An Introduction to the Classification of Amenable C-Algebras

I

World Scientific

An Introduction to the Classification of

Amenable C-Algebras

An Introduction to the Classification of

Amenable C-Algebras

Huaxin Lin University of Oregon, USA

V f e World Scientific wB

New Jersey * London • Singapore • Hong Kong • Bangalore London'Sim

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

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AN INTRODUCTION TO THE CLASSIFICATION OF AMENABLE C*-ALGEBRAS Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN 981-02-4680-3

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To w h o m I love

Preface

The theory and applications of C*-algebras are related to such diverse fields as operator theory, group representations, topology, quantum mechanics, non-commutative geometry and dynamical systems. In light of the Gelfand transformation, the theory of C*-algebras is also regarded as noncommutative topology. Despite the great influence of this subject to other fields, the understanding of C*-algebras itself was very limited. About a decade ago, George A. Elliott initiated the program of classification of C*algebras (up to isomorphism) by their if-theoretical data. It started with the classification of AT -algebras with real rank zero. Since then, great efforts have been made to classify amenable C*-algebras, a class of C*algebras that appears most naturally. Large classes of simple amenable C*-algebras were discovered to be classifiable. With these rapid development, the theory of C*-algebras becomes increasely important to many other fields. For example, the applications of these results to dynamical systems have been well established. The purpose of this book is to introduce some of the recent developments of the theory of classification of amenable C*-algebras to a broad range of readers including non-experts and graduate students. It is an ambitious plan. However, the material presented here has been limited by the author's knowledge as well as the page limitation of this volume. For example, the aspects of classification of purely infinite simple C*-algebras which is quite complete are not mentioned in this volume. The author's effort was concentrated to finite C*-algebras. Even in this case, only simple C*-algebras with tracial topological rank zero are treated in detail. The first three chapters contain the basics of the theory of C*-algebras vii

Vlll

Preface

which are particularly important to the theory of the classification of amenable C*-algebras. References to these three chapters include (but not limited to) [147], [173], [143] and [48]. Chapter 4 offers the classification of the so-called j4T-algebras of real rank zero. The results in Chapter 6 cover the results in Chapter 4, however, the proofs given in Chapter 4 are much more elementary. It is the author's intention to present the classification of simple AT-algebras of real rank zero with limited tools so non-experts and graduate students may be able to read it without advanced knowledge of C*-algebras and if-theory. The first four chapters and first 6 sections of Chapter 5 are self-contained. This part could serve as a text book for a graduate course on C*-algebras. Indeed the author used it for a graduate course in University of Oregon and a lecture series in East China Normal University. The last two chapters contain more advanced topics. In particular, they contain the classification theorem for simple AH-algebras with real rank zero, the work of Elliott and Gong. To achieve these goals in such a limited volume, the author was often forced to give some new proofs (to avoid introducing too much new concepts and materials). Starting Chapter 4, at the end of each chapter, brief remarks are inserted. The intention is to give the reader some rough idea of the development related to the material presented there. They are bound to contain errors. The author asks forgiveness from those experts whose works have not been mentioned. The majority of this book was written when the author was visiting East China Normal University during the summer of 2000, the Mathematical Sciences Research Institute at Berkeley during the fall of 2000 and University of California at Santa Barbara in spring of 2001. It is his pleasure to express his gratitude for all hospitalities he received at these institutes. The exciting environment at the MSRI has left him with a great memories far beyond this book. Even though it is difficult for the author to express his exact appreciation to the people whom he met at MSRI, now is perhaps the only opportunity to record his sincere appreciation. The author thanks Professor N. C. Phillips and Shuang Zhang for their comments and suggestions. He would also like to take this opportunity to thank following persons who were exposed to earlier drafts and made corrections, comments and suggestions: Warren Akers, Shanwen Hu, Bobby Ilapogu, Benjamin Itza-Ortiz, Junping Liu, Shudong Liu, Nancy Livingston, Lisa Oberbroeckling, Michael Raney and Tadg Woods.

Contents

Preface

vii

Chapter 1 The Basics of C*-algebras 1.1 Banach algebras 1.2 C**-algebras 1.3 Commutative C*-algebras 1.4 Positive cones 1.5 Approximate identities, hereditary C*-subalgebras and quotients 1.6 Positive linear functionals and a Gelfand-Naimark theorem . . 1.7 Von Neumann algebras 1.8 Enveloping von Neumann algebras and the spectral theorem 1.9 Examples of C*-algebras 1.10 Inductive limits of C*-algebras 1.11 Exercises 1.12 Addenda

1 1 9 12 16 20 25 32

Chapter 2 Amenable C*-algebras and if-theory 2.1 Completely positive linear maps and the Stinespring representation 2.2 Examples of completely positive linear maps 2.3 Amenable C*-algebras 2.4 if-theory 2.5 Perturbations 2.6 Examples of if-groups

67

ix

38 42 51 60 65

67 72 76 82 89 97

x

2.7 2.8 2.9

Contents

if-theory of inductive limits of C*-algebras Exercises Addenda

103 108 Ill

Chapter 3 AF-algebras and Ranks of C*-algebras 3.1 C*-algebras of stable rank one and their if-theory 3.2 C*-algebras of lower rank 3.3 Order structure of if-theory 3.4 AF-algebras 3.5 Simple C*-algebras 3.6 Tracial topological rank 3.7 Simple C*-algebras with TR(A) < 1 3.8 Exercises 3.9 Addenda

113 113 120 127 133 140 146 154 160 162

Chapter 4 Classification of Simple AT-algebras 4.1 Some basics about AT-algebras 4.2 Unitary groups of C*-algebras with real rank zero 4.3 Simple .AT-algebras with real rank zero 4.4 Unitaries in simple C*-algebra with RR(A) = 0 4.5 A uniqueness theorem 4.6 Classification of simple AT-algebras 4.7 Invariants of simple AT-algebras 4.8 Exercises 4.9 Addenda

165 165 170 177 182 186 192 196 204 208

Chapter 5 C*-algebra Extensions 5.1 Multiplier algebras 5.2 Extensions of C*-algebras 5.3 Completely positive maps to Mn(C) 5.4 Amenable completely positive maps 5.5 Absorbing extensions 5.6 A stable uniqueness theorem 5.7 if-theory and the universal coefficient theorem 5.8 Characterization of if if-theory and a universal multi-coefficient theorem 5.9 Approximately trivial extensions 5.10 Exercises

211 211 217 221 227 233 243 250 255 259 265

Contents

xi

Chapter 6 Classification of Simple Amenable C*-algebras 6.1 An existence theorem 6.2 Simple AH-algebras 6.3 The classification theorems 6.4 Invariants and some isomorphism theorems

269 269 279 288 295

Bibliography

307

Index

317

— 759AD — 4 6 (701 - 762)

Chapter 1

T h e Basics of C*-algebras

1.1

Banach algebras

Definition 1.1.1 A normed algebra is a complex algebra A which is a normed space, and the norm satisfies ||ab|| < ||a||||6|| for all a,be

A.

If A (with this norm) is complete, then A is called a Banach algebra. Every closed subalgebra of a Banach algebra is itself a Banach algebra. Example 1.1.2 Let C be the complex field. Then C is a Banach algebra. Let X be a compact Hausdorff space and C(X) the set of continuous functions on X. C(X) is a complex algebra with pointwise operations. With ||/|| = sup^gjr |/(x)|, C(X) is a Banach algebra. Example 1.1.3 Let M„ be the algebra o f n x n complex matrices. By identifying Mn with B(C n ), the set of all (bounded) linear maps from the n-dimensional Hilbert space C n to C™, with operator norm, i.e, ||ir|| = su P£€Cn,||£|| 0. Hence the sequence {A n /(a n )} converges to zero for each A G ft, and therefore is bounded. Since this is true for every / G A*, by the principle of uniform boundedness, {Xnan} is a bounded sequence. So we may assume that |A n |||a n || < M for all n > 0 and for some positive number M. Hence Mlln

IKH1/noo

n-+oo 1

This implies that r{a) = l i n i n g Ha"!! /™.

•

/1/2 1 0 \ 0 1/2 0 . Then ||a|| > \ 0 0 1/3/ 1 and sp(a) = {1/2,1/3}. So r(a) = 1/2. It follows that \\an\\l'n -J- 1/2. Let T £ B(L2([0,1])) be a bounded linear operator defined by

Example 1.1.17

Let A = M 3 and a =\

T(f)=

[tf(x)dx.

Jo

The reader can compute that ||T n || < ^ . Hence r(T) = 0. Note that T ^ 0 (see Exercise 1.11.3). We now establish the holomorphic functional calculus for elements in Banach algebras (1.1.19). The first application appears in 1.2.9. Later, we will establish continuous functional calculus for commutative C*-algebras (1.3.5) and Borel functional calculus for normal elements in von Neumann algebras (1.8.5). Definition 1.1.18 Let a; be a fixed element in a unital Banach algebra A. Let / be a holomorphic function in an open neighborhood Of of sp(:r), and C be a smooth simple closed curve in Of enclosing sp(a;). We assign the positive orientation to C as in complex analysis. For each (j> £ A*, we consider a continuous function which maps A to f(\)(j>((\ — x)~l) on the curve C. Set L{4>) = ^~ f /(A)#(A 2m Jc

x)~l)d\.

The map (f> H-» L{(j>) is a linear functional on A* and \L{)\ 0, there is b G A with ||6|| = 1 such that ||7r(a)+Al|| 2


(yx))

1 > ||(6-^V/2)(aa/26-Q/2)|| = ||r^V a+/9)/2 *~ a/2 ]ll > = =

r(6-' 3 / 2 [a( Q +^/ 2 6-"/ 2 ]) = r ( a (a+/3)/2 6 -(«+«/2) r([a ( a + / 3 ) / 2 6 _ ( a + / 3 ) / 4 ]6'" ( " + / 3 ) / 4 ) = r (6-( Q +/ 3 )/ 4 a ( a + / 3 )/ 2 6" ( a + / 3 ) / 4 ) || 6 -(a+/3)/4 a (a+ / 3)/2 6 -(a+/3)/4||

Therefore we have 6-( 0, we have (6 + £ ) Q - ( a + e ) a > 0

(0 0 and ||a; — xpn\\ —>• 0

as n —> oo.

The sequence {pn} plays a role which is similar to that of identity in a unital C*-algebra. Let X be a non-compact but cr-compact and locally compact Hausdorff space X. Then CQ{X) = A is non-unital. Moreover, X — \J^=xXn, where each Xn is compact and Xn+i contains a neighborhood of Xn. It is easy to produce an increasing sequence of positive functions / „ € Co(X) such that 0 < fn < 1, fn{t) = 1 on Xn and fn(t) = 0 if t ^ X n + i . One checks that, for any g e C 0 (X), \\gfn -g\\->0

as n ->• oo.

Definition 1.5.2 An approximate identity for a C*-algebra A is an increasing net {e\}\eA of positive elements in the closed unit ball of A such that a = limae^. Equivalently, a = lim^ e\a for all a £ A. Lemma 1.5.3 Let A be a C*-algebra and denote by A the set of all elements a £ A+ with \\a\\ < 1. Then A is upwards-directed; i.e., if a, b € A, then there exists some c S A such that a,b < c. Proof. Suppose that a e A+. Then 1 + a e GL(A), and a(l + a ) - 1 = 1 - (1 + a ) - 1 . We claim that a,b€ A+ and a< b -» a{\ + a)'1 < 6(1 + b)~1.

(e5.4)

Approximate

identities,

hereditary C* -subalgebras and quotients

21

In fact, if a < 6, then 1+a < 1+6 which implies that (1+6) * < (1+a) x by 1.4.10 (4). Consequently 1 - (1 + a ) _ 1 < 1 - (1 + 6 ) _ 1 . This is the same as a(l + a ) - J < b(l + 6)" 1 . So the claim is proved. We note also (by 1.3.6) that if a G A+, then a(l + a ) - 1 G A. Now suppose that a, b G A. Put x = a(l - a ) - 1 , y = 6(1 - 6)" 1 and c = (x + y){\ + x + y)~l • Since x + y G yl + , c G A. We note that ( 1 + a ; ) - 1 = 1 — a and x{l + x)~l = a. So, by e5.4, a = x(l + a;) - 1 < c, since x < x + y. Similarly, b < c. This proves the lemma. • The following positive continuous functions will be used throughout this book. Definition 1.5.4

Let ( 1 fs{t) = < linear

{ 0

iie 0, and 0 < fs < 1. Note that if a > P > 0, then fa > fa. Also f^fi = fi. In

n

n

Let a G A+. Then it follows from 1.3.6 that fe{a)a —> a if e —¥ 0. Theorem 1.5.5 Every C*-algebra A admits an approximate identity. Indeed, if A is the upwards-directed set of all a G A+ with \\a\\ < 1 and e\ = A for all A G A, £/ien {e\)\e^ forms an approximate identity for A. Proof. From 1.5.3 {e\} is an increasing net in the closed unit ball of A. We need to show that lim^ ae\ = a for all a £ A. Since A spans A, it suffices to assume that a G A+. Since {||a(l — eA)a||} is decreasing, it suffices to show that there are un G {e\} such that ||a(l — u„)a|| —¥ 0 (as n -> oo). Note that un = (1 - ^)fi(a) G A. We have i(l - (1 - ^)fi(t))t ->• 0 as n —>• oo uniformly on [0,1]. It follows from 1.3.6 that ||o(l — Mn)a|| —> 0 as n —> oo.

•

22

The Basics of

Corollary 1.5.6 mate identity.

C-algebras

If A is separable, then A admits a countable approxi-

Proof. Fix a dense sequence { i „ } of A. Let {ex} be an approximate identity for A. Choose ek G {e^} such that \\xiek-Xi\\

{b*a)\2 < <j)(b*b) 0. We m a y assume t h a t <j>(b*a) ^ 0. W i t h A = ^fc*^ a n d t G R, this gives t2(f>(a*a) + 2t\<j>(b*a)\ + 0(6*6) > 0 (<j>(b*a) = (a*b)). If cp(a*a) = 0, t h e above becomes 2t\(b*a)\+cf>(b*b) > 0 for all ( 6 R . By taking negative t with large |t|, we conclude t h a t (f>{b*a) = 0. T h e inequality (e6.6) holds in this case. If <j){a*a) + 0, choose t = - J $ g ^ i . We obtain

<j>{a*a)

(j>{a*a)

v

;

_

T h u s (e6.6) follows.

D

T h e o r e m 1 . 6 . 7 An element in A* is positive if and only if \im\ 4*{e\) = ||0|| for some approximate identity {e\} in A. * Proof. If 0 is a positive linear functional on A a n d {e\} is any approximate identity for A, t h e n 4>(e\) is increasing. Hence it h a s a limit, say I. T h e n I < \\\\. For each a € A with ||a|| < 1, by 1.6.6, \4>(exa)\2 < (el)4>(a*a) < 0(e A )||0|| < Z||0||. Since <j> is continuous (1.6.5), we obtain |0(a)| 2 < Z||0||, whence ||0|| 2 < Z||0|| a n d / = ||0||. To prove the converse, suppose that {{e\)} converges to ||0||. We first show that 4>(Asa) C R. Let a e Asa with ||a|| < 1 and write 0(a) = a + if3 with a, (3 € R. By multiplying by —1 if necessary, we may assume that f3 > 0. Choose e\ so that \\e\a — ae,\|| < 1/n. Then \\ne\ - ia\\2 = \\n2e\ + a2 - in(ae\ - e\a)\\ < n2 + 2.

28

The Basics of C* -algebras

On the other hand lim \(nex - ia)\2 = {n\\<j>\\ + f3)2 + a2. Combining these two inequalities, we obtain (nU\\+(3)2

+ a2\\P + f32 + a2\\2. for all n. Hence /? = 0. So <j>(Asa) C R. Now if a G .4+ with ||a|| < 1, then e\ — a € Asa and e^ — a < e\ Therefore

{ex-a)\\. Taking the limit, we obtain ||^>|| — <j>(a) < \\\\. It follows that 4>{a) > 0 which implies that <j> > 0. D Remark 1.6.8 We actually proved that \im.\<j>(e\) = \\<j>\\ for any approximate identity {e\} of A. If A is unital then Theorem 1.6.7 also implies that <j> > 0 if and only if 0(1) = \\ on A define an extension on A by setting 0(1) = \\4>\\- Then <j> is positive on A and

WW = WWProof. Let {e,\} be an approximate identity for A. For each a € A and a G C, by 1.4.10 (3), limsup \\ae\ + a\\2 x

=
+ a ) | = l i m | 0 ( a e A + a ) | < | | a l + a||||0||. Hence ||0|| = ||0||. Since 0(1) = ||0||, by 1.6.7, 0 > 0.

•

Positive linear Junctionals

and a Gelfand-Naimark

theorem

29

Proposition 1.6.10 Let B be a C* -subalgebra of a C*-algebra A. For each positive linear functional 0 on B there is a positive linear functional 4> on A such that 4>\B = 0 and ||0|| = \\4>\\. If furthermore B is hereditary, then the extension is unique. Proof. We may assume that A has a unit by replacing A by A if necessary. Extend 0 to C*(B,1A) = B by setting 0(1) = ||0||. It follows from 1.6.9 that so extended 0 is positive and preserves the norm. Therefore, without loss of generality, we may assume that A has a unit 1 and 1 £ B . By the Hahn-Banach theorem, there is a linear functional 0 such that 4\B = and ||0|| = \\\\. Since 0(1) = ||0||, by 1.6.7, 0 > 0. To prove the second part, assume that B is hereditary and let {eA} be an approximate identity for B. Suppose that ip is a positive linear functional on A with tp\B = 0 and ||V>|| = ||0||. Then \\ip\\ = limA 0(e A ) by 1.6.7. So limA ip{lA ~ ex) = 0. It follows that V ( ( 1 A - e A ) 2 ) < (U ~ eA) ->• 0. By 1.6.6, for any c € A, |V((1 - e A )ac)| 2 < V ( ( U - ex)2)^{c*a*ac)

->• 0.

Therefore ip(a) — lim-0(e A ae A ). Since e\Ae\ C B, we have ip{a) = limV'(eAaeA) = lim0(e A ae A ) for every a S A. Thus •0 = 0.

D

Corollary 1.6.11 Let a E A be a normal element. Then there is a state 0 on A such that |0(a)| = ||a||. Proof. Let B = C*{a). Since r(a) = \\a\\, by 1.2.7, there is A G sp(a) such that |A| = ||a||. Identify B with C 0 (sp(a)), by 1.3.6, we define a state 0i on B by \(g) = g(\) for g e C 0 (sp(a)). In particular, |0i(a)| = ||a||. By 1.6.10, there is a state 0 on A such that 4>\B = 4>id Definition 1.6.12 Let A be a C*-algebra and / € A*. Define f*(a) = f(a*). Denote by fsa and fim the self-adjoint linear functionals ( l / 2 ) ( / + / * ) and ( l / 2 i ) ( / — / * ) , respectively. We also use R e / for the real part of / , i.e., Re/(a) = (1/2)[/(a) + /(a)] (for a G A). R e / is a real linear functional on A (regard A as a real Banach space). One should note that fsa and R e /

30

The Basics of C* -algebras

are different. It is standard that ||/|| = ||Re/||. This is usually used in the proof of the Hahn-Banach theorem (to pass from the real version to the complex version). It will be used later in these notes. We have the following non-commutative Jordan decomposition theorem. Proposition 1.6.13 Let A be a C*-algebra and f e A*. Then f is a linear combination of states. More precisely, we have the following decomposition: J

=

Jsa + Ifimj Jsa

=

\Jsa)+

\fsa)— and

fim

= \fim)+ ~ \fim)—i

where \\fsa\\ < \\f\\, \\fim\\ < ||/||, (fsa)+, positive and

{fsa)-,

(fim)+

and (fim)-

WfsaW = «(/„)+1| + ||(/.„)-|| and ||/ i m || = | | ( / i m ) + | | + | | ( / l m ) _ | | . Proof. It is clear that fsa and fim are self-adjoint and ||/ s a ||, ||/im|| < ll/H are obvious. For the rest of the proof, we may assume that / is selfadjoint. Let f2 be the set of all positive linear functionals g with \\g\\ < 1. With the weak*-topology, ft is compact. View A as a closed subspace of C(ft). Note that A+ C C(ft). Let f £ A*. Then / can be extended to a bounded linear functional / on C(ft) with the same norm. Therefore there is a complex Radon measure f i o n f ] such that

f(x) = [ xdfi (x G C(ft)). Let fx = v\ + 1V2, where V\ and v^ are signed measures. So Vj (j = 1,2) gives self-adjoint bounded linear functionals on A which will be denoted by fj (j = 1, 2). Since / is self-adjoint, (/2)U — 0- S° / i extends / (necessarily they have the same norm). Therefore we may assume that / is self-adjoint. So we may assume that fi is a signed measure. Then by Jordan decomposition, we have positive measures \ij (j = 1,2) such that /z = / ^ — /X2 and \\fj,\\ = ll/xiH + 11/J-211- Each fij gives a positive linear functional fj on A (j = 1,2). We have the following

11/11 = ll/UII < ll(/i)UII + ||(/2)UII < IIMI +1|/ 2 || = 11/11 = ll/llThus 11/11 = ii(/i)mi + ii(/ 2 )uii.

•

Definition 1.6.14 A representation of a C*-algebra A is a pair (H,TT), where H is a Hilbert space and 7r : A —> B(H) is a homomorphism. We say

are

Positive linear Junctionals

and a Gelfand-Naimark

theorem

31

(H, 7r) is faithful if w is injective. A cyclic representation is a representation (H, 7r) with a vector v £ H such that ir(A)v is dense in H; and the vector v is called cyclic. Theorem 1.6.15 For each positive linear functional 4> on a C* -algebra A there is a cyclic representation {H^,^^) of A with a cyclic vector v^ such that (7r^(a)i)^, v^) = 4>(a) for all a £ A. Proof.

Define the left kernel of <j> as the set Nrj> = {a£A:

<j>(a*a) = 0 } .

Since is continuous, Nj, is a closed subspace. It follows from the CauchySchwarz inequality (1.6.6) that Nj, is a closed left ideal. On A/N^ define (a,b)=<j>(b*a).

(e6.7)

It is easy to see that the above is well defined on A/N^ x A/Nj,. By the Cauchy-Schwarz inequality (1.6.6) again, A/N(a*)(j/)) which shows that ir^a)* — n^a*). So -K^ : A ->• B(H^) gives a representation. To complete the proof, let {e^} be an approximate identity for A. Then for A < fi, ||eM - e A || 2 = ((eM - e A ) 2 ) < ^(e„ - ex). Since ^(e^) -» ||0||, the net {e^} is convergent with a limit v^ £ Hj,. For each a £ A we have 7r(a)(v0) = limn^(a)(ex) A

= limaeX = a, A

32

The Basics of C*-algebras

since a —• a is continuous. Hence v$ is cyclic. Moreover, since {^^{a* a){v^),v^)

= (a, a) = <j>{a*a),

we have (7r^(a)(u^),u^) = (a) for all a £ A + (by 1.4.8) and by linearity for all a € A. • Definition 1.6.16 Let (H\,ir\)\£A be a family of representations. Let H — ®\H\ be the Hilbert space direct sum. Define w(a)({v\}) = {n\(a)(v\)}. One verifies that 7r : A —>• B(iJ) gives a representation of A. This representation is called the direct sum of {ir\}Let S be the state space of A. By 1.6.15, for each t £ S, there is a representation irt oi A. The direct sum ~K\J of {7i"t}tgs is called the universal representation of A. Theorem 1.6.17 (Gelfand-Naimark) If A is a C* -algebra, then it has a faithful representation. In other words, every C*-algebra is isometrically ^-isomorphic to a C*-subalgebra of B(H) for some Hilbert space H. Proof. We will show that the universal representation TTU is faithful. Let a G A be nonzero with ||a|| < 1. It follows from 1.6.11 that there is r S S such that |r((a*a) 2 )| = ||(a*a) 2 || = ||a|| 4 . Then HMa)||2

>

K(a)||2 > ||7rr(a)((^)^)||2

-

r((a*o) 1 / 2 (a*a)(a*a) 1 / 2 ) = r((a*a) 2 ) = ||a|| 4 .

Thus 7r[/ is injective.

1.7

•

Von N e u m a n n algebras

Definition 1.7.1 Let H be a Hilbert space and B(H) be the C*-algebra of all bounded operators on H. The strong (operator) topology on B{H) is the locally convex space topology associated with the family of semi-norms of the form x i-» ||a:(£)||> x £ B(H) and £ £ H.ln other words, a net {x\} converges strongly to x and only if { Z A ( 0 } converges to x(£) for all £ E H. The weak (operator) topology on B(H) is the locally convex space topology associated with the family of semi-norms of the form x i->- | (rc(^), 77) |, x e B{H) and £,?? G H. In other words, a net {x\} converges weakly to x G B(H) if and only if ((xA - x){$),v) -* ° for a11 £> V G ^ -

Von Neumann

algebras

33

E x a m p l e 1.7.2 Let H be an infinite dimensional Hilbert space with an orthonormal basis {en}^=1. Define a„(£) = (£, e n )ei for £ G # (n = 1, 2,...). Then a„ G B(H). We have lim^^oo IknCOII = limn-*oo |(^,e n )| = 0. Thus an strongly converges to zero. However, ||a„|| = 1 since ||a„(e n )|| = 1 for all n. Thus the strong topology is weaker than the norm topology. Define 6„(£) = (£, e\)en (n = 1,...). Then, for any £, 77 G H, > \Q nU),f])\ = \{£,ei)(en,v)\ ->• 0, as n -> 00. So 6„ converges weakly to zero. However, ||6„(ei)|| = ||e n || = 1. So bn does not converge to zero in the strong topology. The reader may want to take a look at exercises (1.11.18-1.11.22) for some additional information about the weak and strong (operator) topologies. Proposition 1.7.3 Let {a\} be an increasing net of positive operators in B(H) which is also bounded above. Then {a\} converges strongly to a positive operator a G B{H). Proof. Suppose that there is M > 0 such that \\a\\\ < M. For any ( e f f , {(aA(£)> 0 } i s a bounded increasing sequence. So it is convergent. Using the polarization identity 3

M 0 > r?) = (1/4) 2i k (a x (Z + ikV): £ + ikV), fc=0

we see that {(OA)(£)> V}} is convergent for all £, 77 G H. Denote by L(£, 77) the limit. The map (£,77) i->- L(£,rj) is linear in the first variable and conjugate linear in the second. We also have |L(£,77)| = l i m | M O , r 7 ) | < M | | £ | | | M | for all £, 77 G H. By the Riesz representation theorem, there is a G B(H) such that (a(£),7?) = L(£,rj) for all £,77 € H. Clearly ||a|| < M and a\ < a. Moreover, H0-aA(0H2

=
{a){xi,...,xn) = ( J ^ a i ^ X j ) , . . . , 3=1

j=i

for all (xu...,xn) GffK It is easy to verify that the map • B(H^),a^

4>{a),

is a *-isomorphism. We call (f> the canonical ^-isomorphism of Mn(B(H)) onto B(H(n>), and use it to identify these two algebras. We define a norm on Mn(B(H)) making it a C*-algebra by setting ||o|| = ||0(a)||. The following inequalities for a G Mn(B(H)) are easily verified: n

IK-II < \\a\\ < J2 IMI (*,J = l.-,«)

(e7.8)

k,l=l

For each i < n let Pi be the projection of H^1 onto the ith copy of H. Each element x G B(H^) has a representation (ciij)i) by setting /o(a) = diag(a, ...,a) (where a repeats n times). Lemma 1.7.5 Let (j) be a linear functional on B{H). Then the following are equivalent: (i) 4>(a) = Sfc=i( a (^)»%) for some£i,-~,tn,Vi,--,Vn G H and for all a e B(H); (ii) 4> is weakly continuous; (iii) <j> is strongly continuous. Proof. It is obvious that (i) =>• (ii) => (iii). To prove (iii) => (i), suppose that (f> is strongly continuous. Therefore, there exist vectors £i,...,£ n a n d S > 0 such that \(a)\ < 1, whenever maxfc{||a(£fc)||} < S for all a € B{H). For any a G £ ( # ) , put 6 = ( S „ = i ^ | | 2 ) l / 2 . Then \4>{b)\ < 1. Thus |^(a)|{a) = {p(a)(£),r})

Corollary 1.7.6 closed.

= ^(0^,77*,). fc=i

i?ac/i, strongly closed convex set in B(H)

Q

is weakly

We leave this to the reader for an exercise (1.11.23). Definition 1.7.7 mutant of M, i.e.,

For each subset M C B(H), let M' denote the corn-

M' = {a G B(H) :ab = ba for all b G M } . It is easy to verify that M' is weakly closed. If M is self-adjoint, then M' is a C*-algebra. We will write M" for (M1)'. The following is von Neumann's double commutant theorem. Theorem 1.7.8 Let M be a C* -subalgebra of B(H) containing the identity. The following are equivalent: (i) M = M". (ii) M is weakly closed. (iii) M is strongly closed. Proof. The implication (i) => (ii) (iii) follows from 1.7.7 and 1.7.6. We will show (iii) => (i). Fix £ G H let P be the projection on the closure of {a£ :a£ M}. Note that P£ = £ since 1 G M. Since PaP = aP for all o £ M , Pa* = P a * P for all a G M. Therefore P G M'. Let a; G M " . Then Px = xP. Hence x£ G P.ff. Thus for any e > 0 there is an a G M with

||(*-a)£||< e .

36

The Basics of C*-algebras

To show that x is in the strong closure of M, take finitely many vectors £1, - . fn € # . Set £ = & © • • • © £„ € HW. It is easily checked that p(M)'

= {a G B(H^)

: ay G M ' } .

Therefore p(x) G p(M)". From what we have proved in the first part of the proof, we obtain a G M such that n

£ | | ( z - a ) a | | 2 = |l(p(z)-Ka)KI| 2 B(Hu) be the universal representation. Then (7T[/(A))" is called the enveloping von Neumann algebra (or universal weak closure) of A and will be denoted by A". A representation IT : A —»• B(i?) is said to be non-degenerate if {7r(a)i7 : a G A} is dense in H. P r o p o s i t i o n 1.7.11 Every non-degenerate representation is a direct sum of cyclic representations The proof is an exercise (see 1.11.14). L e m m a 1.7.12 Let f G A* be self-adjoint. Then there are vectors £,rj G Hv with HCII2, |M| 2 < ll/H such that / ( a ) = (nu(a)^),ri) for all a e A. Proof. To save notation, we may assume that ||/|| — 1. By 1.6.13, there are positive linear functionals fj, j = 1,2, such that f = fi — f2 and ll/H = \\f\\\ + H/bll- Each fj is a positive scalar multiple of a state on A. Therefore there are mutually orthogonal vectors £j G HJJ (j = 1, 2) such that ll^ll 2 = H/,-11 and ft (a) = fat/(a) (&)>&>> J = !>2- S e t £ = 6 ® 6 and JJ = ^ © (-£ 2 )- Then / ( a ) = (7T(/(a)£,?7) for all a G A. Furthermore,

INI 2 = IKH2 = IKill2 + ll&ll 2 < 11/11-

Von Neumann

37

algebras

Theorem 1.7.13 The enveloping von Neumann algebra A" of a C*algebra A is isometrically isomorphic, as a Banach space, to the second dual A** of A which is the identity map on A. Proof. Let j : A —» A** be the usual embedding. Let [Hu,^u) be the universal representation of A. For each / £ A*, by 1.6.13, there are Cf^Vf e Hu such that / ( a ) = {iru(a)Cf,Vf)- Define J : A" -»• A** by J{x){f) = {x(£,f),rif) for each state / . It follows from 1.7.12 that J(x) defines an element in A**. In fact (using 1.7.12), we have \\J{x)\\=

sup |/(x)|< sup \(x(0,v)\ II/II ||z||. Suppose that x £ A"a and {ax} C A such that TTU(O,X) converges to x weakly in A". By replacing a\ by (l/2)(a\ + a*x) (see Exercise (1.11.18)), we may assume that a\ are self-adjoint. This implies that, for any self-adjoint 4> £ A*, J{x){<j>) is real. Fix f £ A* with ||/|| < 1, x £ A"a, and assume that \f{x)\ = ei9f(x). Define F = ei9f. Note that 1 > ||F|| = ||ReF||. We have | / ( i ) | = F(x) = ReF(x) = Fsa(x) =

\(x(0,v)\

for some £,77 £ Hu with ||£||, \\r]\\ < 1 (by 1.7.12). Thus \f{x)\ =

\{x{t),r,)\• M2(A)** be as above. Fix f £ A* and x € A". There are £/,?7/ € i?c/ such that J(x)(f) = (x(£f),r)f). In H^2\ set £' = 0 © £y and 77' = 77/ © 0. Define a linear functional on Q, where Q = {(aa) G M 2(A) : a n = a 22 = 0},

38

The Basics of C* -algebras

by cf>(z) = {z{i'),r]') for z G Q. So \\4>\\ = \\f\\. Extend <j> further to M2(A) such that \\<j>\\ = ||/||. By 1.6.13 and 1.7.12, there are ^,771 G Hv such that J2(z)((t>) — (z(£i),77i) for all z G M 2 (-4)". Suppose that z = (z^) G (M 2 (^4))" such that z\\ = Z22 = 0. Since z is in the weak (operator) closure of nuiQ), J2(z)(<j>) = (z(t'),r/)

= / ( z 1 2 ) . Put b = f °

* ) . Then 6 G

Af 2 (yl") so . Therefore, from what we have shown, ||J 2 (6)|| = IHI = ||z||. So

\J{x){f)\ = \J2{b){ct>)\

for all a G A. Conversely, define a linear map u from u(ni(a)£i) = 7r2(a)£2. Since

TTI(A)£I

onto

TT 2 (A)£ 2

ll«M«)fc)l| 2 = (7r 2 (o*a)6,6> = M a ' a ) £ i , 6 ) = IKi(aKil| 2 ,

by

Enveloping

von Neumann

algebras and the spectral theorem

39

we see that u extends to an isometry from the closure of iti(A)£i onto the closure of ^(A)^Since £j and £2 are cyclic vectors, u is an isometry from H\ onto Hi. We have wri(a)7n(&)£i = 7r2(a&)£2 = 7T2(a)w"i(b)£i for all a, b £ A. Thus wri(a) = 7T2(o)u since {7Ti(6)£i : b £ A} is dense in H\. The lemma follows. • Theorem 1.8.2 Let A be a C*-algebra and TT : A —5- B{H) degenerate representation. Then there is a unique ir" : A" —> that TT"\A = 7T and it is (a-) weak-weak continuous, i.e., if {x\\ convergent net in the von Neumann algebra A" then {-^"(^A)} convergent net in B(H).

be a nonir(A)" such is a weak is a weak

Proof. First assume that (7r, H) is cyclic with cyclic vector £1 and ||£i|| = 1. Then <j>(a) = (7Ti(a)£i,£i) is a state. It follows from 1.8.1 that we may assume that {n,H) = (TT^,H^). Let p^ be the projection of Hu onto H4,. Then p^u{o) = ^u{a)P4, — fl>(a) for all a € A. Thus p : C(X) —> B(H) be a unital homomorphism. Then there is a unique homomorphism 4> : B{X) —> B{H) such that 4>\c(X) = 4> and {/A} converges weakly to f in B(X) implies that {4>{f\)} converges weakly to <j>{f) in B(H).

40

The Basics of C-algebras

Definition 1.8.5 Let a e B(H) be a normal element. Then A = C*(l, a) is a C*-subalgebra. Denote by j : A —> .B(-ff) the embedding. Then by 1.8.4, let j : B(X) —> B(H) be the extension, where X = sp(a). We will use / ( a ) for j ( / ) , / € # ( ^ 0 - Thus the following is called the Borel functional calculus. Corollary 1.8.6 Let M be a von Neumann algebra and a £ M a normal element. Then there is a weak-weak continuous homomorphism from #(sp(a)) —>• M which maps z —> a. Definition 1.8.7 Let X be a compact Hausdorff space and H be a Hilbert space. A spectral measure E relative to (X, H) is a map from the Borel sets of X to the set of projections in B(H) such that (1) E( B{H) extends to a weak-weak continuous homomorphism j : B(X) —• B(H). For each Borel subset of sp(a), j(xs) = E(S) is a projection. It is easy to verify that {E(S) : S Borel} forms a spectral measure. Since the identity function on sp(a) is Borel, it follows from the Borel functional

Enveloping

von Neumann

algebras and the spectral theorem

41

calculus that

a=

f

XdEX-

•/sp(a)

We leave it to the reader to check the uniqueness of E.

•

Recall the range projection p of an operator a S B(H) is the projection on the closure of {a(£) : £ & H}. Proposition 1.8.9 If M is a von Neumann algebra, then it contains the range projection of every element in M. Proof. Let a e M. It is clear that we may assume that ||a|| < 1. Since (oa*) 1 / 2 and a have the same range projection, we may assume that a > 0. Note that {alln} is increasing and bounded. It follows from 1.7.3 that {a 1 /"} converges strongly to a positive element, say q€ M. Let / = X(o,||a||] • Then, by 1.8.6, a 1//n converges weakly to / ( a ) , which is a projection. Therefore q = f(fi)- Since a1'™ G C*( a ) and the polynomials are dense in C(sp(a)), q(H) C a(H). One the other hand, qa = f(a)a = a. Hence a(H) c q{H). Therefore q is the range projection of a. D Definition 1.8.10 An operator u on H is called a partial isometry if u*u is a projection. Since (uu*)3 = u(u*u)(u*u)u* = (uu*)2. sp(uu*) = {0,1} by the spectral mapping theorem. Therefore, uu* is also a projection. Proposition 1.8.11 Each element x in a von Neumann algebra M has a polar decomposition: there is a unique partial isometry u £ M such that u*u is the range projection of \x\ and x = u\x\. Proof. Set un = x((l/n) + l^l) - 1 and denote by p the range projection of |x|. Since x = xp, we have un = unp. We compute that \Un

t i m J [Un

Um)

= K£ + N) _1 -(^ + wr1)}*'*^ + \x\rl-(^ + Mr1} llj

Tib

1

Tli

1 2

2

= [(^ + N r - ( ^ + N)- )] N -

ifi

(es.io)

42

The Basics of C* -algebras

This converges weakly to zero (as n, m —> oo) by the Borel functional calculus. So, for any £ G H,

UK - «m)(£)H2 = I(K - Um)*K " UmXO.OI ^ ° as n , m -> oo. This implies that {«„} converges strongly to an element u G M with up = u. As in (e8.10), (un\x\ - u m |x|)*(u„|a;| - um\x\) = [(A + H ) - 1 - ( 1 + |^j)" x )] 2 1^| 4 By the continuous functional calculus (see 1.3.6), the above converges in norm (as n,m-+ oo). On the other hand, by the Borel functional calculus, ((l/n) + |:r|) -1 |a;| converges weakly to p. Thus {u n |x|} converges in norm to x. Hence x = u\x\. Since pu*up = (u*)(u) = u*u, p(u*u) = (u*u)p = u*u. On the other hand, x*x = \x\u*u\x\. For any £,77 G H, ((u*u - p)(\x\(£)), \x\rj) = (\x\(u*u - p ) ( M ( 0 ) , V) = 0. Regarding u*u—p as an operator onp(H), the above implies that u*u = p. To see the decomposition is unique, let x = v\x\ and v*v = p. Then v\x\ = u\x\, or (v — u)\x\ = 0. Therefore, (v — u)p = 0. This implies that v = u. • 1.9

E x a m p l e s of C*-algebras

In this section we give some examples of C*-algebras. More will be presented later. We first give more information about the C*-algebras K, and B{H). If H is a Hilbert space, an operator x € B(H) is said to have finite-rank, if the range of a; is a finite dimensional subspace. Denote by F(H) the set of finite-rank operators on H. It is easy to check that F(H) is a *-subalgebra of B(H) and is an ideal (not necessary closed) of B(H). Clearly every operator in F(H) is compact. It is also easy to see that F(H) is a linear span of rank-one projections. Lemma 1.9.1 If H is a Hilbert space and K(H) is the C*-algebra of all compact operators on H, then F(H) is dense in K(H). Proof. Since F(H) and K{H) are both self-adjoint, it suffices to show that every self-adjoint element x G K{H) is in F(H). We may even further

43

Examples of C* -algebras

assume that x > 0. We use the fact that 0 is the only possible limit point in sp(x). Thus there is a sequence tn G (0,1] such that tn decreases to zero and ftn{x) (see 1.5.4) are projections and are in K(H). Put pn = ftn(x). Then pnx —> x in norm. Since pn are in K(H), they can only have finite dimensional ranges. Thus pn, whence pnx is in F(H). • Lemma 1.9.2 If H is a Hilbert space and IQ is a non-zero algebraic ideal (not necessary closed) in B(H), then IQ D F(H). Proof. Let x ^ 0 be in Jo- Then there is r\ € H such that x(rj) ^ 0. Let p G F(H) be a rank-one projection. We may write p(£) = (£, e)e (for all £ £ H) for some unit vector e € H. Define y(£) = (l/||a:(77)||2)(£,:E(?7))e and z(£) = (£,77)77 for all £ G if. Then y, z G i^ff). We see that p = yx(z)x*y*. Hence p G Jo- Since /^(/J) is spanned by rank-one projections, this implies that F(H) Cl0. D Definition 1.9.3 Recall that an ideal of a C*-algebra is always assumed to be a C*-subalgebra. A C*-algebra is said to be simple if it has no proper ideals. From Lemma 1.9.2 we immediately obtain the following. Theorem 1.9.4

For any Hilbert space, K(H) is a simple C*-algebra.

Theorem 1.9.5 Let H be a separable Hilbert space. Then B(H)/K, simple C* -algebra.

is a

Proof. Let I be an ideal. It follows from 1.9.2 that I D K. Suppose that I contains an operator x which is not compact. We may assume that x > 0. For any e > 0, let pE be the spectral projection of x associated with the Borel function X[e,||x||]- Then fs/2 > X[e,||x||] ( s e e 1-5.4). Therefore Pe < fe/2{x) G I. By 1.5.12, p£ G / . Moreover, \\pex — x\\ < e. So, for some £ > 0, pe is not compact. Thus pe is a projection on an infinite dimensional subspace of H. Therefore, there is a partial isometry v G B(H) such that v*v = pe and vv* = 1# since all separable infinite Hilbert spaces are unitarily equivalent. This says vp£v* = 1#. Hence In G I. In other words, K is the only ideal of B(H). • Definition 1.9.6 A C*-algebra A is said to be finite dimensional if A is a finite dimensional vector space. A representation 7r of A is said to be finite dimensional if IT (A) is finite dimensional Theorem 1.9.7

If A is a finite dimensional C* -algebra, then A = Mni © • • • © M,

44

The Basics of C*-algebras

Proof. First we assume that A is simple. By the proof of 1.6.15, A has a (non-zero) finite dimensional representation 7r. Since A is assumed to be simple, 7r is an isomorphism. So, without loss of generality, we may assume that A C B(H) for some finite dimensional Hilbert space H. It is obvious that in finite dimensional Hilbert space, the weak operator topology is the same as the norm topology. Therefore A is a von Neumann algebra. Let A' be the commutant of A. A' is a von Neumann algebra. Let p be a (nonzero) minimal projection of A'. Since pA'p is a von Neumann algebra, by the spectral theorem, pA'p = Cp. Note that A = pAp © (1 — p)A(l - p) and pAp is a von Neumann algebra. So p € A. Therefore pAp is an ideal of A. Since we assumed that A is simple, p = 1. Thus A' = C Hence A = A" = B{H), by the Double Commutant Theorem. In other words, A = Mni, where n\ is the dimension of H. For the general case, note A** = A. Thus A is a von Neumann algebra. In particular, A is unital. Let / be a (non-trivial) maximal ideal of A. Then A/1 is nonzero and simple. So the dimension of / is smaller than that of A. We have shown that A/1 = Mni. Since / is also a von Neumann algebra, I is unital. Thus A = Mni © I. Since / has a smaller dimension than that of A, an induction argument shows that A = Mni © • • • © Mnk as desired.

• Remark 1.9.8 Theorem 1.9.7 characterizes finite dimensional C*algebras. Let A be a finite dimensional C*-algebra. Then A = Mni © • • • © Mnk. Let ejj be the (n x n) matrix which has 1 for its (z,j)-entry and is zero elsewhere. We will call {e^} the set of matrix units. Let {e\j} be the set of matrix units for Mni. We will call {e\j} the canonical generators for A. Definition 1.9.9 If A is an algebra, Mn{A) denotes the algebra of all n x n matrices with entries in A. The operations are defined just as for scalar matrices. If A is a *-algebra, so is Mn(A), where the involution is given by (a^)* = (a*J. Thus Mn(A) may be identified with (the algebraic tensor product) A® Mn. Let eij be the matrix units for Mn. Then a ® e^ is the element in Mn(A) which has a for its (i, j)-entry and is zero elsewhere. Let b e Mn and b = £)",- = 1 A^-ey, where A^- € C. Then a®b= ^ ™ J = 1 Ay a ® e^. Thus (a ® b)* = a* b*. If a' G A and b' G M„, then (a ® b)'(a' b') = aa' bb'. Theorem 1.9.10 If A is a C* -algebra, then there is a unique norm on Mn{A) making it a C*-algebra.

Examples of C*-algebras

45

Proof. Let the pair (H, it) be a faithful representation of A so that the *homomorphism n^ : Mn(A) —• Mn(B(H)) is injective. We define a norm on Mn(A) by setting ||a|| = ||7r(n)(a)|| for a S Mn(A). Since Mn{B{H)) is a C*-algebra, ||a|| is a C*-norm. To see that Mn(A) with this norm is a C*algebra it remains to show that it is complete. But this follows immediately from the inequalities (e7.8). • To study the C*-algebra A it is often important to study

Mn(A).

Example 1.9.11 Let X be a locally compact metric space and A be a C*-algebra. A map / : X —>• A is said to vanish at infinity, if for any £ > 0, there is a compact subset Y C X such that ||/(t)|| < £ for all t £ X \Y. Define Co (X, A) to be the set of all continuous maps from X into A vanishing at infinity. Using pointwise operations, Co(X, A) becomes an algebra. Set /*(*) = /(*)* f° r e a c n t E X. Then C0(X, A) is a *-algebra. Let ll/H = sup{||/(t)|| : t <E X}. It is routine to check that C0(X,A) is a C*-algebra. Example 1.9.12 In Example 1.9.11, let A = Mn. Then one obtains a C*algebra C0{X,Mn). Fix t G X, then f(t) € Mn for every / e CQ(X,Mn). Write f(t) = (fij(t)) (an nxn matrix). Then fij(t) £ C0(X). This gives an identification C0(X,Mn) = Mn(C0(X)). Both are C*-algebras. By 1.2.8, the identification is an isomorphism of C*-algebras. This identification will be frequently used. Example 1.9.13 Let X be a locally compact Hausdorff space. Denote by Cb(X) the set of all bounded continuous functions and define ll/H = s u p t 6 X \f(t)\- It is a (non-separable) unital C*-algebra which contains CQ(X) as an ideal. With Gelfand representation, Cb(X) = C((3(X)) for some compact Hausdorff space (3(X). This compact space f3(X) is the same as the Stone-Cech compactification of X. Let A be a unital C*-algebra. We denote by Cb(X, A) the set of all continuous maps from X into A. Clearly, C0(X, A) is a an ideal of Cb(X, A). Definition 1.9.14 Let B be a subset of B(H). A closed subspace Hi of H is said to be invariant under B if x(Hi) C H\ for all a; G B. Let A C B{H) be a C*-subalgebra and let H\ be invariant under A. There are two trivial invariant spaces: {0} and H. Put p the projection on H\. Then for any a € A, ap = pap. In particular, a*p = pa*p and ap = pap = (pa*p)* = (a*p)* = pa. Thus ap = pa for all a € A.

46

The Basics of C*-algebras

A representation 7r : A —¥ B(H) is said to be irreducible if ~K(A) has no non-trivial invariant subspace. A C*-algebra A is said to be liminal if n(A) = K{Hn) for every irreducible representation 7r of A. A C*-algebra A is said to be type 7, or postliminal, if TT(A) D £(77^) for every irreducible representation 7r of A. Example 1.9.15 Every commutative C*-algebra is liminal (see 1.11.48). B(l2) is not a type I C*-algebra(see 1.11.49). Every finite dimensional C*algebra is liminal. If A is liminal (postliminal), then Mn(A) is liminal (postliminal) for every integer n, and C0(X, A) is liminal (postliminal) for every locally compact Hausdorff space X. Proposition 1.9.16 A separable simple C* -algebra is of type I if and only if it is isomorphic to K or Mn for some integer n > 0. Proof. Let -K : A —>• B{Hir) be an irreducible representation. Then n(A) D K(HT,:). So A has an ideal 7 so that ir{I) = £(77^). Since A is simple, A S 7r(vl) and I = A. Therefore A ^ 7r(A) = /C. • Definition 1.9.17 A classical dynamical system consists of a compact Hausdorff space X together with a homeomorphism a. Define a(f) = foa~x for / € C(X). For any C*-algebra A, an isomorphism from vl onto itself is called an automorphism. Denote by Aut(A) the automorphism group of A. Then a G Aut(C(X)). Also Z ->• Aut(A) given by n H> a n is a group homomorphism. Given a dynamical system (X,a), a covariant representation is a pair (7r, w), where 7r is a representation of C(X) on a Hilbert space 77 and u a homomorphism from Z to the unitary group in 7?(77) such that u(n)7r(a)(u(n))*=7r(a n (o)). for all a e C(X) and n G Z. A finite Borel measure (j, on X is said to be translation invariant if ^(cr~1(E)) = fi(E) for every Borel subset of X. We need the following fixed point theorem of Markov-Kakutani. L e m m a 1.9.18 Let T be a family of commuting continuous linear maps from a topological vector space X into itself. Suppose that C is a compact convex subset of X such that L(C) C C for all L E T. Then there exists £ G C such that L{£) = £ for all LeT.

47

Examples of C* -algebras

Proof.

For any integer n > 1 and L e T, put L(n> = - ( i d + L + L2 + • • • + Ln~l) n

and CUtL = L^C. Then Cn^ commutes, we have

is a compact convex subset of C. Since T

n?=iCni,LiD(n£in°)(C). 1=1

Denote by ^4 the algebra generated by T. Then the family {Cn,L '• n > 1 , 1 6 ^ } satisfies the finite intersection property. So by compactness, there is a point £ € C\{CntL : n > 1, L G .4}. Let G be any open neighborhood of the origin of X, and let L G T'. Since C is compact so is C — C. Thus there is an integer n so that C - C c nG. Now £ € Gn)£,. There is a point r] £ C such that L ^ r j = £. Hence L£ _ £ = I (id + L + • • • + Ln~l){Ln n

-rj) = \lnr) n

- v) G -{C - C) C G. n

Since this holds for any such G, L£ = £.

D

Theorem 1.9.19 Let (X, <x) be a classical dynamical system. Then there is a Borel probability measure \i on X which is translation invariant for a. Proof. Let a ( / ) = / o o~~l be an automorphism on C(X). It is a norm one linear operator. Let a* be the adjoint on the dual space M{X) of all regular Borel measures on X (a*(/x)(/) = fx a(f)dfi). Then a* has norm 1. Since a > 0, a*(/x) > 0 if /x > 0. If \i > 0, then ||a*( M )|| = (a*(/x))(X) = ^{?-\E)) = (a*M)(£) = no(E) for all Borel sets E. Thus (J,Q is a translation invariant measure. Definition 1.9.20 Let (X,a) be a classical dynamical system and let fj, be a translation invariant measure. Let H — L2(fi). Define 7r(x)(/)(t) =

•

48

The Basics of C* -algebras

x(t)f(t) for / G L2(fi), x G C(X) and t G X, and define U(f) = / o a~\ By the translation invariance of fi, we compute (for x, y G C(X)) (Ux,Uy) = / a(y*)a(x)dfi

= / y*xd(a*(fj,)) = / y*a;o^ = (a;,?/).

Since [/ is clearly invertible, it is a unitary. Finally, (U*(x)U*)(f)(Q

=

=

(Tr(x)U*)(f)( S1 by cro{z) = e _ J z, where z is a complex number with \z\ = 1. The homeomorphism is called irrational rotation. Clearly, the usual normalized Lebesgue measure m on the circle is GQ-invariant. It is easy to see that ag is minimal. So we may write C*(S1,ag,m) = C(SX) xae Z. This is also called the irrational rotation algebra and may also be denoted by Ag. Note the left multiplication maps CiS1) into C*(S1,a0,m) injectively. We identify C(5 X ) with its image m 1 1 C(S )x(TgZ. Let v G C(S ) be the unitary so that v(z) = z for z G S1. Let u be the unitary induced by ag, i.e., ux(f(z)) = x{f\e~"a'n6'z)) for x G C(S1), f G L2{Sl,m) and z G S1. Therefore uv = e~i27v9vu. Since C(S1) is generated by v, Ag is generated by u and v.

(e9.11)

49

Examples of C -algebras

Proposition 1.9.23 Let B be a C*-algebra and v, u be two unitaries with sp(v) = S1. Suppose thatv andu satisfy (e9.11). Then there is a surjective homomorphism from C*{v,u) onto Ag which maps v to the function z : z H-» z and u to the unitary so that uzu* = e~z2nez. Proof. Identify C*(v) with C(5' 1 ) so that v(z) = z for z € S1. Define an automorphism a : C^S 1 ) ->• C^S 1 ) by a(v) = u(v)u* = e " i 2 7 r V So for any polynomial p(z) = Y^k=-N ckz>c (ck e c )> N

JV

a(p(v))=

ck(uvu*)k=

^ k=-N

] T e-i2*keckvk

= (po<j0)(v),

k=-N

where (poag)(z) = p(e~l27r6 z). Let m be the normalized Lebesgue measure on S1. Define ^ m : C(5 X ) -»• B(L2(S\m)) by m(a)(/)(z) = a(z)f(z) for / G L2{S\m) and z € S1. Define Vm(«)(/)(z) = / M * ) ) = /(e" i 2 7 r e z) for / G L2(S\m) and 2 e S1. Note that C*(V' m (C(S' 1 )), Vm(«)) = Ag. Thus we can define a surjective homomorphism 7rm : C*(v,u) -+ Ag. • Theorem 1.9.24

TVie irrational rotation algebra Ag is simple.

Proof. Put Afl = C(Sl) xae Z. Let 7 be an ideal of Ag and J = 7nC(S' 1 ). There is a closed subset F C S1 such that J = {f € C(Sl) : f\F = 0}. Since uJu* C J, we have ag(F) C 71. Since erg is minimal, J = {0} or J = C(X). Note that 1A e C(X). Therefore 7 is trivial if J ^ {0}. Suppose that I is an ideal and 7 ^ Ag. Let ir : Ag ^- Ag/I be the quotient map. Then we have by (e9.11) 7r(u)7r(v) = e-i2n0Tr(v)Tr{u). Furthermore, C*(w(v), 7r(u)) = Ag/I. Since we have shown that 7nC(S' 1 ) = {0}, sp(-7r(t))) = S1. It follows from 1.9.23 that there is a surjective homomorphism 7rm : Ag/I —• Ag such that 7rm(7r(v)) is the identity unitary in C(5 1 ) and 7rm(7r(u))7rm(7r(v))7rm(7r(u*)) = e-i27T0nm(n(v)). Thus 7rm o 7r = id on Ag. Hence 7 = ker7r = {0}. This proves that Ag is simpleD Example 1.9.25 Let G be a group which is also a topological space. G is said to be a topological group if the mappings (x, y) K-> xy and a: i->- a; -1 are continuous. We will consider locally compact (Hausdorff) groups only. An abelian example is R. A unitary representation of a locally compact group is a homomorphism from G to the unitary group of 7? (77) for some Hilbert space 77 which is continuous in the strong operator topology, i.e., g 1-4- n(g)£

50

The Basics of C* -algebras

is continuous for every vector £ € H. A representation 7r : G —> B(H) is irreducible if 7r(G) does not commute with any proper projections in B(H). This is equivalent to requiring that G*(7r(G)) has no non-trivial invariant subspaces. Let II be the set of unitary representations of G. The group C* -algebra of G, denoted by C*(G), is the closure of the universal unitary representation ©^en of G. Every locally compact group G has a regular Borel measure HG such that (laidE) = I^G(E) for all Borel subsets of G and g G 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 /x(G) = 1. If G is infinite and discrete, then we normalize it so that /x(e) = 1, where e is the unit of the group. G has a distinguished representation called the left regular representation on L2(G, fia). This is defined by A(s)(/(i)) = fs(t) = / ( s _ 1 i ) (s,t G G and / e L2(G,fia)- Since HG is invariant under left translation,

- I h{s-H)f{s-H)d^G{t) = J hfdfiG = (f,h). JG

JG

So A(s) is a unitary. It is then clear that A is a unitary representation. The reduced group C*-algebra is G*(A(G)) and is denoted by C*{G). Definition 1.9.26 For any locally compact group G, there is a surjective homomorphism from C*(G) onto C*(G). A locally compact group G is said to be amenable if this homomorphism is an isomorphism, i.e. C*(G) =

c;{G). Every abelian group is amenable (see Exercise (1.11.31)). Definition 1.9.27 Let I be an indexing set (discrete) and {Ai : i G 1} be a family of G*-algebras. The set of functions from x : I —>• UiAi such that x(i) e Ai and i \-* \\x{i)\\ is bounded is a G*-algebra with pointwise operations. It will be denoted by JJieI Ai and called the direct product of C*algebras {Ai : i e I}. The G*-subalgebra of Y\i€l Ai such that ||a;(i)|| -> 0 is called the direct sum of {Ai : i G 1} and denoted by @ieiAi. If I is finite, then Yliei ^* = ®i€iAi and is denoted simply by A Let ®i£jAi be the algebraic direct sum of G*-algebras Ai. Then ®ieiAi is the G*-algebra closure of ®f^IAi. When I = N, U™ An is the set of all bounded sequence {an} with an G A„ and ®^LiAn is the set of all

Inductive

51

limits of C -algebras

sequences {an} such that an G An, and ||a n || -> 0. If A = Ai for all i £ I, then Y[Ai

= Cb(I,A)

and ®i€l Ai =

C0(I,A).

iei

1.10

I n d u c t i v e limits of C*-algebras

Definition 1.10.1 Let {Gn} be a sequence of groups and <j)n : Gn —» G„+i be homomorphisms. We write 4>n,n = idGn and (if n < m) n,m — 4>m-i ° • • • 4>n • Gn -> Gm. If k < n < m, t h e n feim = 4>n<m ° n,n+i(g) for all i > 0. then