to find positive
homomorphisms <t\ : Z* -» Zni and <pi : Zni -» G so that y> = (pi<Ti and ker cri
contains a\. For then, m repetitions will construct a with all m generators of ker (p
in its kernel.
Let a = a\. Rename the standard basis of Z* according to the sign of the
coefficients in the expansion of a as t\,..., er for those with positive coefficients,
/i,..., fs for negative coefficients, and g\,..., gt for zero coefficients. Thus there
are positive integers m; and rij such that
a = 2^ m;e; - ^ ni/i•
i=i j=i
Letp = p(a) = max{m;, n^} and d(a) = |{i: m; = p} U {j : rij = p}|. Replace
a by -a if necessary and re-order the basis so that mi = p. We further reduce our
task to the construction of a such that
(p(v(a)),d(*(a)))<(p(a),d(a))
in the lexicographic order. That is, we decrease d keeping p fixed or we decrease
p. Again, a repeated application of this procedure will eventually produce a homo-
morphism a\ so that (p(ai(a))1 d(ai(a))) = (0,0), meaning that <T\(a) = 0.
Let Si := <p(ei) and r/j := (p(fj). Then
r 8
0 = (p(a) = J^m^ - ^njTjj.
*=i i=i
Hence
r s 8
*=i i=i i=i
Since G is unperforated, we obtain 0 < S\ < ]Cj=i Vj- Bythe Riesz interpolation
property, there are elements 0j in G+ with 0j < tjj such that e\ = ]Cj=i fy-

IV.7. The Effros-Handelman-Shen Theorem 121
Set n = r - 1 + 25 + t and define maps a : Zk -> Zn and rp : Zn -> G
as follows. Denote a basis for Zn by fc£,..., h'8, e'2,..., e'r, f[,..., f8, g[,..., g't.
Then set
8
<r(e1) = Y,h'i ^h'j) = ei for 1<J<*
3=1
a(ei) = ej V(ei) = £t for 2 < i < r
"(/;) = /; + **; *Kfj) = 1j-9j for l<i<*
^(5fc) = fii ^(fli) = v(5fe) for 1 < fc < <
Clearly, both <r and rf> are positive. Compute
m«o = +(£,%) = !>=* = ^i)
if><r(ei) = -0(ei) = £t = ¥>(e») f°r 2 < i < r
Mfi) = *{fi + hj) = toi-*i) + Oj = Vi = *{fj) ^ l<i<5
H(9k) = 4>(g'k) = <P(9k) for l<fc<*
Thus ^(7 = v?.
Finally, we compute
*=i i=i
« r
j=i t=2 i=i
i=i t=2 j=i
Since p — rtj < p for all j, the total number of coefficients equal to p has been
decreased by one.
Thus the first stage of the construction has been successfully completed. The
lemma follows from repeated application. ■
The next theorem shows that Riesz groups are precisely the groups which are
direct limits of groups isomorphic to (Zfc, Z *). Together with Proposition IV.6.4,
this is most of the way to our main result.
Theorem IV.7.2 If G is a countable Riesz group, then G is the direct limit of a
sequenceGk c± (Zn*,Z+*).
Proof. We may assume that G ^ {0}. Let 51,52, •• • be an enumeration of
G+ \ {0}. Let Go = {0}, and let <po : Go -* G be the zero map. For the purpose

122
IV. K-theory for AF C*-algebras
of induction, suppose that Gk ^ (Zn*, Z+*) and (pk : Gk -+ G is a positive ho-
momorphism such that £1,..., £* belong to y>(G£). We will construct an ordered
group Gk+i — (Z^+^Z"*"1"1) and positive homomorphisms cr* : Gk -> Gfc+i
and (pk+i : Gfc+i -* G so that (pk+i<rk = ¥>*, 5fc+i belongs to ^+1(GjJ"+1) and
ker(7fc = kery>fc.
Let .K" = Gk © Z and define positive homomorphisms a : Gk -± K and
y> : K -> G by a(g) = (5,0) and (p(g, n) = (pk(g) + ngu+i. By Lemma IV.7.1,
there is a group Gfc+i ~ (Zn*+1, Z+*+1) and there are positive homomorphisms
<rf : K -> Gfc+i and y>*.+i : Gfc+i —► G so that y>fc+iV = (p and ker(j' = ker (p.
We have the commutative diagram
Gk ^K = Gk © Z--^Gfc+1
Set <rk •= fl'V. Then
Vfc+i^fc = (^fc+icr'JcT = (pa = <pk.
and since
(j(ker y>fc) = ker ^$0C ker (p = ker &\
it follows that ker (pk C ker ak. Hence ker (pk = ker ak.
Let JEf = lim Gk and let the maps from Gk to H be denoted by ^. Let p be
the positive homomorphism from H to G obtained from the universal property of
direct limits so that the following diagram commutes.
We will verify that p is an isomorphism. Notice that p is surjective because for
each gk, there is an element ak in GjJ" so that (pk(ak) = gk. Hence A* := ^k(ak)
belongs to H+ and
/o(frfc) = P^k{ak) = Wk(ajk) = 5fc.
Thus p{H+) = G+ (as containment follows from the positivity of p). Hence
p(H) = p(H+) - />(#+) = G+ - G+ = G.

IV.7. The Effros-Handelman-Shen Theorem
123
Now suppose that p(h) = 0. Then h = ^fc(a) for some element a in Gk for k
sufficiently large. Therefore
0 = p(h) = pipk{o) = (pk(a).
Hence a lies in keryj*. = ker cr*. Thus h = il>k+i<7k{a) = 0. So /o is an
isomorphism. ■
Now we are prepared to collect on the hard work. Since it is often easier to
verify that a group is Riesz than to build an AF algebra, this will supply us with a
potent tool for finding AF algebras with desirable properties.
Theorem IV.7.3 If G is a countable Riesz group with scale T(G), then there is a
unital AF algebra 21 such that
(K0(*)>K+{*),T{*)) ~ (G,G+,T(G)).
Proof. Using Theorem IV.7.2, write G as a direct limit of groups Hk isomorphic
to (Zn*, Z+*) with connecting maps (pkt such that ker (pk = ker y>fc,fc+i.
First suppose that G has an order unit uq such that T(G) = [0, uq]. We may
suppose that uq lies in the image of H\, say uq = ¥>i(ui) for a certain element
t*i. Define Uk = ¥>ifc(ui). Let Gk denote the subgroup of elements of Hk given by
Gk= [J {h e Hk : -nuk <h< nuk}.
n>l
Then Gk ^ (Zm*, Z™*), where m* < n* is the number of non-zero coefficients
of Uk in the standard basis for Hk- Moreover, Uk is an order unit for Gfc.
Since any g in Gk satisfies —nuk < g < nuk, it follows that
-nut < <Pkt{g) < nut.
So (pu(Gk) is contained in G/. Thus limGfc is the subgroup Ufc>i <Pk{Gk) of G.
However, if g belongs to G, there is an integer n so that —uuq <g < nuo. Since
nv>G — g and g + uuq are positive, there is an integer k and positive elements «, y
in H£ so that y>fc(s) = nuG - 9 and y>fc(y) = g + nuG- Thus a := nuk - s and
b:=y- nuk satisfy y?fc(a) = (pk{b) = g. Let z = (pk,k+i(a)- T^11 z = ¥>fc,fc+i(&)
as well because a- b belongs to ker (pk = ker (pktk+i • Hence
z = nufc+i - (pktk+i(v) < nuk+i
and
z = -nuk+i + ¥>fc,fc+i(2/) > -nuk+i.
Therefore z belongs to Gfc+i; and consequently g = y>fc+i(z) belongs to liryGfe.
Thus G = Hm Gk as desired.
Let Uk = (njki,..., nkmkY- Define

124
IV. K-theory for AF C*-algebras
Then K0(9ik) = G*. as a scaled dimension group. By Lemma IV.4.1, there is
a homomorphism a*. : 21*. -> 21&+1 so that a*.* = p*. Let 21 = lim2ln. By
Theorem IV.3.3,
i(To(2l) = lir^i(:o(2lfc) = lirr^G* = G
as scaled dimension groups.
In the general case, enumerate T(G) = {# : i > 1}. Since this is a
directed set, choose elements vn in T(G) such that # < vn for 1 < i < n. From
the arguments above, we can find an integer kn and element un in H£ so that
<Pkn (v>n) = Vn and so that tpn^nk {v>k-i) < ^fc. Then proceed as in the unital case
to construct an AF algebra 21 = lir^2ln such that i^0(2ln) ~ (Gfcn, Gj^, [0, ttn]).
The only additional thing to notice is that in the limit,
Uv*n([o,t*]) = U[°'v»] = r(G)-
n>l n>l
This shows that T(2l) = T(G) as desired. ■
IV.8 Blackadar's Simple Unital Projectionless C*-algebra
In this section, we will provide an application of the ideas presented about AF
algebras to construct a simple C*-algebra that has no projections except for zero
and the identity. This example, constructed by Blackadar, was the first C*-algebra
known to have this property. Of course, it is not an AF algebra, but it is built up
from one in a clever way.
The 2-adic numbers. Let Q denote the group of 2-adic numbers consisting of
elements which are formal sums q = ^2^Lm £fc2* where m 6 Z and Sk G {0,1}.
Addition and multiplication are defined in the usual way. This makes Q into a ring.
The subring Z of 2-adic integers consists of those elements with m = 0. This
subring contains a copy of the integers corresponding to sequences (ek) which are
eventually constant. For example, the formal sum ]Cfc>o ^k represents -1. In the
same way, Q contains the diadic rational numbers. Because 2 is a prime, it can be
verified that Q is a field. (We will not need this fact.)
There is a metric on Q given by the norm ||g|| = 2~m, where m is the least
integer such that em = 1 (and ||0|| = 0). It is easy to verify that
||?i - ?3|| < max{||gi - g2||, ||?2 - ftll}.
00
This is the topology inherited by Q as a subset of the Cantor set J] {0> 1}n-
n=—00
From this, it is evident that Z is compact; and so Q = Un>i2~nZ is cr-compact. It
is also evident from the definition of the metric that Z is a dense subset of Z.
Suppose that G is a compact open subgroup of (Q, +). Then it is contained
in 2"mlE for some m and it contains a basic open neighbourhood of 0, say 2*Z.

IV.8. Blackadar's Simple Unital Projectionless C*-algebra
125
It is not difficult to see that 2~mZ/2fcZ is the cyclic group on 2m+* elements. So
we deduce that the compact open subgroups have the form 2nZ for n in Z. Every
compact open set will be the union of cosets of these subgroups. Indeed, each
point a will have a neighbourhood a + 2*Z contained in the clopen set. A simple
argument allows one to reduce to a finite union of disjoint cosets.
As Q is a locally compact topological group (under addition), there is a Haar
measure n on Q which is unique if we normalize it so that //(Z) = 1. We
summarize these facts in the following theorem. Details may be found in Hewitt and
Ross, Abstract Harmonic Analysis, §H. 10.
Theorem IV.8.1 Q is a locally compact, a-compact, totally disconnected
topological field. Z is a compact open subring which contains Z as a dense subset. The
compact open subgroups of (Q, +) are 2nZfor n 6 Z. Every clopen subset ofQ
is a finite disjoint union of cosets of these subgroups. There is a (unique) Haar
measure fionQ such that fi(Z) = 1.
An AF algebra. Let G = (CC(Q, Z), +) be the additive group of continuous,
integer valued, compactly supported functions on Q. Define a state 0 on G by
%) := / 9 dp.
It is easy to verify that 0 takes values in the diadic rational numbers. Define a cone
onGby
G+:= {geG: 0(g) >0}U{0}.
Clearly G is a countable torsion free group. Let X^ denote the characteristic
function of Z; and define this to be the order unit of G.
Proposition IV.8.2 (G, G+, X^) is a simple dimension group with unique state 0.
Proof. That G+ + G+ is contained in G+ is clear. And if g is in G with \0(g) \ < n,
then 0(g + nXf) = 0(g) + n > 0. Hence g = (g + nX^) - nX^ belongs to
G^ -G^. It also follows similarly that
—nX^ < g < nX%
so that X% is an order unit.
To see that G is unperforated, suppose that ng > 0, meaning that
0 < 0{ng) = n0(g).
Then 0(g) > 0, and hence g > 0. Also G is a lattice ordered group with the
operations of min and max. So it is also a Riesz group.
Suppose that if is a non-zero order ideal of G. Then it contains a positive
element h. So for some integer n, 0(h) > 2~n. Hence 0(2nh - X^) > 0. That is,
0 < X^ < 2nh. By the hereditary property of order ideals, Xg belongs to H and
thus if = G. So G is simple.

126 IV. K-theory for AF C*-algebras
If r is a state on G and 0(g) = 0, then for every n in Z, 0(X% + ng) = 1 > 0
and hence X% + ng > 0. Thus
0 < t(X% + ng) = 1 + nr(flf) for all n G Z.
Hence r(g) = 0.
Consider X2ng for n > 0. For each integer 0 < a < 2n,
0(*2nz ~~ ^2nz+a) = 2 ~ 2 =0.
Hence r(X2ng+a) = f(X2ng). And since
2n-l
1 = r(xz) = r( ]T X2nZ+a) = 2nr(X^),
a=0
we deduce that r(X2ng+a) = 2~n = 0(X2nz+a) for all n > 0 and 0 < a < 2n.
Since these characteristic functions span G, it follows that r = 0. ■
By the Effros-Handelman-Shen Theorem IV.7.3, there is an AF algebra 35
such that (K0(<B), K$(<B), [I]) is order isomorphic to (G, G+, X^). As G is
simple, Corollary IV.5.2 implies that 35 is also simple. By Theorem IV.5.3, 35 has a
unique trace associated to the unique state 0 on G.
Define two automorphisms of G by
<r*9(*) = g(x - 1) and a*g(x) = g{x/2).
It is easy to see that both (7* and a* are positive since Haar measure is translation
invariant and is scaled by 2 on multiplying by 1/2. Set u = X% and v = X2g. Then
we have,
a^(u) = u a*(u) = v and <r*(v) = X1+2g = u - v.
In addition, (7* and a* are related by <rla+ = a*(7*. Indeed,
<rl<x*9{x) = <**5(* - 2) = flf(f - 1)
= ^*5(«/2) = OL+<T+g(x).
We also point out a fact that will be needed later: u is a minimal fixed point
of cr* in G+. Indeed, if cr*^) = g, then 5 is constant on cosets of Z, and hence
by continuity, it is constant on cosets of Z. Hence 0(g) is a positive integer. As
0(u) = 1, this is a minimal fixed point.
By Elliott's Theorem IV.4.3, there is an automorphism a of 35 which induces
cr*. There is a projection P in 35 with [P] = v. Since
Mt*])-w-[*]-[j-n
there is a partial isometry {/ in <B such that
U*(P)U* = I-P.

IV.8. Blackadar's Simple Unital Projectionless C*-algebra
127
Likewise, there is a partial isometry V so that Va(I - P) V* = P. Let
W = {I-P)U + PV.
This is a unitary operator such that Wa(P)W* = I - P. Replace a by Ad W a.
Then a(P) = I - P. Because (Ad W)* = id*, this doesn't affect a*.
Since a* is not unital, it does not come from an automorphism of 35. Define
35n = -P35P. Then using Exercise IV. 13, we may compute K0(<B0). It has order
unit [P] = v = X2^. However, this is an order unit for all of G by the same
argument that shows that Xg- is an order unit. Consequently,
(^o(»o),^o+(»o),[%]) * (<3,G+,t;).
It follows that a* is an order isomorphism of K0{*B) onto K0 (35n) • By Elliott's
Theorem IV.4.3, there is an isomorphism a of 35 onto 35n which induces a* on
the K0 groups. Since <r2(P) = P, we have <72(350) = 2Jo- Let p = cr2|«Bo
belong to Aut(2J0). Because K0(<B0) = G, it follows that /&* = cr2. Consider
(3 = oTlpa<T~l. This is an automorphism of 35 such that
ft* = a~l cla^c'1 = id* .
By Theorem IV.5.7, ft lies in the path component of the identity automorphism.
Let fa be a continuous path of automorphisms such that /3q = id and /?i = /?.
Then define the path at := a/3t of isomorphisms of 35 onto 35n from a0 = a to
ai = pa<T~l.
The Mapping Cone. Now define a C*-algebra T known as the mapping cone
of cr by
r = {/ec([o,i],<B):/(i) = <7(/(o))}.
This is a unital C*-algebra that is neither simple nor AF. It will be used to build
our example. Let us identify the space obtained from [0,1] by identifying 0 and 1
with the unit circle T. This is necessarily in the spectrum of T because the value of
/(l) is determined by /(0).
Proposition IV.8.3 T is a unital C*-algebra with no proper projections. The ideals
of T are precisely 3a := {/ G T : /\a = 0} for closed subsets A of T = [0,1]
mod 1.
Proof. If / = /* = /2, then f(t) is a projection for every t. By the homotopy
property of K0 (Proposition IV.1.2), we have that [/(*)] = [/(0)] for all t in [0,1].
Thus
[/(0)] = [/(I)] = W/(0))] = M[/(0)]).
Hence 0 < [/(0)] < [/] is a fixed point of a*. By our earlier comments, this
implies that /(0) is scalar; and thus / is scalar.
Suppose that 3 is an ideal of T. Let
A = ker3 = {teT: f(t) = 0 for all / G 3}.

128
IV. K-theory for AF C*-algebras
If t £ A, choose / in 3 such that f(t) = B ^ 0. Then since 2J is simple, there are
X{ and Y{ in 2J such that ]T^=1 X{BY{ = I. Pick any elements X{ and yf- in T such
that X{(t) = X{ and jfc(£) = Y{. (When t = 0, the corresponding value at £ = 1
is determined.) Then 3 contains h = ]C?=i xify%> which satisfies /&(£) = I. Set
kt = F(Re /*) where F(s) = a; A 1 V 0. It follows that 0 < kt < 1, kt belongs to
J and kt(t) = I.
The proof is completed by a partition of unity argument. Fix # in 3a and e > 0.
The set B = {£ G T : ||<7(*)|| > e} is compact and disjoint from A. For t in B,
there is a neighbourhood
0* = {s e T : kt(s) > (1 - e)I}.
By compactness, there is a finite open cover Ot|. of S for certain points £.$ in T.
Then
belongs to 3, 0 < fc < 1 and k(t) = I for all t in B. Thus
teb(g,3)<\\g-gk\\< sup 11^)11 11/-*(*)ll<e-
teT\B
Hence 3 = 3,4. ■
Define an endomorphism if; of T by
Note that at maps 35 into 2J0 = P2JP; while a~xai = cracr"1 maps 2$ into
PL 93P1. Consequently, this is a direct sum. It follows that ^ is an endomorphism.
Moreover, it is readily seen to be injective. To verify that tpf actually lies in T,
compute
<rtf/(0) = a (a(/(0)) + <7"1a1 (/(§)))
= ax (/(§))+ <m(<7-1/(l))
The map ^ is unital since i(>l(t) = P + PL = I.
Let Tn = T for n > 1, and set t(>n = -0 : Tn -> Tn+i. Then define a
C*-algebra 21 = lirn(rn, i(>n). We identify Tn with its image in 21. This map is
injective because each rpn is injective. So 21 = Un>i.Tn.
Theorem IV.8.4 The C*-algebra 21 is a simple unital projectionless C*-algebra.
Proof. If E is a projection in 21, then there is an integer n sufficiently large that
dist(i£, Tn) < 1/4. Hence by Lemma HI.3.1, there is a projection F in Tn such
that ||E - F|| < 1/2. By Proposition IV.8.3, F is 0 or 1. Thus E is also scalar. So
21 has no non-trivial projections.

Exercises
129
Suppose that 3 is an ideal of 21. By Lemma HI.4.1,
j= |Janrn.
n>l
As each 3n = 3 D Tn is an ideal of Tn, it has the form 3Un for a certain closed
subset of T by Proposition IV.8.3. Moreover, the nature of the imbedding of Tn
into rn+i yields
^PAJ = ^n+1n^(rn).
This implies that
An = 2^n+l + (2^+1 + 2)'
So An is invariant under translation by |. Inductively, we deduce that An is
invariant under translation by 2~k for all k > 1. Since An is closed, this implies that it is
empty or equal to all of T. If An is empty, then 3n contains the identity and hence
3 = 21. On the other hand, if An = T for every n > 1, then 3n = {0} for all n;
whence 3 = {0}. Thus 21 is simple. ■
Exercises
IV. 1 Show that if two projections are homotopic in the set of idempotents, then
they are homotopic in the set of projections.
IV.2 Show that equivalent projections need not be homotopic.
HINT: Consider the projections
1 0
0 0
and Q
t y/t(l-t)\
[0(1 - t) 1-t J
in M2 (£?[(), 1]). Show that they are contained in a common C*-subalgebra
in which they are equivalent but not homotopic.
IV.3 Suppose that 21 is contained in B(H). Show that if E is an idempotent, then
the equivalent projection computed in Proposition IV. 1.1 is the orthogonal
projection onto the range of E.
IV.4 Two homomorphisms of 21 into 2$ are homotopic is there is a pointwise
continuous path of homomorphisms connecting them. Show that if a is
homotopic to /3, then a* = /?*.
IV.5 A C*-algebra is contractible if the identity automorphism is homotopic to
the zero map. Show that the cone of 21,
C2l={/GC([0,l],2l):/(0) = 0}
is contractible.
IV.6 Show that if 21 is finite and unital, then left invertible elements are invert-
ible.

130 IV. K-theory for AF C*-algebras
IV.7 Show that a UHF algebra 21 has a unique trace r. Then show that r* is an
isomorphism of .K"o(2l) int0 R-
IV. 8 Find all the order ideals of K0 of the GICAR algebra. Compare this
description of the ideals of the algebra with the Bratteli diagram approach.
IV.9 Show that if 21 has a faithful trace, then it is stably finite.
IV.10 Show that Inn(2l), Inn(2l), Aut0(2l) and Id(2l) are all normal subgroups
ofAut(2l).
IV. 11 In a finite dimensional C*-algebra 21, show that [P] < [Q] if and only if
r(P) < t(Q) for every trace on 21.
IV.12 Suppose that P{ and Qj are projections in an AF algebra such that
m n
£*<£<&■
ShowthatE^[^]<Ei=1[Qi]-
HINT: Reduce the question to a finite dimensional subalgebra, and use the
previous exercise.
IV. 13 If 21 is AF and Pisa projection in 21, show that
K0{PZP) = |J [-n[P], n[P]], K+(PZP) = K0(P*P) 0 tf0+(2l)
n>l
and the order unit is [P].
IV. 14 Show that every countable torsion free abelian group G is the direct limit
of a sequence of Zn*. Show that G can be given an order to make it into a
dimension group.
IV15 If0->Cf->(£-i>2l->0isan exact sequence of AF algebras, show that
0 ->> Kofi) -> K0(<£) -> K0(9l) -» 0 is an exact sequence of ordered
groups.
IV. 16 Show that G = Z2 with the order
G+ = {(m, n) : m = n = 0 or m > 0, n > 0}
is an ordered group that fails to have the Riesz interpolation property.
Show that 0 ->> (Z,Z+) ->> (G,G+) ->> (Z,Z+) -> 0 is an exact
sequence of ordered gtoups. Thus extensions of Riesz groups need not be
Riesz groups.
IV.17 (a) If 21 is AF, consider the imbeddings of .Mn(2l) into .Mn+i(2l) given
by an(A) =[£§]. Show that 21® j? := lir^A^n(2l) is AF; and show that
{K0{% ® R), K+{% ® R)% T(2l ® R)) ~ (^o(2l), K+{%), K+{%)).

Exercises
131
(b)If 21 and 35 are AF, show that 21 <g> £ and 2J <g> £ are isomorphic if and
only if (tf0(2l), *tf(2l)) and (tf0(<B), Kf(<B)) are isomorphic.
IV. 18 Let 21 be Blackadar's projectionless C*-algebra.
(a) Show that every trace on T has the form J* t(/(£)) d\i for some
probability measure n on [0,1], where r is the unique trace on 2*.
(b) Define traces on each Tn by rn(f) = J* r(f(t)) dt. Then verify that
rn+i^ = rn. Thus deduce that r = limrn is a trace on 21.
(c) Show that 21 has a unique trace.
HINT: Show that the restriction to each Tn corresponds to a measure fin
which is invariant under diadic translations; and thus is Lebesgue measure.
Notes and Remarks.
Equivalence of projections was first introduced for von Neumann algebras by
Murray and von Neumann [1936]. K-theory comes from algebraic topology. A
classical introduction to topological K-theory is Atiyah [1967]. It was first
introduced to C*-algebras by Elliott [1978], although his paper Elliott [1976]
classifying AF algebras by their equivalence classes of projections makes no reference to
the enveloping group. The basic reference for Riesz groups is Fuchs [1965]. Elliott
[1978] pointed out that dimension groups have the Riesz property. Effros, Handel-
man and Shen [1980] proved Theorem IV.7.3. Blackadar [1980] constructed this
first known example of a simple projectionless C*-algebra. Blackadar's paper also
contains Theorem IV.5.7.

CHAPTER V
C*-algebras of Isometries
In this chapter, we will study several important classes of C*-algebras built
from isometries and weighted shifts.
V.l Toeplitz Operators
The most important non-normal operator is the unilateral shift. It is much
studied because it displays many interesting phenomena that occur in infinite
dimensions but not in finite dimensional spaces. This operator is defined on an or-
thonormal basis {en : n > 0} by
S 22 an^n = 2J an^n+l •
n>0 n>0
It is immediately evident that 5 is an isometry with range equal to the closed
span of {en : n > 1}. This is a proper subspace of codimension 1. Thus S is
a proper isometry, meaning that it is isometric but not unitary. Moreover, it is
Fredholm with index
ind(5) = null(5) - null(5*) = -1.
There is a convenient representation of S which allows us to utilize an
important connection with function theory. Let L2(T) denote the square integrable
functions on the unit circle with respect to Lebesgue measure. This has an or-
thonormal basis en = zn for n G Z; where z is the identity function on T. Let H2
or H2(T) denote the subspace of L2(T) spanned by {en(z) : n > 0}. This is the
subspace of L2 functions of analytic type. Indeed, the expansion h = ]Cn>o a**en
is the Fourier series of h. The Poisson extension of h to the unit disk is ]Cn>o an*n>
which converges uniformly on compact subsets of the unit disk D; and therefore
determines an analytic function h(z) on D.
It is an easy exercise to verify that
lim ||Mre")||2 = 11%.
It also follows from Fatou's theorem that the radial limits exist almost everywhere
lim h(rei$) = h(ei$) a.e.
r-»l-
132

V.l. Toeplitz Operators 133
While we will not make use of this fact here, this shows that there is an intimate
connection with analytic function theory.
If g is a bounded measurable function on the circle, g G L°°(T), define a
multiplication operator Mg on L2(T) by Mgf = gf. This is easily seen to be a
normal operator with norm \\Mg\\ = \\g\\oo- Define the Toeplitz operator Tg on
if2 by
Tgh = PH2gh.
This is just the compression of Mg to the subspace H2. In particular, the operator
Tz acts on the basis en by Tzen = en+1. Thus Tz is unitarily equivalent to the
unilateral shift.
Proposition V.1.1 For g in L°°, T; = Tgand \\Tg\\ = \\Tg\\e = M^.
Proof. The first statement follows from a simple calculation:
(21;/, h) = (/, PH29h) = (/, gh) = (gf, h) = (Tgf, h)
for all/, hin H2. Now
\\Tg\\* < \\Tg\\ < \\Mg\\ = \\g\U
Let e > 0 be given. Since the trigonometric polynomials (polynomials in z and
z"1) are dense in L2(T), there is a polynomialp = ^2^=-n a*e* w^tu ll-Plh = -
such that ||5p||2 > Htfllco - £• Then for all integers n > N, znp belongs to H2.
Suppose that gp = EfcL-oo h^k- Then
oo
Tgznp = PH2Zngp= ]T bkek+n.
k=-n
Hence
lim\\Tgznp\\2 = \\gp\\2>\\g\\O0-e.
n—too
Moreover, the sequence znp tends weakly to zero (and in fact, zZN*p are pairwise
orthogonal for s > 1). Hence we deduce that \\Tg\\e = \\g\\oo- ■
Let H°° denote the subalgebra of L°° of functions of analytic type.
(Consequently, H°° = H2 n L°°.) An operator is subnormal if it is the restriction of a
normal operator to an invariant subspace.
Proposition V.1.2 For h in H°°, the space H2 is invariant for M^; and thus
analytic Toeplitz operators are subnormal Th = Mh\H2. For every g in L°° and h in
H°°f one has
TgTh = Tgh and T^Tg = T^.

134 V. C*-algebras of Isometries
Proof. Let h = £n>o hnzn. Then Mhzk = £n>o hnzn+k belongs to H2 for
all k > 0. Thus if2 is invariant for Mh, and so T/J = hf for all / in H2.
Consequently,
TgThf = T,ft/ = Pj^fc/ = T,fc/.
Finally,
A key fact is that the shift Tz commutes with all Toeplitz operators modulo the
compact operators.
Proposition V.1.3 For every g in L°°, the commutator TzTg - TgTz has rank at
most one.
Proof. Notice that TgTz = Tgz by the previous proposition. So working in L2,
TzTg - Tgz = {PH2MzPH2MgPH2 - PH2MzMgPH2)\H2
= PH2MzP£2MgPH2\H2.
Since PH2 MZP£2 = £o£*-i is rank one, the commutator has rank at most one. ■
Corollary V.1.4 For all g in L°° and f in C (T), the semi-commutators TfTg - Tfg
and TgTf - Tgf are compact.
Proof. Given e > 0, there is a trigonometric polynomial p = J2%=-n a*z* suc^
that ||/ - p\\oo < e. As in the proof of the previous proposition, we obtain
TgTf - Tgf = PH2MgP£2MfPH2.
So it suffices to show that P£2MfPH2 is compact. The range of P^2MPPH2 is
contained in span{z~* : 1 < k < N}; and thus it is finite rank. Since
||P^(M/-Mp)PH2|l<ll/-p||oo<e,
it follows that P^2 MfPH2 is the norm limit of finite rank operators and thus is
compact. The other term is handled by taking adjoints. ■
Now we are prepared to compute the C*-algebra generated by Tz.
Theorem V.1.5 The C*-algebra generated by Tz has the form
T(C(T)) := {Tf + K : / G C(T) and K G £},
where C(T) is the space of continuous functions on T and& is the ideal of compact
operators &. This algebra is irreducible and contains & as its unique minimal ideal
The map s(f) = Tf is a continuous section of the exact sequence
0 ^£—l-+C*(Tz)^=ZC{T) ^0

V.l. Toeplitz Operators
135
Proof. Suppose that P is a projection commuting with Tz. Then P commutes with
I - TZT£ = ene*,. So Pen = e0 or Pe0 = 0. In the former case,
Pen = PT£eo = T£Peo = en
for all n > 0. Thus P = I. While in the second case, we similarly find P = 0.
Hence C*(TZ) is irreducible.
Since C (Tz) contains a non-zero compact operator, it must contain all of & by
Corollary 1.10.4. It also clearly contains Tp for every trigonometric polynomial p
(as these are polynomials in Tz and Tz-i = T*). As the trigonometric polynomials
are dense in C(T), C*(TZ) contains all Tf for / in C(T).
Any non-zero ideal containing a non-zero operator X will contain non-zero
compact operators of the form XF for some finite rank operator F. Therefore,
by Lemma 1.9.15, this ideal is irreducible; and hence contains all the compact
operators by Corollary 1.10.4. So & is the unique minimal ideal.
On the other hand, Corollary V.l.4 shows that products of Toeplitz operators
with continuous symbol are Toeplitz operators (with continuous symbol) plus some
compact operator. Thus T(C(T)) is a *-algebra. To see that it is norm closed,
suppose that T/n + Kn converges to an operator X, where fn belong to C(T) and
Kn are compact. Then
||/n " /mlloo = \\Tfn ~ T/Jle < \\T,m + Kn - (T,m + Km)\\.
So fn is a Cauchy sequence with uniform limit / in C(T). It follows that Kn is
also Cauchy with compact limit K. Hence X = Tf + K. Therefore T(C(T))
is closed. So T(C(T)) is a C*-algebra containing Tz and contained in C*(TZ);
whence they are equal.
Let 7T be the quotient map of C*(TZ) onto C*(Tz)/£. By Corollary V.1.4, this
quotient algebra is abelian. Moreover, the map ns : C(T) -> C*(Tz)/£ is
isometric by Proposition V.l.l, and an algebra *-homomorphism by Corollary V.1.4.
Hence it is a *-isomorphism. With this identification, it is clear that s is a
continuous section of the quotient map. ■
Since an operator is invertible modulo the compact operators precisely when it
is Fredholm, we obtain the following sharp spectral theorem for Toeplitz operators
with continuous symbol.
Theorem V.1.6 For f in C(T), Tf is Fredholm if and only ifO is not in the range
off. In this case, ind(T/) = - wind(/), where wind(/) is the winding number of
the oriented curve /(T) about 0.
Proof. Since C*(Tz)/£ ~ C(T), it follows that
<Te(Tf) = *C(T)(f) = /(T).
So if ft is a homotopy of non-vanishing functions in C(T), the continuity of the
Fredholm index and winding number shows that
ind(T/o) = ind(TA) and wind(/0) = wind(/i).

136
V. C*-algebras of Isometries
Every non-vanishing function / in C(T) is homotopic to zn where n is the winding
number of / about 0. Thus
ind(T/) = ind(2>) = -n = - wind(/). ■
Lemma V.1.7 Either ker Tg = 0 or ker T* = Ofor every non-zero g in L°°. So if
Tg is Fredholm of index 0, then it is invertible.
Proof. Suppose, to the contrary, that h and k are non-zero functions in H2 such that
Tgh = 0 = Tgk. Then 'gh and gk lie in (if2)1 = H^ the space of H2 functions
such that h(0) = 0. Now HqH2 is contained in Hq, the space of L1 functions on
T such that f(n) = 0 for n < 0. Thus / = (gh)k and / = (gk)h lie in H&. But
then f(n) = 0 = f(n) = 0 for n < 0; whence /(n) = 0 for all n in Z. It follows
that / = 0, since the Fourier transform is injective on L1.
To complete the proof, we need to quote a result of F. and M. Riesz from
function theory that asserts that a non-zero function h in H2 does not vanish on a subset
of T of positive measure (see Exercise V.4). Thus hk ^ 0 a.e., and consequently
g = 0 a.e., contrary to fact. Hence at least one of ker Tg or ker T* must be zero. A
Fredholm operator Tg of index 0 satisfies nall(Tg) = null(T£). Thus both kernels
are 0, and therefore Tg is invertible. ■
The following complete spectral picture for a Toeplitz operator with
continuous symbol is now immediate.
Corollary V.1.8 /// belongs to C(T), then <re(Tf) = /(T) and
a(T/) = /(T)U{A:wind(/-A)#0}.
V.2 Isometries
In this section, we consider the C*-algebra of a general proper isometry. The
first step is the Wold decomposition that provides a structure theory for isometries.
Theorem V.2.1 If S is an isometry on a Hilbert space H, then there is a cardinal
number a and a unitary operator U {possibly vacuous) such that S is unitarily
equivalent to TJa) 0 U.
Proof. Let M = ker 5* = (Ran5)1; and let a = dimM. Notice that SnM is
orthogonal to SmM forn > m > 0. Indeed,
{Snx1Smy) = (Sn-mz,y) = 0 for all z,yeM.
Fix an orthonormal basis (4 : 0 < i < a} for M. Define e[ = Ske{0 for k > 1
and 0 < i < a. For fixed k, this forms an orthonormal basis for SkM. Together,
they form a basis for £ ®k>0 SkM =: K. It is evident that the restriction of S to

\3. Bunce-Deddens Algebras
137
Mi := span{ej, : k > 0} is unitarily equivalent to the unilateral shift Tz for each
0 < i < a. Hence the restriction to K is unitarily equivalent to TJ \ In particular,
SK = J2®SkM = K 0 M = Kn Ran5 C K.
**>i
Now SKL is orthogonal to SK + (Ran5)1 = K. So SKL is contained in
KL. On the other hand, /C1 is contained in Ran S and is orthogonal to SK. Hence
5/C1 must contain /C1. So 5/C1 = /C1. This shows that the restriction U = S\KL
is unitary. Thus S is unitarily equivalent to TJa' © 17 as desired. ■
Note that 5 is a proper isometry exactly when a > 1. We are now prepared to
prove Coburn's Theorem.
Theorem V.2.2 If S is a proper isometry, then there is a unique ^-isomorphism (p
ofC*(Tz) onto C*(5) such that <p(Tz) = S.
Proof. By the Wold decomposition, S is unitarily equivalent to TJa^ © 17, where
U is unitary and a > 1 since 5 is proper. Using the notation of the previous proof,
define a subspace A/o = span{ejj! : k > 0}. Then A/o is a reducing subspace for
S and 5|A/o is unitarily equivalent to Tz. Let xj) be the *-homomorphism of C*(5)
into C*(TZ) given by restriction to jf0; that is tf;(A) = il|A/b. Since ^(S) = Tz
and the range of tp is a C*-algebra by Theorem 1.5.5, this map is surjective.
Define a map <p of C*(TZ) = T(C(T)) into C*(5) by
^(T/ + K) = (T/ + i(T)(a) © /(IT) for / G C(T), K e £.
Since the map taking Tf + K in T(C(T)) to / in C(T) is a *-homomorphism,
and the normal functional calculus shows that the map taking / in C(T) to f(U)
is a *-homomorphism, the composition is a *-homomorphism from T(C(T)) onto
C*(17). Thus <p is readily seen to be a *-monomorphism such that (p(Tz) = 5. As
the image is a C*-algebra, this map is also surjective. So it is a *-isomorphism.
Moreover, (p = ^_1. ■
V.3 Bunce-Deddens Algebras
A weighted unilateral shift is an operator T for which there is an orthonormal
basis {e* : k > 1} and weights an such that Ten = anen+i for all n > 1. T is
a periodic weighted shift if there is an integer n such that a*.+n = a* for all
k > 1. We will study a class of C*-algebras which are the direct limit of certain
C*-algebras of these weighted shifts.
Proposition V.3.1 The C*-algebra 2U(n) of all weighted shifts of period n with
respect to a fixed basis {ek : k > 1} is unitarily equivalent to Mn(T(C(T))); and
it is singly generated.

138
V. C*-algebras of Isometries
Proof. Consider the shift T with weights a; = 1 when i is a multiple of n and
a,i = 1/2 otherwise. Decompose the Hilbert space as % = Hi © • • • © Hn where
Hi = span{efcn+; : k > 0} for 1 < i < n. With respect to this decomposition, the
operators T and T*T have the matrix forms
T =
i'
0
0
.. 0
.. 0
.. 0
'■
.. \I
Tx]
0
0
0
oj
and T*T
V
\i
0 0
0 0
0 0
0 0
0 I
Since T*T belongs to C*(T), so does Enn = /(r*r) = diag(0,..., 0, /),
where /(l) = 1 and f{\) = 0. It follows that T*<n-'>£nnTn-' = 2-*-»E'<,-
belongs to C*(T), where J5y denotes the n x n matrix with all zero entries except
for an / in the i, j entry. These form a set of matrix units for Mn- Now C*(T)
also contains C*(TEni):
TEnl =
\TX 0 ••
0 0 ••
0 0 •■
[0 0 •■
• 0
• 0
• 0
• 0
0]
0
0
oj
and C*{TEnl) =
T(C(T)) 0
0 0
0
0
0
0
0 0
0 0
0 0
0 0
Moving this around by the matrix units shows that C*(T) contains the C*-algebra
Mn(T(C(T))). As T clearly belongs to this algebra, this is an equality.
Finally, any n-periodic weighted shift with weights oimodn) • • •, °nmodn has
the form
0 0
ail 0
0 a2I
0
0
0
On-ll
OnTg
0
0
0
0
Clearly this lies in A*n(T(C(T))). Hence 2TT(n) = C*(T) = A4„(T(C(T))). ■
The following corollary is immediate from Theorem V.1.5.
Corollary V.3.2 The sequence 0 —> A —> Q£(n) —> .M„(C(T)) —► 0 is
exact.

V3. Bunce-Deddens Algebras
139
ro
i
0
I
[0
0 .
0 .
1
2 ••
:
0 .
. 0
. 0
. 0
;
1
• 2
z\
0
0
0
OJ
It also follows that the C*-algebra Mn{C(T)) is generated by the operator
T =
Suppose that n*. is an increasing sequence such that n*. divides rik+i for fc > 1.
The Bunce-Deddens algebra 25({nfc}) is the quotient by the compact operators
.£ of the C*-algebra generated by all weighted shifts (with respect to a fixed basis)
of period n*. for k > 1. Let
X(nk) = W(nk)/&~Mnk{C{T)).
Then, since n* -periodic weighted shifts are also rik+i -periodic, there is a natural
injection /?*. of 2J(nfc) into 2J(nfc+i). It follows that 2$({nfc}) is the direct limit of
Mnh (C(T)) with respect to the imbeddings /?*..
To understand the limit algebra, we need to understand the imbeddings /?*.
So consider the imbedding /? of 2J(n) into *B(nm). Notice that the generator Tn
of 2J(n) from the corollary above is mapped to the mn x mn matrix An (which
should be considered as a m x m matrix with n x n matrices as coefficients)
in Figure V.l below. An easy computation shows that /3(T*Tn) = A„An is the
An
[00.
1 o .
0 | .
0 0.
0 0..
0 0..
0 0..
0 0..
..0 0
..0 0
..0 0
.. | 0
. 0 1
. 0 0
. 0 0
. 0 0
0 0..
0 0..
0 0..
[00..
. 0 0
. 0 0
. 0 0
. 0 0
0
0
0
0
0
1
7,
0
0
0 ..
0 ..
0 ..
0 ..
0 .
0 .
1
2
0 .
. 0 0
. 0 0
. 0 0
. 0 0
..0 0
..0 0
..0 0
.. 1 o
:
0
0
0
0
0 ..
0 ..
0 ..
0 ..
. 0 0
. 0 0
. 0 0
. 0 0
...
...
••.
...
0 0 ...
0 0 ...
0 0 ...
0 0 ...
0 0 ...
0 0 ...
0 0 ...
0 0 ...
0 z ]
0 0
0 0
0 0
0 0
0 0
0 0
0 0
: 1
0 0 ...
| 0 ...
o i ...
0 0 ...
0 0
0 0
0 0
\ oj
Figure V.l

140
V. C*-algebras of Isometries
nm x nm diagonal matrix Dn = diag(ck) where dkn = 1 for 1 < fc < m, and
di= \ otherwise.
Let E\f be the canonical matrix units for <B(n). Since E$ = f{T*fn) when
/(l) = 1 and f(\) = 0, we obtain that
From the relation
■'"'tj — — ^nn —
it follows that for 1 < i, j < n
m-l
Aj=0
Thus a scalar matrix X in .Mn is sent to P(X) = diag(X,..., X) c_ X ® Im.
Consider fi(zE^) = ^(^f^) = (^ <g> JmJ-A^ff ® Jm) which has
the matrix form
0
E{n)
0
0
0
E{n)
0
0
0
0
0
11
E
0
0
0
0
Perform the canonical shuffle by permuting the basis vectors from /i,..., fmn
tO A, /„+!, . . ., /(m-i)»+l, /2, /n+2, • • •, /(m-l)n. /™n. This Converts ftE®) tO
the n x n matrix with i, j entry equal to Im and all other entries equal to 0. Thus a
scalar matrix X = [*„•] in Mn is sent to the matrix [zijlm]; and the matrix zE±{'
is sent to
0
0
0
0 0
where Z„
nxn
0 0..
10..
0 1..
: :
0 0..
. 0 z
. 0 0
. 0 0
. : 0
. 1 0
I mXm
Thus /? is unitarily equivalent to In <g> am, where am is the monomorphism
of C(T) into Mm(C(T)) given by am(z) = Zm. So now we concentrate on the
analysis of am.
Identify T with R/Z via the function z(t) = e2irit, t G R. The unitary matrix
Zm(t) has the characteristic polynomial^(A) = Am - z(t), which has roots
W£). *(£*). • • • >^(4±Ti)} = W) : n* = * mod 1}.

V3. Bunce-Deddens Algebras
141
Let t{ = *±*f 0 < i < m be the solutions of mtf = £ mod 1. The corresponding
eigenvalues are
6W =
•v/ra
- z((m-l)t,)'
z((m-2)ti)
1
^m6(«)
/ *(*) \
' z((m-l)t.) X
\/"l
*(2t.)
= *(*.)&(<)•
/
Consider the unitary matrices
«i=[&(0 6(0 ...
<—w] = fcM1—s1—■)] •
This is clearly continuous as a function on R. And since &(1)
that
&+i(0), we see
%=[&(0) 6(0) ... £m-2(0) £m-i(0)]
and
[0 0..
1 0 ..
0 1 ..
• •
[o o ..
. 0
. 0
. 0
*
1-1
1-1
0
0
0
°J
I7i=[*i(0) 6(0) ... *m-i(0) 6(0)]
Thus Ui = UqV (or equivalently UqUi = V), where V is the permutation shift
V =
Let A = diag(z(£)fz(tt-)f.. .fz(«±a=l)). It follows that
zm(t) = fftA0? = Uo{usut)Dt{usutyus.
Notice that U^Ut is a continuous path from I to V on [0,1] and VD\ V* = .Do-
So the diagonal entries of Dt sweep out the m disjoint arcs of the circle between
the m-th roots of unity. The twist obtained from V matches the endpoint of each
arc to the initial point of the next. Hence, in effect, the values on the circle T are
being wrapped around m times, with the gradual shift from I to V implementing
the continuous matching of the cuts at the m-th roots of unity. For this reason, this
imbedding of C(T) into Mm{C(T)) is called an m-times around imbedding.
Since z(t) is the generator for C(T), we obtain the formula
«m(/)(«) = Ut diag (/(£), /(£L),..., /(*£=*)) U; = Utf(Dt)U;. (1)
With this preparation, we are now ready to analyze the Bunce-Deddens
algebras.
Theorem V.3.3 The Bunce-Deddens algebra ©({n*.}) is simple.

142
V. C*-algebras of Isometries
Proof. Let 3 be a non-zero ideal of 35. By Lemma m.4.19
3=\j3C)<B(nk).
Let 3k := 3 n 35(nfc). The ideals of 35(nfc) ~ A4njfe (C(T)) have the form
j£*> = {fe C(T, Mnk) : /(*) = 0 for all t G E}
for some closed set E C T. Let i?*. be the closed sets such that 3k = ^jj^.
Let m*. := nfc+1njjr *. Then the imbedding (3k of 35* into 35*.+i is unitarily
equivalent to Injfe <g> amjfe. Since /3fc(3fc) = /3fc(35nJ n 3fc+i, we see from (1) that
#* = {£f : « 6 #fc+i and 0 < i < mk}.
Repeated use of this observation (or consideration of the imbedding of 35* into
35/ for large t) shows that Ek is invariant under translation by riknj1 for every
£ > k. So Ek is either empty or dense. In the first case, 3k contains the identity
and 3 = 35. In the second case, 3k = {0}. Since this holds for all k, it follows that
3 = {0}. Therefore 35 is simple. ■
Theorem V.3.4 Bunce-Deddens algebras are notAR
Proof. The way to distinguish Bunce-Deddens algebras from AF algebras is to
use another topological invariant—this one associated to the invertible elements.
In any finite dimensional C*-algebra, every invertible operator is triangularizable
with invertible diagonal entries. Thus it can be connected to the identity by a
continuous path of invertible elements in the algebra. This property readily extends
to an AF algebra because every invertible element T in the algebra is close (within
||r~1||""1/2) to an element A in a finite dimensional subalgebra. Every element
this close to T is invertible; so the straight line from T to A lies in the set of
invertible elements. Since A is connected to the identity in the invertibles, so is T.
On the other hand, consider the Bunce-Deddens algebra as a subalgebra of the
Calkin algebra Q{7i) = B(H)/R. It always contains the image Tz of the unilateral
shift Tz. This is an element of Fredholm index — 1. The set of invertible elements
of the Calkin algebra of fixed index form a component of the set of invertibles. In
particular, index is constant on any continuous path. So Tz is not connected to the
identity even in Q(H), and hence is in a separate component in the Bunce-Deddens
algebra. It follows that Bunce-Deddens algebras are not AF. ■
It turns out that the distinguishing invariant between Bunce-Deddens
algebras is the supernatural number we used earlier to classify UHF algebras
(Example IH.5.1 and Theorem ffl.5.2).
Theorem V.3.5 Two Bunce-Deddens algebras 35({mfc}) and 35({nfc}) are *-iso-
morphic if and only if their supernatural numbers 8 ({rik}) and S({mk}) are equal

\3. Bunce-Deddens Algebras
143
Proof. If the two sequences represent the same supernatural number, then one may
drop to subsequences so that mk\rik and nfc|mfc+1. Then the imbedding of 2J(mfc)
into 2J(mfc+i) factors through SB(nfc); and vice versa. So it is apparent that the
limits are equivalent.
Conversely, notice that Mmk is imbedded unitally into 2J(mfc). Therefore
if 2$({mfc}) is *-isomorphic to 2$({nfc}), then Mmk is imbedded unitally into
2*({nfc}). By Lemma HI.3.2, there is an integer £ such that this imbedding can
be conjugated by a unitary to an imbedding cp into 2J(n/) which is isomorphic to
Mnt(C(T)). Compose <p with the point evaluation Si of Mnt{C(T)) into Mnt
given by S\ (/) = /(l). This map is a unital homomorphism, and thus Si(p is a uni-
tal homomorphism of Mmk into Mnr As Mmje is simple, this is an isomorphism.
This can only occur when m^rti. Likewise, n/|mfc/ for some kf sufficiently large.
So these two algebras have the same supernatural number. ■
Theorem V.3.6 A Bunce-Deddens algebra has a unique trace, which is faithful
and unital
Proof. Let 7* be the trace on <B{nk) ~ Mnk (C(T)) given by
Tk(A)= f nllrhA(t)dt.
Jo
Clearly this is a faithful unital trace on 2J(nfc). Moreover, by (1)
Tk+MA) = / nj^TrjMW*
Jo
= f n£x Tr U(t) diag (A(±), A{*g)%..., A{****)) U(t)* dt
Jo
r\ ™*-l
= / »& £-*(¥)* = *(*)•
Jo j=o
Thus r = lim7>. exists, and is a faithful unital trace on ©({n*.}).
Conversely, suppose that a is a trace on 2$({nfc}). Let (7*. denote the restriction
of cr to 2J(nfc). Since there is a unique trace onMnjk, there is a probability measure
Hk on [0,1) such that
*k(A)= f n^TrAWdfik.
Jo
(See Exercise IV. 18(a).) But since <Tk = ^fc+iAfe, it follows as in the proof of
simplicity that Hk is invariant under translation by riknj1 for every £ > k. Thus
Hk equals Lebesgue measure for all k\ whence a = r. ■

144
V. C*-algebras of Isometries
V.4 Cuntz Algebras
In this section, we examine another class of simple infinite C*-algebras
generated by isometries. This time, the isometries will have orthogonal ranges. For
n > 2, the Cuntz algebra On is the universal C*-algebra generated by isometries
5i,...,5n such that
i=i
It is clear that such isometries exist.
Construct a universal C*-algebra On with generators S\,..., Sn satisfying ($)
with the property that whenever T\,..., Tn form another set of isometries
satisfying (|), there is a (unique) homomorphism p of On onto C*({Ti,..., Tn}) such
that p(Si) = T{ for 1 < i < n. This can be done by forming a maximal
collection {ira} of inequivalent irreducible representations of the relation ($). These
representations are necessarily on a separable Hilbert space. So we can form the
representation n = X^©7ra. From the GNS construction, it follows that every
representation is *-equivalent to the direct sum of irreducible ones. So it follows
that On = C*(7r(5i), 1 < i < n) has the desired universal property.
To calculate in On, we work with words in the generators. Let n denote the
set {1,..., n}. For a sequence p = (ii, i2,..., ik) in nfc, set
,b/i = o^ &i2 • • • &ik.
Also let \p\ = k denote the length of the word.
Notice that S?Si = I and SjSi = 0 when i ^ jf. Let v = (ju...,;'/).
The product S£SU will be non-zero only if there is perfect cancellation. That is, if
\p\ = k and \u\ = I, then
implies that i, = jf, for 1 < s < min{fe, £}. We record this as a lemma for future
reference:
Lemma V.4.1 Suppose that S*SU ^ 0 with k = \p\ and£= \v\.
(i) If\p\ = |i/|, then p = u andS*Su = I.
(ii) If \p\ < \v\, there is a word v1 in nl~k such that v = pv1 and S*SU = 5„/.
(iii) If \p\ > |i/|, there is a word pf in nk~l such that p = vp1 and S£SU = 5*,.
Thus any word in the Si's and Sj 's has a reduced expression with no Si's to
the right of any Sj's. Hence every non-zero word in the set {5», 5? : 1 < i < n}
has a unique reduced expression of the form S^S^.
Let W£ denote the set of words in n*; and let Wn = \Jk>0 W£. Define
^ = span{5M5*:^,i/GW?} and y1 = {jS£.
fc>i

V.4. Cuntz Algebras 145
Proposition V.4.2 #£ is isomorphic to Mnk and $" is the UHF algebra of type
n°°.
Proof. #£ is spanned by {S^S^ : p> v € W£} which form a set of matrix units for
Mnk because of the identity
(5M5*)(5M/5*/) = S^S^S^S*, = Sun'S^S*,
which follows from Lemma V.4.1. The imbedding of 3£ into 3£+1 is unital, and
thus has multiplicity n. Moreover,
i=l i=l
This shows that the matrix units for #£+1 are compatible with those of #£.
It follows that 511 is a direct limit of full matrix algebras of size nk under unital
imbeddings. Hence it is the UHF algebra of type n°°. ■
This subalgebra #* plays the role of a coefficient space for a Fourier type
series for elements of On. The first step is to identify the 0-th Fourier coefficient.
An expectation of a C*-algebra onto a subalgebra is a positive, unital idempotent
map. Expectations occur frequently in the study of operator algebras, and have
many nice general properties.
Theorem V.4.3 There is a faithful expectation $o ofOn onto y\
Proof. For any complex number A in T, the isometries AS; for 1 < i < n satisfy ($)
and generate On. Thus there is an automorphism p\ of On such that p\(S{) = AS;
for 1 < i < n. Note that p\{Si) = AS,* for each i.
A simple calculation shows that px(SllS*) = X^^^S^S*. Thus it is apparent
that the function /x(A) = p\(X) is continuous when X is in the algebraic span of
the words in the S;'s and Sj"s. Since these words are dense in On and
automorphisms have norm 1, an easy estimate shows that fx is continuous for each X in
Define
*o(*)= f fx(e2"u)dt= f pe2«n(X)dt
Jo Jo
which makes sense as a Riemann integral. This is easily seen to be a unital positive
contractive map on On. Now
So, on the algebraic span of words, it is apparent that $0 maps into 3™ and is the
identity map on a dense subset of 5™. So $o is a contractive projection of On onto
3™; in other words, it is an expectation.

146 V. C*~algebras of Isometries
If X is positive and non-zero, then pe2*u (X) is positive and non-zero for all t.
Thus the integral $o(X) is also positive and non-zero; whence $o is faithful. ■
A more algebraic way to compute this expectation is needed. The following
simple calculation will help.
Lemma V.4.4 Suppose that \i and v are words in Wn such that \fi\ ^ \u\ and
max{|ji|f M} < k. Set 57 = S^ Then S;(S^)Sy = 0.
Proof. It follows from Lemma V.4.1 that S^S^ = 0 unless 5M = S^ , in which
case S*Sn = 5£51*(*~l/i|). Similarly, S^ = 0 unless S„ = S^9 in which case
S*Sy = Sx _|l/'52. So, even when both are non-zero, their product is
fc^'y*^/i*^i/*^'y — *^2*^1 *^1 *^2 — *^2*^1 *^2 — ^j
where we make sense of S™ as S^ ' when m < 0. ■
Lemma V.4.5 For each positive integer m, there is an isometry W in On
commuting with ^n such that *oQO = W*YW for every Y in
span{5M5* : max{|^|, |i/|} < m}.
Proof. Let 57 = SlmS2 and define W = Y^\s\=m ssS<yS%. Then
w*w= E E sss;(s*ss€)s^s;
\S\=m \c\=m
= E sss;s^s = 2 5'5' =J-
|<J|=m |<J|=m
So W is an isometry. Moreover, if \fi\ = m = |i/|, then
WS„ = ]T SsS^SjSJ = S^
\S\=m
and similarly, S£W = S^S*. Hence the matrix unit S^S^ of 3^ satisfies
WS^St = 5M575* = S^SJW.
So W commutes with all Y in 5£. Consequently, W*XW = X for all X in ££.
On the other hand, when \n\ ^ \u\ and max{|^|, |i/|} < m, Lemma V.4.4
shows that
S^{SZSnSZSs)S; = 0 forall |J| = rn;
so WmStAS*W = 0. Consequently, W*YW = *0(^) for Y in the span of all
words S^S* such that max{|^|, \u\} < m. ■

V.5. Simple Infinite C*~Algebras
147
Theorem V.4.6 IfX ^ 0 belongs to Ont then there are elements A, B in On such
thatAXB = L
Proof. Since X ^ 0 and the expectation $0 onto #" is faithful, it follows that
*o(***) ^ 0. Multiply X by a scalar so that ||$0(***)|| = 1. Then pick
a self-adjoint element Y = Y* in the algebraic span of the words S^S^ so that
\\X*X - Y|| < 1/4. Then ||#o0OH > 3/4. Let m be the maximum length of the
words involved in the expansion of Y.
By Lemma V.4.5, there is an isometry W in On such that W*YW = $o{Y),
which lies in 3^. By Proposition V.4.2, 3^ is a full matrix algebra. So there is a
minimal idempotent E (from the diagonalization of $oQO) such that
E*0(Y) = MY)E = \\MY)\\E > f £7.
Choose a unitary U in ££ such that {7£7{7* = £?n = 537l5J,m; and set
Z = \\$0(Y)\\-1/2SZmUEW*.
Wehave||Z|| < 2/^3 and
ZYZ* = \\*0iy)\\-1S^UE(W*YW)ETrS?
= ii*o(ir)ir15rm«7ii*o(ir)iisEr,5Ti = 5r#nsr = /.
A simple calculation now shows that
||J - ZX*XZ*\\ < \\Z\\> \\Y - X*X\\ < J J = §.
Thus ZX*XZ* is invertible. Let B = ^(ZX'XZ*)"1/2. Then
(S*X*)XS = J. ■
The following important corollary is immediate.
Corollary V.4.7 On is simple. Thus ifTi,..., Tn are any n isometries such that
E?=i TiTi = 7> then C*(Ti,..., Tn) if isomorphic to On.
V.5 Simple Infinite C*-Algebras
A projection in a C*-algebra 21 is called infinite if it is equivalent to a proper
subprojection of itself. A projection P is properly infinite if there are orthogonal
projections Q\ and Q2 such that P ~ Q\ ~ Q2 such that Qi + Q2 < P- A
C*-algebra is (properly) infinite if it contains a (properly) infinite projection. This
definition is consistent with the definition of finite C*-algebra given in section IV.2.
We will show that infinite simple C*-algebras are properly infinite, and in fact
always have a subalgebra which has On as a quotient.
Theorem V.5.1 If 91 is a simple infinite C*~algebra, then 21 contains a projection
Q and partial isometries Tit i > 1, such that TfTi = Q > J27=iT*Ti foral1
n > 1. In particular, 21 is properly infinite.

148 V. C*~algebras of Isometries
Proof. Let 5 be a partial isometry in 21 such that P = SS* < Q = S*S. By
working in 35 = Q21Q, we may arrange that 2$ is unital and Q = I. It is easy to
verify that 35 is simple; and it is infinite because Q is infinite.
Since 35 is simple and I - P ^ 0, there are elements X{ in 21 such that
(Prove it! See Exercise V.9.) LetTi = £*=1 Si-1{I-P)Xi. Notice that 5* (I-P)
have pairwise orthogonal ranges for i > 0. Therefore
k k
t;tx = ]T ]Tx; (/ - p)s^-1si~1(i - p)X;
i=i j=i
= Y>;(i-p)x. = i
*=1
and
riTr < Z)5*-1^" p)5*i"1 =J - sks*k-
Set T; = 5*(*-1)T1 for i > 2. Then
mm* efc(»—l)/r» /t»* e*fc(i—1)
< qM*—l)fr_ cfec*fc\o*fe(t-l)
qk(i—1) e*fc(i—1) _ qki q*ki
Hence Ti2y are pairwise orthogonal projections. As each projection is equivalent
to the identity, it follows that 35 and 21 are properly infinite. ■
Lemma V.5.2 Let <£n be a C*-algebra generated by n isometries Si,..., Sn such
that Y17=i 3*3? = P < I- Then the ideal (PL) generated by PL is isomorphic to
the compact operators and <£n/£ c_ On.
Proof. Since S$PL = 0 = PLSi, it is easy to verify using Lemma V.4.1 that (P1)
is spanned by
{S^St : |p| < oof \u\ < oo}.
Moreover it also follows from this that
(5MP SZ)(SaP Sp) = SvaSuP 5^.
Thus this set forms a set of matrix units for an algebra isomorphic to £. In
particular, this shows that PL is a minimal projection in <£n.
In the quotient algebra <£n/£, the images Si of the Si's are isometries such that
.££=1 SiSi = J- Thus by Corollary V.4.7, this quotient is isomorphic to On. ■

V.5. Simple Infinite C*-Algebras 149
Combining the last two results, we obtain
Corollary V.5.3 If'21 is a simple infinite C*-algebra, then On is the quotient of a
subalgebra of '21 for all n > 1.
A minor modification of the argument yields the following useful lemma.
Lemma V.5.4 If P and Q are projections in a simple C*~algebra 21, and P is
infinite, then Q is equivalent to a subprojection of P.
Proof. First we show that there are elements Z{ such that Q = ]CjLi %*?%{
(which is not immediate in the non-unital case). As 21 is simple, there are elements
X{ and Yi in 21 so that \\Q - J^Li xipYi\\ < V2- Hence
m
Q < Y,QxtpYiQ + QYCpxiQ
»=i
m
< Y^QxipxiQ + QYfPYiQ =:A<cQ
%=i
where c = ]C£Li ll-^ll2 + ll^ill2- By the functional calculus, there is a function /
in Co(<t(A)) such that f(x) = x~ll2 on [1, c]. Hence
m
Q = f(A)Af(A) = £ f(A)QXiPX?Qf(A) + f(A)QY?PYiQf(A).
»=i
By Theorem V.5.1, there are partial isometries Si in 21 for 1 < i < n so that
E"=i SiSf <P = St Si. Let T = YJ7=1 ZiPSl Then
n n n
TT* = E E ZiPSiSjPz; = ]T jg-pjg? = q.
In particular, T is a partial isometry and therefore T*T = PT*TP is a
subprojection of P. ■
Now consider an even stronger notion. A C*-algebra is purely infinite if every
hereditary subalgebra contains an infinite projection. The following result together
with Theorem V.4.6 implies that On is purely infinite.
Theorem V.5.5 If% is a simple unital C*-algebra of dimension at least 2, then the
following are equivalent:
(i) 21 is purely infinite.
(ii) for every non-zero element A in 21, there are elements X and Y such that
XAY = /.
(iii) for every non-zero positive element A in 21 and e > 0, there is an element
X in 21 with \\X\\ < ||4|r1/2 + e such that XAX* = /.

150
V. C*-algebras of Isometries
Proof. Suppose that (iii) holds and A ^ 0. Then there is an element X in 21 such
that X(A*A)X = I which proves (ii).
If (ii) holds and A ^ 0 is positive, find X and Y so that XAll2Y = I. Then
/ = XAll2YY*Axl2X* < \\Y\\2XAX\
Thus Z = XAX* is invertible, whence / = {Z-ll2X)A{X*Z~ll2). This proves
(iii) without the norm estimate. Suppose that 2$ is a hereditary subalgebraof 21 and
B is a non-scalar positive element of 2$ which is not invertible. Then there is an
element X in 21 so that XBX* = I. Let 5 = BX/2X*. Then S*S = I and 5 is
not invertible; so 5 is a proper isometry. Moreover,
P = SS* = Bll2X*XBll2
belongs to 25. This is an infinite projection in 2$ since SP belongs to 2$ and
(SPySP = P\ while SPS* is a subprojection of P orthogonal to S{I - P)S*.
This shows that 25 is infinite, proving (i).
Finally suppose that (i) holds. Let A in 21 be a positive element of norm 1, and
let 0 < e < 1/2. Let c = 1 - e. Define a function
/ft) = /° 0<*<l-e
M; [l-e-^l-*) l-e<*<l"
Let 25 be the hereditary subalgebra f(A)9lf(A). By (i), it contains an infinite
projection P. From the definition of 2$, one obtains P < Ea{[1 - £, 1]); and hence
iMP > (1 - e)P. By Lemma V.5.4, the identity is equivalent to a subprojection
of P. Thus there is an isometry S in 21 such that 55* < P. So calculate
S := S*AS = S*PAPS > 1 - e5*P5 = (1 - e) J.
Therefore,
(s-i/25*)i4(5s-i/2) = 7
Finally, I^S"1/2!! < (1 - e)"1/2 < 1 + e\ which establishes (iii). ■
Corollary V.5.6 The Cuntz algebras Onfor 2 < n < oo are purely infinite,
V.6 Classification of Cuntz Algebras
Lemma V.5.2 suggests studying the possible unital C*-algebras <£ such that
is exact. These algebras are extensions of the compact operators by On. Two
extensions <£ and <£' are called strongly equivalent if there is a *-isomorphism rp

V.6. Classification of Cuntz Algebras
of <£ onto <£' such that
0 -£
151
commutes. This turns out to be too fine an invariant. We will say that two
extensions are equivalent if there is a *-isomorphism if; of <£ onto <£' such that if>(&) = .£
and
-^j?-^<£
+ On ^0
0
*
On
-^0
commutes. The collection of equivalence classes of extensions will be denoted by
Ert(On).
An important observation is that we may represent <£ as a subalgebra of B(H)
containing £. This will allow us to reformulate the equivalence question. Recall
that the Calkin algebra is the quotient Q(H) of B(H) by the compact operators;
and let the quotient map be denoted by 7r.
Lemma V.6.1 Every automorphism <p of the C*-algebra of compact operators &
has the form Ad U for some unitary operator in B(H).
Proof. Fix a unit vector e in %. Since ee* is a minimal projection in £, so is y>(ee*)
which therefore has the form //* for some unit vector / in H. Define
Ux := (p(ze*)f.
Then
\\Ux\\2 = Mze*)fM*'*)f) = (fK^)/,/)
= (||x||V(ee*)/,/) = ||x||2(/r/,/) = ll^||2.
Thus U is isometric.
If y belongs to W, choose a compact operator K such that <p(K) = y/*. Then
U(Ke) = <p(Kee*)f = v(K)<p(ee*)f = (»/*)(//*)/ = y.
Hence U is unitary.
Finally, for each K in £ and y in H,
UKU*y = <p((KU*y)e*)f = V{K)V{{iry)^)f
= <p(K)U(U*y) = <p(K)y.
Therefore <p = Ad U. ■

152
V. C*-algebras of Isometries
Theorem V.6.2 Let <£ be an extension of£ by On. Then the identity representation
on £ extends to a faithful representation on H of the extension <£ of & by 0n.
This induces a *-monomorphism r ofOn into the Calkin algebra. Two extensions
<£ and <£' are equivalent if and only if the associated monomorphisms r and r1
are unitarily equivalent in the sense that there is a unitary U in B(H) such that
T' = Adir(U)T.
Proof. Let id denote the identity representation on £. By Lemma 1.9.14, this
extends uniquely to a representation a of <£ on H. It is unital because <r(I)K = K for
every compact K. Define the map r from On into Q(H) by t(A) = 7ra(p^1(A)).
This is easily verified to be a well defined unital *-homomorphism. Since On is
simple, r is injective.
To see that a is injective, suppose that <r(E) = 0. Then
T(p(E)) = ir<T(E) = 0;
whence p(E) = 0. But then E is compact, so E = a(E) = 0.
Let r and r' be two monomorphisms of On into the Calkin algebra, and let
<£ = 7r_1r(On) and <£' = 7r~1r,(On) be the corresponding extensions. Suppose
there is a unitary U in B(H) such that r' = (Ad itU)t. Then let ip = Ad U. This
is a *-isomorphism of <£ which takes & onto itself and such that
Adtf(<£) = (AdC^Tr-VC^n)) = ir-^AdirlOMOn))
= ir-1(r'(On)) = <£'.
Conversely, suppose that <£ and <£' are equivalent via an isomorphism rp. By
Lemma V.6.1, there is a unitary U in B(H) such that t{>\£ = Ad 17|j?. For any E
in <£ and .K" in £,
4>(E)(UKU*) = rl>(E)i/>(K) = V(#*0
= tftfittT = (UEU*){UKU*).
As the ranges of 17£17* are dense in %, we deduce that tp(E) = UEU*. Hence
^(AdTrCOr. ■
This theorem suggests another notion of equivalence using unitaries in the
Calkin algebra instead of unitaries in B(H). Say that two monomorphisms r and
t' of On into the Calkin algebra are weakly equivalent if there is a unitary element
u in Q(H) such that r9 = (Ad u)r. The collection of all weak equivalence classes
will be denoted by Extw(On).
We define an addition on Ext(On) by setting [a] + [r] := [a © r] where [a]
and [r] are two equivalence classes of monomorphisms of On into the Calkin
algebra. This defines a monomorphism of On into Q(H) © 2(%) which is contained
in M2{Q{U)). But, since M2{B(H)) c_ B(H) and the restriction to the
compacts identifies M2(£) with £, this induces an isomorphism of A^2(2(^)) onto
Q(H). To see that this addition is well defined, suppose that <rf = (Ad ttU)<t and

V.6. Classification of Cuntz Algebras
153
r' = (AdirV)T. Then W = U ®V defines a unitary in M2{B{U)) such that
(Ad7rW)(cr © r) = a1 © r'. Because of the properties of direct sum, it is
immediately evident that this operation is abelian and associative. Hence Ext(On)
becomes an abelian semigroup.
An extension r is called trivial if there is a *-monomorphism a of On into
B(H) such that r = 7rcr. This means that the exact sequence for the extension
splits:
0 ^£ ^(ST^On ^0
We will need the following result of Voiculescu.
Theorem V.6.3 If% is a separable C*-algebra, then all trivial extensions of& by
21 are equivalent; and they form a zero element for Ext(2l).
Proof. Let T{ be trivial extensions of 21 for i = 1,2; and let <ri be *-monomorphisms
of 21 into B(H) such that 7T(t; = r;. Then
ker(7i = ker7T(7i = ker(72 = ker7T(j2 = {0}.
Therefore by Corollary EL5.6 of Voiculescu's Weyl-von Neumann Theorem, a\
and (72 are approximately unitarily equivalent. Thus there is a unitary operator U
such that cr2 - Ad U <t\ has compact range; whence r2 = (Ad7rl7)Ti. Thus the
trivial elements belong to a single equivalence class.
Now suppose that r is an extension of 21; and let <£ = 7r-1r(2l). If 7rcr is a
trivial extension of 21, then the representation of <£ given by p = ar^n
annihilates the compact operators. Therefore by another Corollary n.5.5 of Voiculescu's
Theorem, idg ~# idg ©/o. Equivalently, r and r © 7rcr are unitarily equivalent in
B(H)/£. Hence [r] + [7rcr] = [r]. So [7rcr] is the zero element. ■
While Ext(2l) is not always a group, this is the case for nice C*-algebras. The
proof for On will emerge from our analysis. We will compute both Ext(0n) and
Extw(On). This latter group will turn out to distinguish the Cuntz algebras from
one another.
Lemma V.6.4 Let v in Q(Hi, H2) be a partial isometry in the Calkin algebra; and
suppose that P in B(H2) and Q in B(Hi) be projections such that 7r(P) = tw*
and 7r(Q) = v*v. Then, there is a partial isometry V in S(%i, ^2) of the form
V = PVQ such that 7r(V) = v. Moreover, the integer
dim(Q - V*V) - dim(P - VV*)
is defined independent of the choice ofV.
Proof. Lift v arbitrarily to an operator T, and let V be the partial isometry in the
polar decomposition of PTQ = VA where A = \PTQ\. Then 7r(V)n(A) is the
polar decomposition of n(PTQ) = v. Hence n(V) = v.

154 V. C*-algebras of Isometries
Considering V as an operator in B(PH, QH), one sees that since the choice
of V is unique up to a compact operator, its Fredholm index is well defined
independent the the choice. This index is
null(F) - null(F*) = null(rV) - null(W*)
= dim(Q - V*V) - dim(P - VV*). ■
If r is a unital monomorphism of On into Q(H), define a partial isometry vT
in Q{U(n\U) by
tV=[r(5i) t(S2) ... r(5n)].
Define an integer valued function on extensions by f(r) = ind vT. By the lemma
above, this function is given by
/(r) = dim(/(n) - V;Vr) - dim(J - VTV?),
where VT is any partially isometric lifting of vT.
If r and t' are equivalent extensions, then there is a unitary operator U such
that r' = Ad n(U) r. Thus tv = ttU vt irtjW*. Hence
/(t') = indtv = indTrC/ + indvT + mdnU(n)* = /(r).
Therefore / induces a well defined map / from Ext(On) into Z.
Theorem V.6.5 The function f is an isomorphism ofExt(On) onto Z. In
particular, Ext(On) is a group.
Proof. First we show that / is additive. For if a and r are extensions, then
\a(Si) 0 <r{S2) 0 .. <r{Sn) 0
*V0T =
Hence
L 0 riSJ 0 t(S2) ... 0 r(5n)J
c_ va © vT
/[cr © r] = indtv ®vT = indtv + ind vT = f[a] + f[r].
Therefore / is a semigroup isomorphism.
Suppose that f[r] = 0. Choose a lifting VT of vT. As ind FT = 0, there
is a finite rank isometry W of kevVT onto ker T^. So VT + W is unitary and
n(VT + W) = vT. We could have taken this as our choice for VT. So we may
suppose that VT is unitary. Write it as a 1 x n matrix in B(H^, H) as
K=[Vi V2 ... Vn].
Then I<n) = VT*VT = [VfVj]. This implies that VfVi = I for 1 < i < n, and thus
each VJ is an isometry. And I = VTV? = £?=1 K*VJ*. Hence by Corollary V.4.7,
the map a taking Si to VJ is a faithftil representation of C?n such that 7rcr' = r.
Therefore r is trivial.

V.6. Classification of Cuntz Algebras
155
Conversely, if r = ira is a trivial extension, define V* = <r{Si). The operator
VT = [Vi V2 ... Vn] is a unitary lifting of vr. So /(r) = indFT = 0.
Theorem V.6.3 shows that the trivial elements form a zero element for Ext(0n).
Thus we have shown that ker / = {0}.
We will show that the range of / is Z. Clearly, it suffices to construct two
extensions r± such that f(r±) = ±1. Let Vi, V2,... ,Vn be isometries satisfying
(t), and let S = SViV? + (J - ViV?) be an isometry in B(H) of index -1. Then
let
r±(Si) = 7r(S)^17r(V1) and r±(Si) = tt(K) for 2 < i < n.
For this choice, we have the partially isometric liftings
V+ = [5*Vi V2 ... Vn] and V_ = [SVX V2 ... Vn] .
Then it is readily checked that f(r±) = ind V± = ±1.
This shows that / is an isomorphism and that Ext(On) is a group. For if [r]
belongs to Ext(On) and f[r] = n, let [rf] be chosen so that /[r'] = -n. Then
/(M + M) = n - n = 0.
Hence [r © rf] is the trivial element. Whence [rf] is an inverse for [r]. So / is an
isomorphism of Ext(On) onto Z. ■
So far, this does not differentiate the Cuntz algebras. However, with a little
more work, we obtain the result that we want.
Theorem V.6.6 The weak extension group Extw(On) ofOnfor n > 2 is
isomorphic to Zn_i.
Proof. Extw(On) is the quotient of Ext(0n) by the subgroup of elements weakly
equivalent to the zero element. Suppose that r = ira is a trivial element with
V{ = <r{Si). Let U be an isometry of index -1. Then let r' = Ad n(U) r. Then
vri has the lifting
VT* = [UViU* UV2U* ... UVnU*] = UVTUW*.
Thus
/(r,) = indt/ + indVr + indC/(n)* = -l + 0 + n = n-l.
If a; is a unitary in Q(H) with ind(«) = -fc, then there is a unitary operator
W in B{U) such that 7r(^)7r(C/)fc = x. Thus (Ads)r = (Ad7r^)(Ad7rC/)fcr.
So
f((Adx)r) = f((AdirU)kT) = k(n - 1).
Thus the extensions weakly equivalent to the trivial extensions are identified with
the subgroup (n - 1)Z. Therefore Extw(On) is isomorphic to Zn_i. ■
Corollary V.6.7 Ifn ^ m, then On and Om are not isomorphic.

156
V. C*-algebras of Isometries
V.7 Real Rank Zero
A unital C*-algebra 21 is said to have real rank zero if the invertible Hermitian
elements are dense in 2l,a. When 21 is not unital, say it is real rank zero if the
unitization 21" is real rank zero. This is a property that parallels zero-dimensionality
in the commutative case. The following elementary proposition justifies this view.
Proposition V.7.1 IfX is totally disconnected, then C(X) is real rank zero. IfX
contains an arc, then C(X) is not real rank zero.
Proof. When X is totally disconnected, C(X) is spanned by its projections. The
subspace of finite real linear combinations of characteristic functions is therefore
dense in the subspace of real valued functions. Clearly, a function taking a finite
set of values is easily perturbed slightly to a non-zero function. So C(X) is real
rank zero.
On the other hand, if X contains an arc J, consider any real valued function
/ in C(X) which takes both values ±1 on J. Any real valued function within
distance 1 of / takes both positive and negative values on J. Hence it has a zero
by the intermediate value theorem. So the ball of radius 1 about / contains no
invertible elements. ■
It is clear that every von Neumann algebra is real rank zero. Indeed,
approximate A within e by A + eEA(—e/2, e/2). In particular, finite dimensional C*-
algebras have real rank zero. It is also very easy to see that inductive limits of real
rank zero C*-algebras are still real rank zero. In particular:
Proposition V.7.2 AF algebras have real rank zero.
In fact, many other algebras also have real rank zero including Cuntz algebras,
Bunce-Deddens algebras and irrational rotation algebras. The following theorem
provides several useful formulations of this property.
Theorem V.7.3 For a C*-algebra 21, the following are equivalent:
[RRO] 21 has real rank zero.
[FS] The elements of%8a with finite spectrum are dense in 2l,a.
[HP] Every hereditary subalgebra of% has an approximate unit of projections
{not necessarily increasing).
Proof. Suppose that [RRO] holds. Fix an element A in 2l,a of norm 1 and an
e > 0. Choose an increasing subset t\ = -1,..., tn = 1 of non-zero points in
[-1,1] which forms an e/2 net. Let e\ = e/4. By the real rank zero property,
there is an element A\ in 2l,a such that A - t\I is invertible and ||.A - .Ai|| < e\.
Then choose e2 < e/8 sufficiently small so that [ti - e2, h + £2] does not intersect
<t(Ai). Using [RRO] again, choose A2 in 2l,a such that A2 - t2I is invertible
and H.A2 — -AiN < ^2- T"en neither t\ nor t2 lies in <t(A2). Repeated use of this
argument produces an element An in 2l,a such that *i,..., tn are in the resolvent

V.7. Real Rank Zero 157
of An and
n n
||il-.An||<53«<532-'-1e<e/2.
i=i t=i
Therefore the Riemann sum
n
B = -EAn(-l - e/2, -1] + ^UE^U-i, U] + EAn{\, 1 + e/2).
*=2
belongs to 21. Evidently 2? has finite spectrum and
\\B - A\\ < \\B - An\\ + ||An - A\\ < e/2 + e/2 = e.
Now assume [FS] and suppose that <8 is a hereditary subalgebra of 21. Given
Bi,..., Bn in <B and 0 < e < 1, it suffices to find a projection P in <B such that
\\Bi(I - P)\\ < e for 1 < i < n. Let B = YJ?=1 BfBi. Then
IW " P)||* = ||(/ " P)BtBi{I - P)\\ < \\B(I - P)\\.
So we work with B, which we may assume to have norm 1.
Pick a positive number 8 with 8 < (e — e2)/6. Then choose an integer n so
large that 82ln > 1 - 8. By property [FS], there is a positive element C in 21 with
finite spectrum such that || JB1^ - C\\ < 8/n and ||C|| < 1. Thus A = Cn satisfies
n-l
||4 - fl|| = || £ Cn^^k(C - S^S^H < nys1^ - C|| < 8.
k=o
The projection Q = Ea[8, 1] belongs to 21. From the functional calculus, we get
the estimates
||;1(I - Q)|| < 8 and \\Al'nQAlln - Q|| < 1 - 82'n < 8.
Since 35 is hereditary, the element X = B1lnQB1ln belongs to 35. An easy
estimate shows that
||X - Q|| < 2||S1/n - Al'n\\ + WA^QA1'" - Q|| < 38.
Therefore (compare with Lemma III.3.1)
II* - X2|| = ||(I - Q)(X -Q)-(X- Q)X\\ <66<e-e2.
Hence <r(X) is contained in [0, e] U [1 - e, 1]. So the projection P = Ex[l - e, 1]
lies in 35 and ||P - X|| < e. Finally we compute
\\B(I - P)\\ < ||P - Q|| + ||A - B\\ + \\A(I - Q)\\ <e + 58<2e.
To complete the circuit, assume [HP] and let A be a self-adjoint element of
norm 1. Write A = A+ - A- be the Hahn decomposition of Corollary 1.4.2.

158
V. C*-algebras of Isometries
Set 2$ = A+WLA+. By property [HP], there is a projection P in 35 such that
WA+P11| < e. Since A- A+ = 0, it follows that A_P = 0. Let
S = PAP + 2eP + PXAPX - 2ePL
= A - (PAP1 + PLAP) + 2e(P - P1).
Then
\\A - B\\ < \\PAPL + PLAP\\ + 2e||P - PL\\ < e + 2e = 3e.
Notice that PEP = PA+P + 2eP > 2eP and
PXBPX = PLA-PL + P^A+P1 - 2ePx
< 0 + ePL - 2ePL = -ePL.
As B commutes with P, it follows that B is invertible. So 21 is real rank zero. ■
Now we show that there is a non-trivial class of real rank zero algebras. In fact
the definition of purely infinite algebras suggests something of the [HP] condition.
Theorem V.7.4 If $1 is a purely infinite simple C*-algebra, then it has real rank
zero.
Proof. Suppose that A belongs to 2laa and e > 0. Define
*0 if|<|<£
fe(t) = {
t-e
t + e
ift>e
if* < -e
and ge(t) = max{e - |*|, 0}.
Since 21 is purely infinite, there is an infinite projection P in the hereditary subal-
gebra 35 = ge(A)%Lge(A). By Lemma V.5.4, the projection / - P is equivalent to
a subprojection of P. Thus there is a partial isometry S in 21 such that
S*S = I-P and SS* = Q<P.
Notice that fe(A) - (I - P)fe(A)(I - P). Define
B = fe(A) + e(S + S*) + e(P - Q) ~
MA) e 0
e 0 0
0 0 e
The matrix comes from the decomposition (J - P)H © QH © (P - Q)U with
the matrix unit E2i = S. From this matrix form, it is evident that B is invertible.
Finally,
||B-A||<||/e(A)-A||+e||5 + 5* + (P-Q)||<2e. ■
Corollary V.7.5 The Cuntz algebras Onfor n > 2 are real rank zero.

V.7. Real Rank Zero
159
We consider the class AT of a limit circle algebras which are direct limits
of algebras of the form £) ©*=1 MPi(C(T)). There is an obvious parallel with
AF algebras. However, as the Bunce-Deddens algebras have this form, they can
exhibit different properties from AF algebras. Since T is connected, C(T) is very
far from being real rank zero. Nevertheless, it is possible for AT algebras to have
real rank zero. A subset V of a C*-algebra separates the traces of 21 if two traces
T{ satisfying ti(P) = T2(P) for all P in V are equal.
If F = F* is in MP(C(T)), let Aj(F, t) denote the jth smallest eigenvalue of
F(t). The variation of the eigenvalues of F is
sup max \\j(F, s) - A^F, t)\.
•ft€T1_.J<P
The variation of the trace is
sup p-1] Tr(F(5)) - Tr(F(t))| = sup jT1^ A^F, s) - A^F, t)
Lemma V.7.6 The set of self-adjoint elements ofMk(C(T)) which have k distinct
eigenvalues at each t G T is dense in Mk{C(T))8a.
Proof. Every continuous function from T into Mk may be approximated by a
continuous piecewise linear function such that the "corner" points have distinct
eigenvalues. Therefore it suffices to approximate a linear function f(x) = A + xB
on [-1,1] such that /(±1) has distinct eigenvalues. In this case, the characteristic
polynomial p(x, A) = det(f(x)-XI) is algebraic. It follows that the k eigenvalues
and corresponding eigenprojections are analytic, even at the finitely many points
where the eigenvalues cross. It is easy to make a small perturbation, say at /(0)
so that for each 2, at most one pair of eigenvalues coincide. Thus by splitting the
interval into smaller intervals if necessary, we may suppose that there is a single
point x0 in (-1,1) at which a single pair of eigenvalues Xj{xo) and Aj+i(«0)
coincide.
Indeed, we may suppose that x0 = 0 and Aj(0) = 0. As the eigenprojections
are analytic, after a unitary equivalence, it may be supposed that the corresponding
eigenvectors ej(i) and eJ+i(f) are constant. Then ignoring all the other
eigenvalues, / is (up to a scalar) just the diagonal function with roots Xj(t) and Xj+i(t)
which are distinct except for t = 0. Let e(t) be a real valued function on [-1,1]
such that \\e(t)\\ < e, e(-l) = e(l) = 0 and e(0) > 0. Then write
/(*) =
0 xj+1{t\
and g(x) =
[e{t) xj+1(t)
A simple calculation shows that g(x) has no repeated roots for any t, and it takes
the same values as / at the endpoints. ■
Now we obtain a useful condition implying real rank zero for this class of
algebras. Let us write 21 = lir^2ln where 2ln = £©?=! MPni{C(T)), and let

160
V. C*-algebras of Isometries
<Pn,m denote the homomorphism of 2ln into 21™ for n < m. For a self-adjoint
element F = Fi © • • • © Fkn in 2ln, the variation of the eigenvalues of F, or of the
trace of F, is the maximum of this quantity for each summand.
Theorem V.7.7 If'21 is a simple AT algebra, then 21 is real rank zero if and only if
the projections of% separate the traces of%.
Proof. If 21 has real rank zero, then every Hermitian element is in the closed real
span of the projections. Thus the span of the projections is dense in 21. Hence the
projections separate traces.
So suppose that the projections separate traces. We first show that if F = F*
in 2ln and e > 0, then there an integer m > n so that the variation of the trace of
<Pn,m(F) is less than e. Suppose, to the contrary, that there is an F and an e > 0
so that for every m > n, there is a summand MPmi(C(T)) of 21™ and points sm
and tm in T such that
*mil-(Pn.m^M'm)) " Tt(<pntm(F)i(tm))\ > E.
For each m > n, let <rm be any state of 21 extending the trace on 21™ given by
crm(G) = pm^ Tr(Gi(sm)); and let rm be a state extending the trace on 2lm given
by rm(G) = p^ Tr(Gi(fm)). Choose a subnet A of N such that a = lim^A ^Vn
and r = lirrVn^A ^Vn exist as weak-* limits. These are easily seen to be traces on 21
such that \<t(F) — r(F)\ > e. In particular, cr ^ r.
Now suppose that P is a projection in 21. By Lemma 111.3.1, P is the limit of
a sequence of projections Pk in 21*. For any m > fc, P* is a projection in 21™,
and thus has constant rank on each component. As the trace of a projection in Mp
depends only on rank, it follows that <Tm(Pk) = Tm(Pk). Therefore a and r agree
on each P*, and hence on every projection in 21. This contradicts the hypothesis
that projections separate traces.
We claim that if F and G are positive contractions in 2ln such that FG = F,
then there is an integer m > n so that for every summand MPmi(C(T)) of 21™
and 5, t e T,
rank(^n,m(F)i(5)) < rank(Vn,m(G)i(*)).
Since FG = F implies that this inequality holds for s = t, the result holds if the
right hand side is constant; in particular, this is the case when G is a projection.
So we may suppose that <r(G) intersects (0,1). Let k be a continuous function on
[0,1] such that 0 < k(x) < z, k(0) = k(l) = 0 and K = k(G) ^ 0. Then since
F < FG{1}, we have KF = FK = 0. Define H = G-K.
Since 21 is simple, there are elements Aj and jBj in 21 such that
3=1

V.7. Real Rank Zero
161
t-l
We can arrange that the Aj's and B/s lie in some 21™,, for m0 > n sufficiently
large. Indeed, choose A!i and Bj in some %n0 so that
H4 - 4|| < (4*||iq ll^ll)-1 and ||b;. - B,-|| < (4ft||JT|| H^ll)"1
for 1 < j < k. Then a routine estimate yields \\I - E*=1 A'jKBj\\ < V2- So
X = EjU A'jKB'j is invertible in SUo. Therefore J^^X^A'^KB'j = L
If r is any trace on %n0, it follows that
T{K)>e:=fc\\AMBi\\)'
i=i
Take m > m0 so large that the variation of the trace of <pnim(H) is less than e. For
any s in T, the identity FH = F shows that <pn,m(F)i(s) < ^„,m(H)«(#){-} ^d
hence for any t in T
rank(^,m(F)<(*)) < Tr(^,m(.ff )<(*)) < Tr^^^W) +e
< It(^n,m(if),(«)) + Tr(^n,m(i<:)i(0)
= Tr(^n,m(G)i(0) < rank(^n,m(G)i(0).
We use this result to show that if F = F* is in 2ln and e > 0, then there is an
m > n so that the variation of the eigenvalues of <pn,m(F) is less than e. Fix an
integer r > 3e_1. For 0 < j < r, define
[ 1 x < j/r
fj(x) = lj + l-rz j/r<x< (j + l)/r .
(o ti + 1)/r<x
Set Fj = fj(F). From the functional calculus, we see that FjFk = Fj for jf < k.
Hence we have an integer m > n so that for every summand A4Pm $(C(T)) of 21m
ands,*GT,
rank(^n,m(Fi)i(5)) < rank(^n,m(Fi+1);(*)) for 0 < j < r - 1.
Fix i and a point s0 in T; and set dj = rank(y>njm(FJ)i(50)). Then this is an
increasing sequence such that dr = pm^. Then
rfj < max rank(v?n,m(Fj)i(5)) < min rank(y?n,m(Fj+i );(*)) < dj+i.
Now rank(y>njm(Fj)j(£)) is the number of eigenvalues of <pn,m{F)i(t)) in the
interval [0, (jf + l)/r). This figure lies in the interval [dj_i, dj+i]. Conversely, if
<*j-i < * < rfj, then Afc(Vntm(F)<(*)) lies in [(j - l)/r, (j + l)/r]. So the
variation of the eigenvalues is at most 3/r < e.
Now given A = A* in 21 and e > 0, approximate .A to within e by an F = F*
in 2ln. By replacing n with a larger integer mi, we may suppose that the variation
of the eigenvalues of F is at most e. By Lemma V.7.6, approximate F within e
by an element G of 21™ which has distinct eigenvalues for every 1 < i < km and

162
V. C*-algebras of Isometries
t in T. Clearly, the eigenvalues of G have variation at most 3e. Let -Pm,*,j for
1 < i < km and 1 < j < pmii be the eigenprojection of G corresponding to the
eigenvalue Aj(Gi) (which is well defined because the eigenvalues never intersect
each other). Then define
km Pm,i
*=i j=i
Clearly if has finite spectrum and \\G-H\\ < 3e. Hence \\A-H\\ < 5e. Therefore
21 has real rank zero. ■
Corollary V.7.8 The Bunce-Deddens algebras have real rank zero.
Proof. The Bunce-Deddens algebras are simple by Theorem V.3.3 and have a
unique trace by Theorem V.3.6. Therefore the projections separate the traces;
whence the algebra has real rank zero. ■
Exercises
V.l Verify that for h in if2, linv^i- ||Mre**)||2 = ||/*||2.
V.2 Show that H°° is a weak-* closed subalgebra of L°°.
V.3 (Beurling's Theorem) Suppose that M is a proper invariant subspace for
Tz. Show that there is an inner function u (i.e. w G H°° and |u;| = 1 a.e.)
such that M = u>H2.
HINT: Pick a unit vector w in M Q TZM. Consider (a;, wzk) for k > 1.
V.4 Show that if h in if2 is non-zero, then \h\ > 0 a.e. on T.
HINT: Let E = {t e T : h(t) = 0}. Apply Beurling's Theorem to the
subspace M = {g G if2 : g\s = 0}.
V.5 (a) Show that if Tg is invertible, then g is invertible in L°°.
(b) Hence show that <r(Mg) C <r(Tg) for all g in L°°.
V.6 (a) Show that if g is in L°° and Re g > 1, then Tg is invertible.
HINT: Find c> 0 such that \\c - g\\oo < c.
(b) Hence show that a(Tg) C conv(a(Mg)).
(c) If g = (7, show that ^(T^) = [ess inf g, ess sup 5].
V.7 (a) Show that if /, g belong to L°° and at every point t 6 T one of / or g
is continuous, then T/T^ - Tfg is compact.
HINT: Use a partition of unity pi and consider ]T^=i TfTPiTg.
(b) Let PC denote the C*-algebra generated by the piecewise continuous
functions. Show that T(PC)/i? is abelian.
(c) Show that the maximal ideal space of T(PC)/£ is a cylinder T x [0,1]
with a non-standard topology.

Exercises 163
V.8 (a) Show that an automorphism a of T(C(T)) induces an automorphism
5(/) = / o h of C(T), where h is an orientation preserving homeomor-
phismof T.
HINT: Consider the Fredholm index.
(b) If h is an orientation preserving homeomorphism of T, find a compact
operator K such that Th + K is unitarily equivalent to Tz. Hence show
that there is an automorphism of T(C(T)) which induces h as in (a).
V.9 Show that if a unital C*-algebra 21 has no proper closed ideals, then it
has no proper algebraic ideals either. Then show that if A is positive and
non-zero in a simple C*-algebra, then there are elements X{ in 21 such that
y^V_ X*AX- = I.
HINT: Show that XAY + 7MI* < XAX* + Y*AY.
V. 10 Show that Bunce-Deddens algebras are stably finite.
HINT: Exercise IV.9.
V. 11 Show that T(C (T)) is infinite but not properly infinite.
HINT: Consider the symbol of projections in T(C(T)).
V. 12 Prove that the set of invertible elements of T(C (T)) is connected.
V.13 Consider the weighted shift Sen = anen where an = 2 - gcd(n, 2n)_1
for n > 1.
(a) Show that C*(5) contains the unilateral shift, and hence all compact
operators.
HINT: The weights are bounded below.
(b) Show that C*(5) contains all 2n-periodic weighted shifts.
HINT: Use the functional calculus on S*S.
(c) Show that 5 is a limit of 2n-periodic weighted shifts. Hence deduce
that C*(5) is a Bunce-Deddens algebra.
(d) Show that each Bunce-Deddens algebra is singly generated.
V.14 Define maps from On onto 5" by
tK } \*0(5r*-Y) fori<0'
(a) Show that if Y in On is in the algebraic span of words, then
y = S**(ir)^ + 535*w*<(y).
i>0 i<0
(b) Show that X in On satisfies **(X) = 0 for all t € Z if and only if
X = 0.
HINT: Represent On on a Hilbert space. Consider the Fourier series of the
function /(A) = (p\(X)x, y) for arbitrary vectors x and y.

164
V. C*-algebras of Isometries
V.15
V.16
V.17
V.18
V.19
V.20
V.21
V.22
The C*-algebra #«, is the universal C*-algebra generated by countably
many isometries S*. such that J2k=i SkS^ < I for all n > 1. Show that
Ooo is simple.
(a) Show that {Si, S2S1, Sf} generate a copy of ©3 as a subalgebra of O2 •
(b) Generalize this to show that Ok(n-i)+ican be unitally imbedded in On
for all Jb > 1.
If k divides n, show that Mk{On) is isomorphic to On.
HINT: For 0 < j < n/k and 1 < i < k, consider the operator Tkj+i which
is a k x k matrix with [Skj+i Skj+2
all other entries equal to 0.
Show that Mz(02) is isomorphic to C?2-
HINT: Show that
Skj+k] in the i-th row and
Tx
0
I
0
0 0
0 0
S\ S2
and T2
5X
0
0
S2S1
0
0
5?
0
0
generate ^3(02).
Generalize the previous question to show that Mn(02) is isomorphic to
02 for all n> 1.
Let P be a projection in a C*-algebra 21. Show that 21 has real rank zero if
and only if both P2LP and P121P1 are real rank zero.
HINT: An element X = X* in 21 has a matrix form [^ ^] with respect
to P ® PL. If i4 is invertible, factor this as
BM"1/2
0
A1/2
0
I
If 21 has real rank zero, show that .Mn (21) also has real rank zero.
HINT: Use the previous exercise.
Show that a separable C*-algebra of real rank zero has an increasing
sequence of projections forming an approximate unit.
HINT: Use a strictly positive element A. Given Pi < • • • < Pn, choose a
projection Q < P^ such that ||(P^ - Q)A\\ < 1/n.
Notes and Remarks.
The study of Toeplitz operators is one of the richest areas of study in
operator theory. An excellent introduction to the theory is Douglas [1972].
Theorem V.2.2 is due to Coburn [1967]. Bunce-Deddens algebras were introduced in
Bunce-Deddens [1975]. Cuntz algebras were introduced in Cuntz [1977]. The
basic results about these two classes come from those original papers.
Theorem V.5.5 is due to Cuntz [1981]. The classification of Cuntz algebras by Ext

Exercises
165
is due to Paschke and Salinas [1979], and simultaneously to Pimsner and Popa
[1978]. Cuntz [1981] classifies them as well, but using K-theory. The notion of
a real rank zero C*-algebra is taken from Brown and Pedersen [1991], who prove
Theorem V.7.3. Zhang [1988] showed that purely infinite C*-algebras are real
rank zero. That Bunce-Deddens algebras are real rank zero is due to Blackadar
and Kumjian [1985]. The proof given here for limit circle algebras is taken from
Blackadar, Bratteli, Elliott and Kumjian [1992]. A stronger result due to Blackadar,
Dadarlat and R0rdam [1991] shows that every simple C*-algebra which is the
direct limit of finite sums of matrix algebras over continuous Amotions on spaces of
bounded dimension is real rank zero if and only if projections separate traces.

CHAPTER VI
Irrational Rotation Algebras
In this chapter, we will study a class of C*-algebras that has received a lot of
special attention in recent years. The canonical model acts on the circle T which we
will think of as E/Z via the map taking t to z(t) = e2ntt. Fix an irrational number
0. Let H = L2(R/Z) and consider two unitary operators on H, the operator
U = Mz(t) of multiplication by the unimodular function z(t) and V, the operator
of rotation by 0. That is
Uf(t) = z(t)f(t) and Vf(t) = f(t - 0).
A simple calculation yields
VUf(t) = (Uf)(t -0) = z(t - 0)f(t - 0)
= e-2wi$z(t)(Vf)(t) = e-2™eUVf{t)
Hence
UV = e2**Vtf (t)
We wish to study the universal C*-algebra satisfying (f). A C*-algebra Ae is
universal for the relation (f) provided that it is generated by two unitaries U and
V satisfying (f) and whenever 21 = C*(17, V) is another C*-algebra satisfying (f),
there is a homomorphism of Ae onto 21 which carries UtoU and V to V. Since we
know that there are unitaries satisfying the relation, we may consider the collection
of all irreducible pairs of unitaries (Ua, Va) in B(H) satisfying (f). Then form the
operators
LT = ]Ter/a and V = ^QVa.
LctAe = C*{U,V).
In order to see that Ae is the desired universal algebra, let 21 = C*(17, V) be
another C*-algebra satisfying (f). To verify that there is a well defined
homomorphism cp : Ae -> 21 such that (p(U) = U and (p(V) = V, it suffices to show
that
llP^^^nil < Mu,v,u*,v*)\\
for every non-commutative polynomial in four variables. Fix a polynomial]?; and
let A = p(U, V, U*, V*). By the GNS construction (Theorem 1.9.12), there is an
166

VI. Irrational Rotation Algebras
167
irreducible representation 7r of 21 such that ||7r(i4)|| = ||A||. Consider the pair
U' = tt(U) and V = n(V). Then ({/', V) is an irreducible pair satisfying (f).
Hence by construction, we see that
M&, v, u\ v*)\\ > \\P(u', V, u'\ V)\\ = \\P(u, v, u\ nil-
Therefore <p is well defined and contractive on the *-algebra generated by U and
V into 21. So it extends by continuity to a homomorphism of Ae onto 21.
The C*-algebra Ae for irrational values of 0 is called an irrational rotation
algebra. The argument above produces a universal C*-algebra for any family of
relations.
We apply this universal property to obtain certain special automorphisms of
Ae. For any constants A, \i on the unit circle (|A| = \\i\ = 1), the unitary pair
(XUj fiV) satisfies (f). Thus there is an endomorphism p\^ ofAe such that
P\AV) = W and Px„(V) = pV.
Let cr = PxjtPx^. Since a(U) = U and a(V) = V, we have a = id. Thus p\^ is
an automorphism.
For each fixed A in Ae, the map from T2 to Ae given by /(A, n) = p\A^) ls
norm continuous. To verify this, notice that it is true for all non-commuting
polynomials in U, V, 17* and V*. These are dense and automorphisms are contractive;
so the rest follows from a simple approximation argument.
Define two maps of Ae into itself by the formulae
$i(i4)=/ phe2«it(A)dt and $2(^) = / p^mt^Ajdt.
Jo Jo
These integrals make sense as Riemann sums because the integrand is a norm
continuous function. Some of the nice properties of these maps are captured in the
following theorem.
An expectation of a C*-algebra onto a subalgebra is a positive, unital idem-
potent map. Expectations occur frequently in the study of operator algebras, and
have many nice general properties that will not be developed here. The point of
this next theorem is to show that $i and $2 are expectations. Recall that a map $
is contractive if ||$|| < 1, idempotent if $2 = $, and a positive map is faithful
if A > 0 and $(A) = 0 implies that A = 0.
Theorem VI.1.1 $1 is positive contractive idempotent and faithful, and maps Ae
ontoC*(U). Moreover,
*i(f(U)Ag(U)) = f(U)Z1(A)g(U)
for all /, g in C(T). For any finite linear combination of{UkVe:k,££Z},
k.i k

168 VI. Irrational Rotation Algebras
In addition, for every A in A$,
1 n
#i(il) = lim Y UjAU~j.
iV } n->oo2n + l *-'
3=-n
Proof. Since $i(-A) is a convex combination of {pi^mt{A) : 0 < t < 1}, and
since ||/»lfC2«t(A)|| = ||.A||, it is clear that ||$i|| < 1. As $i(I) = I, this is
equality.
Since Pi^mt(U) = U for all t 6 R, Pi^*it is the identity map on all of
C*(17), which is canonically isomorphic to C(T) by the functional calculus
Corollary 1.3.3. Hence we obtain
*i(f(U)Ag(U)) = /%i^.(/(^)ft^w.(A)ft^.(ff(&))c»
Jo
= f(U) f Pi,e>MA)dtg(U) = f(U)*i(A)g(U)
Jo
On the other hand,
#i(W*) = Uk jf'Pl^u{Vl)dt = Uk £e2^dtVl=l°nk j* °
It follows from examining the (dense) set of polynomials in U±x and V±x that
the range of $i is exactly C*(17). As $i is the identity on C*(t/), it follows that
*? = *i.
If A is positive and non-zero, then p^c2*it (A) is positive and non-zero for all t.
Thus the integral $i (A) is positive and non-zero. Hence $i is positive and faithful.
Again considering a monomial UkVi9
1 n 1 n
lim —--— V" Ui(UkVt)U-i = lim —!— V f*Mfj*Vl
n-»oo 2fl + 1 /—' V ' n->oo 2fl + 1 Z—'
i——n j=—n
= lim 1 Ain(2n + l)7T^\ ~fc~, = ^ = ^ ~k~e
n->oo 2n + 1 \ sin n£0 )
By linearity and continuity, this formula is valid for all A in As- ■
The corresponding results for $2 also hold. Combining them, we obtain:
Corollary VI.1.2 The map r = $i$2 = $2*1 is a faithful unital scalar valued
trace on A$.
Proof. First apply $i$2 and $2*1 to a monomial UkVl. We have
if fc = ^ = 0
^UkVl) = 8ko*i(Vl)=ro
*1-.V~ - , ,vV-xV. ,„ ,,
otherwise

VI. Irrational Rotation Algebras
169
The same formula holds for $2*1, and hence they are equal (to r). Moreover, the
range of r is contained in the scalars.
Since both $1 and $2 are positive, faithful and contractive, r is also positive,
faithful and contractive. And since t(I) = I, ||r|| = 1.
To verify that r is a trace, we again compare on monomials
T((UkVl)(UmVn)) = e-2"UmeT(Uk+mVl+n)
_ (e-2"itme ifk + m = £ + n = 0
lo otherwise
T((UmVn)(UkV1)) = e-2nikneT(Uk+mVl+n)
fe-2*ikne ifk + m = £ + n = 0
0 otherwise
{:
When k + m = £ + n = 0, we also have kn = tm and thus r takes the same value
on these two products. By linearity, we obtain r(AB) = r(BA) for all words in
U±x and V±x. By continuity, this extends to all of Ae. So r is a trace. ■
We have enough structure to show that r is in fact the only trace on Ae.
Proposition VI.1.3 r is the unique trace on Aq.
Proof. Suppose that a is another trace on Ae. Then for any A in Ae, we have
a(A) = aiWAU-J). So by Theorem VI.1.1,
Similarly,
a(A) = <t{$2(a)) = ^(*i*2(^)) = <t{t{A)) = t(A)
because <r(I) = 1 and r(A) is always a scalar. ■
Now we are prepared to prove the main result of this section, which is the
uniqueness of the C*-algebra generated by unitaries satisfying (f).
Theorem VI.1.4 Ae is simple. Thus ifU and V are any unitary elements
satisfying (f), then C*(17, V) is canonically isomorphic to Ae.
Proof. Suppose that 3 is a non-zero ideal of Ae. Then there is a positive, non-zero
element X in 3. Since WXU'* belongs to 3, the limit formula for $» shows that
they map ideals into themselves. Hence r(X) belongs to 3. But since r is a faithful
trace, r(X) is a non-zero multiple of the identity. Therefore 3 = Ae.
If U and V are any unitary elements satisfying (f), then there is a canonical
homomorphism of Ae onto C*(17, V) taking U to U and V to V. Since Ae is
simple, this homomorphism must be an isomorphism. ■

170
VI. Irrational Rotation Algebras
From now on we will drop the tilde, and use the symbols U and V for the
generators of Ae. Because of the simplicity of Ae, this no longer causes any
ambiguity.
VI.2 Projections in A$
Our goal is to decide when two irrational rotation algebras are isomorphic. Of
course, only the value of 0 mod 1 matters. So we restrict our attention to 0 in (0,1).
Also since the pair (V, U) satisfies (f) for -0 when 17, V satisfy it for 0, we see that
Ae and A\-e are isomorphic. To proceed further, we need some invariant to tell
the algebras apart. Our experience with AF algebras suggests trying to compute
the K0 groups.
However, it is not immediately evident that there are any non-trivial projections
at all in As- The two generating subalgebras C*(C/) and C*(V) are isomorphic to
C(T), the space of continuous functions on the circle. Since T is connected, neither
contains any proper projections. The irrational rotation algebras have often been
referred to as non-commutative versions of C(T2), because this algebra is universal
C*-algebra for two commuting unitaries. As the torus T2 is connected, C(T2) does
not contain any projections either. So perhaps we should not expect to find any in
Nevertheless, we have the following result due to Rieffel.
Theorem VI.2.1 For every a in (Z + Z0) D [0,1], there is a projection P in Ae
such that t(P) = a.
Proof. We may assume that 0 < 0 < 1/2. We use the representation of Ae from
the first section with U = Mz and V equal to the rotation operator by 0. We will
write Mf for the multiplication operator on L2(T) by the continuous function / in
C(T). Then C*(U) = {Mf : / G C(T)}. Notice that
VMfh{t) = f(t - 0)h(t -0) = MhVh(t)
where f$ denotes the translated function Vf. As before, we identify T with R/Z.
A dense set of elements of Ae can be represented by a finite sum of the form
A = J2i MfiV%. The functions fc in C(T) can be thought of as Fourier coefficients
of A. Indeed, for any A in Ae, we may define fc by the formula Mf. = $i(.AV~*).
From this formula, we see that the coefficients in a finite sum are uniquely
determined. So we may compare two such terms by comparing coefficients. Also note
that on C*(MZ), the value of the trace r(Mf) (which agrees with ^(Mf)) reads
off the zero Fourier coefficient of /. So
r(Mf)= f1f(t)dt.
Jo

VI.2. Projections in Ag 171
Look for projections of the special form P = MgV + M/ + M/.V*. Since
P = P*, this forces
MgV + M, + MhV* = F* Mj +Mj+ VMK = Af5_, V + M? + Mjj# V.
By comparing coefficients, we see that / — / is a real valued function; and that
fr(f) = ^(t + 0) or equivalently h(t - 0) = g(t). Since P = P2, we also get
= MgVMgV + Af,VAf) + MgVMhV* + Af/flV + Mp +
+ M/hF* + MhV*MgV + MhV*Mf + MhF*Af/.F*
= MggeV2 + My(/+/,)Vr + Mghe+p+hg_e + MM/+/_#)V* + Mfcfc_# V*2
By comparing coefficients and replacing h's with g's using the relation between
them, we arrive at the necessary and sufficient conditions:
g(t)g(t-0) = O (1)
flf(*)(l-/(*)-/(*-')) = 0 (2)
f(t)-f(t)2 = \g(t)\> + \g(t-e)\> (3)
(The other two coefficients yield the same identities.)
These equations can be explicitly solved. (See Figure VI. 1.) Pick any positive
e such that 0 + e < 1/2. Define / to be the piece-wise linear function
(e'H for 0<t<e
fit)
1 for e<t<0
e-l(6 + e-t) for 6<t<6 + e
0 for 6 + e < t < 1
and define
W== Jv7(*)-/(*)2 for 6<t<e + e
* jo otherwise
Clearly (1) holds. Since f(t) + f(t - 0) = 1 on [e, 20] which includes the
support of g, (2) holds. Finally, f(t) - /(f)2 is non-zero on (0,e) and (0,0 + e)
where by design, we have the identity (3). So this determines a projection P in As-
We compute the trace by
t(P) = r(Mf)
f f(t)dt = 0.
Jo
We also get the projection I - P with trace r(I - P) = 1 - 6.
Now notice that UVk = e2nikeVkU for fc in Z. Hence ^ contains a copy of
Ake for every fc. Replacing V by V* and 0 by the fractional part {k0} of kO yields
a projection P in this subalgebra with trace {k0}. Hence we obtain every value in
(Z + Z0) fl [0,1] as the trace of a projection in Ao. ■

172
VI. Irrational Rotation Algebras
/(*)
/*(*)
g{t-o)
g(t)
0 0+€ \
Figure VI. 1
20 20+e
VI.3 An AF Algebra
At this stage, we have identified a large class of projections in Ao which is
directly related to 0. However, we need to know that these are the only possibilities,
and that these numbers are not merely a reflection of the construction. The key step,
due to Pimsner and Voiculescu, is to imbed Ao into an AF algebra with the same
(apparent) K$ group
(G, g+, r(G)) = (z + ze, (z+ze) n r+, (z+ze) n [o, i]).
We need some information about continued fractions. If 0 is an irrational real
number, it has a continued fraction expansion [ao, ai, a2,...] where ao G Z and
ai 6 N are the unique choices so that
0 = lim [ao, ai,..., an] = lim ao H
n-»oo
n->oo
ai +
a2 +
a3 +
1
•• + —
The rational approximations pn/qn = [a0, ai,.
cursion formulae
, an] are determined by the re-
Po = a0
pi = a0ai + 1
Pn = Gn.Pn-1 + Pn-2
?0 = 1
01 = ai
9n = a>nqn-i + 9n-2 for n > 2
Thus
Pn
b>n-l
1]
0
\Pn-l
\Pn-2
?n-l
0n-2j

VL3. An AF Algebra
In particular, we obtain the identity
Pn?n-1 - Pn-lin = <iet
173
Pn 9n
Pn-1 ?n-l
= (-1)"
n-1
From this identity, it follows thaipn/qn is an alternating series satisfying
P2n-2 P2n P2n+1 l>2n-l
?2n-2 ?2n 02n+l ?2n-l
We also can easily deduce that the p's and g's grow geometrically fast, and thus
Z)n>i Qn* < °°- See Hardy and Wright, Theory of Numbers for more details.
The AF algebra we construct will be the increasing union 21$ = Un>i2ln of
subalgebras 2ln = MQn © MQn__l and the partial multiplicities of the imbedding
an_i,n are given by An := [ "" J ].
The computation of -PT0(2l^) follows in the same manner as the Fibonacci
algebra of Example IV.3.6. Thus
K0(%) = ljn^Z2, Z*), An) = (Z2, Pe)
where Pe is the positive cone of the limit. Let
rl = (-irl
1-1
Pn qn
Pn-1 0n-l
J-n •— -^o Al ^n
Thus we get a commutative diagram
Ko{%n) = Z2 -^ KoiXr+i) = Z2
?n-l
-.Pn-1
-5fn
Pn
Tn
*n+l
ifo(a) = z2
So P<j = Un>i Tn^X- Now TnZ^ is the cone generated by the vectors
V (-i)>„-i; and U-irW •
Notice that (TS^I1 ) hves *n the second quadrant, and has slope "F2^1 which
increases to -0; while (-^n) lives in the fourth quadrant, and has slope z^-
which decreases to -0. Therefore
Pe = {(l)eZ2:6x + y>0}.
The order unit of 2ln is (^ ). And so Tn (^ ) = (?) is the order unit of
21*. K0(Qle) = (Z2,.Ffc) is a total order with a unique state ^((y)) = Oz + y. The
map (7* is an order isomorphism of Ko($l$) onto G. By Theorem IV.5.3, there is a
unique trace on 21$, which we denote by a.

174
VI. Irrational Rotation Algebras
VI.4 Berg's Technique
Our plan is to imbed the irrational rotation algebra As into the AF algebra 21$
by approximating the unitaries U and V by their finite dimensional analogues. So
if we take the standard orthonormal basis for L2 to be ek(i) = e2nikt for fc G Z,
the representation of section VI. 1 is seen to have the matrix forms
Uek = ek+1 and Vek = e2nikeek.
The rational number Pn/in approximating 0 suggests defining the two unitaries
on On = L2(Z/Zqn) with basis ejj,n) for k G Z/Z«,n by
K
» _ >)
efc+l
"cfc
and
y p(n) — p2irtfcpn/gnp(n)
It is routine to verify the identity UnVn = e2ldpnlqnVnUn.
We will consider the pairs (Un © J7n-i, Vn © V^-i) in 2ln = MQn © A^^.
The difficulty is to arrange the imbeddings of 2ln into 2ln+i so that these pairs are
Cauchy. Once this is done, the limits will be the desired unitaries.
In order to arrange the imbeddings, we need to know how to approximate shift
operators by direct sums of shifts of smaller order. A method for doing this is
called Berg's technique.
Theorem VI.4.1 Let ej and fj, for 0 < j < n, form an orthonormal set. Suppose
that T in B(U) satisfies Tej = eJ+i and Tfj = /j+i for 0 < j < n - 1.
Then there is an operator S in B(H) such that Sx = Txfor every x orthogonal
to span{ej,/j : 0 < j < n - 1} and 5span{eJ-, fj} = span{eJ+i, fj+i}for
0 < j < n - 1 such that Sne0 = fnt 5n/o = en cind
||S-T||<2Sin£<*.
Proof. We introduce a twist through an angle ir/2n using the matrix
0=rw2 i/\/2i[i o i ri/v/2 imi
l/v/2 -1/V5J |o e^n\ [\/y/2 -l/y/2\
&«il2n [cos(^) tsin(^)]
[isin(£) cos(£)J
Then
0" =
1/^ l/x/21 fl 0
1/V2 -\/y/2\ [0 -1
l/x/2 1/V2 '
\/y/2 -l/\/2
0 1
1 0
Think of T\ span{ej, fj} -+ span{ej+i, fj+i} as the identity matrix with
respect to these bases. Let S\ span{e.,-, fj} -»• span{eJ+i, fj+i} be given by the
matrix 0. Then Sneo = /„ and Snfo = en as desired. Moreover,
||S-T|| = ||0-/|| = |e-/"-l| = 2sin£<£. ■

VIA Berg's Technique
175
Remark VI.4.2 One of the significant points about this approximation method is
that the domain and range of the perturbation is limited to
span{ej, fj : 0 < j < n - 1} and span{eJ:. fj : 1 < j < n}
respectively. So several applications of this method on pairwise orthogonal pieces
will not increase the norm of the perturbation.
Think of T as a right shift acting on two parallel sets of vectors ej and fj for
0 < j < n. The twist introduced in S produces a cross-over so that S is also a
shift on another basis that moves e0 gradually across to fn and /n gradually across
to en. Graphically, we may represent this by the picture in Figure VI.2.
?0
V
s
Po
ei
S
s
h
T
Figure VI.2
en
s
*v
/n
eo * - en /o - * fn
\ : \ j
\ 4
v
\ /
/ ^
Figure VI.3
However, in our application, the two segments of T will be part of a longer
shift. So we may picture this as in Figure VI.3. The interchange between the
vectors {eo,..., en} and {/o,..., fn} is represented by the two sets of crossing
lines. The dashed lines represent the interchange from eo to fn. The result is a leap
in n steps from eo to fn denoted bylhe dotted curve above the diagram. At the same
time, the two dotted lines crossing the dashed lines represent the corresponding
interchange from /o to en. This results in pulling out a direct summand which is a
unitary shift which follows T from en to /o and returns along the dotted lines in n
steps.
In this way, we may repeatedly cut out unitary summands. Provided that the
intertwinings occur on orthogonal pieces, the resultant perturbation is just the
maximum of all the changes. So we will be able to split a long shift into a direct sum
of smaller shifts with a reasonably small perturbation. The example below
demonstrates this in the case of interest to us.
Example VI.4.3 Consider the shift I7n+i acting on L2(Z/gn+iZ) as above. From
the relation gn+i = an+iqn + gn_i, we see that we should approximate I7n+i by
the direct sum of an+i copies of Un plus one copy of Un-i. Let 6 := [gn/2j and
V = qn - 6 = r<Zn/2]; and set s = [gn-i/2]. Perform the interchange between
the vectors {e-kb+j ' 0 < j < s} and {ew+j • 0 < j < s) for 1 < k < an+i.
The following figure illustrates this:

176
VI. Irrational Rotation Algebras
-36
364*
Figure VI.4
Notice that the intertwining between
{e_&+j : 0 < j < s} and {e^+j :0<j<s}
produces a summand Un,i unitarily equivalent to Un. Indeed, the perturbed opera-
,(!)
,(1) ._
tor 5 takes g\ ' := e_6+J- to g^x := e_&+J-+i for s < j < an and takes a vector
,(*)
(i)
,(*)-
5} j in span{e^6+J-, e6/+J} to ^+\ for 0 < j < s starting with gfr } = g^J = eh>
and eventually arriving dXg\ ' = e_6+,.
Similarly, the interchange between
{e_fc6+j : 0 < j < s} and {ekv+j : 0 < j < s}
results in a summand which is a shift on kqn vectors. But out of this, the shift
between {e_(k-i)b+j : 0 < j < s} and {e^-tfb'+j : 0 < i < s} took the middle
(k - l)qn basis vectors out into smaller shifts. What remains is a summand Unjk
unitarily equivalent to Un.
Note that the summand Unjk shifts g\ ' := e-kb+j to g^x := e-kb+j+i for
s < 3' < &• Then because of the (k - l)st intertwining, it shifts a vector
,(*)
g) e span{c.(fc.1)6+y.6)JC(fc.1)6/+y.6)}
,(*)
„(*)
to 5L1 for 6 < j < b + s until arriving at g^8 = e^k^bi+8. Then it shifts
gf] := c(fc.i)6/+(i.6) to ^ for 6 + 5 < j < b + V = qn. Finally, the fcth
interchange produces a shifting from g\ J in span{e_fc6+(j_6), c^z+y.^ } to 5^
for 0 < j < s such that ^fc) = ew and ^fc) = e_fc6+,.
Finally, since the interchange between
{e-an+16+j : 0 < j < s} and {ean+l6/+i : 0 < j < s}
pulls out a shift on an+i*jn vectors, it must leave behind a summand Un-i,o
unitarily equivalent to I7n_i. The basis on which E/n-i.o acts will be denoted as gy
for 1 < j < gn-i, where ^.0) = ean+l6/+J- for s < j < qn-i and g^ belongs to
span{e_an+l6+j, ean+l6/+i} for 1 < j < s.

VI.5. Imbedding Ae into %q 111
The total norm perturbation of these interchanges is the maximum of each
individual perturbation, which are all bounded by n/s < 27r/gn_i.
Let us identify the unitary operator that maps Un © Un-i to
]T©tfn,fc©tfn-l,0.
fc=l
We don't wish to prejudge where each shift begins, so we allow a rotation of each
term. Let e^ ' for j G %/%qn denote the basis on which the kth summand
equivalent to Un acts, and let e\ ' for j G Z/Z9n_1 be the basis for £/n_i. Then define a
unitary Wn from C?n(an+l) © Cgn-1 to 0+1 by
W"e? ='&><*«*) for 0<i<9n
^^=^0-^) for 0<i<fc.x.
The values of the integers c* will be determined later to suit our needs. With any
choice of these c*.'s, we obtain
VI.5 Imbedding ^ into 21$
We follow the plan outlined at the beginning of the last section. Indeed, our
example shows how to imbed %n = MQn © Mqn_x into MqnJtl in such a way
that Un © Un-i is mapped to an operator unitarily equivalent to Unan © Un-\
close to the shift Un+i. The constants c* will now be chosen so that Vn © Vn-i is
simultaneously mapped to an operator unitarily equivalent to VrrJan+1' © Vn-\ close
to Vn+i- The reason that this is possible is that V^+i is a diagonal operator with
eigenvalues that are qn+i periodic, and thus almost qn periodic and qn-\ periodic
due to the nature of the rational approximations. This allows us to match up the
eigenvalues of the V^'s with those of V^+i within a reasonable accuracy. How
close we get is a fairly straightforward calculation.
For convenience, let us write An = e2ir*Pn^n. Now Vn+iej = Xn+\e5 ^d
hence Vn+ie-kb+j = A~^+Je_fc6+J-. Now A™ is close to A™+1. Since we wish
to identify V* with a summand Vn,k acting on the basis {g\ ' : 0 < j < qn}, this
suggests that we send e^ ' to gj+kb- That is, choose c* = kb for the definition of
Wn in Example VI.4.3 for 1 < k < an. For the imbedding of the copy of Vn-i,
choose an integer en so that
|An_i - An+1 | < —.
The key computation is the estimate:

178 VI. Irrational Rotation Algebras
Lemma VI.5.1 With the choices for c*. made above, we have
\\wn{v^) e v^w: - vn+11| < jj + ^.
Proof. Let V^+1 := Wn(yian+l) ®Vn-i)W*. Note that both Fn+i and V^+1 have
eigenvectors e-kb+j = tfj^ for s < jf < 6 and e^jy+j = ^6 for s < j < b'
and 1 < k < an+i; and eigenvectors ean+1&/+j = gy for s < j < qn-i. In
addition, they have common two dimensional reducing subspaces
£kJ := span{e_fc6+i,efc6'+j} = span^,gffi}
for 0 < j < s and 1 < k < an+i and
£QJ := span{e>an+l6+j,ean+l6'+j} = span{^an+1\(7J0)}
for 0 < j < s. Thus it suffices to estimate the difference of norms on all of these
subspaces and take the maximum (as their difference is also a direct sum).
On e-kb+j = 9j for s < j < b we have (using c*. = -kb)
ll(K+i - Vn+1)e.kb+j\\ = |Arfe6 - >Ht\
= 2|«n(.T(,--W)(J-Sgi))|
Similarly,
ll(K+i-^+i)efcy+j||<^.
On £kj9 we have
^n+l5j — An 5j
and
Therefore V^+1 |£fcj = XiTkbh is scalar. Hence as above, we obtain the estimates
IKK+i - fn+i)e-fcb+il| = |A;Tfc6 " A^-f I < fn
and

VI.5. Imbedding Ae into 8$ 179
The estimates relating to the shorter shift l7n-ifo are not quite as good, but still
suffice. On ean+16/+j = g\ ' for s < j < qn-\, we have
||(K+i - vn+l)ean+lb,+j\\ = |a£* - A°;Yb'+i
Pn-1 __ Pn+1
3n-l * 3n-l 3n+l
^i^7 + 2sin^'|
Finally, on Sqj, we have
l|Vn+i|ft„- - Ai;ri6/2H = I W+i - *£ri6| = K;rn -1| < *
and
ii^+iiftj - Ai;ri6/2n=max{|Ar°"+ib - A£ri6i> |a£? - Afcrib|}
Thus we obtain the total estimate
IIK+i-KM-ilKj^ + f.
This establishes the lemma. ■
Theorem VI.5.2 There is a *-monomorphism p of the irrational rotation algebra
Ae into the AF algebra 21$ corresponding to the continued fraction expansion of
0 such that /&* is a homomorphism ofKo(Ae) onto Kq^q). Moreover ifr and a
denote the unique traces on Ae and%e respectively, we have r* = (7*/o* is an order
homomorphism ofKo(Ae) onto Z + Z0.
Proof. Define *-monomorphisms of 2ln into 2ln+i by
<pn,n+i(A © B) := Wn{Ala»+*) © B)WZ © A.
Clearly, this is an imbedding with partial multiplicities An := [ a£ J ]. So the direct
limit is 21^.
From the estimates in Example VI.4.3 and Lemma VL5.1, we see that
\\<Pn(Vn 0 Un-l) - ¥>n+l(tfn+l © Un)\\ < £"
and
llVn^n © V»-l) " ¥>n+l(V„+l © Vn)\\ < £ + £_.
Since 53n>i g^1 < oo, these two sequences are Cauchy. Let
17 := lim <pn(Un © C/„_!) and V := lim ^(K 0 Vn-i).

180
VI. Irrational Rotation Algebras
As UnVn = \nVnUn* it follows that in the limit
UV = lim XnVnUn = e2nieVU.
n—}>oo
Hence there is a *-homomorphism /o which takes the generators of A$ to U and V
respectively. Since As is simple, this is a monomorphism.
If cr is the unique trace on 21$, then clearly ap is a trace on As, and thus
r = (7/o. Hence r* = <T*p* is an order homomorphism of K0(Ao) into Z + Z0. By
Theorem VI.2.1, this map is surjective. ■
In Example VIH.5.2, we will show that r* is an isomorphism. However, we
already have sufficient information to distinguish between different rotation
algebras.
Corollary VI.5.3 Two irrational rotation algebras As and A^ are isomorphic if
and only ifr} = ±0 mod Z.
Proof. Two isomorphic algebras will have the same K$ group and the same unique
trace. Hence we obtain
Z + Z0=Z + Z?7,
from which it follows that 77 = ±0 mod Z. ■
Exercises
VI. 1 Consider the rational rotation algebra An/m the universal C*-algebra for
the relation VU = e2nim/nUV, where gcd(m, n) = 1.
(a) Show that Un and Vn lie in the centre.
(b) Hence show that each irreducible representation acts on an n-dimen-
sional space.
HINT: Find the relationship between V and the spectral projections of U.
(c) Describe all irreducible representations of Am/n.
VI.2 Show that any two irreducible representations of As are approximately
unitarily equivalent.
HINT: Apply Voiculescu's Theorem.
VI.3 Show that there are irreducible representations of As which are not
unitarily equivalent.
HINT: Let U be the bilateral shift, and let V be a diagonal operator.
VI.4 Let $ be an expectation of B(H) onto a von Neumann subalgebra 21.
(a) Imitate the proof of Lemma 1.9.9 to show that $ is self-adjoint and
order preserving.
(b) Show that $(PT) = P$(T) for all T in B(U) and projections P in 21.
Hence show that $(ATB) = A$(T)B for all T in B(U) and A, B in 21.
(c) Show that $(T)*$(T) < &(T*T).

Exercises
181
VI.5 Show that C(T2) is the universal C*-algebra for a pair of commuting uni-
taries.
VI.6 Let 21 be the universal C*-algebra generated by two projections. Show that
every irreducible representation of 21 is at most two dimensional. Hence
show that 21 is isomorphic to the subalgebra of C([0,1], M2) consisting of
those functions F such that F(0) and F(l) are diagonal.
Notes and Remarks.
Irrational rotation algebras were first systematically studied by Rieffel [1981],
who constructed the projections of Theorem VI.2.1. Then Pimsner and Voiculescu
[1980a] imbedded As in the AF algebra of the continued fraction of 0 in order to
classify them. The proof here is taken from Davidson [1984]. Berg's technique
is valid in much greater generality for operators which are block tridiagonal (see
Berg-Davidson [1991]). As we shall see in the chapter on crossed products,
Pimsner and Voiculescu [1980b] show that the map r* is an isomorphism on Ko(A$).
Rieffel [1983b] also showed that the irrational rotation algebras satisfy
cancellation. Putnam [1990b] has shown that the invertible elements are dense in Ae-
Recently, Elliott and Evans [1993] have proven that Ae is a limit circle algebra,
and hence has real rank zero.

CHAPTER VH
Group C*-algebras
Many groups which arise in harmonic analysis yield important examples of
C*-algebras. The C*-algebra of a group encodes all the information about unitary
representations of the group. Groups also arise naturally as subgroups of
automorphism groups of C*-algebras. This in turn leads to a general C*-algebra
construction which will be discussed in the next chapter.
We will be concerned with locally compact (Hausdorff) groups. A unitary
representation of a group G is a homomorphism of G into the unitary group of
B(H) which is continuous in the strong operator topology, meaning that the map
from s in G to n(s)x is continuous for every vector x in H. A representation is
irreducible if the range does not commute with any proper projections. It is easy to
see that this is equivalent to saying that C*(7r(G)) is irreducible.
A locally compact group G supports a regular Borel measure no which is
invariant under left translation; meaning that hg{sE) = Pg{E) for every s in G
and Borel subset E of G. It is unique up to a scalar multiple, and is known as
left Haar measure. This measure is finite when G is compact. In this case, we
normalize it so that pg{G) = 1. If G is infinite and discrete, then we normalize so
that ^g({^})=1» where e is the identity element of G. We will write ds instead of
dno{s) when there is no ambiguity.
In general, left Haar measure need not be right translation invariant. There is a
continuous homomorphism A of G into R+ known as the modular function such
that hg(Es) = A(s)hg{E). A group G is called unimodular if A is trivial. This
occurs when G is abelian or discrete or compact, for example. In particular, we
have the formula
dpait-1) = A^-^gW-
The space L1 (G) of absolutely integrable functions with respect to Haar
measure becomes a *-algebra with the operations of convolution and inversion:
/**(*) = ff{s)g{s"H)ds
f*(t) = aw-1/!^)
The L1 (G) norm is not a C*-algebra norm however.
182

VII. Group C*-algebras 183
The algebra Ll(G) is unital only if G is discrete; and then we write tl(G). In
this case, 8e, the characteristic function of the identity element, is the unit.
Moreover, the group algebra CG consisting of all finite sums J28eG a*&* forms a dense
subalgebra of I1 (G). For many purposes, it suffices to work with the group algebra
rather than ^(G).
In general, Ll(G) has a norm one approximate identity. One may be obtained
as follows—for each open neighbourhood U of e, choose a positive function fjj in
LX(G) supported in U such that /£ = fu and \\fu\\x = 1. This net is ordered by
containment of sets. When G is metrizable, we can obtain a sequential approximate
unit. The details are left as an exercise.
When 7r is a unitary representation of G, it induces a representation of L1(G)
by integration:
Hf) = jf(t)ir(t)dt.
One readily verifies that
l|SF(/)ll</l/WI*=ll/lli-
This is a homomorphism because, by Fubini's Theorem and left invariance,
*(/ *9) = f f f(s)g(s-H) ds ir(t) dt
= J /(«)*■(*) J g(s-H)ir(s-H)dtds
= / /(s)tt(s) / g(u)ir(u)duds = n(f)ir(g).
To see that it preserves adjoints, calculate (with substitution u = t~l)
(*(f)*x,y) = (*,*(/)v) = fWJ(*M*)v) ^
= /TWM*"1)*. V)dt = fl{y^){^{u)x, y) &(u)-ldu
= (I f*(.ti)ir(u)xdu,y) = (ii(f*)x,y).
Conversely, if 5r is a representation of L1(G) which is non-degenerate,
meaning that ^(L1 (€}))% = %, then it determines a unique unitary representation of G.
To see this, let f\, A € A, be a norm one approximate identity for L1 (G). Then
Iim7r(/x)7r(5)i5 = if(g)x
for every g in Ll(G) and x in W. Hence the contractions 7t(/a) converge strongly
to the identity operator. Define
^(s)7f(g)x = Tr(g8)x where g8(t) = g(s~xt).

184
VII. Group C*-algebras
It is easy to verify that ir(s) = SOT-limAeA^CC/x)*)- Moreover, this implies
that 7r is a contractive homomorphism of G into B(7i). Since ||7r(s)|| < 1 and
IkC5)-1 II < -t ^ follows that ir(s) is unitary. These formulae are clearly necessary
if 7? is to be induced by 7r; so this establishes uniqueness. Standard notation now is
to suppress the tilde, and refer to the extension of it to Ll(G) as it as well.
Every locally compact group G has a distinguished representation called the
left regular representation on L2(G). This is defined by
*{s)g{t) = g8(t) = g{s-H).
This map is unitary because left translation is isometric due to the translation in-
variance of Haar measure. The reduced group C*-algebra of G is defined to be
C*r(G) :=X(iJW)-
The group C*-algebra of G is the closure of the universal representation of
L1(G). That is, take iru to be a direct sum of all irreducible representations (up
to unitary equivalence) of G. Then C*(G) is the norm closure of 7rtt(L1(G)).
Equivalently, define a C*-norm on LX(G) by
11/11 = sup{||7r(/)|| : 7r is a ^representation of L1(G)}.
This collection of representations is non-empty because any irreducible
representation of C*(G) provides an irreducible representation of G; and such representations
exist by the GNS construction. Since ||/|| < ||/||i, this supremum is well-defined.
The group C*-algebra C*(G) is the completion of L1(G) in this norm. By
construction, there is a one-to-one correspondence between the irreducible
representations of C*(G) and i1(G), and hence with the irreducible representations of G.
In particular, there is a representation A of C*(G) onto C*(G) extending the left
regular representation.
When G is an abelian group, the dual group G = Hom(G, T) is the group of
characters on G. The following easy result computes C*(G) in this case.
Proposition VII.1.1 IfG is an abelian group, then
C*(G) = Cr*(G) = C0(G).
Proof. For abelian groups, Ll(G) is an abelian Banach algebra. So by the Gelfand
theory, the irreducible representations of L1 (G) are the one-dimensional
representations corresponding to multiplicative linear functionals. These functionals
correspond by the argument above to the one-dimensional unitary representations of G,
namely the elements of G. Thus the Gelfand map sends / in Ll{G) to its Fourier
transform / in C0(G) given by
f(X) = Tf{X) = f f{t)X{t) dt for all X G G.
Jg
The range is self-adjoint and separates points; and thus is dense in C0(G) by the
Stone-Weierstrass Theorem.

VII.2. Amenability
185
The Fourier transform T extends to a unitary operator U from L2(G) onto
L2(G). The left regular representation is now understood by conjugating it by U:
U\(f)U*g = UX(f)g = F{f * g) = fg = Mfg
for all / in Ll(G) and g in L2(G) n Ll{G). Thus each / in C0(G) is sent to
the multiplication operator Ms. This map is isometric; whence A is an isometric
isomorphism. Therefore C*(G) = C*(G). ■
VII.2 Amenability
A group G is called amenable if there is a left translation invariant mean for G.
A mean is just a state m on L°°(G); left invariance indicates that m(g8) = m(g)
for all g in L°° (G) and s in G. Compact groups are amenable—an invariant mean
is given by integration against Haar measure.
Abelian groups are also amenable. The existence of an invariant mean
follows from an application of the following well known fixed point theorem due to
Markov and Kakutani.
Theorem VII.2.1 Let T be a commuting family of continuous linear maps of a
topological vector space X into itself Suppose that K is a compact convex subset
ofX such that TK C K for every T in T. Then there is a point x in K such that
Tx = xforallTinT.
Proof. Expand T to include all finite products of its elements. For n > 1 and
T in T, let T<n> = £(/ + T + T2 + • • • + T""1); and set K(n, T) = T^K,
which is a compact convex subset of K. Since T is commutative, for any subset
{Ti.l <i<k}ofT,
k k k
t=l 1=1 1=1
Thus the family of all K(n, T) for all n > 1 and all T in T satisfies the finite
intersection property. So by compactness, there is a point x lying in the intersection.
Let O be an open neighbourhood of the origin in X, and let T belong to T.
Since K is compact, so is K - K, and thus there in an integer n so that K - K is
contained in nO. Now x belongs to K(n, T), and therefore there is a point y in K
such that T^y = x. Hence
Tx - x = I(/ + T + T2 + • • • + Tn~1){Ty - y)
= k(Tny-y) eHK~K)co.
This holds for every open set O containing 0, and hence Tx = x. ■
Corollary VII.2.2 Every abelian locally compact group G is amenable.

186
VII. Group C*-algebras
Proof. Consider the state space S of L°°(G) endowed with the weak-* topology,
which is a compact convex set. For each s in G, consider the left translation
operator T8(f) = f8 on L°°(G). The dual operator T* acts on the dual of L°°(G) and
is weak-* continuous. Moreover, it is easy to see that if <p is a state, then T*(p is
also a state. Indeed, it is positive because
r>(/) = ?(/.) >0 for all /6I?(G).
Hence
\\t:<p\\ = 11^(1)11 = 1^(1)11 = 1.
It follows that each operator T* maps S into itself. As G is abelian, the family
{T; : s e G} is abelian. Therefore by the Markov-Kakutani Theorem VEL2.1,
there is a common fixed point m in S. Hence
m(/.) = (2>)(/) = m(/) for all / 6 L~(G).
So rn is an invariant mean for G. ■
For discrete groups, many standard group constructions preserve amenability.
Proposition VII.2.3 For discrete groups, amenability is preserved under taking
subgroups, quotients, direct limits and extensions.
Proof. Suppose that H is a subgroup of a discrete amenable group G with invariant
mean mG. Fix a set of elements {s\ : A £ A} with one element from each left
coset of H in G. There is a positive, unital, isometric imbedding rj of L°° (H) into
L°°(G)by
(rjf)(tsx) = /(*) for all teH,\e A.
Notice that this is indeed positive, unital and isometric. Moreover, a simple
calculation shows that r}(ft) = (rif)t for all t in H. Thus mn := mot) is an invariant
mean on H.
If if is a normal subgroup, let G/H be the quotient group. Define a positive,
unital, isometric imbedding q of L°° (G/H) into L°°(G) by qf(s) = f(sH). Then
another computation shows that tuq/h := moq is an invariant mean on G/H.
Indeed, if tH is in G/H and / belongs to L°°(G/H), then
q(ftH)(s) = MHy'sH) = fit-'sH) = (qf)t(s).
Hence
™>GIhU) = mGq{ftH) = mG{{qf)t) = ™,G{qf) = mG/H(f).
Suppose that G = limGn, where each Gn is amenable. By replacing each Gn
by its image in G, we may assume that the imbeddings of Gn into G are injective.
This image is a quotient of Gn, and thus is amenable. Let mn = mGnrn where rn
is the restriction map of L°°(G) onto L°°(Gn), and mGn is an invariant mean on
Gn. Let m be any weak-* cluster point of this net of states. Every s in G lies in
Gn for n sufficiently large. Thus is follows that m is translation invariant.

VII.2. Amenability
187
Suppose that H is an amenable normal subgroup of G with amenable quotient
G/H. Let mn and mG/H be translation invariant means on these groups. Define
a map $ of L°°{G) into L°°(G/H) by
Zf(sH) = mH(f8-l\H),
which is well defined because of the translation invariance of run- Then set
mG(f) = ™>G/H{$f)- This is positive because $ is positive; and is norm one
because $1 = 1, whence moil) = ™>G/H(l) = 1. Finally, it is translation
invariant because
Whence
(Zft)(sH) = mH{f8-H\H) = (Zf)tH(sH).
mG{ft) = mG/H($ft) = mG/H($f) = mG(f).
Example VII.2.4 The free group is a prototypical example of a non-amenable
group. To see this, consider the set Uq (and U{) of elements of F2 which, in reduced
form, begin with an even (odd) power of u followed by 6 or a word beginning with
a power of v. Similarly, define Vo, Vi and V2 beginning with a power of v
congruent to 0,1 or 2 modulo 3. Clearly, U\ = uUq and Vj = v*Vo for j = 0,1,2.
Also, F2 is the disjoint union of Uq and U\, and also of Vo, Vi and V2. Hence a
translation invariant mean must satisfy
1 = m(Xw2) = 2m(Xul) = 3rn(XVo).
However, U\ is a proper subset of Vo. Thus
\ = m{XUl)<m{Xv,) = \
which is absurd. Therefore F2 is not amenable.
We will see examples of C*-algebras for both amenable and non-amenable
groups in this chapter. A deep result that lies at the intersection of harmonic
analysis and C*-algebras is the following, which we will not prove here. We will not
need to quote this theorem for the analysis of our examples because we will be able
to establish the representation theory explicitly in these cases.
Theorem VII.2.5 If G is a locally compact group, then the left regular
representation X ofC*(G) onto C*(G) is an isomorphism if and only ifG is amenable.
We will provide a proof of half of this theorem for discrete groups. This should
give the reader a feeling for the kind of arguments involved relating positive definite
functions on the group with states on the group algebra.
A function <p on a discrete group G is called positive definite if
n n
^2 y^ ^tSJ^^J1*!) > 0 for all n > 1, a; G C, Si e G, 1 < i < n.
i=i i=i

188 VII. Group C*-algebras
Let V{G) denote the set of all positive definite functions on G such that <p(e) = 1.
Any state $ on C*(G) determines a positive definite function in V(G) by setting
<p(s) = $(88) for all s in G. Indeed, <p(e) = 1; and if a; £ C and S{ € G for
1 < i < n, then setting / = J2?=i ai<**,> we obtain
n n
J] J] aiaj^sj'si) = *(/* * /) > 0.
1=1 j=l
Conversely, if y> is a positive definite function on G, define a functional on CG by
n n
*=i i=i
This is a unital positive linear functional since if / = Yh=i ai^. *s m ^^ ^en
n n
*(/* * /) = ££aw(57ls») > o.
i=l j=l
The proof of Lemma 1.9.5 shows that $ is continuous with respect to the C*(G)
norm. Thus $ extends by continuity to a state on C*(G). This establishes a bijec-
tive pairing between V{G) and the state space of C*(G). Consequently, for any
element / = £!=i oti$8i in CG, the norm in C*(G) is determined by
H/||C-(G)= SUp #(/**/)1/2= SUp (j2J2a^^SJlsi))
1/2
Lemma VII.2.6 Ifipi and y>2 belong to V{G), then the product <p\<f>2 belongs to
T(G).
Proof. Let si,..., sn in G be given. Let Ak denote the n x n matrix with
coefficients a\j' = (fkisj1^) for 1 < i, j < n and k = 1,2. This is a positive matrix
since for a in C\ let / = ]£!=i «;<£,,. belong to CG; and note that
(Aia1a) = <p(f**f)>0.
Let A = A\ o A2 denote the matrix obtained from pointwise (Schur) product of the
matrix entries ay = ajj'aj?'. This is a positive matrix (see Exercise VEL3). Thus
n n
*=i j=i
Since <pi<f2(e) = 1, the product lies in V{G). ■
A crucial fact is that positive definite functions with finite support are
determined by states on the left regular representation. This will reduce the problem to
approximating positive definite functions by functions of finite support.

VTI.2. Amenability 189
Lemma VII.2.7 Suppose that <p in V(G) has finite support. Then there is a unit
vector x in £2(G) such that <p(s) = (X(s)x1 x)for s in G.
Proof. Define a linear map on CG by Tf = / * (p. As <p has finite support, it is
evident that T is bounded by ]£,€G \<p(s)\ when CG is endowed with the £2(G)
norm. Thus T may be extended to a bounded operator on £2(G) by continuity.
Furthermore, if / = ££=i oti&8i belongs to CG, then
(r/, /) = £ £ f(tMt-l*)7® = £ £ wM'j'si) > o.
*e<3 tec? t=i j=i
Hence T is positive. Next notice that T commutes with X(G) because
X(s)Tf = S8*(f*<p) = (S8*f)*<p = T\(s)f.
Set z = T1'28e. Then
(X(s)xtx) = {\{s)Tll28e,Tll28e) = (Ss1T8e) = (Ss,<p) = tp{s).
Finally, \\x\\2 = <p(e) = 1. ■
We are now prepared to use amenability to approximate positive definite
functions by positive definite functions of finite support. In this way, we will establish
the following theorem.
Theorem VII.2.8 IfG is a discrete amenable group, then C*(G) = C*(G).
Proof. Let m be an invariant mean on £°° (G). Since £l (G) is the predual of £°° (G),
Goldstine's Theorem states that the unit ball of £l(G) is weak-* dense in the unit
ball of £°°(G)*. Since CG is dense in £X{G), there is a net /7,7 e T, in the £l(G)
unit ball of CG converging weak-* to m. First we show that we may assume
that each /7 is positive and satisfies (/7,1) = 1, where 1 is the constant function
1(5) = 1. (Such functions will be called finite means.) Indeed,
l(/7.i)l-l£AMI<IIAIIi<i-
sEG
But lim7(/7,1) = rn(l) = 1. So it is apparent that each /7 may be replaced with
the function #7 (s) = \ /7 (s) \ / \ \ /7111 which will differ from /7 by small norm for 7
sufficiently large.
Since m is left translation invariant, for every s in G,
w*-lim^ *57-(gf7 = ^*rn-rn = 0
Thus S8*g^- gy converges to 0 weakly in £} (G). By the Hahn-Banach Theorem,
for any finite set s\,..., sn in G, the norm closed convex hull of
{(S*i *9f-9f)i<i<n -7 €?}

190 VH. Group C*-algebras
in ^(G)n contains 0. Therefore we obtain a sequence gn of finite means in the
convex hull of the #7's such that
lim \\S8 * gn - gn\\i = 0 for each s G G.
n—+oo
Then hn = gi are positive functions in £2(G) with ||fen||2 = 1. Note the
simple inequality \a - 6|2 < \a - 6||a + 6| = \a2 - 62| for all a, 6 > 0. Therefore,
since fen is positive,
lim \\\(s)hn - hn\\22 = lim V |ftn(5<) - /.n(<)|2
n—>"00 n—>"00 * *
teG
< lim Y] \gn(st) - 5n(0l = lim ll<** * 9n - 5n||i = 0
n—>>oo *—» n—>>oo
for every s in G. Since (X(s)hni hn) is real, we obtain
lim (X(s)hni hn) = lim (\(s)hn, hn) + ±\\\(s)hn - hn\\\
= inlim(||A(5)M22+IIMi) = l-
In conclusion, <pn{s) = (A(s)ftn, hn) are positive definite functions in V(G) for
n > 1. These functions have finite support because each hn has finite support; and
they converge pointwise to the constant function 1.
Now let <p belong to V{G). Then by Lemma VEL2.6, the functions <p<pn belong
to V(G) and have finite support. Furthermore, they converge pointwise to (p. By
Lemma VH.2.7, there are unit vectors xn in £2(G) such that
<p(s)<pn(s) = {\{s)xn, xn) for all s e G.
Thus if / = J27=i aiS*> belongs to CG, then
*>(/* * /) = E *>«(/* * /)(*) = nfe E *>MM*)(r * /) w
seG *eG
= lim (A(/* * /)*„, *„) < ||A(r * f)\\ = ||A(/)||2.
n—>oo
Taking the supremum over all positive definite functions yields
ll/llc'(G) = l|A(/)|| for all / e CG.
Hence the canonical map of C*(G) onto C*(G) is an isomorphism. ■
VII.3 Primitive Ideals
When studying the representation theory of a C*-algebra, two objects are
normally used to act as a spectrum. The basic elements are the irreducible
representations. The kernel of an irreducible representation is called a primitive ideal. The
set Prim(2l) of all primitive ideals of a C*-algebra 21 is called the primitive ideal
space. This space has a natural topology which we will describe soon. Moreover,

VTL3. Primitive Ideals
191
this space turns out to be a classifying space for the irreducible representations up
to approximate unitary equivalence. The other space that is often used to encode
the representation theory of 21 is the spectrum 21 consisting of all unitary
equivalence classes of irreducible representations. This space contains more precise
information, but is generally intractable when it does not coincide with Prim(2l).
There is a natural map from 21 onto Prim(2l) given by sending each represen-
tation to its kernel. However, the only natural topology put on 21 is obtained by
pulling back the topology on Prim(2l). In fact, there is no good way to topologize
it to separate two approximately unitarily equivalent representations which are not
unitarily equivalent because they each contain the other in the closure of their uni-
tary orbits. Moreover, a deep result of Glimm shows that 21 is unmanageable (in the
sense that it does not have a good Borel structure) except for the very special class
known as GCR algebras, when 21 coincides with Prim(2l). The only simple GCR
algebras are Mn for n > 1 and £. So we will content ourselves in this chapter
with looking at the primitive ideal space; and will note to what extent this classifies
irreducible representations.
The topology on Prim(2l) is known as the hull-kernel topology. The
nomenclature comes from the definitions of the kernel of a collection J of ideals as the
ideal ker( J) := C\^eJ & and the hull of an ideal 3 is the set
hull(J) := {<P e Prim(2l) : % D 3}.
The closure of a set J of primitive ideals is J = hull(ker( J)).
Lemma VII.3.1 Primitive ideals are prime.
Proof. An ideal 3 is called prime provided that whenever the product Z1Z2 of two
ideals is contained in 3, then either Zi Q 3 or Z2 Q 3. Let n be an irreducible
representation with kernel 3. Then by Lemma 1.9.15, n(Zi)" is either 0 or B{%).
In the first case, Z% C 3. While if the second case holds for both Zi and fo, then
*(ZiZ2)H = *$i)"<l2)"U = K;
whence 0»i0>2 is not contained in X ■
Proposition VII.3.2 The hull-kernel topology makes Prim(2l) into a To space.
Proof. First, to establish that this is a topology, we verify the Kuratowski axioms
for a closure operation. The closure of the empty set is evidently empty. It is
also evident that J C J; and that kev(j) = kev(J). Hence J = ~J. The fourth
condition concerns J\ U J2. Let& = kerJi; and note that ker(JiL)J2) = Chnfo-
Thus tp belongs to J\ U J2 if and only if ^3 contains Z1H32 > which in turn contains
Chfo- Then by Lemma VH.3.1, ^3 contains either Zi or ?2> and therefore belongs
to ~J[ or Z2 respectively. It follows that J1UJ2 = 7hvlh-
A To space satisfies the weak separation axiom that given two points tPi and
tp2 in Prim(2l), there is an open neighbourhood of one that is disjoint from the

192
VII. Group C*-algebras
other. Equivalently, it must be shown that if each primitive ideal is in the closure
of the other, then they are equal. But tyi belongs to the closure of {^2} only if <p2
is contained in tyi; and conversely. ■
Example VII.3.3 Quite nice C*-algebras show that one cannot expect the
topology of the primitive ideal space to be better than T0. Consider the Toeplitz algebra
T(C(T)) from section V.l. This algebra contains the ideal of compact operators.
If 7r is an irreducible representation such that 7r(£) ^ 0, then by Lemma 1.9.15,
7r |£ is irreducible. Hence by Lemma 1.10.2, it\R is unitarily equivalent to the
identity representation. Thus by Lemma 1.9.14, it is unitarily equivalent to the identity
representation.
The other irreducible representations of T(C(T)) annihilate £; and thus they
factor through the quotient T(C(T))/£ ~ C(T). As C(T) is abelian, the
irreducible representations are the point evaluations for A in T. They correspond to the
maximal ideals
3a = {Tf + K : / € C(T), /(A) = 0, tf G *}.
So
Prim(r(C(T)) = {£} U {3a : A € T} ~ {1} U T.
The closure of t is the whole space because each 3 a contains £. The circle T
is closed, and the relative topology is the usual one. Indeed, it is evident that
p| 3a = {Tf + K : f € C(T), f\j =0,Ke£}
xes
for any subset S C T; whence we deduce that
hullker{3A : A G 5} = {3a : A G 5}.
This space is not Ti because the point t cannot be separated from any point on
the circle. This leads to the peculiar fact that the constant sequence t converges to
every point.
Nevertheless, this is a useful space. In this case, we see that each primitive
ideal corresponds to a unique irreducible representation up to unitary equivalence.
So T(C(T)) equals Prim(r(C(T)).
Next we establish the connection with approximate unitary equivalence.
Proposition VII.3.4 Two irreducible representations of a C*-algebra 21 with the
same kernel are approximately unitarily equivalent
Proof. Let <r\ and a2 be two irreducible representations with a common kernel
J. Then there is a representation r = a2(r^1 of <ri(2l) onto (7*2(21). Suppose
first that (Ti (21) contains non-zero compact operators. Then by Corollary 1.10.4,
(Ti(2l) contains A. Then by the argument used in the example above, it follows
that r is unitarily equivalent to the identity representation; and therefore a\ and

VII.4. A Crystallographic Group
193
(7*2 are unitarily equivalent. Similarly, we obtain the same conclusion when <r2 (21)
intersects £ non-trivially. Otherwise, we have
kercTi = ker7T(7i = keva2 = ker7T(72,
where it is the quotient by £. Thus by Corollary EL5.6, <i\ ~& a2. ■
As it is more usual to describe a topology in terms of its open sets, we do that
now.
Proposition VII.3.5 For A in 21, the map taking a in 21 to \\a(A)\\ is lower semi-
continuous. If A is a dense subset of% then the sets
0A = {*e&:\\*(A)\\>l} for AeA
form a base for the hull-kernel topology. A net <ra has p in %asa limit point if and
only if
liminf ||<ra(A)|| > ||p(jl)|| for all A e 21.
Proof. Let us show that J = {a e 21 : ||<r(A)|| < r} is closed. Notice that J
consists of all a such that <r(A*A) < r2, which is the same as those sigma such
that a(f(A*A)) = 0, where f(x) = (r2 Vx)- r2. Let Z be the ideal generated by
/(A* A). Then J = hull(3), and therefore it is closed. Consequently the sets Oa
are open.
Suppose that J is a closed set in 21 disjoint from a. Then a does not vanish on
Z = kevj. Hence there is an element B in Z such that ||<r(.B)|| > 2. Clearly the
set Ob is an open set containing a which is disjoint from J. Let A in A be chosen
so that ||A - 5|| < 1. Then \\<r(A)\\ > 1 and \\p(B)\\ < 1 for all p in J. Hence
Oa is also a neighbourhood of a disjoint from J.
Thus p in 21 is a limit point of <ra if and only if for every A in 21 and every
r < ||p(i4)||f there is an a0 so that r < \\aa(A)\\ for all a > a0. Clearly, this is
equivalent to the statement: liminfa ||<7a(A)|| > ||p(A)|| for all A in 21. ■
VII.4 A Crystallographic Group
The first group we will examine is the symmetry group of a tiling pattern of the
plane. Such groups are called crystallographic groups. There are seventeen such
groups, and they are all extensions of an abelian group of translations isomorphic
to Z2 by a finite group. Thus they are amenable by Proposition VII.2.3. The group
C*-algebra admits a very explicit description while displaying some of the features
that can appear in the C*-algebras of nonabelian groups.
The particular example we will focus on is the symmetry group pg of the planar
crystal or wallpaper pattern shown in Figure VEL1. It is assumed to continue
indefinitely in all directions. The origin and the unit square are to fix the coordinates,
and are not part of the pattern. It is evident that pg contains the group of
translations A^Z2 given by rmjn(x} y) = {x + m^y + n) for (m, n) in Z2. This pattern

194
VII. Group C*-algebras
^ ^ ^ ^ ^ ^ SJ-
XXX
/ / /
(0,0)«
f v
>
XXX
t / /
s ^ ^ ^ ^ ^ ^r
t t t f f f f
—5 S ^ * S S ST
Figure VII. 1
is also preserved by a glide reflection <r(x, y) = (x + |, -y). A moment's thought
shows that these elements determine the full symmetry group of this pattern. Every
element has the form Tm?n or armjn for (m, n) in Z2. A simple computation shows
that rm?n(7 = <rrm?_n and a2 = ti?0. In particular, A is a subgroup of pg of index
2. Notice that pg contains no elements of finite order; so this is not a semidirect
product.
The first order of business is to obtain a useful description of the reduced C*-
algebra of pg. We know from Theorem VEL2.5 that this equals the full C*-algebra.
However, we will be able to establish this without recourse to that result. Much
of the detail is the same for any abelian by finite group. So let G be a discrete
group with an abelian normal subgroup A such that D = G/A is a finite group of
order n. Let the quotient map be denoted by 7r. Fix a cross-section 7 : D —> G
so that 7T7 = id and 7(erj>) = e, the identity element of G. Sometimes 7 can be
chosen to be an isomorphism of D onto a finite subgroup of G; in which case, G
is isomorphic to a semidirect product of D with A. In general, 7 cannot be taken
to be a homomorphism. But once 7 is fixed, each element s in G can be uniquely
represented in the form s = 7(d) a, for d in D and a in A.
The quotient group D acts on A by conjugation:
0d(a) := y(d)ay(d)'1 for ^D,a£ A.
This action is independent of the choice of 7 because any other element in the same
coset has the form 7(^)6 for some 6 in A\ and this is readily seen to yield the same
action. The map 0 is a homomorphism of D into Aut(A).
There is a cocycle map a : D x D -> A associated with the cross-section 7.
For c, d in D, the element j(c)y(d) belongs to the coset of cd. Thus, there is a
unique element a(c, d) in A such that
7(c)7(d) = 7(cd)a(c,d).
Calculate for 6, c, d in D,
T(6)(7(c)T(d)) = 7(6)T(cd)a(c,d) = T(6cd)a(6,cd)a(c, d)

VII.4. A Crystallographic Group 195
and
{l(bh(c)h(d) = y(bc)a(b1c)7(d)
= y(hc)jWOd-i(a(bt c)) = y(bcd)a(bCl fyd-i(a(b, c))
Therefore we obtain the 2-cocycle identity
a(6, cd)a(c, d) = a(6c, d)0d-i (a(6, c)).
The group product has a useful expression for two elements S{ = 7(dt)at- in G,
sis2 = y(di)aiy(d2)a2 = 7^^2)01^1, d2)9d-i(ai)a2.
Let U be the unitary taking £2 (A) onto L2 (A) which extends the Fourier
transform. Since A is a discrete abelian group, A is a compact abelian topological
group. Each function h in £2(G) is determined by its restrictions to the n cosets of
A. So define hd in I2(A) by hd(a) = h(y(d)a) for a in A and d in D. Then define
a unitary map * of £2(G) into L2(A)D := L2{A)(n\ the direct sum of n copies of
L2(A) indexed by elements of D, by
«w = (uhd)d€D = (frd)d€D.
The main strategy for describing C*(G) is to compute *A(/)** for / in £l(G).
Let Md{C(A)) denote the C*-algebra ofnxn matrices indexed by D with
coefficients in C(A) acting on L2(A)D. The typical element is a matrix F = [FCjd]
indexed by D x D.
Theorem VII.4.1 Let G be an extension of an abelian group A by a finite group
D. Then C*(G) is isomorphic to the subalgebra o/Md{C(A)) consisting of those
elements F = [FCyd] such that
Fc,d(X) = X(a(cd-1,d))Fcd-i>e(^X) for X G A,
where a is the cocycle onD x D and OdX(a) := X(0d1a).
Proof. Fix c in D, b in A, h in £2(G) and / in £X{G). Then
(A(/)fe)c(6) = (/ * fc)(7(c)6) = £ /MM'-SM*).
*G<3
Make the substitution s = y(c)ay(d)"1 for a in A and d in D. Notice that
jicd-1) = i(c)a{cd-\ d)T(rf)"1 = TCcfrW^Har1. <*)).
Then we obtain
W)h)c(b) = JE^WWIMtW0'1*)
dGl^aGA
= E E^m^^m*-1*)
dG-D aGA

196
VII. Group C*-algebras
= E E^(^(cd"1)^(a(cd"1-d)"la))^(a"16)
dED aEA
= ]T(<7c,d*M(6)
deD
where
9c4{o) = fhicd-^ediaicd-^d)-1^) = /^(^Hcd^d)-1^
Taking the Fourier transform, it follows that
*A(/)** = [S3] G MD(C(A)).
Finally, compute
aEA
= ^2 5c,d(«(cd_1, d)a)x(a(cd_1, d)a)
aEA
= £ fed-1 Ma))X(a(cd-\ d))X(a)
aEA
= X(a(cd-\d)) £ /cd-i(a')X(0jV))
= X^cd"1, <0)E-7(*5X) = X^cd"1, d))Fcd-i,e(^X) ■
This result will be applied to the group pg. For each z in T, let %\z denote the
C*-subalgebra of M2 consisting of all matrices of the form [g *£] for arbitrary
scalars a, 6 in C.
Corollary VII.4.2 C*(pg) is isomorphic (via A') to
a={FeM2(Tx[0,l]):F(z,t)6!Rl /ora// z G T, *G{0,1}}
The generators of the group are sent to
A'OXM)-
0 2
1 0
A/(n,o)(*,*)-
2 0
0 z
and
A'(r0,i)(2,*) =
o^**
0
o— Kit
Proof. As we noted at the beginning of this section, pg is generated by an abelian
subgroup A ~ Z2 and an element a which has non-trivial image in the quotient
pg/A ~ Z2. Define a section 7 by 7(1) = 6 and 7(-l) = <r. We compute the
cocycle
a(l,l) = a(l,-l) = a(-l,l) = e and a(-l, -1) = a2 = n,0.

VH.4. A Crystallographic Group
197
The dual group of Z2 is T2, and 0*_x{z, w) = (z, w). Therefore Theorem VH4.1
shows that C*(pg) is isomorphic to the subalgebra of .M2(C(T2)) consisting of
elements F = [Fij].. z such that
Fi,_i (*,«;) = (ztw)(a(-l,-l))F-l,l(6tl(z,w)) = zF-l,l(z,w)
and
F_i,_i(*,u;) = (^^(atl.-l))^,!^* !(*,«;)) = FM(z,W).
That is,
for /,</€C(T2).
ci;f(z,«;)c;z =
0 1
z 0
0 z
1 0
= F(z,w)
Note that F(z, ±1) belongs to 9*z for all z in T; and setting Uz = [ ? g],
_ [/(z>™) #.»)]
Thus F is determined by its values on the cylinder {(z, w) G T2 : Im w > 0}. This
cylinder is homeomorphic to T x [0,1] by the identification of (z, t) with (z, e1™*)
for 0 < t < 1, which completes the identification with 21.
If f = St is the characteristic function of s in G, the formulae for A(/) from
the proof of the previous theorem may be applied. For s = a = ?(—l)e and
X = (z, w), one computes that i^T = /m^ = 0,
££i(X) = X(a(-1, l)e) = X(e) = 1
and
jCTi(X) = X(a(-1, -1)M«)) = X(r1)0) = z.
Similarly, for r0,i = t(1)t0,i, one obtains jfcZi = g~£i = 0,
5TT(X) = X(a(l, l)^i(r0,i)) = X(r0,i) = to
and
£^(X) = X(o(l, -l)t9_i(7b,i)) = X(r0,_i) = W.
Therefore
A(<r)(z,u;) =
0 z
1 0
A(t0,i)(z,u;) =
w 0
0 W
and
A(tii0)(z, w) = A(<r2)(z,w) =
The identification with 21 merely replaces w by e1™'.
z 0
0 z

198
VTL Group C*-algebras
P±a =
The structure of the algebras on the two boundary circles is interesting, and it
will be reflected in the primitive ideal structure. The algebra 9\z is spanned by a
pair of orthogonal projections
f 1/2 ±a/2]
|±a/2 1/2 J
where ±a are the two square roots of z. The map taking a in T to Pa in 9ta2 is a
continuous map into the boundary circle of 21 that wraps twice around. Moreover,
there is a unique continuous path of projections in the boundary circle starting at
Pi in JHi. This path arrives at P_i when it first returns to VK\ again.
Now it is possible to establish that the full C*algebra of pg is the same as the
reduced C*-algebra, and to compute the primitive ideal space and spectrum.
Theorem VII.4.3 Ifir is an irreducible representation ofpg, then either
(i) 7r is a two-dimensional representation determined up to unitary equivalence
by points z G T andt G (0,1) given by
ir(v) =
and tt(to,i) =
a'KXt
0
0
o—'KXt
or
(ii) 7r is a one-dimensional representation uniquely determined by a character
ofpg given by u(a) = a for some a G Tandu(royi) = ±1.
In particular, two irreducible representations with the same kernel are unitarily
equivalent
Proof. Let ir(a) = U and 7t(t0,i) = V. Since a2 = Ti>0 lies in the centre of pg,
it follows from Lemma 1.9.1 that U2 is scalar, say U2 = a21 for some a in T.
Thus the spectrum of U is contained in {±a}. Consider the spectral projection
E = Eu(a), and note that U = aE - aE1. Write V = [£*] with resPect
to the decomposition of % = E% © EL/H. Since cr^To^a = t0,_i, one has
V* = U*VU; whence
\A* C*
[B* D*
=
al 0
0 -al
A B
C D
al 0
0 -al
=
'A -B]
-C D\
Therefore A = A*, D = D*, and C = -£*. Moreover,
A2 + BB* = / = B*B + D2.
Write B = W\B\ in its polar decomposition. Then it is easy to see that the ranges
of the projections E - WW*, E1 - W*W and WW* + W*W are invariant for
both U and V, and thus for 7r(pg). As they sum to the identity and 7r is irreducible,
one equals the identity.
In the first two cases, U is scalar and V is a self-adjoint unitary. As it is
irreducible, V = ±1 and U = a (in the first case) or U = -a (in the second case)
are one-dimensional scalars determining a character u.

VII.4. A Crystallographic Group
199
In the remaining case, W is a unitary operator from EL% onto E%. So with
this identification on the two subspaces, we may suppose that B > 0 (since it
one-to-one with dense range). Then this reduces to
\al 0
L ° ~aI.
and V =
'A B]
-B D\
Since A2 = D2 = I - B2, it is clear that A and D belong to {£}'; and since
AB = BD, it follows that A = D. If F is a projection in {£}", it is evident that
the range of F © F is invariant for 7r(pg). Hence B is a positive scalar. So if F is
now any projection in {A}', then again F © F is invariant for 7r(pg). So A is scalar
and one-dimensional. As B > 0 and A2 + B2 = J, there is a unique 0 < t < 1
so that A = cos(7r£) and B = sin(7r£). With respect to the basis (\), (_£;), we
obtain the desired matrix forms
[0 a2"
[l 0
and V ~
"«*** 0 1
0 e-^'J
When t = 1, this pair is reducible, and splits into two one-dimensional
representations already noted above. ■
Corollary VII.4.4 The canonical map A of C*(pg) onto C*(pg) /.y an
isomorphism; and the canonical map of the spectrum C*(pg) onto Prim(C*(pg)) is a
homeomorphism. The primitive ideal space consists of an cylinder T x (0,1) with
the usual topology together with two circles that are glued two to one onto the
boundary circles.
Proof. It is evident from the explicit descriptions of the irreducible representations
in Theorem VEL4.3 and of C*(pg) in Corollary VEL4.2 that the two-dimensional
irreducible representations are obtained by evaluations of 21 at (z, i) for z in T and
0 < t < 1. The one-dimensional representations corresponding to u(a) = ±a
and u;(to,i) = ±1 correspond to the characters of 2l(a2, ±1) = *Ra2 on the two
boundary circles. The norm in C*(pg) of / in ^(pfl) is the supremum of ||7r(/)||
as 7r runs over all irreducible representations of pg. Hence it follows that A is
isometric; whence C*(pg) = C*(pg).
Note that each point (a2, ±1) of the closed cylinder corresponds to two points
in Prim(C*(pg)). Proposition VEL3.5 and consideration of functions in 7r(Tlt0)
and 7r(r0,i) show that the topology on the open cylinder is the standard one, and
that the projection onto the closed cylinder is continuous. Consideration of ir(a)
shows that the relative topology on each boundary circle is the standard one, but
that the neighbourhood of a point (a, 0) contains a neighbourhood of the point
(a2,0) intersected with the open cylinder. A similar statement is true for the other
boundary circle. ■
Another notable property of pg is that it is torsion free. There is an important
conjecture concerning the C*-algebras of torsion free discrete groups.

200
VII. Group C*-aIgebras
Conjecture VII.4.5 (Kadison-Kaplansky) IfG is a torsion free discrete group,
then the reduced C*-algebra C*(G) has no projections other than 0 and 1.
This conjecture remains open; although, it has been verified for large classes
of discrete groups. Using our knowledge of the primitive ideal structure in C* (pg),
we can give a simple proof of the conjecture in this case.
Theorem VII.4.6 There are no proper projections in C*(pg).
Proof. Let us work in the concrete realization 21 of C*(pg). Let P in 21 be a
projection. Since T x [0,1] is connected and P(z> i) is continuous, rank P(z, t) is
a constant of value 0,1 or 2 on the cylinder. A constant rank of 2 implies that P is
the identity in 21 and rank 0 implies that P = 0. So assume that rank P(z, t) = 1
for all (z, t) in T x [0,1]. Since 9\z is two dimensional, P(z, 0) is one of the two
projections P±a. But P(z, 0) is continuous as a function of z. If P(l, 0) = Pi
say, then following the projection once around the boundary circle leads to the
conclusion that P(l, 0) = P_i instead. This is a contradiction. Therefore, P is
either zero or the identity.
VII.5 The Discrete Heisenberg Group
Our next example is a little more complicated. The Heisenberg group is still
amenable. It will follow from direct computation that the left regular representation
is faithful on the full C*-algebra. However, the spectrum is not homeomorphic to
the primitive ideal space.
The discrete Heisenberg group is the multiplicative group H3 of matrices of
the form
a, 6, cG Z.
1-H
0
L°
a c]
1 6
0 lj
This group is generated by
[1 1
u =
0
0 10
0 0 1
and
v =
1 0 0
0 11
0 0 1
Notice that
w := uvu 1v 1 =
\1 0
0 1
[o 0
■"■
0
1J
A simple calculation shows that w commutes with u and v, and generates the centre
Z(Eq) which is isomorphic to Z. It follows that H3 is an extension of Z by Z2.
Hence it is amenable by Proposition VII.2.3.
To understand C*(H33), it suffices to know enough about the irreducible
representations.

V7I.5. The Discrete Heisenberg Group
201
Theorem VII.5.1 Let tc be an irreducible representation 0/EI3. Then there is a
real number 0 € (-1,1] such thatir(w) = e2*i$I.
(i) When 0 is irrational, C*(7r(u), n(v)) is canonically isomorphic to the
irrational rotation algebra Aq. The representation is determined up to
approximate unitary equivalence.
(ii) When 0 = k/n andgcd(k,n) = 1, then C*(7r(u),7r(t;)) is isomorphic to
Mn. Let 7 = e2™e. There are complex numbers a and (5 in T such that
ir(u) and it {y) are (simultaneously) unitarily equivalent to the pair
a
1-H
0
0
L°
0 ...
7 •••
0 ...
0 ...
0
0
yU-2
0
0
0
0
.n-1
and (5
0 0
0
0
0
. 1 OJ
and this pair deter-
The pair (an, (5n) is uniquely determined in T2 by 7r;
mines it up to unitary equivalence.
Proof. Since tt(w) lies in the centre of an irreducible C*-algebra, it is a scalar
multiple of the identity by Lemma 1.9.1. Thus there is a unique number 0 in (-1,1]
such that that ir(w) = e2*i6I =: 7 J. Let U = ir(u) and V = tt(v). They satisfy
UVU*V* = tt(w) = 7/. (1)
Let 3$ denote the ideal of C*(H33) generated by the image of w - 76. Then
C* (H3 )/3$ is generated by two unitaries satisfying (1). Hence by Theorem VI. 1.4,
when 0 is irrational, there is a canonical isomorphism of A$ onto C*(H3 )/3$ taking
the generators onto U and V. As A$ is simple, every irreducible representation
7r such that tt(w) = 7/ has kernel 3$. So by Proposition VII.3.4, they are all
approximately unitarily equivalent.
Now suppose that 0 = k/n is rational. Then
Jjny = ynyjjn = yjjn
Thus Un lies in the centre of 21, and so is scalar. Similarly, Vn is scalar; and the
pair (Un, Vn) is clearly an invariant of the representation. Choose scalars a and (3
in T so that Un = anI and Vn = (3nI. Since 7 is a primitive n-th root of unity,
the possible eigenvalues of U are ay* for 0 < j < n. Let Ej — Eu(ay*) be the
corresponding spectral projections. We may write
n-l
i=o
It follows (using the finite Fourier transform) that

202
VTI. Group C*-aIgebras
Therefore, for 0 < i < n,
n-l n-l
vEi=i j2wykvuk=n j2^Tk^~kukv
fc=0 fc=0
n-l
fc=0
Thus we may define partial isometries Eij = {pV^'^Ej for 0 < i, j < n.
Since (fiV)n = /, it is routine to verify that these form a set of matrix units for a
copy of Mn. Moreover,
j=0 j=0
where we interpret 2?n,n-i as i?o,n- Hence C*(7r(H33)) is isomorphic to Mn> By
Theorem 1.10.6, the only irreducible representation of Mn is the identity
representation. Thus E{ are all one dimensional. The matrix forms follow from the
formulae derived above for U and V in terms of the matrix units.
It is also apparent that conjugation of this representation by U leaves U fixed
and sends V to jV. Similarly, conjugation by V fixes V and sends U to yU. Thus
the constants a and (3 are only determined up to a multiple by an n-th root of unity;
which is to say that the pair (an, /3n) is the complete unitary invariant. So it has
been determined up to unitary equivalence by 0 = k/n and the pair (an, (5n).
Conversely, given 0 = k/n, a and (3, it is evident that these formulae yield an
irreducible representation of H3. So we have a complete description of C*(7r(lHl3))
for every irreducible representation 7r. (Note that we have not classified the
irreducible representations of A$ up to unitary equivalence.) ■
Remark VII.5.2 This theorem allows us to describe the primitive ideal space.
There is a canonical projection p of Prim(C*(Irl3)) onto T given by evaluation
at w. The inverse image of irrational points is a singleton; but the inverse image
of each rational point is a copy of T2. The basic neighbourhoods of any irrational
point is the inverse image under p of a neighbourhood on the circle. While for a
point on a torus over a rational point, the neighbourhoods consist of a
neighbourhood on the torus together with the inverse image under p of a punctured
neighbourhood of the rational point on the circle. The details will be left as an exercise.
The spectrum of C*(Bl3) projects onto Prim(C*(Irl3)). The fibres are
singletons over each point on the torus at each rational point of the circle. However, over
each irrational point of the circle, there is a very large space corresponding to the
unitary equivalence classes of representations of Aq. (See Exercise VI.3.)
Next we will show that C*(H3) has the same irreducible representations. This
will imply that C;(%) is isomorphic to C*(Hi).

VII.6. The Free Group
203
Theorem VII.5.3 The set of irreducible representations of C*(Eq) are the same
as the set of irreducible representations of C*(Ba).
Proof. First we identify W = X(w). For convenience of notation, let us write 8afbjC
for the basis vector of £2(Eq) corresponding to on . Notice that
WSafbfc = <$a,6,c+l for a11 Ct, 6, C G Z.
Hence W is a bilateral shift of infinite multiplicity. In particular, its spectrum is
the whole unit circle. For each 7 in T, let 37 be the ideal of C*(H3) generated by
W - 7 J. Since W lies in the centre, this is a proper ideal. Consider the quotient
map g7 of C^Efe) onto 2ly := CyfHaJ/Oy. Then qy(W) = 7/. So any irreducible
representation p of 2ly determines an irreducible representation it = pqyX of H3
such that tt(w) = 7/. The proof of the previous theorem now applies, and we
deduce that C*(H3) has the same irreducible representations as C*(H3). ■
Corollary VII.5.4 The canonical homomorphism of C*(Eq) onto C*(H33) is an
isomorphism.
Proof. From the GNS construction Theorem 1.9.12, it follows that the norm of A
in a C*-algebra 21 is determined by sup \\k(A) || as the sup runs over all irreducible
representations of 21. Since the quotient algebra C*(H33) has the same irreducible
representations as C*(H3), it follows that the quotient map is isometric. ■
VII.6 The Free Group
A very important example of a non-amenable group is the free group F2 on two
generators u and v. In the next few sections, we will examine this C*-algebra in
detail. This group is discrete, and therefore ^(F2) is spanned by the characteristic
functions of elements of F2. Hence C*(F2) is generated by a pair of universal
unitaries (^(ti), ^(v)). Any representation 7r takes u and v to unitary operators
U = 7r (u) and V = ir(v). Since there are no relations on u and v, every unitary pair
U, V determines a representation of F2. The irreducible pairs determine irreducible
representations. Let {(Ua, Va) : a G A} be a collection of irreducible unitary
pairs, one from each unitary equivalence class. (Irreducible representations of F2
act on separable Hilbert spaces. See Exercise VII.6.) Then the group C*-algebra is
isomorphic to C*(U, V) where
V = J2®Ua and V = J]eVra.
This is the C*-algebra which is universal with respect to the property that if U, V
is any unitary pair, then there is a *-homomorphism it of C*(U, V) onto C*(17, V)
such that 7r(U) = U and 7r(V) = V.
We will establish the existence of some separating families of irreducible
representations to obtain some interesting faithful representations of C*(F2).

204
VII. Group C*-aIgebras
Proposition VII.6.1 C*(F2) has a family irn, n > 1, of finite dimensional
representations such that J2 ©n>i nn is faithful.
Proof. First notice that there is a pair (U, V) of unitaries on a separable Hilbert
space H such that the canonical map a of C*(F2) onto C*(?7, V) is an
isomorphism. Since F2 is countable, £1(F2) has a countable dense subset. Hence the
image under iru yields a countable dense subset of C*(F2). So this is a separable
C*-algebra; and therefore it has a faithful separable representation which we will
also denote by a. Let U = a(u) and V = c(v).
Let Pn be an increasing sequence of finite rank projections in B(H) tending
strongly to the identity. Let An = PnU\pn% and Bn = PnV\pn%. Let In denote
the identity in B(PnU), and define operators on Un = PnU 0 PnU by
An (In - AnAn)^]
Un =
(In-A^An)1'2
-a:
and
Vn =
Bn (In - BnBtfl2
[(In - B*nBnY'2 ~B*n
These are unitary operators acting on a finite dimensional Hilbert space. Let nn be
the representation determined by 7rn(U) = Un and ?rn(V) = Vn.
Set 7r = Yl ©n>i *n- ^ we &**& °f &n and V^ as acting on a subspace of
%®H, then it is evFdent from the definition of An and Bn that
SOT-lim?7n =
n—5>oo
u
0
0
and
SOT-limyn =
n-t-oo
V
0
0
-V*
Therefore, if f(s) = ^ a*^« is a finite sum (in the group algebra CF2 as a subset
of£1(W2)), it follows that
k(/) ol
SOT-lim7rn(/) =
n-*oo
0
Therefore
||7r(/)||-sup||7rn(/)ll>H/)ll = IM/)ll-
Since CF2 is dense in C*(F2), it follows that it is isometric. ■
A C*-algebra 21 is called quasidiagonal if it has a faithful representation it
for which there exists an increasing sequence Pn of finite rank projections tending
strongly to the identity such that
lim WPnvU) - nU)Pn\\ = 0 for all A G 21.
n-j>oo
So we immediately obtain:
Corollary VII.6.2 The group C*-algebra C*(F2) is quasidiagonal
We also obtain some information about traces.

VII.6. The Free Group
205
Corollary VII.6.3 The group C*-algebra C*(F2) has a faithful trace.
Proof. Let n be the representation constructed in Theorem VEL6.1. Let rn be the
normalized trace on the matrix algebra B(PnH). Then
r(A):=£2-"rnMA))
n>l
is a trace on C*(F2). If A > 0, then nn(A) is non-zero for n sufficiently large;
whence rn(7rn(A)) > 0. Therefore r(A) > 0, and thus r is faithful. ■
Next we twist this representation a bit to get a faithful irreducible
representation. A simple preliminary lemma is needed.
Lemma VII.6.4 Suppose that D = diag(dn) is a diagonal operator with respect
to a basis {en} with distinct eigenvalues dnfor n > 1. If A is any operator such
that (Aei, en) ^ 0 for all n > 2, then C*(D, A) is irreducible.
Proof. The commutant of the diagonal operator D is the masa of all diagonal
operators (see Exercise VEL7). Clearly, A does not commute with any proper diagonal
projection. Thus C*(D, A) is irreducible. ■
Theorem VII.6.5 The group C*-algebra C*(F2) has a faithful irreducible
representation.
Proof. Let U and V be a pair of unitaries of joint infinite multiplicity acting on
a separable Hilbert space such that C*(?7, V) ~ C*(F2). Let a be the canonical
isomorphism of C*(F2) onto C*(?7, V). (One can use U^ and V^ where U
and V were those used in the previous theorem.) By the Weyl-von Neumann-
Berg Theorem (Corollary n.4.2), there is a compact perturbation Ui = U+K of U
which is diagonal with respect to a basis {en}n>i. With a further compact diagonal
perturbation, it can be arranged that U\ = diag(txn) where un are distinct points
on the unit circle. Let V\ be a unitary operator which is a finite rank perturbation
of V such that (Viei, en) ^ 0 for all n > 2. For example, one can change V on
the two dimensional subspace span{ei, V*x}, where x = Y^k>i 2~k^2ek, so that
V\ei = x.
By the previous lemma, C*(C/i, V\) is irreducible. If <r\ is the canonical ho-
momorphism of C*(F2) onto C*(C/i, V\) and it is the quotient map of B(H) onto
B{U)/^ then for each A in C*(F2),
||oi(A)|| > ||«n(A)|| = ||ir<r(A)|| = ||<r(A)|| = ||A||.
Therefore a\ is an isomorphism. ■
We finish this section with an easy argument showing that C*(F2) is projec-
tionless. This is also the case, as we shall see, for the reduced C*-algebra. But the
proof in that case is much more difficult.
Theorem VII.6.6 The group C*-algebra C*(F2) contains no proper projections.

206
VH. Group C*-algebras
Proof. We will imbed C*(F2) into the C*-algebra
21 = {$ € C([0,1] ,£(«)) : *(0) € CI}.
Let 17 and V be a universal pair of unitaries on a separable space, and let a be the
corresponding isomorphism of C* (F2). By the spectral theorem, there are bounded
Hermitian operators A and B such that U = etA and V = etB. So one may define
a pair of unitary elements in 21 by U(i) = ettA and V = ettB. Therefore there
is a *-homomorphism (p of C*(F2) into 21 such that y>(U) = U and <p(V) = V.
If Si is the homomorphism on 21 of evaluation at 1, we see that S\(p = a is an
isomorphism. Hence <p is also an isomorphism.
The C*-algebra 21 has no proper projections. To see this, note that $ is a
projection in 21 if and only if $(£) is a projection for all £ £ [0,1]. However, $
is evidently homotopic to the constant projection $o(0 = $(0) which is scalar.
The only projection homotopic to 0 (or J) is 0 (or J). Hence there are no proper
projections in 21; and a fortiori, there are no projections in C* (F2). ■
Corollary VII.6.7 Ifir is a faithful representation tf/C*(F2), then there are no
non-zero compact operators in the range ofir.
Proof. A C*-algebra containing a non-zero compact operator contains a non-zero
finite rank projection. ■
VII.7 The reduced C*-algebra of the free group
We write £2 (F2) instead of L2 (F2) because the Haar measure just assigns mass
one to each point. Let St denote the characteristic function of t in F2 as an element
of ^2(F2). Clearly, these vectors form an orthonormal basis for ^2(F2). Notice that
X(s)St(r) = Stis^r) = S8t(r) for r, s, t € F2.
Thus X(s)St = S8t implements a permutation of these basis vectors. It is evident
that this left Haar measure is also right translation invariant. So F2 is unimodular.
Consider the right regular representation on ^2(F2) given by
{p{s)g)(t) = g(ts) whence p(s)St = 8t8-i.
These two representations commute since
p(s)X(t)Sr = p(s)Str = Str8-i = \(t)Sr8-i = X(t)p(s)Sr.
Proposition VII.7.1 The map r(A) = (ASeiSe) is a faithful trace on C*.(F2).
Thus the map $(A) = r(A)I is a faithful expectation o/C*(F2) onto the scalars.

V7I.7. The reduced C*-algebra of the free group
207
Proof. Consider / = J28e¥2 a*^* and 9 = £*eF2 PsS* in CF2 (i.e. finite sums).
Then t(A(/)) = ae and r(X(g)) = 0e. Calculate
*6F2 t€F2
= J2a'&-1 = £ p*a»-1 = t(a(^ * /))•
*€F2 *6F2
So by continuity, r is a trace on C*(F2).
Suppose that A in C*(F2) is positive and r(A) = 0. Then
(AS81S8) = (AX(s)Sei X(s)Se) = (X(s-1)AX(s)Se} Se)
= T(\(8-l)A\(8)) = T(A) = 0.
Hence
\(ASs1St)\ < {AS8,S8)ll2{ASuStfl2 = 0
for all s, t in F2. Therefore A = 0; and so r is faithful.
It is then evident that $ is a faithful expectation onto the scalars. ■
We wish to show that C* (F2) is simple with a unique trace. As for the irrational
rotation algebras, we first construct an explicit formula for the expectation $.
Lemma VII.7.2 Let U = M 0 Mx. Suppose that B has the form [° *] and U{
are unitary operators such that U{Uj have the form [ * q] when i ^ j. Then
U^UiBUtW <+\\B\\.
1=1
Proof. First suppose that B and C have the forms
TO 0
B =
and
C =
* *
0 0
Then
||£ + C\\2 = ||(B* + C*){B + C)\\ = \\B*B + C*C\\ < \\B\\2 + \\C\\2.
Notice that when i ^ j, the operator (U{Uj)*B(UiUj) has the form
* *
* 0
[..][:;]-
* *
0 0
Hence
n
t=l
t=2
< \\Bf+ |E(TO)*(W)|f = ||B||2+ lE^^II2-
t=2
t=2

208 VII. Group C*-algebras
By induction, it follows that \\J2i=i UiBU?\\2 < n\\B\\2. Therefore,
{{kituiBUfW < js\\B\\.
1=1
The same result holds when B has the form [g *]. For the general case, one
may split B as a sum of two terms in the forms [ 2 2 ] + [ § o ] • ™s y *elds *e desired
estimate. ■
Corollary VII.7.3 For s in F2,
i. i v^ w t\ w \ w -t\ I H8) if 8 = uk for some k £ Z
lim £■ > X(ux)X(s)X(u l)= { K J J J
n->oo"*-f v / w i ) |Q otherwise
Proof. An element of F2 is a product of terms in u, v9 u~x and v~x. It is in reduced
form if no adjacent terms cancel. If s is not a power of u, then it has some non-zero
power of v in its reduced form. Thus there are integers k0 and £q (possibly 0) such
that s = v!00 sou*0, where s0 has the form v^Hv*1 or v±x.
Let
M = span{<S, : s = 1 or s = ti^1* in reduced form}.
Then
M1 = span{<S, : s = v±lt in reduced form}.
So X(s0)M is contained in ML\ while X(uk)ML is contained in M for & ^ 0.
Hence with respect to this decomposition, A(s0) = [2 *] and A(izfc)A(iz**) has the
form [ * o ] when k ^ L By the previous lemma, for s = uk° s0ul°,
i=l
= A(t**») (jimi|]A(tt»)A(«o)A(t«-»)) A(t/°) = 0.
On the other hand, if s = uk, then evidently
n
i^A.y)A(»A(u-*) = A(» for all n > 1.
»=i
Now we can construct a useful method for computing the expectation $.
Theorem VII.7.4 For a// A in C;(F2),
m n
JSSo ,&> ™ E E A(«V)ilA(«-i«-) = r(A)J.
*=i j=i

VII.7. The reduced C*-algebra of the free group 209
Proof. Consider / = J28ew2 a*^s m ^2' By *e Previ°us corollary,
£& iE^JWM = £<vA(**) =: /o.
j=l
Hence the desired limit equals
m
J!™ ± £A(«i)A(/o)A(«-i) = ae/ = t(A)/ = *(ii).
m-*oo
t=l
By continuity, this identity extends to every element of C*(F2). A routine norm
estimate shows that this limit exists as m and n tend to infinity independently of
each other. ■
Corollary VII.7.5 The reduced group C*-algebra C*(F2) is simple.
Proof. If A 56 0, then A* A ^ 0 is positive. Hence t(A*A) > 0. By the previous
theorem, the ideal generated by A contains
lim lim ^^^A(uV)A*AA(t;-iO = r(A*A)/.
m *=i j=i
Hence this ideal is all of C*(F2).
Corollary VII.7.6 C£(F2) has a unique trace.
Proof. If t' is a trace on C*(F2), then by linearity and the trace property,
m n
AA) = r' (^ £ £ A(«V) AA(tT V)) •
t=l j=l
But the right hand side converges to t'(t(A)I) = t(A). ■
The non-amenability of the free group is evidenced by the following simple
consequence.
Corollary VII.7.7 The reduced group C*-algebra C* (F2) is not isomorphic to the
full group C*-algebra C*(F2).
Proof. C*(F2) is simple, while C*(F2) has many quotients including finite
dimensional ones. ■
Another property distinguishing C*(F2) from C*(F2) is quasidiagonality. This
also depends on non-amenability.
Proposition VII.7.8 A countable discrete group G such that C*(G) is quasidiag-
onal is amenable.

210 vll. Group C*-algebras
Proof. Suppose that there are finite rank projections Pn in B(£2(G)) increasing to
the identity such that
lim ||PnA(«) - A(«)PJ| = 0 for all s e G.
n-J>oo
Let £°°(G) act on £2(G) by multiplication: Mfx(s) = f(s)x(s) for / in £°°(G),
x in £2(G) and s in G. Notice that for t in G,
\(t)Mf\(t)*x(s) = M/Xityxir1*)
= fit-'s^Xityx^r's) = f(r1s)x(s).
Hence \(t)Mf\(t)* = Mft.
Let dn = rank(Pn); and let Tr be the usual trace on the finite rank operators.
Choose any free ultrafilter U on N, and define a state on £°°(G) by
m(/) = limd-1Tr(PnM/Pn).
Then
\m(ft) - m(/)| = lim<1|Tr(PnA(«)M/A(«)*Pn) - Tr(PnMfPn)\
= limd-1|Tr(PnA(t)M/A(«)*Pn) - Tr(X(t)PnMfPnX(t)*)\
= limcC1|'&([PnA(*) - X(t)Pn]MfX(tyPn)
+ Tr{X(t)PnMf[PnX(t) - X(t)Pn])|
<lim2||PnA(«)-A(«)P„||||/||oo = 0.
Thus we have constructed a left invariant mean on £°°{G)\ and therefore G is
amenable. ■
Corollary VII.7.9 C*(F2) is not quasidiagonal.
VII.8 C;(F2) is projectionless
We have already seen an example of a simple unital projectionless C*-algebra
in section IV.8. However, C*(F2) provides a "naturally occurring" example of
this phenomenon. Nevertheless, the first known examples were constructed
"artificially". This proof uses a dense subalgebra which is an analogue of the C°°
functions in C(X). Recall that every element of F2 can be uniquely written as a
reduced word in u±l and v±x.
To define this subalgebra, we first need another representation of C*(F2). Let
Vl be a Hilbert space spanned by orthonormal basis vectors £{Xly} where {z, y}
runs over all the unordered pairs V of elements of F2 such that x~xy has length 1
in reduced form (i.e. x~ly G {ti*1, v*1}). Define a representation a of F2 on V!
by
a(8)Z{x,y} = Z{sx,sy}-

VII.8. C;(F2) is projectionless
211
This is easily seen to be a unitary representation.
Lemma VII.8.1 The representation a is unitarily equivalent to A © A. Hence a
extends to a *-representation 0/C*(F2).
Proof. Let H[ = span{€{x,y} • {^iV} € 7\*}, where Vu consists of those pairs
{x,y} such that x~ly = u±l. Then^ := H'f = span{f{a:>y} : {x,y} G Vv},
where Vv = V\VU consists of those pairs {a, y} such that x~xy = v±x. It is clear
that H[ and H'2 are invariant for a. Let a{ = a|^/ for i = 1,2, so that a c_ <t\ 0 <t2 .
For each pair {x,y} in Pu, one of the words has the reduced form su±x
such that the other equals s in reduced form. Let W\ be the unitary map in
B(£2(F2), H[) given by W\St = £{tytu}- It is routine to verify that this is indeed
unitary. Moreover,
W1X(s)St = Wi&8t = Z{8tl8tu} = <ri(s)Z{t,tu} = <r\{s)WiSt.
Thus <r\ = Ad W\ A. Similarly, there is a unitary operator W2 of ^2(F2) onto
H'2 such that a2 = AdW2 A. Hence we may extend a to the *-representation
Ad Wi 0 Ad W2 of C;(F2) into B(ft')- ■
The sneaky part of our argument is to use another map which almost
intertwines A and a. For each element s in F2 \ {e}, define 7(5) to be the element of
F2 with the last term deleted from the reduced word representing s . Define a map
Sin£(^2(F2),ft')rjy
SSe = 0 and SS8 = ^{s^(s)} when s ^ e.
It is easy to verify that 5 is a co-isometry of index 1.
Lemma VII.8.2 With the definitions above, X(s) - S*a(s)S is finite rank for every
s inF2.
Proof. Note that y(si) = sy(t) unless s completely cancels t, which occurs only
if s = anan_i • • • a\ and t = a^1 • • • ajl for 0 < i < n; where a* belong to
{ti*1, v±x}. So for s = anan-i • • • au compute
10 ift = e
S*t{stMt)} = Sst if7(st) = s7(t)
C*C X iff n~ 1 n—1
<-> S{a„an_i."ai+i,a„a„_i-o,} — °a„a„_i-a,- II I — Oj_ • ■ • a^ ,
1 < i < n.
Therefore
1<J, if* = e
0 if 7(st) = s7(t)
X X ift-n-1 „-l
°a„a„_i-a,+i — Oa„an_i-o,- II t — Oj • • • af ,
1 < i < 71.
Consequently, A(s) - S*a{s)S is finite rank. ■

212 vll. Group C*-algebras
Thus A - S*a(A)S is finite rank for every A in A(CF2). Define Aoo to be
the set of elements A in C*(F2) such that A - S*a(A)S is in the trace class. Let
Tr denote the trace on the space C\ of trace class operators. The following lemma
outlines the main properties of Aoo-
Lemma VII.8.3 Aoo is a dense unital *~subalgebra 0/C*(F2) which is inverse
closed. The function
fA(a) = (al - A)"1 - 5V((aJ - A)-X)S
is continuous from the resolvent of A into C\. For A in A<x>,
Tr(A-S*a(A)S) = r(A).
Proof. Clearly Aoo is a unital self-adjoint subspace of C*(F2). It is an algebra
because if A, B belong to Aoo, then
AB - S*cr(AB)S = {A - S**(A)S)B + S*a(A)S(B - S*a(B)S)
is trace class. It is dense because A(Qr2) is already dense. Aoo is inverse closed
because of the identity
A-1 - S*<r(A-l)S = A~1(S^a(A)S - A)S*<r(A)-lS + A'1 {I - 5*5)
since J - S*S = 8e8* is rank one.
Replacing A by al - A in this identity yields
(al - .A)"1 - 5V(aJ - A)~lS
= (al - A)-1 ((5V(aJ - A)S - (al - ,4))5V(aJ - A)-XS + (I - 5*5))
= (a/ - A)"1 (A - SV(A)S - a<Ue*)5V(aJ - A)~lS + (al - A)"l8e8*
which is continuous in the trace norm.
From the formula computed in the previous lemma, one may deduce that
((A(.)-5VW5)M«) = {»m = t(aw) [tl-
This uses the observation that ar1 does not equal at- or at-+i because s is written in
reduced form. By linearity and norm continuity,
((A-SV(A)S)M,) = {°(4) \t\-
for all A in C* (F2). Thus if A is in Aoo > it makes sense to compute the trace in two
ways
Tr(A - S*<r(A)S) = £((A - SV(ji)S)Mt)
= ((A - SV(j4)S)*c, <*e) = r(A). ■

VII.8. C; (F2) is projectionless 213
Lemma VII.8.4 If P is a projection in C*(F2) and 0 < e < 1, then there is a
projection Q in «A» such that \\P - Q\\ < e.
Proof. Pick A = A* in Aoo such that \\P - .A|| < e/2. Then the spectrum of A is
contained in (-e/2, e/2) U (1 - e/2,1 + e/2). Let Q = EA(1 - e/2,1 + e/2).
Then Q is a projection such that \\P - Q|| < \\P - A|| + ||j4 - Q|| < e. From the
Riesz functional calculus,
Hence Q - S*a(Q)S is given by the expression
-^ J* ((1 + e*/2) J - A) -1 - 5* ((1 + e*/2)/ - il) "*5 *,
which is the Riemann integral of a continuous Ci valued function, and therefore is
also trace class. So Q belongs to A<x>. ■
We have now reduced the problem to analyzing the projections in Aoo* We
need a simple lemma from operator theory.
Lemma VII.8.5 IfP and Q are projections such that P - Q is trace class, then
Tr(P - Q) is an integer.
Proof. Notice that
P(P - Q)2 = P - PQP = (P - Q)2P.
Similarly, Q commutes with (P - Q)2 which is a positive trace class operator. By
the spectral theorem for compact operators, there are distinct positive eigenvalues
Xi and finite rank projections E{ such that (P - Q)2 = ]T\ Ati%. The kernel of
(P - Q)2 is E0 = (£\ Ei)1. Thus P#0 = <?#o = #o and
P = 5^P£i + .Ro and Q = ^QEi + R0.
i i
Therefore there is a convergent sum
Tr(P-Q) = X)Tr((P-Q)^).
i
But (P - Q)#» is the difference of two finite rank projections. Hence its trace is
rank PS; - rankQi?; which is an integer. As the sum converges, this difference
equals 0 except finitely often; and Tr(P - Q) is an integer. ■
Combining all these results, we obtain the main result of this section.
Theorem VII.8.6 C*(F2) contains no proper projections.

214
VII. Group C*-algebras
Proof. Suppose that there is a proper projection in C*(F2). By Lemma VEL8.4,
there is a proper projection P in Aoo. Hence Q = S*a(P)S is also a projection;
and P - Q is trace class. Therefore by Lemma VEL8.3, r(P) = Tr(P - Q).
By Lemma Vn.8.5, it follows that r(P) is an integer, necessarily 0 or 1. But
by Proposition VEL7.1, r is faithful. Thus if r(P) = 0, then P = 0; while if
t(P) = 1, then r(I - P) = 0 and so P = I. Hence there are no proper projections
inC:(F2). ■
Exercises
VII. 1 Verify that L1 (G) has a norm one approximate identity. When G is metriz-
able, show that there is a sequential approximate identity.
VII.2 Use the spectral theorem to prove Stone's Theorem that every unitary
representation of R is associated to a spectral measure E on R such that
n(t) = / ei8tE{ds).
Jr
VII.3 Show that the Schur product of two positive matrices is positive.
HINT: Write each positive matrix as a sum of positive rank one matrices.
VII.4 Identify the topology on the primitive ideal space of the discrete Heisen-
berg group.
VII.5 The continuous Heisenberg group H is the set of all matrices of the form
[1 X Z 1 0
oiyfor&9y9z€R with the usual topology on R .
(a) Show that unitary operators on L2(R) determined for each real number
s by U8(x, y, z)f(t) = ei8(xt+z)f(t - y) yields an irreducible
representation of H.
HINT: The operators of multiplication by exxt generate a masa.
(b) Show that C*r(H) = C*(tf).
VII.6 Show that any countable family {Tn}n>i of operators on a Hilbert space
has a non-zero separable reducing subspace.
HINT: Fix a vector x ^ 0 and consider C*({Tn})z.
VII.7 Show that if D is a diagonal operator with distinct eigenvalues, then {D}f
is the diagonal algebra.
Vn.8 Show that A(F2)' = p(F2)".
HINT: Show that T in {A(F2), p(F2)}' must be scalar.
VII.9 Verify that the operators W\ and S defined in Section VII.8 are a unitary
and a co-isometry of index 1 respectively.
VH.10 Let Ho be a subspace of % of infinite dimension and co-dimension. Let
U be a unitary such that UUq = %q and U2 = J; and let V be a unitary

Exercises
215
operator such that H& = VUQ 0 V2Uq and V3 = L Consider the C*-
algebraC*(l7,F).
(a) By mimicking the proof for C*(F2), show that C*(17, V) is simple, and
has a unique trace.
(b) Let E be the projection onto H0. Show that C*(17, V, E) is isomorphic
to .M2(02), which is isomorphic to 02 by Exercise V.17.
HINT: Think of U as [J J] andV as [5° 5f|*] wxtWoe^-
(c) Show that C*(U, V) is not quasidiagonal.
HINT: Use the identities I = E + UEU = E + VEV* + V*EV. If P
is a finite rank projection almost commuting with U and V, show that the
trace of PEP should be close to both rank(P)/2 and rank(P)/3.
VEL11 Let G = Z2 * Z3 denote the free product of Z2 and Z3. This is the group
generated by two elements u and v subject to the relations u2 = e = vz.
Let K be the Cantor set consisting of all reduced infinite words in u and
v. That is, x = vf^v^uv^uv^u • • • for k = 0,1. Then G acts on K
by left multiplication. Pick any element x in K and define S to be the set
{t e G : tx begins with a u}. Show that G = SUuS = SUvSUtT1.?.
Hence show that C*(G) is isomorphic to the C*-algebra of the previous
exercise.
Notes and Remarks.
The classical source for general information of the C*-algebraic approach to
group representations is Dixmier [1964], which also contains a good treatment of
the primitive ideal space and spectrum of a C*-algebra. A proof that
amenability of G is equivalent to C*(G) = C*(G) may be found in Pedersen [1979] or
Paterson[1988]. The proof for discrete groups given here was provided by Alan
Paterson (private communication). The observation that the primitive ideal space
classifies representations up to approximate unitary equivalence is due to Hadwin
[1980]. The treatment of the crystal groups was shown to me by Keith Taylor
(private communication). The Kadison-Kaplansky conjecture has been verified for
groups of polynomial growth by Ji [1992]. The special representations of the free
group were found by Choi [1980]. The simplicity of C*(F2) and its unique trace
are due to Powers [1975], but the proof given here is taken from Choi [1979] where
he actually discusses the group Z2 * Z3 (see the exercises). Cohen [1979] showed
that C*(F2) is projectionless. Pimsner and Voiculescu [1982] showed that C*(F2)
is projectionless by K-theoretic methods. The proof given here is due to Connes
[1986], which was exposited by Cohen-Figa-Talamanca [1988] and Effros [1989].
That quasidiagonality implies amenability is due to Rosenberg [1987].

CHAPTER Vm
Discrete Crossed Products
Crossed product C*-algebras were introduced as a tool for making a
systematic study of groups acting on C*-algebras as automorphisms. They provide a
larger algebra which encodes the original C*-algebra and the group action. The
consequence has also been to produce new classes of interesting algebras, and new
ways of looking at old ones.
A C*-dynamical system (21, G, a) consists of a C*-algebra 21 together with a
homomorphism a of a locally compact group G into Aut(2l). We will denote by
a8 the automorphism a(s) for s in G. Given a C*-dynamical system, a covariant
representation is a pair (n-, U) where n is a ^representation of 21 on a Hilbert
space % and s —■> U8 is a unitary representation of G on the same space such that
U8n(A)U; = n{a8(A)) for all A G 21, s G G.
In these notes, we will restrict our attention to crossed products by discrete
groups. Because the Haar measure is then just counting measure, the notions
simplify in this setting. We will outline the general situation later. So let G be a
countable discrete group, and let (21, G, a) be a dynamical system. The space of
continuous compactly supported 21-valued functions on G is just the algebra 21G of
all finite sums / = SteG ^tt w^^ coefficients in 21. Multiplication is determined
by the formal rule tAt~l = at(A). Whence if g = ]Ct*eG B*u *s a"0^1* finite
sum, then
fg = E E A*tB"u = E E M*But- x)tti
t€Gu€G t£Gu€G
= ]T VJ Atat(£u)<t. = VJ(Y^ AtatOB,-!,))*
which is just a twisted convolution product. The adjoint is determined by the rule
s* = s_1, so that
(As)* = s*A* = 5"1A*55"1 = a;1^*)*"1.
Hence
teG
216

VIII. Discrete Crossed Products 217
Notice that a covariant representation (7r, U) of (21, G, a) yields a
♦-representation of 21G by
*(f) = J2*(At)Uf
teG
Indeed,
tec? tec?
*6<3
and
*(/)*(*) = E E '(^)W)«.
=E E ^(^)(^w^*)^^u
= EE7r(A*)7r(a*(B«))^
= E(E7r(^a*(B*-i«)))^=*(fa)-
seG teG
Conversely, when 21 is unital, a *-representation of 21G yields a covariant
representation of (21, G, a) simply by the restrictions
tt(A) = a(Ae) and U8 = <r(s).
For indeed,
U8tt(A)U; = a{s)*{Ae)a{s)* = ^As"1) = a{a8{A)e) = ir(a#(;i)).
When 21 does not have a unit, let En be an approximate unit, and let
U8 = lim a(Ens).
n-*co
The proof that this converges is left as an exercise.
The crossed product 21 xa G is the enveloping C*-algebra of 21G. That is,
one defines a C*-algebra norm by
= suPH/)||
as a runs over all *-representations of 21G. The supremum is always bounded by
ii/iii-Ew-
tec?
This supremum is taken over a non-empty family of representations because certain
representations can be explicitly constructed. Indeed, let it be any ^representation
of 21 on %. Then we form the tensor product of this representation with the left

218
VIII. Discrete Crossed Products
regular representation of G. To this end, form the Hilbert space £2(G, %) of all
square summable functions x of G into % with the norm
NlS-5>Wlf-
teG
Define a covariant representation (7?, A) of (21, G, a) by
(n(A)x)(s) = ir(«:\A))(x(s))
(Atx)(s) = x(r1s)
for all A in 21, x in ^2(G, %) and s, * in G. It is easily verified that 7? is a *-
representation of 21 and A is a unitary representation of G. To verify the covariance
condition, compute
(At*(A)A*tz)(s) = mA)A;x)(r1s) = iria^AM^Mt-1*))
= ir(a:lat(A))(x(s)) = (i(at (A))x) (s)
In particular, since there are faithful representations of 2l by the GNS
construction, we obtain an isometric imbedding of 2l into 2l xa G by sending A to Ae.
And in the unital case, 2l xa G contains a unitary subgroup isomorphic to G. This
crossed product has the universal property: if (n*, U) is any covariant representation
of (2l, G, a), then there is representation of 2l xa G into C*(7r(2l), 17(G)) obtained
by setting
*(/) = I>0W for / = V>t*e2lG,
tec? tec?
and extending by continuity. In the unital case, this map is surjective.
For general discrete groups, the collection of representations constructed above
using the left regular representation of G are not sufficient to determine the norm
in 21 xa G. This restricted class of representations determines the reduced crossed
product 21 x ar G. For example, if G is any discrete group, C x G is just the group
C*-algebra. (The action of G on C is the trivial one.) The reduced crossed product
is just the reduced group C*-algebra. It is a theorem that when G is amenable,
the reduced crossed product equals the full crossed product. The cases we will
examine are crossed products by the integers, and usually we will be able to show
that the crossed product is simple. Since there is a canonical homomorphism of the
crossed product onto the reduced crossed product, equality will then follow.
Example VIIL1.1 The irrational rotation algebra is a good example of a crossed
product. Recall that this is the universal C*-algebra A$ generated by two unitaries
satisfying the relation UV = e2*i$VU. In particular, VUV* = e"2™**/. This
implies that the spectrum of U is invariant under rotation R$ through the irrational
angle 27T0; whence the spectrum is the whole unit circle T. For any polynomial

VIII. Discrete Crossed Products
219
p{z) = EjkL-w akzk>one has
N N
Vp(U)V*= £ ak(VUV*)k= £ e-^ik$akUk = T$(p)(U)
k=-N k=-N
where r$ (/)(*) = f(e~2^i$z) = foR^x(z) is the automorphism of C(T) induced
by the rotation homeomorphism R$.
We claim that A$ is the crossed product C(T) xT0 Z. Indeed, the crossed
product is generated by the image Z of the coordinate function z and a unitary W
implementing the automorphism r$, whence
WZW* = re(z)(Z) = e~2*i$Z.
Therefore by the universal property of A#, there is a homomorphism of A$ onto
C(T) xT$ Z taking U onto Z and V onto W. Conversely, we saw above that A$
provides a covariant representation of (C(T), Z, r$). Therefore by the universal
property of the crossed product, there is a homomorphism of C(T) xT0 Z onto A$
taking Z to U and W to V'. Clearly these maps are inverses.
Later in this chapter, we will use this connection to extract some additional
information about A$.
Example VIII.1.2 Recall from Proposition V.4.2 that the Cuntz algebra On
contains a UHF subalgebra $" of type n°°. Consider the direct limit C*-algebra
21 of the algebras 2l_fc := Mnk($n) given by the multiplicity one imbeddings
ot-k{A) = E\\® A of 2l_fc into 2l_fc_i. Then it is evident that the limit algebra is
just k ® y*, where A is the algebra of compact operators. We will show how this
can be extended to a doubly infinite sequence. Since $" is the direct limit of Mnk
with multiplicity n imbeddings, define 21&, k > 1, to be the compression 21& of #"
to the range of the rank one projection e[x' in Mnk. Evidently, these algebras are
all isomorphic to y\ Thus the direct limit sequence for 21 extends to the doubly
infinite sequence
► 2l2 -> 2li -> 2l0 -> 2l_i -> 2l_2 -> ...
where all the imbeddings of 21& into 2l^_i are given by the multiplicity one maps
ak(A) = En® A. L^ afc?00 denote the maps from 21* into 21 determining the
limit.
Define an automorphism a of 21 by "shifting to the left" in this sequence. In
other words, since every 21& is isomorphic to $" and all the maps ak are equivalent,
there is a sequence of isomorphisms ak from 21& onto 21&+1 for k in Z such that
afc+i<7fc = <7fc_iafc. Therefore there is a (unique) automorphism a of 21 such that

220
VIII. Discrete Crossed Products
<roLkl0o = afc+i,co0"fc for all k £ Z. This yields the commutative diagram
T
\<Tk
<*k-l
-St* ■
<**
■Sl*-i
-*2l
-a
^2lfc_2
Consider the crossed product 21 x<r Z. This algebra contains a unitary element
U such that 17 AU* = a (A) for all A in 21. Let P& denote the unit of the algebra
2lfc. Then UPkU* = Pfc+i. Consider the algebra <B = P0(2l x<, Z)P0. This
contains 2l0 = P021P0 and V = UP0 (since J7P0 = Pi 17 = P0l7Po). Notice that
V is a proper isometry in 2*.
We claim that 2* is generated by 2l0 and V. First note that for k > 0,
P0tf*P0 = tf*p0 = yfc and thus P0U-kP0 = P0^"fc = F*fc. A dense set
of elements of 21 are given by the finite sums of the form
N
N
N
X = J2 A^k = £ AkUk + J2 U~kA'_k
k=-N fc=0 fc=l
where A'_k = UkA-kU*k. Hence the elements of the form
N N
PoXPo = J2 PoAkUkP0 + J2 PoU-kALkPo
fc=0 fc=l
N N
= J2 PoAkPoUkP0 + J2 PoU-kP0A'_kP0
fc=0 fc=l
N N
= J2 PoAkP0Vk + J] V*kPoA'_kP0.
fc=0 fc=l
provide a dense subset of 2*. So 2ln and V generate 2J.
Think of 2l0 as -Mn(2li). Let E{j be set of matrix units for Mn(CPi) be
chosen so that En = Pi. Let Si = EnV for 1 < i < n. Then Si are isometries
in <B such that 5»5t* = £k. Therefore C*(Si,..., Sn) is isomorphic to On. We
claim that Si,..., Sn generate all of 2J. Since V = Si, it suffices to show that
2l0 is also generated by Si,..., Sn. Indeed, 2l0 may be thought of as nk x nk
matrices over 21&. Those matrices with scalar entries from 21& form a copy 9Jlk of
Mnk in 2lo. The union of these subalgebras is dense in the full UHF algebra 2lo.
Recall from Section V.4 that the elements of the form 5M5*, for words /z, v in W£
where Wk are the words of length k in the symbols 1,..., n, generate a copy 2Ufc
of Mnk. To see that Wk = VJlk, notice that the matrix units for Pfc_i9JtfcPfc_i are
just
*k-\Eij) = Uk-\SiS])Uk-1 = (Sk-1Si)(Sk-1Sj)* for l<ij<k.

VIII. Discrete Crossed Products
221
The rest of the matrix units are obtained in a similar fashion. Hence 2l0 belongs to
C*(5i,..., 5n); whence <B is isomorphic to On.
Likewise <Bfc := P^Pk is isomorphic to On for all k el. The imbedding of
2$fc into 2Jfc_i is just the multiplicity one imbedding fik{B) = En ® 5. The full
crossed product 21 is the direct limit of the <Bfc; so the crossed product (£03™) x ^ Z
is seen to be isomorphic to £ ® On. ■
For the interested reader, we now outline how the crossed product is
constructed in the general locally compact case. Form the algebra L1(G, 21) as
follows. First consider all continuous, compactly supported functions / from G into
21 and complete with respect to the norm
ll/lli=/ll/MII4M').
Jg
Multiplication is defined by a convolution twisted by a:
f*g(*)= f fWcttigit^s^dfiGit).
Jg
An adjoint operation is obtained by the formula
where A is the modular function relating left and right Haar measure. It will be
left to the interested reader to verify that this is a *-algebra.
The crossed product of 21 by G is the enveloping C*-algebra 21 xa G of
LX(G, 21). That is, define a C*-norm on LX(G, 21) by
||/||= sup H/)||
where this supremum is taken over all the non-degenerate ^representations of
L1(G, 21). The crossed product is the completion of L1(G, 21) in this norm.
As above, covariant representations always exist. Let n be any
♦-representation of 21 on a Hilbert space H. Form the Hilbert space L2(G, %) of all square
integrable ^-valued functions on G. Then define a covariant representation (5r, A)
of (21, G, U) on L2(G, U) as follows:
(n(A)f)(s) = n(a:\A))(f(s))
(Atf)(s) = f(t-*s)
for all A in 21, / in L2(G, %) and 5, t in G. It is easily verified that yields a co-
variant representation. The completion of L1(G, 21) with respect to this restricted
class of representations is called the reduced crossed product.
Any *-representation of 21 xa G yields (by restriction) a representation of
L1(G,2l); and by definition, every representation of L1(G,2l) is a
representation of 21 xa G. By an argument essentially identical to the group C*-algebra
case, there is a bijective correspondence between the covariant representations of

222
Vm. Discrete Crossed Products
(21, G, a) and ^representations of L1(G, 21). Thus the representation theory of
21 xa G encodes the covariant representation theory of (21, G, a).
VIII.2 Crossed Products by Z
Consider a single automorphism a in Aut(2l). This gives rise to an action of
the integers by an := an. For convenience, assume that 21 is unital so that there is
a unitary u in the crossed product 21 xa Z such that uAu* = a(A) for every A in
21. We shall consider 21 as a subalgebra of 21 xa Z.
There is always a faithful expectation from 21 xa Z onto 21 analogous to
computing the zero-th Fourier coefficient of a function on the circle. This will prove to
be a very useful tool.
Theorem VIII.2.1 Let (21, Z, a) be a C*-dynamical system. Then there is a
canonical faithful expectation $ 0/21 xa Z onto 21.
Proof. For every scalar A of modulus 1, Xu also determines a unitary representation
of Z (by n -> Xnun) such that
{\u)nA(\u)-n = unAu'n = an(A).
Thus (id, Xu) is another covariant representation of (21, Z, a). Clearly, 21 and Xu
generate 21 xa Z. From the universal property of the crossed product, there is a
homomorphism p\ of 21 xa Z onto itself such p\(A) = A for all A in 21 and
pA(tx) = Xu. Evidently p\ is an automorphism.
It is easy to check that for X in 21 xa Z, the function /(*) = pe2ir;t (X) is norm
continuous. Indeed, one may verify it on the dense subalgebra 21Z, and extend it to
the closure by a simple approximation argument. Then define a map $ on 21 xa Z
by
$(*)= / Pe2«u{X)dt.
Jo
Notice that since each p\ is a faithful (completely) positive isometric map, it
follows that $ is a faithful (completely) positive contraction. Next, consider the effect
on AXB for A, B in 21.
$(AXB) = f Pe2*n(AXB)dt = A f Pe2«it{X)dtB = A$(X)B.
Jo Jo
So $ is an 2l-bimodule map. In particular, $(A) = A for all A in 21. Also,
$(uk)= [ pe2nn{uk)dt= f e2*iktukdt = 0 for k ^ 0.
Jo Jo
Hence on an element of 21Z such as a finite sum J2n Anun, it follows that
*(Y^Anun) = YfAnZ(un) = A0.
n n

VHI.3. Minimal Dynamical Systems 223
This lies in 21. By the density of 21Z and the continuity of $, it follows that the
range of $ lies in 21. Since $ is the identity map on 21, it follows that $ is an
expectation. ■
Theorem VIH.2.1 permits the definition of a Fourier series for an element of
2lxaZby
$n(X):=$(Xu-n).
For every element X = ]£n Anun in 21Z, it follows that *nPO = An; and hence
X = J2n $n(X)un. In fact, we obtain the following analogue of Fejer's Theorem.
Define the Cesaro sums
Sn(X) = £ (1 - ;#r)*i(*y.
Theorem VIII.2.2 For every X in 21 xa Z, the Cesaro sums Sn(X) converge in
norm to X.
Proof. The proof is the same as the scalar case. Indeed,
j=-n
= fD1-!)6"^^)*5 f p^u{X)an{t)dt
Jo .±?n Jo
where <rn(*) = E^-nC1 - ^+r)e""27rijt is the FeJ^r kernel function. It is well
known that (rn(t)dt are probability measures on T = R/Z converging weak-* to
the point mass at 0. In particular, each En is a contractive (completely) positive
map.
The proof can be completed in the standard way, but since we already know
that 21Z is dense, it follows more simply. A routine calculation shows that for Y in
21Z, Y = limn-»oo Sn(y). Thus for any X in 21 xa Z and e > 0, choose Y in 21Z
with ||X - Y|| < e and an integer N so that ||Y - En(Y)|| < e for n > N. Then
for n> N,
\\x - sn(x)|| < \\x - y\\ + \\y - sn(r)|| + ||sn(y - x)|| < ze. m
VIII.3 Minimal Dynamical Systems
A classical dynamical system consists of a compact Hausdorff space X
together with a homeomorphism a of X onto itself. This determines a C*-dynamical
system (C(X), Z, a) where
<rn(f)'-=fo*-n.
In this section, we will develop some properties of the crossed product C(X) xa Z.

224
Vm. Discrete Crossed Products
We start with a basic result from ergodic theory. A finite Borel measure /z on X
is translation invariant if ^{a"l{E)) = fi(E) for every Borel subset of X. The
existence of such measures follows from an application of the Markov-Kakutani
fixed point theorem.
Theorem VIII.3.1 Let (X, a) be a classical dynamical system. Then there is a
Borel probability measure on X which is translation invariant for a.
Proof. Let Ca denote the automorphism Caf = f o a"1 of C(X). Clearly this is
a norm one linear operator. Thus the adjoint C* is a norm one linear map on the
dual space M(X) of all finite regular Borel measures on X. Moreover, since Ca
is positive, /z > 0 implies that C*/z > 0. Finally, if/z is a positive measure, then
IIOH = <7>(x) = n(*-\x)) = MX) = |H|.
Combining these observations shows that C* maps the weak-* compact convex set
V = {/x € M(X) : /x(X) = ||,.|| = 1}
of all probability measures into itself. Thus by the Markov-Kakutani fixed point
Theorem VH.2.1, C* has a fixed point //n. In other words,
So no is a translation invariant measure. ■
Certain of these invariant measures have an additional minimality property.
The system (X, a, /z) is ergodic if /z is translation invariant, and whenever E is a
translation invariant measurable set, then n(E) = 0 or 1.
Proposition VIII.3.2 Every dynamical system (X, a) has an ergodic measure.
Proof. By Theorem VIII.3.1, there are invariant probability measures for (X, a).
It is easy to check that the set T of all invariant probability measures is convex and
weak-* closed. So by the Krein-Milman Theorem, T is the closed convex hull of
its extreme points. We will show that every extreme point of this set is ergodic.
This will establish the proposition.
If /z in T is not ergodic, there is a translation invariant set E such that
0 < fi{E) < 1.
Evidently Ec is also essentially translation invariant. Let
fi1(A) = fi(E)'1(AnE) and /z2(A) = ^'^{AnE0).
Then both m belong to T; and they are distinct as they have disjoint supports.
Since
/z = fi(E)m + (1 - /z(£7))/z2,
it follows that /z is not extreme. ■

VIII3. Minimal Dynamical Systems
225
These measures allow us to construct a useful covariant representation of the
dynamical system (X, a). Let /z be a translation invariant measure, and form the
Hilbert space £2(/z). Represent C(X) on L2(/z) as multiplication operators Mi1,
and define an operator Up by U^h = h o a"1. Then the translation invariance of /z
implies that Up is isometric: for g, h in the dense subspace C(X),
(U»g, U^h) = / Ca{gh) dfi= ghdC^fi = / ghdp = (g, h).
Since U^ is clearly invertible, it is a unitary operator. Finally,
for / in C(X), h in L2(/z) and f in X. Thus (M^, U^) is a covariant representation
of (C(X), a). Let A£ denote the algebra generated by this covariant
representation.
A dynamical system (X, a) is minimal if X has no proper closed cr-invariant
subsets. This notion corresponds to simplicity of the crossed product. In this case,
we will obtain a lot of additional structure. We begin with the easy direction.
Proposition VIII.3.3 If F is a proper closed invariant subset of(X, a), then the
ideal <£f = {/ G C(X) : f\p = 0} generates a proper ideal 3f qfC(X) x^ Z.
Proof. Let Jjp be the set of elements in C(X)Z consisting of all finite sums of the
form J2n JnUn for Jn in dp. This is an ideal in C(X)Z because of the invariance
of F. Hence its closure 3f is the ideal of C(X) xa Z generated by ftp. Let $ be the
expectation of C(X) xa Z onto C(X) of Theorem Vm.2.1. Since ${3p) = <£f,
continuity shows that $(3p) = £f as well. Thus 3p is a proper ideal. ■
We need another basic result from ergodic theory known as Rohlin's Lemma.
Lemma VIII.3.4 Let (X, <r, /z) be an ergodic dynamical system such that the
support of ft is not finite. Then for every e > 0 and positive integer n, there is a Borel
subset F ofX such that <rl(F) are disjoint for 0 < i < n - 1 and
.«(U *'(*■)) >l-e.
Proof. Clearly the set
Xn = {x e X : an(x) = x^ <r>{x) for 1 < j < n}
is translation invariant for each n > 1. By ergodicity, each has measure 0 or 1. If
n(Xn) = 1 and cannot be divided into invariant subsets of proper measure, then
it is easy to see that Xn must consist of precisely n point masses of /z contrary to
hypothesis. Thus we may assume that off a set of measure zero (which we discard),
<rn(x) ± x for all n > 1.

226
VIII. Discrete Crossed Products
Choose an integer k > e_1. We will construct a measurable subset E of X
of positive measure such that a*(E) are pairwise disjoint for 0 < j < kn. This
is done by induction. Suppose that E is a set of positive measure such that o*(E)
are pairwise disjoint for 0 < j < m. Ergodicity and the previous paragraph imply
that there is a non-null subset E1 of E such that E1 and am(E/) are disjoint. Thus
a^(Ef) are disjoint for 0 < j < m. Eventually we obtain m = kn.
Now apply Zona's Lemma to obtain a maximal set (up to a null set) with this
property. If E\ is an increasing chain of measurable subsets with this property, it
is easy to extract a sequence E\n such that supn>1 fi(E\n) = supxii(E\). Then
E = Un>iE\n contains all the E\ modulo nulfsets. Therefore Zona's Lemma
provides a maximal set E such that <r*(E) are pairwise disjoint for 0 < j < kn.
Let G = akn"1(E). For each x in G, let N(x) denote the smallest positive
integer such that aN^x belongs to E and set it equal to oo if it never returns to
E. Notice that a{(x) does not belong to Ufe1 ^KE) for * < N(x)- For if **(*)
belongs to a*(E) and i > j, then a%~*(z) lies in E\ and if i < j9 then x lies in
a^"%(E), neither of which is possible. Define
Gj = {xeG: N(x) = j} for 1 < j < kn
and
G' = {x e G : N(x) > kn}.
The set G1 is a null set; for otherwiseE' = EU <r(Gf) determines a larger set with
ai(E') pairwise disjoint for 0 < j < kn. The disjoint union
kn-l kn j-1
|J <r^E) U |J |J otiGj)
j=o i=i *=i
is invariant for a, and thus has full measure by ergodicity.
The idea is to form a set F from every nth translate of E and each Gj. Define
k—1 kn
F=\Ja'n(E)U [j [j a^-^n+iGj.
*=0 j=n+l sn<j
It follows by construction that <rl(F) are disjoint for 0 < i < n. Moreover,
t=0 *=1 l<i<j<n
This has measure
k k
E E ^(G(-i)n+i) < E E MG{s-i)n+i) = nKG) = W(E).
8=1 l<i<j<n s=l l<j<n
The sets a*(E) for 0 < j < kn are disjoint and all have the same measure. Thus
nfi(E) < k"1 < e as desired. ■

V7II.3. Minimal Dynamical Systems
227
Now we are in a position to show that the algebra AS constructed above is in
fact isomorphic to the crossed product when a is ergodic (and X is infinite). The
key is showing that the expectation $ onto C(X) factors through AS-
Lemma VIII.3.5 Let (X, a, p) be an ergodic dynamical system such that the
support of p is not finite. On the algebra AS, the map
*(EM£(W)=M/o
j=-n
is contractive, and thus extends to an expectation of AS onto C(X).
Proof. Fix an element A = Sj=~n Mf- £# and an e > 0. Let
E = {xeX:\f0\>\\fo\\oo-e}.
Apply Rohlin's Lemma VIII.3.4 (for 2n + 1) to obtain pairwise disjoint non-null
sets Fj for -n < j < n such that <r(Fj) = Fj+i for -n < j < n. By the
ergodicity of a, there is an integer m such that 0-m(Fo) n E has positive measure.
By replacing each Fj by am(Fj), we may assume that m = 0.
Let g be any unit vector in L2(p) supported on E D F0. Then M^.U^g is
supported on Fj for all \j\ < n. In particular, they are pairwise orthogonal. Thus
l|A||2 > \\Ag\f = J2 \Wl9\\2
j=-n
>||/o«7||2>(II/o||oo-£)2 = (||m;o||-£)2.
Therefore ||*|| = 1. It is now evident that the map * extends by continuity to an
expectation of AS onto C(X). ■
Corollary VIII.3.6 When (X, <r, p) is ergodic and X is infinite, the algebra AS is
isomorphic to the crossed product C(X) xa Z.
Proof. There is a canonical homomorphism <p of C(X) xaZ onto AS which carries
each element E?=_n fjuj in C(X)Z to Ej=-n MfjU^ K is evident that tfy>
agrees with $ on this dense subalgebra, whence $ = Vtp. However, $ is faithful
and therefore <p is faithful and so must be an isomorphism. ■
Now we will use Rohlin's Lemma again to establish uniqueness of the
expectation and simplicity when a is minimal.
Lemma VIII.3.7 If(X, a) is a minimal dynamical system on an infinite compact
Hausdorff space X, then for each element A in C(X) xa Z and e > 0, there are
unimodularfunctions #i,..., 0m m C(X) such that
m
*=i

228
VTII. Discrete Crossed Products
Proof. Let /z be an ergodic measure for (X, a). First we show that if e > 0 and
k ^ 0, there is an open subset F of X with /z(F) > 1 - e and a unimodular
function V> in C(X) such that Re V>(V> ° °"~k) = 0 on F. Clearly we may suppose
that k > 0. To this end, let N > 2ke~l and apply Rohlin's Lemma Vm.3.4
to obtain pairwise disjoint sets Fj for 0 < j < N such that <r(Fj) = Ff+i for
0 < j < N and n(uf=0Fj) > 1 - e/2. Since /z is regular, we may replace each
Fj by a closed subset with the same properties. Then using the continuity of <r and
the disjointness of the Fj, we may find an open neighbourhood of the F/s with
these same properties and disjoint closures. Let these open sets be renamed Fj for
convenience.
Set F = ufj0kFj. Since n(Fj) < N'1 < e/2k, it follows that
/z(F) > 1 - e/2 - ke/2k = 1 - e.
Let A = e7"/2*. Define ip to equal A-7 on each Fj for 0 < j < N, and extend
it to a unimodular function in C(X). Then i/)(i/)oo—k) = -i on each Fj for
0 < j < N - & which is all of F. Notice that
V^V"1 + V*"1^ = 2(Retp(tpo ak))uk
vanishes on F.
Consider an element A = £"=_n fjuj in C(X)Z such that $(A) = /0 = 0.
For each 1 < |k\ < n, let V>fc be constructed as above so that Re tl>k{i>k ° °"~k) = 0
on an open set Fk such that fi(Fk) > 1 - (2n)~1. This implies that the open set
F = Cii<\k\<nFk has positive measure, and thus is non-empty. Let
*•= n *? f°r *z n fr-1}-
For each |fc| < n, consider 5 and s' which agree except on the fcth coordinate,
which differ; so that there is a function <p such that 08 = <pi/>k and 0,/ = (ft/*^1.
Then
0,/fct*fc07 + 08,fkvk6^ = 2<pfk (Re Vfc(V>fc o (7-fc)) tzV
which vanishes on F&. Thus summing over all values of s yieldsJhe same
conclusion. Applying this to each k in turn shows that 2~2n J2S ^sA08 vanishes on
F.
Now the sets a* (F) form an open cover of X by the minimality of a, and hence
there is an integer M so that the first M of these sets cover X. Let 08 := 0, o a'*
fori < j < M. Consider the family of functions
^,..,.M=n^ f°r s^ n {-i,i},i<j<Af.
j=i i<l&l<"

V7II.3. Minimal Dynamical Systems
229
Then the calculation of the previous paragraph now shows that
2-2nM J2 08A08
S=(*i,...,*Af)
vanishes on U^L^F = X.
Finally, C(X)Z is dense in C(X) xa Z and both $ and the averaging operator
are contractive and unital. Therefore, a simple approximation argument completes
the proof. ■
Corollary VIII.3.8 When (X, a) is minimal and X is infinite, the crossed product
C(X) xa Z has a faithful trace which is obtained from the expectation $ onto
C(X) followed by integration with respect to any ergodic measure /z. If(X, a) has
a unique invariant measure, then the trace is unique.
Proof. It is easy to verify that r(A) := fx $(A) d/z is a trace on C(X) xa Z.
Indeed, if A = £m fmum and B = £n gnun are elements of C(X)Z, then
T(AB) = J2 [ fn9-ndfl = T(BA).
By continuity, the trace condition extends to C(X) xa Z.
We show that any trace r1 must factor through $. By the previous lemma, for
any e > 0 and A in C(X) xa Z, we obtain unimodular functions 68 in C(X) such
that
m
e>|/(*(A))-r'(fn-l2M*.)|
*=1
m
= |r'(*(A)) - m"1 YV(4)| = |r'($(A)) - r'(A)|.
*=1
Hence r' factors as r' = y>'$, where <p' is the restriction of r1 to C(X). By
the Riesz Representation Theorem, there is a unique probability measure p! on X
such that t'Q) = fx f dp!. Moreover this measure is translation invariant because
r'(/) = r'{UfU*) = r'(a(f)).
Conversely, any invariant probability measure yields a trace. Thus the
uniqueness of this measure implies the uniqueness of the trace. ■
Theorem VIII.3.9 Let (X, a) be a dynamical system on an infinite compact Haus-
dorff space X. Then the crossed product C(X) xa Z is simple if and only if a is
minimal
Proof. Suppose that a is minimal and J is a non-zero ideal of C(X) xaZ. Let A be
a non-zero positive element of 3. By Lemma Vm.3.7, it follows that $(A) = /0

230
VIII. Discrete Crossed Products
belongs to X Since $ is faithful, /0 is non-zero. Hence the ideal contains
/n:-E^(/0y*-2^(/0).
Now /o > 0 on a non-empty open set O. As in the proof of Lemma VTO.3.7,
there is an integer n sufficiently large that the sets <t*(0) for 0 < j < n cover
X. Therefore fn is strictly positive, and thus invertible. So the ideal 3 is all of
C(X) xa Z.
The converse is contained in Proposition VEH.3.3. ■
We remark that if X is finite, then a minimal action of Z on X is a cyclic
permutation. In this case, the crossed product C(X) xaZ is isomorphic to Mn(C(T)),
where n = \X\. This is not simple.
VIII.4 Odometers
In this section, we will consider a well known homeomorphism on the Cantor
set studied in ergodic theory. Perhaps surprisingly, the crossed product algebra
turns out to be one we have already studied.
Let {rii} be a sequence of integers n» > 2. Let X{ = {0,1,..., n; - 1}, and
let //; be the measure on X{ assigning mass njl to each point. Form the Cantor set
X = n»>i Xim* ^d ^t /z be the product measure of the //;. Think of each element
a = {a;} as a formal sum S*>i aiN% where Ni = 1 and iVj+i = riiNi. A group
operation may be defined on X by formal addition with carries. That is, the sum
{a;} + {&;} = {c{} where c» are the unique integers in X{ such that
n n
]T(ai + bi)Ni = ^ ^iV; mod JVi+1 for all i > 1.
i=i i=i
In particular, b = -a is obtained as follows. Let i0 be the least integer such that
a,i0 ^ 0 (or oo when all a» = 0). Then 6; = 0 for i < i0, 6»0 = n;0 - a»0 and
bi = nt- - 1 - a; for alii > i0. It is readily verified that the group operations
are continuous. Also Z imbeds as a dense subgroup by sending each n in N to the
corresponding finite sum of the same total. Then the negative integers are sent to
sequences such that a; = n» - 1 for all sufficiently large i.
The odometer action a on X is obtained by addition of 1. The name comes
from the parallel with the way a car odometer works. When the ith digit rolls over
from rii -1 to 0, the i + 1st digit in incremented by 1. Since addition is continuous,
this is a homeomorphism. Consider the cylinder sets
J(*i,..., Xk) = {a : a; = X{ for 1 < i < k}.

Vin.4. Odometers
231
Then it is easy to see that <r( J(xi}..., Xk)) = J(yi,..., yu) where
k k
1 + £ *iN* = YlViNi m°d Nk+* *
t=l t=l
Therefore any invariant measure for a must agree with /z on every cylinder set. As
these cylinders generate the Borel sets as a <r-algebra, it follows that /z is the unique
invariant measure for a. Hence it is also the Haar measure for the group X.
Form the crossed product 2l = C(X)x<rZ. We wish to determine the structure
of this algebra. Let u represent the universal unitary element implementing a. The
action is minimal because the semi-orbit of any point a is a + N which is dense
in X. Therefore by Theorem VIH.3.9 and Corollary VHI.3.8, this is a simple C*-
algebra with a unique trace.
Theorem VIII.4.1 Let X = n»>i X% where each X{ has n» > 2 points, and let a
be the odometer action. Then the crossed product C(X) xaZis isomorphic to the
Bunce-Deddens algebra corresponding to the supernatural number n = II»>i n»-
Proof. The proof is accomplished by identifying a nested sequence of subalgebras
isomorphic to Mnu (C(T)) in the crossed product 21, and verifying that the imbed-
dings are the right ones. Map C(T) into 21 by the functional calculus of the unitary
u inducing a. That is, pi(f) = f(u). Then for each &, let 21* be the subalgebra
generated by the unitary u and the characteristic functions X(Xlj_jXk) of the
cylinder sets J(*i,..., Xk)* Since <r(J(xu • • • i xk)) = J(vu • • • i Vk) with notation as
above, it follows that
UX(xly...yxk)=X(yil...yyk)U.
Thus E}Q' := t^X(o,...,o) are partial isometries with initial projection X(0,...,o) m(*
range projection X(Xi,...jxk) Wnere i = Si=i xiN%- T"ey generate a set of matrix
units for Mnu given by E\y = E>Q'Efc*. With respect to this decomposition,
the span of the cylinder sets of size n is the set of diagonal scalar matrices. And
the unitary u is given by the matrix Uk with copies of the identity in the (j + 1, j)
entries for 1 < j < Nk, in the (0, Nk - 1) entry there is a coefficient Vk which we
must analyze; all other entries are 0.
To understand V&, note that it represents the matrix entry given by
^o!N*-i = X(o,...,o)uX(ni-i,...,n*-i)-
Thus
Vk = E<gtJ$l_lfi = X(0,...,o)^X(0,...,o).
This is a unitary element in the algebra X(0 o)2lX(o,...,o)- Hence there is a homo-
morphism pk which carries Mnic(C(T)) onto 2lfc = ^({E^}, Vk) by
/»*([/«]) = UuiVk)].

232
VIII. Discrete Crossed Products
Let ak denote the injection of 21*. into 2lfc+i. Likewise recall the injections fa
of *8(Nk) = MNk{C(T)) into 5S^+1 in the construction of the Bunce-Deddens
algebra 2$(n). We need to verify that otkPk = Pk+ifa> Once this is established,
it follows that there is a well-defined homomorphism p of 2J(n) onto 21 given
by p(A) = pk{A) for every A in 2J(iVfc). Moreover, p must be an isomorphism
because ©(n) is simple.
Let us determine the structure of the map a*.. Notice that each cylinder set
J(xi,..., Xk) splits as the disjoint union of J(x\,..., xni j) for 0 < j < rik+i.
Thus we have the decomposition
Eik) -
XX
njk+i-1
= E
i=o
Ei(*+1)
<*+1> _x t0 Sg^) by 14,.
The unitary u is represented as Uk and C4+i respectively in the two subalgebras.
If we think of Uk as an Nk x Nk matrix whose entries are rik+i x rik+i matrices
obtained from this decomposition in the natural order, then the entries on the first
subdiagonal are equal to Jnjk+1, most entries are 0, and the (0, Nk - 1) entry V*
maps each ^V+i^^-i+i^ onto Elitl)Nk,ti+i)Nh for ° ^ ^ n*+i " 2 by
the canonical identification (/) and it takes S// _x ^
Thus
V5k =
Now recall that the imbedding C(T) into Mnu(C(T)) in the Bunce-Deddens
construction takes z to the matrix Ak with l's on the first subdiagonal, z in the
(0, Nk - 1) entry, and O's elsewhere. The map pk takes z to Vfc. Hence it is evident
that pfc(Afc) = C7fc. Hence
otkPk(Ak) = Vfc+i = Pfc+i(Afc+i) = Pk+ifa(Ak)-
The restriction of pk to the iV*. x Nk matrices with scalar entries is the identity
map. Moreover, the imbeddings pk and fa agree on these scalar matrices. The
scalar matrices and Ak generate Mnu (C(T)). Thus the two mappings agree. ■
0
I
0
0
0 ..
0 ..
I ..
0 ..
. 0
. 0
. 0
. J
Vfc+l
0
0
0
VIII.5 K-theory for Crossed Products
There is an important tool for computing the K-theory of crossed products.
Unfortunately, the proof of this fact requires the development of much more general
K-theory than is contained in these notes. As its use in these note is limited, we
have decided to state them without proof.

VIII.5. K-theory for Crossed Products
233
We will need to define the Ki group of a C*-algebra 21. Let GLn(2l) denote
the group of invertible elements of -Mn(2l); and let GL£ (21) denote the connected
component of the identity, which is a clopen normal subgroup of GLn(2l). There is
a natural imbedding of GLn(2l) into GLn+i (21) given by sending A to [ £ J ]. It is
clear that GL£(2l) is carried into GL°+1(2l). Thus this induces a homomorphism
of the quotient GLn(VL)/GL°(21) intoGLn+1(2l)/GLQ+1(2l). The direct limit of
this sequence of groups is denoted by Ki(2l). It is always an abelian group even
though the quotient groups need not be abelian. (See the exercises.)
When 21 is not unital, we define GLn(2l) to be the group of those invertible
n x n matrices with coefficients in 2l~ such that the off-diagonal entries lie in 21
and the diagonal entries belong to 7+21. (That is, matrices congruent to In modulo
A4n(2l).)
In a C*-algebra, every invertible element is connected in the invertibles to the
unitary operator in its polar decomposition. Thus it is also easy to see that one
could use the unitary groups U(Mn($l)) instead of GLn(2l) in the definition of
K\. Notice that the direct limit of .Mn(2l) under multiplicity one imbeddings is
£ ® 21. An element of GLi((£ ® 21)") is connected to a unitary in the image of
GLn(2l) for n sufficiently large. It isn't difficult to show that
#i(2l) = #i(£® 21) = GLx((£® 2l)~)/GL?((£® 2l)~).
In any Banach algebra, it is elementary to check that when \\A - I\\ < 1, there
is an element L in the algebra such that A = eL. So GL° (21) is the set of all finite
products of exponentials eL for L in -Mn(2l). In the commutative case, the product
of exponentials is the exponential of the sum. So GL^(C(X)) = e°(xh
Any invertible element with spectrum in the right half plane is connected to
the identity by a straight line. More generally, any invertible element with 0 in the
unbounded component of the resolvent is connected to the identity by using the
functional calculus to construct a path to an element with spectrum in the right half
plane. In particular, any invertible element with finite spectrum is in the connected
component of the identity.
It follows that A"i(2l) = 0 for every finite dimensional C*-algebra. Since
GL£(2l) is open and closed in GLn(2l), it follows that when 21 = liir^2ln, then
#i(2l) = lir^i^1(2ln). Hence tfi(2l) = 0 for every AF-algebra.
If X is a totally disconnected space such as the Cantor set, every element of
GLn(C(X)) can be approximated by a function taking only finitely many values
in Mn- Such functions have finite spectrum and consequently lie in GL^(C(X)).
Therefore #i(C(X)) = 0. On the other hand, #i(C(T)) = Z. The map is
obtained by sending each element of GLn(C(T)) to the winding number of its
determinant function.
As in the case of K0, a homomorphism a from 21 into 2* induces a
homomorphism a* of A"i(2l) into Ki(*B). This makes Ki a covariant functor from the

234
VIII. Discrete Crossed Products
category of C*-algebras into the category of abelian groups. This is barely
sufficient for us to state the Pimsner-Voiculescu short exact sequence. In fact, we do
not give sufficient information here to define the vertical maps. However, in our
applications, this extra detail will not be needed.
Theorem VIII.5.1 Suppose that a is an automorphism of a C*-algebra 21. Then
there is a cyclic six term exact sequence
#o(2l) ^-^ #o(2l) ^- #o(2l xa Z)
K& xa Z) ^— Kl{*) ■*£--- tfi(2l)
Example VIII.5.2 Consider the irrational rotation algebra A$, which is shown to
be a crossed product in Example VIE. 1.1. First compute the K-theory for C(T).
Since T is connected, every projection valued function in C(T, Mn) has constant
rank. It is easy to find a homotopy of projections that connects any projection P
in C(T, Mn) to one with a fixed unit vector in the range of P(i) for every t in T.
Thus by induction, one shows that the rank is the only homotopy invariant. Hence
K0(C(T)) = Z. As noted above, KX(T) = Z as well.
Now consider the automorphism 0 on C(T) induced by rotation through the
angle 27T0. This does not effect the rank of a projection, nor the winding number
of the determinant of an invertible function. Therefore 0* = id*. Plugging this into
the P-V sequence, we obtain
0 -> K0{C(T)) = Z-> K0(Ae) -> #i(C(T)) = Z -> 0
and
0 -> ffi(C(T)) = Z -> Ki(At) -> K0(C(T)) = Z -> 0.
Therefore K0(A$) ~ K\(Ae) ~ Z2. It follows that the order homomorphism t*
of K0(Ae) onto Z + Z0 of Theorem VI.5.2 induced by the unique trace r is an
isomorphism.
Example VIII.5.3 Consider the crossed product representation of £ ® On from
Example Vffl.1.2. This is the crossed product (i?®^) xa Z where $" is theUHF
algebra of type n°° and a is a left shift. We will use the P-V exact sequence to
compute the K-theory of the Cuntz algebras. As noted above, K* (£®2l) = K* (21).
Thus K0(£ ® On) = K0{On) and Ki{& ® On) = #i(0n). Because tf1 is an
AF algebra, Ki(& ® #*) = K^tf1) = 0. As in Example IV.3.4, we may easily
compute that K0{& ® $") = Koffi1) = Z [£]. By Exercise IV.7, the map r*
induced by the unique trace is an isomorphism. The automorphism a satisfies
a(Pk) = Pfc+i. Since r(Pk) = n~fc, we deduce that a* is multiplication by n"1.

V7II.6. AF subalgebras of Crossed Products
235
Putting these facts together, we obtain the exact sequence
vi<U-<r«.
KitS") — ffi(S®0„) —■ ^o(r) ^-X(r) — K0(£®On) -* tfiGT1)
K^On)-
I]
[£] -^0(O„) -0
Consider the group homomorphism on Z [£] of multiplication by 1 - n_1. This
consists of the automorphism of multiplication by n_1 followed by multiplication
a by n -1. Since Z [£] is an integral domain, multiplication is injective. We claim
that the range of a consists of all elements of the form n~ka such that n -1 divides
a. Indeed, if a = (n - 1)6, then a(n~kb) = n~ka. Conversely, suppose that
n
-*„_
a = a(n 6) = n (n - 1)6.
Then since n and n - 1 are relatively prime, unique factorization in Z shows that
n - 1 divides a. We conclude that
K0(On) ~ ^(ryimOd* -a,) = Zn_x
and
JFT1(On) = ker(id*-^) = 0.
This provides another method of distinguishing Cuntz algebras from one another.
VIII.6 AF subalgebras of Crossed Products
In this section, we will show that certain natural subalgebras of crossed
products for minimal dynamical systems on Cantor sets are AF. Then we will extend this
analysis to obtain several useful structural results about these algebras including an
imbedding into one of these AF algebras analogous to the Pimsner-Voiculescu
imbedding of A$ into an AF algebra.
For this section, let X be a Cantor set and let a be a minimal homeomorphism.
We will call (X, a) a minimal Cantor system. A partition of X is a finite
collection V of pairwise disjoint clopen subsets of X which cover X. Let 0(7*) denote
the span of the characteristic functions of the members of V. Let u denote the
unitary element of C(X) xa Z which implements a on C(X).
Lemma VIII.6.1 Let Y be a non-empty clopen subset ofX and let V be a
partition ofX into clopen sets. Then the C*-subalgebra ofC(X) x^ Z generated by
C(V) anduXx\y is finite dimensional.
Proof. Define the^m return function for Y by
nY(y) := min{n > 1 : <rn(y) G Y} fory G Y.
Since a is minimal, {<rn(y) : n > 1} is dense in X. Thus since Y is open, ny
is finite for every y in Y and is upper semi-continuous. Indeed, if ny (y) = n,
then <rn(y') belongs to Y for all y1 in a sufficiently small neighbourhood of y\ and

236
Vm. Discrete Crossed Products
thus ny(y') < n on this neighbourhood. Similarly, since Y is closed, ny is lower
semi-continuous. Hence ny is continuous on Y. Then because Y is compact, ny
takes only finitely many values, say {ni,..., n#}.
For 1 < k < K and 1 < j < nk, define Y(kJ) = <rj(riy1 {nk)). Then the
sets Y(&, j) are pairwise disjoint clopen sets (why?). Notice that the sets Y(&, n*.)
must form a cover of V because each y in V is the first return of the first point
<r~8(y), s > 1, to lie in Y. It follows that the union of all the Y(&, j) is invariant
for a because
Y = u£=1y (*, nfc)f <r(y) = u£=1 Y(*, 1)
and
^(y(ft,j)) = y(*lj + l) for l<j<nfc.
As a is minimal, the sets Y(k, j) form a partition ^ of X. The collection
% = {lr(*,j):l<J<nfc}
is a tower of height n*. for each 1 < k < K.
It is a routine matter to further subdivide these towers vertically in order to
arrange that {Y(&, j)} refines V. Indeed, if Pis'mV and A = PC) Y(k, j0) is a
proper subset of Y(&, jo), construct the sets A(&, j) = d^""* (A) for 1 < j < n*..
This forms a tower of height nk, as do the sets Y(k, j) \ A(k, j); splitting the
tower into two of the same height. Continuing this process for each such proper
intersection yields a finer partition y' of X than y\ZV with the additional feature
that it consists of towers Y'(k, j) for 1 < j < n'k and 1 < k < K1 so that
a(Y(kJ)) = Y(kJ + l) for 1 < j < n'k,
uCiY'(k, 1) = a(Y) and u£x Y'{k% n'k) = Y
We will work with this partition, but will suppress the primes for notational
simplicity.
Now we define a finite dimensional subalgebra 2l(y, y) of C(X) xa Z
isomorphic to J2%=i ®Mnk determined by the matrix units
E& = u{-jXY(kJ) = Xy^ut-i
for 1 < k < K and 1 < i, j < nk. These do form a system of matrix units because
of the identities
uXY(kj)V* = X>*{Y{kj)) = Xy(fc,i+i) for 1 < J < ™fc-
The diagonal elements Ejy span C(y) which contains C(P). The off diagonal
elements E\y have the form (uXjr\y)*~JXy(fcj) when 1 < j < i < nk. They

V7II.6. AF subalgebras of Crossed Products
237
evidently lie in the algebra generated by uXx\y ^d C(y). Conversely,
K nk-l
fc=i i=i
Therefore 2l(Y, y) is the C*-algebra generated by C(y) and uXx\y• Since the
C*-algebra generated by C(P) and uXx\y is contained in this one, it is also finite
dimensional. ■
Theorem VIII.6.2 Let (X, a) be a minimal Cantor system; and let Y be a closed
subset ofX. Then the C*~subalgebra 2ly ofC(X) xa Z generated by C(X) and
uCo(X \ Y) is AF. Moreover the injection j ofC(X) into 2ly induces a surjec-
tive unital order homomorphismj* ofKo groups which carries Kq(C(X))^ onto
#o(2ly)+
Proof. Start with a decreasing sequence Yn of clopen sets with intersection Y; and
with an increasing sequence of partitions Vn whose union generates the topology of
X. Thus Un>iC(7>n) is dense in C(X). We then construct an increasing sequence
of partitions yn into towers built from Yn as in the preceding lemma so that yn
refines both Vn and yn-i. Then form the sequence of finite dimensional algebras
2ln = 21(1^, yn). Notice that this is an increasing chain of algebras because C(yn)
is contained in 0(3^+1) and
u*x\Yn = (^x\Yn^)Xx\Yn £ 2U+i.
The closed union of the 2ln contains C(Vn) for all n > 1, and thus it contains
C(X). Also any function / in Co^ \ Y) (thought of as an element of C(X)
vanishing on Y) can be approximated by a function g in some C(yn) which vanishes
on Yn. But then ug belongs to 2ln and approximates uf. Conversely, it is evident
that each 2ln lies in 2ly. Therefore 2ly is AF.
The injection jn of C(yn) into 2ln maps onto the diagonal algebra Vn spanned
by the diagonal matrix units. Every projection in 2ln is unitarily equivalent to a
diagonal projection. Therefore j'n* maps
tfo+(c(:»n))i = {Meow •p=p2=p* e c(yn)}
onto the corresponding set
*o+(2Ln)i = {\p]*n :p = p2=P^e^}.
By Theorem IV. 1.6, these sets are the order intervals [0,1] in the two Kq groups.
Hence jn* is an order isomorphism. Let an be the injection of 2ln into 2ly; and let

238
VIII. Discrete Crossed Products
/3n be the injection of C(^n) into C(X). Then
KZ(KY)l = IJ Ofn^+tan)! = \J O^j^K^(C(yn))l
n>l n>l
= |J MnK+wynVt = y^+cc^))!.
n>l
Thus j* is a surjection which takes the unit order interval of K0(C(X)) onto the
unit order interval of A"o(2ly). ■
Example VIII.6.3 Recall the crossed product representation of the Bunce-Ded-
dens algebra from Theorem VIII.4.1. Take Y to be the single point 0 = (0,0,...).
In the construction above, take Y*. = J(0,..., 0) where the sequence has k zeros;
and let the partition be
yk = {J(xu.. .,xh) : 0 < xi < nu 1 < i < k}.
From the proof and notation of that theorem,
N*-2
i=o
It is evident that the subalgebra 21(1^, yn) is isomorphic to Mnu . The imbeddings
are all unital, and therefore the algebra 2ln is isomorphic to the UHF algebra of type
n = IIt>i ^i-
It will be shown in the next section that the injection j of 2l0 into 21 induces an
isomorphism j* of K0 groups.
VIII.7 Crossed Product subalgebras of AF algebras
We will show that for a minimal Cantor system (X, a) and an arbitrary point y
in X, there is an imbedding of the crossed product C(X) xaZ into 2l{yj in a way
which induces an isomorphism of Kq groups. This is analogous to the Pimsner-
Voiculescu imbedding of the irrational rotation algebra A$ into the AF algebra
of the continued fraction of 0. In fact, our imbedding can be used to obtain the
Pimsner-Voiculescu imbedding.
Fix an arbitrary point y in X. As in the previous section, construct an
increasing sequence 2ln = 21(1^, yn) of finite dimensional subalgebras corresponding
to a decreasing sequence Yn of clopen sets with intersection {y} and increasing
partitions yn of X. By dropping to a subsequence if necessary, we may suppose
that the sets <r*(Yn) Bit pairwise disjoint for 1 < j < 2n and so that each is
contained in a single element of yn-i. (This is easily achieved because the points
a^(y) are distinct and each is contained in some element of the partition yn-i for
1 < j < 2n. Hence these properties remain valid for a sufficiently small
neighbourhood of {y}.) Following the notation of the previous section, let E\y for
1 < k < K, 1 < i, j < nk be the matrix units of 2ln.

VIII.7. Crossed Product subalgebras of AF algebras 239
Define a unitary vn in 2ln by
vn = uXXXYn + £ u^XY{k,nk) = £«>, + "g1 *#> ,).
fc=i fc=i j=i
Notice that vnXx\Yn = uXx\Yn- Also vn mimics the role of u in an important
way.
Lemma VIII.7.1 Iff in C(yn) is constant on Yn, then vnfv„ = a(f). In
particular, vnXYnVn = X<T{Yn) andvnfvl = <r(f) for every f in C(^n_i).
Proof. If fXYn =0,then
Vnfr* = (VnXX\Yn)f(vnXX\Yny = Xlfu* = *(f).
Also from the definition of vn, we obtain
K K
VnXYnV„ = Vn^2XY(k,nk)Vn = ^2XY(ktl) = X<r(Y)-
fc=l fc=l
Thus the result follows. Since Yn is contained in a single element of the partition
yn-i, every function in 0(3^-1) is constant on 1^. ■
Now vn and vn+i are direct sums of weighted shifts. We are looking for a
unitary wn in 2ln+i which commutes with C(3^n-i) such that wnvn^iw^ is close
to vn. In the case of the irrational rotation algebra, this type of approximation was
managed by Berg's technique. The same principle is used here, but it will be done
in a more algebraic way. Notice that
Vn+lXx\Yn = uXX\Yn = vnXX\Yn
= Xx\*(Yn)V = XX\<r(Yn)Vn+l = XX\v(Yn)vn-
Therefore
X*V(y„)<>n<+i = *X\<r(Yn) = vnVn+lXx\*(Yn)-
Thus we also obtain
Since 2ln+i is finite dimensional, the unitary element vnVn+i "as finite
spectrum in T. Hence by the continuous functional calculus, we obtain a unitary
z = fc(tinti*+1) where h(ei$) = ei2~n* for -ir < 0 < ir. Therefore
z2n=v»<+i and lk-l||<2-»ir
zX<r(Yn) = Xv(Yn)Z zXX\<r(Yn) = XX\a(Yn) = XX\<r(Yn)z
Recall that o*(Yn) are pairwise disjoint for 1 < j < 2n. From the fact that
z commutes with Xa(Yny we see that vP^ztvl'* commutes with X9i^yny Set

240 Vm. Discrete Crossed Products
Z = X \ U?=i °*(Yn)- Define a unitary element
Wn = XZ + f)u*-V,+1-V-^(y1|).
i=i
The unitary z can be thought of as a unitary twist through 2~nth of the unitary
*>n*>*+1. The unitary wn is the identity on Xz and acts by z* on X^+i-jvyv Thus
there is a gradual shifting from J to vn+i*>n ^ one shifts by a along the blocks.
This is the analogy to Berg's technique.
The key approximation lemma is the following.
Lemma VIII.7.2 The unitary wn belongs to the algebra 2ln+i, commutes with
G(yn-i) and satisfies
IK*>n+l<-*>n|| <2"n7T.
Proof. Since z belongs to 2ln+i and Xz and X9^yn) belong to C{yn), the
containment of wn in 2ln+i follows from the fact that
also lies in 2^. So
-.j-l^+l-j' l-jv . _ „,i-ly2n+l-iv „l-j
— *,i-iv ^2n+i-i,,i-i _ „,i-iv _2n+i-i„,i-i
- K X<r(Y)Z JVn J = VJn Xa{y)Z JVn
belongs to 2ln+i. Now wnXz = Xz, and wn commutes with each Xaj(yn) f°r
1 < j < 2n. As any function in C(yn-i) is constant on each X<ri(yn), it will
commute with wn.
Notice that for 1 < j < 2n, Lemma VIH.7.1 shows that
(vnWn - WnVn+^X^lYn) = ^nX^y^"1*2"*1"^1-' - t/;nX^+i(yn)ti
Thus
\\(vnWn - WnVn+O^i^jH < \\Z - l|| < 2"n7T.
Also
(vnwn - ^nVn+iJXy,, = vnXYn - WnX^y^Vn+i
= (vn - z2nvn+i)Xyn = (t> - vn<+1t>n+i)Xyn = 0
and since
a{Z \Yn)=(X\\J <r*(Yn)) \ a{Yn) = Z \ o«"+1(YB),
i=i

V7II.7. Crossed Product subalgebras of AF algebras 241
it follows that
(vnWn - WnVn+i)XZ\Yn = VnXZ\Yn ~ ™nX*(Z\Yn)V>XZ\Yn
= V>XZ\Yn - wnXz\<T2nJti(Yn)uXZ\Yn
= (U - u)XZ\Yn = 0.
Because the terms Xaj+i (Yn)(vnwn - WnVn+i)X<r*(Yn) have pairwise orthogonal
domains and ranges, one can compute
2n
\\vnWn ~ V>nVn+l\\ = II X)X^+1(^)K^n - ^nVn+l)X^(yn)||
i=i
= l<i<2« ^VnWn " ^^+l)X^(Vn)ll < 2"n7r- "
Now we are ready to state the main result.
Theorem VIII.7.3 Let (X, a) be a minimal dynamical system on a Cantor set X,
and let y be a point in X. Then there is a unital imbedding a ofC(X) xa Z into
2l{yj such that a* is an ordered group isomorphism of the Kq groups.
Proof. For n > 1, define automorphisms an of 2l{yj by
an(A) = Ad(wx.. .wn)(A) = an-i(wnAw*n).
Each g in C(yn) commutes with Wk for k > n. Hence ctk(g) = an(flO for all
k > n. So for an arbitrary / in C(-X") and e > 0, there is an integer n sufficiently
large andg in C(yn) such that ||/ - g\\ < e. Hence for k,£> n,
IM/) - «£(/)|| < ||o*(/ - g)\\ + Kb) - a£t»|| + IM/ - 5)|| < 2e.
Thus 0!fc(/) is Cauchy. So we may define an isometric *-homomorphism on C(X)
by ot0(f) = limn -J>oo
Next notice that by Lemma VHI.7.2
||On-l(«n) - OCn(vn+i)\\ = ||an_i(vn - WnVn+1W^)\\ < 2_n7T.
Therefore this is a Cauchy sequence. So define a unitary v = lim„_j.oo «n(vn+i)-
Now compute for / in Ujt>i C(lVfe) using Lemma VIII.7.1
va(/)v* = lim an(vn+i)an(f)an(vZ+i)
= JlHL a»(viK+i) = «(o-(/))-
n—>>oo
This is valid on a dense subset of C(X), and thus extends by continuity to the
whole algebra.
Therefore the pair (ao, v) is a covariant representation of the dynamical system
(X, a). So we may define a representation of the crossed product C(X) xa Z by
a(f) = a0(f) for / € C(X) and a(u) = v.
Since C(X) x^ Z is simple by Theorem VIEL3.9, a is injective.

242
VIII. Discrete Crossed Products
To compute the K-theory, let j be the injection of C(X) into 2l{y}, let i be the
injection of 2l{yj into C(X) xa Z, and let j0 = *J. Consider the commutative
diagram
C(X)
JO
a0
%}^C(I)x,Z^%}
Now a0 is the pointwise limit of inner automorphisms composed with j. So by
Remark IV.5.6, a0* = J*- Therefore
J* = <*o* = <**i*j*.
By Theorem VIII.6.2, j* is surjective. Therefore ct*i* = id^g^ })- We now need
to quote the Pimsner-Voiculescu exact sequence (Theorem VIII.5.1) only to obtain
the information that jo* is also surjective. Thus i* is surjective and consequently
a* = t"1. Both a* and i* are unital order homomorphisms, so they are order
isomorphisms. ■
There is another immediate consequence of the construction used in this last
proof. Recall from section V.7 that a limit circle algebra (AT algebra) is an
inductive limit of algebras of the form ]£ (B^i «Mpn,i(C3(T)). Notice that if some
of the summands are replaced with scalar matrices, the limit algebra is still an AT
algebra.
Corollary VIII.7.4 Let (X, a) be a minimal dynamical system on a Cantor set X.
Given a finite partition V ofX and ane > 0, there is a subalgebra of the crossed
product C(X) xa Z of the form <B ~ MPnl (C(T)) 0 J2 ©*=2 MPn,i containing
C(V) and a unitary uf such that \\uf - u\\ < e. Thus any finite set of elements
of C{X) x<r Z may be simultaneously approximated to any given accuracy by
elements of a subalgebra of this form.
Proof. We use the notation developed throughout this section. We may suppose
that V is contained in 3^. Let
Notice that
^ = c*{c(y„-i),<^n} = <c*{c(yn-i),«ii}wn
is contained in w* SlnWn which is finite dimensional, say
K
£n-J2®MPk'

V7II.7. Crossed Product subalgebras of AF algebras
243
So this algebra has a tower structure corresponding to a partition in between 3>n_i
and yn. Moreover, t£+1tiXy\yn+1 = Xjr\yn+1. So there is a unitary element z
acting on the support of Xyn+1 such that
<+1tx = zXyn+1 + XX\yn+1.
Since Xyn+1 is dominated by a minimal projection in C(yn), it is also dominated
by a diagonal matrix unit, say E{\\ in (£„. As z is a unitary supported on this
matrix unit, it follows that
K
®n = C*(£», <+1u) ~ MPl (C*(z)) 0 J] ®MPk.
fc=2
Moreover this algebra contains uf = (Wn^nWnH^n+i1*)- Compute
Hi*' - U\\ = ||«Vnt0nt£+l " 1)tiH = IK ~ Wn*>n+l<|| < 2"n7T
by Lemma Vin.7.2.
Now C*(z) is isomorphic to the continuous functions on its spectrum, which is
a subset of the circle. Let us show that it is the full circle. The unitary z corresponds
to the dynamical system (l^+i, <p) where <p is the first return map induced on
Yn+i by a. From the tower structure of 2ln+i> it is evident that z is a unitary in a
covariant representation of (1^+1. ¥>)• ^ *s easYt0 check that <p is minimal. For if
C is a closed non-empty <p invariant subset of yn+i> then C" = Ufcez °k{p) ls a
non-empty closed invariant subset of X. Thus Cf = X. But then
C = C n yn+i = ^n+i •
Therefore the crossed product C(Yn+i) x^, Z is simple, and so z is the universal
unitary in this algebra. So by Exercise VIII.3, it has spectrum equal to the whole
circle.
Any element of the crossed product may be approximated to any desired
accuracy by elements of the form YaL-n fiU% where fc belongs to C(V) for some
sufficiently fine finite partition. It is clear that if yn contains V and \\u - uf\\ is
sufficiently small, the corresponding polynomials in v! provide the desired approx-
imants. ■
Theorem VIII.7.5 Let (X, a) be a minimal system on a Cantor set X. Then the
crossed product 21 = C(X) xa Z is a limit circle algebra.
Proof. The remaining details are routine. It suffices to show that for an appropriate
subsequence of the algebras <Bn constructed above, there are homomorphisms fa
of *Bnu into 2Jn*+1 which asymptotically are almost isometric. Suppose that a
circle algebra <Bn and an e > 0 are given. Let S = 8(n, e) be given by Lemma IEL3.1,
where n is the dimension of the matrix algebra of scalars in <Bn. Choose m > n
sufficiently large so that the matrix units E\j' and the unitary v! in <Bn are within
S of elements of <Bm. By Lemma IEL3.1, there is a unitary element Wn in 21 such

244
VIII. Discrete Crossed Products
that || Wn - J11 < e so that each matrix unit E\j' is contained in Wn^BmW^ Recall
the element z in <Bn given by the formula v! = zE[\> + Eft'1. There is an element
U' = ZE[\] + E^1 in Wr&nWZ such that \\u' - U'\\ < 2(8 + e) < 3e. Since
the spectrum of uf is the whole circle, there is a homomorphism of C*(u') onto
C*(U/) taking uf onto U'. Combining this with the injection on the matrix algebra,
one obtains a homomorphism (5n of <Bn into Wn*BmW* which is the identity on
the matrix units and satisfies ||/3(^/) - uf\\ < 3e.
Choose an increasing sequence of such algebras and maps such that en
converges rapidly to 0. There is a natural sequence jn of injections of these circle
algebras in 21 and automorphisms an = Ad Wn of 21 such that jn+i Ai ^d otnjn
are asymptotically close on the generators. Some routine estimates show that the
limit map is an isomorphism of the limit circle algebra onto 21. The details are left
as an exercise. ■
Example VIII.7.6 We wish to show how the Pimsner-Voiculescu imbedding of
the irrational rotation algebra into the AF algebra of the continued fraction can be
obtained in another way. Consider the Cantor set X obtained from the unit circle
T = R/Z by introducing cuts at the points nO for all n in Z. More precisely, replace
each such point by a left and right limit point. The topology on this space is the
order topology induced from the order on [0,1]. This space is a totally disconnected
compact metric space; so it is homeomorphic to the Cantor set.
Let a be the homeomorphism of rotation by 6 on X, and consider the crossed
product <B = C(X) x* Z. Since C(T) sits inside C(X), there is a natural
imbedding j of A$ into 21. Follow this with the imbedding a of 25 into the AF algebra
2*{y}. To show that this is the right one, it suffices to compute the K-theory. See
Exercise VIII.7.
VIII.8 Topological stable rank
For a unital C*-algebra, let Lgn($l) and Rgn($l) denote the n-tuples of
elements of 21 which generate 21 as a left or right ideal respectively. The smallest
positive integer n such that Lgn(ty) is dense in the set 2ln of n-tuples of 21 is called the
topological stable rank of 21; write *sr(2l) = n. If no n exists, then tsr (21) = oo.
Similarly, we could define a right topological stable rank using i2<7n(2l)- However,
it is evident that Rgn{^) = £<7n(2l)*; so the two notions coincide.
This notion is another algebraic measure of something akin to dimension. We
have already seen the notion of real rank in section V.7. As with real rank, we
concentrate on the lowest case—topological stable rank one, and our analysis will
be fairly superficial. As a simple example, note that if X is a closed subset of
the circle T, then C(X) has topological stable rank one because every function on
the circle can be approximated by a non-vanishing one. Another simple example
is that AF algebras are topological stable rank one. Indeed, every element can

Vm.8. Topological stable rank
245
be approximated by an element in a finite dimensional C*-subalgebra. But the
invertibles are dense in every Mn>
Proposition VIII.8.1 A C*~algebra has topological stable rank one if and only if
the invertible elements are dense.
Proof. If the invertible elements are dense, then the set £<7i(2l) of left invertible
elements are also dense. Conversely, suppose that L0i(2l) is dense in 21. Let A be
a left invertible element with left inverse B. Find a left invertible element C such
that \\B-C\\< || A\\-\ Then
||CA-J||=||(C-B)A||<||C-B||||A||<1.
Thus CA is invertible. Therefore C is right invertible with right inverse A(CA)~l.
Since it is also left invertible, C is invertible. So A is also invertible with inverse
(CA)-XC. Thus 21"1 = L(7i(2l) is dense in 21. ■
Theorem VIII.8.2 Let %be a unital C*~algebra, and let n be a positive integer.
The invertible elements are dense in 21 if and only if they are dense in .Mn(2l).
Proof. Suppose that the invertible elements are dense in 21. We will show by
induction that the invertibles are also dense in Mn{%)> Suppose that the result is
valid for n, and consider a matrix T in Mn+i($L) and a positive real number e.
Write T in the form T = [cd] where a € 21 and .D G A^n(2l). By the induction
hypothesis, there is an invertible element an in 21 such that ||a — an|| < e and an
element D0 in .Mn(2l) such that D0 - Ca^lB is invertible and \\D - D0|| < e.
Hence
T0:=
C
B
D0
Oo
C
'n][0
0 Do-Ca^B
-i
B
In
The three matrices on the right are all invertible, and thus To is invertible. Since
||T - T0|| < e, it follows that the invertibles are dense in Mn+i (21).
Conversely, suppose that for a certain n, the invertible elements are dense in
■M„+i(2l). Given a in 21 and 0 < e < 1, form the matrix T = [% £ ] in.Mn+i(2l).
LetTo = [$ £] bean invertible element in Mn+i (21) such that ||T-T0|| < e/2.
In particular, ||£|| < e/2, \\C\\ < e/2 and ||I> - 7n|| < e/2. It follows that D is
invertible and ||-D-11| < (1 — e/2)-1. Thus we construct an invertible matrix
-BD
-l
Oo
C
B
D
1
-D~XC
0
In
Co - BD-^C
0
0
D
Hence oq - BD~lC is invertible in 21. Finally,
\\a - (a0 - BD-'Cn < \\a - ao\\ + llfllMiT'llllCII
< e/2 + (e/2)2(l - e/2)"1 = e(l - e/2)_1/2 < £•
So the invertible elements are dense in 21.

246 VIII. Discrete Crossed Products
The following consequence is immediate.
Corollary VIII.8.3 A C*-algebra which is a finite direct sum of algebras of the
form Mn(C(Xn)) where Xn are homeomorphic to closed subsets of the circle has
topological stable rank one.
This next result is easy, but is significant because we have been able to show
that many C*-algebras have this form for non-obvious reasons. The proof consists
merely of approximating by elements of a special subalgebra and applying the
corollary.
Theorem VIII.8.4 Every limit circle algebra has topological stable rank one.
Corollary VIII.8.5 Let (X, a) be a minimal system on a Cantor set X. Then the
invertible elements are dense in the crossed product C(X) x^ Z.
Proof. This follows from Corollary Vffl.7.4 and the observations above. ■
Corollary VIII.8.6 The invertible elements are dense in every Bunce-Deddens
algebra.
The following connection with K-theory indicates the usefulness of this notion.
Theorem VIII.8.7 If the invertible elements are dense in 21, then 21 has
cancellation.
Proof. First we show that if P and Q are equivalent projections in Mn($l), then
they are unitarily equivalent; and hence P1 is equivalent to Q1. Indeed,
suppose that V is a partial isometry such that P = V*V and Q = VV*. By
Theorem Vm.8.2, there is an invertible element A in -Mn(2l) such that ||V - A\\ < 1/6.
Then
IIP - A'A\\ < \\V{V - A)\\ + \\(V - A)*A\\ < 1/2.
Similarly, ||Q - AA*\\ < 1/2. Let A = U\A\ be the polar decomposition of A.
Then U is a unitary element since A is invertible. Then since U(A*A)U* = AA*,
we obtain
\\UPLU* -QL\\ = \\UPU* - Q\\ < \\U{P - A*A)CT|| + \\AA* - Q\\ < 1.
By Proposition IV. 1.2, Q1 is equivalent to UPLU* which is equivalent to P1. Let
W be the partial isometry such that W*W = P1 and WW* = Q1. Then V + W
is a unitary implementing the equivalence between P and Q.
We must show that if P and Q are stably equivalent projections in A/tn(2l),
then they are equivalent. Suppose that P and Q are projections in A/tn(2l) and
that R is a projection in A/lm(2l) such that P © R and Q © R are equivalent in
A*n+m(2l).Then
P©/m = (P©i2) + (0©i21)-(Q©i2) + (0©i21) = Q©/m.

VIII.9. An order 2 automorphism 247
Hence by the previous paragraph, PL © 0 is equivalent to QL © 0. By compressing
the implementing partial isometry to Mn{%) (which contains its domain and range
projections), one obtains an equivalence of PL and QL. A second use of the
previous paragraph yields the equivalence of P and Q. ■
VIII.9 An order 2 automorphism
In this section, we construct a crossed product of the Bunce-Deddens algebra
by Z2 to obtain an AF algebra. This shows that there is an AF algebra and an order
2 automorphism which has a fixed point algebra that is not AF. The first example
of such a phenomenon was considered very surprising. The constructions involved
are by now familiar to the reader.
The first step is to represent the 2°° Bunce-Deddens algebra as a crossed
product in a natural way. Let D denote the group of diadic rationals modulo 1. Think
of the circle as T = R/Z. Then let r be the action of D on T by translation. We
claim that the crossed product of C(T) by this action is the 2°° Bunce-Deddens
algebra.
Since the action of D is minimal, the crossed product 21 = C(T)xTD is simple
by the analogue of Theorem VHI.3.9. It will be convenient to use the particular
representation of 21 on L2(T) by multiplication and rotation operators. The group
D is the direct limit of the subgroups Z2" generated by 2~n. Let un = R2-n
denote rotation by 2~"n.
We claim that C*(C(T), un) is isomorphic to A^2«(C(T)). This can be seen as
follows. The spectrum of un consists of the 2nth roots of unity A* for An = e2 "n7r*
and 0 < k < 2n. Let the corresponding eigenspaces are
UnM = span{z*+i2n : j e Z}.
Let Ek denote the spectral projection of un onto %n,k- Consider the action of
multiplication by z on these subspaces. It is evident that
Mzzk+in = zk+1+i2n for jGZandO<Jb<2n-l
and
MzZ(2»-l)+j2» = 20+(i+l)2» f()r iGZ
This makes it evident that with respect to this decomposition, Mz has the matrix
decomposition
[0
\I
0
[0
0 ..
0 ..
/ ..
0 ..
. 0
. 0
. 0
. I
u]
0
0
oj

248
Yin. Discrete Crossed Products
where U is the bilateral shift. Thus it is clear that E{j = E{Ml~*Ej forms a set
of matrix units for M2n which are matrices with scalar entries with respect to this
basis. Then 2ln = C*(t/n, Mz) is seen to be isomorphic to
2l» = M2»(C*(U)) ~ AMC(T)).
Next consider the imbedding of 2ln into 2ln+i. A moment's reflection shows
that the operator U becomes the 2 x 2 matrix [J ^] when Hn,k is split into
%n+i,fc©Wn+i,fc+2n • This is just the two times around imbedding from section V.3.
Thus 21 is the 2°° Bunce-Deddens algebra as asserted.
The next step is to consider the order two automorphism of 21 given by
a(f)(t) = f(-t) for / € C(T) and a(Rd) = R-d for all d e D.
Covariance is preserved because
(Ad a(Rd))a(f)(t) = R.da(f)Rd(t) = a(f)(t + d)
= f(-t -d) = (RdfR.d)(-t) = v((AdRd)f)(t).
Consider the algebra 2* = 21 x^ Z2; and let v represent the order 2 unitary
implementing a. For 0 < k < 2n, let 9nyk = 2-n_1(2A: + 1). Then ulk+1v implements
the automorphism associated to the homeomorphism of the reflection which sends
OnM + x to 0nik - x. Indeed,
uf+lvfvu-2k-\0n,k + x) = (vfv)(x - 6nM) = f{0nM - x). (1)
Fix a positive integer n and a real number 0 < 8 < 2~n_1. We will define
projections in 2ln x^ Z2 which are analogous to the Rieffel projections in A$. Let
<Pn,k{x) be the linear function defined on the line segment [6n^ - £,0n,fc + &] such
that (pn,k(0n,k - S) = 0 and ¥>n,fc(^n,fc + S) = l. Define functions fUik and gUik in
C(T) by
for 0n,*._i - S < x < 0n,fc_i + S
for 0nik-i + 6 <x <0n,k-S
for enik-s<x<enik + s
otherwise
fn,k{X) =
and
a
(x)= I (V>n,fc(s)(l - <Pn,k(X)))1/2 for °nM ~ S < X < 0n,fc + -
10 otherwise
Then define
Pn,k = fn,k + ("I) W+1<><7n,fc + ^"^n.fc-l).
The calculation parallels section VI.2 for the irrational rotation algebra. Since
9n,k is symmetric about #„,*., it follows from (1) that

VIII.9. An order 2 automorphism 249
Thus
PnJ. = fn,k + (-1) W*^*-1 + 9nJk-lVttf»l)
= fnjk + (-l)k(9n,k«lk+1V + g^k-iuf^v)
= fnjk + (-l)k(ulk+1V9ntk + uf -'^n.Jk-l) = Pn.fc
Notice that
(u2nk+1vfnjkix) + fn,kulk+1v)gn,k = u^vg^f^x) + fn,k(20n<k - x))
= uf^vgn,k.
and similarly
tf*-lvfnj,(z) + fnjk^-lv)gnjk-l
= u^vg^k.iiU^x) + fn,k(29n,k-i - as))
= t»^fc-1V5n,fc-l-
Then calculate
l*.fc = flk + (u2* W + (uf-\gn,k_{f +
+ (-l)fc(/«,rf+1t><7n,fc + u2nk+1vgn<kfn<k) +
+ (-l)fc(/n,fc«^-1V<7n,fc-l + «nfc_1V</„,fc-l/n,fc)
= flk + glk + *t*-l + W'ti*1"** + (-W^Vgn* = Pn,fc.
Thuspn?fc are projections. A simple calculation shows that
!>,* = 2£ u+2£ ((-*)*+(-i)fc+Vfc+W = i
fc=0 fc=0 fc=0
So the pnik form a partition of the identity. Note that
Un-lPn,kUn-l
= ulfn,ku? + {-l)kuf+\gn,kuZ2 + (-l)fct4fc+W-i<2
= fk+2 + (-l)fc+2«^+3v„-2i7n fc+2 + (_l)fc+2u2fc+lvu-25n fc+1
=/fc+2+(-l)fc+2«f+5^n,fc+2+(-l)fc+2«f+3^n,fc+1 = p„)fc+2.
Also vpn,fc<> = pn,2»-fc and vt/n_iv = u^. So the algebra
»n = C*(tln-1, *,**,*, 0 < AJ < 2n)
is finite dimensional. In fact, it is isomorphic to the direct sum of four copies of
M2n-i (see the Exercises).
To see that the union of the 2Jn is dense in 2J, note that the set of polynomials in
v and un for n > 1 with coefficients in C(T) is dense in fB. Clearly v and un belong

250
VIII. Discrete Crossed Products
to 2$n+i. So it suffices to approximate each function / in C(T). This follows from
an easy partition of unity type estimate. By uniform continuity, given e > 0, there
is an integer n so that \f(x) - f(y)\ < e when \x - y\ < 2~n. The support of the
functions /n^, gn^ and gn,k-i are contained in {x : \x - 2~nk\ < 2~n}. Hence
we see that / - f(£;)pn,k = £kPn,k where e*. is a function in C(T) supported
on the interval [(k - l)2~n, (k + l)2"n] with norm at most e. Thus eiek = 0 if
\£-k\ > 1. So
2n-l
fc=0
)Pn,fc
|2
<
2n~l
||E^n*|
fc=0
2n-l
I 2J PnM€k
fc=0
2
|2Pn,
2n-12n-l
|| XJ y^Pn,/g7gfePn,fe||
fc=0 /=0
2n-l
k\\+ 2 2J PnlJfe+l5Jfe+lfejfePnlfc
fc=0
<3e2.
Therefore <Bis AR
It follows that Ad v is an inner automorphism of 2* of order two with 21 as
its fixed point algebra. Since this is a Bunce-Deddens algebra, it is not AF by
Theorem V.3.4.
Exercises
VIH.l Suppose that (21, G, a) is a discrete dynamical system. Show that if 21 is
not unital and En is an approximate unit for 21, then any representation a
of 21G yields a covariant representation of (21, G, a) by
w(A) = a(Ae) and U8 = lim (7(2?ns).
v ' ' n-fco
VDI.2 Show that if U, V are in GLn (21), then U®In and Jn © [/ are connected in
GL2n(2l) • Hence show that UV © In and VC/ © In are connected. Deduce
that Ki(2l) is always abelian.
VIII.3 Show that if (C(X), u) is a covariant representation of a minimal system,
then the spectrum of u is the full unit circle.
VDI.4 If X is an n element set and a is a cyclic permutation, show that C(X) x^Z
is isomorphic to Mn(C(T)).
VIH.5 Use the Pimsner-Voiculescu exact sequence to compute Ki of the Bunce-
Deddens algebras.
VIIL6 Show that for a minimal system (X} a) on a Cantor set and a closed subset
Y of X, there is an exact sequence of groups
0 —> Z -i> C(y, Z) A iCo(2ly) -ii> X0(C(X) x„ Z) —> 0
where a(n) = nXy, i is the imbedding of C(X) into 2ly, i is the
imbedding of 2ly into C(X) xa Z and /3(g) = j*(g - a(g)).

Exercises
251
VIIL7 Use the Pimsner-Voiculescu exact sequence to compute K$ of the
algebra 21 of Example Vffl.7.6. Hence show that the imbedding of A$ into 21
induces an isomorphism of K0 groups. Hence deduce that this is an
imbedding into the same AF algebra as the Pimsner-Voiculescu imbedding.
VDI.8 Suppose that 2J = liir^(fBn, /3n), and that there are homomorphisms (pn of
2Jn into a C*-algebra 21 such that for every element B in ?Bn, the sequence
VnPk,n{A) is Cauchy. Show that this limit yields a homomorphism of 2*
into 21. Hence verify the details of Theorem VHI.7.5.
VIH.9 Show that 21 has topological stable rank one if and only if £ ® 21 has.
(Recall that £ <g> 21 is the direct limit of ^fn(2l).)
VIII.10 Show that the algebra fBn = C*(i/n-i, v,pnyk,0 < k < 2n) of the
section VIII.9 is isomorphic to the direct sum of four copies of M2n-i.
HINT: First show that the sum Pe of the pnik for even k commutes with
t/n_i and v\ and that the even (odd) projections and Peun-i (P^un-i)
generate a copy of M2n-i. Then note that the action of v on M2n-\ is
equivalent to an inner automorphism by a unitary w. Show that the
projection Q = (wv + J)/2 commutes with both v and M2n-i.
Notes and Remarks*
The best comprehensive text on crossed products of C*-algebras is Pedersen
[1979]. The simplicity of C*-algebras of minimal actions by amenable groups is
due to Zeller-Meier [1968]. The proof for Z actions is taken from Power [1978].
The crossed product representation of Bunce-Deddens algebras goes back to their
original 1975 paper. Pimsner and Voiculescu [1980b] established the short
exact sequence for K-groups of crossed products, and computed these groups for the
irrational rotation algebras. Cuntz [1981] computed the K-groups of On. The
analysis of AF subalgebras of minimal crossed products of Cantor sets is due to Putnam
[1989]. Versik [1981] establishes the imbedding of these algebras into AF algebras,
but the proof here is taken from Putnam's paper which establishes that it yields an
isomorphism of Ko groups. That this leads to the limit circle algebra structure is
due to Putman [1990a]. Pimsner [1983] established a more general criterion for
imbedding a crossed product into an AF algebra. His construction does not yield
an isomorphism of Ko groups. Topological stable rank was introduced by Rieffel
[1983a] where the theory is developed at length. It was independently introduced
in Corach and Laratonda [1984]. Putnam [1989] showed that the minimal crossed
products on Cantor sets are topological stable rank one; and used this to prove that
the irrational rotation algebras are also stable rank one in Putnam [1990b]. Elliott
and Evans [1993] showed that irrational rotation algebras are limit circle algebras.
Blackadar [1990] gave the first example of an order two automorphism of an AF
algebra with non-AF fixed point algebra. The example given here is due to Kumjian
[1988].

CHAPTER IX
Brown-Douglas-Fillmore Theory
An operator T in B(H) is called essentially normal if T*T - TT* is compact.
Let 7r be the quotient map of B(U) onto the Calkin algebra Q{U) := B{U)/&.
Then we can restate this as saying that ir(T) is a normal element of Q(H).
The problem is to classify essentially normal operators. As we shall see, this
is handled by solving a C*-algebra problem about extensions of the compact
operators. The first step is to decide on the right notion of equivalence to use to
identify two operators which are essentially the same. A natural choice for
operators is unitary equivalence, that is A ~ B if there is a unitary operator U such
that B = UAU*. However, in this case, there is an extra ingredient. If T is an
essentially normal operator, then T + K is also essentially normal for any K in .£
(a compact perturbation of T). So we will include this as a trivial change. We say
that two operators A and B are compalent if there is a unitary operator U such that
B — UAU"" is compact.
Another natural choice would be to use unitary equivalence in the Calkin
algebra. The notion of compalence of A and B may be reformulated as saying that
there is a unitary operator U such that ir(B) = ir(U)ir(A)ir(U)*. So we might
think to use unitary elements of Q{H) instead, since the elements n(U) are
precisely the unitary elements in Q(H) of index 0. Say that A and B are weakly
compalent if there is a unitary element u in Q(H) such that ir(B) = uir(A)u*.
For essentially normal operators, these two notions coincide. However, as we saw
in § V.6, this is not the case in general.
If T is essentially normal, then C*(7r(T)) is an abelian subalgebra of Q(H)
which is canonically isomorphic to C(ae(T)) by the functional calculus
(Corollary 1.3.2), where cre{T) = a(ir(T)) is the essential spectrum of T. Let
<S(T) = tt^CXT)) = C*(r) + £.
Then this algebra contains ^ as an ideal, and has quotient equivalent to C(<re(T))
obtained by identifying ir(T) with the identity function z. We obtain the exact
sequence
0 _» £ -!> (8(T) -^> C((Te(T)) —> 0
This is an extension of the compact operators by the abelian C*-algebra C(<re(T)).
More generally, when X is a compact metric space, we say that an extension
of A by C(X) is a C*-subalgebra <£ of B(H) containing the compact operators £
252

IX. Brown-Douglas-Fillmore Theory
253
such that
<H A C(X) —» 0
is exact. Two extensions (<Hi, <pi) and (<H2, ^2) are said to be equivalent if there
is a *-isomorphism i/> of (Hi onto <H2 such that (pi = (p2if>. That is, the following
diagram commutes:
^C(X)
^C(X) -0
By Lemma V.6.1, there is a unitary operator U such that i>\&= AdU. Thus
V> = Ad U because of the identity
1>(A)1>(K) = j>(AK) = AdU{AK)
= Ad U(A) Ad U(K) = Ad U(A)i/>(K).
Since £ is an essential ideal of 3(71), this implies that tp(A) = Ad f/(A) for all
A in (Hi. So ((Hi, <pi) and (£2, (f2) are equivalent if and only if there is a unitary
operator U such that <H2 = U<£iU* and <pi = (p2 Ad [/.
Returning to the case of a single operator, the metric space X — <xe(T) is
contained in C. Thus C(X) has a single generator z(t) = t. It follows that two
extensions <H(Tj) are equivalent if and only if Tx and T2 are compalent. Indeed,
when T2 = UTiU* + K for a unitary operator [7 and compact operator if, the
map V> = Ad [7 is the desired map since
¥>2V>(-\) = ¥>2(r2 + If) = z = pi(2\).
As T; generate <£,• modulo £, this implies that <pi = il>(p2. Conversely, if (H; are
equivalent via ip = AdUy then it follows that T2 - UTiU* lies in ker <p2 = £. So
Ti and T2 are compalent.
Associated to an extension ((H, (p) of ^ by C(X) is a *-monomorphism t of
C(X) into Q(%) given by r(f) = 7ry>_1(/), which is the inverse of the
identification of the subalgebra <£/& of Q{%) with C(X) via (p. Conversely, if r is a
*-monomorphism of C(X) into Q{H), let (H = 7r~1r(C(X)) and set <p = T_17r.
Then it is clear that (H is a C*-subalgebra of #(%) containing the compact operators
such that <£/& is isomorphic to C(X) via <p. So (H is an extension of R by C(X).
With this formulation, tx and t2 are equivalent if there is a unitary operator U such
that t2 = Ad 7r([7) ti. In the context of planar sets, an essentially normal operator
T with <re(T) = X is identified with the monomorphism r(f) = f(ir(T)) by the
functional calculus for the normal element n(T).
We will work with both formulations of an extension as is convenient. The
reader should note that the C*-algebra C(X) may be replaced by any C*-algebra in

254
IX. Brown-Douglas-Fillmore Theory
the discussion above. Indeed, in chapter V, this was considered for the C*-algebra
Two extensions ti and t2 are called weakly equivalent if there is a unitary
element u in Q(H) such that r2 = Ad u n.
We denote by Ext(X) the collection of all equivalence classes of extensions.
The goal of this chapter is to show that Ext(X) is an abelian group, and that
the map Ext is a functor from compact metric spaces into abelian groups that
has the properties of a homology theory. This will enable us to compute Ext(X)
for compact subsets of the plane, and hence classify essentially normal operators.
These results are known as the Brown-Douglas-Fillmore (BDF) Theory.
This is a rather unusual approach to a problem in operator theory that sounds
much more pedestrian at first sight. However, twenty years after the original proof,
there still is no easy way to obtain a proof (although there is an "operator theoretic"
proof). But the real impact of BDF came from the introduction of topological
methods into C*-algebras. This connection was one of the most important developments
in C*-algebras in this century.
IX.2 An Addition and Zero Element for Ext(X).
The group operation + on Ext(X) is most easily defined for the monomor-
phism version. Set
[ti] + [t2] = [ti0t2].
It is implicit in this definition that we may identify M2{Q{'H)) with Q(H) because
we may identify H®% with % for any separable Hilbert space H, and this
identification yields an isomorphism of .M2(#(%)) onto B(H) which takes .M2(.£) onto
£. It is evident that this is a well defined abelian and associative operation. The
details of this routine verification will be left to the reader. In the case of essentially
normal operators, the generator of the sum of [T\] and [T2] is [Tx © T2]. See the
exercises for the other formulation of sum for extensions (<£, <p).
The first step is to identify a zero element. An extension ((£, (p) is called trivial
if there is a *-monomorphism a of C(X) into <£ such that (pa = idc(x)- -0 other
words, we obtain the split exact sequence
o—^£—^cd^rCpO—^0
In the planar case, N = a(z) is a normal operator with spectrum and essential
spectrum equal to X such that <£ = C*(7V) + £. Conversely, given such a normal
operator N, the map a(f) = f(N) is the desired splitting of the extension defined
byiV.
The main step in showing that the trivial extensions form a single equivalence
class which acts as a zero element for Ext(X) is the Weyl-von Neumann
Theorem (Theorem H.4.6) for representations of C(X). This result also follows from
Theorem V.6.3, but we give a more elementary proof here.

IX.2. An Addition and Zero Element for Ext(X).
255
Theorem IX.2.1 Let X be a compact metric space. Then C(X) has trivial
extensions, and all trivial extensions of & by C(X) are equivalent.
Proof. Existence is easy. Take any countable dense subset {fn} of X and define
<r(f) = J2n>i ®f(€n)I- This is a *-isomorphism of infinite multiplicity, so the
range contains no compact operators. Thus ira is also a monomorphism. It
determines an extension which, by construction, has the section a. Thus it is a trivial
extension.
Two trivial extensions of C(X) are given by *-isomorphisms p and a of C(X)
into B(H) such that irp and ira are still injective. Thus the ranges of p and a
contain no compact operators, whence rank/>(/) = rank(7(/) = oo for every
non-zero function / in C(X). Hence by Theorem H.4.6, p and a are
approximately unitarily equivalent relative to the compact operators. In particular, there
is a unitary operator U such that Ad U p - a has range in the compact operators.
Hence Ad ir(U) irp = ira and so p and a determine equivalent extensions. ■
To show that the trivial class represents a zero element, we need to show that
every extension essentially contains a big diagonal representation as a summand.
This is accomplished by pulling out lots of approximate eigenvectors. Again, this
follows from Voiculescu's Theorem as in Theorem V.6.3. But we will show that
it is more elementary in the commutative case. Recall that if {ai, a2,...} are
elements of a commutative Banach algebra #, then their joint spectrum is the set
of sequences
{(¥>(ai)>¥>(«2),...) '<pe M8}]
where Mb is the maximal ideal space of B. In a C*-algebra, it is easy to see that
A = (Ai, A2,...) is in the joint spectrum of (Ai,A2}...) if and only if
is not invertible. In particular, if T = (Ti, T2,...) is an essentially commuting
family of essentially normal operators, then the joint essential spectrum of T is
just the joint spectrum of 7r(T) = (7r(Ti), n"^), „..). So this consists of those A
such that
is not Fredholm.
Lemma IX.2.2 Let T = (Ti, T2j...), be an essentially commuting family of
essentially normal operators. Suppose that An = (An?i, Ani2,...) is a sequence
of points in the joint essential spectrum of the T. Then there is an orthonormal
sequence xn such that
Km \\Tkxn - \n,kXn\\ = 0 for all k>l.
n—>oo

256 IX. Brown-Douglas-Fillmore Theory
Proof. Suppose that we are given a finite dimensional subspace L of H, e > 0 and
a point A = (Ai, A2,...) in the joint essential spectrum of T. Then
^ k(Tk-\kI)*(Tk-\kI)
is not Fredholm. Hence either A has infinite dimensional kernel or it is not bounded
below on the complement of its kernel. In either case, there is a unit vector x
orthogonal to C so that (Ax, x) < e2. Thus
\\Tkx-\kx\\2<2h\\Tk-\kI\\2(Ax}x)
< 2k4\\Tk\\V = (21+*/2||rfc||£)2.
Now proceed recursively. At the n-th stage, we suppose that we have
constructed an orthonormal set xi,..., xn-i so that
\\TkXi - \i,kxi\\ < 2"1 for all 1 < k < i and 1 < i < n - 1.
Then set C = span{aji,..., «n-i} and choose e sufficiently small so that
max 21+k'2\\Tk\\e < 2~n.
l<fc<n
Then applying the previous paragraph, one obtains a unit vector xn orthogonal to
C so that \\Tkxn - Xnik^n\\ < 2~n for 1 < k < n. This is the desired sequence. ■
Theorem IX.2.3 Let X be a compact metric space. The class of trivial extensions
for C(X) forms a zero element for Ext(X).
Proof. Suppose that r is a *-monomorphism of C(X) into Q(H) representing
an element of Ext(X). Let /lf /2,... be a countable family of norm one
positive functions in C(-X") which separate points of X. Then the joint spectrum
A of (/i, /2,...) is homeomorphic to X via the map which takes f in X to
(/i(0./*(0.-).
Let Tk be positive elements of norm 1 in B(H) such that n(Tk) = r(fk). Then
(2i, r2,...) is an essentially commuting family of essentially normal operators
with joint essential spectrum A. Apply the previous lemma to a sequence An which
is dense in A and repeats each isolated point infinitely often to obtain orthogonal
unit vectors xn so that ||Tk;cn - An>fc;cn|| < 2~n for 1 < k < n. Let P be the
projection onto span{«i, x2j...}. We will show thatp = ir(P) commutes with
every r (/) and that prp is a trivial extension into Q(PH).

IX.2. An Addition and Zero Element for Ext(X). 257
Indeed, for each k, let Dk denote the diagonal operator on RH given by
DkXn = An,A.ajn. Then
\\TkP-Dk\\l = J2\m-Dk)xn\\2
n>l
<(k-l) + J2\\Tkxk-Kkxk\\2
n>k
<Jfe-l + ]T4-n< Jfe.
n>fc
Hence TkP — Dk is a Hilbert-Schmidt operator for every A:, and thus is compact.
Therefore, PLTkP = PL{Tk - Dk)P is compact; whence
PTk - TkP = PTkPL - PLTkP = -2iIm(PLTkP)
is also compact. That means that p commutes with each r(fk). As they generate
C(X),p commutes with the range of r. Moreover, Tk = (Dk 0 PLTkPL) + Kk
where Kk is compact.
Let fn be the point in X corresponding to the point An in A. Evidently, the
map a(f) = diag(/(fn)) with respect to the basis xn of PH is a trivial extension
by C(X) because {fn : n > 1} is dense in X and isolated points have infinite
multiplicity. From the preceding paragraph, we see that
r(/)-=7T(T(/)e(l-p)r(/)(l-p).
By Theorem IX.2.1, ira is unitarily equivalent to ira © ira. We have
r(f) = na(f)®(l-p)T(f)(l-p)
- ira(f) 0 ira(f) 0 (1 - p)r(f)(l - p)
= 7T(T(/)er(/).
Hence [r] + [a] = [r]. That is, [a] is a zero element for Ext(X). ■
We can use this result to show that weak equivalence and equivalence coincide
in Ext(X). Recall that this is not the case for the Cuntz algebras.
Corollary IX.2.4 Weakly equivalent extensions by C(X) are equivalent.
Proof. Let r be an extension by C(X). It suffices to show that Adt/r and r are
equivalent for every unitary element of Q(H). However, if s is any particular
unitary element of index 1, then every unitary u in Q(H) has the form ir(U)sk for
some unitary U in B(H) and integer k in Z. Thus it is enough to find one particular
s of index 1 so that [Ad st] = [t].
Let a be the diagonal representation constructed above; and let S be the
backward shift with respect to the basis xn. Then
a'(f):= Ad S a(/) = diag(/(£n+i))

258
IX. Brown-Douglas-Fillmore Theory
is another diagonal representation. The density of the sequence f n shows that ira'
is another trivial extension. By Theorem IX.2.1, [ira*] = [ira]. Hence
[r] = [r 0 ira] = [r 0 ira*] = [Ad7r(J0 S) r 0 ira] = [Adsr]
where s is the unitary of index 1 corresponding to ir(10 S) under the equivalence
between r®ira and r. ■
IX.3 Some Special Cases
Before going on to the general theory, we show how to compute a few
important examples.
Theorem IX.3.1 IfC(X) is generated by a single real valued function, then every
extension is trivial and so Ext(X) = {0}. In particular, this is the case ifX is a
subset ofR.
Proof. Let a denote the real generator of C(X), and let Y := a(X) = a(a).
Then C(X) = {/(a) : / G C(Y)}. For any extension r of £ by C(X), r(a) is
Hermitian. So there is a Hermitian operator A such that irA = r(a). Replace A
by a compact perturbation if necessary so that a(A) = ae(A) = a(r(a)). Define
a representation p of C(X) by the functional calculus of A:
p(f(a)) = f(A) for /€C(y).
It is clear that r(f(a)) = irf(A) = irp(f(a)). So r is a trivial extension.
In particular, if X is contained in R, then the coordinate function z(t) = t
generates C(X) by the Stone-Weierstrass Theorem. ■
Recall that a topological space X is totally disconnected if the topology is
generated by a family of clopen sets.
Corollary IX.3.2 IfC(X) is generated by its projections, then Ext(X) = {0}.
This occurs when X is totally disconnected.
Proof. From the proof of Corollary H.4.5, we saw that if C(X) is generated by
a family £ = {En} of commuting projections, then C(X) is generated by the
Hermitian operator A = J2n>i 3~n#n- Projections in C(X) are characteristic
functions of clopen sets. By the Stone-Weierstrass Theorem, they generate C(X)
if and only if they separate points. They separate points exactly when the clopen
sets generate the topology. ■
The case of the unit circle T is the problem of classifying unitary elements of
Q(H) with spectrum equal to the whole circle up to compalence. Non-trivial
extensions may be recognized by non-zero Fredholm index. In particular, the Toeplitz
extension determined by r (/) = irTf is generated by the unilateral shift Tz = r(z)
which has index -1. As unitary equivalence and compact perturbations preserve
Fredholm index, this extension is not trivial. Similarly, there is an extension
r'(/) = /(ttT;) = wTj

IX.4. Positive Maps
259
where f(z) = f(z) for / in C(T). This yields the same C*-algebra T(C(T)) since
C*(T;) = C*(TZ), but the maps onto C(T) are different. One can see that t' is a
different extension because the Fredholm index ind(r'(2)) = ind(T*) = +1.
Theorem IX.3.3 Ext(T) = Z is generated by the Toeplitz extension. The Fred-
holm index y([p]) = ind p(z) is an isomorphism from Ext(T) onto Z.
Proof. First note that 7 ([/>]) is well defined. For if p and p' are two representatives
of [/>], then there is a unitary operator U such that pf = Ad U p. Hence
indp'(z) = indUp(z)U* = indp(z).
This map is a homomorphism because
T(l>i] + fa]) = ind(pi(z) 0 p2{z))
= mdPl(z) + ind p2(z) = 7([/)1]) + 7(^2]).
Let p be any extension by C(T), and let W be chosen so that irW = p(z),
where z is the identity function on T. Let U be the partial isometry in the
polar decomposition of W. Then irCW^W)1^2 = 7rJ, so ttU = irW. Moreover,
n(U*U) = irl = 7r(C/Z7*), so I - U*U and I - UU* are finite rank projections
and U is Fredholm. Let k = ind U.
When k = 0, U has a finite rank perturbation [/' which is unitary obtained by
adding a finite rank partial isometry from (J - U*U)H onto (J - UU*)H. Thus
d(/) = /({/') provides a representation of C(T) such that ira = p. Hence p is
trivial. It follows that 7 is injective.
Let r denote the Toeplitz extension. Then
7([r]) = indr(z) = indTz = -1.
Likewise, the extension r' generated by the backward shift satisfies y[rf]) = 1.
Since ±1 generate Z as a semigroup, 7 is surjective and thus an isomorphism. ■
IX.4 Positive Maps
A positive linear map between C*-algebras is just a linear map which takes
positive operators to positive operators. Hence it is a map which preserves the
order structure. Let V\ (21,2J) denote the space of all positive unital maps from 21
into 05. If (p is a map between C*-algebras 21 and 93, it induces a map <p(n) from
Mn(%) into Mn(&) by
Say that (p is n-positive if <p(n) is positive and completely positive is it is n-positive
for all n > 1. These maps were introduced briefly in section H.5. In this section,
we will develop some of their basic properties.

260
IX. Brown-Douglas-Fillmore Theory
A positive map cp on a unital C*-algebra 21 is bounded by ||<p(l)|| on 2l,a.
Indeed every self-adjoint element of norm 1 is the difference of two positive
contractions, say A = Pi - P2. Since 0 < <p(P;) < <p(I) < \\<p{I)\\L it follows
that
-Ib(/)||/<V(A)<||V(/)||/.
So the norm is always bounded by 2||y>(/)|| on 21. In particular, (p is continuous. (In
the commutative case, these maps always have norm ||p(l)||; but the 2 is needed
in general.)
Proposition IX.4.1 Let X be a compact Hausdorff space. Every positive map (p
from C(X) into a C*-algebra 2* is completely positive.
Proof. There is a canonical isomorphism between Mn(C(X)) and C(X, Mn)> the
space of continuous .Mn-valued functions. An element F in C(X, Mn) is positive
if and only if F(£) > 0 for all f in X. First note that if F(£) = /(£)-"> where /
belongs to C(X) and T is a positive scalar matrix in Mn, then
<P{n)(F) = [<p(f)tij] = <p(f)T.
In particular, if / > 0, then (p(f) > 0 and so (pW(F) > 0.
Let F in C(X, M+) and e > 0 be given. By continuity, there is a finite open
cover Ou • • •, Ok of X such that ||F(f) - F(C)|| < e when f, C G £>;. Fix a point
& in Oi for each 1 < i < k and set T» = F(&)- Let p» be a partition of unity for
this open cover. That is, pi > 0, «(£) = 0 for f £ C\-, and £*=1 p; = 1. Then
lino - E»(c)iiii < IXouno - ^ii <«
because ||F(C) - Ti|| < e when#(C) > 0.
Thus F is uniformly approximated by a sum of the type considered in the
first paragraph. As (p^ is continuous, (p^ (F) is uniformly approximated by the
positive maps X^Li P& and thus <p(n) is positive for all n > 1. ■
The usefulness of positive maps on C(X) lies in their rigid structure described
in the following dilation theorem of Naimark. It says that every positive unital
map is the corner of a * -representation.
Theorem IX.4.2 Let (pbea positive unital map ofC(X) into B(H). Then there is
a ^-representation a ofC(X) on a Hilbert space K containing % such that
<p(A) = P<H<r(A)\H.
The proof follows immediately from the complete positivity of (p and the
following more general dilation theorem due to Stinespring. The proof is a souped
up version of the Gelfand-Naimark Theorem.

IX.4. Positive Maps 261
Theorem IX.4.3 Let (pbea unital completely positive map from a unital C*-alg-
ebra 21 into 3(71). Then there is a Hilbert space K containing % and a *-repres~
entation a of 01 on K such that (p(A) = Pu<r(A)\u.
Proof. Form the algebraic tensor product 21 ® H. Define a form on it by
n ra ra n
<E *f ® *i, E 5» ® *»> = E I>(*Mi)«i, *)•
i=i »=i t=i i=i
Evidently, this form is sesquilinear. Moreover, it is positive semidefinite because if
x =(*!,..., xn)\ then
n n n n
<E Aj <g> as,-, E Ai ® *•> = E EMA?^i)** **)
i=i »=i »=i i=i
= (^)([^Ai])x,x)>0.
The Cauchy-Schwarz inequality implies that |(t*, v)\ < (u} u)ll2{v, v)ll2 for
all u, v in 21 ® %. Thus the set
M = {veOi®H:(v,v) = 0}
= {v G 21 ® ft : (v, u) = 0 for all u G 21 ® ft}
is a subspace of 21 ® ft. Let /C be the Hilbert space completion of 21 ® ft/W in the
positive definite inner product induced by (•, •). Define a representation of 21 by
n n
t=l t=l
This is well defined because if v is in Af, then pv(X) = (Xv, v) is positive; whence
0 < (Xv,Xv) = pv{X*X) < \\X\\2Pv{I) = 0.
So a (01) ff is contained in Af.
Clearly a is linear and multiplicative. To see that it is self-adjoint, compute
n n
i=i »=i
n
= <(£ am,. ® *,•+jsn, C ** ® »+^>
i=i t=i
ra n
-EE(^M*iii)«i,i«)
= ([Y,Aj®xj + tf],[Y,ABi®yi + Ar])
i=i »=i

262 IX. Brown-Douglas-Fillmore Theory
n n
= <[£ Aj ® zj + Af], <r(A)[Y,Bi ®Vi + Af])
i=i »=i
= (<r(Ay[f^Aj<2>xj + tf],[JTBi®yi + /S]).
i=i t=i
So a(A*) = (j(A)*.
To imbed H into /C, define an operator V from H into /C by Vaj = [I® « + M].
This is isometric because
\\Vx\\2 = ([I®x+MUl®x+M]) = (V>(I)xJx)=\\x\\2.
Moreover,
(VV(A)V*,y) = <cr(A)[J® x+^],[J® y + ^]> = fo>(A)*fy).
Thus V aV* = <p. M
If (p is a positive unital map of a C*-algebra 21 into a quotient algebra 9$/3,
then say that y> has a positive lifting if there is a positive unital map tp of 21 into 2*
such that y> = irtp. We will show that every positive map on C(X) is liftable. We
begin with an elementary but somewhat tricky result of general interest.
Lemma IX.4.4 Let Z be an ideal in a unital C*-algebra 21, and let ir be the
quotient map. Let Abe a positive element of% and let y be an element of 01/3 such
that yy* < n(A). Then there is an element Y in 21 such that n{Y) = y and
YY* < A.
Proof. Let a = ?r(A), and let Y be an arbitrary lifting of y. Set
B = YY* + \A-YY*\.
Note that B > YY* + (A - YY*) = A, B > YY* and
7r(J5) = yy* + \a- yy*\ =yy* + a- yy* = a.
Define Yn = All2(B + £I)~ll2Y. We verify that Yn is a Cauchy sequence. Set
Dnm = (B+±irV2-(B + ±I)-V2.
Then for m < n
\\Yn - Ym\\2 = WA^D^YY'D^A^W < \\A^2DnmBDnmA^2\\
= {{B^D^A^W2 = WB^D^AD^B1^
< WB^D^BD^B^W = \\fnm(B)\\

IX.4. Positive Maps
263
where
/nm(a!)-a;2[(a;+I)-1/2_(a; + i)-l/2]2
(*+£)(*+£)(y^+\A+£)2
Hence Yn is Cauchy. Let Y^ be the limit. Since
YnY* = Al'2{B + ±I)-l'2YY*{B + ^I)-l'2A1'2
< Al'2{B + ±I)-l'2B{B + ^J)-1/2^1/2 < A,
it follows that Y^Y^ < A- But ^Q^) = a1/2(a + £l)~1/2y converges to y.
Therefore ^(1^) = y; and so Y^ is the desired lifting. ■
Corollary IX .4.5 Suppose that A is a positive element of a C*-algebra 21 with an
ideal 3; and that 0 < b < n(A) in $1/3. Then there is an element B in 21 such that
ir(B) = bandO<B < A.
Proof. Take y = 61/2, and apply Lemma IX.4.4. ■
We take the first small step towards lifting positive maps on C(X).
Corollary IX.4.6 Let 3 be an ideal in a unital C*-algebra 21; let ir be the quotient
map; and let X be a finite discrete set. Suppose that p is a positive unital linear
map ofC(X) into 21/3. Then there is a positive unital linear map <rofC(X) into
21 such that p = ira.
Proof. Let 8X. for 1 < i < n denote the minimal idempotents in C(X). Then the
positivity of a linear map (p on C(X) is equivalent to (p(SXi) > 0 for 1 < i < n.
Let a; = p(SXi). Since p is unital, we have ]C£=i a% = 1- ft suffices to construct
positive operators A{ in 21 such that ir(Ai) = a; and Yl?=i ^* = ^-
Proceed by induction. Use the previous corollary to find 0 < Ai < I so
that ir(Ai) = a\. Then since a2 < 1 - «i = n(I - Ai), a second use of the
corollary yields a positive element A2 < I - A\ such that 7r(A2) = a2. At the
♦fe-th stage, similarly choose a positive element Ak such that ir(Ak) = a*, and
Ak < I - YaZi a%- For the last term> set An = I - ^2?=i A{. Then the map
a(f) = ]C£=i f{xi)Ai is the desired lifting. ■
Next we show that the identity map on C(X) is the limit of (completely)
positive maps which factor through finite dimensional C*-algebras. A C*-algebra with
this property is nuclear (although this is a theorem, not the definition).

264
IX. Brown-Douglas-Fillmore Theory
Theorem IX.4.7 Let X be a compact metric space. Then the identity map id is
the point-wise limit of a sequence of maps akfik where (3k is a *-homomorphism of
C(X) onto C(Yfc), where Yk is a finite discrete space, and a*, is a positive unital
map ofC(Yk) into C(X).
Proof. Let Yk be a finite subset {yk% : 1 < i < rtk} of X such that the balls of
radius £ about the yk% cover X. Let (3k be the restriction homomorphism of C(X)
onto C(Yjk). To obtain a*., construct a partition of unity fk% subordinate to the cover
ofthe^-balls. Then set
«fo) = $]0(v«)A« for 9£C(Yk).
Clearly this is a unital positive linear map of C(Yk) into C(X).
If h belongs to C(-X") and e > 0, the uniform continuity of h implies that there
is an integer k so that dist(«, y) < £ implies \h(x) - h(y)\ < e. Hence
\h(x) - akfik(h)(x)\ = \h(x) -JTh(yki)fki(x)\
»=i
<JT\h(x)-h(yki)\fki(x)<e
t=l
since when fk%{x) > 0, dist(«, yk%) < £ and thus \h(x) - h(yk%)\ < e. It follows
that lirrifc^oo otk(3k{h) = h uniformly on X for every h in C(X). ■
The point-norm topology of pointwise convergence in 7*1(21,2J) is metrizable
when 21 is separable. Indeed, take any countable dense subset Ai, A2,... of the
unit ball of 21. Then define a metric on 7*1(21,55) by
dM) = Y^*-nMAn)-4>(An)\\.
n>l
It is clear that a net (pa in 7*1(21,25) converges in the d metric if and only if
<fa{Ak) converges for each k. The boundedness of the net and the density of the
sequence Ak implies that this is equivalent to pointwise convergence in the norm
topology.
We will be concerned with maps into more than one image C*-algebra.
However, if we fix our sequence Ak, there should be no serious confusion if we use d
to denote the distance function for all range C*-algebras 25.
Now consider whether it is possible to lift a convergent sequence of positive
maps into a quotient algebra to a convergent sequence "upstairs".
Lemma IX.4.8 Suppose that (p and tf> belong to 7*i(2l, 25) and 3 is an ideal of<B
with quotient map irof*B onto 25/X Then
d(7np, ttV>) = inf <%>,V0-
irrp '=7ry

IX.4. Positive Maps 265
Proof. It is clear that d(ir(p} nip) < d{(p, ip') whenever ip1 in V\{%2$) satisfies
nip' = nip. So consider the other direction. By Theorem 1.9.16, there is a quasi-
central approximate identity E\ for X Define
MA) = E]!\{A)E]I2 + (I - Exfl^{A){I - Exfl\
It is routine to verify that this is positive, unital, and that nipx = nip.
Notice that for B in 2J, the sequence
Bx = e]I2BE]I2 + (J - EX)1/2B{I - Ex)112
converges to B since ||JB - JBa|| is dominated by
\\E]I2B - BE]!2\\ \\E]!2\\ + ||(7 - Exfl2B - B{I - Ex)^2\\ \\(I - Exfl\
and by Exercise H.8 or II.9, \\E]j2B - BE]j2\\ converges uniformly to 0 as a
function of \\EXB - BEX\\. The same is true for \\(I - EX)1/2B - B(I - Ex)1/2\\.
So the right hand side converges to 0.
Hence if we set
<px{A) = El/2<p(A)El/2 + (I - Exfl2<p{A){I - E,)1'2,
we see that (px converges to (p in the point-norm topology. Thus
inf d((p,ipx) < Hminf d((px,ipx).
However,
d(<PX, 4>x) = £ 2-l(/ - Etfl^K) - ^(An))(7 - Exfl2\\.
n>l
From the proof of Theorem 1.5.3, it follows that for every B in 2J,
lim ||(/ - ^a)1/2^^ - £7a)1/2||
= lim \\(I - EX)1/2B - B(I - Ex)1/2\\ + \\B(I - Ex)\\ = \\*B\\.
Consequently, if e > 0, choose N so large that 2~N < e/4. Then choose Ao so
that for all A > A0 and 1 < n < Ny
\\(I-Ex)^2(AAn)-^(An))(I-Exn < \W<p{An)-iP{An))\\+e/2.
Then it follows that
d(<pxM < f;2-(||7r(¥>(An) - 4>(An))\\ +e/2) + £ 2-»(|M| + ||V>||)
n=l n>N
N
< £ --n|k(¥>(An) - iP(An))\\ + e/2 + 21-" < d{n<p, nip) + e
n=l
for all A > Aq. This establishes the non-trivial inequality. ■

266
IX. Brown-Douglas-Fillmore Theory
Corollary IX.4.9 The set of positive unital maps from 21 into V&/3 which have
unital positive liftings to 2* is closed in the point-norm topology.
Proof. Suppose that (pk in 7^(21,2J/3) for A: > 1 is a sequence of liftable
unital positive maps which converge pointwise to (p. Drop to a subsequence so that
ditfkjtfk+i) < 2~fc for all A: > 1. Choose any positive lifting fa in 7*1(21,2J) such
that 7r^i = <pi. Then use Lemma IX.4.8 to recursively choose tpk in 7*1(21,2J)
such that
inpk = <pk and d(ipk, ifo+i) < 2~\
Then tpk is Cauchy, and thus has a positive unital limit ip. Evidently, nip = (p. So
(f is also liftable. ■
Now we are ready for the main result on liftings.
Theorem IX.4.10 Let X be a compact metric space. Every positive unital map of
C(X) into a quotient *B/3 has a positive unital lifting.
Proof. Fix a map (p in 7>i(C(X), 2*/3). Consider the maps (pk = <po-kfa where a*.
and fa are constructed as in Theorem IX.4.7. They converge to (p pointwise.
Moreover (pak are positive unital maps from C(Yjk) into 2*/X So by Corollary IX.4.6,
there is a positive unital map tpk in 7^(0(1^), 2J) such that inpk = <Pafc- Since fa
is a unital positive map (in fact a homomorphism), it follows that ipkfa is a positive
unital lifting of <pfc. Hence by Corollary IX.4.9, (p is also liftable. ■
IX.5 Ext(X) is a Group
In this section, we establish the non-trivial fact that Ext(X) is a group. This is
now an easy consequence of the material developed about positive maps.
Theorem IX.5.1 Let X be a compact metric space, and let r be an extension of &
byC(X). Then there is another extension a such that r © a is trivial. Hence [a] is
an inverse for [r] in Ext(X). Therefore Ext(X) is a group.
Proof. Think of r as a *-monomorphism of C(X) into the Calkin algebra Q(H).
This map is, a fortiori, a positive unital map. Thus by Theorem IX.4.10, there is a
positive linear map 7 of C(X) into B(H) such that r = ttt. Then by Naimark's
Dilation Theorem IX.4.2, there is a ^representation p of C(X) on a Hilbert space
K containing % such that, with respect to the decomposition K = H(b HL, p has
the form
/»(/)=
r(f) PuM
[P2l(f) P22(f)\ '
Next we show that pu{f) and pz\(f) are compact. Indeed, since
P(\f\2) = pV)pQ)

IX.5. Ext(X) is a Group
267
and r is a *-homomorphism, the 1,1 entry modulo compacts yields
<\f?) = t(/)t(/) + n(p12(f)p21(J)) = r(|/|2) + n(pl2(f)pl2(f)*).
Thus npi2(f) = 0, whence puif) is compact. Likewise, P2i{f) = P12UY is
compact. It follows that <x'(/) := 7r/>22(/) is a *-homomorphism. Indeed,
*'(fg) = *P22(fg) = n(p22{f)p22{g) + P2i{f)pi2{g)) = <rf{f)<Tf{g).
Note that it now follows that the trivial extension irp decomposes as r © a'.
However, there is no reason that a' need be a *-monomorphism. To ensure that
it is injective, we add on a trivial extension 7. So define a(f) = <r'(f) © t(/).
Since 7 is injective, this is a monomorphism into the Calkin algebra. Moreover,
r © <r = 7T/>©7.
This is a trivial extension. Therefore [r] + [a] = 0. It follows that the semigroup
Ext(X) has inverses, and thus is a group. ■
If / : X —> Y is a continuous function between compact metric spaces, we
may define a map /* from Ext(X) into Ext(y) as follows. For a monomorphism
r of C(X) into Q(%), it is natural to consider r (gof) forg in C(y). This is easily
seen to be a *-homomorphism on C(y), but in general it is not injective. This is
easily remedied by adding on a trivial extension. So we set
f*(T)(g) = r(gof)(B<r(g) for all 9eC(Y)
where a is any trivial extension by C(y). If r and r' are equivalent extensions, then
there is a unitary operator U such that r' = Ad itUt. Likewise, if <rf is another
trivial extension by C(y), then by Theorem IX.2.1 there is a unitary operator V so
that a' = Ad itV a. Hence
Adir{U®V){f*r){g) = Adir(U® V)(r{g o /) © a(g))
= (AdirUr)(g o f) ® AdirVa(g)
= Tl{gof)®cr\g) = U{Tl){g).
So we have a well defined operation /*([r]) = [/*(r)] on Ext(X).
Corollary IX.5.2 Ext is a covariant functor from the category of compact metric
spaces to the category ofabelian groups.
Proof. We have seen from Theorem IX.5 A that Ext(X) is always an abelian group.
Let / be a continuous map from X into Y (a morphism in the category of
compact metric spaces). To see that /* is a group homomorphism (a morphism in the
category of abelian groups), notice that
(AW + A[r'])(<7) = r(g o f) © a(g) © r\g o f) © v'(g)
= (r © r')(<7 o /) © (a © a'){g) = /*([r] + [r>])(g).
Hence /*([r] + M) = /*[r] + fjf\.

268
IX. Brown-Douglas-Fillmore Theory
It is routine to verify that if / is a continuous map from X to Y and g is a
continuous map from Y to Z, then (gf)* = 5*/*. And if id* is the identity map
on X, then idx* is the identity homomorphism on Ext(X). ■
IX.6 First Topological Properties
In this section, we establish some important topological properties of Ext.
These properties are the beginning of showing that Ext is a homology theory.
Theorem IX.6.1 Suppose that q is a continuous surjection of X onto Y. Let B
be a closed subset ofY containing all points with multiple preimages in X, and
let A = q~1(B). Let j denote the injection of A into X, and let q1 denote the
restriction ofq to A.
q
X—^Y
A A
j\ \i
I q' I
A—-*B
Then ker g* is contained in j* ker <j£.
Proof. Suppose that [r] in Ext(X) satisfies q*[r] = 0- Because q is surjective,
we may define q*r(g) = r(g o q) without adding a trivial representation as this is
already injective on C(y). So without loss of generality, g*r is trivial. Thus there
is a homomorphism a of C(y) into B(H) such that ire = g*r. By Corollary H.4.5,
we may suppose that a is a diagonal representation. Let
A = {/ e C(X) : f\A is constant}.
For each / in A, there is a unique function g in C(y) such that / = g o q. Thus
we can define a diagonal representation of A by a(f) = <r(g). This satisfies
T(f) = T(9 °q) = nv{9) = *■?(/) for / € A
Let £> be the diagonal algebra containing the range of a. There are points £ * in
X such that a(f) = diag(/(f*.)) for all / in A. Thus the representation a extends
to a representation a of the bounded Borel functions on X which are constant on
A into the diagonal algebra 1) given by a(h) = diag(fe(^)).
Let
In = {^el: dist(£, A) > 2"n} for each n > 1,
and set Pn = <r(Xn) where Xn is the characteristic function of the set Xn.
Now r(/) belongs to 7rD when / in A So r(/) commutes with irPn for each
n > 1. We will show that r(/) commutes with irPn for all / in C(X). To this end,
define functions
Jfe(*) = 2(* A1V|)-1, *€R,

IX.6. First Topological Properties
269
and
Pn(0 = M-ndist(e,A)), £ex.
Then pn belongs to C(X), equals 1 on Xn, and is 0 on X \ Xn+i. Notice that
irPn = ira(pnXn) = r(pn)irPn
and
r{Pn) = ™{PnXn+i) = r(pn)irPn+1.
Split / in C(X) as / = fpn+1 + /(l - Pn¥1). Then
r(/)7rPn = r(fpn+1)irPn + r(/(l - pn+i))r(pn)7rPn
= *PnT{fPn+l) + T(/(l - Pn+l)Pn)nPn = ^Pnr{fpn+l)
= 7rPnr(/pn+i) + *PnT{pn)T{f{l - Pn+i)) = *rPnr(/).
Also note that we have shown that 7rPnr(/) = 7rPnr(/pn+i).
Let {/»} be a dense subset of the unit ball of C(X). Choose operators T; so
that ttT; = r(/j). We have 7r(TtPn - PnTi) = 0, so T;Pn - PnT{ is compact for
all i and n. Let #i = Pi and En = Pn - Pn_i for n > 2. Thus T;J57n - #nT;
and (from the previous paragraph) En(Ti - a(fipn+i))En are compact. Therefore
there are diagonal projections E'n in £> of finite codimension in En so that
\\TiE'n-E'nTi\\<2-n and ||<(T. - <t(Aah-i)XII < 2"n
for 1 < i < n. Let P' = X)n>i ^ T^11
TiP'-P'Ti = J2TiK-KTi
n>l
and
P'TiP'-^£;>(/l.pn+1)£?;
n>l
= ]T ^(2* - cr(/iPn+1))K + P' YjP&n ~ KTi)K
n>l n>l
are norm convergent sums of compact operators, and thus are compact. We deduce
that 7rP' commutes with the range of r, and irP'r(f) lies in irP"D.
Define extensions r'(/) = 7rPV(/)|p/^ and r0(/) = nPfLT(f)\pij_<H. It
follows that r' = 7T(7', where a'(f) = ^n>i #n<T(/Pn+i)#n- Thus r' is trivial.
We next show that to depends only on /l^. Indeed, let / in C(X) be such that
f\A = 0, and let e > 0. Then there is an integer n so that ||/|x\x„ll < e- Since
r(/pn) = 7rPn+ir(/pn) = irPfPn+1r{fpn)}
it follows that T0(fpn) = 0. Thus
IM/)|| = ||r0(/(l ~ pn))\\ < 11/(1 - Pn)\\< e.
Since e > 0 is arbitrary, it follows that r0(/) = 0.

270
IX. Brown-Douglas-Fillmore Theory
Hence we may define an element [p] in Ext(A) into B{PfL% © %) by
P(f) = ro(/)0M/)
where / is any continuous extension of / in C(A) to a function in C(X) and p, is a
trivial extension of C(A). This definition is well defined because to depends only
on f\A = /. Moreover, it is evident that if p! is a trivial extension in Ext(X), then
j*p = to © (p © p!) is equivalent to r0®rf = r. ■
The following important corollary providing a short exact sequence for Ext
will be used extensively.
Corollary IX.6.2 Suppose that A is a closed subset of a compact metric space X.
Let j be the canonical injection of A into X, and let p be the quotient map ofX
onto X/A. Then
Ext(A) ^ Ext(X) --^ Ext(X/A)
is exact.
Proof. In Theorem IX.6.1, take B to be the point {A/A} in X/A. Since pj is
constant, it is evident that p*j* = 0. Likewise, pf = p\a is constant, so that
kerp* = Ext (A). Therefore Theorem IX.6.1 shows that kerp* is contained in
jf* Ext (A). Thus kerp* = jf* Ext (A). ■
If Xn is a sequence of compact metric spaces andpn is a sequence of
continuous maps from Xn+i to Xn for n > 1, then the projective limit X = proj limXn
is defined as the subset of ELm -^n consisting of those sequences (xn) such that
Pn(«n+i) = «n for all 7i > 1. There are natural maps qn of X into each Xn via
the coordinate maps, and they satisfy pn<Zn+i = 9n- This space satisfies the
universal property that whenever Y is a compact metric space and fn is a sequence
of continuous maps from Y into Xn such that pn/n+i = fn for all n > 1, there
is a unique map fofY into X such that qnf = /n for all n. Likewise, we may
define the projective limit of groups. In particular, {Ext(Xn),pn*} determines a
sequence of groups and group homomorphisms so that proj limExt(Xn) is
defined. There is a sequence of homomorphisms gn* of Ext(X) into Ext(Xn). Thus
there is an induced homomorphism k of Ext(X) onto proj limExt(Xn) such that
qn*K = Pn*- The following result will be helpful.
Theorem IX.6.3 Suppose that X = proj limXn is the projective limit of a family
{Xn}pn}. Then the induced map
k : Ext(X) -> proj limExt(Xn)
is surjective.
Proof. First suppose that each pn is surjective. Let ([rn]n>i) be an element of
the group proj limExt(Xn). Choose rn be monomorphisms of C(Xn) into Q(H)
such that pn*Tn+i = Tn. This is accomplished recursively. Once rn is defined,

IX.6. First Topological Properties
271
choose t£+1 representing [rn+i]. Thenpn*r^+1 is equivalent to rn. So there is a
unitary operator Un so that AdirUnPn*Tn+1 = rn. Thus rn+i = AdirUnTn+1
will suffice.
Define r on the union of the subalgebras >U = {foqn : / e C(Xn)}ofC(X)
by r(f o qn) = rn(/). This is well defined because if fnoqn = /m o qm for fn in
C(Xn) and fm in C(Xm) and m < n, theng = fm°Pm° -"°Pn-i belongs to
C(Xn). Since g o qn = /m o qm = fn o qny it follows that fn and g agree on the
range of qn> which is all of Xn by the surjectivity of each pn. Thus
Tn{fn) = Tn(/mOPm°,,,0Pn-l) = Pm* • • - Pn-l*Tn{frn) = Tm(/m).
As Un>iAi is dense in C(X), the definition of r extends to C(X) by continuity.
Clearly r is a homomorphism since it is a homomorphism on a dense subalge-
bra. Moreover, on each Any the map r is injective and hence isometric. Therefore
r is isometric and thus injective on the closed union of the An% C(X). Finally, it
is readily apparent by construction that gn*r = rn; so /c([r]) = ([rn]n>i).
Now consider the general case. It is easy to construct a continuous surjection
from the Cantor set onto any compact metric space. Thus if Yn = Xn V Cn is the
disjoint union of Xn and a Cantor set Cn, one may define a surjection pfn of yn+i
onto Yn such that j4U„+1 = Pn- Let Y = proj liml^, and let qn be the induced
maps of Y ontoyn.
For each n, let jn be the injection of Xn into Yn. Because the jn commute with
them's and gn's, there is a map j = proj limjfn injecting X into Y. In particular,
qn = qnj for n > 1. Also
Yn/Xn = CnV{Xn/Xn) = C£,
the disjoint union of Cn and a point, which is totally disconnected. Because
p'n(Xn+i) is contained in Xny there are induced maps pn of yn+i/Xn+i onto
Yn/Xn. Therefore
y/X = projlimyn/Xn.
This is homeomorphic to a subspace of IIn>i ^n»^d thus is totally disconnected.
Corollary IX.3.2 shows that Ext(yn/Xn) = 0 because Yn/Xn is totally
disconnected. Thus applying Corollary IX.6.2 to the injection jn yields an
isomorphism jjn* of Ext(Xn) onto Ext(l^). Similarly, jf* is an isomorphism of Ext(X)
onto Ext(y). Therefore j = proj lim jjn* is an isomorphism of proj lim Ext (Xn)
onto proj lim Ext(yn). Thus we have the commutative diagram
Ext(X) -^ proj limExt(Xn)
Ext(y) -^*- proj limExt(yn)
Since «y is surjective, it follows that «y = j nyj* is also surjective. ■

272
IX. Brown-Douglas-Fillmore Theory
One key tool for computing the Ext groups is the Fredholm index map. If irT
is an invertible element of the Calkin algebra, then one can compute the Fredholm
index ind(7rT) = ind(T). This determines the connected component of Q{%)~1
in which irT lies. Every normal operator N has index zero, and thus 7riV lies in
the connected component of the identity, Q{H)q *. Indeed, ind(iV - A J) = 0 for
every A £■ ae(N).
Suppose that X is a compact metric space and / is invertible in C(X). Then for
[r] in Ext(X), one may compute ind(r(/)). This is easily seen to be independent
of the choice of representative r. Indeed, this follows because index is preserved
under unitary equivalence; and if a is a trivial extension, then md(a(f)) = 0 for
every invertible /. As index is a homomorphism, it follows that adding a trivial
extension does not change the index. It is also important that index is a homotopy
invariant due to the fact that it is a continuous integer valued function. This means
that ind(r(/)) = ind(r(<7)) whenever / and g lie in the same connected
component of C(X)-1. Thus ind r(/) depends only on the equivalence class [r] and the
homotopy class [/].
The group C(X)~1/C(X)q x of homotopy classes of invertible functions is
denoted by 7r1(X). Thus there is a map 7 from Ext(X) into Eom(ir1(X)JZ)
defined by
T[r]([/]) = indr(/).
It has been shown that this map is well defined. To verify that y[r] is a
homomorphism, compute:
ind r(fg) = ind (r(f)r (g)) = indr(/) + indr^).
Finally, we must verify that 7 is a homomorphism. This follows from
T([n] + [r2])([/]) = ind(r1(/)er2(/))
= indntf) + indr2(/) = TN([/]) + tN([/]).
When necessary, we will write yx to indicate that 7 is acting on Ext(X).
A more sophisticated variant makes use of matrix algebras over C(-X") and
introduces a connection with K-theory. If r is a *-monomorphism of C(X) into
Q(H), then r<n) is a monomorphism of Mn{C{X)) into Mn{Q{U)) defined
by taking annxn matrix [fa] to the matrix [r(/y)]. Since we may identify
Mn(Q(H)) with B{UW)/&{UW) which is the Calkin algebra, this defines an
extension of & by Mn(C(X)). As above, we may evaluate ind r(n) ([fij]) for every
invertible element of Mn(C(X)). This suggests looking at the group Ki(C(X))>
which in the commutative case is the group K1^). There is a natural imbedding
in of GLn(X) = MniC(X))-1 into GLn+i(X) by sending F to F 0 [1]. This
takes the connected component of the identity GLn(X)o into GLn+i(X)o, and
thus induces a map from GLn(X)/ GLn(X)0 into GLn+i(X)/ GLn+i(X)0. The

IX.7. Ext for Planar Sets
273
group KX(X) is the direct limit
K\X) = lirriGLn(X)/GLn(X)o.
There is an index map 7^ from Ext(X) into Hom(GLn(X)/GLn(X)0, Z)
given by
7^M([[/ii]])=indrW([/ti]).
This is well defined on equivalence classes, and is a homomorphism by an
argument identical to the n = 1 case. It is evident that
indr(n+1)(F 0 [1]) = indr(n)(F) + indTrJ = indr^F).
These index maps are compatible in the sense that 7£+1*n = 7x • Therefore there is
a direct limit homomorphism 7^ = liru7x ^rom Ext(-X") into Eom(K1(X)} Z).
These constructions are functorial. That is, suppose that h is a continuous map
of X into Y. Then this induces the endomorphism an of C(Y) into C(X) by
a(f) = f oh. Clearly, this takes invertibles to invertibles and maps the connected
component of the identity C(Y)q x into C(X)q *. Hence it induces a map h* of
7r1(lr) into tt1(X). This in turn induces a homomorphism fe# of Hom(7r1(X), Z)
intoHom(7r1(y),Z)by
h#(eW]) = e(h*[f]) = W°h]).
This construction is compatible with 7 in the sense that
7r(MTD = Vrx(M).
For if [/] is in T1(y), [r] is an element of Ext(X) and a is any trivial element of
Ext(y), then
lY{K[r]){\f\) = indr(foh) 0 «r(/) = indr(/ o h)
= 7x([r])([/ofc]) = M7x[r])([/l).
We will show that 7 is injective when X is homeomorphic to a planar set.
This will complete the analysis in that case. However, in general, this map has
non-trivial kernel. We will see some examples later.
IX.7 Ext for Planar Sets
In this section, we will completely analyze Ext(X) for X a subset of the plane.
The key is to show that the index map 7 is an isomorphism for subsets of the plane.
This yields a simple, computable set of invariants in this case. For planar sets,
C(X) is generated by the identity function z. The group 7r1(X) has a convenient
description in this case.
Theorem IX.7.1 When X is a subset of the complex plane Q the group tt1(X)
is the free abelian group generated by {[z - Xn]} where A = {An} is a set with
exactly one point Xn in each bounded component ofC \ X.

274
IX. Brown-Douglas-Fillmore Theory
Proof. Suppose that X is bounded by finitely many disjoint piecewise smooth
Jordan curves Ti,..., Tn. Given an invertible function / in GL(X), one can compute
the winding number indr,(/) of / about rt* for 1 < i < n. We assume that the
reader is familiar with this concept. There is a continuous logarithm for / if and
only if / has winding number 0 about each bounding curve. In this case, one
may express / as an exponential eg\ and thus / is in the connected component
of the identity. Since winding number is a homotopy invariant in GL(X), this
provides a necessary and sufficient condition for [/] = 0. The map i from [/] to
(indrx(/),♦.., indrn(/)) in Zn injective. So irx(X) is a finitely generated free
group. (In fact, deleting those curves forming the exterior boundary of each
component will yield an isomorphism.)
Pick a point fij in each bounded component 0\,..., Om of C \ X. We will
show that every function in GL(X) may be factored as a finite product
m
/ = II(* " H?' ^
i=i
for a unique choice of integers plf... ,pm and a continuous function g in C(X).
Hence [z - fij] for 1 < j < m freely generate ^(X). This can be established
inductively on the number of bounding curves.
For n = 1, the interior of the curve Ti either lies in X or in Xc. In the first
case, X is simply connected, there are no bounded components of the complement
and 7T1 (X) = 0. If the interior O lies in Xc, then there is one bounded component
of the complement and indrfc - fi] = -1, which freely generates Z = ^(X).
Assume the result is established for n and consider X with n + 1 boundary
curves. At least one of these curves, say Ti, does not contain any other 1^ in
its interior component O. When O is contained in XyY = OuTi is a simply
connected component of X. Thus nl{X) = irx(X \ Y), and X \ Y has one
fewer boundary component. So by the induction hypothesis, given / in GL(X),
the restriction f\x\y factors uniquely as IIjLiC* ~ Pj)Pj ed- ^s evei7 function
in GL(y) is an exponential, the definition of g can be extended to Y to obtain
equality on X. The uniqueness of the pj's follows from the uniqueness on X \ Y.
Otherwise, O = 0\ is one of the components of C \ X. So ind^ [z - fix] = 1
and indrj* - fij] = 0 for j > 1. Given / in GL(X), letpi = indrj/]. Then
the function h = (z - pi)~Plf has winding number 0 around Ti. Hence h may
be extended to an invertible continuous function on Y = X U 0\. Now Y is
bounded by T2,..., rn+i and the components of C \ Y are just C?2, • • •, Om. By
the induction hypothesis, there are unique integers P2, • • •, Pm so that h factors as
n£=2(z ~ H)Pi eff- Hence / factors as rijLi(2 " N)Pi e9- Now
m
3=1

IX.7. Ext for Planar Sets
275
This shows that the integer pi = ind^ [/] is uniquely determined as the exponent
ofz-Xi in the product for /. The uniqueness of the others now follows from the
induction hypothesis.
It is easy to show that every planar set X is the intersection of a
decreasing sequence of sets Xn which are finitely connected and have piecewise smooth
boundary. For example, one may cover X by a finite union of disks of radius 2~n
with centres in X. Given / in GL(X), let / be any extension of / to a
continuous function on a large disk containing X. By the uniform continuity of /, it
will be invertible on Xn for some sufficiently large integer n. Thus / factors as
n^LiC* "" w)k' ed- Let Xj be the chosen point in the same component Oj of
C \ X as hj. Then IIjLiO2 ~ H)hj(z ~~ *i)~*J' has winding number 0 around
every component of C \ X and thus is an exponential eh. So
m
f = H(z-\jpeS+h.
This shows that [z - Xj] generate irx(X).
To see that [z - Xj] are free generators, let us suppose that there is a relation
YXLi Pj[z - Xj] = 0. That is, there is a continuous function g in C(X) so that
ra
l[(z-Xj)*=e°.
i=i
As above, there is an integer n sufficiently large so that this relation extends to Xn.
But then the freeness of the generators for it1 (Xn) shows that pj = 0 for all j. ■
The main theorem for Ext of subsets of the plane can now be stated.
Theorem IX.7.2 If X is a compact subset of the plane, then y is an isomorphism
of Ext(X) onto Hom(7r1(X), Z).
Before proving it, we note two important operator theoretic corollaries that
classify essentially normal operators.
Corollary IX.7.3 Two essentially normal operators T\ and Tz are compalent if
andonlyif<Te{Tx) = ae{T2) andmd^-XI) = 'md(T2-XI)forallX $ <re{Ti).
Proof. The only if direction is easy, so we prove the other direction. Since Ti
and Ti have the same essential spectrum X, they each determine an extension
7i(/) = f(irTi) of C{X) fori = 1,2. Then
7M([* " A/]) = ind(r! - XI) = ind(T2 - XI) = y[r2]([z - XI]).
By Theorem EX.7.1, this implies that 7(71] = y[r2]. So by Theorem IX.7.2, rx
and r2 are equivalent. Thus there is a compact operator K and unitary U so that
T2 = UTXU* + K as desired. ■

276 IX. Brown-Douglas-Fillmore Theory
Since ind(iV - XI) = 0 for every normal operator and A £ ae(T)9 the
following is an immediate consequence of the first corollary.
Corollary IX.7.4 An operator T has the form "normal plus compact" if and only
ifT*T - TT* is compact and ind(T - XI) = 0 for every X $ ae(T).
Another consequence that surprisingly has yet to be given a simple direct proof
is the following. Perhaps the reason it is difficult is that the corresponding result
for pairs of operators is false (see the examples at the end of the chapter).
Corollary IX.7.5 The set of all operators of the form "normal plus compact" is a
norm closed set.
Proof. Suppose that Tk are "normal plus compact" and converge to an operator T.
Then
t*t - tt* = Hm r*rn - rnr*
n—foo
is compact. Moreover, since the Fredholm index in continuous and the set of Fred-
holm operators is open, it follows that if A £■ <re(T), then A £ <re(Tn) for large n
and
ind(T - XI) = lim ind(Tn - XI) = 0.
So by the previous corollary, T is also normal plus compact. ■
In preparation for our proof, we need a few lemmas. The first is a useful partial
result.
Lemma IX.7.6 Suppose that A is a compact subset of a closed interval J. Then
the map 7 on Ext(J/A) is injective.
Proof. It is evident that J/A is homeomorphic to the union X of countably many
disjoint smooth curves Ck with disjoint interiors which meet at a common point
fo and have diameters decreasing to 0. Each curve is homeomorphic to a circle
or line segment. Let Xn = U£=1Cfc and Yn = Ufc>nCfc. Let in and jn denote
the injections of Xn and Yn into X. Let pn denote the retraction of X onto Xn
obtained by sending Yn to f0; and similarly, let qn denote the retraction of X onto
Yn obtained by sending Xn to fo-
in ^ jn
Xn ^ X = Xn U Yn >■ Yn
Pn Qn
Fix r in ker7. Note that id* = (inPn)* + {JnQn)*- In particular, consider
n = 1. Thenpi*[r] belongs to Ext(Ci) and
7Pi*M=Pi#7M = 0.
By Theorems IX.3.3 and IX.3.1, Ext(Ci) = Z or 0 depending on whether C\ is
homeomorphic to a circle or line segment; and 7^ is an isomorphism in either

IX.7. Ext for Planar Sets 277
case. Sopi*[r] = 0. Hence
[r] = iupu[r] + ji*gi*M = Ji*[ri]
where [ti] = qu[r] belongs to Ext(Yi). Therefore there is a faithful representation
(7i ofC(X) such that
r(/) ~ 7T(7i(/) 0 (jigi)*r(/).
Thus there is a projection E\ so that <j\ acts on B(EiH), and irEi commutes with
the range of r.
Repeated use of this argument produces pairwise orthogonal projections Ek
and ^representations ak of C(X) on B(Ek'H) so that
r(/) = ]T <&n<Tk(f)Ek 0 (Jn?n),r(/)(^^)J-
fc=l fc=l
on U = ££=i 0£?^ 0 (££=i tf*)1^ Let Ax = {/ o in : / € C(Xn)}. Then
it follows that for / in Any
n n
r(f) = J2®ir<Tk(f)Ek 0 fm(Z,Ek)L.
fc=i fc=i
Define
a(f) = YJe<Tfc(/)£?fc e f(£0)(£Ek)\
k>l k>l
Then r(/) = ira(f) for all / in Un>i Ai> which is dense in C(X). Therefore r is
trivial. ■
The key step involves cutting the spectrum.
Lemma IX.7.7 Suppose that X is a compact subset ofQ and that [r] in Ext(X)
satisfies 7[r] = 0. Let L be a line splitting C into two half planes H+ and H~.
Define X* = X C\ H±, and let i± be the injections ofX^ into X. Then there are
elements [p*] in Ext(X±) such that [r] = i+[/>+] + K[p~] andylp*] = 0.
Proof. Let J be a closed line segment of L sufficiently long to contain the
orthogonal projection nx(X) of X onto L. Define injections j#± of X± U J into X U J and
j of X into X U J, and injections ft* of X* into X* U J. Also define a retraction
of XL) JontoX+U J by
{:
+ ( \- jz for z€X+UJ
T ^Z'~KL(z) for zeX-UJ

278
IX. Brown-Douglas-Fillmore Theory
The following diagram may be helpful.
»+
+ X +
x-
k~
x+uJ^=;iuJ^rruJ
First we show that [j*r] can be decomposed as j*[t] = Jt[T+] + J* [r ]
for [r*] in Ext(X± U J) satisfying ^[r^] = 0. Indeed, r+ = r+j*[r] lies in
Ext(X+ U J). Let [/>] = jm[r] - j+[r+]. Then
rt[p] = r+j.M - r+j+[r+] = (r+jJ.M - (r+j+r+j).M = 0
because r+j+r+ = r+. Consider the quotient q of X U J onto
(X U J)/(X~ U J) = (X+ U J)I J.
Since g factors as g'r+, it follows that q*[p] = 0. So by Corollary K.6.2 for the
sequence *
ruJ-^iuf^
{x u j)/(x- u j)
it follows that there is an element [r ] in Ext(X U J) such that [p] = j*r .
Clearly, ind r+ (2 - A) = 0 for all A in if ~ and ind r ~ (z - A) = 0 for all A in
ff+.SoforAGff+\XUL,
indr+(2 - A) = ind j+t+(z - A)
= indr(z — A) — indj~r~(z — A) = 0.
Hence 7(7+] = 0; and similarly, j[t~] = 0.
Now consider the sequence
X
+
^I+UJ-^
(i+uJ)/i+ = J/(inf).
Since yq+[r+] = q#y[r+] = 0, Lemma K.7.6 shows that g+[r+] = 0. Hence by
the short exact sequence of Corollary K.6.2, there is an element [p+] in Ext(X+)
such that [r+] = k+[p+]. An easy calculation shows that y[p+] = 0 as well.
Similarly, there is an element [p~] in Ext(X~) such that k~\p~] = [r~] and
7[p~] = 0. So
j*(it[p+]+K[p-]) = (i+*+)*[/>+] + (rn*[/>-]

IX.7. Ext for Planar Sets
279
To complete the proof, it must be shown that j* is injective. Consider
xnj-^+x—^x/ixnj)
i
—7-— XUJ —^ {X U J) IJ
By Theorem IX.3.1, Ext(X n J) = 0 = Ext(J). By Corollary K.6.2, it follows
that p* and g* are injective. Since g* = p*j*, it follows that j* is also injective.
Therefore t+ [/>+] + i" [/>"] = [r]. ■
Following the notation of the previous lemma, let Y = X+ V X~ denote the
disjoint union of X+ and X ~; and let p denote the natural surjection of Y onto X.
In a natural way, [p+ ©/>"] becomes an element [p] in Ext(y) such that p* [/>] = [r]
and7[/>] = 0.
We are now prepared to complete the proof of the main result.
Theorem IX.7.8 For a compact subset X of the plane, the map yx is injective.
Proof. Suppose that [r] in Ext(X) satisfies y[r] = 0. Think of X as contained
in a large square S. Cover the square with a countable number of horizontal and
vertical lines Lk which chop S into a grid of smaller rectangles whose diameters
decrease to 0 as A: tends to infinity. Repeatedly apply the previous lemma to cut the
spectrum along Lk for each k. At the A:-th stage, let Xk be the disjoint union of the
intersection of X with each closed rectangle in the &-line grid, and let pk be the
canonical surjection of Xk onto Xk-i (with Xo = X) and let r*. be the surjection
of Xk onto X. By the formulation preceding this theorem, there are elements [pk]
in Ext(Xfc) such that y[pk] = 0 andPk*[pk] = \pk-i], whence rk*\pk] = [r].
Let Y denote the projective limit of the sequence {X^Pk} and let qk denote
the maps from Y onto Xk. In particular, go maps Y onto X. Since the diameters
of the grids decrease to 0, the space Y is totally disconnected. Thus Ext(y) = 0
by Theorem IX.3.2. By Theorem IX.6.3, there is an element [p] in Ext(y) such
that [pk] = qk*[p] for all k > 0. Therefore [r] = g0*|>] = go*0 = 0. ■
It remains to prove surjectivity. In the case of nice spectrum, we use Toeplitz
operators to write down explicit generators. Actually, explicit generators can be
given for arbitrary subsets of the plane. However, the proofs are considerably more
difficult.
Lemma IX.7.9 If X is a compact subset of the plane bounded by finitely many
disjointpiecewise smooth Jordan curves, thenyx is surjective.
Proof. Let [z - Xk] for 1 < k < m be a set of free generators for tt1(X)
corresponding to the bounded components Ok of C \ X. Thus Hom(7r1(X), Z) is
generated by homomorphisms hj>l < j < m such that hj[z - Xk] = Sjk- Hence
it suffices to construct extensions tj so that ind tj(z — Xk) = ±&/fc.

280 IX. Brown-Douglas-Fillmore Theory
The boundary of Oj is given by T0 - X)Li ^*» which is a sum of oriented
Jordan curves from the boundary of X. It is well known that
mdd0j{z - A) = indrj* - A) - ^mdTi{z - A) = <
o zec\oj
Let 50 be a homeomorphism of the unit circle T onto To with positive
orientation, and for 1 < i < p, let gi be homeomorphisms of T onto I\ with reversed
orientation. Let
p
where Tgi are Toeplitz operators and N is normal with <r(N) = ae(N) = X. By
Theorem V.1.6, each Tgi is essentially normal with cre{Tgi) = I\- and
ind(Tgi - XI) = - indT(gi - A) = - mdgi{T)(z - A).
Hence
p
ind(Tj - XkI) = - ^2 indgi{T)(z ~ A*0 = - ind^C* - Xk) = -Sjk.
As Tj is essentially normal with cre{Tj) = X, we may define an extension tj by
Tj(f) = f{nTj). By the calculation above, 7(7^) = -hj as desired. ■
The proof of Theorem IX.7.2 can now be completed by establishing
subjectivity by a projective limit argument.
Theorem IX.7.10 IfX is a compact subset of the plane, then yx is surjective.
Proof. It is easy to write X an the intersection of a decreasing sequence of closed
subsets Xn which are finitely connected and are bounded by a finite number of
disjoint piecewise smooth Jordan curves. Letpn be the injection of Xn+i into Xn,
and qn the injection of X into Xn. Then X = proj limXn.
Suppose that h belongs to Hom(7r1(X),Z). Then hn = qn#h defines an
element of Hom(7r1(Xn), Z) for each n and pn#^n+i = hn. By Lemma IX.7.9,
there is an extension rn in Ext(Xn) such that yxn [fn] = hn. Also
7X„([rn] - Pn*[rn+i]) = ixn[rn] -Pn#7x„+i [rn+i]
= hn -pn#hn+i = 0.
Since yxn is injective by Theorem IX.7.8, this shows that [rn] = pn*[rn+i]- Thus
the sequence {[rk]) determines an element of proj limExt(-X'n).
By Theorem EX.6.3, the map k of Ext(X) onto proj limExt(Xn) is surjective.
So we may find [r] in Ext(X) such that qn*[r] = [rn] for all n. Finally if / is in

IX.8. Quasidiagonality
281
GL(X), extend / to a continuous function F on a large disk containing X\. Then
fn = F\xn will belong to GL(Xn) some n sufficiently large. So
lx[r]{f) = 7x[r]{fn o qn) = <Zn#7xM(/n)
= 7*n<Zn*M(/n) = M/») = fa#M/») = *(/)•
Hence 7x is surjective. ■
IX.8 Quasidiagonality
An operator T is quasidiagonal if there is an increasing sequence Pn of finite
rank projections tending SOT to the identity such that lim^oo \\PnT - TPn\\ = 0.
In this case, it is possible to drop to a subsequence so that ||PnT - TPn\\ < 2~ne
for a given e > 0. Then the projections En = Pn - Pn_! form a partition of the
identity into finite rank projections such that
T = J2EnTEn + K where K = ]TEnTP^ + P^TEn
n>l n>l
is compact with norm
\\K\\<J2\\EnTPt + P£TEn\\
n>l
< J2max{||(PnT - TPn)P£\\, \\P^(PnT - TPn)\\)
n>l
<J]2-ne = e.
n>l
Thus T is a block diagonal operator plus a compact operator. The converse is
readily apparent.
Similarly, a subset S of B(H) is (jointly) quasidiagonal if there is a sequence
{Pn} of finite rank projections so that every element of S is quasidiagonal with
respect to {-Pn}- If T is quasidiagonal with respect to a sequence Pn, then so is
T + K because lim^oo ||if-P^"|| = ||P^lf || = 0 for every compact operator K.
Moreover, C*(T) is quasidiagonal. Indeed, write T = D + K where D is block
diagonal with respect to a sequence Pn and K is compact. Then
C*(T) C C*(D) + X C {Pn}' + £•
So the sequence Pn implements the quasidiagonality of every element of C*(T).
On the other hand, every semi-Fredholm quasidiagonal operator has index 0.
To see this, suppose that T = D + K where D = ]£n>1 ®Dn is a block diagonal
operator and K is compact. Since Dn is finite rank, fcer Dn and ker D* have the
same dimension. The kernels of D and D* are just the direct sum of these kernels.
Since at least one of ker D or ker D* is finite dimensional, only finitely many
of kerDn and kerD* are non-zero. Moreover, null(D) = null(D*); whence
ind(T) = ind(D) = 0.

282
IX. Brown-Douglas-Fillmore Theory
The Weyl-von Neumann-Berg Theorem H.4.2 shows that every normal
operator is quasidiagonal. Corollary IX.7.4 shows that the only obstruction to an
essentially normal operator being normal plus compact is the Fredholm index. Thus an
essentially normal operator is quasidiagonal if and only if it is normal plus
compact. We shall see that this is no longer the case for extensions of C(X) when X
has higher dimension.
Since quasidiagonality is not affected by compact perturbations, we say that a
subalgebra of the Calkin algebra is quasidiagonal if its preimage is quasidiagonal in
B(H). So define an extension [r] in Ext(X) to be quasidiagonal if 7r~1r(C(X))
is quasidiagonal. This is independent of the choice of r because quasidiagonality is
invariant under unitary equivalence. Let Extqd{X) denote the set of quasidiagonal
extensions in Ext(X).
By Theorem H.4.1, every trivial extension of .£ by C(X) is quasidiagonal.
On the other hand, since quasidiagonal operators have index 0, Extqd(X) must
lie in kevyx- As it is easy to verify that if 21 is quasidiagonal, then Mkty) is
quasidiagonal for all k > 1, it also follows that Extqd(X) must lie in ker7^. So
{0} C Ext^X) C ker7^ C ker7x.
In fact, we will see in the examples at the end of this chapter that all of these
containments may be proper.
Our definition of quasidiagonality of a C*-subalgebra of B(H) is somewhat
different from the definition for the quasidiagonality of an abstract C*-algebra 21
given in section VH.6. However, if a is a faithful, quasidiagonal representation of
21, then <r(°°) is a faithful, quasidiagonal representation with the additional property
that (t(°°)(21) Ci £ = {0}. By Corollary H.5.6 of Voiculescu's theorem, any other
faithful representation p of 21 such that />(2l) fl £ = {0} is approximately unitarily
equivalent to <r(°°) relative to £; and thus is also quasidiagonal. So the
quasidiagonality of one faithful representation implies the quasidiagonal of all essentially
faithful representations.
First we establish a simple lemma that shows that the requirement on
increasing sequences is unnecessary.
Lemma IX.8.1 A separable C*~subalgebra 21 of B(H) is quasidiagonal if and
only if for every e > 0, finite dimensional projection P and finite subset A of%
there is a finite rank projection Q such that \\PQL\\ < e and \\QA - AQ\\ < efor
all A in A.
Proof. If Pn implements the quasidiagonality of 21, then
lim||PP^|| = 0 and lim ||PnA - APn\\ = 0 forall A e 21.
n-foo n—foo
Conversely, suppose that the technical property holds. Choose a sequence An
dense in the unit ball of 21, and fix a basis {e*.} for H. Suppose that we have
constructed an increasing set of projections Pi < Pi < • • • < Pn so that i^e; = 6;

IX.8. Quasidiagonality
283
and
\\AkPi - P%Ak\\ < 2_i for all 1 < k < i and 1 < i < n.
Set e = 2-n-V5, P = Pn V en+1e;+1 and A = {Ai,..., An+i}; and apply
the technical property to obtain a finite rank projection Q so that H-PQ11| < e and
H-AfcQ - QAk\\ <eforl<k<n + l.
Following the argument of Lemma HI.3.1, there is a projection Pn+i > Psuch
that ||Pn+1 - Q|| < 2e. To recall the ideas, X = PQP + PLQPL is a positive
contraction such that
\\X-Q\\ = \\PQP^ + P^QP\\
= msz{\\PQLP% \\PLQLP\\} < \\PQL\\ < e.
Thus the spectrum of the finite rank operator X is contained in [0, e] U [1 - e, 1].
So the projection Pn+i obtained from the functional calculus on X for the
characteristic function of [1 - e, 1] satisfies
||Pn+1 - Q\\ < ||Pn+1 - X\\ + \\X - Q\\ < 2e.
Since X commutes with P, so does Pn+i- Moreover, PX = PQP > (1 — e)P;
whence Pn+i > P. Finally, for 1 < k < n + 1,
WAkPn+i - Pn+iAk\\ < \\AkQ - QAk\\ + 2\\Ak\\ ||Pn+i - Q\\
<5e = 2"n~1. ■
There is a natural topology on Ext(X). This is obtained by taking the topology
of point wise norm convergence on the collection of monomorphisms of C(X) into
Q(H) and taking the quotient norm on the equivalence classes.
Theorem IX.8.2 The closure of the zero element in Ext(X) (in the quotient
topology) equals Extqd(X).
Proof. Suppose that r^ are quasidiagonal monomorphisms of C(X) into Q{%)
converging pointwise to r. To verify that the limit r is quasidiagonal, we apply
Lemma IX.8.1. Suppose that e > 0, a finite rank projection P and finitely many
operators T\,..., Tn in 7r_1rC(X) are given. Let /i,..., fn be functions in C(X)
such that r(/») = 7rTj for 1 < i < n. Choose k sufficiently large that ||r(/t-) -
7fc(/t-)|| < e/3 for 1 < i < n. Then choose operators Si in 7r~1rife(/;) so that
\\T{ - S{\\ < e/Z. By the quasidiagonality of 7*., there is a finite rank projection Q
such that \\PQL\\ < e and \\SiQ - QSi\\ < e/Z for 1 < i < n. Then
\\TiQ - QTi\\ < \\SiQ - QSi\\ + 2\\Si - T{\\ < e.
Hence r is quasidiagonal. Therefore Extqd(X) is closed. It has already been
observed that the zero element of Ext (X) is quasidiagonal. Hence Extqd(X)
contains the closure of the zero element.

284 IX. Brown-Douglas-Fillmore Theory
Now suppose that r is quasidiagonal, and let Pn be a sequence implementing
the quasidiagonality. By dropping to a subsequence if necessary, we may suppose
that £n>JPnT-TPn||<oo for evei^
be the associated partition of the identity. It follows that T - ]£n>i EnTEn is
compact for all T in 7r~1r(C(X)). By Theorem IX.4.10, there is a positive linear
map (p from C(X) into B(H) such that r = inp. Replace (p by the positive map
Z)n>i ^nP(*)^n' This differs from the original by a map into the compacts. So
we assume that (p already has this form. Let <Pn(*) = ^nVKOUr/H-
By Naimark's Dilation Theorem IX.4.2, each <pn has a *-dilation an into the
bounded operators on Un = EnH 0 U'n such that an(f) = [j£((^ ^jjjj] with
respect to the given decomposition of %n. Let Pn be the projection of %n onto
EnH\ and let P = £n>1 ©Pn> which is the projection of K = £n>1 ©%i onto
X^n>i ®EnH. Define a representation into #(/C) by a = £)n>1 ©^n-
Notice that for any / in C(X), we have
?a/i2) = jmi/iv=p<f)°{f)*p=E®p^«(/w(/r-Pn
n>l
= SeP»«rn(/)Pn(rn(/)*Pf, + 2eJ,„«rn(/)^.!.M/),,^»
n>l n>l
= EaM/w/r+£ep„«tf2(/K2(/rp„.
n>l n>l
Therefore,
0 = r(|/|2) - r(/)r(/r = *r(y>(|/|>) - ?(/M/)*)
= T(VJePn(Tll2(/)^2(/)*Pn).
n>l
Consequently, it follows that
lim|K2(/)||= lim|K(/)||-0 forall / € C(X).
n—foo n—foo
To show that [r] is in the closure of the trivial elements, consider a^°°\ the
direct sum of countably many copies of a. Clearly, this is a representation of
C(X) which contains no compact operators. Moreover, since r(/) = ir(Pa(f)P)
is a monomorphism, so are ira and 7ra^°°\ Hence ira(°°) is a trivial extension. It
suffices to show that there are unitary operators Un so that
lim ||r(/) 0 7r<7<°°> - *Un<r{oo)(f)UZ\\ = 0 for all / G C(X).

IX.8. Quasidiagonality
Notice that
^°°)(/) =
=
+
>n(/)
Oflif)
0
0
0
0
>«(/)
0
0
0
0
0
r o
ki(/)
0
0
0
0
*?*(/)
*&(/)
0
0
0
0
0
»&(/)
»&(/)
0
0
0
*?*(/)
0
0
0
¥>«(/)
*?l(/)
0
0
0
*?l(/)
?n(/)
0
0
0
0
-*21
-*&(/) o
0
0
0
0
0
*&(/)
«&(/)
0
0
0
0
0
*&(/)
*?!(/)
0
(/)
0
0
0
0
¥>»(/)
*&(/)
0
0
0
"?!(/)
*>»(/)
0
0
0
*&(/)
«?l(/)
0
0
—0
0
Ti(/)
0
0
0
0
0
*&(/)
»&(/)
0
0
0
0
0
"&(/) •
0
0
0
-*&(/)
0 <7
^l(/)
2
. .
-j
0
0
0
0
&(/) -
0
285
Moreover, ||en(/)|| < 2 max{||cr^2(/)||, ||cr^1(/)||} which tends to 0 as n tends to
infinity for each / in C(X). Let Vn be the unitary implementing this equivalence,
so that
VncrtHw; = ¥>„(/) e •£■>(/)+*»(/)•
Then set W„ = j(n) e £fc>n ©^- Notice that
Wn*(°°\f)Wn = it(B<rtHf) © E ©M^/W
fc=l fc>n
= E©^/) e E ©(**(/) © ^(Z)) + «*(/)
fc=l fc>n
^E©^(/)©"(00)(/)+E©£'fc(/)
k>n
k>n

286 IX. Brown-Douglas-Fillmore Theory
where the unitary equivalence obtained by shuffling the terms is implemented by a
unitary Sn, and e'k(f) = SnSk{f)S*. Then Un = SnWn is a unitary operator such
that
£ e«k(/) e <r(oo)(/) - un*^(f)K = - £ ©**(/)•
k>n k>n
The unitary Un maps £n>i ©ft^ into £fc>n ©#fcft©En>i ®^n°°) which
is a subspace of H © X)n>i ®Wn° °f fin*te codimension. So we may think of Un
as a Fredholm isometry into this larger space such that
*>(/) © *(oo)(/) - ^a(~)(/)c;n* = J2®Mf) - £ <**(/)•
fc=l fc>n
By Corollary IX.2.4, weak equivalence and equivalence coincide for extensions of
C(X). Therefore it Ad Un a^ is a trivial extension. Moreover
||r(/) 0 tt<t(~)(/) - tt Ad Un „l-){f)\\ = Utt VJ eei(/)||
fc>n
<2max||£fc(/)||.
fc>n
This tends to 0 pointwise as n increases. Hence r is the pointwise limit of trivial
extensions. ■
IX.9 Homotopy Invariance
Recall that two maps / and g from X to Y are said to be homotopic if there
is a continuous function H from X x [0,1] into Y such that #(f, 0) = /(£) and
H(£, 1) = g(£). The important topological property of Ext proved in this section
is known as homotopy invariance, which says that if / and g are homotopic, then
/* = 9*-
First we push the quasidiagonality argument a bit further to obtain:
Corollary IX.9.1 If ft is a homotopy of maps from X toY and r is in Ext(-X'),
then [/i*t] — [/o*t] is quasidiagonal.
Proof. We will show that /i*r is the limit of extensions of the form Ad Un /o*r.
Let 7 be a monomorphism representing the extension -[r], and let a denote a fixed
trivial extension by C(Y). Then pt(g) = y(g o ft) © a(g) is an inverse for fur.
Moreover, the map taking t to pt(g) is continuous for g in C(Y). Define trivial
extensions
n n—1
fc=l fc=0

IX.9. Homotopy Invariance
287
Then
n
(/0«T 0 <Tn)(<7) = X)®(Afc-
l)/n*^ © Pk/n
fc=i
= «(f)®fur)(g) + en(g)
where
n
M<?) ^ 2©(/>(*-l)/nfo) " Ak/nfo)) 6 0.
fc=l
By the continuity of pt(g), it follows that limnH>00 ||en(.flOII = 0 for all g in C(y).
This shows that [/i*r] is in the closure of [/o*t]. Hence [fur] - [/o*t] is in the
closure of the zero element, and thus is quasidiagonal by Theorem EX.8.2. ■
The key step is to prove injectivity for a special map.
Lemma IX.9.2 Let 80 be the map ofX x [0,1] into itself given by J0(f, *) = (f, 0).
Then <$o* is infective.
Proof. Suppose that $o*[r] = 0- Then we will show that [r] = 0. The maps
r,(f, i) = (f, min{s, t}) for 0 < s < 1 define a homotopy from 80 = r0 to the
identity map id = n. So by Corollary IX.9.1, [r] = id*[r] - <$o*[r] is
quasidiagonal. Let Pk be a sequence of finite rank projections implementing the quasidiago-
nality of r, and let Ek = Pk~ -Pfc-i-
As in Theorem IX.8.2, (after replacing the Pk by a subsequence if necessary)
there is a positive map (p = J2k>i ®Vk of C(X) into 3(H) such that r = 7r<p,
where <fk{f) = Ek<p{f)\EkH- Also recall from that proof that there are finite rank
linear maps a^2 and a^i such that
Vn(f9) - <Pn(f)<Pn(g) = ^(/KlG?)
and
lim K2(/)|| = lim ||<r»(/)|| = 0 for all / G C(X x [0,1]).
n—foo n—foo
It is worth noting that this limit is uniform on any compact set S of functions
because \\an\\ = 1. Indeed, for any e > 0, there is a finite e/2 subnet {/i}^
of <S. For n sufficiently large, both ||(Tj*2(/i)|| < e/2 and ||<T2i(/*)ll < e/2 for
1 < i < N. Hence an easy estimate shows that ||0J*2(/)|| < 6 ^ ||<r^1(/)|| < e
for all/ in <S.
Nowr5*r(/) = 7r<p(/ or,) for every 0 < s < 1. Since {/or, : 0 < s < 1} is
the continuous image of [0,1], it is a compact set. So the previous remarks apply.
That is, ||(tJ*2(/ ° r*)|| converges to 0 uniformly for s in [0,1].
We will define elements of Ext(J.T x [0,1]) by
k
(T(/) = 7r2Z)©^(/oriA)
k>i y=i

288
IX. Brown-Douglas-Fillmore Theory
and
k
°V) = * ^2 2 ®?k(f ° r(i-i)A)-
k>l i=l
To see that a is a *-monomorphism, notice that a is positive and
k
<r(f9) ~ <r{f)<r{9) = * ^ S ®Vh{fg ° ri/fc) - <pfc(/ o rj/k)<pk(g o r,yfc)
fc>i i=i
= * Z) S ©^(Z ° rj/k)*2i(9 o ri/k).
k>\ i=l
By the previous paragraph, we see that this sum is a direct sum of finite rank
terms converging to zero in norm. Hence it is compact, and thus is annihilated
by 7r. It follows that a is a homomorphism. It is self-adjoint because of positivity.
Injectivity follows from the fact that the restriction to the sum over k of the &-th
block of the fc-th sum is it Ylk>i ©Vfc = r which is already one-to-one.
Next notice that af = a. Indeed,
k
*(/) - *'(/) = * J2 2 ®**(/ ° ri/* - / ° rtf-i)/*)-
fc>i i=i
By the uniform continuity of /, ||/ o r^y* - / o ry_!)/fc|| tends to 0 as A: increases
independent of j. Thus the sum is again a direct sum of finite rank terms tending
to 0 in norm, and so is compact and killed by 7r.
Finally, we will show that [ro*r] + [a] = [a] + [r]. To this end,
k
ro*r(/) 0 <r(f) = n ^ ®<Pk(f ° ro) © * 2 ^ ®**tf ° ri/*)
fc>i fc>i i=i
fc>i i=i k>i
= ^(/)er(/) = <r(/)er(/).
Since Ext(X) is a group, this implies that [r] = [>o*t] = 0 as claimed. ■
The lemma above handled all the technical difficulties. It remains to mop up
in order to establish homotopy invariance.
Theorem IX.9.3 Iffo and f\ are homotopic maps from X into Y, then /o* = /i*.
Proof. Let j,(£) = (f, s) be imbeddings of X into X x [0,1] for 0 < s < 1.
First let us show that j0* = ji*. Indeed, let [r] belong to Ext(J.f). Note that
<W'o = <Wi- Hence
$>*(ji*M - io*M) = (<W'i)*M - (<Wo)*M = o.

IX.10. The Mayer-Vietoris Sequence 289
By the lemma above, this shows that ju[r] = jo*[r].
More generally, let F be a function from X x [0,1] to Y implementing the
homotopy; that is, F(f, i) = /;(£) for i = 0,1. Then if [r] is in Extpf),
/o*M = ^*Jo*W = F*h*[r] = /i*[>]-
So homotopy invariance is established. ■
Recall that a topological space is contractible if it is homotopic to a point. The
cone on X is the space CX = X x [0,1]/X x {0}, which is always a contractible
space.
Corollary IX.9.4 IfX is contractible, then Ext(X) = 0. Thus Ext(CX) = Ofor
every compact metric space X.
IX.10 The Mayer-Vietoris Sequence
In this section, we will develop the algebraic topology of Ext a bit more to
obtain another computational tool. If / is a continuous map from X to Y\ the
mapping cylinder Z(f) of / is the space X x[0}l]vY modulo the identifications
(f, l) = /(£). The mapping cone is the space
C(f) = Z(f)/(X x {0}).
The suspension of X is the space SX = CX/(X x {1}).
Lemma IX.10.1 Let f be a continuous map from X into Y, and let i be the
injection ofY into C(f). Then there is an exact sequence:
Ext(X) A Ext(y) -S Ext(C(/)).
Proof. Define j to be the inclusion j(f) = (f, 0) of X into Z(f)\ q the quotient
of Z(f) onto C(f)\ k to be the injection of Y into Z(f)\ and p the projection of
Z(f) onto Y given by p(f, t) = /(£) for (f, t) in CX and pfa) = 77 for 77 in Y.
Consider the diagram
X_^£(/)-^C(/)
y
Then f = pj,i = qkypk = idy and gj is constant. By Corollary IX.6.2,
Ext(X) -£> Ext(£(/)) -?-> Ext(C(/))
is exact. The map kp is homotopic to id^/) via the map from Z(f) x [0,1] to
Z(f) given by
H((Z}t),s) = (Z,max{s}t}) for (£ft) € X x [0,1]
iif(77,5) = 77 for 77 G y

290
IX. Brown-Douglas-Fillmore Theory
By the homotopy invariance Theorem IX.9.3, k*p* = idz(f)*- A$P*h = idy*, it
follows thatp* and K are reciprocal isomorphisms. Hence
Ext(X) ?-4 Ext(y) -^> Ext(C(/))
is exact. ■
Lemma IX.10.2 If A is a contractible closed subset ofX, then the quotient map
q ofX onto X/A induces an isomorphism of Ext(X) and Ext(X/A).
Proof. Consider the sequence
Ext(A) ^-> Ext(X) -^> Ext{X/A) -S Ext(C(g)).
This is exact at Ext(X) by Corollary IX.6.2. Exactness at Ext(X/A) follows
from Lemma IXAO.L By Corollary IX.9.4, Ext (A) = 0. So it suffices to show
that Ext(C(g)) = 0 as well.
There is a natural imbedding k of SA into C(q) given by k(a} i) = (a, t) for
a in A and 0 < t < 1 since this extends by continuity to &(*, 1) = £o where
fo = A/A in X/A. The quotient
C(q)/SA = C(X/A)/(£ox[0,l])
where C(X/A) is the cone on X/A. Thus both SA and C(q)/SA are contractible.
By Corollary IX.6.2,
0 = Ext(SA) —> Ext(C(g)) —> Ext(C{q)/SA) = 0
is exact. Therefore Ext(C(g)) = 0 as claimed. ■
Now we are able to obtain a long exact sequence for Ext.
Theorem IX.10.3 Suppose that A is a closed subset ofX. Then there is a natural
long exact sequence
Ext{A) A Ext(X) ^-> Ext{X/A) A Ext{SA) ?h Ext{SX) ^ ...
Proof. The maps Snj are the injections of SnA into SnX, and Snq is the quotient
map of SnX onto SnX/SnA (which is homotopic to Sn(X/A)).
The sequence is exact at Ext(X) by Corollary IX.6.2, and similarly it is exact
atExt(5,nX)foralln> 1.
LctXUCA := (Xx{l})uCA denote the union ofXandCA with Ax {1}
identified with the corresponding subset of X x {1}. To define the connecting
homomorphism d from Ext(X/A) to Ext(SA), consider the quotient mapsp from
XUCA to (XUCA)/X = SA and r from XUCA onto (XUCA)/CA = X/A.
Since CA is contractible, Lemma IX.10.2 shows that r* is an isomorphism. Hence
we may define d = p^r"1 from Ext {X/A) into Ext (S A).

IX.10. The Mayer-Vietoris Sequence
291
If i is the injection of X into X U CA, we have a commutative diagram
Ext(X U CA)
Ext(X) —--* Ext(X/A) > Ext(5'A)
This is exact at Ext(X U CA) by Corollary IX.6.2. Since the diagram commutes
and r* is an isomorphism, it follows that our sequence is exact at Ext(X/A).
Let CX U CA denote the disjoint union of these two sets modulo the
identification of A x {1} with the corresponding subset of X x {1}. Let k be the injection
of X U CA into CX U CA\ and let s be the quotient map of CX U CA onto
(CX U CA)/(X U CA) = SX. Also let t denote the quotient of CX U CA onto
(CX U CA)/CX = SA. And lastly, we define the flip map of SX onto itself by
/(£, t) = (f, 1 — t). Consider the diagram
Ext(X U CA) -^ Ext(CX U CA) ^—^ Ext(SX)
Ext(X/A) * Ext(SA) s, > Ext(SX)
The top row is exact by Corollary IX.6.2. Since CX is contractible, t* is an
isomorphism by Lemma DC 10.2. Since / is ahomeomorphism, /* is an isomorphism.
To prove exactness at Ext(SA), we need to prove that /*s* = Sj*t*. This will
follow from Theorem IX.9.3 if we can show that fs and (Sj)t are homotopic.
Let £a denote the point A/A and let £x denote X/X. The homotopy H from
(CX U CA) x [0,1] to SX is given by
Hca(<x} u\ v) = (a, (1 - v)u) ae A
Hcx(Z,u;v) = (Z}l-uv) £eX
for 0 < u < 1 and 0 < v < 1. This definition identifies the points (a,0) with
(£4,0) in CA, the points (f, 0) with (fx, 0) in CX and SXy and the points (f, 1)
with (fx,l) in SX. Thus
i?CA(eA,0;t;) = ircA(a,i/;l)=(eA,0) a G A
#cx(£x,0;t;) = HCx(Z,u;0) = (fr,l) £ G X
i?cx(£,i;i) = (£x,o) £ex.
Notice that this is well defined because at u = 1,
Hca(ol, 1; v) = (a, 1 - v) = HCx(*> 1; t/) <* e A.

292
IX. Brown-Douglas-Fillmore Theory
It is evidently continuous even at the endpoints. Moreover, at v = 0,
Hca(<*, u; 0) = (a, u) = (Sj)t(a, u) a£A
Hcx(Z,«; 0) = (tx, 1) = (Sj)t{t,«) £ 6 X
and at v = 1,
tfc^a,«; 1) = (&, 0) = fs(a, u) a € A
#cx(£, u; 1) = (£, 1 - u) = /*(£,«) e € X.
All the other terms are exact by applying these results to the imbedding of 5" A
into SnX for all n > 0.
Naturality refers to the fact that the sequence commutes with maps. Suppose
that B is a closed subset of another compact metric space Y, and / is a continuous
map from X to Y such that /(A) C B. Then / induces homomorphisms
Ext(i4) -
(/U).
Ext(X)
u
Ext(X/A)
7.
Ext(SA) —^ Ext(SX)
(5/U).
(«/).
Ext(5) —^ Ext(F) —^ Ext(y/J3) -£*• Ext(SB) —^ Ext(SF) —»- • • •
To verify that this diagram is commutative, it suffices to verify it on the square
involving d. This follows from
X/A
1
Y/B
*J— XuCA —*-+ SA
fucfi
YuCB
\Sf\A
+ SB
As this diagram commutes, it follows that the induced maps commute with d.
Remark IX.10.4 This sequence isn't as long as it looks. An important theorem
about Ext is periodicity which states that there is a natural isomorphism from
Ext(5,2X) onto Ext(X). Thus this sequence turns into a 6 term exact cycle. We
will not prove periodicity here.
A fairly straightforward consequence of the long exact sequence is the Mayer-
Vietoris sequence.
Theorem IX.10.5 Suppose that B and C are two closed subsets ofX such that
B U C = X and B n C = A Then there is a natural long exact sequence
Ext(A) (*B""*C*)) Ext(J3) 0 Ext(C) JB*+jc*) Ext(X) -> Ext(SA) -> ...

IX.10. The Mayer-Vietoris Sequence
293
Proof. We denote the injections of A into B and C by %b and %c respectively; and
the injections of B and C into X by jb and jc- Let q be the quotient map of X
onto X/B. Then the long exact sequences for the injections of A in C and of B in
X are intertwined by the maps induced by jc'-
*c
Ext(A) -^W Ext(C) ^ Ext(C/A)
Ext(SA) ^-i Ext(SC7)
*B*
JC*
5»B*
5ic*
Ext(J3) -^- Ext(X) --^ Ext(X/B) —* Ext^JS) -^ Ext(SX) -j— • • •
The map from Ext(X) to Ext(S A) is defined by 9g*.
The exactness of the Mayer-Vietoris sequence at Ext(X) and onwards to the
right now follows from standard "arrow chasing" arguments. For example,
consider Ext(X). It is evident that dq*(JB* + jc*) = 0. Suppose that [r] in Ext(X)
satisfies dqjj] = 0. Exactness at Ext(C/A) above yields [ac] in Ext(C) such
that qc*[<rc] = 9*[r]> where qc is the quotient of C onto C/A. Since qc* = g* jc*,
we see that g*([r] -jcJi^c]) = 0. Hence by exactness at Ext(X) above, we obtain
an element [jb] in Ext(J3) such that jb*[vb] = [t] - jc*[vc]> whence
[r] = {Jb* + ]c*)Wb © *c]-
It remains to demonstrate exactness of the diagram at Ext(JB) © Ext(C).
Notice that JBtB = JA = jcic, and therefore
(JB* + jc*)(iB* © -ic*) = ]A* - ]A* = 0.
So we suppose that [as] in Ext(J3) and [ac] in Ext(C) are extensions satisfying
Jb*[&b] + jc*[<rc] = 0 in Ext(X). By Exercise IX.3, there is a natural
isomorphism between Ext(J5) 0 Ext(C) and Ext(J3 V C) where B V C is the disjoint
union. Let p = js V jc be the canonical map of JB V C onto X\ and let pf be its
restriction to A V A. Consider the commutative diagram
Ext(A) 0 Ext(A) iB***cl Ext(B) 0 Ext(C)
Ext(A V A)
Ext(A)
*B*Vic*
Ext(J5 V C)
p*
M*
Ext(X)
Since
P*[<tb V ac] = {Jb* V jc*)[<tb V ac] = JbA^b] + jc*[<rc] = 0,
Theorem IX.6.1 shows that there are extensions [ps] and [pc] in Ext(A) such that
(*B* V ic*)\pB V pc] = [<?B V ac]

294
IX. Brown-Douglas-Fillmore Theory
and
0=A\P1VP2] = \PB] + \PC].
Hence [pc] = ~[pb]- So
(is*, -ic*)[pB] = {iB*[pB], ic*\pc\) = (Ob], OC]). ■
IX.ll Examples
In this section, we will examine some examples of higher dimensional spaces
which exhibit interesting topological and analytic aspects that have been alluded to
in this chapter.
IX.11.1 Spheres. A direct computation of Ext(5'n) is difficult because the
useful topological sequences we have developed are also valid for homology
theories that are not periodic. So we quote the periodicity theorem of Remark IX. 10.4
to obtain that Ext(5'2n) = Ext(S°) = 0 and Ext^2"*1) = Ext(5'1) = Z for
n> 0.
For the odd spheres, one can use the fact that S2n~x is the unit sphere in C1 to
construct a Toeplitz extension which is a generator for Ext(5'2n""1). We consider
Sz here. Another interesting feature of this example is that 753 is the zero map, but
the higher order map on 2 x 2 matrices over C(5'3) yields an isomorphism. The
reason is the topological fact that tt1(5'3) = 0, meaning that every invertible
function on Sz is homotopic to the constant function 1. Hence the map 7 of Ext(5'3)
into Hom(7r1(5'3), Z) = 0 is necessarily the zero map.
Think of Sz as S = {{z,w) : z}w e C, \z\2 + \w\2 = 1}. Let m be the
normalized surface Lebesgue measure on S. Set jfiT2(5') to be the closure of the
polynomials in z and w in the Hilbert space L2 (m), and let P denote the orthogonal
projection onto jfiT2(5'). For / in 0(5), define a Toeplitz operator by Tfh = Pfh.
It is easily seen that Tj = Tj. Also if / is analytic (uniform limits of polynomials
in z and w), then H2(S) is invariant for T/. Thus if g is in 0(5), then
TgTf = Tgf and TjTg = TJg.
Note that {zkwl : k} I > 0} is an orthogonal set spanning H2(S). Thus
ekt = \\zkwt\\-1zkwt=y/^^^zkwt for k}£ > 0
forms an orthonormal basis for H2(S). Notice that
\zz w }z w ) = iz w }z w ) — < , /ll9 .
v ' v ' I \\zkwl\\2 m = k-landn = £
Hence
ry- k l \\zkwl\\2 k-1 t k k-1 I

IX.11. Examples
295
(Note that k = 0 is a special case.) Compute
(T;TZ - T\z?)ekl = P\z\2eu - PzPzeki
= llzVir^Pz^+V - zPzZkWl)
= \k+l+2 " fc+/+l)Cfc/ = (fc+*+2)(fc+M-l)tkt
Therefore, this commutator is a compact diagonal operator. A similar calculation
shows that T^TW - T\wp and T*TW - Tzw are compact. It follows that TfTg - Tfg
is compact whenever / and g are uniform limits of polynomials in z, 2, w and w,
which is all of C(S) by the Stone-Weierstrass Theorem.
As in the case of the circle, it is routine to check that 111/11 e = 11 /1100 • Therefore
r(/) = irTf is an extension of .£ by C(S). We will show that [r] is a generator
for Ext(S). It would be natural to compute y[r]. However, as noted above, it is
a topological fact that tt1(5'3) = 0, and thus 753 = 0. It is necessary to look at
matrix algebras over C(5).
Consider the unitary valued matrix function F = [ J& 5? ]• Then
A = TF =
Tz Tw
is essentially unitary (irA*A = irAA* = J). We compute 72M(.F) = ind A.
First suppose that (£) belongs to ker A. Then
(0) -A\g) - {p(-wf+zg)J
f+zg)J '
Since zf = —wg, a consideration of the basis expansion shows that there is a
vector h in jfiT2(5') such that / = wh and g = —zh. Thus the second coefficient
yields the identity
0 = P(-\w\2h - \z\2h) = -Ph = -h.
Therefore / = g = 0. So A is injective.
Similarly compute for (£) in ker A* to obtain
(8) = **({) = (353)-
Therefore,
g = P(\w\2 + \z\2)g = PwPzf - PzPwf = (TZTW - TwTz)*f = 0.
Hence 0 = Pwf = Pzf which shows that / is constant. In particular, (J) lies in
ker A*. Hence ind A = -1.
This shows that [r] ^ 0 since 72(0) = 0. It also shows that [r] is not a multiple
of another extension since -1 is indivisible. As Ext(5'3) = Z, it follows that [r] is
a generator.

296
IX. Brown-Douglas-Fillmore Theory
IX. 11.2 The Projective Plane. The real projective plane is introduced to give
an example where Ext(X) has torsion. Clearly the groups Rom(ir1(X)1 Z) and
Rom(K1(X)} Z) are torsion free. So the homomorphisms 7 and 700 cannot detect
torsion elements.
The real projective plane P2 is the 2-sphere S2 modulo the identification of
antipodal points. Perhaps a more useful representation is the unit disk modulo the
identification of antipodal points on the boundary circle. We will also see that it
is a quotient of the Mobius band M. The Mobius band is the unit square [0, l]2
modulo the identification of (s,0) with (1 - s,1) for 0 < s < 1.
Consider two imbeddings of the circle T = R/Z into M. Let
a(t) = (ht) for °<t<1
and
/*(*) =
(0,2*)
(1,2*-1)
0<<< \
Note that a(0) = (§, 0) ~ (§, 1) = a(l) and /3(0) = (0,0) ~ (1,1) = /3{1) and
/3(|) = (0,1) ~ (1,0). These identifications show that a and /3 are continuous
imbeddings of T into M. There is a retraction of M onto the centre circle a(T) by
r((s,t)) = (\,t).
Clearly, ra = idx; and hence r*a* = id*. Therefore a* is injective. However,
rfi(t) = 2t (mod 1) wraps twice around and thus r#/3# = 2 id#. As 7 is injective
on Ext(T), it follows that r*/3* = 2 id*.
Notice that M/a(T) is homeomorphic to the unit disk D. (The concentric
circles given by the first coordinate equal to \ ± s for 0 < s < \ have a limit point
a(T)/a(T).) Thus by the short exact sequence Corollary IX.6.2, we have
Z = Ext(T) -^> Ext(M) -£» Ext(D) = 0.
Therefore a* is an isomorphism of Z onto Ext(M) with inverse r*. Also
/?* = {a*r*)/3* = a* (2 id*) = 2a*.
Hence the range of /?* is 2Z.
Now notice that M//?(T) = P2 (as it is a disk with antipodal points on the
boundary identified). Thus by the long exact sequence Theorem DC 10.3
Ext(T) -^ Ext(M) --U Ext(P2) -^ Ext(52) .
-^Z-
Z/2Z-
■^0
Hence Ext(P2) = Z2 is a two element group. The non-zero element can be seen
tobeg*a*(l).

IX.11. Examples
297
Consider the imbedding of P2 into C2 defined by sending the unit disk D of the
complex plane into C2 by h(z) = (z2, (1 - |^|2)^). One may readily verify that
this is one-to-one on the interior of D and identifies antipodal points of the
boundary. Let (JVi, N2) be a pair of commuting diagonal operators with joint spectrum
equal to h(V). Then they represent the trivial element of Ext(P2). The non-trivial
generator of Ext(T) is given by the Toeplitz extension r. In this representation, the
range of a is homotopic to the map k(z) = (z, 0). So the element [1] in Ext(P2)
is generated by
k.(T)(z) = (TMt0)®(NuN2) = (TM®N1,0®N2).
In operator theoretic terms, this means that the pair (Tz © JVi, 0 © N2) is not
diagonalizable modulo £, but the pair (Tz © Tz © Ni, 0 © 0 © N2) is jointly diago-
nalizable modulo £. In fact, the non-zero element of Ext(P2) is not quasidiagonal
either. So Ext^P2) is properly contained in ker7p?.
IX.11.3 The Suspended Solenoid. Our last example will be used to show
several things. It will provide a subset X of C2 for which Extq(i(X) ^ 0. Hence
we will obtain an example that shows that the set of commuting normal plus
compact pairs is not closed. It also provides an example in which the surjection k of
Theorem IX.6.3 is not injective.
Consider a solid torus Ti in R3 with cross sections of radius 1. Inside Ti,
choose another solid torus T2 which wraps around 3 times inside Ti, and has
uniform cross section of radius 1/3. Continue selecting solid tori Tn+i inside Tn
wrapping around 3 times, and having uniform cross section of radius 3~n. The
intersection T is called a triadic solenoid. The example we want is the suspension
ST. It is easy to imbed STi in C2. So we obtain ST as the intersection of the
decreasing sequence STn. Each STn is homotopic to S2, and thus Ext(5Tn) = 0.
(Alternatively, notice that the injection i of the circle onto the centre circle of the
solid torus has a retract r. The solid torus modulo this circle is a solid ball which
is contractible. Thus i* and r* are reciprocal isomorphisms. The suspended maps
yield an isomorphism of Ext(52) = 0 and Ext(STn).)
It follows that proj limExt(5Tn) = 0 and thus the map k from Ext(ST) onto
this zero group is the zero map. Thus if Ext(ST) contains a non-zero element [r],
we obtain an example where k is not injective. Suppose that (.Alf A2) were an
essentially normal pair representing [r]. Then let (JVi, iV^) be a diagonal normal pair
with joint spectrum equal to ST. Choose diagonal normal pairs (iVni, Nn2) with
joint spectrum equal to STn converging to (JVi, N2). Then if jn is the injection of
ST into STn, it follows that jn*r is trivial. Thus (Ax © NnlyA2 © Nn2) is of the
form (Dni, Dn2) + (KnllKn2) where (Dni, Dn2) is a diagonal normal pair with
spectrum STn and Kn{ are compact. Since
lim (Ax © NnllA2 © Nn2) = (Ax © Nu A2 © iVi),
n—foo

298
IX. Brown-Douglas-Fillmore Theory
it follows that r is quasidiagonal. In fact, this shows that Ext(5T) = Extqd(ST).
However, as [r] is non-trivial, this is not a diagonalizable pair modulo k. This
shows that the analogue of normals plus compacts being closed fails for jointly
normal plus compact pairs, provided that we can find a non-zero element of Ext(5T).
To facilitate calculations, we will specify the imbeddings of Tn+i into Tn more
precisely. Parameterize Tn as {(2, w) : z € T}w €D}. Define an imbedding jn
of Tn+1 into Tn by jn(z, w) = (zz, (w + 2z)/3). Let kn denote the injection of T
into Tn.
We can also express T as a projective limit of circles Tn = T via the maps
pn(z) — zz. Denote this projective limit by Zy and let qn be the induced maps of
Z onto Tn. There are maps irn of Tn onto Tn by projecting the solid torus onto
its centre circle by irn(z} w) = z. It is clear from the formulae that we obtain a
commutative diagram
1
71"!
1
1 <* 7
1 .» ft
' ^
1 < ±2 -«-3
7T2
■2 J-^-T
'3^
... .<—
... .<—
— Tn<
We will show that map 7r = proj lim7rn is a homeomorphism. It is easily seen to
be surjective. On the other hand, suppose that f 1 and & are two points in T such
that tt(£i) = 7r(f2) = (*n)n>i in £. Then
*nM£l) = «n*"(£l) = *n = 9n^(6) = ^ViM^) for n > -•
In particular, gi (f 1) and gi (f 2) both lie in the disk {21} x D. Inside this disk, both
lie in the subset ji{{z2} x D) which is a disk of radius of radius 1/3. Repeating
this for the nth disk, both points lie in a common disk of radius 31_n for all n, and
thus f 1 = &• So 7r is injective. A continuous bijection of one compact Hausdorff
space onto another is a homeomorphism.
Thus we may identify T with Z. This latter space is the subset of the infinite
product group T°° consisting of those sequences z = fa, z2l 23,...) such that
z„+1 = zn. If z lies in Z, then -z = (-21, -z2l -23,...) also lies in Z. Also
notice that each z has two square roots iz1/2 = ±(«J , z\'2,...) because once
a square root z\' is chosen, there is a unique square root of z2 with cube equal to
1/2
2/ , etcetera.
Let P denote the space obtained from CZ by identifying (1, z) and (1, -z) for
all z in Z. Then there is a non-trivial map of Z into P given by
It is evident that P/(p(Z) = SZ. Let J be the quotient map. There is also a map r
of P onto the real projective plane P2 induced by the map r' of CZ onto D given
by r'(t,z) = tz\. Since r'(l,±z) = ±zi which are antipodal points, r is well

Exercises 299
defined. Define an injection V> of T into P2 by tf>(z) = [iz1/2]. It is easy to check
that np = t/>qi. We have a commutative diagram
Since the map pn wraps the circle three times around itself, it follows that
pn* is multiplication by 3 from Ext(Tn+i) into Ext(Tn). Hence the image of
Ext(Tn+i) in Ext(Ti) is 3nZ. As the map qu of Ext(Z) into Ext(Ti) factors
through pn*, the range of qu is contained in 3nZ for all n > 1 and thus qu = 0.
Therefore r*<p* = V**9i* = 0» which means that the range of <p* is contained in
ker r*. By the short exact sequence Corollary IX.6.2,
Ext(Z) -^> Ext(P) -S Ext{SZ)
is exact. So if we can find an element [a] in Ext(P) which is not in ker r*, then
[r] = <$4(7] ^ 0 in Ext(SZ).
Let 1 = (1,1,1,...) in Z. Define a map 0 of T into P by
^(e^) = [(|t|, sgn(t)l)] for - 1 < i < 1.
This is well defined because 0(e±7ri) = [(1, ±1)] is a single point in P. Notice
that r9 maps T onto the diameter of P2. This is the map called qot in the previous
example. Therefore r*0*(l) is the non-zero element of Ext(P2). And consequently
[a] = 0*(1) does not lie in ker r*. This completes the construction.
Exercises
LX.1 Suppose that (<£», (pi) represent two extensions of £ by C(X). Show that
the sum of these extensions is given by
{ [k\\ eII ] •• E* 6 C*, Kn e £, and ^n) = <p2{E22)) .
IX.2 Show that Corollary IX.9.4 and Corollary IX.6.2 imply homotopy invari-
ance.
HINT: Consider the two injections j,(£) = (^, s) for 5 = 0,1 and the
projection p of X x [0,1] onto X. Note that pj3 = id and use the sequence
X H X x [0,1] —> CX
to show that jo* is surjective.
IX.3 Show that if X = Xi V X2 is the disjoint union of two compact subsets
Xi and X2, then Ext(X) - Ext(Xi) 0 Ext(X2).
HINT: There is a natural map of the sum into Ext(X). For the inverse,

300
IX. Brown-Douglas-Fillmore Theory
one may choose a projection corresponding to Xxx to split an extension r.
The uniqueness of this choice depends on Corollary DC2.4.
IX.4 Let T be an essentially normal operator with <xe(T) = X corresponding
to an extension r of ^ by C(X). Let A be a closed subset of X, and let p
be the quotient of X onto X/A. If p*[r] = 0 and e > 0, show that there is
an essentially normal operator R with ae(R) = A and a diagonal normal
operator D with a(D) = ae(D) = X such that T ~(R@D) + K where
if is a compact operator with \\K\\ < e.
HINT: Use Corollary IX.6.2 and swallow most of the compact into R by
moving a finite projection from D.
IX.5 Suppose that X = B U C is the union of closed subsets. Suppose that there
is a retract of C onto A = B n C. Then show that the natural map from
Ext(5) 0 Ext(C) into Ext(X) is surjective.
IX.6 (a) Show that Ext(.Mfc) = Z*. by lifting matrix units in the Calkin algebra
to matrix units in 3(H), and computing the codimension of the lifting of
the identity,
(b) Show that Ext(A^ni © • • • © Mnie) = 2>d where d — gcd(ni,..., n*.).
IX.7 (a) Suppose that j is a unital imbedding of Mn into Mkn> and that r is
an extension of Mkn- Show that if a is a (possibly non-unital) homo-
morphism of Mn into B(H) such that ira(A) = r(j(A))9 then there is a
homomorphism a of Mkn into B(H) such that aj = a and ira = r.
HINT: Lift matrix units.
(b) If r is trivial in part (a) and a is unital, show that there is a unital lifting
a.
(c) Let 21 be a UHF algebra of type k°°. Show that the map k from Ext(2l)
into proj lim Ext (.Mfcn) is surjective by repeated use of (a).
(d) Show that k is an isomorphism by repeatedly using part (b).
IX.8 Show that the Mayer-Vietoris sequence is exact at Ext(5-A).
Notes and Remarks.
The main results of this chapter are due to Brown, Douglas and Fillmore
[1973] and [1977]. A nice treatment of the minimal route to the planar case is
contained in Davie [1976], as well as the examples at the end of this chapter.
Arveson [1974] was the first to point out the role of lifting completely positive
maps in this context. Stinespring [1955] generalized Naimark [1943] to the non-
commutative setting. Paulsen [1986] provides a nice treatment of completely
positive maps. Voiculescu's Theorem II.5.3 shows that the trivial elements form the
zero element for every separable C*-algebra. Arveson [1977] shows how to
combine Voiculescu's results with a lifting result of Choi and Effros [1976] to obtain

Exercises
301
inverses for liftable maps. The proof for homotopy invariance was generalized to
the non-commutative setting by Salinas [1977], and we follow his approach
exploiting quasidiagonality here. O'Donovan [1977] contains a similar argument.
Berg and Davidson [1991] provide an operator theoretic proof of the planar case
Theorem IX.7.2 using a generalization of Berg's technique.

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Index
AF algebra, 75,235
characterization, 82
ideals, 85
algebraically irreducible, 26
amenable group, 185
approximate identity, 11
approximate unitary equivalence, 57
approximate unitary equivalence, relative to
£,57
approximately finite dimensional, 75
approximately inner automorphism, 116
approximately inner derivation, 96
automorphisms of &, 151
Berg's technique, 174,239
Bratteli diagram, 76
Bunce-Deddens algebra, 139,162,231,238,
246,247
C*-dynamical system, 216
cancellation, 99,246
canonical anticommutation relations, 87
canonical shuffle, 140
CAR algebra, 76,87,106
Cauchy-Schwarz inequality, 28
Cesaro sums, 223
character of a group, 184
classical dynamical system, 223
Coburn's Theorem, 137
commutant, 19
compalence, 72
compalent, 252
completely positive, 65,259
concrete C*-algebra, 1
cone, 101,129
cone on X, 289
continuous functional calculus, 8
contractible C*-algebra, 129
contractive homomorphism of ordered groups
103
covariant representation, 216
crossed product, 217
cry stallographic group, 193
Cuntz algebra, 144,219,234
current algebra, 89
cyclic representation, 46
cyclic vector, 46
derivation, 96
diagonal representation, 62
dimension group, 102
discrete Heisenberg group, 200
Double Commutant Theorem, 19
Effros-Handelman-Shen Theorem, 123
Elliott's Theorem, 109
enveloping C*-algebra, 217
equivalent extension, 151,253
equivalent idempotents, 97
♦-equivalent projections, 97
ergodic transformation, 224
essential spectrum, 252
essentially normal, 252
expectation, 145,167,206,222
Ext(On), 154
Ext(X), 254
extension, 91,150,252
Fibonacci algebra, 106,116
finite C*-algebra, 101
finite dimensional C*-algebra, 74
free group, 203
gauge automorphisms, 89
gauge invariant, 89
Gelfand-Naimark Theorem, 33
GICAR algebra, 89,108
307

308
Index
Glimm's Lemma, 64
GNS construction, 29
Grothendieck group, 100
group C*-algebra, 184
Haar measure, 182
hereditary, 13,85,102
homotopic maps, 286
homotopy invariance, 98,286
hull, 191
hull-kernel topology, 191
ideal, 12
index map, 272
inductive, 84
infinite C*-algebra, 147
initial projection, 23
inner automorphism, 116
inner derivation, 96
inner function, 162
inverse closed, 15
irrational rotation algebra, 167,218,234,244
irreducible, 26
Jacobson radical, 33
Jordan decomposition, 43
Ko, 101
#1,233
Kadison Transitivity Theorem, 27
Kadison-Kaplansky conjecture, 200
Kaplansky Density Theorem, 20
Ll(G), 182
L°° functional calculus, 51
lattice ordered group, 119
left regular representation, 184
limit circle algebra, 159,242,246
long exact sequence for Ext, 290
m-times around imbedding, 141
mapping cone, 127,289
mapping cylinder, 289
Markov-Kakutani Theorem, 185
matrix units, 74
maximal abelian algebra, 48
Mayer- Vietoris sequence, 292
mean, invariant, 185
minimal Cantor system, 235
minimal dynamical system, 225
modular function, 182
multiplicity free representation, 48
multiplicity function, 56
n-positive, 65,259
Naimark Dilation Theorem, 260
non-degenerate representation, 34
normal, 2
odometer transformation, 230
operator monotone, 40
order, 9
order ideal, 112
order unit, 101
tt1(X),273
P-V short exact sequence, 234
partial isometry, 23
partial multiplicities, 75
partition, 235
periodic weighted shift, 137
periodicity, 292
polar decomposition, 23
positive, 2
positive definite function, 187
positive homomorphism, 103
positive lifting, 262
positive linear functional, 27
positive map, 65,259
prime ideal, 191
primitive ideal, 190
primitive ideal space, 190
projectionless C*-algebra, 124,200,210
projective limit, 270
projective plane, 296
proper isometry, 132
properly infinite, 147
pure state, 30
purely infinite, 149
quasi-similarity, 72
quasicentral approximate unit, 35
quasidiagonal C*-algebra, 204,209
quasidiagonal extension, 282
quasidiagonal operator, 281
quotient algebra, 13
range projection, 23
real rank zero, 156
reduced crossed product, 218
reduced group C*-algebra, 184
Rieffel projections, 170
Riesz group, 118

Index
309
Rohlin's Lemma, 225
Russo-Dye Theorem, 25
scale, 102
self-adjoint, 2
semi-commutator, 134
semi-simple, 33
separating vector, 48
simple dimension group, 114
spectral measure, 51
Spectral Theorem, 47
spectrum, 3
spectrum of a C*-algebra, 191
stably equivalent, 99
stably finite, 101
state, 27
state of a dimension group, 114
state space, 30
Stinespring Dilation Theorem, 260
strong operator topology, 16
strongly continuous, 20
subrepresentation, 44
supernatural number, 86,142
suspended solenoid, 297
symmetry, 19
2-adic numbers, 124
Toeplitz operator, 133
topological stable rank, 244
topologically irreducible, 26
trace, 42,114
trace class, 41
translation invariant measure, 224
triadic solenoid, 297
trivial extension, 153,254
UHF algebra, 86,219,238
uniform multiplicity, 54
uniformly hyperfinite, 86
unilateral shift, 132
unimodular, 182
unitarily equivalence, 46
unitary, 2
unitary representation, 182
universal C*-algebra, 166
universal representation, 33
unperforated, 118
weak approximate unitary equivalence, 57
weak operator topology, 16
weak-* topology, 42
weakly compalent, 252
weakly equivalent extension, 152,254
weighted unilateral shift, 137
Weyl-von Neumann-Berg Theorem, 59
Wold decomposition, 136
vector state, 33
Voiculescu's Theorem, 68
von Neumann algebra, 19

C*-Algebras by Example
Kenneth R. Davidson
The subject of C*-algebras received a dramatic revitalization
in the 1970s by the introduction of topological methods through the
work of Brown, Douglas, and Fillmore on extensions of C*-algebras
and Elliott's use of if-theory to provide a useful classification of
AF algebras. These results were the beginning of a marvelous new
set of tools for analyzing concrete C*-algebras.
This book is an introductory graduate level text which presents
the basics of the subject through a detailed analysis of several
important classes of C*-algebras. The development of operator
algebras in the last twenty years has been based on a careful study of
these special classes. While there are many books on C*-algebras
and operator algebras available, this is the first one to attempt
to explain the real examples that researchers use to test their
hypotheses. Topics include AF algebras, Bunce-Deddens and Cuntz
algebras, the Toeplitz algebra, irrational rotation algebras, group
C*-algebras, discrete crossed products, abelian C*-algebras
(spectral theory and approximate unitary equivalence) and extensions.
It also introduces many modern concepts and results in the subject
such as real rank zero algebras, topological stable rank, quasidiag-
onality, and various new constructions.
These notes were compiled during the author's participation
in the special year on C*-algebras at the Fields Institute of
Mathematics during the 1994-1995 academic year. The field of C*-
algebras touches upon many other areas of mathematics such as
group representations, dynamical systems, physics, if-theory, and
topology. The variety of examples offered in this text expose the
student to many of these connections. A graduate student with
a solid course in functional analysis should be able to read this
book. This should prepare them to read much of the current
literature. This book is reasonably self-contained, and the author has
provided results from other areas when necessary.
ISBN 0-8218-0599-1
9»780821ll805992