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,||£||
2

The Basics of C* -algebras

A commutative Banach algebra is a Banach algebra A with the property that ab = ba for all a, b £ A Examples 1.1.2 and 1.1.4 are of commutative Banach algebras while Example 1.1.3 1.1.5 are not commutative. Definition 1.1.6 In a unital algebra, an element a £ A is called invertible if there is an element b £ A such that ab = ba = 1. In this case b is unique and written a - 1 . The set GL(A) = {a £ A: a is invertible} is a group under multiplication. We define the spectrum of an element a to be the set sp(a) = spA(a) = { A e C : A l - a 0

GL(A)}.

Whenever there is no confusion, we will write Al simply as A. The complement of the spectrum is called the resolvent and -R(A) = (A — a ) - 1 is the resolvent function. E x a m p l e 1.1.7 Let A = C(X) be as in 1.1.2. Then sp(/) = f(X) for all / £ A. In other words, the spectrum of / is the range of / . Let A — Mn. If a = ( a y ) £ A, then the reader can check that sp(a) is the set of eigenvalues of the matrix a. Proposition 1.1.8

For any a and b in A, sp(ab) \ {0} = sp(ba) \ {0}.

Proof.

If A ^ sp(ab) and A ^ 0, then there is c £ A such that c(A — ab) = (A — ab)c = 1.

Thus c(a6) = Ac — 1 = (ab)c. So we compute that (1 + bca)(X — ba) = =

A — ba + Xbca — bcaba A — ba + b(X — ab)ca = A — ba + ba = A,

which shows that A _ 1 (l + bca) is the inverse of A — ba. Hence A ^ sp(6a) and sp(6a) \ {0} c sp(a6) \ {0}. • Definition 1.1.9 A Banach algebra A is said to be unital if it admits a unit 1 and ||1|| = 1. Banach algebras in 1.1.2, 1.1.3 and 1.1.4 are unital.

3

Banach algebras

Lemma 1.1.10 Let A be a unital Banach algebra and a be an element of A such that ||1 - a\\ < 1. Then a G GL(A) and oo

1

a" = 5 ^ ( 1 - a ) " . n=0

Moreover, ||<x—x[| < yrpr^ir Proof.

and

I'1

- a_1

H - i-l|I-l|l •

Since oo

oo

1

E iia - ° n * E in - «ir = a-..I- 0 |D < °°'

n=0

the series J2^1 0 (l (l-P-a||)

V

n=0

—a

"

"'

s

)™ * convergent. Let b be its limit in A. Then ||6|| <

aIld

00

Ml

ni-6ii<Eiii-«ir n=l

II

1 -"

l -"o "

'

One verifies k

a ( E ( l - <*)") = (1 - (1 " a ) ) ( E ( l ~ ° n = ! - (1 " a ) f c + 1 n=0

n=0

and that it converges to ab = ba = 1 as k -> oo. Hence 6 is the inverse of a.

• Definition 1.1.11 A function / from an open subset Q, c C to a Banach algebra is said to be analytic, if for any Ao G fi there is an open neighborhood O(A0) such that /(A) = J2^=0an(X - A0)™ converges for every A G O(Ao). To include the case that Ao = oo, we say / is analytic at infinity if /(A) = X^^Lo an^~n for all A in a neighborhood of infinity. Theorem 1.1.12 In any unital Banach algebra A, the spectrum of each a G A is a non-empty compact subset, and the resolvent function is analytic on C \ sp(a). Proof.

If |A| > ||a||, then ||A- n a"|| < (^)n.

E n=0

d

So the series

4

The Basics of C* -algebras

converges (in norm). Similarly to 1.1.10, \n+l \k+l' (*-)E^i^-(? K

n=0

which converges to 1. This shows that sup A6sp ( a ) |A| < ||a|| and R(X) is analytic in {A : |A| > ||a||}. Moreover, lim P ( A ) | | < |A|->oo"

V

'"

lim

|A|

|, I, = 0 .

|A|->oo 1 _ ( M l )

(el.l) V

'

Similarly, if A0 — a is invertible and |A — AQ| < . A _',.!,,, then (A-ar^^Ao-AraAo-a)-1)" This also shows that the resolvent is open. Since sp(a) has been shown to be bounded, sp(a) is compact. We have also shown that the resolvent function is analytic on the complement of the spectrum. In particular, f(R(X)) is a (scalar) analytic function for every bounded linear functional / € A*. If sp(a) were empty, then f(R(X)) would be an entire function for every f £ A*. However, ( e l . l ) shows that f(R(X)) is bounded on the plane. Thus Liouville's theorem implies that f(R(X)) is a constant. But (e 1.1) also implies that f(R(X}) = 0. So, by the Hahn-Banach theorem, R(X) = 0. This is a contradiction. Hence sp(a) is not empty. • Corollary 1.1.13 isC.

The only simple commutative unital Banach algebra

Proof. Suppose that A is a unital commutative Banach algebra and a € A is not a scalar. Let A € sp(a). Set I = (a — X)A. Then I is clearly a closed ideal of A. No element of the form (a — X)b is invertible in the commutative Banach algebra A. By 1.1.10, ||(a-A)6-l||>l. So 1 ^ / and / is proper. Therefore, if A is simple, a must be a scalar, whence A = C. • Lemma 1.1.14 If p is a polynomial and a is an element of a unital Banach algebra A, then sp(p(a)) =p(sp(a)).

Banach

5

algebras

Proof. We may assume that p is not constant. If A 6 C, there are c, 0\,..., [3n G C such that n

p(z)-A = cJI(2-ft). i=l

and therefore

p(a)-A = cJJ(a-/3 i ). i=i

It is clear that p(a) —A is invertible if and only if each a—ft is. It follows that A G sp(p(a)) if and only if A = p(a) for some a G sp(a). Thus sp(p(a)) = p(sp(a)). D Definition 1.1.15 Let A be a unital Banach algebra. If a G A, its spectral radius is defined to be r(a) =

sup |A|. Agsp(a)

Theorem 1.1.16

If a is an element in a unital Banach algebra A, then r(o) = lim H a l 1 / " . n—foo

Proo/. If A G sp(a), then A™ G sp(a") by 1.1.14, so |An| < ||a n ||. Therefore, r(a) < liminfra_>.00 Ha™!!1/™. Let ft be the open disk in C with center 0 and radius l/r(o) (or oo if r(a) = 0). If A G ft, then 1 - Xa G GL(A). If / G A*, then / ( ( l — Aa) _1 ) is analytic. There are unique complex numbers zn such that oo

/((l-Aa)"1) = ^ z n A "

(AG ft).

71=0

However, if |A| < l/||a|| < l / r ( a ) , then ||Aa|| < 1, so (1 - Xa)-1 = E~=o A n ° " . a n d therefore, / ( ( l - Xa)'1) = £ ~ = 0 Xnf(an). It follows that z»» = /(a™) f° r a ll « > 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/n<^,

n = 0,l,....

6

The Basics of C* -algebras

Consequently, limsupllal1/™^/^!. n—too

This implies that if r(a) < - i , then limsup„^ 0 0 ||o."-||x/" < 1/|A|. It follows that limsupyj^^ Ha™!!1/™ < r{a). From what we have shown at the beginning of this proof, we obtain l i m s u p l l a l 1 / " < r(a) < liminf ||a"|| 1 / n . n->oo

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{)\<^U\\^V{\f{X)\\\{X-x)-'\\:\€C},

Banach

7

algebras

where / is the length of the curve C. Hence there exists an F G A** such t h a t F(4>) = L(4>).

On the other hand, the function A i-> /(A)(A — z ) - 1 is a continuous function from C into A. So the limit of n

^2f(^i)(^i

~ a;)~1(Ai - A i+ i) (as maxi|Aj - A i + i| ->• 0),

i=0

where {Ao,..-, An, A„+i = A0} is a partition of the curve C, converges in norm in A to y. By the continuity of <j>, we know that (y) = L{) for all
/(l)=

2^X / ( A ) ( A _ a 0 _ l d A '

We denote by Hol(sp(a;)) the algebra of all functions which are holomorphic in a neighborhood of sp(aj). Theorem 1.1.19 Fix an element x in a unital Banach algebra A. The map f H4- f(x) from Hol(sp(a;)) into A is a homomorphism which sends the constant function 1 to the identity of A and the identity function on C to the element x. If f{z) = ^Z^Lo c".z™ * n a neighborhood of sp(x), then f(x) = I2™=ocnXnProof. Linearity is clear. Let / and g be functions holomorphic in neighborhoods Of and Og of sp(a;), respectively. Set O = Of D Og and let Ci,i = 1,2 be smooth simple closed curves in O enclosing sp(x) such that C\ lies completely inside the curve C^. Then

f(x)g(x)

=

= =

{^-ff{X){X-x)-H\){~jg{Z){z-X)-'dz)

~ i / [ i fWdi^-xr'iz-xr'd^dx 1

1 IIWg[z) I » m , , [ ( ^ - » ) ~ ' - ( A - J ) ' dzdX ] , ,,

'^kk

A^

The Basics of C* -algebras

- . I f(\)9{\){\-x)-ld\ 2ni J Ci The second to last equality holds because ^ Cauchy formula) and because Q^

=

{f-g){x).

Jc ^z\dz

= g(X) (by the

is holomorphic inside the curve Ci if

A e C , (so that JCi j^dX = 0). To complete the proof, pick a circle C with center at 0 and large radius. We note that oo

oo

A " " 1 ^ - A " 1 ^ - 1 = A™"1 £ V A ~ * = n=0

^xkXn-k~\ k=0

where the convergence is in norm and is uniform on C. Therefore f xk\n~k-1d\ = ( f Xn~k-1d\)xk = 0 • xk = 0 (e 1.2) Jc Jc unless k = n, in which case the integral is 2irixn. This implies that xn = — f Xn(X - a;)_1dA. (e 1.3) 2ni Jc Now suppose that f(z) = X ^ o c"z™ m a neighborhood of sp(a;). Then it converges in an open disk with center 0. Let C be a circle with center at 0 contained in the open disk. Then the series converges uniformly on C, so from (el.3),

Proposition 1.1.20 If I is a closed ideal in a Banach algebra, then A/1 is a Banach algebra with the quotient norm ||a|| = ||a + /|| = inf||a + 6||. Proof. It is well known that A/I, as a normed space is complete. It remains to show that A/I is a normed algebra. Let e > 0 and a, b £ A. Then, there are i, j £ I such that \\a + i\\<\\a + I\\+e

and \\b + j \ \ < \\b +I\\ + e.

Hence, for c = ib + aj + ij £ I, \\ab + c\\ < ||a + *||||& + j | | < (||a + /|| +e)(||6 + 7|| + e ) .

C* -algebras

9

Thus, ||a6 + / | | < ( | | o + / | | + e ) ( | | 6 + / | | + e ) . Letting e —¥ 0, we obtain ||ob + J | | < | | a + /||||6 + /||. In other words, A/1 is a normed algebra.

1.2

•

C*-algebras

Definition 1.2.1 An algebra A is called a *-algebra if it is a complex algebra with a conjugate linear involution * which is an anti-isomorphism, i.e., for any a, b E A and a E C, (a + 6)* = a* + b*, (aa)* = aa*,a** = a and (afe)* = 6*a*. li a £ A, then a* is called the adjoint of a. Let A be a *-algebra which is also a normed algebra. A norm on A that satisfies

Kail = HI2 for all a E A is called a C*-norm. If, with this norm, A is complete, then A is called a C*-algebra. Since ||a;*x|| < ||x*||||a;|| we have ||x|| < ||z*|| for all x E A. Thus ||a;*|| =

INA closed *-subalgebra of a C*-algebra is also a C*-algebra. Such a *subalgebra will be called a C*-subalgebra. E x a m p l e 1.2.2 (a) Let A be as in 1.1.3 and let a = (a^) E A. Define a* = (Sji). Then A is a C*-algebra. (b) Let X be a locally compact Hausdorff space and CQ(X) be the set of all continuous functions vanishing at infinity. Define f*(t) = f(t) (for t E X). Then C0(X) becomes a *-algebra. With ||/|| = s u p t 6 X |/(t)|, CQ(X) is a C*-algebra. CQ(X) is unital if and only if X is compact. E x a m p l e 1.2.3 Let X be a compact Hausdorff space and B{X) be the set of all bounded Borel functions on X. With ||/|| = s u p x e X | / ( x ) | and f*(x) = f(x), B(X) becomes a C*-algebra. For any complex Borel measure

10

The Basics of C*-algebras

fi on X, there is a bounded linear functional F^ G C(X)* such that F^f) Jx fd\x. It is clear that (by considering point-evaluation)

=

sup | ^ ( / ) | > ll/H IMI
IW)I< / l/|dH<||/HIH|. Jx Thus the C*-algebra B(X) is the same Banach space when we regard it as a subspace of the second dual of C(X) as described above. Definition 1.2.4 Let A be a C*-algebra. An element x G A is normal if xx* = x*x and is self-adjoint if a; = x*. The self-adjoint part of A is denoted by Asa. For each x G A, the element A (a: + x*) is in Asa, and is called the real part of x, and the element — | ( x — x*) is in v4sa and is called the imaginary part of x. It follows that Asa is a closed real subspace of A and each element x G A has a unique decomposition as x = y + iz with y, z G v4Sa • An element p G Asa is called a projection if p 2 = p. When A admits a unit, we denote the unit (or identity) by 1A, or 1 when there is no confusion. If A has a unit, then A 7^ 0. Since 1^ = 1^, 2 an ||1A|| = l|l/i|| d ||1A|| = 1- Thus a C*-algebra with unit is a unital Banach algebra. We will call it a unital C*-algebra. An element u in a unital C*-algebra is unitary if u*u = uu* = 1. Since i* = i , H = i. One can always unitize a C*-algebra. Proposition 1.2.5 For each C*-algebra A there is a C*-algebra A with unit containing A as a closed ideal. If A has no unit, A/A = C. Proof. Let B(A) be the set of all bounded linear operators on A. Consider the map n : A -» B{A) denned by n(a)b = ab for all a, b G A. (This map is often called the left regular representation of A ) It is clear that TT is a homomorphism. Since ||7r(a)6|| < ||a||||6||, ||7r(o)|| < ||a||. Also ||a|| 2 = llaa'H = ||7r(a)al < |k(a)||||a*|| = ||7r(a)||||a||. Hence n is an isometry. Let 1 denote the identity operator on A and let A be the algebra of operators on A of the form -rr(a) + A • 1 with a £ A and A G C. Since TT(A) is complete and A/n(A) = C, A is also complete. Define

C*

11

-algebras

an involution on A by denning (n(a) + Al)* = 7r(a*) + Al. For each e > 0, there is b G A with ||6|| = 1 such that ||7r(a)+Al|| 2

<

£ + | | ( a + A)6||2 = £+||&*(a*+A)(a + A)&||

<

e+\\(a*+

A)(a + A)6|| < s + ||(7r(a*) + A)(7r(a) + A)||.

So A becomes a C*-algebra.

D

Definition 1.2.6 For a non-unital C*-algebra A, the spectrum oix G A, denoted by sp(a;), is defined to be the spectrum of x in A. Lemma 1.2.7

If A is a C*-algebra and x G A is a normal element, then \\x\\ = r{x).

Proof.

If x G Asa, then ||a;2|| = ||:r||2 implies that r(a;)= lim Ha;2"!!1/2" = ||x||. n—>oo

If in general, x is normal, we obtain r(xf

<

||a;||2 = ||a;*x|| = lim ||(a;*x) n || 1/n n—¥oo

<

lim(||(i*)"||||a:n||)1/noo

Hence r(x) = \\x\\. Corollary 1.2.8 C* -algebra.

D There is at most one norm on a *-algebra making it a

Proof. If || • ||i and || • ||2 are two norms on a *-algebra A making it a C*-algebra, then ||a|| 2 = ||a*a||i = r(a*a) = sup{|A| : A £ sp(a*a)} (t = 1,2). Thus ||o|| 1 = ||o|| 2 .

•

Lemma 1.2.9 Let A be a C*-algebra and a e Asa. Then sp(a) C R. / / u € A is a unitary, then sp(u) is a subset of the unit circle. Proof. Let u be a unitary in a unital C*-algebra A and A G sp(u). Since ||u|| 2 = ||u*u|| = ||1|| = 1, |A| < 1. However, A" 1 G s p ^ " 1 ) . Since u~l = u* is also a unitary, we conclude that |A| = 1. For a G Asa, by considering A if necessary, we may assume that A is unital. The function exp(iz) = X]^Lo(*z)™/n! is an entire function. By applying 1.1.19, we see

12

The Basics of C*-algebras

that u = exp(ia) is a unitary (with u* = exp(-ia)). ° = £ £ L i ( * ) n ( a - A)""1/™!, then by 1.1.19,

If A S sp(a) and

exp(ia) - eiX = (exp(i(a - A)) - l)e i A = (a - X)beix. Since b commutes with a (by 1.1.19), and a —A is not invertible, exp(ia) — elX is not invertible. Hence \elX\ = 1, and therefore A € R. In other words, sp(a) CM. D

1.3

Commutative C*-algebras

The C*-algebra Co(X) in 1.2.2 (b) is a commutative C*-algebra. In this section, we will show that every commutative C*-algebra has this form. Definition 1.3.1 A multiplicative linear functional on a Banach algebra A is a nonzero homomorphism of A into C The set of all multiplicative linear functionals on A is called the maximal ideal space of A and will be denoted by Cl(A). If A is a unital algebra, and / is a proper ideal of A, then an application of Zorn's lemma shows that there is an ideal M D / such that A/M is simple and nonzero. Such ideals are called maximal ideal. We leave this to the reader as an exercise. Theorem 1.3.2 Let A be a unital commutative Banach algebra. (1) If£il{A), then \\\\ = 1. (2) The space £l(A) is non-empty, and the map defines a bijection from Q,(A) onto the set of all maximal ideals of A. Proof. Suppose that <j> £ Q,(A) and a G A such that ||a|| < 1 = (a). Let b = YlnLi a™- Then a + ab = b and 4>(b) = <}>{a) + (a)(b) = 1 + ^ ( 6 ) . This is not possible. So ||^|| < 1. Since (f)(1) = 1, we proved that \\4>\\ = 1. For part (2), let <j> € fl(yl). It follows that M = ker is a closed ideal of codimension 1 in A, and thus is maximal. If <j>\,4>2 G 0(A) and ker^i = ker2, then for each a G A, a - 4>2{a) £ kerx. This implies that <j>i(a — 4>2(a)) = 0 or i(a) = ^ ( a ) - This shows the map is one-to -one. Conversely, if M is a maximal ideal, then dist(M, 1) > 1 because the open unit ball with center at 1 consists of only invertible elements (by

Commutative

C* -algebras

13

1.1.10). It follows that the closure of M still does not contain 1. It is easy to see that the closure is an ideal. So it is a proper ideal. We conclude that M itself is closed. So the quotient A/M is a simple commutative Banach algebra. By Lemma 1.1.13 A/M = C. So this quotient map (j> gives a continuous homomorphism from A —¥ C with ker = M. The map is therefore bijective. To see that Q(A) is non-empty, we may assume that y l ^ C , otherwise the identification of A with C gives a nonzero homomorphism. So A is not simple (by 1.1.13). Let 7 be a proper ideal of A. Since A has an identity, there is a maximal proper ideal of A containing I (see 1.3.1). • If A is a unital commutative Banach algebra, it follows from 1.3.2 that Cl(A) is a subset of the (closed) unit ball of A*. So £l(A) with the relative weak *-topology becomes a topological space. Theorem 1.3.3 If A is a unital commutative Banach algebra, then Cl(A) is a compact Hausdorff space. Proof. It is easy to check that il(A) is weak* closed in the closed unit ball of A*. Since the unit ball of A* is weak* compact (Banach-Alaoglu theorem), Q(A) is weak* compact. • Definition 1.3.4 Suppose that A is a unital commutative Banach algebra and il(A) is its maximal ideal space. If a £ A, we define a function a by a(t)=t(a)

(ten(A)).

The set { £ il(A) : \(/>(a)\ > a} is weak* closed (for every a > 0) in the closed unit ball of ^4*. Thus it is weak* compact by the Banach-Alaoglu theorem. Hence a e C(ft(A)). Thus we define a map T : a —• a from A into C(Q(A)). This map is called the Gelfand transform. It is clear that T is a homomorphism. Assume A is a non-unital C*-algebra and A is its unitization. If <j> : A —• C is a nonzero homomorphism, then <j>{a + A) = <j>{a) + A (a G A and A 6 C) is a homomorphism from A into C. Thus Q.{A) is a subset of Cl(A). In fact Cl(A) = £l(A) U {
14

The Basics of C*-algebras

The following is the Gelfand theorem for commutative C* -algebras. Note that if A is unital, il(A) is compact and C0(Cl(A)) = C(fl(A)). Theorem 1.3.5 Suppose that A is a commutative C* -algebra. Then the Gelfand transform a \-+ a is a *-preserving isometry from A onto Co (fl(A)). Proof. Suppose first that A is unital. Let t G fl(A) and o G A then t(a) — a£ ker t. Since kert is a maximal ideal, t(a) G sp(a). Therefore, if a G Asa, then t(a) G K by 1.2.9. By writing a = (l/2){a + a*) + i[(l/2i)(a-a*)], we conclude that t(a*) = t(a) for each a G A This shows that T(a) = a is a *-preserving homomorphism from A into C(fl). If a is invertible, then T(a _ 1 ) = r ( a ) _ 1 , since T is a homomorphism. Conversely, if a is not invertible, then the ideal / = aA is proper and thus is contained in a maximal ideal M (we still assume that A is unital). If 4> G Q,(A) and kerc/> = M, then 4>{a) = 0. Thus d is not invertible in C(fl(A)). This implies that a and a have the same spectrum and this coincides with the range of a (see 1.1.7). Since the norm in C(Cl(A)) is the supremum norm and the range of & is sp(a), we conclude that ||d||oo = r{o) — ||a||, since a is normal (see 1.2.7). This says that T is a *-preserving isometry from A into C(Q,(A)). Since points in £l(A) are multiplicative linear functionals which can be only distinguished by elements in A, {a : a € A} separates points in Cl(A). It follows from the Stone-Weierstrass Theorem that F is surjective. For the non-unital case, from the above, T : A —> C(Q,(A)) is a surjective ^-preserving isometry. Since A/A = C, T(A) is a maximal ideal. Note as in 1.3.4, we may write tt(A) = Cl(A) U {IT}, where IT is determined by the quotient map A —> A/A = C. Then Co(fi(A)) is the (maximal) ideal (of C(£l(A))) of continuous functions vanishing at 7r, which is identified naturally with C0(Q(A)). • Let S be a subset of a C*-algebra A. Then the C*-subalgebra of A generated by 5, denoted by C*(S), is the smallest C*-subalgebra of A containing S. Corollary 1.3.6 If a is a normal element in a unital C* -algebra A, then there is an isometric ^-isomorphism from C*(a) to Co(sp(a) \ {0}) which sends a to the identity function on sp(o). Proof. Since a is normal, C*(a) is commutative. Note that G £l(C*(a)) is determined by (j)(a) = A. Thus the map from fi(C*(a)) into C taking to 4>{a) is a homeomorphism onto a{U{A)). From the Gelfand transform, a(Cl(A)) = sp(a). This map identifies d with the identity function z on sp(o)

Commutative

C*-algebras

15

as desired. It follows from 1.3.5 that this map is an isometric *-isomorphism

• Definition 1.3.7 If a is a normal element of A and / € C 0 (sp(a) \ {0}) we denote by / ( a ) the element of A corresponding to / via the isomorphism given in 1.3.6. Corollary 1.3.6 is known as the continuous functional calculus for normal elements. The following corollary is often called the spectral mapping theorem. Corollary 1.3.8 Let a be a normal element in a unital C* -algebra A and f € C(sp(a)). Then /(sp(a)) = sp(/(o)). Proof.

This follows immediately from the above corollary.

•

Theorem 1.3.9 Let A = C(X) and B = C(Y), where X and Y are two compact Hausdorff spaces. Then A = B if and only if X and Y are homeomorphic. Proof. Let <j> : A —>• B be an isomorphism. Define r : Y —• X by (y)(f) = J/(0(/))- Here we identify X with the maximal ideal space (multiplicative states) of A and Y with the maximal ideal space of B. The continuity of <j> implies that r is continuous. Since
I = {ftC(X):f\F

= 0}.

Then ker D I. Thus F = X. In other words r is onto, whence r is a homeomorphism. Conversely if r : Y —> X is a homeomorphism, define 4>{f)(y) = f(r(y)). To verify that <j> is an isomorphism one can simply reverse the above argument, and we leave this to the reader. • Remark 1.3.10 The Gelfand representation theorem for commutative C*-algebras is fundamentally important. Even in a non-commutative C*algebra A, we often obtain useful information of A via the study of certain commutative C*-subalgebras of A. So Theorem 1.3.5 certainly plays an important role in studying non-commutative C*-algebras as well. Theorem 1.3.9 shows that to study commutative C*-algebras, it is equivalent to study their maximal ideal spaces. Therefore the commutative C*-algebra theory is

16

The Basics of C -algebras

the theory of topology. Much of general C*-algebra theory can be described as non-commutative topology. 1.4

Positive cones

Example 1.4.1 Let H be a Hilbert space and let B^H} be the space of all bounded operators from H to H. If TUT2 G B{H), we define (TiT 2 )(v) = Ti(T2(v)) for v e H. With the operator norm ||T|| = s u p N | = 1 ||T(v)||, B(H) is a Banach algebra. If T e B{H), define T* by (Tx,y) = (x,T*y) (x,y £ H), where (•, •) is the inner product on H. Then one easily checks ||T*T||=

sup \(T*Tx,y)\ Il*ll=i=lli/ll

=

sup \(Tx,Ty)\ IMI=i=NI

= ||T|| 2 .

Thus B(H) is a C*-algebra. Every closed *-subalgebra oiB(H) is a C*-algebra. Later in this chapter we will show that every C*-algebra is a closed *-subalgebra of B(H) for some Hilbert space. Example 1.4.2 Let H be a Hilbert space. An operator T € B(H) is said to be compact if it maps bounded sets to precompact subsets. If the range of T is finite dimensional, then T is compact. Denote by IC(H) the set of all compact operators in B(H). )C(H) is clearly a closed subspace of B{H). If T e K{H), L e B(H), then TL, LT e K(H). Therefore fC(H) is a closed ideal of B(H). Moreover if T G )C(H), then T* e K,{H). Therefore K is a C*-subalgebra of B{H). When H is separable infinite dimensional Hilbert space, we will use K. for K.(H). K. is a very important example of a (simple) C* -algebra. Example 1.4.3 To show that any C*-algebra is a C*-subalgebra of B(H) for some Hilbert space H, we need to study the order structure of C*algebras. Recall that an operator T € B(H) is called positive if (Tv, v) > 0 for all v € H. In the case that H is finite dimensional, T is positive if and only if it is self-adjoint and all eigenvalues are nonnegative. If A = C{X), an element / € A is positive if f(t) > 0 for all t G X. We now introduce the following definition. Definition 1.4.4 An element a in a C*-algebra A is positive if a € yl sa and sp(a) C R+. We write a > 0 if a is positive. The set of all positive elements in A will be denoted by A+.

Positive

cones

17

A projection p is always positive, since p2 = p and p E Asa implies that sp(p) = {0,1}, by 1.3.6. Lemma 1.4.5 Let A be a C* -algebra and let a £ A. Then the following are equivalent: (i) a > 0; (ii) a = b2 for some b E Asa; (hi) a = a* and \\t — a\\ < t for any t > \\a\\; (iv) a = a* and \\t — a\\ \\a\\. Proof, (i) => (ii): In C*(a), by 1.3.6, we set b = a1'2. (ii) => (i): Consider the C*-subalgebra C*(b). Then a E C*(b) and, by 1.3.6, C*(b) = C 0 (sp(6)). We see that a = a* since 6 € Asa. (i) => (hi): Since t — a is normal, from 1.3.5 we have (with t > \\a\\), \\t - a\\ = sup{|t - A| : A G sp(a)} < t. (hi) =>• (iv) is immediate. (iv) => (i): If A e sp(a) then t - A £ sp(i - a). Thus |* — A[ < | | t - a | | 0 since A < t. • Definition 1.4.6 Let a e A s o . Then a 2 G A + . Set |a| = (a 2 ) 1 / 2 (by 1.3.6). Then a+ = (l/2)(|a| + a) and a_ = (l/2)(|a| - a). By the Gelfand transform, \a\, a+ and a_ are positive. Moreover, by 1.3.6, a+a^ = 0 and a = a_|_ — a_. Corollary 1.4.7 unitaries. Proof.

If A is a unital C* -algebra, then A is the linear span of

Let a G A s a and ||a|| < 1. Then 1 - a 2 G A+. Put u = a + i(l-

a2)1'2

and v = a - i(l - a 2 ) 1 / 2 .

Then u* = v and w*u = uu* = 1. So u and v are unitaries. On the other hand, we have a = (l/2)(u + v). D Theorem 1.4.8 The set A+ is a closed cone (a + b G A+, if a,b G A+ and J4+ n A- = {0} ) and a G A+ if and only if a = x*x for some x G A. Proof. It follows from (hi) of 1.4.5 that A+ is closed and, if a G A+, then Xa G A+ for any A G R+. To see that A+ is a cone, take a and b in A+. By

18

The Basics of C* -algebras

(iii) of 1.4.5, ||(IH| + | | 6 | | ) - ( a + 6)||

=

| | ( | | a | | - a ) + (||fe||-6)||

< IIIH|-a|| + lll|6||-6|| \\a + b\\, from the above and (iv) of 1.4.5, a + b G A+. Now suppose that a = x*x for some x £ A. Then a* = a. Set a = a+—aas in 1.4.6. We have /

l/2\*/

l/2x

1/2 «

1/2

- ( a ; a j ) (xaj. ) = —aj x xa_ l li

Put xa_

1/2/

\ 1/2

2

,.

= ~a_!_ (a+ - a,-)a_! = a_ G A+.

= 6 + ic with 6, c G A s a . Then

{xa\[2){xal_!2Y

= 2(b2 + c2) + [ - ( z a ^ 2 ) * ^ 2 ) ] e A+

since A+ is a cone. It follows from 1.1.8 that sp(a2_) C R + fl R_ = {0}. Therefore a_ = 0 and a > 0. • Definition 1.4.9 In Asa, we write 6 < a if a—6 > 0. Since Asa = J 4 + - A + and A+ (~) (-A+) — {0}, Asa becomes a partially ordered real vector space. Theorem 1.4.10 Let A be a C* -algebra. (1) A+ = {a*a : a G A). (2) / / a , 6 G vl s a and c £ A, then a < b implies that c*ac < c*bc. ( 3 ) / / 0 < o < 6 , ||o||<||6||. (4) / / A is unital and a, b G A+ are invertible, then a < b implies that 0 0 and invertible, then fr-1 > 0 (by 1.3.6). It follows from (2) that b-l'2ab~1'2 < b'1'^^1'2 = 1. Hence ||a 1 /2 6 -i/2|| < 1 2 1 1 1 (by (3)). Thus ||a / 6- a /2|| < i which implies that al/2b~la1/2 < 1. By (2), we have 6-1 = (a- 1 / 2 )(a 1 / 2 6- 1 a 1 / 2 )(a" 1 / 2 ) < (ar1'2)!^1/2)

= a" 1 .

g

Theorem 1.4.11 Let A be a C* -algebra. If a, b £ A+ such that a < b, then aa < ba for any 0 < a < 1.

Positive cones

19

Proof. By considering A, we may assume that A is unital. Fix 0 < a < b in A+. Let C = {a £ R+ : aa < ba}. Then 0, 1 £ C. Since the function fn(t) = ta" -» i a if a n -» a for a„ > 0 uniformly on [0, if] for any if > 0, x a " —>• xa for any positive element x £ A+ if a„ —> a (and a n > 0). Since A+ is closed, if an £ C and an —>• a, then 6 a — a a S -A+. This shows that C is closed. To prove the theorem, it suffices to show that C is convex. We first consider the case that both a and b are invertible. Let a, 3 £ C. Then b-a/2a«b-a/2

<

:

a n d

b-P/2afib-fi/2

<

L

Hence, by 1.4.10, \\b~a/2aab-a/2\\ < 1 and | | & - ' 3 / W 3 / 2 | | < 1. So || a o/2 6 -a/2||2 < 1 a n d || a ^/2 6 -/3/2||2 < L T n e r efore (using the fact that sp(zy) = sj>(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-(<*+/3)/4a(a+/3)/26-(a+/3)/4 < L H e n c e a(a+/3)/2 < 5(a+/3)/2_ Therefore C contains (l/2)(a + /3). Consequently, C 3 [0,1]. In general, for any e > 0, we have (6 + £ ) Q - ( a + e ) a > 0

(0
Since .A+ is closed, letting e —> 0, we obtain a a < ba.

D

Example 1.4.12 It is not true that 0 < a < b implies a2 < b2 in general non-commutative C*-algebras. For example, let A = M^ and let

Then both p and q are projections and p < p + q. But p2 = p ^ (p + q)2 = p + q + pq + qp, since

q+pq + qp = 1/2 P is not positive.

JJ

20

1.5

The Basics of C*-algebras

Approximate identities, hereditary C*-subalgebras and quotients

To deal with non-unital C*-algebras, and avoid the troubles caused by the absence of unit, one can often embed A into A. However, more often, one has to work in the original non-unital C*-algebra. Therefore the notion of approximate identity is essential. Example 1.5.1 Let H be a Hilbert space with an orthonormal basis {vn}^Li- The C*-algebra /C, the set of all compact operators on H, is a nonunital C*-algebra. Let pn be the projection (from H) onto span{vi, ....,vn}. Then {pn} is an increasing sequence of projections in K. The reader easily checks that, for any x € /C, \\pnx — x\\ —> 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
(e5.5)

if 0 < t < s/2

so that fe G Co((0,if]) for any K > 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\\


{l
k)

and efc+i > ek, k = 1, 2,...,. Thus {efc} forms an approximate identity for

A.

•

Definition 1.5.7 A C*-algebra is said to be a-unital if A admits a countable approximate identity. The above corollary (1.5.6) shows that every separable C*-algebra is cr-unital. Definition 1.5.8 A C*-subalgebra B of A is said to be hereditary if for any a G A+ and b G B+ the inequality a < b implies that a G B. Obviously, 0 and A are hereditary C*-subalgebras of A. Any intersection of hereditary C*-subalgebras is hereditary. Let S C A. The hereditary C*-subalgebra generated by S is the smallest hereditary C*-subalgebra of A containing S. Such a hereditary C*-subalgebra is denoted by Her(S'). Lemma 1.5.9 Let A be a C*-algebra and a G A+. Then aAa is the hereditary C*-subalgebra generated by a. Proof. Let B = aAa. It is easy to see that aAa is a *-subalgebra. It follows that B is a C*-subalgebra. Claim (1): If c G B+ then cAc C B. In fact, if axna -> c, then, for any y £ A, axnayaxna —> eye. Therefore cAc C B. This proves the claim. Claim (2): Let e„ = f\/n{a). Then {e„} forms an approximate identity for B. In fact by 1.5.4, \\ena — a\\ —>• 0 as n —¥ oo. Suppose x = aba for some b € A. Then we have \\enx — x\ —> 0 and \\xen - x\\ ->• 0 as n -> oo. Since aAa is dense in B, we conclude that {en} is an approximate identity for B. Since ae„a -» a 2 , a 2 G 5 . Hence a € B and C*(a) C -B. In particular, from Claim (1), fi/n{a)Afi/n{a) C B. Now suppose that 0 < c < b with 6 G B+. We will show that c G B. By 1.4.11, c 1 ^ < fci/2. Thus ||(1 - e„)c(l - e„)|| < ||(1 - e„)6(l - e„)|| < ||(1 - c n )6|| -»• 0.

Approximate

identities,

hereditary C* -subalgebras and quotients

23

Therefore | | ( l - e n ) c 1 / 2 | | -» 0, or e^c 1 / 2 -s- c 1 / 2 . Hence e„ce„ -s- c. By Claim (1), e„ce n G B. Therefore c £ B. So B is hereditary. We have shown that a £ aAa. It follows that Her(a) = aAa. • Theorem 1.5.10 A C*-algebra A is a-unital if and only if there is a £ A+ such that Her(a) = A. Proof. We have shown (from Claim (2) in the proof of 1.5.9) that A is (T-unital if A = Her(a) for some a G A+. For the converse, let {e„} be an approximate identity for A. Put a = £ ~ = 1 ( l / 2 n ) e n and B = Her(a). Since e„ < 2na, by 1.5.9, e„ G B for all n. As in the proof of 1.5.9, enben G B for any b G A + . Since enbxl2 —• 61//2, enfoe„ ->• 6. Therefore 6 G 5 . It follows that A = B. • Corollary 1.5.11 If A is a a-unital C* -algebra, then A admits a countable approximate identity {en} satisfying ^n^-n+l

=

^n ~ ^-n+l^n

^

=

-Lj A —

Proof. Let yl = a^4a for some a G yl+. Set e„ = A/2"( a )- Then {e„} is an approximate identity. By 1.5.4, /i/2»/i/2("+1> = A/2" f° r a u n ^ Theorem 1.5.12 / / J is an ideal in a C*-algebra A, then I is a hereditary C* -subalgebra of A. If {ex} is an approximate identity for I, then for each a £ A, \\a\\ = inf \\a + b\\ = lim \\a — e\a\\ = lim \\a — ae\\\. 66/

A

A

Proof. Let B = I* (~)I. Then B is a C*-subalgebra of A. Let {ex} be an approximate identity for B. Note that ex G B C / . If a G / , lim}, a*a(l — ex) = 0. Hence lim \\a - aexf

= lim ||(1 - ex)a*a(l - ex)\\ < lim \\a*a(l - e A )|| = 0.

Therefore a = limA o,ex- Hence a* = lim A exa*. Since exa* G / , we conclude that a* G L Therefore / is C*-subalgebra. To see it is hereditary, let 0 < c < a, where c G A+ and a G i+. By 1.5.9 c G Her(a) = aAa. But aAa C / . Hence / is hereditary. Now suppose that {e^} is an approximate identity for I. By definition, ||a|| = infb£/ \\a + b\\ for all a G A. Fix a £ A. Put a = infbgj \\a + b\\ and /3 = limA ||a(l — eA)a*|| (it exists since the net is

24

The Basics of C* -algebras

decreasing). We have a2 < ||a(l - e A )|| 2 = ||o(l - e A ) V | | < ||o(l - eA)a*|| -> (3. Hence a 2 < f3. Thus it suffices to show that a2 > f3. To do this, let e > 0 and take b G I such that a + e > ||a + 6||. Then (a + £ ) 2 > \\a + b\\\\l - ex\\\\a* + b*\\> \\(a + 6)(1 - ex)(a* + b*)\\ -> 0 since both ||6(1 - e A )|| and ||6*(1 - e\)\\ tend to zero. Thus we have

(a + s)2>/3 for every e > 0. Hence a 2 > (3.

D

Corollary 1.5.13 If I is a closed ideal of A, then A/1 equipped with its natural operation is a C* -algebra. (In particular, x* = x*.) Proof. It follows from 1.1.20 that it suffices to show that \\x*x\\ — \\x\\2. Let {e\} be an approximate identity for I. Then, ||S*S||

=

lim||x*x(l-eA)||>lim||(l-eA)x*a;(l-eA)||

=

lim ||rr(l - e A )|| 2 = ||x|| 2 . A

D

Definition 1.5.14 From now on, an ideal of a C*-algebra is always closed, and a homomorphism from a C*-algebra to another is always a *-homomorphism. Theorem 1.5.15 Each homomorphism h : A —>• B is norm decreasing, and h(A) is always a C*-subalgebra of B. If h is injective, then it is an isometry. Proof. If a G Asa, then sp(/i(a)) \ {0} C sp(a) \ {0}. Since ||a|| = r(a) by 1.2.7, we conclude that for each a S A,

HMa)||2 = ||M^)ll• B if necessary, we may assume that A and B

Positive linear junctionals

and a Gelfand-Naimark

theorem

25

are unital. Thus, by 1.3.5, we may assume that A — C(S), where S is a compact Hausdorff space. Since h(A) is commutative, by taking its closure, we may also assume that B = C(T) for some compact Hausdorff space T. Define h* : T —• S by h*(t) = t o h. As in 1.3.9, since h is injective, h*(T) = S. Therefore, for each g G C(S), \\g\\ = sup\g(s)\=

sup \g(h*(t))\ = \\h(g)\\.

Thus h is an isometry. To complete the proof, we note the J — ker/i is an ideal of A (closed). So h induces an isomorphism from A/I into B. Therefore A/I is isometrically •-isomorphic to h(A). It follows that h{A) is a C*-subalgebra. • Definition 1.5.16 From now on, an isomorphism h from a C*-algebra A to another C*-algebra B is always a *-isomorphism. Furthermore, it is an isometric ^-isomorphism. If A is isomorphic to B, we write A = B. Corollary 1.5.17 Let I be a closed ideal of a C*-algebra A and let B be a C* -subalgebra of A. Then B + I is a C* -subalgebra of A. Moreover, B/{Br\I)^(B + I)/I. Proof. Clearly B + I is a *-subalgebra of A containing / and B. Let 7T : A —> A/I be the quotient map. By 1.5.15, ir(B) is closed in A/I, whence the preimage of TT(B) is closed, i.e., B + I is closed. Note that the map 6 + JBn/h->6 + / i s a *-isomorphism from B/(Bf)I) to (B + I)/I. Thus, by 1.5.15, it is a C*-isomorphism. •

1.6

Positive linear functionals and a Gelfand-Naimark theorem

Definition 1.6.1 A linear map 4> : A —»• B between C*-algebras is said to be self-adjoint if (Asa) C Bsa, and positive if <j>(A+) C B+. It follows that if <> / is positive then is self-adjoint. Every homomorphism h : A —> B is positive. If B = C, then a positive linear map <j> : A -4 C is called a positive linear functional. If, in addition, <j> is bounded and \\<j>\\ = 1, then <j> is calle a state on A. If is a linear functional, we write > 0 if it is positive.

26

The Basics of C*-algebras

A positive linear functional r : A —> C is called a £race if 4>{u*au) = <j>(a) for all a G A and all unitaries u £ A. A trace which is also a state is called a tracial state. Example 1.6.2 Let A = C{X), where X is a compact Hausdorff space, and let / i b e a positive Borel finite measure. Then the linear functional r defined by

r(f) = JfdfM is positive. Since A is commutative, r is also a trace. Example 1.6.3

Let A = Mn. Define a linear functional Tr on A by n

Tr((a,ij)) =

y^ag. i=l

This is a trace. It is called the standard trace on Mn. The normalized trace tr on A is defined to be tr = (l/n)Tr. It is a tracial state. Example 1.6.4 Let A be a C*-subalgebra oiB(H), where H is a Hilbert space. Let v G H be a nonzero vector. Define f(a) = (a(v),v) for a G A. Then / is positive. To see this, note that for a > 0, there is x G A such that a = x*x. Thus f(a) = (x*x(v),v) = {x(v),x(v)) = \\x(v)\\ > 0. This functional is, in general, not a trace. Lemma 1.6.5 bounded.

Every positive linear functional

on a C*-algebra A is

Proof. Let be a positive linear functional on A. If is not bounded, then there is a sequence {xn} C A with ||xfc|| < 1 such that |(xfc)| —>• oo. Since {Asa) C M, by considering the real part and imaginary part of x^, without loss of generality, we may assume that Xk G Asa. By considering Xk — {xk)+ — (xk)-, we may further assume that Xk G A+. By passing to a subsequence if necessary, we may assume that <j>(xk) > 2k, k = 1,2,...,.

Set x = x X i f £ - T h e n

x GA

+-For

OO

an

y n' OO

fc

n<X)2- ^ife)= CE%)<
This is impossible. Thus <j> must be bounded.

D

Positive linear Junctionals

and a Gelfand-Naimark

theorem

27

T h e following is t h e Cauchy-Schwarz inequality. L e m m a 1.6.6 for all a, b € A

If<j> is a positive linear functional

on a C* -algebra A, then

\4>{b*a)\2 < <j)(b*b)
(e6.6)

Proof. For each complex number A, we have 0((Aa + b)*(Xa + 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<(n2

+ 2)U\\2

for all n. Thus 2n\\4>\\P + f32 + a2<2\\<J>\\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

<1.

ct>{ex-a)<\\4>\\. 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

= <

limsup |||a| 2 e^ + ae\a + aa*e\ + a*a\\ x limsup | | | Q | 2 1 + ae\a + aa*e\ + a*a\\ — ||al + a\\2. x

It follows from 1.6.7 that | 0 > + 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{x*a*ax) < ||a||2(a:*x) = ||a|| 2 ||z|| 2 (the inequality holds because a*a < \\a\\2 and is positive). Thus n^ can be uniquely extended to an operator (still denoted by 7r<^(a)) on Hj, such that ||7r0(a)|| < ||a||. Also, (7i>(a)(£),y) = 4>{y*ax) = (z,7i>(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-aA)1/2(a-aA)1/2(OI!2 \\a - ax\\\\(a - a A ) 1 / 2 (£)l| 2 < 2M <(a -

ax)(£),£),

and ((a — a\)(£,), £) —¥ 0, so a(£) = lim,\ a^(£)- Thus a\ converges strongly to a. •

34

The Basics of C*-algebras

Definition 1.7.4 If H is a Hilbert space, we write H^ for the orthogonal sum of n copies of H. lia G Mn(B(H)), we define 4>{a) G B(H^) by setting n

n

^2anj(xj))

<j>{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 <£ : Mn(B(H))

->• 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• B(H(n>) 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)|<(l/<5)(^||aa||2)1/2. fc=i

(e7.9)

Von Neumann

35

algebras

for all a £ B(H). With notation as in 1.7.4, we define £ = £1 © • • • © £„ in ij( n ). On the vector subspace V = {p{a)£ : a £ B(H)}, define ^(p(a)O = 4>{a). From the definition (in 1.7.4), ^ is a linear functional (on the \^{p{o)i)\ < (l/($)||p(a)£||. So it is a bounded linear functional. to a bounded linear functional on HQ, the closure of V. By the resentation theorem, there is a vector 77 = r/i © • • • © r]n £ H0 that

span) and It extends Riesz repC H^ such

n

<j>{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 <£ 2 . fc=i

It follows that x is in the strong closure of M. Thus x G M. In other words, M " C M. Since clearly M cM",M = M". D Definition 1.7.9 A weakly closed C*-subalgebra M C B(H) is called a von Neumann algebra. In other words, a C*-subalgebra M C B{H) is a von Neumann algebra if M = M" and 1 G M. Definition 1.7.10 Let A be a C*-algebra and nu : A —> 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<1,/€All€ll,NI<2

<4||x||.

On the other hand, if ||x|| < 1, and £ £ Hu with ||£|| < 1, {nu(a)£,x(£)) for a £ A defines a linear functional £,x(£) o n A with ||<^,a;(£)|| < llx(OII- So

\J(x)(hM0)\

= K*(0>*(0>l = IWOIIIWOII-

Hence ||J(x)|| > \\x{£)\\ for all £ £ Hv with ||£|| < 1. This implies that ||J(x)|| > ||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,)\<\\x\\.

This implies that || J(a;)|| = ||a;|| for all x £ A"a. In general, note that M2{A") = (M2{A))". Let J 2 : (M 2 (A))" ->• 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>)\<\\b\\U\\ = \\x\\\\f\\. This implies that ||J(a;)|| < ||ir||. Therefore J is an isometry. Since A is weakly dense in A", J{A) is dense in J(A") in the weak*-topology as a subset of (A*)*. However, j(A) is dense in (A*)* in the weak*-topology. Therefore, since J (a) = 3 (a) for a G A, J {A") = (A*)* = A**. D The following corollary follows from the above theorem and the uniform boundedness theorem. Corollary 1.7.14 Let {a\} be a net in a C*-algebra A". If {a\} converges in the weak operator topology in A", then {||a\||} is bounded. Remark 1.7.15 The weak operator topology for A as a subalgebra of A" acting on H\j is called the a-weak topology. So the weak operator topology for A" is sometimes called cr-weak topology. 1.8

Enveloping von Neumann algebras and the spectral theorem

Lemma 1.8.1 Let {TT\,HI) and (7r 2 ,iJ 2 ) be two cyclic representations of a C*-algebra A with cyclic vectors £1 and £2 (||£i|| = H&H)- Then there exists an isometry u : H\ —>• H2 such that m = U*TT2U with u(£i) = £2 if and only if {ni(a)£i,£i) = (7r2(a)£2,£2) for all a £ A. Proof.

If u£i = & then (i"i(a)£i,6) = (u*7T 2 (aXi,fi) = (7r 2 (a)6,6>

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 p$x is weak-weak continuous from A" into i^{A)" which extends 7r (from A to n(A)). By 1.7.11, in general, it = (B\ir\, where each ir\ is a cyclic representation. Therefore, there are states cj>\ of A and an isometry U : H —>• @xH^x such that U*TVU = ®\n^,x. So we may assume that n = ©ATT^- Let p be the projection from Hu onto ®\H^,X. Then the map x i-> px is weak-weak continuous and extends -K. • Definition 1.8.3 Let X be a compact Hausdorff space. Let B(X) be the set of all bounded Borel functions on X. B(X) is a C*-algebra containing C(X) as a C*-subalgebra. It follows from 1.2.3 that B(X) is a subspace in C(Xy*. By 1.7.13, C(X)" = C{X)**. So B(X) is a C*-subalgebra of C(X)". We say {/A} converges weakly to / in B{X), if they do so as elements of the von Neumann algebra C(X)". If {/A} is bounded and converges pointwise everywhere on X, then, by Lebesgue's Dominated Convergence Theorem, {/A} converges weakly. Corollary 1.8.4 Let <J> : 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(
f g{X)dEx = Y^akE{Sk) ^x

fe=i

is an operator in B(H) (Sk are Borel sets). Let / 6 B(X). There is a sequence {gn} of simple functions such that \\gn — f\\ -4 0 a s n -4 oo. Since fx(S) = {E(S)£, rj) defines a regular Borel complex measure, fx gndE\ converges weakly to an element (in B{H)). Denote by Jx fdE\ the weak limit. Clearly this limit does not depend on the choice of gn. We have the following corollary otherwise known as the spectral theorem. Corollary 1.8.8 Let a be a normal operator on a Hilbert space H. Then there is a unique spectral measure E relative to (sp(a),H) such that

a=

f

XdEx.

Jsp(o)

Proof. By the Borel functional calculus, the embedding j : C*(l,a) —> 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) = ^
= /i(X) = ||/i||.

Thus a* maps the state space of C{X) into itself. Since the state space is compact and convex (C(X) is unital), by 1.9.18 a* has a fixed point fi0 in the state space. In other words,

m>{?-\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)(
x((T-\0)f(0=Hx°°-1)(f(t))

for all x G C(X), f G £ 2 ( M ) a n d £ € ^ - Thus (ir, U) is a covariant representation of (C(X), a). Denote by C*(X, a, n) the C*-subalgebra generated by n(C(X)) and U. Definition 1.9.21 Let (X, a, n) be a dynamical system with a translation invariant measure fi. The triple is said to be ergodic if cr(E) C E (for Borel sets E) implies n{E) — 0 or n{E) = 1. If (X, a, //) is ergodic, we often denote C*{X,a,ii) by C{X) xaZ. A dynamical system (X, a) is said to be minimal if X has no proper closed er-invariant subset. One can show that minimality implies ergodicity (1.11.30). Furthermore, if (X, a) is minimal and X is infinite, then C(X) xa Z is a unital simple C*-algebra. Definition 1.9.22 Let X = S1 be the unit circle and 6 be an irrational number. Define a homeomorphism a : 5 1 —> 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 ° <^fc,n- If F is a

group and we have homomorphisms tpn '• Gn —>• F such that the diagram <Jn

>

^n+l

F commutes for each n, then ipn = 4>m ° 0n,m for all m > n. The product flnLi ^ n with pointwise operation is a group. Let oo

^ = {{0n} G J J Gn • ffn+i = ^n(fln) for all sufficiently large n } . n=l

Then F is a subgroup of fl^Li ^n- ^ e « *s the u m t °f ^ m then the set N of all {gn} G n ^ L i ^ n s u c n that gn = en for all sufficiently large n is a normal subgroup of F. Denote the quotient group F/N by G. We call G the inductive limit of the sequence {(Gn, .00(Gn, 4>n). When there is no ambiguity, we may also write G — limn-Kx, Gn. If g G Gn, define 4>'n{g) to be the sequence {gk} where g^ = e^ for k < n and g„+i = 4>n,n+i(g) for all i > 0. then 'n(g)N is a homomorphism. This is called the homomorphism induced by the inductive limit. One verifies that the diagram

G


n

V

> n.co

f-i

^n+l I N"Pn+l,ex>

G

52

The Basics of C* -algebras

commutes for each n and that G is the union of the increasing sequence {n,oo{Gn)}- From the definition, if n,oo(g) = e, the unit of G, then for some m>n, 4>n,m(g) = em. T h e o r e m 1.10.2 Let G = lim„_ >00 (G n , (/>„), where Gn are groups. (1) if g & Gn and f G Gm and 4>n,oo{g) = m,oo(f), then there exists k>n,m such that n,k(g) — m,k(f)(2) If G' is a group and for each n there is a homomorphism tpn : Gn —• G' such that the diagram

r

^\

^n

r >

\

"'n+l

"

4-i/Jn+l

G' commutes, then there is a unique homomorphism ip : G —>• G' such that the diagram Gn

^ \

n

G 4-0 G'

commutes for each n. Proof. Part (1) follows from the definition immediately. Suppose that G' and tpn : Gn —• G' are as in (2). If g G Gn and / G Gm and cj>n,oo(g) = 4>m,oo{f), then by (1), there is k > n, m such that n,k{g) = m,k{f)- So ipn(g) = i>k ° <£n,fc(fl) = ^fc ° m,k{f) = i'mif)- Therefore the map ip : G —»• G' defined by ip((f>n,oo(g)) — ^n{g) is well defined. It is then easily checked that ip is a homomorphism and, by definition, V°^>n,oo = VViUniqueness of ip is also clear now. D E x a m p l e 1.10.3 Let Gn = Z and <j>n : Z —> Z be defined by 4>n(k) = Ik for k G Z. Let G = lim n ^ 00 (G n ,<)6 n ). One can check that G = Z[l/2] (see 2.7.4). Definition 1.10.4 Let p be a seminorm on a *-algebra A satisfying p(ab) < p(a)p(b), p(a*) = p(a) and p(a*a) = p(a)2. Set N = p _ 1 (0). It is a self-adjoint ideal of A. Then A/N is *-algebra with norm \\a + N\\ = p(a). Let B be the Banach space completion of A/N with this norm. It is easy to check that B is a G*-algebra. The map j : A -» B defined by j(a) = a + N is called the canonical map from A to B. Note that J(J4) is dense in B.

Inductive

53

limits of C*-algebras

Let {An} be a sequence of C*-algebras and hn : An —»• An+\ sequence of homomorphisms. Set

be a

oo

A' = {{an} £ TT An : a n +i = hn(an)

for all sufficiently large n}.

n=l

Then A' is a *-subalgebra of n ^ L i -^n- It is important to note that lkfc+i|| < llafc||

for

k> N

for some integer N > 0. So {||afc||} converges. We define p(a) = lim n ^oo ||an||- It is easy to check that p is a seminorm on A' satisfying p(ab) < p(a)p(b), p(a*) = p{a) and p(a*a) = p(a)2. We define A to be the completion of A' /p~1{Q). As above, A is a C*-algebra. We call it the inductive limit of the sequence (An,hn) and write A = lim„_ >00 (A rl , hn) (when there is no ambiguity, we may simply write A = linin^oo An.) Remark 1.10.5 As in the group case, if a € An, define h'n{a) = {an} in A' such that a\ = a-i = • • • = a n _i = 0 and an+j = /i n j „ + J (a) (j = 0,1,...,). If j : A' —> A is the canonical map, then the map /i„>00 : An —> A defined by a — I > j(h'n(a)) is a (C*-algebra) homomorphism. It will be called the homomorphism from An to A induced by the inductive limit. One checks routinely that the diagram An+i

•"•n

A commutes for each n, and the union of the increasing sequence of C*subalgebras {hn!oo{An)} is dense in A. Furthermore, HVoo(a)|| =

lirn

ll^n,n+m(a)||

(e 10.12)

m—too

if o G AnTheorem 1.10.6 Let A = lim„_ >00 ( J 4„, hn) be an inductive limit of C*algebras. (1) If a £ An, b S Am, £ > 0 and /i n]0 o(a) = hmiCO(b), then there is ko > n,m such that \\hn,k(a) - hm>k{b)\\ < s for all k > k0.

54

The Basics of C*-algebras

(2) If B is a C*-algebra and there is a contractive positive linear map tpn : An —>• B such that the diagram ^n

A

*

A ™

N'V'n + l

B commutes for each n, then there is a unique contractive positive linear map ip : A —»• B such that the diagram An

~¥

A B

commutes for each n. Moreover, if ipn is a homomorphism for each n, then ip is a homomorphism. Proof. Part (1) follows from the equation (e 10.12) immediately. For part (2), let B and ipn be as in (2). Suppose that a £ An, b G Am and hnt00(a) = hmi00(b). If e > 0 is given, then by (1), there exists ko > n,m such that ||/in,fc(a) — hmtk(b)\\ < e for all k > ko- Therefore, HV'n(a) - "
£

for all e. Thus ipn(a) = ipm(b). This shows that the map ip : Un B defined by tp(hnt00(a)) = ipn{a) is well defined. Since II^n(a)|| = | | ^ + i 0 ^ n > n + i ( a ) | | < ||/i n>n+ i(a)||, we have \\^{hn,oo{a))\\ = \\ipn(a)\\ < lim ||/i n > n + m (a)|| = ||/i„ i0O (a)||. m—•oo

Thus \\ip\\ < 1 and it is easy to see that it is a contractive positive linear map. Since Unhn,oo(An) is dense in A, ip can be uniquely extended to a contractive positive linear map ip : A —> B such that ip o / i n o o = ipn for each n. It is also clear that, if ipn is homomorphism for each n, ip is a homomorphism. Corollary 1.10.7 Let A be a C*-algebra and let {An} be an increasing sequence of C* -subalgebras of A whose union is dense in A. Set B = \imn^00(An, i), where %: An —>• A is the embedding. Then A — B.

Inductive

Proof. reader.

limits of C*-algebras

55

The proof is an easy application of (2) in 1.10.6 and is left to the •

Example 1.10.8 Let An = M^. Define hn : An —>• An+i by defining hn(a) = diag(a, a). A = lim„^ 0 0 (A n , hn) is called the UHF-algebra of 2°°type. It is infinite dimensional and unital. A is a simple C*-algebra by the following proposition, which we leave to the reader as an easy exercise. Proposition 1.10.9 then A is simple.

If A = limn^>00(An,hn),

where each An is simple,

Definition 1.10.10 An /IF-algebra is an inductive limit of C*-algebras of the form M ^ © Mk2 © • • • © Mkt. In other words, an AF-algebra is an inductive limit of finite dimensional C*-algebras. The C*-algebra A in 1.10.8 is a simple AF-algebra.

Definition 1.10.11 Let A and B be two C*-algebras and fa : A —• B (i = 1,2) be two maps. Let e > 0 and J- C A. We say fa and fa are approximately the same within e on J- if ||0i(a)-^2(o)|| < £ for all a £ J7. When this happens we write fa ~£ fa on T. Let u G B be a unitary. We denote adw : B —> B the automorphism a i-> u*au. We write fa —e fa on T, if there is a unitary u G B such that adw o fa- w£ fa on J-, and say fa and fa a r e We say fa and fa are ^l — £ fa on any finite are unitarily equivalent

approximately unitarily equivalent within e on J-. approximately unitarily equivalent if for any e > 0 subset J7 C A and write i ~ fa. We say 0i and fa if there exists a unitary u £ B such that adu o fa = fa.

56

The Basics of C*-algebras

For two elements a and b in B we write a wE b if \\a — b\\ < e. We write a ~ £ 6 if there is a unitary u £ A such that u*au « £ b and we write a ~ b if there is a unitary u G A such that u*au = b. Definition 1.10.12 Let L : A -> B be a positive linear map. Given e > 0 and a subset J 7 C A, L is said to be JF-e-multiplicative if \\L(ab) - L(a)L(b)\\

<s

for all a,b £ T. Definition 1.10.13 Let B\, B2, Ai and A2 be C*-algebras. Consider the (not necessary commutative) diagram: V

Bi

B2 4-L2

4-ii V-

A!

A2

Let J- C B\ and e > 0. We say the above diagram approximately on J 7 within e if •J/J

commutes

o L i K,e L2 ° o n .?-".

Note that this includes special case that A\ = A2 and -ip = i d ^ . Let A = \imn^.00(An,hn) and B = lim.n->.00(Bn,4>n) be two inductive limits of C*-algebras. Suppose that we have the following (not necessary commutative) diagram: Bl

-H B2 H

4-Z/i

ArL^

Al

%

A2

B3 H ••• B "~rZ/3

%

A3

%

...

A

where Li is a contractive positive linear map for each n. Suppose that there are finite subsets T\ in the unit ball of B\, J-2 in the unit ball of B2, ..., with hni^n) C J-n+i such that the closure of Unhn:00(J-n) contains the unit ball of B. Suppose also that there is a decreasing sequence of positive numbers {rn} with X ^ i r„ < oo such that the nth-square of the above diagram approximately commutes on J-n within rn, i.e.,
intertwining.

Inductive

57

limits of C -algebras

If each square is actually commutative, then we say the diagram is intertwining. The following theorem will be used in Chapter 4 and Chapter 6. Theorem 1.10.14 Let A = lim n _ >00 ( J 4„,/i 7l ) and B = \imn->.00(Bn, (f)n) be two inductive limits of C*-algebras. Suppose that there are contractive positive linear TTicvps Ln '. Bn —y An such that the didgvenn B1

H

•VL\

Al

B2

H

B3

H

A3

~v~L/2

%

A2

H

•••

B

H

•••

A

YL3

is one-sided approximately intertwining. Then there is a contractive positive linear map L : B —• A such that the diagram Bn

^

An

B ->

A

approximately commutes on J-n within ^ibLnrfc> and L o hni00(b) = lim k,co ° £fc ° hntk{b) k—>oo

for all b £ Bn. If furthermore, Ln is Fn-en-multiplicative for some en S n L i e " < °°! then L : B —> A is a homomorphism. Proof.

> 0 with

For each b £ Bj consider the sequence {k,oo oLfc

ohjtk(b)}.

We claim that the above sequence is Cauchy. We assume that 6 ^ 0 and set b' = 6/||6||. Given e > 0, there is a £ U^ = 1 /i n i 0 0 (^ r n ) such that \\hJi00(b') - a | | < £ / 4 ( | | f c | | + l). By 1.10.6, choose a' £ T% (i > j) and s > i such that hit00(a') = a and \\hi,aohj,i(b')-hiiS(a,)\\<e/A(\\b\\

+ ^)-

58

The Basics of C* -algebras

By assumption, k+l

||^fcifc+; o Lk o ha!k(hiiS(a'))

- Lk+t o hitk+i(a')\\

< y"Vt. t=k

Therefore Uk,k+i olioh Stk (h jta (bf))

k+l - L k + l oh tik+l (hj tS (V))\\ < e/2(\\b\\ + l ) + £ r t . t=k

This implies that {4>n,oo ° Ln o hjtn(b')} is Cauchy in B. So {4>ni00 ° Lno hj!n(b)} is Cauchy. This proves the claim. Define V'j : Bj —¥ A by •0j(^) — hmfc-s-oo <^fc,<x> ° Lk o hjtk(b). It is clear that V'j is positive and contractive. It is also clear that it is linear. From the definition, we obtain the following commutative diagram (for each n) E>n

> \

"

-Dra+1 -M'n+1

Thus by 1.10.6, there is a unique contractive positive linear map L : B —• A such that Bn

^

V"

B

It A

commutes for each n. Now if Yln°=i £n < oo (en > 0) and Ln are .F n -£„-multiplicative, then <

lim ||<^n,oo ° Ln ohjn(ab)

- n,oo ° Ln o hjiTL(a)
n—>oo

=

lim \\<j)n,oo(Ln(hj
< en

n—>oo

if /ij, n (a), /ij,n(&) € J"„. So i/>j(a6) = ipj(a)ipj(b) if hjtn(a), hj 1, let Qn C An be a finite subset of the unit ball such that 4>n{Qn) C £7n+i and U„<£„i0o(£n) is dense

Inductive

59

limits of C* -algebras

in the unit ball of A, and let sn > 0 with Xl^Li sn < °°- If m addition, we have contractive positive linear maps: Hn '• An —>• -Bn+i such that Bn 4-L„ An

"n

5„+l

/*„

YLn+i

^71^

A n +1

is approximately commutative on J>j and on £„ within rn and within s n , respectively, i.e., Hn o L„ w r n /i„ on ,F n

and Ln+\ o iJ n « S r l „ on £ n ,

then we say the diagram

4-Li

-^ /V,

IL2

/H2

IL3

Al

J^

A2

A

A3

Bl

B2

±h

B3

^

•••

B

A

•••

A

is two-sided approximately intertwining. If each triangle above is actually commutative, then we say the diagram is intertwining. Theorem 1.10.16 Let A = lim rl _ >00 (A„,/i n ) and B = lim n _ > . 00 (5„, n) be two inductive limits of C*-algebras. Suppose that there are contractive positive linear maps Ln : Bn -» An and Hn : A, • i? n +i such that the diagram B1

-^

Bi

4-Li

/'Hi

IL2

/H2

IL3

A1

-^

A2

A

A3

—>

h3

Bz

3^

B ••

A

is two-sided approximately intertwining. Then there are contractive positive linear maps L : B —»• A and H : A —> B such that L o H = id A and H o L = ids and such that the diagram Bn

^

B

An

^

A

approximately commutes on J-n within rn and approximately commutes on Qn within sn-

60

The Basics of C*-algebras

If, furthermore, Ln are J-n-£n-multiplicative, Hn is Qn-8n-multiplicative with X^n=i £ n < oo and X^„=i $n < oo, then L and H are isomorphisms. Proof. It follows from 1.10.14 that there are L : B —>• A and H : B —>• A such that Lohn>00(b)

= lim 4>ktXoLkohntk(b)

and Ho<j>noo{a) = lim

fc—»oo

hk+lt00oHko(f>nk(a)

A:—•oo

for all a 6 An and b e Bn. Therefore \\hk+i,oo ° Hk o(f>n
->• 0 as k ->• oo.

Also ||
o (l>ntk(a)) - L o hk+ltOC(Hk

o ^„ i f c (a))|| -> 0

as /s —>• oo. Since ||L fc+ i(iJfc o <j)nik(a)) - <j>nik+1(a)\\ ->• 0

as fc —>• oo (this can be verified in a similarly manner as in the argument used in the proof of 1.10.14 using the fact that Dnn,oo(Qn) is dense in the unit ball of A and Lk+i o Hk mSn k on Qn), we conclude that \\L o H(ni00(a)) - 0 as fc —> oo. This implies that Lo H — id^. Similarly, H o L = id^.

1.11

D

Exercises

1.11.1 Prove that in a unital algebra, every proper ideal is contained in a proper maximal ideal. 1.11.2 Let T be a bounded operator on a Hilbert space H. If sp(T) is disconnected, then T has a non-trivial invariant subspace, i.e., there is a proper closed subspace Ho C H such that T{HQ) C HQ. 1.11.3 Let T be in 1.1.17. Show that lim„_>oo HT^I1/™ = 0. Is zero an eigenvalue of T? 1.11.4 Let H be a Hilbert space with orthonormal basis { e ^ l ^ j . Define a bounded operator 5 by S(en) = e n +i. This operator 5 is called the shift operator. Show that S has no square roots.

61

Exercises

1.11.5 Let A be a unital C*-algebra and u G A a unitary. (1) Show that if sp(u) 7^ S1, then there is a self-adjoint element a G A such that u = exp(ia) and au = ua. (2) If \\u —1|| < 2, then u = exp(ia) for some a G A s a such that au = uo. 1.11.6 Finish the proof of Theorem 1.3.9. 1.11.7 Let T be a normal operator on a Hilbert space H with d\mH > 1. Show that T has a non-trivial invariant subspace. 1.11.8 Every irreducible representation of a commutative C*-algebra is one-dimensional. 1.11.9 Show that every hereditary C*-subalgebra of a simple C*algebra is simple. 1.11.10 Let A and B be C*-algebras and let h : A —> B be a surjective homomorphism. Suppose that b G -B+. Show that there is a G A+ such that h(a) = b. If b G Bsa, then there is a G Asa such that h(a) = b. 1.11.11 Let A = Co(X), where X is a locally compact Hausdorff space. Then for every ideal of A there is a closed subset F C X such that I = {fGA:f\F

= 0}.

Every quotient of A has the form Co (F) for some closed subset of X. 1.11.12 Let x and y be elements in a C* -algebra A and a G A+. Suppose that x*x < aa and yy* < a13 with a + f3 > 1. Then the sequence {x(^ + a)~1y} converges in norm to an element b G A with H&l < Ha^+Z5-1)/2!!. 1.11.13 Let x and a be elements in a C*-algebra A such that x*x < a. If 0 < a < 1/2, there is an element u G A with ||u|| < ||a x / 2 ~ a || such that x = uaa. 1.11.14 Prove 1.7.11. 1.11.15 If / i and ji are positive linear functionals on a C*-algebra, then t l / i + / 2 | i = H/ill + ! j / 2 | | . 1.11.16 Let a G Asa- Then a > 0 if and only if f(a) > 0 for all positive linear functionals. 1.11.17 Let r be a positive linear functional on a C*-algebra A. Then r is a trace if and only if T(X*X) = T(XX*) for all a G A.

62

The Basics of C* -algebras

1.11.18 Show that the mapping a M- a* of B(H) into B{H) is weak operator continuous but not strong operator continuous. 1.11.19 Show that the mapping a — i >• ab and the mapping a ^ t e o f B{H) into B{H) are weak operator continuous for each b G B(H). 1.11.20 Let H be an infinite dimensional Hilbert space. Show that the map B{H) x B{H) ->• B{H) defined by (a;, y) —> xy is not continuous in the weak operator topology. 1.11.21 Let H be a Hilbert space and B(H)r = {T G B{H) : ||T|| < r}. Show that (a, b) »->• ab : (B(H)r x B(H) —• B(H) is continuous when (B(H)r x B(H) is provided with the product of the weak operator topology on B(H)r and the strong operator topology on B(H), and the range B(H) is given the weak operator topology. 1.11.22 Let if be a Hilbert space and R be a von Neumann algebra on H. Show that (R)i is weak operator compact. 1.11.23 Complete the proof of Corollary 1.7.6.

1.11.24 Let T G B(H) be a normal operator and A $ sp(T). Show that

^ - ^ ' ^ d i s t ^ r ) ) -

1.11.25 Let if be a separable infinite dimensional Hilbert space with orthonormal basis {£n}> a n d let A = B(H). Denote by en the operators defined by e„(£) = Y2=i <£,£k> 6fc(1) Show that {e„} forms an approximate identity for /C. (2) Show that {en} converges strongly to IB(H) i n ^{H). (3) Show that {e„} does not converge weakly to 'i-B(H) m the von Neumann algebra B{H)". 1.11.26 Let M be a von Neumann algebra and u G M be a unitary. Show that there exists a G Msa such that u = exp(ia) and au = ua. 1.11.27 Let A be a C*-algebra acting irreducibly on a Hilbert space H. Suppose that A contains a nonzero compact operator. Then KL{H) C A. 1.11.28 Let S be the shift operator on the infinite dimensional separable Hilbert space and A — C*(S). (1) Show that ICcA.

Exercises

63

(2) If 7T : B{H) -> B(H)/K is the quotient map, then n(A) ^ C^S 1 ). (3) There is no unitary u E A such that ir(u)(z) = z, where z G S1 and n(A) is identified with CiS1). (4) A is a type I C* -algebra. 1.11.29 Let (X,a) be a classical dynamical system and let F C X be a closed subset such that F is invariant under a. Show that {/ G C(X) : / | F = 0} generates a proper ideal I C C* (X, er, /i), where \i is a translation invariant measure. 1.11.30 Show that a minimal dynamical system (X, a) must be ergodic. 1.11.31 Let G be an abelian group. The dual group G = Hom(G, Sx) is the group of characters. Show that C*(G) = CP*(G) = Co(G). 1.11.32 A state / on a C*-algebra A is called pure if p is a positive linear functional on A and p < f implies that p — tf for some t G [0,1]. (1) / is pure if and only if (Hf,iTf) is irreducible. (2) If A is commutative, then / is pure if and only if / is multiplicative. 1.11.33 Let a be a normal element in a nonzero C*-algebra A. Then there is a pure state f on A such that | / ( a ) | = ||a||. 1.11.34 Let A be a C*-algebra and PS(A) be the set of pure states. Then ©/ePSfA)71"/ is faithful. 1.11.35 Let A be a C*-algebra and be a pure state. Suppose that Lj, = {a G A : (f>(a*a) = 0} and iV^ = {a G A : (a) = 0}. Show that Nf = L

64

The Basics of

C-algebras

1.11.38 Let A be the UHF-algebra of type 2°° and T = {r(p) : p G A, projections}, where r is the unique normalized trace. Show that T = {r G Z[l/2] : 0 < r < 1}. 1.11. 39 Define 4>n : Mj,n —>• M3r>+i by „(a) = diag(a,a,a) for a G M^n. Let B = limn_+00(M3'»,^>n). Show that B is not isomorphic to the UHF-algebra of type 2°°. 1.11.40 Let a : A —¥ A be an automorphism. An automorphism a is said to be inner if there is a unitary u G A such that a(a) = u*au for all a G A. Show that any automorphism on B(H), where H is a separable Hilbert space, is inner. 1.11.41 Give an example of an automorphism which is not inner. 1.11.42 Let A be a unital C*-algebra. Then every ideal / of Mn(A) has the form Mn(J) for some ideal of J of A. So Mn(A)/I = Mn(A/J). In particular, if A is simple, then Mn(A) is also simple. 1.11.43 Show that every infinite dimensional C*-algebra contains two nonzero orthogonal positive elements a and b, i.e., ab = ba — 0. 1.11.44 A C*-algebra is said to be elementary if A == /C or A = Mn. Let 4 b e a non-elementary simple C*-algebra. Show that every hereditary C*-subalgebra A has infinite dimension. 1.11.45 Let i b e a non-elementary simple C*-algebra. Show that for any nonzero hereditary C*-subalgebra B, there is a £ B+ such that sp(a) contains infinitely many points. 1.11.46 Prove Corollary 1.10.7. 1.11.47 Let A = lim„_).00(A„, „), where each An is a simple C*algebra. Show that A is simple. 1.11.48 Show that every commutative C*-algebra is liminal. 1.11.49 Show that B(l2) is not a type I C*-algebra. 1.11.50 Let Ai* be direct sum of 2 n copies of C. Define a map hn : An ->• An+i by hn(a) = (a,a). Show that A = lim„_,.oo(^4n,^n) is a commutative C*-algebra. So A S* C(X). What is XI 1.11.51 Let A be a C*-algebra and L be a closed left ideal of A. Show that L fl L* is a hereditary C*-subalgebra of A. 1.11.52 Let A be a C*-algebra and I be an ideal of A. Then / admits

65

Addenda

an approximate identity {ea} such that \\eaa — aea\\ —• 0

a S A.

Such approximate identity is called quasi-central. 1.11.53 Let A be a C*-algebra and I be an ideal of A such that I is a cr-unital C*-algebra. Then J admits a quasi-central approximate identity {e n } satisfying e„ + 1 e„ = en (n = 1,2,...). 1.11.54 For each C*-subalgebra B of A the strong closure of B in A" is isomorphic to B".

1.12

Addenda

Theorem 1.12.1 Let H be a separable infinite dimensional Hilbert space. Then K.** = B(H). Definition 1.12.2 A W-algebra M is a C*-algebra which has a faithful representation IT : M -> B(H) such that ir(M)" = tr(M). Theorem 1.12.3 A C* -algebra A is a W*-algebra if and only if A is a dual space of some Banach space A*. Theorem 1.12.4 Let (X, a, fj.) be an ergodic dynamical system on a compact infinite Hausdorff space X. The C(X) xCT Z is simple if and only if a is minimal. Theorem 1.12.5 Let ¥2 be the free group on two generators. Then C*(F2) has a family of irreducible finite dimensional representations 7rn, n — 1,.... Furthermore, ©n>i7r„ is faithful. Theorem 1.12.6 The group C*-algebra C*(F2) admits a faithful trace and a faithful irreducible representation. Theorem 1.12.7 The reduced group C*-algebra C*(F2) is simple C*algebra without non-trivial projections and has a unique normalized trace. Definition 1.12.8 Let A be a C*-algebra. A composition series for A is a family {Ip}s
Theorem 1.12.9

The following are equivalent:

66

The Basics of C* - algebras

(i) A is a type I C*-algebra. ^ ^ ^ J ( i i ) A has a composition series {,},<. (iii) A has a composition series {Ip}i3
f IS

lmmd.

— £ i t * (688 - 742)

Chapter 2

Amenable C*-algebras and iiT-theory

2.1

Completely positive linear maps and the Stinespring representation

Definition 2.1.1 linear map. Then

Let A and B be two C*-algebras and <j> : A -¥ B be a

<j> ® id Mn : M n (A) -> M n (J3) by (aij) ^

(^K))

is also a linear map. We denote it by <j>(n\ It is easy to verify that if cj) is a *-homomorphism, then 4>^ is also a *-homomorphism. Proposition 2.1.2 vl positive linear map from one C* -algebra to another is always continuous. The proof of this is left as an exercise (2.8.1). Definition 2.1.3 Let <j>: A —> B be a linear map. We say <j> is n-positive, if (/>(n) : Mn(A) -¥ Mn(B) is positive. If is n-positive for every n, then <j> is said to be completely positive. Definition 2.1.4

We define |M| c 6 = s u p { | | ^ n > | | : n = l , 2 , . . . } .

A map : A —> B is said to be completely bounded, if \\4>\\cb is finite and is said to be completely contractive if \\4>\\cb < 167

Amenable C* -algebras and

68

K-theory

Theorem 2.1.5 (Stinespring) Let A be a C*-algebra and let <j> : A —>• B(H) be a completely positive linear map. Then there exists a Hilbert space Hi, a homomorphism IT : A —> B{H{), and a bounded operator V : H -> Hi with hm\\\(j)(e\)\\ = ||^|| 2 for some approximate identity {e\} of A such that {a) = V*n(a)V. Proof. Let A ® H be the algebraic tensor product of vector spaces A and H. Define a sesquilinear form on A ® H as follows: if £ = X^™=i ai ® xi a n d tnen V = YJj=ib:®Vh n

m

it,v) = y}y^{4>{Vjai)xuyj)HThe fact that <j> is completely positive insures that so defined (•, •} is positive and semi-definite since n

where x = (xi,...,xn) G # ( n ) and (-,•)#(*») denotes the inner product on n # ( ) , the direct sum of n copies of H. As in the proof of 1.6.15, N =

{££A®H:(U)=0}

is a subspace oi A® H. This gives an inner product on the quotient space (A O H)/N defined by

(Z + N,r, + N) = (£,r,). Let Hi be the resulting Hilbert space (by completion). For every a e A, define a linear map n(a) on A® H by 7r(a)(^J ai
aa

i ® ^t-

t=i

In

M„(J4),

one verifies that

(a*a*aaj) < \\a*a\\(a*aj). We then have, with £ = Y^i=i

a

i®xi,

(el-l)

Completely positive linear maps and the Stinespring representation

69

n

(7r(a)£,7r(a)0

=

^(^(a'^aa^x^x^H i,3 n

<

\\a*a>\\-

=

||a|| 2 «,0.

Yl((aiai)xi'Xi)H (el-2)

This last inequality is justified by n-positivity and (e 1.1). Thus ir(a) leaves N invariant and defines a linear map on (A ® H)/N, which we shall still denote by ir(a). Inequality (el.2) also shows that 7r(a) is bounded with |7T(OE.)II < ||a||. Thus 7r(a) extends to a bounded linear operator on Hi, which we again denote by 7r(o). It is straightforward to verify that the map 7r : A -* B(H\) is a homomorphism. Let {ex} be an approximate identity for A. Then {\\, so {. For each A, define V\x = ex ® x for x G H. We then have \\Vxx\\2

=

((el)x,x)<\\
So

||VA|| <

U(e,)\\1/2.

Moreover, if A < A',

\\(Vy-Vx)x\\2

= <

(((ex,-exf)x,x)H {{e\' -ex)x,x)H

< ||(
Hence {Vx} converges strongly to a bounded operator V defined on H. Since \\Vxx\\2

= {ex®x,ex®x)

< ({ex)x,x)H

< ||4>(e A )||||:r||,

we conclude that | | y | | 2 = l i r n | ^ ( e A ) | | = ||ao||. For each A, we have n

{Vx^,y)Hl

= (LVxy) =

(52ai®xi,ex®y)

70

Amenable C* -algebras and

K-theory

n

=

n

^2((e\a,i)xi, y)H = ( ^ i=l

{e\ai)xi,y)H,

»=1

so that n

n

VxC^ai®Xi) i=i

=

y^4>(e\ai)xi. »=i

Therefore, we have n

n

V*(^2di

®Xi) = '^2
i=l

j=l

Hence, for each a E A and x E H, V*-n(a)V\x

=

^*7r(a)(eA<S>x)

=

F*(aeA <8> x) = cj)(aex)x.

Thus, we obtain

V*n(a)Vx

= =

\imV*ir{a)Vxx limV* n(ae\)x = (j>(a)x.

Finally, we have V*n(a)V = 4>(a) for all a E A.

D

Corollary 2.1.6 Let 4> : A —> B be a completely positive linear map. Then for any approximate identity {e\} for A, W=lim||0(eA)||. Proof. By 2.1.5, \\<j)\\ > limA \\(ex)\\ = ||V|| 2 . By the proof of 2.1.5, one can choose any approximate identity {e^} in the proof (although V may be different). By 1.6.17, we may assume that B c B{H) for some Hilbert space H. Thus, by 2.1.5, there exist a Hilbert space Hi, a representation •K : A -)• B{Hi) and V G B(H,Hi) such that (a) = V*n(a)V. Therefore,

Completely positive linear maps and the Stinespring

representation

71

by 2.1.5, =

||V*7r(a)V|| < ||7r(a)||||V*y||

<

||a||||y|| 2 = lim|^(e A )||||a||.

Thus = lim||^(eA) D

Proposition 2.1.7 Let A be a C* -algebra. If 4> : A -» B is completely positive, then <j> is completely bounded and \\4>\\cb = ||<^||- Consequently, if <j> is also contractive, then 4> is completely contractive. Proof. Let {e\} be an approximate identity for A. Let E\ = diag(e^,..., e^)- Then {E\} forms an approximate identity for Mn(A). Note that 4>{n\Ex) = diag(^(e A ),...,^(e A )). So Un\Ex)\\

= ||diag(
Thus by 2.1.6, if (f) is completely positive, then

We* =||0||. Remark 2.1.8 It is evident that if is a self-adjoint linear map, then <^™) is self-adjoint. One might think perhaps that every positive linear map is a completely positive linear map. The following example shows that this is certainly not the case in general. Example 2.1.9 Let {e»j}j j = 1 denote the system of matrix units for M2. Let c$> : M2 —• M2 be the transpose map. One easily observes that the transpose of a positive matrix is again a positive matrix, and that the norm of the transpose of a matrix is the same as the norm of the matrix. So 4> is positive and \\\\ = 1. Now consider ^ : M2(M2) -> M2(M2). Note that en

ei2\

e2i

C22j

_

/I 0 0 \1

0 0 !\ 0 0 0 0 0 0 0 0 1/

72

Amenable C* -algebras and

K-theory

is positive. However 0(2)(('en e2i

e12V = ^ ( e i l ) e22;; V^( e 2i)

cf>(e12)\ ^(e 2 2 )y

=

1 0 [0 0 | 0 1 0 0

0 0\ 1 0 0 0 0 1/

is not positive. Thus, <> / is not completely positive. Many positive linear maps, however, are complete positive. In the next section, we will give examples of completely positive maps. 2.2

Examples of completely positive linear maps

Proposition 2.2.1 Let A, B and C be C*-algebras. (1) If 4> '• A —¥ B is a homomorphism, then is a contractive completely positive linear map. (2) If 4> • A —> B and ip : B —>• C are completely positive linear maps, then tp o (f> is a completely positive linear map. (3) / / <j> : A —>• B is a completely positive linear map and b G B, then L : A —> B,a H-> b*<j)[a)b, is a completely positive linear map. Proof. The assertions (1) and (2) are rather trivial and are left for the reader to verify. We will show (3). For any n and a = (a^) G Mn(A)+, L^n\a)

=

(L(alj)) = (b*(aij)b) (b®lMny4>{n)(a)(b®lMn)>0,

=

since cf>^ (a) > 0. Therefore L is n-positive.

•

Lemma 2.2.2 An element a £ Mn(A) is positive if and only if it is a sum of matrices of the form (a*aj) with a\,..., an £ A. Proof, lib — (a*a,j), then b = a*a for a = (a*,-) with aXj = a,j, 1 < j < n, and Oy = 0, 2 < i < n, 1 < j < n. So b > 0. If a = (a^) £ Mn{A) is positive, then there is an element c = (cy) G M „ ( J 4 ) such that a = c*c. Hence we have a

ij

~ / k=l

,ckickj-

Examples of completely positive linear maps

Put bk = (CfriCkj) G Mn(A), then have a = Y^k=i ^feLemma 2.2.3 x a x

Z)"j=i i ij j

which is a matrix of the indicated form, we ^

A matrix a = (a^) G Mn(A) or

ever

^ ° f

y

73

x

is positive if and only if

x

i> •••' n G -A-

Proof. Suppose that a > 0. By 2.2.2, we may assume that a^ = a*a, for some ai,..., a n G A. Then we have n

n

n

^2 XiO-ijXj = ^

x'aidjXj

n

= (^2aixi)*(^2a3xj)

^ °-

(e2-3)

We leave the converse as an easy exercise (2.8.6). Corollary 2.2.4

•

A linear map : A —> B is n-positive if and only if n

Yl y*(x*xi)yj ^ ° for every x\,..., xn G A and j/i, ...,y„ G B. This follows directly from 2.2.2 and 2.2.3. Theorem 2.2.5 Let A and B be two C*-algebras. If B is a commutative C* -algebra, then every positive linear map : A —> B is completely positive. In particular, every positive linear functional on a C*-algebra is always completely positive. Proof. Write B — CQ{X) for some locally compact Hausdorff space X. For each xi,..., xn G A and j/i,..., yn G B = Co(X), we have for each t G X, n

n

n

x

(Y, y*( >j)yj)(t) = Yl wCWto *,•)(*)&•(*) = E

^W)<x3y^W)

#(£yi(0*i)(I>(*)z>))(t) 2=1 n

=

j= l n

0-

Hence ^ 3 " = 1 y*4>(x*Xj)yj > 0. Therefore ^ is completely positive by 2.2.4.

•

74

Amenable C -algebras and

K-theory

Theorem 2.2.6 If A is a commutative C*-algebra, then any positive linear map <j> from A to another C* -algebra B is completely positive. Proof. We write A = CQ(X) for some locally compact Hausdorff space X. By 2.2.4, it suffices to show that n

Y

yi^(xixj)yj

^ °

for every xu...,xn e A and yi,...,yn e B. Since ^2^j=xyt4>(^i^j)yj self-adjoint, by 2.2.4, it suffices to show that, for any state / on JB,

is

n

£ m^ixiivi) > o. Fix a state / . Let fJ.ij(a) = f(y*{a)yj), 1 < i, j < n. Then //y is given by a complex Radon measure on X. Let fi = $2™.-=1 \fcj\- For each i,j, there exists /ij G Ll{X,[i) such that Mij(ffl) = / Jx

a(t)fij(t)dfj,.

For any Ai,..., A„ G C, we have n

Y

^ij(x*x)Xi\j

=

^2f(y*(j)(x*x)yj)XiXj n

=

n

/((^Ai2/i)>(^a;)(^Ai2/i))>0; i=l

(e2.4)

i=l

hence 5^^7=1 Vij^-i^j ls a positive measure. So Yllj=i fijif)^i^j > 0 f° r /i_ almost everywhere. Fix finitely many mutually disjoint measurable subsets Fi,..., FN and ty. £ Fk, 1 < k < N. There is a measurable subset ft c X such that J27j=i fij(t)%i(tk)xj(tk) > 0 on alH £ ft such that /x(X\ft) = 0, k = 1,2,..., N. Let Zi = Ylk=i xi(tk)XFk, where XFk is the characteristic function on Fk. Then Y%j=i fij(t)zi(t)zj(t) > 0 //-almost everywhere. Note that Xj € Co(X). From measure theory, we have n

Y,

fiji^x^tjxj^yo

Examples of completely positive linear maps

75

^-almost everywhere. Thus n

f(yi(x*xj)yj)

X

= X / (x^mfijitw J X

• •

D Theorem 2.2.7 Let A and B be two C*-algebras and a contractive completely positive linear map. Suppose that and p £ B is a projection such that <j>(a) < p for all 0 < exists a contractive completely positive linear map : 4>\A

— 4>

4> : A —¥ B be A is non-unital a < 1. Then there A —> B such that

an

d 0(1) = v-

Proof. By considering pBp, we may assume that p = 1B- Let n be a positive integer. We will identify a scalar matrix with a matrix in Mn(B) with entries in C • 1B- For c = a + A • 1^, where a G A, we define <j>(c) = 4>(a) + Xp. We claim that for any scalar matrix m and a = (ay), # n >(m(ay)) = m • 0 ( n ) (a) and

ftn\{aij)m)

= 0(a) • m.

(e2.5)

Suppose that m = (m^). Then by definition,

where C;J = £)fc=i mn-akj- Thus n

0

(n)

(mo)

=

(4>(ci:j)) =

(^2mik<j>(akj)) fc=i

=

(mij)(«A(afej)) = m 0 ( n ) ( o ) .

The second equation in (e2.5) also follows. This proves the claim. Any element 6 e Mn(A) can be written as b = a + m, where a £ M n (A) and m e M n . Suppose that 6 > 0. Since Mn{A) is an ideal of Mn(A), by considering the quotient map -K : Mn(A) —• M n , we conclude that b > 0 implies that m > 0. For any e > 0, let mE = m + e • 1M„(B)- Then mE is invertible. We also have a + mE > 0. Thus — me ame ' < lMnr^\- Let {ex} be an approximate identity for Mn(A). Then - e A m - 1 / 2 a m J 1 / 2 e A < e\.

76

Amenable C* -algebras and K-theory

Since — me ' ame ' -1/2

-1/2

m £ ' 077iE

£ A and <^n) is positive, e\me

ame ' e\ —•

,

and

Therefore ^)(_m£-l/2amrl/2) <

lMri(B)

.

By the claim, we obtain -m:"2^n\a)m^l2

< lMn(B),

or equivalently, (f>l-n\a)+me

>0.

Thus 4>{n)(a + m) = ^n\a)+m>

-elMn(B)

for all s > 0. Therefore <^ n) (a + m) > 0 . This implies that is completely positive. 2.3 Amenable C*-algebras

D

Definition 2.3.1 Let A be a C*-algebra. A is said to be amenable if the identity map on A can be approximated pointwise in the norm topology by contractive completely positive linear maps through finite dimensional C*-algebras, i.e., for any finite subset T C A and e > 0, there exist a finite dimensional C*-algebra B, a contractive completely positive linear map : A —> B and a contractive completely positive linear map L : B —>• A such that \\a- Lo(j)(a)\\ < e for all a € T. An amenable C*-algebra is also called nuclear.

Amenable

C*-algebras

77

Thus A is amenable if only if, for any e > 0 and any finite subset T C A, id « e L o (f> on f for some finite dimensional C*-algebra B and contractive completely positive linear maps 4>: A —J- B and L : B —> A. The following is immediate from the definition. Corollary 2.3.2 Every finite dimensional C*-algebra is amenable. If A is amenable, then Mn(A) is amenable for every n. Definition 2.3.3 Let {ej}" =1 be the canonical orthonormal basis for an n-dimensional Hilbert space. If a G Mn, then let a^- denote the (i, j)-entry of a so that a^ = (aej, e$). If : A —>• M n is a linear map, then we associate a linear functional s^ : Mn(A) -4 C to <j> by 71

s 0 (a) = ( l / n ) ^((a)

= (1/n) ((A(n) (o)x, x),

(e 3.6)

where the inner product is taken in C n . Let s : Mn(A) - > C b e a linear map. Define <j>s : A —> Mn via {4>s{a))ij =n- s{a® e^),

(e3.7)

where a ® e^ is the element of Mn(A) which has a for its (i,j)-entry and is zero elsewhere. We leave it to the reader to verify that s, = and s(j>s = s (2.8.7). Lemma 2.3.4 Let A be a unital C*-algebra and 4>: A —• Mn be a linear map. Then the following are equivalent: (1) 4> is completely positive, (2) 4> is n-positive, (3) s is m-positive, we will apply 2.2.2. It suffices

78

Amenable C* -algebras and

K-theory

to consider a positive element of Mm(A) of the form (a*aj) and show <j>(m\(a*aj))>0. Take x = xx © • • • © xm, where xt = X)fc=i A*fcefc € C "> a n d {ei> •••, e «} is an orthonormal basis for C™. Let bi be the n x n matrix whose first row is (A»i,..., Xin) and whose other rows are zero. We have n b b

iJ

=

X ] ^n^jkeik-

(e3.8)

k,l=l

Since <j)(m\(a*aj)) acts on Cmn, to see that it is positive, it is sufficient to show that

(^m\(a*aj))x,x)>0. Using (e3.8), we have ({m)((a*aj))x,x)

=

Y^((a*aj)xJ^xi)

=

n

$ Z ^jkXus(aiaj i,j,k,l n

=

^ikXiiWala^e^ei)

®elk) = n's^2/s{a*a-i ® b^bj) = i,j n

ns(^2aibi)*(^2aj
since s is positive.

= Yl

>0,

j-i

D

Theorem 2.3.5 Let A be a C*-algebra and B be a C*-subalgebra. Suppose that <j> : B —> Mn is a completely positive linear map. Then there exists a completely positive linear map : A —» Mn such that 4>\B = - Moreover, if (j> is a contractive completely positive linear map, then

as in 2.3.3. Extend it to a positive linear functional s on Mn(A) (by 1.6.10). By 2.3.4 the map ip associated with s is completely positive. Since s extends s^, it is easy to see that ip extends , using 2.3.3. Now assume that is a contractive completely positive linear map. We may assume that B c l s o that the 1^ = 1^. By 2.2.7, there is a contractive completely positive linear map <$>\ : B -4 Mn such that 4>i(lg) = 1M„Then, from the first part of the proof, there exists a completely positive

Amenable C* -algebras

79

linear map : A -> Mn such that \g = \. Since \\<j>\\ = ||(1^)| ||i(lg)|| = 1, 4> is a contractive completely positive linear map. D Corollary 2.3.6 A C* -algebra A is amenable if and only if for any finite subset J- C A and any e > 0, there exist an integer n, contractive completely positive linear maps <j> : A —>• Mn and L : Mn —• A such that id « E L o <j>

on J7.

Proof. One direction is trivial. We assume that A is amenable. Fix £ > 0 and a finite subset T C A. By definition, there is a finite-dimensional C*-algebra B and there are contractive completely positive linear maps <j>\ : A ->• B and L\:B^tA such that id « e Li o 0J

on .T7.

There is an embedding h : B —> M n for some large integer n. Set (j> = ho(j>\. It follows from 2.3.5 that there exists a contractive completely positive linear map V : Mn —• i? such that ip oh = idg. Define L = L\oty. Then id « e L o <> / on ^ Theorem 2.3.7

D Every commutative C* -algebra is amenable.

Proof. We will prove the case in which A is unital. The non-unital case is left to the reader. Let A — C(X), where X is a compact metric space. We may assume that X is infinite. Fix e > 0 and a finite subset T C A. There are open subsets Oi,...,On C X and distinct points t\,...,tn £ X such that U £ Oi and \a(t) -a(ti)\

<s/2

for a G T,

whenever t € Oi, 1 < i < n. Define a homomorphism <j> : A —» J3 = ®" =1 C by (a) = (a(ti), ...,a(t n )). Let gi,...,gn G C ( ^ ) he a partition of unity corresponding to Oi,...,©^. Then X)r=iff»(*) = 1 f° r a n * G X Define L : B -+ A by L((zi, ...,£„)) = E?=i^5<- S i n c e Si > 0 (1 < i < n), L is positive. By 2.2.5 (or by 2.2.6), L is completely positive. Since L(lB) = Y^i=i 9i = 1> by 2.1.6, L is a contraction. For any t € X, if gi{t) ^ 0, then

80

Amenable C* -algebras and

K-theory

t<=Oi. Therefore n

X > W = X>(0 = 1teOi

(e3.9)

i=i

Furthermore, if t € Oj l~l Oj, then |a(ti) - a ( t , ) | < \a(U) - a(t)\ + \a(t) - a(tj)\ <e/2 + e/2 = e

(e3.10)

for all a e J7. Thus, from (e3.9 )and (e3.10), n

|a(t)-^a(ti)ffiWI < I ^2(a(t)-a(ti))gi(t)\ <e i=i

(e3.11)

teOi

for all a £ T. We conclude that ||a-Lo0(a)|| < e

(e3.12)

for all a & J7. Therefore A is amenable.

•

Proposition 2.3.8 Every hereditary C*-subalgebra of an amenable C*algebra is amenable. In particular, every ideal of an amenable C* -algebra is amenable. Proof. Let C be a hereditary C*-subalgebra of an amenable C*-algebra A. Fix ci, ....,c„ G C and e > 0. There are a finite dimensional C*-algebra B, contractive completely positive linear maps <j> : A —»• B and X : B —>• ^4 such that ||ci-Lo0(ci)|| <e/2,

t = l,2,...,n.

By using an approximate identity for C, we obtain an element a G C with 0 < a < 1 such that \\acia - Ci\\ < e/2,

i=

l,2,...,n.

Set <j>i = <j)\c and L\(b) = aL(b)a (for b G i?). Then aca,L\{b) G C for any c £ C and 6 G 5 . In particular Li : B —t A is completely positive. Since L(1B) = aL(ls)a < 1, by 2.1.6, L\ is a contractive completely positive linear map. Now we estimate that \\ci - L\ o<j)i(ci)\\ < \\ci - acia\\ + \\aaa - a(L o <j>(a))a\\ <

e/2 + e/2

Amenable C* -algebras

for i = 1, 2, ...,n. It follows that C is amenable.

81

•

Theorem 2.3.9 Let {An} be a sequence of amenable C*-algebras with An C An+i, n = 1,2,.... Suppose that A is the closure of Un=iAn. Then A is amenable. Proof. Let T = {ai,...,am} C A be a finite subset and e > 0. There exists an integer n and there exists a finite subset Q = {bi, ...,bm} C An such that | | a i - 6 i | | < e/3

i = 1,2, ...,m.

Since J4„ is amenable, by 2.3.2, there are contractive completely positive linear maps 4> : An —> MN and L : MN —»• An such that \\bi-Lo<j>(bi)\\<e/3

t = l,2,...,m.

It follows from 2.3.5 that there exists a contractive completely positive linear map ip : A —> Mjv such that IP\A„ = 4>- Thus Hoi-Lo^aOH

<

K - f e i | | + ||6i-LoV(6i)|| + ||L o V>(6i) - L o ^(ai)|| < e/3 + e/3 + e/3 = e

(i = 1, ...,m). Therefore vl is amenable. Corollary 2.3.10

D

Every AF-algebra is amenable.

Definition 2.3.11 If A = M„(C(F)), where F C 5 1 is a compact subset, then we say that A is a circle algebra. A C*-algebra A is said to be an ATalgebra if A is an inductive limit of An, where each An is a finite direct sum of circle algebras. Corollary 2.3.12

Every AT-algebra is amenable.

Proof. It is easy to see (see 1.11.42 and 1.11.11) that every quotient of Mn(C(S1)) has the form Mn(C(F)), where F is a compact subset of S 1 . Therefore A is the closure of UnBn, where Bn is a quotient of An. Since C(F) is amenable by 2.3.7, Mn(C(F)) is amenable (by 2.3.2). So Bn is amenable. It follows from 2.3.9 that A is amenable. • Finally, we end this section with the following important theorem. Theorem 2.3.13 Let A be a C*-algebra, B be a C*-subalgebra of A and C be another C* -algebra. Suppose that : B —• C is a contractive completely positive linear map. If either B or C is amenable, then for any

82

Amenable C* -algebras and

K-theory

finite subset J- C B and e > 0, there is a contractive completely positive linear map : A —• C such that ~ E <j>

on J-.

Proof. Case (1): C is amenable. Let {&i,..., bm} C B and e > 0. Let Cj = 4>{bi), 1 < i < m. Since C is amenable, there are contractive completely positive linear maps L\ : C —> MN and i 2 : MJV —> C (for some integer TV) such that \\ci- L2°Li(ci)\\

<e,

i=

l,...,m.

By 2.3.5, there exists a contractive completely positive linear map Lj, : A —> M n such that (La)|s = L\ o <j>. Set <(> j = L2 o L 3 . Then ^ is a contractive completely positive linear map and (bi) = L2 o L\ o 4>{bi) = L 2 ° Li(ci). Therefore ll^(6i) - ^(&0ll = 11^2 o ^ ( c * ) - Ci|| < e. for i = 1, ...,m. Case (2): B is amenable. There are contractive completely positive linear maps tp\ : B —>• M ^ and ^2 : MK —> B (for some integer K) such that ll&i- V"2 ° ^ i ( ^ ) | | < £,

« = l,...,rn.

By 2.3.5, there exists ^3 : A —> M^- such that ft oi/j2 ° ip3- Then

||^(M-^0ll

2.4

= V'l- Let 0 =

= 1 1 ^ - ^ 2 0^(60)11 <

for i = 1, ....m.

(V>3)|B

||6i-V>2o^i(60ll<e D

K-theory

Definition 2.4.1 Let A be a *-algebra. Consider the matrix algebras Mn(A),n = 1,2,.... We identify M„( A) with a subalgebra of Mn+i(A) using the embedding a H-> diag(a,0). With this identification, we have Mn(A) C Mn+i(A) and define Moo(i4) = UnMn(A). We denote by Proj(M 00 (,4)) the

K-theory

83

set of all projections in MX(A). If p, q G Proj(M 00 ( J 4)) are two projections, we may assume that p, q G M n (A) for some n. In this section, we will denote I

1 by p ffi q.

Let p and g be two projections in Moo(A). We say p and g are equivalent in MOO(J4) and write p ~ q if there exist n and u G M„(^4) such that p, q G M n (A), U*U = p and uu* — q. In some cases, we also say that p and q are equivalent in Mn(A). It is a straightforward exercise to check that "~" is an equivalence relation on Proj(Moo(A)) (and on projections in Mn{A)). We will write p -< q if p ~ q' for some projection q' < q. Proposition 2.4.2 Let A be a *-algebra andp, q, p' and q' be projections in Proj(Moo(i4)). (1) / / p ~ p' and q ~ q', then p®q~p'®q'. (2) pffiq ~ qffip. (3) If p,q G Mn(A) and pq = 0, then p(Bq ~ p + qProof. Suppose that p ~ p' and q ~ q'. There are u and v such that u*u = p, uu* = p', v*v = q and vv* = q'. Set w = u © v. Then w*w = p © q and ww* = p' © g'. This proves (1). For (2), set

••COThen u*u = p © q and uu* = 5ffip. To show (3), we let

Hi »)• Then v*v = p ffi g and tw* = p + g.

D

Definition 2.4.3 Let A be a unital *-algebra. We say elements p and q in Proj(Moo(j4)) are stably equivalent, and write p ~ s q, if there is a positive integer n such that p © ljv^ ~ g ffi 1M„- It is easy to check that " ~ s " is an equivalence relation on Proj(M 00 ( J 4)). Note that by 2.4.2, if p ~ s p' and q ~ s q', then p © q ~ s p' ffi g'. For p G Proj(M 00 (A)), let [p] denote its stable equivalence class, and denote by P(A) the set of all stable equivalence classes. Define an addition on P(A) as follows: If [p], [q] G P(A),

84

Amenable C* -algebras and

K-theory

define [p] + [q] = [p © q]. It follows from 2.4.2 that this binary operation is well-defined. Lemma 2.4.4 If A is a unital *-algebra, then P(A) is a cancellative abelian semigroup with zero element [0]. Proof. Associativity and commutativity are immediate. It is also evident that [0] is the zero element in P(A) (with the above defined addition). To show that P(A) is cancellative, let \p] + [q] = [p] + [r]. There is an integer m > 0 such that p@ q@ 1MTO(A) ~ P ® r ® ^-Mm(A)- We may also assume that p e Mn(A). Then (1M„(A) But

-p)®P®q®lMm(A)

~

(1M„(A)

-p)®P®r®lMrn{A).

by 2.4.2. Thus we have l M r i ( A ) ©g©l M m ( j 4 ) ~ ®r® lMm(A)- Therefore, by applying 2.4.2 again, q ® lMm+n(A) ~ r ® lM„+m(A)- Hence [q] = [r]. D ( 1 M „ ( A ) - P ) ® P ~ 1M„(A)

1M„(A)

Definition 2.4.5 Let 5 be a cancellative abelian semigroup with a zero element. We define an equivalence relation ~ on S x S by setting (xi, y\) ~ (^2,2/2) if ^l + 2/2 = £2 + 2/1- Denote by [x,y] the equivalence class of (x,y). The set G ^ ) of equivalence classes is an abelian group under the well-defined operation [x, y] + [w, z] = [x + w,y + z]. G(S) may be thought of as the group of (equivalence classes of ) formal differences of elements of S, thinking of [x, y] as x — y. The prototypical example of this construction is the construction of Z from N U {0}. We call G(S) the enveloping or Grothendieck group of 5". The map 4> '• S —> G(S) defined by 1 H> [a;, 0] is a homomorphism and is injective. So S can be identified with its image (by identifying x with [z, 0]). Hence, with this identification, indeed G(S) = {x — y : x,y £ S}. Suppose that G is an abelian group and h : S —> G is a (semigroup) homomorphism. Then there exists a unique homomorphism h : G(S) —• G extending h. The above are standard in group theory and their elementary proofs are left as exercises. If A is a unital *-algebra, we define Ko(A) to be the Grothendieck group

oiP(A). Remark 2.4.6 If A and B are two unital *-algebras and h : A —» B is a *-homomorphism, then h^ : Mn(A) -> Mn(B) is a *-homomorphism.

85

K-theory

We will use h for h^n\ If p and q G Mn(A) are projections and p ~ q, then /i(p) ~ h(q). Suppose that p © 1M„(A) ~ 9 © 1 M „ ( A ) - Then /i(p) © M ! M „ ( A ) ) ~ % ) © /i(l Mn (A))- Let e = 1M„(B) - /I(1M„(A))- Then h(p) © /i(l M „(A)) © e ~ h(q) © /I(1M„(A)) © e - Therefore h(p) ~ s /i(g). Hence h defines a map ft, : P(A) -4 P(B) by setting /i*([p]) = [^(p)]- It is clear that /i*([p]+ [• K0(B) such that /i*([p]) = [h(p)]If / i : ^4 —• 5 and

C are homomorphisms of unital G*-algebras, then ((f) o /i)» = 4>* o h*. Definition 2.4.7 Let ^4 be a unital Banach algebra. Let GLn(A) be the group of invertible elements of Mn(A). We embed GLn(A) into GLn+i(A) by a; H4 diag(a;, 1). Let GLoo(A) = lim n GL n (A). Then G'L00(A) is a group. Let GLn(A)o be the subgroup of invertible elements which are in the same path connected component of 1M„(A)- Let x G GLn(A)o and let {xt : t G [0,1]} be a path in GLn(A)o with XQ = x and x\ = 1M„(A)- Then for any z G GL n (i4), \z~xxtz : £ G [0,1]} is a path in GLn(A). Since z~xx\z = l 1M„(A)) the entire path is in GL n (.4)o. Hence z~ xz G GL n (A)o. Thus GLn(j4)o is a normal subgroup of GLn(A). We define K^A)

= \im[GLn(A)/GLn(A)0]

-

GLoo(A)/GLoo(A)0.

n

If x G GLn(A),

we use [a;] for the image of x in

K\(A).

Definition 2.4.8 Let A be a unital G*-algebra. Let Un(A) be the group of unitaries in Mn(A) and Un(A)Q be the path connected component of the identity of Un(A). Note that Un(A) C GLn(A) and Un(A)0 c GL„(^) 0 . When n = 1, we denote l/i(^4) by J7(J4) and Ui(A)0 by t/(.A)o, respectively. T h e o r e m 2.4.9 Let A be a unital C*-algebra. Then U(A)Q is a normal subgroup of U(A). If u G A, then u G U(A)o if and only if there exist a\, ...,an G Asa such that u — exp(iai) • • • exp(ian). Proof. Let E be the set of all elements u G A of the form u = exp(ia{) • • • exp(ian) for some n and some a\,...,an G Asa. Any such element u belongs to U(A)Q since the function U : [0,1] -» U(A) defined by 11-» exp(ita\) • • • exp[itan)

86

Amenable C* -algebras and

K-theory

is a continuous path in U(A) from 1 to u. It is clear that E is a subgroup of U(A). By 1.11.5, if u is a unitary and \\u — 1|| < 2, then u = exp(ia) for some a G Asa. Therefore, if u G E and v G U(A) such that ||u — v\\ < 2, then v = u • exp(ia) for some a G Asa. This implies E is both open and closed. Since 1 G E, E = U(A)0. If u G £7(A), then, for any w G C/(A)0, uvu-1 G J7(A)o. Therefore f/(^4)o is a normal subgroup. • Proposition 2.4.10

Let A be a unital C*-algebra. Then K1(A) =

\imUn(A)/Un(A)0. n

Proof. For any invertible element x G Mn(A), define a map s : GLn(A) —>• E/„(J4) by a; i-> a;|a;|_1 using the polar decomposition. Then s is a contraction. It is then clear that s\ : GLn(A)/GLn(A)o —> Un(A)/Un(A)o is an isomorphism. • Lemma 2.4.11 Let A be a unital Banach algebra. If x,y G GL(A), then there is a path of invertible elements in GL2(A) from dia.g(xy, 1) to diag(x,y). If A is a unital C*-algebra and x,y are unitaries, the path may be chosen to consist of unitaries. In particular, if z G GL(A), then d i a g ^ . z - 1 ) G GL2(A)0. Proof.

Set u>t = diag(a;, 1) • m • diag(y, 1) • u^ , where costOr/2)

wt

sini(7r/2)

-™*(*/2)y cosi(7r/2) J

(e413) v

One verifies that wt meets all the requirements. Proposition 2.4.12

' •

Ki{A) is an abelian group.

Proof. Let Bn = Mn(A) [diag(x,2/)]. Let

and x,y G Un(A). By 2.4.11, [diag(a;y, 1)] =

- a J)Then z G U2(Bn)0 and z* = z. Note that &\&g{x,y) = z • diag(y, a;) • z. Therefore [xy] = [diag(o;y, 1)] = [diag(a;,y)] = [diag(j/,x)] = [di&g{yx, 1)] = [yx].

K-theory

87

We present one easy example here. More examples will be presented in t h e next section. E x a m p l e 2 . 4 . 1 3 Let p, q G Mn be two projections. It is easy to check t h a t p ~ q in Mn if and only if they have the same rank. So, in P ( C ) , [p] = [g] if and only if they have the same rank. Also, we have r a n k ( p © q ) = rank(p) + r a n k ( g ) . T h u s , we may define a homomorphism rank: P(A) —>• Z by setting rank([p]) = rank(p). This extends uniquely to a homomorphism Ko(C) —• Z. This m a p is surjective, since 1 = rank([lc])- It is injective, since if x £ ker(rank) we can write x = \p] — [q], where p, q are projections of t h e same rank in Mn for some n. So \p] = [q] and x = 0. T h u s we have KQ(C) = Z. In this example, stable equivalence and equivalence are the same. One may think of KQ(A) as a "dimension" group and think of [p] as t h e "generalized dimension" of p. Since Un(C) = U(Mn) = U(Mn)0, ffi(C) - {0}. Definition 2.4.14 Let A be a non-unital C*-algebra a n d A be the unitization of A. If n : A —• C is the canonical quotient m a p , t h e n 7r* : KQ(A) —> Z is a homomorphism. We define KQ(A)

= ker7r».

Proposition 2.4.15 If A and B are two C* -algebras and h : A —>• B is a homomorphism, then h induces a homomorphism h* : K O ( J 4 ) —> KQ{B) which maps \p] to [h(p)] for p £ P{A). Proof. This has been verified in 2.4.6 in the case t h a t A a n d B are unital. If b o t h A and B are non-unital, then h extends uniquely to a unital homomorphism h : A —»• B and it induces a unital homomorphism h : C —>• C. By 2.4.5, h induces h* : K0(A) -4 K0(B). T h u s we have the following commutative diagram

K0(A) H K0(B) I _ I Z -S Z Clearly h* is a n isomorphism. Therefore h*(K0(A)) c KQ(B). We define /i* = (h*)\K0(A)- If ^ is non-unital and B is unital, let C = h(A). By adding 1 B to C, we may identify C with C + C • 1 B - SO h extends uniquely to h! : A —* C. T h u s , from what we have just proved, h induces h't : Ko(A) —>•

88

Amenable C* -algebras and

K-theory

K0(C) c K0(C). But the embedding i : C -> B induces a homomorphism i* : Ko(C) —> KQ(B). Let /i* = », o ^ . Since /i^, maps \p] to [/i(p)] and z, maps [/i(p)] to [i o h(p)], we see that /i*([p]) = [/i(p)]. Finally, if A is unital and J3 is non-unital, then h maps A into B. So, by 2.4.6, h induces K : K0(A) -» ^ 0 ( 5 ) . Since TT o ft = 0, h,(K0(A)) C # o ( £ ) . • Definition 2.4.16

Let A be a non-unital C*-algebra. We define K\ (A) =

KM). Lemma 2.4.17 Let A and B he unital C*-algebras and h : A —• B be a unital homomorphism. Then h induces a continuous homomorphism, h(U(A)0) c U(B)0. Proof. Let u G U(A)0. exp(iai) • • • exp(ian),

By 2.4.9,

we may write that

u

=

h(exp(iai) • • • exp(ian)) = exp(ih(ai)) • • • exp{ih{an)). Remark 2.4.18 Let A be a unital C*-algebra, e e A be a projection and C C A be a C*-subalgebra such that \c = e. Let i : C —> A be the embedding. If u G U(C), then u + (1 - e) G f/(A). Moreover, if u G U(C)0, then u + (1 - e) G U(A)0. Thus z» : A"i(C) -> ifi(A), [M] H->- [u + (1 - e)] is a homomorphism. Without confusion, we may denote [u + (1 — e)] by [u]

in KM)Corollary 2.4.19 Let A and B be two C*-algebras and h : A —> B be a homomorphism. Then h induces a homomorphism /i»i : K\(A) —> K\(B) which maps [u] to [h(u)]. Proof. If both A and B are unital, or both are non-unital, the corollary follows from 2.4.17 immediately. If A is unital and B is non-unital, we view h as a homomorphism from A into B. Then in this case, the corollary follows from 2.4.18. If A is non-unital and B is unital, let C\ = h(A) and C = Cx + C • 1B. Then h extends to h : A ->• C C B. So the corollary follows from the unital case. • Remark 2.4.20 If h : A —>• B, then we will use h*i for the induced homomorphism from KM) t o Ki(B) (i = 0,1). If : B —• C, then *i o /i»j = (<£ o /i) t i (i = 0,1). Therefore we have covariant functors A t-¥ KM)>h

^ h-.

Perturbations

89

from the category of all C*-algebras to the category of abelian groups. 2.5

Perturbations

We present a number of technical lemmas commonly used in C*-algebra theory. Most of them involve a small perturbation of projections or partial isometries. Only a few will be immediately used in the next section. However they are related and will be used more frequently in the later sections. Lemma 2.5.1 Let p and q be projections in a unital C*-algebra A and suppose that \\p — q\\ < 1. Then there exists a unitary u € A such that q = upu* and ||1 —u|| < %/2||p—
U ||

2

= ||(1 -

U )*(l

-

U )||

= 2||1 - i?e(«)|| = 2||1 - | V ||| < 2||1 - M 2 ||.

But 1 - \v\2 = {q- p)2. Therefore ||1 - u|| 2 < 2\\q - p\\2. Finally, ||l-U||
n

Ifp and q are two projections in a C*-algebra A such that

h-<w\\ < i,

90

Amenable C* -algebras and

K-theory

then there exists a partial isometry v £ A such that vv* = q and v*v < p. Moreover vpv* = q. Proof. We have \\q — qpq\\ < 1. Therefore, qpq is invertible in qAq. Set v = (qpq)~1^2(qp), where the inverse is taken in qAq. Then vv* = (qpq)~1/2qppq(qpq)~1/2

= q.

Moreover, v*v = (pq)(qpq)~1{qp) G pAp is a projection. Therefore v*v < p. Since vp = v, vpv* = vv* = q.

•

Lemma 2.5.3 Let p, q £ A be two projections. Suppose that there is an element x £ A such that x = qxp and \\p — x*x\\ < 1 and ||g - xx* || < 1. Then p ~ q. Proof. The inequalities ||p —aj*a;|| < 1 and \\q — xx*\\ < 1 imply that x*x is invertible in pAp and xx* is invertible in qAq. Let w = a:|:r| -1 , where the inverse is taken in pAp. Then w*w = |a;|_1a;*a;|a;|_1 = p and xx*ww* = xx*x\x\~2x*

= xx*.

Since ww* is a projection in qAq and xx* is invertible in qAq, ww* — q. • Lemma 2.5.4 Suppose that a £ Asa and p is a projection in A with \\a—p\\ < 1/2. Then there is a projection q in the C*-subalgebra B generated by a such that \\q-p\\ Furthermore, q ~ p in A.

<2||a-p||.

91

Perturbations

Proof. Let 6 — \\a - p||. If A G E with d = dist(A,sp(p)) > 5, then, by 1.11.24, ll(A

-rt"1||

=

di S t(A!sp(p))-

Therefore, if ||x - (A - p)\\ < d, then x G GL(A). Now ||(A-a)-(A-p)|| = ||o-p||=J
- 4 < $

a n d

/Isp(o) = (X[l-«,l+*])lsp(a)-

Therefore q = f(a) G B, the C*-subalgebra generated by a, is a projection and \\f(a)-a\\<6. Hence ||<7-p|| = | | ( / ( a ) - o ) + ( a - p ) | | < 2 | | f l - p | | . Since | | a - p | | < 1/2, by 2.5.1, 3 ~ p .

•

Lemma 2.5.5 If a £ Asa such that \\a\\ > 1/2 and \\a — a2\\ < 1/4, then there exists a projection p in the C* -subalgebra generated by a such that \\a-p\\<2\\a-a2\\. Proof. The proof of this lemma is contained in that of lemma 2.5.4. We leave it to the reader to check this. • Lemma 2.5.6 If s > 0 and n is a positive integer, there is S(e, n) = 6 > 0 with the property that if A is a C* -algebra and ifpi, ...,Pn are projections in A with HpiPjII < S for i ^ j , then there are mutually orthogonal projections 9i, •••,9n G A such that \\pi - qi\\ < e. Proof. We use induction on n. If n = 1 there is nothing to prove. Suppose that the lemma holds for all n < k. For any 0 < e < 1/3, we define 6{e, k + l) = min{l/3, e/12fc, 6{e/\2k,

k)},

where the right hand side is already denned by the inductive assumption.

92

Amenable C* -algebras and

K-theory

Suppose that we have projections pi, ...,pk+i € A such that \\piPj\\ < S(e,k + 1) if i

±j.

By inductive assumption, we can choose mutually orthogonal projections qi,...,qk e A such that \\pi - qi\\ < e/12k for i = l,...,k. Set q = Yli=iQiThen \\pk+i - (l - q)pk+i(i - q)\\

=

\\ - qpk+iq + qpk+i + Pk+iq\\ k

<

3||gpfc+1||<3^||gipfc+1|| k

<

^y\piPk+1\\+e/12k))<e/2. »=i

By 2.5.4, there is a projection qk+i € (l — q)A(l — q) with \\qk+i—pk+i\\ < £• Hence \\pi - qi\\ < e for i = 1, ...,k+ 1. Moreover, qk+iqt = 0 for all i = 1,..., k, since 5^+1 € (1 — q)A(l — q).

D

Lemma 2.5.7 If {pi,---,pn} and {qi,...,qn} are two families of mutually orthogonal projections in a C*-algebra A, and if \\pi — qi\\ < 1, then there is a partial isometry v £ A such that vpiV* = q± and v*qiV = pi, i = 1,..., n. If s > 0, 5 = y/2~£/4n (< 1) and if \\pi — qi\\ < S then v can be chosen so that \\pi — pivqi || < e/n. If furthermore Y^i=iPi = 1 then v can be chosen so that || 1 — v\\ < £. Proof.

By 2.5.1, there is a unitary u\ £ A such that || 1 - Ui|| < V26 and UiqiU* — pt,

i = l,...,n. Set v = Yji=\Piui
n

C^PiU-iqi)*C^PiUiqi) i=l

i=l

e

A. Then, by mutual orthogonality, we

n

n

n

= ^2,qi and (^2lPiUiqi)(^2lPiUiqi)* i=l

i=l

i=l

So v is a partial isometry. For each i, (Pivqi)* (pivqi) = (piUiqi)* (piUiqi) = qiU*piUiqi = qi

n

=

^Pii=l

93

Perturbations

and (pivqi)(pivqi)*

= piVqiV*pi

— PiUiqiU*pi

= Pi.

Moreover, \\pi~PiVqi\\

< \\pi ~qi\\ + \\pi(l ~ uihi\\

i = l , . . . , n . If X)"=iPi orthogonal,

=

!>

since

{Pi>—>Pn}

< \\Pi ~Qi\\ + ^

S

a n d

are mutually

{qi,—,1n}

<

£ n

/,

n

||l-t;|| = |lX;(Pi(l-^OII<e»=1

n

L e m m a 2.5.8 Let 1 > e > 0 and 6 = m i n { l / 5 , £ / ( 8 — 5e)}. Suppose {pi,P2} a i d {91,92} are two pairs of mutually orthogonal projections, Pi,P2 € A and 91, 92 £ -B, where B is a C*-subalgebra of A. Suppose that \\Pi ~ 1i\\ < 6 (i = 1,2,), and there exists u G A such that u*u = p\ and uu* = P2, and there again b £ B with \\b\\ < 1 such that \\u — b\\ < 5. Then there is v £ B such that v*v = 91, vv* = 92 and ||u — u\\ < e. Proof.

We have 916*92691

«2,5 ms

q\u*q2uqi qiu'uqx

= qipxqi «<s 91.

Set z = 92691. T h e n 0 < 91 — \z\ < 91 — z*z < 46. Therefore \z\ is invertible and if \z\~x is the inverse taken from 91-B91, | | | ^ | _ 1 | | < il^g- As in the proof of 2.5.3, if v — z | z | _ 1 , then v*v = 91 and vv* = 92- Moreover, u ttS q2u «<5 92691 KS z. Also

hi-\zn\<

4<5 1-46

which implies t h a t AS

|-i|| < 11*1111^-1*1-111 <(1 + J ) _45'

94

Amenable C -algebras and

K-theory

Hence „ \\u-v\\<35

+

,„ r , 4J 85 (l + S)Y-I-s
Q

Theorem 2.5.9 For any e > 0 and integer n > 0 there is 5(e, n) > 0 wii/i i/ie following property. Let A be a C* -algebra, B C A be a C*-subalgebra and {eij : i,j = 1, ...,n} be a set of matrix units in A. If there are a^- G B such that ||e»j - a i j || < J,

i/ien t/iere are matrix units fij G B such that \\fij - e y | | <s. Proof. By 2.5.4, without loss of generality, we may assume that projections. If i ^ j , we have ll^ii^jjll

j ^ W^a^jj S

a

|| ii

— e

a jj\\ ' \\eaajj a

ii|| + || jj

—

enejj\\

e

jj\\-

Therefore, by 2.5.6, without loss of generality, we may assume that {an} is a set of mutually orthogonal projections. Furthermore, we may assume that, by 2.5.8, a^ are partial isometries and a*,-ajj = ajj and a^a*,- = an. Note that J™= projection. So {a^} forms a set of matrix units. • Lemma 2.5.10 For any e > 0 and on integer n, there is 6(e, n) satisfying the following: Let B andC be C* -subalgebras of a unital C* -algebra A such that dimB < n and B has a set of canonical generators (1.9.8) {e\j} with dist(e' J , C) < 5. Then there is a unitary u G C*(B, C, 1), the C*-subalgebra generated by C and B, with \\u — 1|| < e such that u*Bu C C. Proof. We first assume that B = Mn. It follows from Lemma 2.5.9, there is S(e/2,n) > 0 such that if the conditions of this lemma hold, there are matrix units {fij} C C such that We^ - fij\\ <

^(2)e/(16n2).

It follows that there is a unitary v G C*(B, C, 1) such that ||u — 1|| < £/4n and v*euv = fa i = 1, ...,n.

95

Perturbations

Set U = YTj=l ejlvflj-

T1:ien

\\ejj - e^vfijW

<e/2n.

Therefore n j= l

Note that ekiu = ekienvfu

= ekivfu

= ekivfikfki

= ufki-

Therefore u*ekiu — fki, k,l = l,...,n. Therefore u*Bu C C. In general, write B = Mni © • • • © Mns and write pi = 1M„- > * = 1, •••,sNote s < n. By 2.5.4 and 2.5.6, we may assume that there are mutually orthogonal projections qi £ C such that ||p* — qi\\ < 5/4, i = l,...,s. It follows from 2.5.7 that we may further assume that Pi = qt G C. Then we can apply the case that B = Mn to each pair PiBpi and PiCpj (i = 1,2, ...,s). Note that there are only finitely many Mk with k < n. The lemma then follows. • Lemma 2.5.11 Let D be the unit disk, X C D be a compact subset and f € C(X). For any e > 0, there exist 5\ > 0, J 2 > 0 and 5z > 0 satisfying the following: Suppose that A is a unital C*-algebra, a, b £ A with \\a\\, \\b\\ < 1 and b is normal with sp(6) C X. (1) / / \\ab - ba\\ < 6X and \\ab* - b*a\\ < 6x then \\f(b)a-af(b)\\<e. (2) If both a and b are normal, sp(a) C X, and \\a — b\\ < 82, then ||/(a)-/(6)||<e. (3) / / ||a&- 6|| < S3, /(0) = 0, and \\ab* -b*\\<

af(b)-f(b)\\<e.

63, then

96

Amenable C* -algebras and

K-theory

Proof. By the Stone-Weierstrass theorem, there exists a polynomial P(z, z) of two variables such that \\f-P\\D<E/Z. Note that \\bn+1a-abn+1\\

<

| | 6 n + 1 a - 6 n a 6 | | + ||6 T l a6-a6 n + 1 ||

<

\\ba-ab\\ +

\\bna-abn\\.

So, by induction, \\bna-abn\\

< n5x

for all n. We also have ||(fo*)na—o(6*)n|| < nS\. Therefore, if <5i is sufficiently small, \\P(b,b*)a-aP(b,b*)\\

< e/3.

Therefore \\f(b)a-af(b)\\

<

\\(f(b)-P(b,b*))a\\ +

<

+

\\P(b,b*)a-aP(b,b*)\\

\\a(P(b,b*)-f(b))\\

e/3 + e/3 + e/3 = e.

The proofs for (2) and (3) are similar and left to the reader.

•

Lemma 2.5.12 For any e > 0 and any integer n > 0, there exists 5{e,n) > 0 satisfying the following: If A is a C*-algebra and ai,...,an,£ A+ with \\ai\\ < 1 (i = l,...,n) such that ||ajaj|| < # when i =£ j , then there are bi, ...,bn £ A+ such that bibj — 0 when i ^ j and \\a,i — 6j|| < e, i = l, ...,n. Proof. We only prove the case in which n — 2. For n > 2, we leave it to the reader. By (3) in Lemma 2.5.11, we obtain 5 > 0 associated with / — fe/2 a n d £/2 so that when ||a2(l — a i ) ~ a2\\ = lla2fli|| < S, ll/ £ / 2 (« 2 )ci|| = ||A/ 2 (« 2 )(1 - ai) - / e / 2 ( o 2 ) | | < e/2. Set bi = (1 - / e / 2 (a 2 ))ci(l - / E / 2 (a2)) and b2 = fE(a2)a2. and b\b2 = 0. Moreover, \\b2 — a2\\ < e and

Then h,b2 £ B

||fci-ai|| < | | ( l - / e / 2 ( o 2 ) ) a i - a i | | + | | 6 i - ( l - / B / 2 ( a 2 ) ) a i | | < e/2 + e/2 = e.

a

Examples of K -groups

2.6

97

Examples of lf-groups

In this section, we present some calculations in K-theory. Once one establishes the so called "six-term exact sequence" and Bott periodicity (see section 5.7), some of the calculations in this section will become trivial. Since we are not going to use much X-theory machinery until the second part of this volume, we will use some elementary tools to do our calculation. Some of the arguments based on the last section will be used again later. Example 2.6.1 For any n, Ki(M„(A)) = Ki(A), i = 0,1. This follows from the definition of Ki (i — 0,1) immediately. Consequently, Ko(Mn) = Z and tfi(M„) = {0}. Example 2.6.2 Let H be a separable Hilbert space and /C be the C*algebra of all compact operators on H. We leave to the reader (2.8.13) to verify that K0{K) = Z and K^JC) = {0}. Example 2.6.3 A non-trivial C*-algebra may have trivial if-theory. For example, if H is a separable infinite dimensional Hilbert space, then B(H) is a (non-separable) C*-algebra. For any projection p G Mn(B(H)) = B(H^), p® ^B(H) ~ 1_B(H)I because both have infinite rank. It follows that [p] = 0. Hence Ko(B(H)) = {0}. By spectral theory, for every unitary U G B(H), there exists a self-adjoint operator S G B(H) (by Borel functional calculus) with U = exp(iS). Therefore U G U(B(H))o- This implies that Ki{B(H)) = {0}. Example 2.6.4 Let H be an infinite dimensional separable Hilbert space and let C(JC) = B(H)/K. It is well known that every projection p G C(K) can be lifted to a projection P G B(H), i.e., 7r(P) = p, where n : B(H) -> C(K.) is the quotient map. This implies that 7r» is surjective. Since K0{B{H)) = {0}, we have K0(C{K)) = {0}. Given a unitary u G C(/C), one can produce a partial isometry V G B(H) such that 7r(V) = u. We leave to the reader to verify that the Fredholm index ind([w]) = [u*u — uu*] : Ki(C()C)) —• Z gives an isomorphism. So Ki{C(tC)) = Z (see 2.8.14). This is the first example of a non-trivial .Regroup presented here. Definition 2.6.5 Two projections p and q are said to be nomotopic and written p ~h q, if there is a projection P G C([0,1], A) such that -P(O) = p and P ( l ) = q.

98

Amenable C* -algebras and

K-theory

Lemma 2.6.6 Let p and q be two projections in a unital C*-algebra A such that p ~/j q. Then there is a unitary u G A such that u*pu = q. Moreover, if P(t) G C([0,1], A) is a projection such that P(0) = p, P ( l ) = q then there exists a unitary u G C([0, l],v4) such that u(0) = 1A and u*(t)P(0)u(t) = P(t) for all t G [0,1]. Proof. Let P G C([0,1],A) be a projection such that P(0) = p and P ( l ) = q. There are 0 = t\ < i 2 < • • • < tm = 1 such that \\P(U) - P(U-i)\\ < 1 (i = 2,...,m.) By 2.5.1, there is a unitary m G A such that UiP(ti)ui* = P(£; + i), i = 1,2, ...,m — 1. Thus u

*m-i • • • «iP(0)wi • • • u m _ i = q.

Set u = U\ • • • w m _i. To prove the last statement, set Vi(t)

= 1 - P(t) - P(ti+1)

+

2P(t)P(ti+1)

and Wi(t) = v»(£)|uj(t)| -1 for t G [£j,£t+i]- It follows from Lemma 2.5.1 that Wi(t) is a continuous path of unitaries and w*(t)P(t)u>i(t) = P(ti+i) (for U < t < ti+1). Now define u(t) = vi(t) if t G [0,i2] and u(t) = ui(tz) • • • Ui-i(ti)ui(t) for t G [ti,tj + i] (i = 2, ...,m — 1). Then we see that u(t) G C([0,1], A), P(t) = u(t)*P(0)u(t) for t G [0,1] and u(0) = 1. D Definition 2.6.7 Let , V* : ^4 —• P be two homomorphisms of C*algebras. We say <j> and ip are homotopic and write <> / ~^ ip, if there exists a homomorphism H : A —> C([0,1],P) such that TTQ O H = and -K\oH = ip, where nt '• C({0,1],B) —>• B is the point-evaluation: 7r*(/) = /(£) for / G C([0,1],P). We say A and B are homotopic and write A ~^ P , if there are homomorphisms : A —>• P and ip : B ^ A such that 0O?/J ~ ft ids and -0 o 0 ~ h id^. Theorem 2.6.8 Let A and B be two C*-algebras and let <j>, ip : A —» B be homotopic homomorphisms. Then *j = ip*i : Ki{A) —¥ Ki(B) (i = 0,1). Proof.

Suppose that H : A —> C([0,1], B) is a homomorphism such that — and -K\oH = tjj, where nt : C([0,1],P) —> B is a point-evaluation at t G [0,1]. Set 4>t = Tt°H. If A is not unital, let <j>t : A -> P or 0 t : A ->• P , depending on whether P is non-unital or unital as in the proof of 2.4.15. Clearly we see that {4>t '• t € [0,1]} is also a homotopy from <j> to •0. Thus it suffices to assume that both A and P are unital. TTQOH

Examples of

99

K-groups

Now let p £ Mn(A), and let pt = cj>[n\p). Then by 2.6.6, [^n)(p)} = [V»(n)(p)]- Therefore fa = V.oIf u € f/„(A), let u* = 4 n ) ( w ) . Then, [<j>^(u)\ = [V»(n)(«)] in Un(A)/Un(A)0. So <^1 = V*i• Corollary 2.6.9 Ki{A) = Ki(B).

Lei A and B be two homotopic C*-algebras.

Then

Proof. Let <j> : A -> B and ip : B -> A such that <^ o ip ~h ids and ip o ~/j id^. It follows from 2.6.8 that 4>*i ° ip*i = id-Ki(B) a n d V^* ° <^*i id«-i(A). * = 0,1. D Example 2.6.10 (1) For any contractive space X and Y — X \ {£}, where £ € X is a point, Ki(C 0 (y, A)) = {0} since C0(Y,A) ~h {0}. (2) Since C([0,1]) ~ fc C, tf0(C([0,1])) - Z and tfi(C([0,1])) = {0}. (3) If X is a contractive space and A is a C*-algebra, then Ki(C(X,A)) = Ki(A), since C{X,A) - ^ A. Lemma 2.6.11 For any e > 0 and d > 0, f/iere ezisis <5 > 0 satisfying the following: Suppose that A is a unital C* -algebra and u € A is a unitary such that S1 \ sp(u) contains an arc with length d. Suppose that a € A with \\a\\ < 1 such that \\ua — au\\ < 5. Then there is a self-adjoint element h € A such that u = exp(ih), \\ha — ah\\ < e and \\exp(ith)a — aexp(ith)\\ < e for all t G [0,1]. //, furthermore, a = p is a projection, we have \\pup-p+\yy ^—'

, n!

<£•

71 = 1

Proof.

Let nd = { e ^ ' ) : - 1 + a < t < 1 - s} C S 1 .

We choose 0 < s < 1 so that the S1 \ fi^ has arc length d. By replacing u by e™9 • u for some 6 € [—1,1], we may assume that sp(u) C ild- There exists a continuous function g : ild —• [—1,1] such that u = exp(ig(u)). Let

=

100

Amenable

C * -algebras

and

K-theory

h = g(u). Choose iV such that oo

J2 l/n!<e/6. n=N+l

It follows from 2.5.11 (u* = w _1 ) that there exists S > 0 such that \\ua — au\\ < S implies that \\hna - ahn\\ = \\g(u)na - o fl (u) n || < e/6 (and \\phnp - (php)n\\ < e/6 ) for n = 1,..., N (if a = p is a projection). Then llezpCt/Oa-aezptii/OH < | | ( £ ^ ) a n=0

- a ( £ ^ ) | | + 2( f )

'

n=0

'

I)

n=JV+l

n=l

for any t € [0,1]. If a =• p is a projection, we have

n=0

n—0

n=l

So the last assertion also follows.

•

Theorem 2.6.12 Let p £ Mn(C(S1)) be a projection such that p(t) has rank m (< n) and q be a constant projection in Mn(C(S1)) with rank m. Then there exists a unitary u £ M^C^S1)) such that u*pu — q. Proof. Let d = ir/n and 0 < e < 1/2. Let 5 > 0 be as in 2.6.11 associated with d and e. For convenience, we may identify Mn{C{S1)) with {/ £ C([0,1],M„) : /(0) = / ( l ) } . Let e = p(0) = p(l). From linear algebra, we may assume, without loss of generality, that q = e. Suppose that 0 < s < 1 such that ||p(t)-p(l)||<<J/2 for any 0 < s < t < 1. By the last part of Lemma 2.6.6, there exists a unitary W{t) £ C([0,s],M n ) such that W(0) = 1 and W{t)*p{t)W(t) = e

Examples of

101

K-groups

for a l l i e [0,s]. So \\W(s)p(s)-p(s)W{s)\\<6/2. Therefore \\W(s)p(t)~p(t)W(s)\\<6 for all s < t < 1. Since W(s) is an nxn-matrix, S' 1 \sp(W(s)) contains an arc of length greater than ir/n = d. By the choice of 6, it follows from Lemma 2.6.11 that there is a self-adjoint element h G Mn such that W(s) = exp(ih) and \\exp(i(^s)h)p(t)-p(t)exp(i(^s)h)\\

< e < 1/2

for all s < t < 1. Set

u(t)-i W

W{t)

if0
\ e^(i(iff)/i)

if s < t < 1

Since W(0) = 1 and u(l) = 1, we conclude that u(t) G Mn(C(S1)).

Now

W(i)*p(i)W(t) = e for 0 < i < s and ||u(t)p(t) - p ( t ) u ( t ) | | < 1/2 for t e [s, 1]. Thus | | u * p u - e | | < sup {\\u*(t)p(t)u(t)-p(t)\\}+ s
sup { | | p ( t ) - p ( l ) | | } < 1. s
By applying 2.5.1, there is a unitary V G Mn(C(S1)) V*u*puV — e. Corollary 2.6.13

K0(Mn{C(S1)))

such that

= Z.

Corollary 2.6.14 Let p and q be two projections in C([0,1], Mn), where p(t) = p(0) for all t G [0,1] and q(t) has the same rank at each point t G [0,1]. Suppose that there are unitaries v\,V2 G Mn such that vjg(0)vi = U2
102

Amenable C* -algebras and

K-theory

We now show that 2.6.13 holds for all connected finite CW complexes of dimension one. Theorem 2.6.15 Let A = C(X, Mn) and q G Abe a projection, where X is a compact connected finite CW complex of dimension 1. Suppose that p is a constant projection with the same rank as q at each point t £ X. Then there is a unitary u G A such that u*qu = p and upu* — q. Consequently, if the rank of q is k, then qAq = C(X, Mk) and any projection e € A with rank < k is equivalent to a subprojection of q. Proof. For each t G X, there is a unitary w{t) € Mn such that w(t)*q(t)w(t) = p{t) and w(t)p(t)w(t)* = q(t). Since q is continuous, there exists a neighborhood 0{t) such that

||g(t)-g(OH
K-theory

and y*(t)q(t)yj(t)

of inductive limits of C* -algebras

103

— p(0) for all t G Ij. Now define Xi(t) u(t)

~\

V]{t)

if

t£V(U) iitelj.

It is well-defined and for any t £ X, u(t) is a unitary. To see it is continuous and in C(X, Mn), we note that u(t) is continuous on V(U), since V(*i) is an open subset of O(tj). It is equally clear that u(t) is continuous on the interior of Ij. The continuity at the end point t* follows from the definition. Clearly, u*qu — p. The rest of the statements in this lemma follow immediately. • Corollary 2.6.16 Let X be a connected finite CW complex of dimension one. Then K0{C(X, Mn)) = Z. Lemma 2.6.17 Let Abe a unital abelian C* -algebra. Then the embedding i : U(A)/U(A)o -» Un(A)/Un(A)o is infective for every n. Proof. Write A = C(X) for some compact metric space X. Let w £ Un(A). Then the determinant det(w)(:r) is a continuous function on X, i.e., det(w) G U(A). Suppose that u G U(C(X)), w = diag(u,l, ...,1) G U„(A) and {wt} G C([0, l],Un(A)) such that wo = w and w\ — 1M„- Then, since n is fixed, {det(wt)} is a continuous path in U(A). Note that det(wi) = 1 and det(ro) = u G U(C{X)). Therefore u G U(C{X))0. This implies that i is injective. D Corollary 2.6.18 .We have that Z C Ki{Mn(C(S1))). X 1 ( M n ( C ( 5 1 ) ) ) ^ { 0 } ( n = l,2,... ). Proof.

In particular,

By considering the winding number, we see that

uicis^yuicis^o^z and is generated by z : Sl —> S1, the identity map. It follows from 2.6.17 that U(C{S1))/U(C(S1))0 may be viewed as a subgroup of JFsTi(C(51)). •

2.7

if-theory of inductive limits of C*-algebras

In this section, we discuss the "continuity" of the if-theory functors, namely whether Ki(\im.n_>.00(An, hn)) = lim n _ i . 00 (ii'j(A rl ), (/i„)») holds in a suitable sense.

104

Amenable C -algebras and

K-theory

Lemma 2.7.1 Let A = limn^oo(An,hn) be an inductive limit of C*algebras and B = lirnn_>00(Mfc(.An),/i„ ), where k is a positive integer. Denote by n,oo : Mk(An) —> B the homomorphism induced by the inductive limit. Then there is a unique isomorphism ip : B —¥ Mk{A) such that Mk(An)

^ x "'

B oo

4-0 4-i/>

Mk{A) commutes for each n. Proof.

Since the diagram Mk(An)

^

Mk(An+1) " n + l.oo

Mk(A) commutes for each n, it follows from Theorem 1.10.6 that there exists a unique homomorphism tp : B —> Mk(A) such that the diagram

Mk{An)

^?

B Mk(A)

commutes for each n. Since U„/i„ ]00 (A„) is dense in A, so is i Jnhn,oo(Mk(An)) in Mk(A). It follows that ip is surjective. To show tp is injective, it suffices to show that it is injective on ^n,4>n,oo{Mk(An)). Let a G Mk(An) and 4>n,oo{c) G ker^>. From the commutative diagram above, we see that hn,oo(a) = 0. Write a = (aij), a k x k matrix with a^ G An. Then, /i n)0 o(oij) = 0 for all i and j . By 1.10.6, for any e > 0, there is an integer mo > n such that \\hn,m(aij)\\

<e/k2

for all i, j and m > m 0 . Hence ||/in,m(a)|| < Z)i,j=i ll^n.mt/Hf)!! < £• Consequently ||^n,oo(a)|| < ||^m,oo ° K,m(a)\\

< ||/l„, m (a)|| < £.

Letting e —> 0, we obtain ||^ n ,oo(a)|| = 0. So ip is injective.

D

K-theory

of inductive limits of C*-algebras

105

Let p G A be a projection, where A = lim„_ >00 (.A rl ,/i n ). It is easy to get a projection q G hnt00(An) for some large n so that ||p - q\\ is small. It should be noted, however, that a projection e G hnt00(An) may not be the image of another projection in An, in general. A similar problem occurs if one tries to find a unitary v G An to be the preimage of a given unitary in hn,oo(A„). Therefore the following may not be as trivial as one might think at the first glance. Lemma

2.7.2

hmn_KX)(./ln,

Let An

be a sequence of C*-algebras and A

—

tin).

(1) If p £ A is a projection, then there exists an integer n > 0 and a projection q G An such that p is unitarily equivalent to hn:00(q) in A. (2) Ifp,qe An are projections such that hnt00(p) ~ hn:00(q) in A, then there is an integer m > n such that hn,m{p) ~ hn,m( e > 0, there exists an integer n > 0 and a unitary v G An such that \\hnt00(v) — u\\ < e. In particular [«] = [fcn,oo(v)] in U(A)/U(A)0. (5) Ifu,ve U{An) such that [/i„)00(u)] = [hnj00(v)] in U(A)/U(A)0, then there is an integer m > n such that [hn^m(u)] = [hntTn(v)} in U(Am)/U(Am)0. Note in (3) and (4) we assume that A is unital. Proof. (1) Let p be a projection in A. There is a sequence {hnki00(ak)}, where afc G A^, such that it converges to p. Since p is self-adjoint, by replacing a^ by ak 2a,i, we may assume that a^ is self-adjoint for each n. Since p2 = p, /in)i.,oo(afc) —• P- Therefore hnkj00(ak — o?k) -¥ 0. Hence there exists an integer m and a G (Am)sa such that \\p-h

m,oo

(a)\\ < 1/2 and ||/i m)00 (a — a 2 )|| < 1/4. It follows from 1.10.6 that there exists n> m such that \\hmyn(a — a?)\\ < 1/4. Set b = hm^n(a). Then b G (^4 n ) so such that \\b-b2\\
106

Amenable C* -algebras and

Note t h a t hmt00(a)

= hUi00(b).

\\p - hnt00(q)\\

K-theory

We have

<

| | p - / i m ] 0 0 ( a ) | | + ||/i„,°o(k) - ^ , 0 0 ( 9 ) 1 1

<

l b " />m,oo(a)|| + \\b - q\\ < 1/2 + 1/2 = 1,

so by L e m m a 2.5.1 p is unitarily equivalent to hn:00(q) as desired. (2) Suppose t h a t p, q G An are two projections such t h a t /i n ,oo(p) ~ hn,oo(q) in A. T h e n there is a partial isometry u G A such t h a t u*u = hn,oo{p) and uu* = hn<00(q). There are vfc G Ank such t h a t hnkiOQ(vk) ->• u. Since u = i i u ' u = hnt00(q)u = uhnt00(p), we may assume t h a t Vk = hn,nk( n and v € Ak such that v = hnii(q)vhnti(p), \\hi,oo(hn,i(p)

-v*v)\\

< 1 and \\hii00(hn>i(q)

- vv*)\\ < 1.

It follows t h a t there exists an integer m > I such t h a t \\hi,m(hn,i(p)

-v*v)\\

If we set x = hiiTn(v),

< 1 and \\hitTn(hnti(q)

- vv*)\\ < 1.

then x £ Am and we have

\\K,m(p)

- x*x\\ < 1 and ||/i„, m (g) - xx*\\ < 1

and x = hn:m(q)xh„!m(p)Hence, by L e m m a 2.5.3, hntTn(p) ~ hn^m{q) in Am. This proves (2). (3) This follows from (1) immediately by considering p = 1. (4) Since A is assumed to be unital in this case (and in (5)), we m a y assume t h a t An is unital and hn is unital for every n. Let u G U(A). It follows t h a t there exists a^ G Ank such t h a t hnk,oo{o-k) —> u. We m a y assume t h a t ||afc|| < 1. Moreover, hnki00(ala.k ~ 1) -> 0. This implies t h a t there exists an integer m and a n element a £ Am with ||a|| < 1 such t h a t \\hm,oo(a) - u\\ < 1/4 and \\hmt00(a*a

- 1)|| < 1/8.

It follows t h a t there exists n > m such t h a t ||/i m>7l (a*a — 1)|| < 1/8. Set x = hm^n{a). T h e n ||i*a;-l|| < l / 8 . Since ||a;|| < 1, by spectral theory, 0 < 1 - (x*x)1/2

< 1 - x*x < 1/8.

K-theory

107

of inductive limits of C* -algebras

Hence

||i - M- l || < HlxrMKi/s) < j-Ti/iti/s) = V7. Let v = i|a;| _ 1 . Then v G U(An) and ||/ln,oo(v) - W||

<

||ftn,oo(w) ~ hn>00(x)\\

<

\Wx\~1 - l|| + l / 4 = 1 / 7 + 1/4 < 1/2.

+ \\h„l00{x)

Therefore [/»„,«,(«)] = [«] in C/(A)/C/(A)0. (5) Suppose that [ft„i00(«)] = [hn,oo(v)] in U(A)/U(A)ow o : w i , - . w s € U{A) such that •u>o = hn,oo(u),ws

= hnt00(v)

- u\\

There are

and jlru, — iu£_ij| < 1/8,

i = 1,..., s. As in the proof of (4), there is m > n and Z{ G U(Am) such that \\hm,oo(zi) - Will < 1/2

i = 1,2,..., s - 1.

Let zo = /in,m(w) and z s = /i„, m (v). Then ||zi-zi_i|| < l / 2 + l / 8 < l , Therefore [z^ = [zi-i] in U(Am)/U(Am)0, [/in,m(v)] in [7(A m )/t/'(A m )o as desired.

t = l,...,«. i = l,...,s. Thus [/i„,m(u)] = D

Theorem 2.7.3 Let A = lim n _ >00 (A n ,/i n ) and d = lim n ^ 0 0 (i('i(A n ),(/i„)»i), i = 0,1. Denote by G* i/ie homomorphism induced by the inductive limit. Then there is a unique isomorphism f/'i : Gi —> /Cj(A) such that the diagram Ki(An)

-^

d

commutes for each n. Proof.

For each n the diagram

KM

(

H*;

i^i(An+1) Ki(A)

108

Amenable C* -algebras and

K-theory

commutes, so there is a unique homomorphism if>i : Gi —> Ki{A) such that for each n the diagram

Ki(An)

%

d Ki(A)

commutes. It remains to show that tpi is an isomorphism (i = 0,1). For each integer k, let Bk = \im.n^too(Mk(An), /i„ ') and let Hn : Mk(An) —• Bk the homomorphism induced by the inductive limit. By Lemma 2.7.1, there is a unique isomorphism ^t, : Bk —>• MfcM) such that, (k) for each n, h„t'oo =^k° Hn. We will only show that tpi : G\ —• K\ (^4) is an isomorphism. The proof for ipo is similar. To show that ipi is surjective, let u G Uk(A). Hence u = ^>k(v) for some v £ U(Bk)- By (4) of 2.7.2, there is a w S Uk(An) for some n such that [v] = [Hn[w)\ in U(Bk)/U(Bk)o. Hence [u] = [ f t i l W ] in E/*^). This implies that i^i(A) C i m ^ i ) . So ^ l is surjective. To show that tpi is injective, let x G ker^i- Suppose that zn G Ki(An) such that (/>^(z„) = a;. We may assume that zn is represented by w G I7fc(An). So, by the diagram (e7.14), [h(n%(u)] = 0 in Ki(A). We may assume that di&g(hi,t'00(u), l m ) G fm+fe(-4)o- It follows from (5) of 2.7.2 that there is I > n such that diag(foni;(u), l m ) G ?/m+fc(-<4z)o- Therefore 4>] ° (^n,/)*i(-Zn) = 0. So x = 0 in G?i. D Example 2.7.4 Let A be the UHF-algebra of 2°°-type. Recall that A = \imn^00{M2",hn), where hn(x) = &\&g(x,x) for x G Af2". By 2.7.3, K0(A) = lim n _ +0 o(Z,(/i n ).o) and K^A) = {0}, since K1(M2^) = 0. Let <7i G Ko(M2) be the element represented by the minimal projection in M2, ..., Qn G i^o(M 2 n) be the element represented by the minimal projection in M2". Note that gt are generators for K0(M2™)- Define a homomorphism tpn : K0(M2n) -> Q by ipn{9n) = 1/2". We obtain a homomorphism V> : i^o(^4) —>• Q- Since each (/i„)* and i/»„ are injective, ip is injective. One then easily checks that KQ{A) = Z[l/2].

2.8

Exercises 2.8.1 Prove 2.1.2.

Exercises

109

2.8.2 Let A be a unital C*-algebra and a £ A. Then ||a|| < 1 if and only if 1 a*

a 1

is positive.

2.8.3 Show that in the Stinespring representation theorem ||T^||2 = ||^||. 2.8.4 Let : A —» B be a completely positive linear map, where A and B are C*-algebras. Show that [aY4>{a) <

UU(a*a)

for all a € A. 2.8.5 Let (j> : A —>• B and ip '• B —> C be completely positive linear maps. Show that tp o (j> is a completely positive linear map. 2.8.6 Complete the proof of Lemma 2.2.3. 2.8.7 In 2.3.3, prove that s^ = (f> and scj>3 = s. 2.8.8 Let A be C*-algebra. Suppose that, for any e > 0 and finite subset T C A, there are an amenable C*-subalgebra B C A, contractive completely positive linear maps : A —>• B and L : B —> A such that id ~£ L o $ on J?7. Prove that A is amenable. 2.8.9 Let {An} be a sequence of unital C*-algebras. Put A = Jl^Li An and i? = ©$£!=! Ai- Let 7r : ^4 —» A / B be the quotient map. Prove that (1) if p G A / 5 is a projection, then there is a projection q € A such that ir(q) = p. (2) if u £ U(A/B) then there is a unitary » £ i such that ir(v) = u. 2.8.10 A unital C*-algebra is said to be finite if x*x = 1 implies that xx* = 1. (1) Show that a unital C*-subalgebra B of a finite C*-algebra A is finite if 1B = 1A(2) Let {An} be a sequence of finite unital C*-algebras. Show that both n ~ = i An and n ~ = i An/ ©~ =1 An are finite. 2.8.11 Let A be a unital C*-algebra and s : GL(A) -> [/(A) be defined by x t-¥ a;|a;|_1. Verify s is a contraction.

Amenable C* -algebras and K-theory

110

2.8.12

Complete the proof of 2.5.5.

2.8.13 Show that K0(K.) = Z and KX{K) = {0}. 2.8.14 Let i b e a C*-algebra. Define a map j n : Mn(A) ->• Mn+i(A) by jn{o) = diag(a,0). Denote by A ® K the inductive limit. Show that Ki{A)=Ki{A®K) (t = 0,l). 2.8.15 Let Ax = C © C, A2 = M2 © C, A3 = M 3 © M 2 , A 4 = M 5 © M 3 , and, if An = Mnk © M„ fc _ i; then An+l = Mnk+n.k_1 © Mnic, n = 1,2,.... Define (f>n : An -> An+i

by matrix f

J , i.e., 4>n{x,y) =

(diagfo y), a;). Put 4 = l i m ^ o o ^ , „)• Show that tf„(4) = Z+Z(±±^2). 2.8.16 Let Ai and A 2 be two C*-algebras. Show that Ki{A\ © ^42) = Ki(A1)®Ki(A2),i

= 0,l-

2.8.17 Let 0 -> I-UA-^A/I

-> 0

be a short exact sequence of C*-algebras. Then the sequence KO(I)HK0(A)^K0(A/I)

is exact in the middle. 2.8.18 Let H be an infinite dimensional Hilbert space and n : B(H) —¥ B(H)/K. be the quotient map. Let p be a projection in B(H)/K. Show that there is a projection q £ B{H) such that 7r(g) = p. 2.8.19 Let A be a C*-algebra and I be an ideal. Let u £ U(A/I). Show that there is a partial isometry v £ M2(A) such that n(v) = u, where 7T : M 2 (A) -> M2(A/I) is the quotient map. 2.8.20 Let A be a C*-algebra and I be an ideal. Suppose that I admits an approximate identity consisting of projections in / . Let u £ U(A/I). Show that there is a partial isometry v £ A such that ir(v) = u. 2.8.21 Let A be a C*-algebra and I be an ideal. Let u £ Then there is a unitary v £ U(A) such that n(v) = u.

U(A/I)Q.

2.8.22 Let H be an infinite dimensional Hilbert space. Use Fredholm theory to compute that Ki(B(H)/)C) = Z. 2.8.23 Complete the proof of 2.5.11 2.8.24 Complete the proof of 2.5.12.

Addenda

111

2.8.25 Let A and B be two C*-algebras and n : A —> B be a sequence of contractive completely positive linear maps such that n(a) converges in norm for every a & A. Show that (f>(a) = limn^>„(a) (a G A) is also a contractive completely positive linear map. 2.8.26 Prove this variation of 1.10.6: if each VVi is a contractive completely positive linear map, then ijj is also a contractive completely positive linear map. 2.8.27 Let X be a connected compact subset of S1, a £ and B = Her(a). Show that tsr(B) = 1. 2.9

C(X,Mn)+

Addenda

Corollary 2.9.1

Inductive limits of amenable C*-algebras are amenable.

Theorem 2.9.2

Every C*-algebra of type I is amenable.

Theorem 2.9.3 Let G be a locally compact group. C*(G) is amenable if and only if G is amenable, i.e., C*(G) = C*(G). Definition 2.9.4 Let A and B be two C*-algebras. Let A ®aig B be the algebraic tensor product (of the two algebras over C). There are C*- norms on A®aig B such that ||a<8>6|| < ||a\\\\b\\. The completion of A®aigB under such a C*-algebra norm gives a C*-algebra. Theorem 2.9.5 A C*-algebra A is amenable if and only if, for each C*-algebra B, there is only one C*-norm on A®aig B.

— 200BC — *] -ty (2S6BC - 195BC)

Chapter 3

AF-algebras and Ranks of C*-algebras

3.1

C*-algebras of stable rank one and their if-theory

Definition 3.1.1 A unital C*-algebra A is said to have stable rank one, and written tsr(v4) = 1, if GL{A) is dense in A, i.e., the set of invertible elements is dense in A. A non-unital C*-algebra A is said to have stable rank one if tsr(j4) = 1. Theorem 3.1.2 Quotients and ideals of C*-algebras of stable rank one have stable rank one. Proof. It is clear that we may assume that A is unital. Suppose that / C A is an ideal and x € A/I. Then there is y e A such that n(y) = x, where n : A —» A/I is the quotient map. Moreover, there is an invertible element a € A such that \\a — y\\ < e. Thus ||7r(a) — a;|| < e. But 7r(a) is invertible. This shows that every quotient of a C*-algebra with tsr(A) = 1 has stable rank one. Now let x e I (= C • 1 + 1). By replacing x by e + x, we may assume that x $ I. By multiplying by a scalar multiple of the identity, we may further assume that x = 1 + b for some b £ I. Since tsr(A) = 1, there exists an invertible element y € A such that ||x — y\\ < e/4. Let IT : A —> A/1 be the quotient map. Then ||1 — n(y)\\ < e/A. Hence with e < 1/2,

HI - TKJT1)!!

< E ii1 - *(y)\\n < r ^ T i < 2£ / 7 113

114

AF'-algebras

and Ranks

of C*

-algebras

Since ||1 - T T ^ " 1 ) ! ! = limA ||(1 - y " 1 ) ^ - e A )|| (by 1.5.12), we choose A so that the following hold: ||(1 - y - x ) ( l - e A )|| < e/3, ||6(1 - e A )|| < e/3 and ||(1 - eA)6|| < e/3. Set z — (1 — eA) + ye\. So z G C • 1 + / and \\z-x\\

<

\\z-y\\

\\y-x\\<\\(l-ex)-y{l-ex)\\+e/A

<

||(1 - ex)(x -y)\\ + ||(1 - eA)6|| + e/A < e/A + e/3 + e/A < e.

+

It remains to show that z is invertible in C • 1 + J. We have \\y-\y-z)\\

=

Hy-^l/Cl - eA) - (1 - eA))|j = H y " 1 ^ - 1)(1 - e A )||

=

||(l-y-1)(l-eA)||<£/3
Thus J2'^Lo[y~1(y ~~ z)\n converges (in norm). Hence oo

W

(J^ly-Hy-^Dy-1 n=0 n=0

is an element in A. However, we verify that oo

wz

= [J20.-y-1z)nEy-1z-V oo

+ i]

oo

= -J^l-y-^r + J^l-jT1*)^!. n=l

n=0

So z is invertible in A. Since C • 1 + / is a C*-subalgebra of A, z is invertible

in C • 1 + I. Proposition 3.1.3

• Both C([0,1]) and C(S1) have stable rank one.

Proof. Let / G C([0,1]). If / = 0, then / « e e • 1. If necessary, we may replace / with g, where ||<; — / | | < e. Hence we may assume that |/(0)| > e and |/(1)| > e. Note that {t G [0,1] : |/(i)| > e} = F is a closed subset. Therefore [0,1] \F = (0,1) \F is an open subset. So (0,1) \F = Un(tn, sn). Note that | / ( i n ) | = l/( s n)| = £• We define a continuous function

a(t) = l yK

'

m

\ a(t)

iHeF

iitn
C* -algebras of stable rank one and their K-theory

115

where a(t) is the continuous path which is the part of the arc of the circle with center at the origin and radius e starting at /(£„) and ending at f(sn). Hence \g(t)\ >e,g£ GL(C([0,1]) and \\9-f\\<e. This shows that tsr(C([0,1])) = 1. To show that tsr(C(5' 1 )) = 1, we set 7 = { / € C ( [ 0 , l ] ) : / ( 0 ) = / ( l ) = 0}. Clearly I is an ideal of C([0,1]). By 3.1.2, tsr(7) = 1. Therefore I has stable rank one. However, C(S1) = I. • Remark 3.1.4 From 3.1.3 and 3.1.2, we see that G{F) has stable rank one for any compact subset of the real line. Also, the same method used in 3.1.3 shows that tsr{C{ft)) = 1 if ft is the figure 8. Later, (Theorem 3.2.12), we will prove that tsi(C(X)) — 1 for any compact metric space X with covering dimension one. This can be also proved directly by using the method in 3.1.3, by first showing that tsr(C(F)) = 1 for any connected finite CW complex with dimension one. Lemma 3.1.5 Suppose that A is a unital C*-algebra, p £ A is a projection, and x £ A such that the element b = (1 — p)x(l — p) is invertible in (1— p)A(l — p). Then x is invertible in A if and only if a — cb~1d is invertible in pAp, where a = pxp, c = px(l — p) and d = (1 — p)xp. Furthermore, if x is self-adjoint, then d = c*, and a, b as well as a — cb~lc* are self-adjoint. Proof.

With matrix notation we have

(a X=

{d

c\

b)

cb~l\

fa-cb^d

{o l-p){

0

(p =

OW

p

b){b^d

0

\

1-p)-

Set

* 1 = ( o 7-p)

a n d z 2 =

(-^

1-p)'

Since pcb'1 = pcb-\l

-p) = cb~l and (1 - p)b~ld = b~*dp = b~ld,

^

AF-algebras

116

and Ranks of C* -algebras

(p we verify directly that Z\ is the inverse of I

cb"1 \

and z-i is the inverse

of

, , , , ) . Since the outer factors in the product in ( e l . l ) are \b a 1 — p J always invertible in A, the invertibility of x is equivalent to the invertibility of the factor in the middle, which is diagonal. The conclusion then follows easily. Q Definition 3.1.6 A unital C*-algebra is said to have real rank zero , written RR(A) = 0, if the set of invertible self-adjoint elements is dense in Asa. A non-unital C*-algebra is said to have real rank zero, if RR(A) = 0. Proposition 3.1.7

If RR(A) = 0, then RR(A/I)

= 0.

The proof is the same as in the first part of the proof of 3.1.2. We leave it to the reader. Theorem 3.1.8 Let A be a C* -algebra and p G A be a projection. If tsr(A) = 1 (or RR(A) = 0 ) , then tsx(pAp) = 1 (or RR(pAp) = 0). Conversely, if tsv(pAp) = 1 and tsr((l — p)A(l — p)) = 1 (or RR(pAp) = 0 and RR((1 - p)A(l - p)) = 0), then tsr(A) = 1 (or RR(A) = 0). Proof. Assume first that A is unital. If tsr(A) = 1, (RR(A) = 0) and x e pAp (x G (pAp)sa), we can by assumption, for each 1 > e > 0 find an invertible element y £ A, (y £ Asa) such that ||a; + 1 — p — y\\ < e. With b = (1 — p)y(l —p), this means that ||(1 — p) — b\\ < e. It follows that b G GL((l - p)A(l - p)). Furthermore, if y G Asa, b G ((1 - p)A(l - p))sa. Also, ||6 - 1 || < j - L . It follows from Lemma 3.1.5 that z = pyp-

py(l - p)b~l(l - p)yp

is in GL(pAp). Moreover, if y € Asa, then z G (pAp)sa. We then have

\\py(l-p)b-lyp\\<^—£\\py(\-p)\\2<^—e. Thus e2 \\x -

Z\\

< \\x - pyp\\ + YZT£ -

e

which shows that tsx(pAp) = 1 (RR(pAp) = 0).

•

C*-algebras of stable Tank one and their K-theory

117

Conversely, if tsi(pAp) = tsi((l-p)A{l-p)) = 1 {RR{pAp) = RR((1p)A(l - p)) = 0) (and A is still unital) we take x G A (x G Asa) and write

as in the proof of Lemma 3.1.5. Given e > 0 we can find b' G GL((l-p)A(l-p)) (with (6')* = b') such that ||6-6'|| < e. Then, there is w G pAp (w G (pAp) sa ) which is invertible such that \\a—c(b')~1d—w\\ < e (\\a - c(6') _1 c* - w\\ < £ )- S e t a' = w + c{b')~ld (a' = w - c(6') _1 c* which is in (pAp)sa). Then llo-a'H = \\(a-c(V)-1d)-w\\

<e (\\a - a'\\ = | | ( a - c(6') _1 c*) - w|| < e).

By Lemma 3.1.5

is invertible in A. But ||x' - a;|| < £. Hence tsr(A) = 1 (i?i?(A) = 0). In the case where A is non-unital, we note that (1 - p)A(l - p) = (1 - p)A{l -p)+ C(l - p). We leave it to the reader to complete the proof for non-unital case. • T h e o r e m 3.1.9 Let A be a C*-algebra. (1) //tsr(A) = 1, then tsi(Mn(A)) = 1 for all n. (2) IfRR{A) = 0, i/ien RR(Mn(A)) = 0 /or aH n. Proof. We prove (1) only. The proof of (2) is exactly the same (by switching to self-adjoint elements) and is left to the reader. We first assume that A is unital. Assume that tsv(Mk(A)) = 1 for all k < n. Let p = 1 (g) en G Mn+i(A), where e„ denotes the projection in M n + i with the form diag(l,..., 1,0). Then pMn+i(A)p = Mn(A) and (1 - p)Mn+i(A)(l — p) = A. By assumption, both of these C*-algebras have stable rank one, whence tsr(Mn+i(A)) = 1 by Theorem 3.1.8. If A is non-unital, Mn(A) is an ideal of Mn(A). We have shown that tsr(M„(i)) = 1. By Theorem 3.1.2, tsi(Mn(A)) = 1. D

118

AF-algebras

and Ranks of C*-algebras

Lemma 3.1.10 If A is a unital C*-algebra with tsr(^4) = 1, then the map from U(A)/U(A)o —> Un(A)/Un(A)o is surjective for all n. Consequently, the map from Un(A)/Un(A)o —* K\{A) is surjective for every n. Proof. As in the proof of 2.4.10 (see also 2.8.11), it suffices to show that GL(A)/GL(A)o -> GLn(A)/GLn(A)o is surjective. Let

be in GLn(A), where a £ A, b' £ Mn-i(A), c is a row of n — 1 elements in A and d is a column of n — 1 elements in A. Choose 0 < S < l / | | x _ 1 | | . Since tsr(A) = 1, by 3.1.9, tsr(M n _i(.A)) = 1. Hence there is an invertible element b £ GLn-i(A) such that ||6 - 6'j| < 6/2. Set

-COThus ||x — y\\ < S. Hence y £ GLn(A) and y and x are in the same component of GLn(A). We will show that there is a continuous path in GLn{A) which connects y to an element in GLn(A) of the form diag(z, 1). By (e 1.1), we may write y

~\0

l n _J {

0

b) [b-'d 1„_J-

Let

As in the proof of 3.1.5 zi(t), Z2(t) £ GLn(A) for each t £ [0,1] and they are continuous paths of GLn(A). Since -Zi(O) = ^ ( 0 ) = l n , y is connected to

(a-cb~ld Z3=

\

o

0\

b)-

If n = 2, it follows from 2.4.11 that z3 ~h z±, where z4 = diag((a — cb-ld)b, l „ _ i ) . This implies that the map from GL(A)/GL(A)0 to GL2{A)/GL2(A)o is surjective. We then use induction on n. Suppose that the map from GL(A)/GL(A)0 to GL„_i(A)/GL n _i(^4)o is surjective. Then

C* -algebras of stable rank one and their K-theory

zz ~/» z'3, where z'3 £

119

GLn(A),

4 = (a

o

y,\

and b

"=

dia

g( c > ln-2)-

Then by applying 2.4.11 again, we have z'3 ~& z'4, where z'4 = diag((a — cb~ld)c, l n _ i ) . Thus the map from GL{A)/GL(A)0 to GLn(A)/GLn(A)0 is surjective. Theorem 3.1.11 Let A be a unital abelian C*-algebra with tsr(A) = 1. Then the map U(A)/U(A)o -¥ Un(A)/Un(A)o is an isomorphism. ThereforeU(A)/U(A)0=K1(A). Proof. By 2.6.17, U(A)/U(A)0 -> Un(A)/Un{A)0 is injective. It follows from 3.1.10 that this map is also surjective, whence it is an isomorphism. It follows that U(A)/U(A)0=K1 (A). • Corollary 3.1.12 Proof.

K^M^C^S1)))

= Z.

This follows directly from 2.6.18 and Theorem 3.1.11.

•

Definition 3.1.13 A C*-algebra A is said to have cancellation of projections if, for any projections p,q,e,f £ A with pe = 0, qf = 0, e ~ / , p + e ~
Every unital C* -algebra of stable rank one has cancel-

Proof. Suppose that p, q G Mn{A) are two projections such that p ~ q. Let v G Mn(A) such that v*v = p and vv* = q. It follows from 3.1.9 that tsi(Mn(A)) = 1. So there is x £ GLn(A) such that | | x - v | | < 1/8. Without loss of generality, we may assume that ||x|| < 1. Let x = u(x*x)1/2 be the polar decomposition (u = x(x*x)~1^2). Then u £ Un(A). We have \\x*x - p\\ < \\x*x - x*v\\ + \\x*v - v*v\\ < 1/4. Therefore upu* « i / 4 u(x*x)u* = xx* « i / 4 q.

120

AF-algebras

and Ranks of C* -algebras

Therefore ||upu* — q\\ < 1/2 < 1. It follows from 2.5.1 that there is a unitary w G Mn(A) such that w*{upu*)w = q. •

3.2

C*-algebras of lower rank

Proposition 3.2.1 (1) Inductive limits of C* -algebras with stable rank one have stable rank one. (2) Inductive limits of C* -algebras with real rank zero have real rank zero. Proof. (1) We may assume that A is the closure of UnAn, where An has stable rank one (by 3.1.2). Furthermore, we may assume that A and An are unital and 1A„ = 1A for all n. If x G A, then there is n and y G An such that x ~e/2 V- Since tsr(^4n) = 1, there is z G GL(An) such that y « £ /2 z. Hence x « £ z. Therefore tsr(A) = 1. We leave the proof of part (2) as an exercise. D Corollary 3.2.2 Let A be an AF-algebra. Thentsi(A)

= 1 andRR(A)

= 0.

Proof. It is evident that tsr(C) = 1 and RR(C) = 0. Then it follows from 3.1.9 that tsr(M„) = 1 and RR(Mn) = 0. Therefore every finite dimensional C*-algebra has stable rank one and real rank zero. We then apply 3.2.1.D Corollary 3.2.3 Every AT-algebra has stable rank one. Moreover, if A is a unital AT-algebra, then U(A)/U(A)o = K\{A). Proof. Write A = \im.n^t(x>(An,(/)n), where each An is a circle algebra. Let Bn = 4>n,oo(An) be the image of An in the inductive limit. It follows from 3.1.3 that tsr(^4n) = 1. Thus 3.2.1 applies. To see that the second part holds, we let u G U(A). It suffices to show that the map from U(A)/U(A)o to U2(A)/U2(A)o is an isomorphism. It follows from 3.1.10 that it is always surjective. So it remains to show that the map is injective. There is x G Bn such that ||a; — u|| < 1/4. It follows that ||v — u\\ < 1, where v = a;|a;|_1. Note that v is a unitary and v and u are in the same path connected component of U(A). So, without loss of generality, we may assume that u G Bn. Suppose that diag(w, 1) G U2(A)0- There are WQ,WI, ...,Wk G Uz(A) such that WQ = diag(u, 1), wk = 1 2 and \\wi - tUj_i || < 1/2,

121

C -algebras of lower rank

i = 1,2, ...,k. From what we have shown above, there is an integer N > 0 such that there are Vi G [^(-BAT) with \\vi -Wi\\ < 1/4 i = l,...,k. By choosing a larger n, if necessary, we may assume, without loss of generality, that Wi G U2{Bn) and \\wi — Wi-i\\ < 1. Therefore, WQ G U2{Bn)oSince B n is a finite direct sum of circle algebras, it follows from 2.6.17 that u G U(Bn)0. Hence u G U(A)0- This shows that the map from U(A)/U(A)0 to U2(A)/U2(A)o is injective. rj Proposition 3.2.4

Every von Neumann algebra has real rank zero.

Proof. Let A be a von Neumann algebra. If x G Asa and e > 0 is given, let p denote the spectral projection of x corresponding to the interval [—e/2, e/2}. Then y = (1 - p)x + ep is invertible in Asa and ||a; — y\\ < e by spectral theory. • T h e o r e m 3.2.5 Let A be a C*-algebra. The following are equivalent. (1) RR(A) = 0. (2) The set of elements in Asa with finite spectrum is dense in Asa, i.e., if x G Asa and e > 0, there are mutually orthogonal projections p\,...,pn and real numbers Ai,..., An such that

^2^iPi\ <e. i=l

(3) For every hereditary C*-subalgebra B c A, b\,..., bn G B and e > 0, there is a projection p G B such that \\biP-hW <e,

i=

l,...,n.

Proof. To show that (1) => (2), we may assume that A is unital. Let x G Asa with ||a;|| < 1 and let e > 0. Choose - 1 =ti

< t2 < •••
= 1

such that \ti+i - U\ < e/2. Let e\ = e/4. Since RR(A) = 0, there is an element a\ G Asa such that a\ is invertible and \\{x-h

•l)-a1|| <£l.

122

AF-algebras and Ranks of C*-algebras

Set xi = ai +1\ • 1. T h e n a\ = x\ — t\ • 1 is invertible and \\x — Xx\\ < £\. Choose 0 < £2 < e / 8 so t h a t [ii - e2, h + £2] n s p ( z i ) = 0. By (1) again, as above, we obtain X2 G Asa such t h a t x2 —12 • 1 is invertible and \\x2 -Xi\\

< e2.

T h e n ^1,^2 G" sp(x2). By repeated use of this argument, one produces elements xn G Asa such t h a t ti,t2, ...,£« ^ sp(a; n ) and n

\\x — xn\\ < ^ £ i <

e/2.

i=l

There is 0 < d < e / 4 such t h a t (ii - d,tt + d) n s p ( i n ) = 0. Set F» — [U-i + d/2,ti — d/2}. T h e n \Fi is a continuous function on sp(a; n ). T h u s Pi = XFi(xn) is a projection in A. Set 6„ = YH=\ UPi- T h e n b n G Asa, bn has finite spectrum and \\x - bn\\ < \\x -xn\\

+ ||x„ - 6„|| < e / 2 + e / 2 = e.

(2) => (3): Let &;, 62,..., bn £ B and c = maxj{||6i||} + 1. Let {e^} be a n approximate identity for B. Suppose t h a t \\he\ -bi\\

< £/3c>

i

= 1, •••,"•

T h u s , if we find a projection p £ B such t h a t \\pe\ — e\\\ < e/3c,

\\pbi-bi\\

<

\\pbi - pexbi\\ + \\pexbi -

+

\\e\bi-bi\\

<e/3

+ e/3 +

then

e\bi\\ e/3c<e.

So it suffices to consider the case t h a t n = 1 and 0 < 6j < 1. Set b = b\. C h o o s e ^ > 0 as in 2.5.11 such t h a t ||6 — z\\ < 52 implies \\fe(b) — fE(z)\\ < e for all 0 < z < 1. By (2), we may choose z = V • Xjej, where e i , . . . , e^ £ A are mutually orthogonal projections and Xj G R, j = l,...,fc. Set q = Eo-:A,i>e}ei-Then ||gz - z\\ < £ and fe(z)q

= q.

C* -algebras of lower rank

123

Thus \\fE(b)qfe(b)-q\\

< \\fe(b)q(f£(b)~

fs(z))\\

+ \\fE(b)q- fe(z)q\\

< S + S = 2e.

With 2e < 1/2, by 2.5.4, there exists a projection p G C, the C*-subalgebra generated by fe{b)qf£(b), such that I|P-«II <4£. Since C C B, p G B. Therefore, \\pb - &II < II (P " #11 + Ub - b\\ < 4s + e = 5e. (3) => (1): Let x € A s a and e > 0. Write a; = :r + — a;_, where x+,X- G A + and a; + x_ = 0. Set B — x+Ax+ = Her(a; + ). By (3), there is a projection p G B such that ||(1 — p)x+|| < e/4. Since a;_ is orthogonal to x+, px- = 0. Set y = pxp+-p+(l-p)x(l-p)

- -(1 - p ) .

Then, we estimate H i - j / | | < 11^(1 - p ) + (1 -p)xp|| + | | | ( p - (1 - p ) | | < £/2 + e/2 = e. Since p commutes with y, pyp = px+p + §p > fP and (l-p)y(l-p) = ( l - p ) a ; + ( l - p ) - ( l - p ) x _ ( l - p ) - | ( l - p ) < - | ( l - p ) , we see that y is invertible in Asa. So i?i?(v4) = 0 .

D

Corollary 3.2.6 If A is a C*-algebra with RR(A) = 0, then RR(B) = 0 /or every hereditary C* -subalgebra B of A. Proof. It follows from 3.1.8 that every unital hereditary C*-subalgebra of A has real rank zero. Let B be a (non-unital) hereditary C*-subalgebra and x G Bsa. We write x = X+b, where A is real and b G Bsa. By 3.2.5, there is a projection p G Bsa such that ||(1 — p)6|| < e/2. Since pAp =• pBp has real rank zero, there is a self-adjoint and invertible element z G (pBp)sa such that ||(Ap + p6p) - z|| < e/2. Set y = e(l - p) + z if A = 0 and y = A(l - p) + z if A ^ 0. Then y G GL(A) n A s a and I k - I / l l < | | ( l - p ) a ; - A ( l - p ) + (Ap + p 6 p - z ) | | < e. Therefore RR(B) = 0.

D

124

AF-algebras and Ranks of C -algebras

Proposition 3.2.7 If A is a a-unital C* -algebra with RR(A) = 0, then it has an approximate identity consisting of projections. Proof. Let a be a strictly positive element in A with 0 < a < 1. By 3.2.5, there is a projection p\ £ A such that ||(1 — pi)a|| < 1/2. Since RR((1 - p\)A(l - pi)) = 0, there is a projection p2 G (1 - pi)A(l - p{) such that ||(1 - P 2 ) ( l - p i ) a 2 ( l - P l ) ( l - p 2 ) | | < 1/42. This implies that 11(1 - (Pi + P 2 ) H 2 = 11(1 - P 2 ) ( l -Pi)a\l

- P l ) ( l - P 2 ) | | < 1/42.

Continuing by induction we obtain a sequence {pn} of mutually orthogonal projections in A such that

||(1 - f > ) a | | < 1/2". fe=i Then

Set en = E L i P f c there is n such that

>

b

y applying (3) of 2.5.11, for any / e C o ((0,1]), ||en/(a)-/(a)||<£.

Since there exists a sequence {fk} C Co((0,1]) such that {/fc(a)} forms an approximate identity for A, by the proof of 3.2.5 (the implication (2) => (3)), {e n } forms an approximate identity for A. • The following are the general definitions of stable rank and real rank for C*-algebras. Definition 3.2.8 For a unital C*-algebra A, we define the stable rank (and real rank) of A to be the smallest integer, tsr(A) (RR(A)), such that for each n-tuple {x\, X2, ••-, xn) of elements in A (in Asa), with n < tsr(A) (n < RR(A) + 1), and every e > 0, there is an n-tuple (2/1,2/2, •••)2/n) in ^4 (in vl sa ) such that ^ky%yk is invertible and n

\\'S£J(xk-yk)*{xk-yk)\\<£fc=l

(e2.2)

If A is non-unital, we define its stable rank and real rank to be tsr(vl) and RR(A), respectively.

C* -algebras of lower rank

125

Definition 3.2.9 Recall that the covering dimension of a compact metric space is the smallest integer n such that every continuous function / from X into R™+1 can be approximated arbitrarily well by another such function g for which 0 £ g(X). So the Cantor set has dimension zero and the unit interval has dimension 1. The unit circle has dimension 1 and the disk has dimension 2. The 2sphere S2 has dimension 2 and the ball in R 3 has dimension 3. Any compact subset of R n has dimension no more than n. For those readers who are not particularly familiar with covering dimension, they can use the above as the definition of the covering dimension. Corollary 3.2.10 dimX

If X is a compact metric space, then RR(C(X))

=

Proof. Suppose that dimX = n and A = C(X). Let / = (xi,...,xn+i) be an n+ 1-tuple in Asa. Then / G C(X,Rn+1). For any e > 0 there exists g G C(X, R™+1) such that 0 £ g(X) and

\\f-g\\<e/yfr. Write g = (glt ...,gn+1) for gk G C(X,R). Then T.lt\9k{x)2 words, X3fc=i 9k ls invertible. We also have

> 0. In other

II ^2(xk-9k)2\\ <£2k

Therefore RR(A) < n. The reader can reverse the argument above and show that if RR(A) = n then dimX < n. • Theorem 3.2.11

If A is a C*-algebra, then RR(A) < 2tsr(A) - 1.

Proof. It is clear that we may assume that A is unital. Suppose that tsr(A) = n < oo. Given {x\,..., x2n), where Xk G Asa, 1 < k < n, let dk = Xk + ixk+n for 1 < k < n. Since tsr(A) = n, for each s > 0, there is an n-tuple (6i, ...,bn), bt G A (i = 1,2, ...,n) such that n

Y^(ak-h)*(ak-bk) fc=i

<e/(n + l)

(e2.3)

126

AF-algebras

and Ranks of C* -algebras

Ysblbk>5

and

(e2.4)

fc=i

for some 6 > 0. Write bk = yk + iyk+n, with yi G Asa, 1 < i < 2n. Then by (e2.4) 2ra

fc = l

fc=l n

TI

= X>iU>/b + MS) >£>£&*> *, fc=i fe=i

so that X^=i 2/fc i s invertible. By (e 2.3) we have (afc - 6fc)(afc - 6fc)* < \\ak - bk\\2 < e/(n + 1). Therefore n

^2(ak ~ bk){ak - bk)* < ne/(n + 1). fc=i

Finally In

n

2Y^{xk-yk)2

= 2^2[(xk-yk)2+ {xk+n-yk+n)2}

fc=l fc=l n

=

Yl^ak

<

e/(n+l)

~ bk)*(ak - bk) + (at - 6fc)(afc - bk)*] +

ne/(n+l)=£.

Thus RR(A) < 2n - 1 as desired. The inequality becomes an equality for the abelian C*-algebras with even dimension. Proposition 3.2.12

If X is a compact metric space, then ,„,..,NN

tsr(C(X)) where [^f^] is the integer part Proof.

• C(X)

r dimX, = [—£-] + 1,

of^sgL.

The proof is similar to that of 3.2.10. We leave it to the reader.

•

Order structure of K-theory

3.3

127

O r d e r s t r u c t u r e of J f - t h e o r y

Definition 3.3.1 An ordered group (G,G+) is an abelian group G with a distinguished subsemigroup G+ containing zero, called the positive cone of G satisfying the properties (1) G+-G+=:G and

( 2 ) G + n ( - G + ) = {0}. G+ induces a partial ordering on G by x < y if y - x £ G+. If x < y and z £ G then x + z < y + z, since (y + z) — (x + z) = y - x £ G+. Also, we write g < f, if g < f and g ^ / . An order ideal I of an ordered group (G, G+) is a subgroup of G such that g < f for some / £ I implies that g £ I. An element u £ G+ is called an order unit if for any g £ G there is an integer n > 0 such that nu > g. In other words, the order ideal generated by u is the whole group G. A triple (G, G + , u) consisting of an ordered group (G, G + ) with a fixed order unit is called a scaled ordered group. We say G is a simple ordered group if G has no proper order ideals, i.e., if every nonzero positive element is an order unit. E x a m p l e 3.3.2 An important example of ordered group is given by Zk. This is an ordered group with positive cone Z+ = {(xi,...,xk)

:xi>0,

i = 1, ...,&}.

Definition 3.3.3 A unital G*-algebra A is said to be finite, if x*x — 1 implies that xx* = 1. A is said to be stably finite, if Mn(A) is finite for all n. From the definition and 3.1.14, we immediately have the following: P r o p o s i t i o n 3.3.4

Every unital C*-algebra A with tsr(^4) = 1 is stably

finite. P r o p o s i t i o n 3.3.5 dered group.

If A is stably finite, then (K0(A),K0(A)+)

is an or-

Proof. We must show that K0(A)+ n (-#0(^4)+) = {0}. Let x be in ^0(^4)+ H (—Ko{A)+). Suppose that x = \p] = —[q] for some projections p, q £ Moo(A). Then [p © q\ = 0 in K0(A). So if e = p © q and e £ Mn(A), then [l n ] = [1„ — e], and therefore, for some positive integer m,

128

AF'-algebras and Ranks of C* -algebras

lm © (In — e) ~ l m © 1„. In other words, there is v G Mn+m(A) v*v = lm+n

such t h a t

and vv* = l m © (1„ - e).

Since A is stably finite, vv* = l m + „ , e = 0. Therefore p = q = 0, and consequently, x = 0. • L e m m a 3.3.6 If p is a projection in A, b G ^4 + and p is in £/te ideal generated by b, then there are xi,...,Xk G A such that k

p=

^Xibx*. i=l

Consequently, if A is a unital C* -algebra and b generates then for any a G A+, there are x\, ...,Xk G A such that a =

A as an ideal,

y^jbx*. i=i

Proof.

There are c\,..., Ck and y\,...,yk

in A such t h a t

k

||p-^

C i

%||
i=\

Let z = p X^i=i cibyiP- T h e n z is invertible in pAp. So p = ]T)i=i -z where the inverse is taken in pAp. To save notation, we 9i,-,9k,yi,-,yk € A such t h a t p = 52i=1gibyi. Set 9i 0 0

92 0 0

fyi

9k\ 0

1

Cibyip, obtain

o 0

y=

2/2

0

0/

0/ \Vk

0

e = diag(p, 0, ...,0), and d = diag(6, ...,&) T h e n e = x'cfy in Mk(A). So e = ex'dyy*d(x')*e. We have e < ||2/2/*||ea;'(icf(a;')*e. Hence z\ = ex'd1/2dd1/2(x')*e has an inverse in eMk(A)e = pAp. Denote by z<x the inverse. Note t h a t

x'd^dd^ix'y

Ys(9idll2)d{9idll2y

= i=l

Order structure of

K-theory

Therefore p = £ i = i ^ ^ p O ? ^ 1 ^ ) ^ ^ 1 / 2 ) * ^ 1 7 2 . Set xt = as desired. In the unital case, from what we have just proved, we have

129

z^pgid1'*

k i=l

for some yi,..., yk € A. Therefore

a={a^)(£yiby*){a1'2). i=i l 2

Set Xi = a l yi as desired.

D

P r o p o s i t i o n 3.3.7 If A is stably finite and simple, then KQ(A) is a simple ordered group. Proof. Suppose that p and q are two non-zero projections in Mn(A). By 1.11.42, Mn(A) is simple. So p = X)i=i xiQxi f° r some xx,..., Xk G Mn(A) by Lemma 3.3.6. Set / xi 0

x2 0

\ 0

0

...

xk \ 0

,e = diag(p,0, ...,0), and / = diag(g,g, ...,g)

0 /

in M fc (M„(A)). Then e = z / z * = exfx*e. Set # = fx*exf. Then 3 is selfadjoint, g2 = fx*exfx*exf = fx*exf = g. If v = xf, then vv* = e and v*v = 3. But g< f. Therefore [e] < fc[/]. D Definition 3.3.8 Let Gi and G? be ordered groups. A homomorphism : G\ —• G2 is said to be positive if 0((Gi)+) C (G2)+. A positive homomorphism <j) is said to be an order isomorphism if <j> is bijective and B be a homomorphism. Then h*o : KQ(A) —»• Ko(B) is positive. Proof.

This follows from 2.4.15 immediately.

•

Corollary 3.3.10 Let A = lim„_ >00 (A ri ,/i„) be an inductive limit. Suppose that An are stably finite, then Ko(A)+ = Un%1(hn)*o(Ko{An)+)• Proof.

This follows from 3.3.9 and 2.7.3.

•

130

AF-algebras

and Ranks of C* -algebras

Definition 3.3.11 An ordered group (G,G+) is imperforated if nx > 0 for some integer n > 0 implies that x > 0; a simple ordered group G is weakly imperforated if nx > 0 for some integer n > 0 implies x > 0. Example 3.3.12 Let .A = M n . Then A is stably finite and (K0(A),K0(A)+) Moreover, i£"o(^4) is unperforated.

= (Z,Z+).

Definition 3.3.13 Let (G,G+) be an ordered group. G is said to have the Riesz interpolation property if given 01,02,61,62 G G with a^ < bj (i, j = 1,2), there exists c G G such that a^ < c < fy (i, j = 1, 2). L e m m a 3.3.14 Le£ (G, G + ) 6e an ordered group. Then the following are equivalent: (1) G has Riesz interpolation property. (2) If 0 < a < b\ + 62 and bi G G+ (i = 1,2), then there exists ai £ G+ such that aj < bi (i = 1,2) and a = a\ + a2. (3) If ai + a2 = b\ + 62, where ai,bi G G+ (i = 1,2^, then there are Cij € G+ (i,j = 1, 2j such that a.j = ca + Ci2 and 6j = cy + C2j, i, j = 1, 2. Proof. (1) =>• (2). Suppose that 0 < a < 61 + 62 and bi, 62 G G+. Since a — 61 < a, 62, we may choose a2 G G such that 0, a — by < a<2 < a, 62. Set ai = a — a2. Then ai G G + and a\ < (a — (a — bi)) = b\. However &\ + 0.2 = a. (2) => (3). Suppose that a1+a2 = h + b2, au bi G G+ (i = 1,2). By (2), «2 = C21 + c22, c2j G G+ and c2j < bj. Let cXj = bj - c2j (j = 1,2). Then cy G G+. We check that ai = bi + b2 - a2 = (&i - c 2 i) + {b2 - c22) = en + ci 2 and 61 = en + C21 and 62 = C12 + c22(3) => (1). Suppose that a^ < bj, i,j = 1,2. We have (61 - ai) + (b2 - a2) = (61 - a2) + (b2 - ax).

Order structure of

K-theory

131

Note that, by assumption, all terms above are in G+. By (3), there are Cij G G+ {i, j = 1,2) such that 6i — ai = en + C12, 62 — 02 = C21 + c22, bi — a 2 = Cn + C21, and 62 - ai = c 12 + c22- Set c = 61 — en G G+. Since c = 61 — 01 — en + ai = C12 + ai > ai and c = 61 - 02 - en + 02 = c2i + a 2 > 02, we have aj < c < 6j (1 = 1,2).

D

Example 3.3.15 Consider ordered group (Z fc ,Z+). It is easy to see that it is unperforated. To see that it also satisfies the Riesz interpolation property, we let 01,02,61,62 G Zfc with a, < bj (i,j = 1,2). Write ai = (nil, •••,'n-ik) and 6i = (m,i, •••,mik) G Zfc. Set Sj = max{riij : i = 1,2}, j = l,...,fc, and c = (si,...,Sfc). Then a^ < c (i = 1,2). Since s^ < rrij^-, i = l,...,fc, c < 6 j (j = 1,2). Lemma 3.3.16 If A is a C*-algebra with RR(A) = 0 and if p, q G A are projection, then there are projections e, / G A such that [f] < [p], [e] < [1 - p] (in A) and q — e + f. Proof.

Write o c c* b

as the matrix decomposition of q with respect to 1 = p + (1 — p), where o = pqp, c = pq{l-p) and b = (l-p)q(l-p). The fact that q is a projection implies that a — a = cc*, b — b = c* c and ac + cb = c. Note that ||a|| < 1 implies ||c|| 2 = ||cc*|| = | | o - a 2 | | < sup | i - i 2 | = l / 4 . 0
Hence ||c|| = ||pg(l - p)|| < 1/2. We have two cases. Case (1): ||q(l - p)\\ < 1. Then \\q - qpq\\ < 1. It follows from Lemma 2.5.2 that q X p. So we let / = q and e = 0 as desired. Case (2): ||o(l - p)\\ = 1. Set x = q(l - p)q. Then ||a;|| = 1.

132

AF-algebras

and Ranks of C* -algebras

Let (with 0 < r < 1/4)

{

1

if 1 - r < t < 1

linear \il-2r
||e - (1 - p)e|| = ||pe|| < ||p(e - ie)|| + ||po;e|| < 2r + ||cge|| < 1. By 2.5.2, e H (1 - p). Set / = q - e. Then II/-/PII2

=

II/X/||< ||/(1

-gr/4(x))x\\+\\fgr/4(x)x\\

<

(1 - r/4) + || (1 - c)ffT./4(a:)|| < 1 - r / 4 + e < 1.

By Lemma 2.5.2, / •< p. Thus q1 = e + / is as desired.

D

Corollary 3.3.17 Suppose that A is a C*-algebra with RR(A) = 0. If P,Pi,P2 S A are projections such that p
Let B = (jpi +P2)A(pi +P2}- Then Lemma 3.3.16 applies.

•

We end this section with the following theorem of S. Zhang: Theorem 3.3.18 If A is a stably finite C*-algebra with RR(A) = 0, then (Ko(A), KQ(A)+) is an ordered group with the Riesz interpolation property. Proof. Since A is stably finite, (Ko(A), KQ(A)+) is an ordered group. The Riesz interpolation property follows from 3.3.17, since RR(Mn(A)) — 0 for all integer n > 0. •

133

AF-algebras

3.4

AF-algebras

We have seen several examples of AF-algebras. In this section we will classify AF-algebras by their scaled ordered group KQ. Theorem 3.4.1 Let A be a separable C* -algebra. Then A is an AFalgebra if and only if the following holds: For any e > 0 and oi, ...,an G A, there exists a finite dimensional C*subalgebra B C A and b\, ...,bn 6 B such that Ho, - &i|| < £,

i = l,...,n.

Proof. Suppose that A is an AF-algebra. Write A = lim where each A n is a finite dimensional C*-algebra. Let Bn = /i n ,oo(A n ). Then Bn is a finite dimensional C*-subalgebra. Since U^ = 1 B r l is dense in A, the "only if part of the theorem follows. To prove the converse, fix a sequence {an}, a dense subset of the unit ball of A. We will construct an increasing sequence of finite dimensional C*-subalgebras {Afc} such that dist(a;, Afc) < £fc for 1 < i < k, where £fc > 0 and £fc —>• 0 as k —> oo. Assume that we have found such A\,..., Afc. Write Afc = Mni ©• • -®Mns and let {e\j : I = l,...,s;i,j = l,...,m} be the standard generators for Afc. Let S = 5(ek+i/3, dimAfc) be as in Lemma 2.5.10. By the assumption, there is a finite dimensional C*-subalgebra B of A so that dist(e' J -,S) < 6 and dist(a m ,B) < £fc+i/3 for i,j = 1, ...,n;, I = 1, ...,s and m = 1,...,fc+ 1. Then by Lemma 2.5.10, there is a unitary u £ A such that uAkU* C B and \\u — 1|| < £fc + i/3. Set Afc+i = u*Bu. Then Afc+i is finite dimensional, Afc C Afc+i and dist(aj, A fc+ i) = dist(uaiU*, B) < 2\\u - 1|| + £fc+i/3 < e fc+1 . Thus we obtain the required increasing sequence of finite dimensional C*subalgebras {Afc}. Since the closure of its union contains the unit ball, U„A n is dense in A. Therefore A is AF, • Lemma 3.4.2 Let A be a C*-algebra and pi,...,pn be projections in MQO(A), and q a projection in A such that q ~ p\®- • -®pn- Then there exist

134

AF-algebras

and Ranks of C* -algebras

mutually orthogonal projections qi,...,qn and pi ~ qi (i = l,...,n).

G A such that q = qi + • • • + qn

Proof. We may assume that p\, ...,pn G Mk(A) for some integer k > 0. There is a partial isometry v G Mk{A) such that v*qv = p\® • • • ®pn and v(pi © ••• ©p„)v* = g. Set qi = vpiV*. Then q = J27=i 9*> 9i>—.9n are mutually orthogonal projections and ^ G ^Ag c A ( i = l,...,n). D Definition 3.4.3 Let A = Mni © • • • © Mnk and {e' •} be its canonical generators. The homomorphism ip : Zfc —> iiTo(^l) defined by fc (mi,...,mfc) M- J ^ z=i is the canonical map from Zfc to i^o(^l)Proposition 3.4.4 If A = Mni © • • • © M nfc , i/*en the canonical map V> : Zfc —> i^o(^4) *s aw order isomorphism. Proof. Let p G Proj(Moo(^4))- Then p = (pi, ...,pk), where Pi G Proj(M 00 (M 7li )). For each /, pi ~ eln © • • • © eln ( the number eln in the sum is the rank of pi). Therefore the elements [e^],..., [efj] generate the group KQ(A), so ip is surjective. Furthermore, ipilJ^) = .Ko(^.)+Let 7T; : A —> Mni be the projection. Suppose that ' 0 ( m i ! •••irnk) = 0, that is, X)z=i m *[ e iil = °- Then, for each j , (7TJ)*(5ZJ=I m ' [ e i i ] ) = m j'[ e iil = 0- This implies that m^ = 0. Hence ip is injective. It is obvious that ip-1 is also positive. • Proposition 3.4.5 If A is a unital AF-algebra, then KQ(A) is an unperforated ordered group with the Riesz interpolation property. Proof. That Ko(A) satisfies the Riesz interpolation property follows from 3.3.18. To see that KQ{A) is unperforated, we note that Ko(A) — lim„^ 0O (G'„, (G„)+, (/i„)»), where Gn = ZfcW and (Gn)+ = Z+ (n) by 3.4.4. Suppose that x G Ko(A) and kx > 0 for some integer k > 0. There is gn £ Gn such that (hnt00)*(gn) = x. Since kx > 0, it follows from 3.3.10 that k(hn!m)t,(gn) > 0 for some m > n. Ko(Am) is unperforated, by 3.3.15. Therefore (/i„,m)*(ff„) > 0 in K0(Am). Thus a; = hmt00)*o(hntm)*(gn) > 0. D The following is sometime called the uniqueness and existence theorem for homomorphisms between finite dimensional C*-algebras.

AF-algebras

135

Theorem 3.4.6 Let A be a finite dimensional C* -algebra and B be a C*-algebra with tsr(B) = 1. (1) Suppose that ip : Ko(A) —> KQ(B) is a unital positive homomorphism. Then there is a unital homomorphism h : A —»• B such that /i* = ijj. (2) If hi,fi2 : A —> B are unital homomorphisms, then (/ii)» = (/12)* if and only if hi = adu o h\ for some unitary u £ B. Proof. Write A = Mni © ••• © Mnk and set At = Mnn I = l,...,k. Let {e\j} be the canonical generators for Ai, and let e; = 1^,. We will use the fact that in a finite dimensional C*-algebra, stable equivalence and equivalence are the same (Example 2.4.13). We have ^([e;]) = \pi] f° r some projection pi £ Moo(B), since V is positive. We also have k

[pi © • • • © pfc] = V>($>«]) = ^([U]) = [1B]. (=1

Therefore, by Lemma 3.4.2, there exists mutually orthogonal projections qi,...,qk G B such that qi + • • • + qk — I s and qt ~ p; for Z = l,...,fc. Since ^([e;]) = [li] a n d B which maps e^ to p,j. It is routine to check that h is a homomorphism. Since h(eu) = p n and ^ ( ^ n ] ) = [Pn]i Zi» = ip. This proves (1). If u G U(B), then (adu)» = idx 0 (s), so (adw o Zi^,,, = (hi)*. Conversely, suppose that (Zii)* = (h?)*. Note that we return to the general case. We will retain the notation in the first paragraph. Set p' • = hi(e\j) and qltj = Zi2(e^) for Z = l,...,fc and i,j — l,...,n/. Then

b!i] = C»i)*([e5«]) = ( M - a ^ D - [<£]•

136

AF-algebras

and Ranks of C* -algebras

Since t s r ( B ) = 1, by 3.1.14, there exist partial isometries vi € B such t h a t P i i = vivl

an

d 9ii = viv*. Set k

ni

i=l i = l

One checks directly t h a t u is a unitary in B and t h a t up\^ = q\,u for all l,i,j. T h u s h2(e\j) = u*/i i (e' J -)u = (adw) o / n ( e [ . ) for all / , i , j , so ft2 = a d u o / i j .

D L e m m a 3.4.7 i e i G i , G 2 and G3 be ordered groups. Suppose that fc (Gi,(Gi)+) = (Z ,Z+) and <j> : G i ->• G 3 anrf ^ : G 2 ->• G3 are positive hornomorphisms such that ((Gi)+) C ^ ( ( ^ 2 ) + ) . T/ien i/tere is a positive homomorphism <j>' : G\ —f G 2 suc/i £/ia£ tp o cf>' = cf>. Proof. Let ej = ( 1 , 0 , . . . , 0),..., e^ = (0,..., 0,1) be t h e canonical positive generators for Z fc . Since 0 ( ( G i ) + ) C ^ ( ( ^ 2 ) + ) , there are g\, ...,gk £ ( G 2 ) + such t h a t ip(gi) = <j>{ei) (i = l,...,k). Define cf>' : G\ —> G 2 by setting 4>'{ei) — gi (i — 1,..., k). Clearly = ip ° <j>'. Since 4>'(ei) > 0 (i = 1,..., A;), ' is positive. • T h e following is a theorem of Elliott. T h e o r e m 3.4.8 ( Elliott) Let A and B be unital AF-algebras. If there is an order isomorphism ip : KQ(A) —> Ko{B) with ^([1^]) = [ 1 B ] , then there exists an isomorphism h : A —> B such that h* = ip. The converse also holds, i.e., if h : A —• B is an isomorphism, then h* : Ko(A) —> KQ{B) is an order isomorphism with ^ » ( [ 1 A ] ) = [1B] • Proof. To simplify notation, we assume t h a t A is t h e closure of ^JnAn and B is the closure of UnBn, where An C An+i and Bn C -B n +i are finite dimensional G*-subalgebras. Moreover, we assume t h a t 1A„ = 1 A and 1B„ = 1 B for all n- We will use in for the embedding from An to A, in,m for the embedding from An to Am (m>n), j n for the embedding from Bn to B and j n > m for the embedding from Bn to Bm (m>n), respectively. Denote by <j> : K0(B) -)• K0(A) the inverse of ip. Note t h a t <£([1B]) = [1A] and b o t h <j> and ip are order isomorphisms. By 3.3.10, Ko(A)+=Un(in)t(K0(An)+), so K0(A)=Un{in)*(K0(An)). Similarly, K0(B)+ = Un(jn)*(K0(Bn)+) a n d K0(B) = Un(jn),(K0(Bn)). P u t ni = 1. Since Ko(.Ai)+ is finitely generated, so is ipo(i1)t,(K0(Ai)+). Therefore, there is raj > 0 such t h a t (ii)*(K0(Ai)+) C (jmi)*(Ko(Bmi)+)-

137

AF-algebras

It follows from Lemma 3.4.7 that there is a positive homomorphism ipi Ko{Ai) -¥ Ko(Bmi) such that the diagram

K0(A!)

(n).

K0(A) 4-i/>

4-1/" i

Ko{Bmi)

(Jmj).

K0(B)

commutes. Since Ko{Bmi)+ is finitely generated, there exists n'2 > n\ such that 4>{{jmi)*{KQ{Bmi)+) C (in'2)„(Ko(An0+). Applying Lemma 3.4.7, we obtain a positive homomorphism <]>[ : Ko(Bmi) —• Ko(Ani) such that the diagram Ko{An,2) Ko{Bmi)

commutes. Let g £

(•-')• ' "2* (i™.)-

i^o(B)

then

KQ(AI),

(x)* 0(t>'i °^i(5) = 0{Jml)* °i>i(g) = n'2 such that ( v J * ° (*n 2 ,n 2 )*(/*) = ( V 2 ) * ° ( * n i , n 2 ) * ( S i ) ,

where /$ = ^ o i / i j ^ ) , i = 1,2, ...,Z and i = (t„ 2 ,„ 2 )* o <£i : K0(Bmi) -> if 0 (A„ 2 ). Then ^ o ^ = (*i,n2)*Therefore we have the following commutative diagram: ffo(^i)

{l1

-^'

i-ipi

(

K0(An2)

H*

/ 0i

K0(A) J

#o(Bmi)

°^*

K0(B)

Similarly, there is m-i > mi and a positive homomorphism tp2 '• Ko(An2) —>• K(,(Bm2) such that the diagram commutes. (t Ko{Al)

tfo(Bmi)

(Jm

L^*

Ko(An2

)

-^ 2 ) *

ffo(B„,a)

< ^

(

H*

KQ{A)

Ko(B)

138

AF-algebras

and Ranks

of C* -algebras

Continuing the above construction, we inductively construct two sequences of integers 1 = n\ < m < • • • and mi < m^ • • •, and positive homomorphisms ipk : K0(Anic) ->• K0(Bmic) and <$>k : K0(Bmk) ->• K0(Ank+1) such that the diagram commutes. ('l.r»2)»

K0(Ai)

A

•H>i

\3rni

Ko{Bmi)

K0(An2) \-lp2

,rn2J*

Ko(Bm2)

(2™2'n3 ) *

^

A

Um2,m3)»

/ <

N

(»n3,„4).

4-1p3

/

^o(om3J

—•

•

i^o(^)

J

03

•

#o(5)

By (1) in 3.4.6, there are unital homomorphisms hi : A\ —> Bmi and H[ : Bmi —> An2 such that (/ii)» = ipi and (.Hj)* = 4>i- Since <j>i o ipi = (li,n2)*, by (2) in 3.4.6, there is a unitary U\ G An2 such that ad«i o H[ o hi = ii,„ 2 . Define Hi = adui ° #{- Then we have the following commutative diagram: Ai

—$ An2

4-/ii / * » ! •"mi

commutes. By applying (1) in 3.4.6, we obtain another unital homomorphism h'2 : An2 -> i? m 2 such that (/i'2)» = ^2- Since i | i 2 ^ i = (jm1,m2)*, by (2) in 3.4.6, there is a unitary Vi g i? m 2 such that advi oti2o Hi = j m i

>m2.

Define hi — advi o h'2. Then we have the following commutative diagram. Ai ihi

•l,n, ^4-712 4-/12

/ • * J - 1 W

Hm\

-£*m2

Continuing in this fashion, we construct inductively unital homomorphisms /ifc : A„fc ->• B mfc and #& : 5 m t ->• ^4nfc+1 such that the following diagram Ai

——f

4-/U

/'Hi JnH.tng '

£>mi

intertwining.

An2 4-/12 _ -Om2

>• / V 2 Jm2,m3 '

An3

——>•

••• A

4-/13

/ ^ S Jm3,m4 '

••• D

-"7713

AF-algebras

139

It follows from 1.10.16 that there is an isomorphism h : A -> B determined by the above intertwining diagram. • To complete the classification of AF-algebras, we have the following theorem. Theorem 3.4.9 Let G — lim„_> 00 (G n ,i/'n), where each (G„,(G„) + ) has the form (Z f c ( n \Z+ ( n ) ) and ipn : Gn -> Gn+i is positive homomorphism. Suppose that gn G (G„)+ is an order unit for Gn and ^„,oo(9n) = u € G+ is an order unit. Then there is a unital AF-algebra A such that (Ko(A),K0(A)+,[lA]) = {G,G+,u). Proof. Since ipnioo(gn) = u, by 1.10.2, for any n' > n there is an integer m > n' such that 4>n,m+i{gn) — Vv.m+iCflw)- Therefore, by passing to a subsequence, we may assume that tpn(9n) = Sn+i- Let gn = (Zi,...,Zfc(„)), where k > 0. Define An = Mh © • • • © M/ k ( n ) . Then (K0(An),K0(An)+,[lAn}) = (Gn,(Gn)+,gn). It follows from (1) in 3.4.6 that there is a homomorphism hn : An —t An+\ such that {hn)* = ipn- Let A = liran^,00(An,hn). One then checks easily (by 2.7.3) that (K0{A),K0{A)+,[1A]) = (G,G+,u). D Definition 3.4.10 Recall that a tracial state is a trace which is also a state (1.6.1). Every finite dimensional C*-algebra has tracial states. Write B = M n i ©- • -®Mnk. Let Ti be the normalized trace on M n . and let a\, ...,ak be non-negative numbers such that Y^i=i afc = 1- Then -r(a) = Yli=i aiTi{a) (a G B) is a tracial state. Let A be a C*-algebra and T be a tracial state on A. One extends r to Mn(A) by r®Tr, i.e., r((aij)) = Y^i=i T(aa)- It i s a trace but not state. In what follows, for any tracial state r, if we write r(a) for some a G Mn(A), we mean (r Tr)(a). Let A be a C*-algebra. We will use T(A) for the space of tracial states on A with weak * -topology. If A is unital, T(A) is compact. Proposition 3.4.11

Every unital AF-algebra has tracial states.

Proof. Write A = LinAn, where An c An+i are finite dimensional C*subalgebras and 1^^ = 1A for all n. Let tn be a tracial state on An. Extend tn to a state on A. Let r be a weak*-limit of {tn}. Fix a G Am, there is a subsequence {nfc} such that r(a) = limfc_>oo tnk{a), r(a*o) = linin^oo tnk(a*a) and r(aa*) = limfc-^oo tnk(aa*). Since tfc(a*a) = tk(aa*) for all k > m, r(a*a) = r(aa*). So r is a tracial state. •

140

AF-algebras

and Ranks of C* -algebras

Theorem 3.4.12 If A is an AF-algebra andp,q are two projections in Mn(A), then r(p) > r(q) for all r G T(A) implies that q 0. We claim that for any tn G T(An), tm(p) > tm(q) for all tm G T(Am) (m> n). Otherwise there is a subsequence {m^} such that tmk (p) < tmk (q) for k = 1,.... Let r be a weak*-limit oi{tmk}. As in 3.4.11, r is a tracial state on A. Since tmk(p) < tmk(q), we conclude that r(p) < r(g), a contradiction. Thus, for some large m, t(p) > t(q) for all t G T(A m ). Since J4TO is finite dimensional, from the linear algebra, we know that q •< p in Am, whence q •< p in A. D

3.5

Simple C*-algebras

Examples of simple C"*-algebras are given in Chapter I (see 1.9.4, 1.9.5 and 1.9.24) In this section we will give some elementary properties of simple C*-algebras and introduce purely infinite simple C*-algebras. Lemma 3.5.1 Let x G A with the polar decomposition x = u\x\ in A" and B — x*Ax. Then ub G B for every b G B. Proof. Since \x\ G B, iip(t) is a polynomial, with p(0) = 0, then up(|a;|) G B. Therefore, for any / G C 0 ((0, |||a;|||]), uf(\x\) G B. For any b G B, there exists / G C 0 ((0, |||x|||]) such that (by the proof of 1.5.9)

11/(1*1)6-6||<£/2. Since uf(\x\)b G B, so ub is in the closure of B. Therefore ub G B.

•

Definition 3.5.2 Let a and 6 be two positive elements in a C*-algebra A. We write [a] < [b] if there exists a partial isometry v G A" such that, for every c G Her(a), v*c,cv G A, vv* = pa, where pa is the range projection of a in A", and v*cv G Her(6). We write [a] = [b] if v*Rei(a)v = Her(6). Caution: We do not write [a] = [b] if we only know [a] < [b] and [b] < [a].

Simple C* -algebras

141

It is easy to see that, if [a] < [6] and [b] < [c], then [a] < [c]. Let n be a positive integer. We write n[a) < [b] if there are n mutually orthogonal positive elements 61,62, •••, bn £ Her(6) such that [a] < [bi], i = 1,2, ...,n. Proposition 3.5.3 Let A be a C* -algebra. (i) IfQ 0, then [a] < [6]. (ii) / / p and q are two projections in A, then \p] < [q] if and only if p (c) = ucu* is an isomorphism from Her(a;*a;) to Her(a;a;*) which maps x*x to xx*. Thus

uUlx^u*

= fa(\x*\)

for every a > 0. This proves (iv). For (iii), let c = b^xx*^/2. By (iv), [a] = [c]. But c < ||a;a;*||6. So by (i), [a] < [6], For (v), choose v = pa, the range projection of a in A". Then v*cv = c for all c e Her(a). Moreover, w*Her(a)v = Her(6). Conclusion (vi) follows from (v) since Her(a) = Her(a 2 ). For (vii), let z = p - (1 - p). Then [a] < [a + zaz] = [2(pap + (1 - p)a{\ - p))} = [pap + (1 - p)a(l - p)].

a

142

AF-algebras

and Ranks of C* -algebras

L e m m a 3.5.4 Let A be a simple C* -algebra and a, b be nonzero elements in A. Then there is a nonzero y G A such that y*y € Her(|a|) and yy* G Her(|6|). Proof.

Clearly we may assume that a, b G A+. Set I = {c : cxa = 0 for all x G A}.

I is a closed linear subspace of A. If z G A, then zc £ I for all c £ I. Moreover, czxa = 0 for any c G / and x G A. So I is an ideal of A. Hence / = {0}. This implies that there is x G A such that y = bxa ^ 0. Note y*y G Her(a) and yy* G Her(6). • Definition 3.5.5 We say a C*-algebra A has the property (SP), if every nonzero hereditary C*-subalgebra of A contains a nonzero projection. Every C*-algebra with real rank zero has property (SP). There are (simple) C*-algebra with property (SP) which have real rank other than zero. L e m m a 3.5.6 Let A be a simple C*-algebra with property (SP), p G A be a nonzero projection. (1) Suppose that a G A+ is a nonzero element. Then there is a nonzero projection q in aAa such that q •< p. (2) If q is a nonzero projection in A, then there are nonzero projections p' < p and q' < q such that p' ~ q'. Proof. (1) By Lemma 3.5.4 there is an element y G A such that y*y G Her(a) and yy* G Her(p). Since A has property (SP), Her(y*2/) contains a nonzero projection q. It follows from the proof of 3.5.3 that [q] = [qi] for some projection 91 G Her(yy*). Then by 3.5.3 q^p. (2) follows (1). • L e m m a 3.5.7 Let A be a non-elementary simple C*-algebra. Suppose that A has property (SP). Then for any nonzero projection p G A and any integer n > 0, there are n + 1 mutually orthogonal projections qi,
143

Simple C* -algebras

projections pi G Bi. By applying (2) in Lemma 3.5.6 repeatedly, we obtain a nonzero projection e G piApi such that

H < MThe lemma then follows.

•

Lemma 3.5.8 Let A be a C*-algebra satisfying the condition that every hereditary C* -subalgebra contains at least two mutually orthogonal nonzero positive elements. Then for any nonzero elements a, b G A+ there are nonzero positive elements a\ E Her(a) and b\ G Her(6) such that aib\ = 0. Proof. We may assume that x = ab ^ 0. Suppose that x — v\x\ is the polar decomposition of x in A". Fix 0 < e < \\x\\. By the assumption, there are mutually orthogonal nonzero positive elements di,d,2 such that fe(\x\)di = di, i = 1,2. Then ct = vdtv* G Her(|ir*|) and f£(\x*\)ci = a, i = 1,2 (see the proof of (iv) of 3.5.3). Since C2vd\d\v*C2 = 0, C2vd\ = 0. Let g(t) = t-lfe{t)- Then g G C 0 ((0, ||i||])+. Set zx = g(\x\)dx. Then xzx = v\x\g(\x\)di = vfe{\x\)di

= vdy.

Therefore C2XZ\ — C2vd\ = 0. Now set b\ = bz\b and a\ = ac2a. Note that b\ G Her(6) + , a\ G Her(a) + and both are nonzero. However, a\b\ = ac2abz\b = a(c2XZ\)b = 0.

• Lemma 3.5.9 Let A be a unital simple C*-algebra with property (SP). Then for any nonzero positive elements a,b G A there is u G U(A) such that uHer(a)u* nHer(6) ^ {0}. Proof. It follows from the above lemma that we assume that ab = 0. By (2) in Lemma 3.5.6, there are nonzero mutually orthogonal projections p G Her(a) and q G Her(6) such that p ~ q. By 3.5.7, there are nonzero mutually orthogonal projections pi,P2 < p and qi,q2 < q such that pi ~ P2 ~ qi ~ 92- Let e = p\ + p2 + q\ + ?2, &\ = Pi + 9i and e 2 = P2 + 92- Let u G e\Ae\ such that w * = pi and v*v = q\. Set B = eAe. If we identify B with M2(C), where C = eiAei, we write

w=(V

q

\Pi

l).

v* J

144

AF-algebras

and Ranks of C*-algebras

Then w*w = e and ww* = e. So w is a unitary in eAe. Furthermore, w*p\w = qi. Set u = w* + (1 — e). Then u e U(A) and wpiu* = 31, whence uHer(a)u* n Her(6) ^ 0. • Lemma 3.5.10 Let A be a C*-algebra and {An} be a sequence of C* subalgebras so that U„An is dense in A. Let I be a nonzero ideal. Then, for some large n, An D I ^ {0}. Proof. Otherwise, An n I = {0} for all n. Let n : A -> A/I be the quotient map. Since ir(An) = An/InAn, ||7r(x)|| = ||x|| for all x & An. Let a G I with ||a|| = 1. For any e > 0, there exists n and x € An such that \\a — x\\ < e. So ||x|| > 1 — £. However, we have \\x\\ = \\ir{x)\\ =

\\n{a)-iT{x)\\<e.

A contradiction, if e < 1/2.

•

Definition 3.5.11 A nonzero projection p is said to be infinite if p ~ q, where q < p and q ^ p. Lemma 3.5.12 Letp be an infinite projection in a simple C*-algebra A. Then, for any integer n > 0, there exist n mutually orthogonal projections <7i, ...,g n such that qi

Simple C* -algebras

145

Definition 3.5.13 A unital C*-algebra A is said to be purely infinite, if A^C and if for any nonzero a € A, there are x, y G A such that xay = 1. Lemma 3.5.14 Let A be a unital simple C*-algebra. Then the following are equivalent: (1) A is purely infinite. (2) Every nonzero hereditary C* -subalgebra contains an infinite projection. Proof. Suppose that (1) holds and that B is a hereditary C*-subalgebra of A and b G B+ which is not a scalar multiple of identity. Then there are x,y G A such that xby = 1. Therefore xbyy*bx* = 1. Set z = byy*b. Then z G B+. Let v = zx/2x*. Then v*v = xzx* = 1 and p = vv* = zll2x*xzll2 G B. Note that p is a projection and v(l — p)v* = p — v*pv is a nonzero projection. Since vp G B, vpv* G -B, (vp)*vp = p and (up)(up)* = upu*, p is an infinite projection in B. This shows (2). Suppose that (2) holds. Take 0 < a < 1/4, then g(t) = t-1ftr/i(t) G Co((0,1]) and g(t)t = 1 for all t G (tf/2,1]. Fix any nonzero element x E A with ||x|| = 1 and let a = x*x. By the assumption, there is a nonzero infinite projection p G Her(/CT(a)). So p < g(a)a. Thus, to show that there are z,y E A such that zxy = 1, it suffices to show that there are v G A such that 1 = vpv*. It follows from 3.3.6 that there is n > 0 such that 1 = wPw*, where w G Mn(A) and P = diag(p, ...,p) (there are n copies of p). By Lemma 3.5.12 there are mutually orthogonal projections q\,..., qn < p such that qi ~ p, i = 1, ...,n. Therefore Q •< p in M„(A). So 1 < p in M„(A). Hence there is u G M n (A) such that uu* = 1 and u*u < p. This implies that u £ A. Moreover, we have upu* = 1. • Recall that a C*-algebra with stable rank one is stably finite. While 3.2.10 suggests that the notion of real rank may be closer to noncommutative dimension for C*-algebras than stable rank, we have, however, the following theorem of S. Zhang. Theorem 3.5.15 RR{A) = 0.

If A is a unital purely infinite simple C*-algebra, then

146

AF-algebras

Proof.

and Ranks of C*-algebras

Let a G Asa and e > 0. Define

r 0 f(t)=< t-e/A [ t + e/4

if \t\ < e/4 \it>e/4 if t < - e / 4

and g(t) = max{e/4 - |£|,0}

Since A is purely infinite, there is an infinite projection p £ B, where B = g(a)Ag(a). By 3.5.12, we may assume that 1 — p is equivalent to a subprojection of p. Thus there is a partial isometry v £ A such that v*v = 1 - p and vv* = q < p. Notice that f(a) = (1 — p)f(a)(l — p). Define b = f(a) + (e/4)(v + v*) + (e/4)(p - q). It has the following matrix decomposition corresponding to the decomposition 1 = (1 — p) ©
b=

//(<*) e/4 \ 0

e/4 0 0

0 \ 0 . e/4/

It is clear that b € A s a and 6 is invertible. Finally, ||6 - a|| < ||/(o) - a|| + (e/4)\\v + v* + (p- q)\\ < e.

3.6

Tracial topological rank

The last theorem (3.5.15) of Zhang suggests that in order to obtain finite and "lower rank" C*-algebras, both real rank and stable rank need to be considered. In this section we introduce another rank which we think is even closer to the rank for commutative C*-algebras. However, due to several complications, we only study the cases in which C*-algebras are simple. The prototype of C*-algebras of "dimension" n surely should be the following classes of C*-algebras. Definition 3.6.1 We denote by 1^ the class of all finite dimensional C*algebras, and denote by 1^ the class of all unital C*-algebras which are unital hereditary C*-subalgebras of C*-algebras of the the form C(X) ® F, where X is a A;-dimensional finite CW complex and F £ 1^. We think that C*-algebras in 1^ have zero rank and those in I^k' have rank k. In what follows if S is a subset and dist(a;, 5) < e, we will write x S e S. For simple C*-algebras, we have the following:

147

Tracial topological rank

Definition 3.6.2 A unital simple C*-algebra A is said to have tracial topological rank no more than k if for any £ > 0, any finite subset J- and any nonzero element a G A+, there exist a nonzero projection p € A and a C*-subalgebra B G X ^ with 1B = P such that (1) \\px — xp\\ < e for all x G !F, (2) pxp G£ B for all x G T and (3) [1 — p] < [a], or equivalently, 1 — p is equivalent to a projection in Her (a). If A has tracial topological rank no more than k, we will write TR(A) < k. If furthermore, TR(A) % k - 1, then we say TR{A) = k. E x a m p l e 3.6.3 Let A = Hmn^>00(An,hn) be a unital simple C*-algebra, where each A„ G l^k\ Then TR(A) < k. We leave this to the reader to check (see 3.8.19). An example in the next section shows that there are C*-algebras A with TR(A) < k which are not direct limits of C*-algebras in 1^. In fact, direct limits of C*-algebras in 1^ may be regarded as C*algebras that can be approximated in norm (pointwise) by C*-subalgebras in l( fc ). However, we will explain that TR(A) < k means that A can only be approximated by C*-subalgebras in 1 ^ "in measure". R e m a r k 3.6.4 Tracial topological rank is also defined for non-simple C*algebras. In particular, if A G X
(with very small size of q).

fqAq\

\

V

/

148

AF-algebras

and Ranks of C*-algebras

If we think that AF-algebras are C*-algebras that can be approximated by finite dimensional C*-subalgebras in (pointwise) norm, C*-algebras with tracial topological rank zero are C*-algebras that can be approximated by finite dimensional C*-subalgebras in trace (or in "measure"). While C*algebras which are inductive limits of C*-algebras in 1^ may be viewed as C*-algebras that can be approximated by C*-algebras in X^> in (pointwise) norm, C*-algebras with tracial topological rank k are C*-algebras that can be approximated by C*-subalgebras in I^> in trace (or in "measure"). This justifies the term. Lemma 3.6.5 Let A be a unital simple C*-algebra with TR(A) Then for any unital hereditary C* -subalgebra B of A, TR{B) < k.

< k.

Proof. Write B = qAq for some projection q £ A. Let 6 > 0, a =/= 0 in B+ and T C B be a finite subset. Without loss of generality, we may assume that J 7 is a subset of the unit ball of B. Since TR{A) < k, there is a projection p £ A and a C*-subalgebra C £ 1 ^ with lc = p such that (1) \\px - xp\ < 6 for all x £Q, (2) pxp £s C for all x £ Q and (3)[l-p]<[a], where Q = T U {q}. So \\(qpq)2 — qpq\\ < 26 and there is c £ C+ such that \\c-qpq\\

< 5.

With small 5, by 2.5.5, we obtain a projection e £ C, a projection q\ £ qAq — B and a unitary u £ A such that Iki - QPlW < £ A Iki - ell < e A u*eu = 9i and ||u - 1|| < e/3. Let Ci = u*eCeu. Since eCe £ I ^ , Cx = u*eCeu £ j( fc ). For each x £ F, (1')

\\q\x-xqi\\

< \\qpq - q\\\\\x\\ + \\qpqx - xqpq\\ + M\hP1-
< e/3 + S + e/3 < e

for all x £ !F, provided that 6 < e/3. By (2), for each x £ T, there is c £ C such that \\pxp — c\\ < 5. Thus, qiWl

« e / 3 PXP « 5

c

«e/3

w

*cu-

Tracial topological rank

149

Therefore we have (2') qixqi G£ C\ for all x G T. Since \\(q — qpq) - (q - qi)\\ < 1, if S is sufficiently small, (9 ~ 9i)(9 ~~ QPl)(l ~ 9i) is invert ible in (g — gi)/l(g — 91). Hence [q ~ 9i] = [(q - ?i)(l - p)(« " 9i)] < [9(1 " P)q}. Also, it follows from (iv) of 3.5.3 that [q(\ —p)q] < [1 —p]- Finally, we have (3') [9 - 91] < [9(1 - p)q] < [1 - p] < [a]. Thus TR{B) < k. D L e m m a 3.6.6 ^4 simple unital C*-algebra which satisfies conditions (1) and (2) in 3. £..2 /las property (SP ). Proof. Let ^4 be a simple unital C*-algebra satisfying (1) and (2) in 3.6.2 and B be a hereditary C*-subalgebra of A. If i? is finite dimensional, then B contains a nonzero projection and we are done. Otherwise, we obtain nonzero positive elements 01, 0-2, 0.3 G B+ such that 0 < as < 1, (1 < s < 3) a\a<2 = a
XjjejXjj.

»=1 1 /2

By the assumption, with .T7 = {as,as ,ej,Xi,x*,i = 1, ...,n(j),j = 1,..., 2/c + 1,1 < s < 3}, for any S > 0, there is a projection p G A and a C*-subalgebra C G 2 ^ with \ c = p such that (1) ||pz - zp\\ < 5 for z G J7, (2) pzp G<s C for z G T. Let 6S, dj G C+ such that 0 < bs < p, \\pasp — b„\\ < S and \\pejp — dj\\ < 5. This implies that ||6 s o s + 1 — 6 S+1 || < 36 (s = 1,2). It follows from 2.5.12, for any 0 < 77 < e, that there are c\, C2, C3 G C + such that \\c[ - (p - 61)|| < 77, ||c 2 - 62)|| < 7? and ||c 3 - 63|| < 77 and cic2 = 0 if 5 is sufficiently small. Let ci = p — c[. Then ||ci — 6i|| < e and C1C2 = C2. In c^Cc-x, \\c2C3 — C3j| < 3(5 + 3?7. Thus, by replacing C2 by /i/2n( c 2) and C3 by fi/n{c2)czfi/n{c2) with sufficiently small <5 and 77 and sufficiently large n, we may further assume that C2C3 = c 3 .

150

AF-algebras

and Ranks of C*-algebras

We now show that there is a nonzero projection e G C such that c\e = e and ecz = C3. If k = 0, dimC < 00. This is evident. If k = 1, it is also easy show such e exists. For k > 2, we note that, with a sufficiently small 5, we may assume that, using 2.5.12, without loss of generality, dj are mutually orthogonal and dj < C3. Furthermore, we may assume that n(j)

||p-$>ydiy£||
(e6.5)

for some yij e C, i = l,...,n(j),j = 1,2, ...,2k + 1. Write C — ®Lj=1PjC(Xj,Mn:J)Pj, where Pj 6 M„3.(C(.X"j)) are projections and Xj are connected compact finite CW complexes. The inequality (e6.5) implies that, for each t € Xj, C3(i) has rank at least 2k + 1. It follows from 4.8.17 that there is a nonzero projection e such that c\e = e and ec3 = C3. We have \\a\/2pa\/2

-

Cl\\

<

2*+|balP-Cl||

<

2 J + | | p a 1 p - 6 i | | + | | 6 i - c i | | < 3 ( J + e.

So Ha.]/ pa/

ea-L pa/

— e|| <3(3<5 + e).

With 6 < 1/18 and s < 1/6, by 2.5.4, Her(ai) contains a nonzero projection. • Lemma 3.6.7 Every simple unital separable C*-algebra A satisfying conditions (1) and (2) is stably finite. Proof. It follows from 2.8.10 that it suffices to show that there is an injective homomorphism h : A —> OnLi Cn/ffi^LiCn; where Cn is a sequence of unital stably finite C*-algebras. Let {xm} be a dense sequence of A. Let pn be projections and Bn be C*-subalgebras of A which are in I^k' such that lBn =pn, \\xipn -pnXi\\

< 1/2™ and dist(p

Bn) < l / 2 n (1
(e6.6)

Let 6j(n) G Bn such that ||6i(n) - p„Xipn\\ < 1/2™ (1 < i < n). Let irn : Bn —> Fn be a finite dimensional representation so that ||7rn(&i(n))|| > (1 —l/n)||6i( n )|| (i = 1, ...,n). It follows from 2.3.5 that there are contractive

151

Tracial topological rank

completely positive linear maps L'n : pnApn —• Fn such that (L'n)\Bn = nn. Let Ln(a) = L'n(pnapn) for o G A. It follows from (e6.6) that ||L n (a;2/)-L n (a:)L n (j/)||-yO.

(e6.7)

Define * : A -» rj^Li ^n by *(a) = {£„(«)} for a G A Let n : n r = i ^ n ->• n r = i ^ n / ©~=i ^ be the quotient map. By (e6.7), {Ln(xy) - Ln(x)Ln(y)} G 8~ = 1 F„ for all x, y G A Therefore n o * : -^ "*• n ^ L i -^n/ ®5S=i ^»» is a homomorphism. Since A is simple, II o * is injective. This ends the proof. • Remark 3.6.8 The embedding of A into L J ^ i F*/ ®S£=i ^ a l s o s h o w s that A is quasidiagonal, an important notion that we will only mention here. Lemma 3.6.9 Assume A is a unital simple C*-algebra and x G A is not one-sided invertible. Then, for any e > 0, there exist y G A, u G U(A) and a nonzero a G A+ such that \\x — y\\ < s and a{uy) = (uy)a = 0. Proof. Let x be an element of A which is neither left nor right invertible. So neither \x\ nor \x*\ are invertible. Let e > 0 and set g G C([0, |||x|||])+ such that g(0) — 1 and g(t) = 0 if t > e/2. Let b = fl(|x*|), c = g(\x\) and y = vf£/2(\x\)\x\, where x = v\x\ is the polar decomposition of x in A". Hence ||a; - y\\ < e. Since 0 G sp(|a;|) and 0 G sp(|a;*|), b ^ 0, c ^ 0. Then (see the proof of (iv) in 3.5.3) by =

fl(|i*|)i;/e/2(|x|)

= 0 = yc =

vfe/2(\x\)g(\x\).

If A = Mn, there is a non-zero projection e\ G Her(6) which is unitarily equivalent to a non-zero projection e2 G Her(c). In other words, there is u G U(A) such that u*e2u = e\. Note that e\y = 0 = ye2- Therefore u*e2uy = 0. Since u* is a unitary, e2uy = 0. Set a = e2. Otherwise, by 3.5.7, every hereditary C*-subalgebra of A contains at least two mutually orthogonal nonzero positive elements. By Lemma 3.5.9, there are nonzero a G Her(c) + and a unitary u G U{A) such that u*au G Her(6). Therefore u*a(uy) = 0. Hence a{uy) = 0. We also have (uy)a = 0.

• Theorem 3.6.10 Every unital TR(A) < 1 has stable rank one.

separable simple

C* -algebra

with

Proof. We may assume that A is not finite dimensional. Let x G A We will show that for any e > 0, there exists an invertible element y G A such

152

AF-algebras

and Ranks of C* -algebras

that \\x-y\\

<e.

We may assume that ||a;|| < 1 and x is not invertible. By 3.6.7, A is stably finite. Therefore x is not one-sided invertible. To show that x is a norm limit of invertible elements in A, it suffices to show that ux is a norm limit of invertible elements (for some unitary u G A). Thus, by 3.6.9, we may assume that there exists a nonzero positive element c\ G A such that xc\ = c\x = 0. Since A is simple and TR(A) < 1, by 3.6.6, there are non-zero mutually orthogonal and mutually equivalent projections pi,pi, G Her(ci). Put Ax = (1 - pi)A(l - pi) and let T = {x\. By 3.6.5, TR(AX) < 1. So, for any S > 0, there is a projection p G Ai and a C*-subalgebra C G X^1) with \c = P such that (1) \\px - xp\\ < 5, (2) pxp G<5 C and (3) lAl -p
Hx-Ori +x2)\\ <26. By 3.2.12 and 3.1.9, tsr(C) = 1. Thus there is an invertible element y\ G C C pAp such that | | a ; i - y i | | < 5 + e/4. Let v G A such that v*v = lAl — p = 1 — p\ — p and vv* < pi. Set j/2 = a;2 + (e/16)v+ (e/l&)v* + (e/4)(p! - vv*). Note that z 2 + (£/16)u+ (£/16)v* has matrix representation / a;2 ^£/16

£/16\

0 ) '

In particular, a;2 + e/16u + e/16v* is invertible in ((1— p) + vv*)A((l-p)+vv*). Therefore y2 is invertible in (1 — p)A(l—p). Hence y\ + y2 is invertible in A. Finally, if d is sufficiently small, ||3-(yi+J/2)||<e. Theorem 3.6.11 Every unital simple C* -algebra A with TR(A) = 0 has real rank zero and stable rank one.

Tracial topological rank

153

Proof. The proof is similar to that of 3.6.10. Fix a self-adjoint element x £ A and s > 0. We will show that there is an invertible self-adjoint element z £ A with \\z - x\\ < e. Let / £ C([-||x||, ||a;||]) with 0 < / < 1 and f(t) = 1 if |i| < e/8, and f(t) = 0 if |i| > e/4. If 0 £ sp(:r), then x is invertible. So we assume that 0 £ sp(ir). Thus f(x) ^ 0. Let B = Her(/(:r)). Then, by Lemma 3.6.6, there exists a nonzero p\ £ B. Let y = fe/2(x)x. Then p\y = yp\ = 0 and \\y — x\\ < e/2. Set Ax = (1 -pi)A{l - p i ) . Then TR((1 -pi)A{l - p i ) ) = 0 by 3.6.5. There is a projection p £ A\ and a finite dimensional C*-subalgebra C C A with lc = P such that (1) \\py - yp\\ < e/4, (2) pxp GE/4 C and (3) lAl-plPiSet yi = pyp and y2 = (lAl ~ p)y(lAl ~ p)- Then ||l/-(j/i+lft)||<e/2. Since C is finite dimensional, there is an invertible element y\ £ Csa such that \\pyp-yi\\

<e/2.

Let v £ A such that v*v = lAl - p = I — p\ — p and vv* < p\. Set z2 = y2 + (£/16)v + (e/16)w* + (e/4)(pi -vv*). Note that y2 + {e/16)v + (e/16)^* has matrix representation: ( Vi

£/4\

Ve/4

0) •

In particular, j / 2 + (e/16)i; + (£/16)u* is invertible in ((1 — p) + w*)A((l p)+vv*). Therefore z2 is an invertible self-adjoint element in (l-p)A(l-p). Moreover, \\z2 — 2/2II < e/4. Hence j/i + z 2 is invertible in Asa. Finally, if 6 is sufficiently small \\x - (yi + z 2 )|| < ||x - y\\ + \\y - (j/i + z 2 )|| < e. D

154

3.7

AF-algebras

and Ranks of C* -algebras

Simple C*-algebras with TR{A)

< 1

Proposition 3.7.1 If A is a unital separable simple C* -algebra satisfying conditions (1) and (2) in 3.6.2, then A admits at least one tracial state. Proof. We retain the notation used in the proof of 3.6.7. For each n, let r„ : Fn —> C be a tracial state (recall that Fn is a finite dimensional C*-algebra). Set tn = r„ o Ln. Each tn is a state on A. Let r be a weak*limit of {tn). We claim that r is a tracial state. The weak*-limit is a state on A. To see that it is a trace, fix a G A. For any e > 0, there is an integer N > 0 such that \\Ln(a*a) - Ln(a)*Ln(a)\\

< e and \\Ln(aa*) - Ln(a)Ln(a)*\\

hold for all n > N. Set bn — Ln(a). Tn(bnbn) = T „ ( 6 „ 6 * ) . Hence we have

Since r„ is a trace on Fn, we have

\\tn(a*a)-tn(aa*)\\ for all n>N.

<e

< 2e

Therefore r(a*a) = r(aa*).

D

Theorem 3.7.2 Let A be a unital separable simple C*-algebra with TR{A) < 1 and T(A) the tracial space. Suppose that p, q € A are projections. If r{q) < r(p) for all T G T(A), then q •< p. Proof. We may assume that A is non-elementary. It is easy to see that T(A) is a (weak*-) compact subset of the state space. The map x M- x(r) = T{X) for r £ T(A) gives a positive affine map from Asa to Aff(T(^4)). Set a = p — q. Then d is a continuous function. Let a = inf{d(r) : r e T{A)}. Since r(p) > r(q) for all r G T(A), o > 0. It follows from Lemma 3.5.7 that there is a nonzero projection eo € pAp such that r(eo) < cr/A and for all r G T(A). Let {xi, ...,xn,...} be a dense subset of A. Set po = p —eo- Then T(PO) > r(g) for all r G T ( J 4 ) . Let Tn = {po,q,xi,x2, ...,xn}. Since TR(A) < 1, there is, for each n, a projection Pn E A and a C*-subalgebra C„ G 1^ with lc„ = Pn such that (1) ||P„a; - xPn\\ < 1/4 for all x G ^ „ , (2) PnxPn Gi/4n C for all x G !Fn and (3) 1 - Pn < e 0 .

Simple C* -algebras with TR(A)

< 1

155

By applying lemmas in section 2.5, to save notation, without loss of generality, we may assume that there are projections pn,qn S Cn and p'n, q'n G (1 - Pn)A(l - Pn) such that ||po - (Pn + p'n) || < £ n , ||? ~ fan + 9 n ) l l <

£

«-

for all n = 1,2,... (1/2 > £„ ^ 0 as n -> oo). The above inequalities together with 2.5.1 imply that r(p 0 ) = 7"(Pn +P„), T"(g) = T(qn +q'n) for all T G T(A). Claim (for some n> N): t{pn) > t(«n) for all t G T(C„).

(e7.8)

Otherwise, there are £„ G T(Cn) such that *n(pn) <

tn(qn).

Denote by tn the state extensions of tn on PnAPn and still write by tn for tn{Pn • -P«)- Let T be a weak limit of {tn}. As in the proof of 3.7.1, r is in T{A). Since |*n(Po) - * n ( P n ) | < £ n ,

|
and also i„(p„) < tn(qn) for all n > JV + 1 , r(p 0 ) < r(q). However, r(p' n ) < r(eo) < CT/4. Therefore T

(Po)

=

T(pn + p'n) < T{qn) + r(e 0 )

<

r(q) + (7/4.

This implies that T(P)

=

T{PO)

+ r(e 0 ) < r(q) + a/2.

This contradicts the fact that r(p) —r(q) > a for all r G T"(A). This proves the claim. Now assume (e7.8) holds. It follows from 2.6.15 that in Cn there is a partial isometry v G Cn such that v*v = qn and vv* < pn for some n> i> N. Note that q — qn < e 0 . Therefore q ^ qn © e 0 -< p.

D

156

AF-algebras

and Ranks of C* -algebras

Theorem 3.7.3 Let A be a unital simple C*-algebra. Then TR{A) < k if and only if TR{Mn{A)) < k for every integer n. Proof. If TR(Mn(A)) < k, since A may be regarded as a hereditary C*subalgebra of Mn(A), by 3.6.5, TR(A) < k. Now we assume that TR(A) < k. We first show that Mn{A) has property (SP). We note that Mn(A) is also simple (see 1.11.42). Given any nonzero positive element 6 in a hereditary C*-subalgebra B of Mn(A) and a nonzero element a £ A (regarded as a hereditary C*-subalgebra of Mn(A)), there is nonzero y such that y*y £ Her(a) and yy* £ Her(6). Since Her(y*y) contains a nonzero projection, by (the proof of) (iv) of 3.5.3, Yiei{yy*) has a nonzero projection. This actually shows that every hereditary C*-subalgebra of Mn(A) contains a nonzero projection which is equivalent to a projection in A. Now let £ > 0, Q C Mn(A) and a nonzero a £ Mn(A)+. From above and 3.5.7 there are mutually orthogonal and mutually equivalent projections ei,...,e„ in Her(a) such that each of them is equivalent to a nonzero projection eo £ A. For each x £ Q, we write x = (xij) (n x n matrices with Xij £ A). Let J- be the (finite) set of those entries (in A). So {(ay) : ay £ J7} D Q. Since TR(A) < k, there is a projection q £ A and a C*-subalgebra C £ X^> with \c = q such that

(!') hv - yq\\ < e/™2 for all ye J7,

(2') qyq £e/n2 C for all y £ J7 and (3') 1A - q r< e 0 . Then, we set p = diag(g, • • • q) (q repeats n times) and C\ = Mn(C). Then C\ £ 1^ and I d = p. Furthermore, (1) \\px — xp\\ < e for all x £ Q, (2) pxp ££ C\ for all x e Q and (3) l - p = d i a g ( ( l - g ) , - - - , ( l - q ' ) ) •< e i © e 2 - - - e „ , whence [1-p] < [a].

a Corollary 3.7.4 If A is a unital separable simple C*-algebra TR(A) < 1, then Ko(A) is weakly unperforated.

with

Proof. By 3.7.3, TR(MS(A)) < 1 for all integers s > 1. To show that i^o(j4) is weakly unperforated, it suffices to show that if m\p] > m[q] for any projection p, q £ MS(A) then \p] > [q], where m > 0 is an integer. But m\p] > m[q] implies that r(p) > r(g) for all r e T(A). By 3.7.2, [p] > [g].

Simple C*-algebras with TR(A)

< 1

157

Theorem 3.7.5 If An are unital simple C*-algebras with TR(An) < k, then, for any unital inductive limit A = lim n _ ) . 00 (A n , hn), TR(A) < k. Proof. It follows from 1.11.47 that A is simple. Without loss of generality, we may assume that An c An+i and A is the closure of \J^=1An. Fix a finite subset T C A, e > 0 and a nonzero a G A+. We may assume that

Ml = iFix e > 0. We may assume, without loss of generality, that T C Am for some integer m > 0. Since TR{Am) < k, we see that A has property (SP). To complete the proof, we let e be a nonzero projection in Her(a). Therefore, we may assume that there is a projection eo G Am such that ||eo — e|| < 1/2, whence e 0 ~ e. Since TR(Am) < k, we obtain a projection p G Am and a C*-subalgebra C C Am with lc = P and C G 1^ such that (1) \\px — xp\\ < e for all x G J-, (2) pzp G£ C for all x G J 7 and (3) 1 - p ~ q < e 0 . Since eo ~ e in A, (3) implies that [1 — p] < [a]. D Definition 3.7.6 A separable C*-algebra A is said to be residually finite dimensional, (we will write A is RFD), if there is a sequence of finite dimensional representations {7rn} of A such that for any a ^ 0, there is an integer n such that 7r„(a) ^ 0. Every separable commutative C*-algebra A = CQ(X) is RFD, where X is a locally compact Hausdorff space. This is because a dense sequence {£n} C X gives a separating sequence of one-dimensional representations. It follows that A = Mn(C0(X)) is RFD. There are RFD C*-algebras that are not even amenable. Example 3.7.7 Let B be a unital separable RFD C*-algebra. Let {7Tn} be a sequence of finite dimensional irreducible representations of B such that, for any nonzero element b G B, there exists irn with nn(b) ^ 0. Suppose that 7rn has dimension k(n). We assume that in the sequence {7rn} each irreducible representation repeats infinitely many times. For each n, define V>„ : B -»• Mk(n){B) by the composition: B % Mk{n) idHB Mk{n){B). Consider the following sequence of homomorphisms. Fix {ITI, ..., 7f;(i)}. We define a homomorphism hi : B —¥ M/(2)(-B), where 1(2) = 1 + X)j=i M*)> by b !->• diag(6,V'i(^), •••, ipi(i)(b))

158

AF-algebras

and Ranks of C* -algebras

for all b € B. Suppose that hm : M/( m ) (B) ->• M/( m + 1 ) (5) has been defined. Choose {7Ti, ...,7T/(m+1)} and let i/>„,m+i = ipn® l / ( m + i ) : M / ( m + 1 ) ( B ) -*• M f c ( 7 l ) / ( m + 1 ) (B). Define / i m + 1 : M / ( m + 1 ) ( B ) ->• M / ( T O + 2 ) ( B ) by 6i->-diag((id B <8)id / ( m + 1 ) )(6),'0i i m + 1 (6),...,V; / ( m + 1 ) i m + 1 (6)) for all b e M / ( m + 1 ) ( 5 ) , where I(m + 2) = / ( m + 1)(1 + £*L™+1) fc(*))We consider the C*-algebra A = lim n _ >00 (M/( m )(.B), /i m ). In what follows, we will use /i„, m : MI{n)(B) ->• M / ( m ) ( S ) and / i ^ : MI{n)(B) -> A for the monomorphisms induced by this inductive system. Furthermore, we let Bn = /ioo(M /(n) (B)). Proposition 3.7.8 A is always simple.

Let A be the C*-algebra constructed in 3.7.7. Then

Proof. We use the notation defined in 3.7.7. By Lemma 3.5.10, it suffices to show that / f~l Bn = {0} or J n Bn = Bn. Let x € I D Bn be a nonzero element. There is y e M/( n )(B) such that hao(y) = x. By the construction of {7rm}, there is 7rm such that 7rm ® id.MHnAy) ^ 0. Let n + I > m. By considering the composition hn+i o hn+i-i o • • • o hn+i, we may write hn,n+i(y) = H(y) © {irm ® id M/(Tl) ® id M j )(y) for some positive integer J and some homomorphism H. Since ipm
Lei A be as in 3.7.7. Then TR(A) = 0.

Proof. Let T be a finite subset of A. Without loss of generality, we may assume that T C /ioo(M/( m )(5)). There is a finite subset Q c M/( m )(B) such that hCK>(Q) D T. Since J(m+1)

J(m + 2) = J(m+l)(l + J 3 fcW) ^ 2 7 ( m + 1)> i=\

for any integer n > 0, we can choose I large enough so that I(m + l)/I(m) n + 1. Let hm,m+i = hm+i-i ° hm+i-2 ° • • • ° hm. Then we have hm,m+l(x)

=o(x)

®${x)

>

Simple C* -algebras with TR(A) < 1

159

for all x € MI(m)(B), where a : MI{m)(B) -»• (1 - p)M / ( m + i ) (J3)(l - p) is a unital injective homomorphism such that 1 — p = X V e"> where {e^} is the set of matrix units for Mr^m+i), and $ : M^m)(B) ->• F, where F is a finite dimensional subalgebra of pMj^m+i)(B)p with lj? = p. Note that I(m + l)>{n+ l)I(m). So n[l-p]<[p]<[lBJ We see that C = h^F) and

(inB„).

is a finite dimensional subalgebra with \c — hOQ(p)

n[/ioo(l - p ) ] < [/ioo(p)] in A. We also have [!/,MP)]=0 for all y £ J-. The above also shows that the simple C*-algebra A has property (SP). To complete the proof, it remains to show that for any nonzero projection e, we can choose p so that /ioo(l — p) is unitarily equivalent to a subprojection of e. To do this, without loss of generality, we may assume that e G Bk- So there is a projection q £ B ® Mj^k) s u c n that /ioo( k. We may write hk,m(l) = d i a g < y > 0. In the above we could choose I so large that I(m)/I(m

+

l)
also holds. Then it is clear that 1 — p is unitarily equivalent to a subprojection of hmtm+i(q") (in M/ ( m + /)). Since (q") < hk (q), we conclude that hx{l — p) is unitarily equivalent to a subprojection of e. • Proposition 3.7.10

A has a unique tracial state.

Proof. It follows from 3.7.1 that A admits at least one normalized trace. To show that there is only one normalized trace, we let t\,t2 6 T(A) and take a £ Bm. In what follows, to save notation, we will identify Bn

160

AF-algebras

and Ranks of C -algebras

with B ® M/(„). For any integer k > 0, choose a large I so that I{m)/I(m

+ l)
+ l).

Then we may write a = diag(a',a") in B M/( m + i) such that a1 & qB ® M/( m + 1 )g and a" G M/( m + ;), where g = 5^i=T ea a n ( l { e ij} i s the set of matrix units for M/( m + /). Note that U\MHm+[) = tr (i = 1, 2), where tr is the normalized trace on M/( m + ;). We also have ti = Si ® tr, where «i = i i | s (z = 1,2). Therefore we have | 0, we conclude that ii(a) = ^ ( a ) - Thus there is only one normalized trace on A. • Example 3.7.11 Let us consider a very special case. Let B = C(S1). Fix a dense sequence {z n } of the unit circle S1. Define h\ : B —>• M2(B) by /i 1 (/) = diag(/,/(z 1 )). If /ife : M( fc )i(5) ->• M( fc+ i)i(B) is defined, define /i fe+ i : M( fc+1 )|(B) -> M ( f c + 2 ) ,(5) by ^fc+i(/, /( z 0> / ( ^ ) , •••, f(zk)), where /(ZJ) is the evaluation of / at the point Zj (i = l,...,k), where we identify Mk\(B) with C(S\Mk\). Set A = lim„^ 00 (M fc! (C(6' 1 )),/i fc ). This is a very special case of algebras discussed in Example 3.7.7. In particular, this A is simple and has a unique tracial state. It is an AT-algebra.

3.8

Exercises

3.8.1 Let F be the Cantor set in [0,1]. Show that C{F) is an AFalgebra. 3.8.2 Let A be a C*-algebra with RR{A) = 0 and tsr(-A) = 1. Show that, for any hereditary C*-subalgebra B
= 0.

Exercises

161

3.8.5 Prove part (2) of Theorem 3.1.9. 3.8.6 Show that inductive limits of C*-algebras with real rank zero have real rank zero. 3.8.7 Complete the proof of 3.2.10. 3.8.8 Prove Proposition 3.2.12. 3.8.9 Prove Proposition 3.3.4. 3.8.10 Let A be a unital C*-algebra. Show that RR(A) = 0 if and only if for any e > 0 and for any pair of orthogonal positive elements a,b G A+ there is a projection p € A such that \\pa — a\\ < e and (1 — p)b = b. 3.8.11 Let A be a C*-algebra with RR(A) = 0. Show that RR(A®K)

=

0. 3.8.12 Let A be a C*-algebra and / C A be an ideal. Suppose that RR(A) — 0 and p G A/I is a projection. Show that there is a projection q G A such that n(q) — p, where n : A —>• A/1 is the quotient map. 3.8.13 Let A be a C*-algebra and J C A be an ideal. Suppose that tsr(A) = 1 and u G U(A/I). Show that there is a unitary v € A such that 7r(i>) = u. 3.8.14 Let A and B be two unital AF-algebras. Suppose that there is an order homomorphism a : KQ{A) —> KQ(B) such that a([l^]) = [1B]. Then there is homomorphism h : A —>• B such that /i* = a. 3.8.15 Construct a unital simple AF-algebra such that KQ(A) = Q © Z and ifo(;4)+ = {(r, z) : r > 0 and z 6 Z } U {(0,0)}. 3.8.16 Give an example to show that in a unital simple C*-algebra A, r(p) = r(q) for all normalized traces T does not imply that p ~ q. 3.8.17 Let A be a simple C*-algebra and a € A with ||a|| = 1. Suppose that z\, Z2,..., zn G A+ and 0 < e < 1/4. Then there is b G A+ such that 0 < 6 < / £ ( a ) , H&l = 1 and [b] < [Zi], i = 1, 2, ...,n. 3.8.18 Let A be a separable C*-algebra. Show that there exists a separable simple C*-algebra B such that there is an embedding j : A —» B. 3.8.19 Show that every simple AT-algebra A has TR(A) < 1. Moreover, if A is an (simple) inductive limit of C*-algebras in ZW, then Ti?(A) < k.

162

AF-algebras

and Ranks of C -algebras

3.8.20 Let A = lim„_>.00 An be an inductive limit of C*-algebras An. If for any n and every a, b, c,e,f£ (An)+ with ab = a, be = b, dc = c, ed = d and fe = e, there is a projection p € A such that ap = a and cp = p, then RR{A) = 0. 3.8.21 Let A be a unital simple C*-algebra with TR(A) that TR(B) = 0 for every hereditary C*-subalgebra B of A.

= 0. Show

3.8.22 Construct a unital separable simple C*-algebra with TR{A) = 0 such that Ki(A) = Z, K0(A) = Q © Z and K0(A)+

= {(r,z) :r > 0 and z £ Z} U {(0,0)}.

3.8.23 Let A be a unital C*-algebra with real rank zero. Let 0 < a, b < 1 in A such that 06 = a. Then there is a projection p £ A such that such that ap = a and frp = b. (L. G. Brown). 3.8.24 Let A be a non-elementary simple C*-algebra with TR(A) = 0 and p £ A be a projection. Suppose that eo S A is a nonzero projection and n > 0 is an integer. Then there are mutually equivalent and mutually orthogonal projections pi,...,pn < p such that (p — Y^=iPi) — Pi a n ^ (p — ^2i=iPi) — eo- (S- Zhang showed that the same is true for simple C*-algebras with real rank zero).

3.9

Addenda

Theorem 3.9.1 (S. Zhang, [182] and [27]) / / I is an ideal of a C*algebra A, then RR(A) — 0 if and only if RR(I) — RR(A/I) = 0 and every projection in A/1 lifts to a projection in A. Definition 3.9.2 Let A be a C*-algebra. A quasitrace r on A is a function from Moo (A) —• C which is linear on commutative subalgebras and satisfies 0 < T(X*X) = T(XX*) for all x £ M^A). Denote by QT(A) the normalized quasitraces. Equipped with the weak*-topology, QT{A) is a compact metrizable space. Denote by Aff(QT(A)), the set of all continuous affine functions on QT(A)). Let A be a, stably finite C*-algebra. Then the map p : -Ko(A) -> AS(QT(A)) defined by p(\p\) = T(J>) is a homomorphism. Theorem 3.9.3 Let A be a simple C* -algebra with TR{A) = 0. Then the range of p : KQ(A) —> Aff(QT(A)) is uniformly dense.

163

Addenda

Definition 3.9.4

Let 0 < o\ < o2 < 1 be two positive numbers. Define

{

1

if t > a2

linear if ax < t < a2 (e9.9) 0 if 0 < t < (7i Definition 3.9.5 A unital C*-algebra is said to have tracial topological rank no more than k if for any e > 0, any finite subset T containing a nonzero element b > 0, any 0 < 03 < 04 < o\ < a2 < 1, any integer n > 0, there exists a nonzero projection p G A and a C*-subalgebra B G 1^ with lB = p such that (1) ||[z,p]|| < £ for alia; G .F, (2) pxp G£ B for all x € J- and

(3) [/£((! -P)b(l -P))] < [/£(ptp)]. If A has tracial topological rank no more than A;, we will write TR(A) < k. If furthermore, if TR{A) % k - 1, then we say TR(A) = k. A non-unital C*-algebra A is said to have TR(A) = k if A has TR(A) = k. Theorem 3.9.6 Let A be a separable commutative unital C* -algebra. Then the following are equivalent. (i) A = C(X) for some compact HausdorfJ space X with dimX = n. (ii) TR(A) = n. Proposition 3.9.7 Let A be a unital C*-algebra with TR(A) = k. Then TR(A/I) < k for every (closed) ideal I of A. If A is separable and TR(A) = k, then TR(A/I) < k for every ideal I of A. Theorem 3.9.8 Let TR(A) = k and B be a unital hereditary C*subalgebra of A. Then TR{B) < k. Theorem 3.9.9 Let k > 0 be an integer. A unital C*-algebra A has TR{A) = k if and only ifTR{Ms{A)) = k for every positive integer s. Theorem 3.9.10 Every unital simple C*-algebra A with TR(A) = k has the the following Fundamental Comparability: if p, q G A are two projections with r(p) < r(q) for all tracial states r on A, then p ^< q. Theorem 3.9.11 Every unital simple C*-algebra A withTR(A) stable rank 1 and real rank no more than 1.

= k has

164

AF-algebras

and Ranks of C* -algebras

Theorem 3.9.12 (L. Brown,[l6]) Let B be a a-unital hereditary C*subalgebra of a a-unital C* -algebra A. Suppose that the ideal generated by B is A. Then A®K,^B®K.. Theorem 3.9.13 Let Abe a separable C*-algebra with tsr(A) = 1. Suppose that B is a hereditary C* -subalgebra of A. Then tsi(B) = 1. Theorem 3.9.14 (Effros, Handelman and Shen, [51]) Suppose that G is a countable unperforated ordered group satisfies the Riesz interpolation property. Then there is an AF-algebra A such that (KQ(A),KO(A)+) = (G,G+). Deflnition 3.9.15 For n > 2, the Cuntz algebra On is the universal C*-algebra generated by isometries si,..., sn such that ^Z™=i sisi = 1Theorem 3.9.16 (Cuntz, [35]) (1) On is a unital purely infinite simple C*-algebra. (2) K0(On) = Z/(n - 1)Z and Ki(On) = {0}. Theorem 3.9.17 ([8]) Let Abe a unital amenable simple C*-algebra with TR(A) = 0. Then A is an inductive limit of amenable RFD-algebras.

— 759AD - 4 it (701 - 762)

Chapter 4

Classification of Simple AT-algebras

In this chapter we will classify unital simple AT-algebras of real rank zero up to isomorphism. The isomorphism invariant of such simple C*-algebras is the scaled ordered i^o-group together with the -ftTi-group. This is a very important result of George A. Elliott. In section 2 we will show that unitaries in U(A)o for a unital C*-algebra with real rank zero are norm limits of unitaries with finite spectrum. From this we will establish a so-called uniqueness theorem (4.5.5) which plays the key role in the proof of the Elliott theorem. We begin with some basics about AT-algebras. 4.1

Some basics about ^4T-algebras

Lemma 4.1.1 Let e > 0. Then there is 5 > 0 satisfying the following: For any unital C*-algebra A andx 6 A if \\xx* —1|| < <5 and \\x*x — 1|| < S, then there is a unitary u £ A such that \\u — a;|| < e. Proof.

The proof is similar to that of 2.5.3. We leave it to the reader. •

Lemma 4.1.2 Let A be a finite direct sum of circle algebras. Then there are C* -subalgebras Ak such that (i) each Ak is a finite direct sum of C*-algebras of the form M m (C(X)), where X is either Sl, a (compact) arc of the circle or a point in S1, (ii) Ak C Ak+i, fc = 1,..., and Uk%1Ak is dense in A. Proof.

We leave this to the reader to verify (4.8.3). 165

•

166

Classification

of Simple

AT-algebras

Definition 4.1.3 Let B = Mni{C{Xl))®---®Mn,{C{Xa)), where Xt is a compact subset of S1. By a set of standard generators of B, we mean the set of standard generators {e' • : i,j = 1,2, ...,ni,l = 1, ...,s} of Mni ©• • •© Mnt and standard unitary generators of vi of elnMni(C(Xi))elu (= C(X{)) (vi(z) = z for z G X;). Lemma 4.1.4 i e i A be a C*-algebra, A0 = M ni (C(A"i)) © ••• © M„a(C(JsCs)), where Xi is a connected compact subset of S1, i.e. the whole S1, a (compact) arc of the circle, or a point (I = 1, ...,s), and h : Ao —• A be a homomorphism. Then for every e > 0, and every finite subset J- C AQ containing the standard generators of Ao, there exists 6 > 0 such that, whenever Ai is a C*-subalgebra of A with dist(a;,i4i) < 5 there exists a homomorphism

for x G h(!F),

<j) : AQ —>• A\ such that 4> w e h

on T•

Proof. We may assume that C = h(A0) = Mni(C(Yx)) © ••• © MnmC(Ym), where m < I and Yj is a compact subset of X[. Let {e\j, ui : i,j — l,...,ni,l = l,...,m} be a set of standard generators for C. Let 5\ = S(e/6) as in 4.1.1 and 82 = 5(min(Ji,e/6),dimjB) as in Lemma 2.5.10 (associated with B = Mni © • • • © M„ m ) and Let 6 = min(<5i/2,52/2). By assumption, dist(e' J -,i4i) < S and dist(ui,Ai)

<S

for i,j = l,...,n/, Z = l,...,m. Then by Lemma 2.5.10, there is a unitary w € A such that wBw* C J4I and ||w — 1|| < min(<51,e/6). Let b[ = welnw* G Ai. Then there is c; G b[Aib[ such that ||wu;w* — c;|| < (J2/4. It follows from 4.1.1 that there is a unitary vi G U(b[Aib[) \\wuiw* — vi\\ < e/3. Therefore sp(^) c f z e S 1 : dist(z,sp(w;)) < e/3}. Thus sp(w;) c {z G 5 1 : dist(z,X;)) < e/3}.

such that

Some basics about

167

AT-algebras

Since Xi is S1, an arc or a point, there is fi £ C(sp(i>/)) such that \fi(z) - z\ < e/3 and /i(sp(v/)) C Xi

for z <E sp(u/),

Z = l,...,m. Set it;/ = fi(vi). Then ||W/ - M / | | < £ .

There is a homomorphism /i; : C(X;) -> C*(w;) which maps ui to «;/. From this we obtain a homomorphism (f> : AQ —• A'0 C A\ which maps e\ • to we\ -w* and the standard unitary generators of e 1 1 C(X/)e 1 1 to wi, i,j = 1, ...,ni,l = 1, ...,m (and i£m < s, maps the rest to zero). • Theorem 4.1.5 yl unital separable C*-algebra is an AT-algebra A if and only if for any e > 0 and any finite subset J- C A, there exists a finite direct sum of circle algebras B C A such that x e£ B

for all

i6f.

Proof. By the definition 2.3.11, if A is an AT-algebra, then A = limn_>oo(j4n, hn), where An is a finite direct sum of circle algebras. Set Bn = hniOC(An). Then each Bn is a circle algebra. Since U^L 1 i? n is dense, we see that the "only if " part of the theorem follows. Moreover, from 4.1.2, we may assume that each B in the statement of 4.1.5 is of the form Mni (C(Xi)) ® • • • © Mn,(C(Xs)), where each Xt is S \ an arc, or a point. Let {a^} be a dense sequence of elements in the unit ball of A and let {e/t} be positive numbers such that Y^kLi £k < °o. Furthermore, there exists A\ = Mni{C(X\))®• -®Mn3(C{Xs)), where Xi is a connected compact subset of S1, such that a\ G£l A\. Suppose that we obtain A\,...,Ak, each of which is a finite direct sum of circle algebras with connected spectrum, and homomorphisms fc : Ai —>• Ai+i, i = 1,..., k — 1 such that there are bm G Ai (m = 1,..., i) with dist(a m ,&W) < £ j / 2 , m = l,...,i, and \\4>i{x) - x\\ < et,

i=

l,...,k-l

for all x in a finite subset Ti which contains the standard generators of Ai and contains bm (m = 1, ...,i). There is Ak+\ such that dist(a m ,A f c + i) < £ fc+ i/2 and dist(z,Afc +1 ) < £ fc+ i/2

168

Classification

of Simple

AT-algebras

for m = 1,..., fc+1 and for all x £ Tk which contains the standard generators of Ak and {bm> : m = 1,..., k}. Applying 4.1.4, we obtain a homomorphism 4>k '• Ak —>• Ak+i such t h a t

k(x) « £fc+1 id(a;)

on J^.

Let C = lim71_>00(yl„, <j>n). Denote by ji : Ai ->• A the embedding. Thus (see 1.10.13) we construct the following diagram, which is (one-sided) approximate intertwining: A\

—>

vjl

A2

—>•

4-J2

A3

—>

• ••

C

4-J3

A Jl> A A

... ^

,4 A

This gives an homomorphism h from C to A. Since <j)n(bm') —• a m as n —> 00 for m = 1,..., we see that the map is also surjective. Therefore U^_1/i(Afc) is dense in A, whence A is an AT-algebra. • Lemma 4.1.6 For any e > 0 and any integer m > 0, i/iere ezists 5 > 0 satisfying the following: If A = A\© • • • © ^4 m , where each Ai is a unital C*-algebra, if L : A —» B is an J- -5-multiplicative contractive completely positive linear map, where B is a unital C*-algebra and T D IAI, • ••, ^-Am, then there is an J--s-multiplicative contractive completely positive linear map 4> : A —> B such that ^ ( l ^ ) , . . . , \\<£. Proof. Let L : A -> B be an J^-iJ-multiplicative contractive completely positive linear map and let a^ = L(l J 4 i ) (i — l,...,m). Then ai &s Q-i and ajOj < 5

if i ^ j .

Fix 77 > 0. It follows from 2.5.5 and 2.5.6, with a sufficiently small 5, that there are mutually orthogonal projections pi, ...,pm G B such that \\L{lAi)-pi\\


i=

l,...,m.

Define Li : Ai —> PiBpi by Li{a) = piL(a)pi for a € Ai, i = 1, ...,m. Then Lj are contractive completely positive linear maps. With 77 < 1/2, 2^(1^)

Some basics about

AT-algebras

169

is invertible. Let bi = (p i L(l > i i )pi) * mpiBpi and let L'^a) = bj for a G Ai. Then \\Li{a) - L[{a)\\ < \\PiL(a)Pi for all aeAi.

Lj(a)&/

- 6 l 1/2 L i (a)6 l 1/2 || < 2|| P i - 6! / 2 ||||i(a)||

Define 4>{a) = YT=i L'i(a)

for a e A

-

Then

||L-0||
•

T h e o r e m 4.1.7 Let A = Mni{C{X{))®- • -®Mni(C(Xi)), where Xt c S1 is connected and compact. For any e > 0 and any finite subset T C A, there exists S > 0 and a finite subset Q C A satisfying the following: If L : A —> B (where B is a unital C* -algebra) is a Q-5-multiplicative contractive completely positive linear map, then there exists a homomorphism h : A —¥ B such that h ~e L

on T'.

Proof. It follows from 4.1.6 that we may assume that A = Mn(C(X)), where X is S1, an arc or a point. Let {«, e^ : i, j = 1,..., n} be the standard generators of A. Let x = L(u) and a^ = L(eij), i,j = 1, ...,n. As in the proof of 4.1.4, for any 77 > 0, if 5 is small enough, there is a system of standard generators {v, e't • :, i, j = 1,..., n} of Mn(C(Y)) for some compact subset Y c S1 in B such that \\e'ij ~aij\\


\\v-x\\


sp(u) C {z G S 1 : dist(2i, X) < 77}. Since X is a connected subset of S1, there is / G C(sp(v)) such that |/(z) — z\ < 77 for z G sp(v) and /(sp(v)) c Y. Set w = f(v). Let u and {e^}™ • be the standard generators of Mn(C(X)). Now define a homomorphism h : Mn(C(X)) —>• S by /i(w) = w and h(eij) = e[j (1
L ~e h

on T.

•

170

Classification

of Simple AT-algebras

Proposition 4.1.8 Assume that A = lim n ^. 00 ( J 4 n , „) is a nonelementary unital simple C*-algebra, where An = (BrjZiC(Xnj,Mn^)) and Xnj is a connected CW complex such that dimXnj < K for all n, j and for some K > 0. Let a G (An)+ be nonzero. Then there is 6 > 0 and mo > n such that for all m > mo, every j = 1,..., rm and t G Xmj we have (l/m(j))Tr(<j>^m(a(t)) Proof. that

> 6.

Because A is simple there are mi > n, N G N and Xj G Ami

so

N

(see 3.3.6). Let 5 =

1 N

U—^.

It follows that for m > m\, all j and

Tr(^m(a(t))

4.2

> m(j)S.

Q

Unitary groups of C*-algebras with real rank zero

By 3.2.5, a C*-algebra A has real rank zero if and only if every selfadjoint element can be approximated by selfadjoint elements with finite spectrum. What about unitaries? Can unitaries in a unital C*-algebra with real rank zero be approximated by unitaries with finite spectrum? Exercise 3.8.22 shows that a unital AT algebra A with real rank zero may have a nontrivial K\{A). Proposition 4.2.1 shows that if u £ U(A) \U(A)o, then u can not be approximated by unitaries with finite spectrum. So the appropriate question is: can unitaries in U(A)o in a unital C*-algebra A with real rank zero be approximated by unitaries with finite spectrum? In this section, we will give an affirmative answer to this question. Proposition 4.2.1 Let A be a unital C*-algebra and u G U(A) \ U(A)Q. Then u can not be the limit {in the norm topology) of unitaries with finite spectrum. Proof. Suppose that v G U(A) with finite spectrum such that v = S r = i ^iPi> where Aj G S1 and pi,p2, ••-,pn are mutually orthogonal projections. Then there are t\,...,tn G [—7r,7r] such that Xj = eltj. Set

Unitary groups of C* -algebras with real rank zero

171

h = Y^i=itjPj- Then ||/i|| < 7r and h £ Asa. Furthermore, exp(ih) = v. By considering exp(ith) (t £ [0,1]), we see that v £ U{A)Q. If u g' U(A)o, then u can not be the limit of unitaries with finite spectrum since U(A)o is closed in U(A). • Definition 4.2.2 Let : [0, a] -4 X be a continuous map (a path), where X is a normed space. Let V = {0 = to < t\ < • • • < tn = a} be a partition of [0,o]. Put L ( M P ) = £ ? = i || W O - 0(ti_i)||. We define the (rectifiable) length of (j> to be L(^) = sup{L(^)("P) : V,partition

of [0,a]}.

Note the length of could be infinite. Let A be a unital C*-algebra and u £ U(A)o- Define cel(u) = mi{L{4>M • [0,1] -* U(A)0, (0) = 1, (!) = u}

Lemma 4.2.3 Let A be a unital C*-algebra, u, v £ U(A). Suppose that A £ sp(u). Then there is fi £ sp(i>) such that \\ — fx\ < \\u — v\\. Proof. By replacing u by Xu and v by Xv, we may assume that A = 1. Thus it suffices to show that dist(l,sp(i;)) < ||u — u||. Otherwise, ||(1 _ „ ) _ ( ! _

U )||

= ||U _ „|| < dist(l,sp(i;)) = 1/||(1 - v)- 1 !!

(see 1.11.24 for the last equality). It follows from 1.1.10 that 1 — u is invertible. We reach a contradiction. • Lemma 4.2.4 Let A be a unital C*-algebra and t H4 u(t) (for t £ [0, a]) be a continuous path in U(A) such that u(0) = 1. Let L be the length of the path. Then sp(u(a)) C {eie :-L<9
n

Y,\Xi~ ^-i\<^2\HU)-u(ti-i)\\ i=l

i=l


(e2.1)

172

Classification

of Simple

AT-algebras

Thus n

limsup

VJ |A; — Aj_i| < L.

max; | t i - t i _ i | - > 0

i = l

If max, \ti — U-i\ —> 0, then max; |Aj - Aj_i| < max; \\u(U) -u(U_i)\\ —> 0. Then the supremum of the left side of (e 2.1) (taking among all partitions) is at least the length of the arc from A0 to 1. Thus the length of the arc from A0 to 1 is bounded by L. It follows that A0 = el9 for some 9 G [-L, L\. D Lemma 4.2.5 Let Abe a unital C*-algebra and u G U(A). For any e > 0 there exists h G M2(A)sa such that \\u(Bu* — exp(ih)\\ < e. Moreover ce\(u © u*) < IT. Proof.

Define , . _ /u ^'~\0

U

0 \ / cos t l) V - s i n i

sin t \ ( u* 0 \ / cos t c o s t / l^ 0 l j ^ s i n t

— sin t cost

for t G [0,7r/2]. By the product rule, we compute that v(t)' = z\(t) + Z2(t), where . . Z l ( i ) =

*2^

(u U

=

(u V0

0 \ / — sin t cos t \ ( u* 0 \ / cos t 1 I -cost -sint 0 1 sint 0 \ ( cost 1/ V-sint

s i n t \ (u* cost/ ^ 0

0 \ / — sint I / V cost

— sin A costJa

n d

—cost -sint

Both z\(t) and Z2{t) are unitaries for all t. Therefore ||zi(t) + ^2(t)|| < 2. Hence \\u(t)-u(s)\\<2\t-s\ for all t, s G [0,7r/2]. Thus the length of v|[o,Q] is less than n for any a < TT/2. It follows from 4.2.4 that - 1 £ sp(v(aj) for all a < TT/2. By 1.11.5, u(a) = exp(ih) for some h G M2(A)sa with ||/i|| < 7r. This ends the proof.

•

Unitary groups of C -algebras with real rank zero

173

Lemma 4.2.6 Let F be a proper closed subset of the unit circle S1. Then for any e > 0, there is S > 0 satisfying the following: For any unital C*'algebra A with real rank zero, two nonzero orthogonal projections p, q £ A and u £ U(pAp), if there exist v £ U(qAq) with sp(v) C F (as an element in qAq) and w £ U((p + q)A(p + q)) with finite spectrum such that \\u®v -w\\ < S, then there is w' £ U(pAp) with finite spectrum such that \\u-w'\\

< £.

Proof. Suppose that S1 \ F contains an arc with length at least d > 0. Fix a closed subset ft of S1 \ F which contains an arc with length > d/2. Let / and g be two continuous functions defined on S1 such that 0 < / < 1, f(z) = 0 if z £ F, f(z) = 1 if z £ ft, and 0 < g < 1 if z £ F, g{z) = 0 if z £ ft, and f{z)g(z) = 0 for all z £ S1. For any 6 > 0, by (2) in 2.5.11, there is 0 < T] < S such that for any two unitaries u', u" in A if \\u' — u"\\ < r], then | | / ( U ' ) - / K ) | | < * a n d \\g(u') - g(u")\\ < 5. Now suppose that A is a unital C*-algebra with real rank zero, p and q are mutually orthogonal projections in A, u £ U(pAp), v £ U(qAq) and w £ U((p + q)A(p + q)) such that ||u©w — w\\ < r), where sp(v) C F (as an element in qAq) and sp(w) has only finite many points. We may assume that n i=l

where {pi : 1 < i < n} is a set of mutually orthogonal projections in A and and |Aj| = 1, i = 1,2, ...,n. Set wi = ^ A . e n A j p i , w2 — J2x^a^iPi r = J2\t£nPi- Since r commutes with w, q(u © v) = qv = vq and g(v) = q, we have f(w)r = rf{w) = r and g{u © v)q = qg(u © v) = q

174

Classification

of Simple

AT-algebras

Therefore \\rq\\ = <

||r/(io)fl(u©i;)g|| \\rf(u © v)g(u © v)q\\ + \\r(f(w) - f(u © v))q\\ < 6.

Consequently, \\r — prp\\ < 25. If 6 < 1/4, by 2.5.4 and 2.5.1, there exists a projection r' < p such that ||r'-r|| <2J and there exists a unitary u\ £ (p + q)A(p + q) such that ll u i ~ {p + l)\\ < 4(5 and u^rui = r'. Thus ||uiU>iWi —wi|| < 8(5 and ||ujW2"i — w2\\ < 8(5. Then ||pw2 - w2p\\

<

||pwi - wip|| + \\pw - wp\\

<

\\prwi — wirp\\ +2r) < 2(6 + 77).

We estimate \\p(ulw2ui) - (ulw2ui)p\\

< 166 + \\pw2 - w2p\\ < 32(5 + 2rj.

Put p' = p — r'. Since r'u[w2ui

= u\w2u\r'

= 0,

We obtain \\V'{u\w2Ul)

- (uItU2Ui)p'|| < 32(5 + 2??.

Since Q. contains an arc with length d > d/2, by 2.6.11, if 6 is small enough (and depends only on d), there is h £ Asa such that

Hp'K™2Uly - (p' + f; te^)n < e/4. 71. n=l

Unitary groups of C* -algebras with real rank zero

175

Then (with 6 < e/2),

71=1

Since p'Ap' has real rank zero (see 3.2.6), there are mutually orthogonal projections qi,...,qm in p'Ap' with J^iLili = P' anc * complex numbers a\, ...,am with \ai\ = 1 such that

n=l

i=l

Hence (note that p' = p — u\ru\) m u

^i(. \Pi l) ~ X ] ai9i H

||U - ^

u

AiGfi

< £

-

i=l

[J

Lemma 4.2.7 Let A be a unital C* -algebra with real rank zero and u € U(A)o- Then for any e > 0, there is an integer k > 0, and there are unitaries v e Mk(A) and w € Mk+i(A) with finite spectrum such that \\u®v - w\\ < e. Proof. For any e > 0, there are u = uo, u\, ...,u m + i = 1 in U{A)Q along a path connecting u to the identity such that \\ui - Ui + i|| < e/3, i = 0, l,...,m. Let u' = diag(ui, ui, u\, u2,..., < , , u m , 1) be a diagonal unitary in M2TO+i(A). It follows from 4.2.5 that there is hi £ M 2 m + i ( A ) s a such that \\u' — exp(ihi)\\ < e/6. Since RR(M2m+i{A)) = 0, hi can be approximated by selfadjoint elements with finite spectrum. Therefore there are mutually orthogonal projections qi, ...,9s € M2m+i{A) and complex numbers ot\, ...,as, |a»| = 1, i = 1, ...,s, such that s \\u' -^caqiW i=l

< e/3.

176

Classification

of Simple

AT-algebras

Put u" = diag(u, u*,uu u\,..., umi, < , - i , um, u*m), a diagonal unitary which is in M 2m+ 2(^4). Then, by the above construction, s

s

\\u" - u © 5 3 a ^ i | | < | | w " - u © u ' | | + \\u' -^atqiW i=l

< e/3 + e/3 = 2e/3.

*=1

By 4.2.5 again, there is /i 2 G M2m+2(>l)sa such that ||u"-ea;p(t/ia)|| < e/6. Since i?i?(M2m+2(^4)) = 0, there are mutually orthogonal projections di,...,d,N G M2m+2(^4) and complex numbers /3I,...,/3JV with |/3j| = 1 such that N

||u"-^/3^||<£/3. .7=1

Therefore s

AT

a

iiu©53 **-53^'djii<£i=l

j=l

So we can take k = 2m + 1, v = 5Z*=1 ccjQi and w = X)i=i Pjd.

D

The following is the main result of this section. Theorem 4.2.8 Let A be a unital C* -algebra. Then RR(A) = 0 if and only if every unitary u G U(A)o can be approximated (in norm) by unitaries in A with finite spectrum. Proof. We will show that if RR(A) — 0 then every unitary in U(A)o can be approximated in norm by unitaries with finite spectrum. The other direction is easy and is left to the reader as an exercise (4.8.4). Suppose that u G U{A)Q. By 4.2.7, for any S > 0, there exist an integer k and unitaries v G Mk(A), w G Mk+i(A) with finite spectrum such that | | u © v - - w | | < 6. Let F be the part of the unit circle which lies in the closed right half plane. Since the spectrum of v has only finitely many points, there is a projection p G Mk(A) such that p commutes with v. Moreover, sp(pv) C F if we view

Simple AT-algebras with real rank zero

177

pv as an element in pMk(A)p, and sp((l —p)v) C S1 \F, if we view (1 —p)v as an element in (1 - p)Mk(A)(l — p). So by 4.2.6, for any r\ > 0, if S is small enough, there is a unitary w' G [i-Mk © (1 ~ p)]^k+i(A)[lMk © (1 — p)] with finite spectrum such that ||lt© (1 ~p)v -w'\\

< T}.

Since sp((l -p)v) C Sl\F, we may apply 4.2.6 again. Thus, for any s > 0, if both S and rj are small enough, there is a unitary u' G A with finite spectrum such that | | u - u ' | | < e. D Corollary 4.2.9 Let Abe a unital C*-algebra with RR(A) = 0 andp G A be a projection. Suppose that u G U(pAp) and u + (1 — p) £ U(A)o. Then u G U(pAp)0. Proof. By 4.2.8, u + (1 — p) can be approximated by unitaries in A with finite spectrum. It follows from 4.2.6 that u can be approximated by unitaries in pAp with finite spectrum. Thus u G U(pAp)o. • Corollary 4.2.10 Let A be a C*-algebra with RR(A) = 0. Then the map U(A)/U(A)o -¥ Ki(A) is injective. Proof. Without loss of generality, we may assume that A is unital. Let u G U(A). It suffices to show that if z = diag(u, ljvffc) G U(Mk+i(A))o, then u G U(A)0. But this follows from 4.2.9. • Corollary 4.2.11 Let A be a C*-algebra with RR(A) = 0 and tsi(A) = 1. Then U(A)/U{A)0 = K^A). Proof.

4.3

This follows directly from 3.1.10 and 4.2.10.

D

Simple AT-algebras with real rank zero

Lemma 4.3.1 Let A be a simple C*-algebra of stable rank one. For any x G K\(A), and projection p G A, there exists u G U(pAp) such that [u + ( 1 - p ) ] =x inKi(A).

178

Classification

of Simple

AT-algebras

Proof. By 3.1.10, there is a unitary z G U(A) such that [z] — x. Since A is simple, by 3.3.6, there are xi, ...,xn G A such that IA =y^Xjpx*.

We may assume that Xip = Xi (i n times) and

1, ...,n). Set P = diag(p, ...,p) (p repeats

I xx 0

x2 0

\ 0

0

...

xn\ 0 0 /

Then vv* = 1A and v*v < P. Set e = v*v. Since tsr(^l) = 1, there is a unitary V 6 Mn(A) such that VeV* = 1A and V*1AV = e. Set w = V*(z + (1M„(A) " U ) ) ^ = ^**V + (1M„(A) - e). Then by 2.4.11, [u; © !M„(A)] = [z © ( 1 M 2 „ ( ^ ) - U)] in t M ^ / t M ^ V It follows from 3.1.10 that there is u G U(pAp) such that [it © ( P - p)] = [F*2:V + ( P - e)] in Un(pAp)/Un{pAp)0. Therefore [u® (1M„(A) - e)] = [w] in t/„(yl)/C/„(A)o. Hence u © (1M„(A) - e) © 1M„(A) and z © (1M 2 „(A) — 1A) are in the same connected component of U(M2n{A)). Therefore [u+(l— p)} = [x] in K\{A). vspace0.15in D Remark 4.3.2 L. G. Brown has shown that if B is a cr-unital full hereditary C*-subalgebra of a cr-unital C*-algebra A, then B K = A ® K (see 3.9.12). The above lemma follows from this stable isomorphism theorem of Brown. Lemma 4.3.3 Let A be a unital C*-algebra and <j> : C{X) —> A be a homomorphism, where X is a compact metric space. For any £ > 0 and any finite subset T C C(X), there is S > 0 such that if (1) Si, S2,..., Sn are disjoint open subsets of X such that dist(x,x') < 5 for any x, x' G Si (i = 1, •••,n); (2) Aj G Sit i = 1,2, ...,n; (3) pi is a projection in Her(ai), where at — (xi) and Xi G C(X)+ such that Xi(t) — 0 if t ^ S^ (4)j// = (i-Er=iPi)^(/)(i-Er=iPi)>

Simple AT-algebras

with real rank zero

179

then n

n

iw/)-(y/+Ew*)p*)ii <

e

and

n

iia-EttM/j-^/xi-Epoii < £

i=l

i=l

i=l

/or a/Z / S T. Proof.

Let ci > 0 such that l / ( 0 - / ( O I < e / 8 . if d i s t ( ^ 0 < d

for all / £ J 7 . To save notation, without loss of generality, we may assume that ll/H < 1 for all / g f . Since x\,...,xn are mutually orthogonal and Pi € Her(aj) (i — l,...,n), pi,...,pn are mutually orthogonal. Now we work in A". Let qi = lim„_yoo a / n (converges strongly). Therefore qi commutes with (/) for all / e C(X). Note that qi is a projection in A". We have

i=l

i=l

n

n

i=\ i=l

i=l i=l

i=l

n

= nE^(/)-/( A ^)ii<£/8 i=l

for all / 6 f .

Similarly,

A <£ 8 (E^(/)-E/( ^n / i=l i=l for all / G T. Therefore, ( X > M / ) " # / ) ( ! > ) II < e/4 and i=l

iKi -

i=l

EPO
- 0(/)(i - X>)ii < £ / 4

i=l

i=l

for all f & J-. Moreover,

(i - X>w)(X>)ii < £ / 4 and i=l

i=l

IKX>M/XI i=l

- X>)ii < e / 4 i=l

180

Classification

of Simple

AT-algebras

for all / G T. Thus n

n

A

n

1

\\
=1

n

n

+IKE^(/XI i=l

n

i=\ n

n

- EP*)II + IKEP*M/)(E*) - E/(A»)^)ii =1

=1

=1

i=l

< £/4 + e/4 + e/2 = £ for all / € JF. D Lemma 4.3.4 Let A be a unital simple C*-algebra of real rank zero and stable rank one. Let u G U(A). Then, for any nonzero projection e G A and e > 0, there is a projection q € A such that

(i) [q] < H; (2) u « e u\ © u2, w/iere «i G U(qAq), u2 G C((l — g)vl(l - q))Q and [ui®(l-q)] = [u] inKi{A). Proof. We may assume that sp(w) = S1. Otherwise, the lemma is trivial (by taking q — 0). Fix s > 0, let d\ = e/4. By applying 4.3.3 and 4.1.1, we may write n

U « £ / 4 Z0 © E ^iPU

(e 3-2)

i=l

where pi,.-.,pn are mutually orthogonal projections in A, zo is a unitary in pAp and p = 1 — X)?=i Pi- Furthermore, we may assume that {Ai,..., A„} is e/16-dense in S1. It follows from 3.5.7 that there are (nonzero) mutually orthogonal projections ei and e2 in (1 — p)^4(l — p) such that e\ and e2 are equivalent and [ei + e2] < [p»] for i = l,...,n and [ei + e2] < [e]. By 4.3.1, there is a unitary z\ G e\Ae\ such that [zi + (1 — ei)] = [it] = [zo + (1 — p)} in JK"I(A). Since [ei] = [e2], let z2 € U(e2(Ae2) such that zi + z 2 £ f/((e! + e2)A(ei + e2))o. It follows from 4.2.8 that there are mutually orthogonal projections q[,...,q'm G {e\ 4- e2)A{e\ + e2) and ai,...,Q! m G S"1 such that 771

lki + 2 2 - E a ^ l l < £ / 8 -

Simple AT-algebras

with real rank zero

181

Since {Ai,...,A n } is e/8-dense, it is easy to see that there are mutually orthogonal projections q\, ...,qn G (e\ + e2)A(ei + e2) such that m

n <£ 4

W^ajq'j-^^QiW j=i

/ -

t=i

Since [ei + e2] < [pi] for i = l,...,n and tsr(A) = 1, there is a unitary w G (1 — p)A(l — p) such that w*qiW
i=

l,...,n.

Set di = pi — w*qiW. Then n

n

y^AjPi « e / 2 w*ziw + w;*Z2^ + ^ A i d j . i=l

(e3.3)

t=l

On the other hand, by 2.4.11, [w*z2w + (1 - w*e2w)] = [z2 + (1 - e2)] in #i(,4). Therefore, by 4.2.9 and 4.2.11, z0 + w*z2w G U((p + w*e2w)A(p + w*e2w))o. Now set q = w*eiw, u\ = w*z\vj and u2 = ZQ + w*z2w + S r = i ^idi- Then from (e3.2) and (e3.3), we obtain (1) [q] < [e] and (2) u K,E m + u2 and u2 G U((l - q)A(l - q))0. • Theorem 4.3.5 Let A be a unital simple AT-algebra. Suppose that RR(A) = 0. Then TR(A) = 0. Proof. We may write A = (U™=1An), where An C An+i, An ^ B1n )(C(Zj in )) and X J>n is a compact subset of S 1 . Let e £ A be a nonzero projection and e > 0. Fix a finite subset F . Without loss of generality, we may assume that J- C An for some large n. By applying 3.5.7, there are k(n) (non-zero) mutually orthogonal projections e±, ...,e,t(n) G eo^eo, where eo < e and r(n)[eo] < [e] for r{n) = max{s(j,n) : j = 1, ...,/e(n)}. Let e3'n G M s (j in )(C(Xj )7l )) be a constant rank one projection. We identify e^nB^ne^'n with C(Xjtn). Let z^n be the identity function in C(Xj:Tl) which is the standard unitary generator of C(X, jTl ). Note we assume that Zjn is a unitary in e^nAe3,n. Let 5 > 0. By applying 4.3.4, we obtain projection pj < e J ' n such that [pj] < [ei] and v^n G U(pjApj) and Uj>n G U((ej'n - pj)A(ej'n - pj))0 such that z

j , n ««S/2 vj,n

+ Uj,n-

(e3.4)

182

Classification of Simple AT-algebras

It follows from 4.2.8 that there is a finite dimensional C*-subalgebra Cjn G (ejn - Pj)A(ejn -pj). Such that Uj,n G5/2 Cj,n-

(e3.5)

From (e3.4) we also have \\pjZj>n - ZjtV,pj\\ < 5 and PjZjtTlpj <=s CjtJl.

(e3.6)

Set C'jn = M s (j in )(Cj )n ), p'j = dia,g(pj,...,pj) (pj repeats n times and we view p'j as a projection in Ms(j>)(e-7'™Ae-7'™) C l s ^ A l ^ . n ) . Since Zj>n together with a set of matrix units generate Bjn, with sufficiently small 6, we have (1') | | p ^ - 6 ^ . | | < £ a n d (2') p'jbp'j G£ C'j>n Als for all&G lBjnFlBjn°

(3')[^.]
Unitaries in simple C*-algebra with RR{A)

D

= 0

Lemma 4.4.1 Let A be a unital C*-algebra with RR(A) = 0 and u, v G U(A). Suppose that [u] = [v] G U(A)/U(A)0. Then, for any 5 > 0, there is a continuous path 7 : [0,1] —• U(A) such that 7(0) = u, 7(1) = w and £(7) < 7T + 5. Proof. First we show that, for any 2 > <5 > 0, if w G C/(A) with | | i o - l | | < 8/TT, there is a continuous path a : [0,1] —• f/(A)o such that a(0) = ru, w(l) = 1 with L(a) < 5. In fact, let f(eu) = £ (for - 5 < t < 6) be a continuous function on u {e : -5 < t < 6}. Set h = /(w). Then h G A, a and \\h\\ < S. Define a(s) = exp(i(l - s)h) for s G [0,1]. Then ||o(s) - a(s')\\ < \\h\\\s - s'\ < 6\s - s'\.

Unitaries in simple C* -algebra with RR(A)

= 0

183

Therefore L(a) < S. Now set z = uv*. Then z € U(A)Q. It follows from 4.2.8 that there is a unitary z' £ U(A)o with finite spectrum such that \\z-z'\\

<6/n.

Since z' has finite spectrum, there is a € Asa such that ||a|| < ir and z' = exp(ia). Let 7'(t) = exp{i(l — t)a). Then 7'(0) = z', 7'(1) = 1 and (as above) £(7') < 7r. From the first part of the proof, we obtain a path a' : [0,1] -> C/(A) such that a'(0) = z'z*, a ' ( l ) = 1 and L(a') < 5. Let a"(s) = a'(s)z. Then a"(0) = z', a"(l) = z and L(a") < <5. Connecting 7' and a", we obtain a path 7" : [0,1] ->• t/(A) such that 7"(0) = z, 7"(1) = 1 and L(-y") < TT + S. Set 7(f) = 7(i)v. Then 7(0) = u, 7(1) = v and L(7) < 7r + 6. • Theorem 4.4.2 For any £ > 0, and integer mo, there exists an integer n > 0 satisfying the following: If u and v are two unitaries in a unital simple separable C*-algebra of stable rank one and real rank zero such that [u] = [v] € U(A)/U(A)o if z = 5Zi=i \di, where {Ai,..., A;} is s/8-dense in Si, {di, ...,di} are mutually orthogonal projections such that mo[di] > [1A] for i = 1, ...,l, then there there is a unitary W € Mn+i(A) such that ||W*dia,g(u, z,..., z)W - diag(w, z,..., z)|| < e, where z repeats n times on both diagonals. Proof. It follows from 4.4.1 that, for 6 > 0, there is a path 7 : [0,1] —> U(A) such that 7(0) = u, 7(1) = v and £(7) < n + 6. Let N > ^ ^

and 0 = t0 < ti < • • • < tN = 1 with |K-Ui+i|| <e/4, i =

0,l,...,N,

where u\ = 7(ij). Let wx = dia,g(uQ,ui,u*,- • • ,u*N_2,uN-i)

and

w2 = diag(uJ,ui,U2,---,u|v-i> u JV-i)Thus diag(u,u)i) « s / 4 dia,g(ui,ul,- • •

,uN-i,u*N_i,v)

184

Classification

of Simple

AT-algebras

Therefore there is a unitary W\ G M2N-i(A) \\W*(u®wi)Wi

such that

- u © w 2 | | < £/4.

Since

we obtain a unitary W2 G

M2JV-I(^4)

such that

||W 2 *(ueu>i)W 2 -v®wi\\

<e/2.

It follows from 4.2.8 such that there is a unitary w3 G spectrum such that \\wi ~w3\\

M2N-2(A)

with finite

<e/A

Without loss of generality, we may assume that 103 = YLi=i ^iPi> where Pi,.-,pi are mutually orthogonal projections in M 2 AT_ 2 (J4). Now let n = mo(2N — 2) and let z^> = diag(z, ...,z), where z repeats n times. So z^ = 5Z i = 1 AjZPj, where D{ = diag(di, ...,di) (di repeats n times). Since mo[d»] > [1A], there is a unitary W3 G Mn(A) such that Pi<w;Dtw3,

i = i,...,i.

Set ei = W^DiWj, — pi and x = ]T}»=i ^iei- Then W3*z^W3=w3®x. Let W4 = 1 A © W 3 . Then W4 (u 0 z(n))W4 = u®w3®x

and W4 (v © z ( n ) )W 4 = u © w3 © x.

Therefore w © z(n)

~ ~£/2

u © w 3 © a; ~ e / 4 u © wi © a; v©u>i ©a; ~ £ / 4 v@w3 ®x ~ w©z ( r i ) .

L e m m a 4.4.3 Lei A be a unital simple C*-algebra with RR(A) = 0 and tsv(A) = 1. For any w G £/(J4) wiiA sp(u) = S 1 , 5 > 0 and any integers mo > 0 and n > 0, (i) there is a nonzero projection p and a unitary vi G U(pAp) such that [(1 - p) + Vl] = [u] inKi(A);

Unitaries in simple C*-algebra

with RR(A)

= 0

185

(ii) there are n mutually equivalent and mutually orthogonal projections ei, ...,e„ € (1— p)A(l— p) and a unitary z £ U(eiAei) with z = X)i=i ^»9»> where {\\,..., X^} are S/2-dense in S1, {qi,..., qjv} are mutually orthogonal projections such that mo[ft] > [p] (i = 1, ...,N); (in) i/iere is a unitary v2 € C/((l — p — ]£™=i e i ) ^ ( l - P — X)"=i e») w * ^ finite spectrum such that \\u - vi ®v2 ©diag(z,z, ...,z)\\ < 6, where z repeats n times on the diagonal (we also identify (%2i=i e *)-^(2i=i e») with Mn(eiAei)). Proof. Since sp(u) = S1 and TR(A) = 0, we see, by 4.3.3 that there are mutually orthogonal projections qi,...,qN in A and X\,...,XN which are 5/2 -dense in S1 such that N

\\u-v®^2xiqi\\ <5/2, i=\

where v G U(qAq) and q = 1 — Yli=i Ii- Now fix mo > 0 and n > 0. It follows from 3.5.12 that there are nonzero mutually equivalent and mutually orthogonal projections qn,...,qin < qi, i — 1,2,...,N. Set di = qi = YJj=i liji D — Z)i=i dii ei = Qii, i = 1, 2 , •••, N and JB = X)t=i e»- Denote z = 2Zi=i ^iei- Choose a nonzero projection eo < q such that mo^ii] > [eo] for each i. Note that tsx(qAq) = 1 and RR(qAq) = 0. By applying 4.3.4, we obtain a projection p < q with [p] < [eo], a unitary ui G U[pAp) and a unitary Vj G £^((<7 — p)^(? ~ p))o with finite spectrum such that \\v-vi

®v'2\\ < (5/2.

Then N

u

K5/2

v © diag(z,..., z) ® ^

Xjdj

i=l JV

«5/2 " l S ^ S diag(z,..., z) © y ^ A^j. i=i

Let v2 = v'2 © 5Zi=i -Mi-

186

4.5

Classification of Simple

AT-algebras

A uniqueness t h e o r e m

The purpose of this section is to prove the uniqueness theorem of 4.5.5. Remark 4.5.1 Let A be a unital C*-algebra and B be another C*algebra. Suppose that L : A -> B is a contractive completely positive linear map. We will use L for £ ® i d : Mk(A) —• Mk(B) for every A; > 0. Let V = {pi, ...,p;} be a finite subset of projections in M00(A). We may assume that V C Mk(A). There is a finite subset T C A such that Qk = {(«„•) G Mk(A) : a,ij G J-"} contains V. For any e > 0, there is 6 > 0 (2.5.1) such that if L : A —• B is a ^"-^-multiplicative contractive completely positive linear map, then, there are projections qi,...,qi G Mk{B) such that \\L<j>i) - Qi\\ < £

{i =

l,...,l).

Suppose that e < 1/2. Let [L{pi)\ = fa], i = 1,2, ...,l. Then, by 2.5.1, we see that |X(pi)] is well defined. We will use [L]\-p for the corresponding finite subsets of KQ(B). Fix a finitely generated torsion free subgroup Go C KQ(A). Let gi,...,gi be the free generators of Go. There are projections pi,...,pi,qi,...,qi G Mk(A) for some large k such that gi = [p»] — fa]. Suppose that L : A —• B is a .^-^-multiplicative contractive completely positive linear map so that [Ii(pi)] and [£(• KQ{B). Suppose that P C Mfc(j4) is a finite subset of projections so that \p] G Go in Ko(A) for each p G "P. Then there are integers fc» such that X)i=i([P»] ~ [?»]) = fa]> o r £ 1 = 1 ^ M = fa] + Yli^ikfa]Let di = diag(pj, ...,pi) and d'{ = diagfe, ...,&), where pi and 9i repeatfcjtimes, i — 1,..., /. There is a partial isometry v G MN1+N2+I(A) for some integers N\ and A^2 such that v*v = @\=ldi ® ljv2 and vv* = ©-=1d- © p © 1 JV2• We let <7 = {v*,v,di,p,di,lN,v*v,vv*} U J-. If J is sufficiently small, by applying the perturbation lemmas in section 2.5, we see that, if L is also Q(5-multiplicative, [L(q)] = [L]([q]). Suppose that A and B are stably finite. We say that [L]\Go is positive if [L](G0 n K0(A)+) C K0(B)+. If u e Mfc(j4) is a unitary, then ||L(«)*L(u) - lMfc(B)|| < £ and \\L{u)L(u)* - lMfc(B)ll < e,

A uniqueness

theorem

187

provided that T is large enough (to contain u for example) and S is small enough. Thus, as in 2.5.3, there is a unitary v € Mk{B) such that \\v-L{u)\\ 0 and a finite subset T c A such that any T-5-multiplicative contractive completely positive linear map L : A —> B induces homomorphisms [L] : Gi -> K^B) with [L{p)\ = [L]{[p]) and [L{u)} = [L]([u]) for all p £ V and u €U. In what follows, when we write [L](G) we mean that [L](G) is welldefined. Lemma 4.5.3 Let A be a finite dimensional C*-algebra. Then, for any e > 0 and any finite subset T C A, there exists 6 > 0, a finite subset V C Ko(A) and a finite subset Q
[L2]\G(V),

where G(P) is the subgroup generated by V, then there is a unitary w € B such that adw o L\ « e L2 on T. Proof. G(P) =

Since KQ(A) is finite generated, it is convenient to assume that K0(A).

188

Classification

of Simple

AT-algebras

It follows from 2.5.9 that, for any 5i > 0 and finite subset Q\ C B, with sufficiently small 6 and sufficiently large Q, there are homomorphisms hi : A-> B (i = 1, 2) such that i» ««! /ij on Q\ (i — 1,2). Therefore, (/n)» = (/i 2 ). (on ifo(i4)). Thus the lemma follows from 3.4.6.

•

Lemma 4.5.4 Let A be a unital simple C*-algebra with TR(A) — 0 and u £ U(A) be a unitary. For any e > 0, there exists 5 > 0, a finite subset V C KQ(A) and a finite subset Q C A satisfying the following: Suppose that Li : A —>• B are two unital Q-6-multiplicative contractive completely positive linear maps, where B is a unital C* -algebra with tsr(B) = 1 and RR(B) = 0. If [LI]\GIV)

= [L2]\G(-P)

and [ii]([«]) = [L2]|([U]),

then there is a unitary w £ U(B) such that adw o L\{u) wE L^iu). Proof. We first consider the case where sp(w) ^ S1. In this case, there are nonzero mutually orthogonal projections d\,...,di £ A and Ai,...,A/ € S1 such that | | u - u | | < e/4. We then choose V which contains [d\],..., [dj]. So, with sufficiently large Q and sufficiently small 6, we obtain mutually orthogonal projections d'x, ...,d[ and d",...,d[' such that ||L!(dO - dl|| < £/{41 + 1), \\L2(di) - <'|| < e/(4Z + 1) and [<] = [<] for i = 1,2,..., Z. Since tsr(S) = 1, we obtain a unitary w £ B such that w*diW = d'i for i = 1,..., I. Thus adiu o L\(v) s=se/4 L2(v).

A uniqueness theorem

189

Therefore, adw o L\(u) « £ /4 adit; o L\(v) « £ / 4 L2(v) « £ / 4 adw o L2(u). Now we consider the case where sp(u) = S1. Fix e > 0, and 77io > 1. Let n be the integer in 4.4.2 associated with e/2 and mo. Fix an ?7 > 0. Applying 4.4.3, we obtain a nonzero projection p £ A, a unitary vi € f/(pAp) with [(1 —p) + vi] = [u], z and i>2 satisfy the conditions (i), (ii) and (iii) in 4.4.3 associated with r] (in place of 5), mo and n. Note that z = 2 » = i ^»9i (with ^ofe] > [p]) a n d u2 = Ylj=iaj9j, where ax,...,a; € S 1 and gi,...,gi are mutually orthogonal projections. Let T3 € i^o(^4) be a finite subset which contains {diagfe, 0,..., 0), diag(0, qu 0,..., 0),..., diag(0,..., 0, &) : i = 1,..., N} [p], [1 - Pi. {91, - , ff;}> Q = E j = i flj. [^ + Q] a nd [1 - P - Q]. Let Ti be a finite subset of A which contains diag(z, 0,..., 0)),..., diag(0,..., 0, z), d i a g ^ , 0,..., 0),..., diag(0,..., 0, &), (i = 1,..., JV)and u,v1>v2,g1,...,gt. Suppose that L\ and L2 are .TV^-multiplicative contractive completely positive linear maps from A to B such that [L1}\v = [L2}\v

(e5.7)

Without loss of generality, by 4.1.6, we may assume that L\ and L2 are unital and that there are projections P', P", Q', Q" G B such that ||P' - i!(p)|| < V, \\Q' - Lx{v*2V2)\\ < r,, \\Q" - L2(v*2V2)\\ < V and \\P"~L2(P)\\ < V- By (e5.7), [P'} = [P»] and [l£] = [1-Q"] in K0(B). Since tsr(B) = 1, by replacing L\ by a d W o L\ for some suitable choice of W, without loss of generality, we may assume that P' = P", P'Q' = 0 and Q' = Q". With a sufficiently small rj, by 4.5.1, there are unitaries v^v" S U{P'BP'), v'2, v'2< e U(Q'BQ') and z', z" e U{(1 - P'- Q')B(1 - P'- Q')) such that v[ « e / 1 6 Ll(vi), v" « E / 16 L2(^l), f2 «e/16 £1(^2), V2' We/16 ^2(^2), z' «£/i6 Li(z),

2" we/i6 £2(2),

190

Classification

of Simple

AT-algebras

diag(z',..., z') « e / 1 6 Li(diag(z,..., z)), diag(z",..., z") « e / 1 6 L 2 (diag(z,..., z)),

z' = f > ^

4 = EA^',

i=l

i=l

where {<j^ : 1 < i < /} are mutually orthogonal projections, {g" : 1 < i < 1} are mutually orthogonal projections, [qt] = [q"], and / v

/ a

2 = 2^ i9v

v

2 = 2 ^ ai9i '

j=l

j=l

where {<^ : 1 < j < Z} are mutually orthogonal projections, {g'J :< j < 1} are mutually orthogonal projections and [g^] = [g"] in KQ(B). Since tsr(B) = 1, we obtain a unitary Wi G £/((l - P')B(1 - P')) such that W 1 *diag(z',...,z')W 1 = diag(z",..,z") and W > 2 W i = v2'. Set W2 = P ' © Wx. Then, we have W2*L1(u)W2 « 3 e / 1 6 «i © diag(z",..., z") © i#. We also have v'2 © diag(z",..., z") © v2' « 3 g / 1 6

L2(u).

Note that (with sufficiently small 77) [^] = [1//] = [Li(u)] = [L2(w)]- We also have mo[g[] > [P'] by (e5.7). By the choice of n and applying 4.4.2, we obtain v[ © diag(z",..., z") © v'2' ~e/2 v'2 © diag(z",..., z") © v'2\ Therefore, Li(u) ~ e L 2 (u). Theorem 4.5.5 Le£ i lie a unital simple AT-algebra. For any e > 0 and any /im£e subset J- C A, there exists a 6 > 0, /rniie subsets Q C A, V C KQ(A) and U C -^i(^4) satisfying the following. If Li : A ->• i? (i = 1,2) is a Q-6-multiplicative contractive completely positive linear maps, where B is a unital C* -algebra with tsr(P) = 1 and RR{B) = 0, such that [L1)\V = [L2\\V and [Ll\\u = [L2]\u,

A uniqueness

191

theorem

then there is a unitary w G U(B) such that &dw o L\

K,£

L2

on T.

Proof. Without loss of generality, we may assume that T C C, where C = Mni(C(Xi)) © • • -Mnk(C(Xk)), where Xj is a connected compact subset of S1, 1 < j < k. We first consider the case in which C = Mn{C(X\)). Let {e^} be a system of matrix units and let u G e\\Ce\\ be the unitary generator (given by the function z t-¥ z on S 1 ). Then C is generated by {e^} and u. So, without loss of generality, we may assume that T = {u} U {ejj : i,j = l,...,n}. Suppose that V contains {en, . . . , e n n } . So [Li](en) = [L2](en). Therefore, since tsr(B) = 1, without loss of generality, by replacing L\ by adwi o L\ for some suitable w, we may assume that L i ( e n ) = L2(en) = q is a projection in B. By 4.5.4, there is a ^i > 0, a finite subset V\ C KQ{A) and a finite subset Q\ C e\\Ae\\ satisfying the following: if LJ : e\\Ae\\ —> qBq is a £/i-<5i-multiplicative contractive completely positive linear map such that [L\]\Vl = [L2]\Vl and [L[}(u) = [L'2](u), then there is a Wi G U(qBq) such that adwi o L'^u) K,e/i

L^u).

Thus, by applying 4.5.3, choosing a small 0 < <5 < 5\/n2 and a finite subset Q D Gi, we obtain a unitary w2 £ U(B) such that adw 2 ° Li wE/„2 L2

on T\,

where T\ = {e^, i, j = 1, ....n}. Without loss of generality, we may assume that adu>2 o L\(eu) = Z/2(en) is a projection q G B. Then, we consider L\ = adu>2 o Li\eilJ\eil and L'2 = L2\eilAeilFrom what we have shown above, there is a unitary W G U(B) such that ad o W o Adw o L\ w£ L2

on T.

For the general case in which C = Mni(C(Xi)) 8 • • • Mnk(C(Xk)), we let qi,..., qk be the identities of each summand. We may assume (by either 4.1.6 or 4.1.7) that Lj(qi) are projections (j = 1,2). We may reduce the general case to the case in which L\(qi) = L2(qi)- By considering each

192

Classification

of Simple

AT-algebras

summand, we further reduce the general case to the case in which C has only one summand.

• 4.6

Classification of simple AT-algebras

Lemma 4.6.1 Let A = Mnx(C(Xi)) © • • • ®Mns(C(Xs)), where Xi is a 1 connected compact subset of S (I = 1,..., s) and let B be a unital simple C*algebra with stable rank one. Suppose that at : Ki(A) —t Ki(B), i = 0,1, is a homomorphism and OLQ is positive. Then there is a homomorphism h : A —>• B such that hi„ = oti

i = 0,1.

Proof. Let {ui,e\j : i,j = l,...,ni,l = l,...,s} be a system of standard generators of A. Let F = Mni © ••• © M n , . We will identify {e\j : i,j = l,...,ri[,l = l,...,s} with a set of standard generators of F. Note that (K0(A),K0(A)+) = (K0(F),K0(F)+). Thus, by 3.4.6, there is a homomorphism (f> : F —> B such that ^„o = <*o- Set q|- = ^>(e'), i,j = l,...,ni,l = l,...,s. It follows from 4.3.1 that there are vi, ...,vs such that vi G U(qlnBqln) with [vi + (1 - qlu)} = ai([ut + (1A - eln)}), I = l,...,s. Note that if sp(u;) ^ S1, a\{[ui + (1A — eii)]) = [Is]- Therefore we may assume that sp(vi + (1 - qln)) — sp(u/ + (1 - e' n )). By sending ui to vi and e1^ to q1^, i,j = 1,..., ni, I — 1,..., s, we obtain a homomorphism h : A —> B. By the construction above, it is evident that h*i = a»j, i = 0,1. • Lemma 4.6.2 Let A be a unital AT'-algebra and B be a unital C* -algebra with stable rank one and let cti : Ki(A) —> Ki(B) (i = 0,1) be homomorphism such that OCQ is positive. Then, for any e > 0, any finite subset J- C A, any finitely generated subgroup Vi C Ki(A) (i = 0,1), there is an J-'-e-multiplicative contractive completely positive linear map L : A —^ B such that [L]\Vo = (a0)\p0

and [L]\Vl = ( a i ) | P l .

Proof. Fix !F, Vo, and V\, and e > 0. Since A is an ^T-algebra, we may assume that A is the closure of the union of increasing sequence {Bn} of finite direct sums of circle algebras. Denote by j n '• Bn —• A the embedding.

Classification

of simple

193

Al-algebras

We may assume that, for some N0 > 0, {jn)*i(Ki(Bn)) D Vi for all n> N0 (i = 0,1) Thus we may also assume that x G e / 4 Bn for all x G T. By 4.1.2, we may assume that there are An C Bn, where An is a finite direct sum of C*-subalgebras of the form Mk(C(X)), where X is a compact connected subset of S1, such that (jn)*i ° {Jn)'*i(Ki(An)) 3 Vi and x G£/2 An for all x G T, where j ' n : An -4 Bn is the embedding (i = 0,1). Let in = in ° j„ a n d Qi be a finitely generated subgroup such that (in)*i{Qi) = Vi (i = 0,1). By 4.6.1, there is a homomorphism h : An —» B such that h*i\Qi = ai o (in)ti\Q.

(i = 0,1).

Since A is amenable, by 2.3.13, for any 6 > 0 and any finite subset Q C A, there is a contractive completely positive linear map L : A —• B such that L ftS{/2 h

on
We choose Q large enough so that x G5/2 5- Since ft. is a homomorphism, L is ^-"-e-multiplicative if Q is large enough and S is small enough. It is also clear that we can assume that [L]\-pi is well defined and [L)\vt=ai\V,

(» = 0,1),

provided that Q is large enough and 5 is small enough.

•

Definition 4.6.3 Let A and B be two unital stably finite C*-algebras. If there are homomorphisms aj : Ki(A) —>• Ki(B) (i = 0,1), we may write a : (KQ(A), KI(A)) —>• (A"O(JB), X I ( B ) ) . If Qo is positive, then we say a is positive and write a : (K0(A),K0(A)+,K1(A)) -»• (K0(B),K0(B)+,K1{B)). If furthermore, a([lyi]) = [1B], then we write a: (K0(A),K0(A)+,[lA],K1(A))

-> (K0(B),K0(B)+,

[1 B ], ^ ( B ) ) .

Theorem 4.6.4 (Elliott) Let A and B 6e iwo wniiaZ simple AT-algebras with real rank zero. Suppose that there is a positive isomorphism a : (ffo(A),tfo(A) + ,[l A ],tfi(A)) - • (K 0 (B), ^ 0 ( B ) + , [1 B ], # i ( B ) ) . Then there is an isomorphism h : A —>• B such that h,i = a, (i = 0,1). Proof. Define (3 = a - 1 . Let {a n } and {&„} be dense subsets of A and B respectively. Let {rn} be a sequence of positive numbers so that X^nt=i rn < 00.

194

Classification

of Simple

AT-algebras

Let T\ be a finite subset containing a,\. Let <^i = 8{ri/2,!F\) > 0, Q\ = G{n/2,F\) C A be a finite subset, V\ = V(r1/2,Fi) c #o(-4) be a finitely generated subgroup and U\ = U(r\j2,J:\) C -f^i(^4) be a finite subset corresponding to r i / 2 and T\ as required in 4.5.5. We may assume that T\ C Q\ and Si < ri/2. If follows from 4.6.2 that there is a Qi-Si/2multiplicative contractive completely positive linear map L\ : A —• B such that [Li\\vi =ao\vi

and [ L i ] ^ = a i | W l .

Let S\ be a finite subset containing 6i and L\(Qi). Set ii = m i n j r ! ^ , ^ ^ } . Let r)X = J(t 1 /2,5i) > 0, U\ = Q{tx,S{) C A be a finite subset, Qj = ^ ( i i , ^ ! ) be a finitely generated subgroup of KQ{B) and Vi = U{t\,S\) be a finitely generated subgroup of K\(B) corresponding to ti and Si be required as in 4.5.5 ( and interchange A and B). We may assume that rji < h/2, Hi D Si and [Li](Vi) C Qi and [ii](Wi) C Vi. It follows from 4.6.2 that there is an ^1-771-multiplicative contractive completely positive linear map A[ : B —>• A such that [A'i]| C i=A>lei and [A'i]\Vl = (3i\Vl. Therefore A[ o Li is ^i-Ji-multiplicative and [Ai oLi]\Vl

= [idyt]!^ and [Ai o Li]\Ul = [icUlk.

It follows from 4.5.5 and the choice of Si, Qi, V\ and lAi that there is a unitary Ui € A such that ad?7i oA'j 0 I 1 ~ri/2 idyi

on T\.

Set Ai = a d t / i o A i . Let J~2 be a finite subset containing T\, 0,2 and Ai(%i). Set s2 = min{r 2 /2,771/2). Let 62 = ^ ( s 2 / 4 , ^ 2 ) > 0, Q2 = 0(s 2 /4,.F2) be a finite subset of A, V2 = Visil^T?) be a finitely generated subgroup of K0(A) and U2 = U(s2/^,J-2) be a finitely generated subgroup of Ki(A) corresponding to S2/4 and Ti as required in 4.5.5. We may also assume that S2 < s2/2 and G2 D F2 U tti, P2 D [Ai](Qi) and W2 D [Ai](Vi). It follows from 4.6.2 that there exists a (/2-^2/2-multiplicative contractive completely positive linear map L2 : A —> B such that [L2\\v2 = ao\v2

and [L'2]\u2 = ai|w 2 .

Classification

of simple

AT-algebras

195

Then L'2 ° Ai is "Hi-jyi-multiplicative and [L'20^I]\QI

= [idsllci

and

[i' 2 ° Ai]| V l = [idsllvi-

It follows 4.5.5 that there is a unitary W\ € B such that adWi o L2 « S 2 / 2 id B

on Hi-

Set L2 — s,$Wy o L2. Thus the following (not necessary commutative) diagram

A

iH

A

X-L\

/ Ai

4-1/2

is approximately commutative on J-j within r\ and on <Si within s\, respectively. Let S2 be a finite subset of B containing 63, <Si and L2(Cj2) and let t2 = min{r 2 /2,<5 2 /2}. Let r\2 = 6(t2,S2) > 0, H2 = G(t2,S2) be a finite subset of B, Q2 = V(t2,S2) be a finitely generated subgroup of KQ(B) and V2 = U(t2, S2) be a finitely generated subgroup of Ki(B) corresponding to t2 and S2 as required by 4.5.5 (and exchange A and B). We may assume that 772 < t2/2, H2 D S2, [L2]{V2) C Q2 and [L2](U2) C V2. It follows from 4.6.2 that there exists an ^2-^2/2-multiplicative contractive completely positive linear map A2 : B —• A such that [A2]|Q2=/?O|Q2

and [A' 2 ]| V2 =/3i|V 2 .

Therefore A'2 ° L2 is 52-^2-multiplicative and [A'2 ° L2]\v2 = a0\V2 and [A'2 o L2]\u2 = a i | « 2 . It follows from 4.5.5 and by the choice of S2, Q2, V2 and U2 that there is a unitary U2 £ A such that &dU2 o A2 o L2 »t 2 id^

on T2.

Set A2 = ad£/"i o A2. Continuing in this fashion, we obtain a sequence of ^-"n-rn-niultiplicative contractive completely positive linear maps Ln : A —)• B and a sequence of

196

Classification

of Simple

AT-algebras

-multiplicative contractive completely positive linear maps A „ : B 4 A such that the following (not necessary commutative) diagram A 4-Li

B

^ / Ai

^

A

^4

B3

4-L2

/ A2

4-Z/3

B

- ^

B

^4

•••

A

^

•••

B

is two-sided approximately intertwining. It follows from 1.10.16 that there is an isomorphism h : A —• B (and h~1 : B —¥ A) such that A li„

^4 /'A,,

A 4-^

is approximately commutative on J-n within rn and on Sn within sn, respectively. Therefore h*i = a*i {i = 0,1). •

4.7

Invariants of simple AT-algebras

To complete the classification of unital simple ^4T-algebras of real rank zero, we need to describe the invariants, namely the ordered A'o-groups and AVgroups of such unital simple j4T-algebras. We also need to show that given such an invariant there is a (unique) unital simple AT-algebra whose invariant is exactly the given one. Finally, in this section, we will also give a criterion which determines when a unital simple AT-algebra has real rank zero. Definition 4.7.1 Let Gi = ©™=1GM and G2 = ®f=iG2tj, where Giti and G2J are groups. Let iti : G\ —> Gu and ipj : G2 —• G2j be the quotient maps. Suppose that <> / : G\ —> G2. A partial map of from Gu to G2j is the map ^ ^ = ipj o <j>\Gl_. induced by . If Gu S G2J =• Z and if <^(IJ')(1) = rn, then we say the partial map <j>^1'^ has multiplicity m. Let A = \imn^oc(An, 4>n) be an inductive limit of C*-algebras. Suppose t h a t An — ®ll1An,i and ntTn = m o • • • o (j>n ; An -> A m ( m > n). Let 7Tj : Am —> Amj be the quotient m a p . We will write 4>n,m = ^j ° 4>n,m-

Proposition 4.7.2 Let Abe a unital simple AT-algebra with RR(A) = 0. Then K\{A) is a torsion free countable abelian group and KQ(A) can be written so that (K0(A),K0(A)+) = lim n _ +00 (G rl , (n), where each

Invariants

of simple

AT-algebras

197

Gn is a finite direct sum of Z and each partial map of 4>n has (positive) multiplicity at least 2. Proof. It follows from 4.1.2 that we may assume that A = \im.n^oc(An,hn), where each An is a finite direct sums of C*-algebras of the form Mfc(CpQ), where X is S 1 , an arc or a point. Since K0(Mk(C(X))) = 1 Z and Ki(Mk(C(X))) = Z, or = {0}, if X is S , or an arc (or a point). The only thing requires further proof is the statement about the multiplicity of partial maps of each <j>n. Let G\ = Ko(Ai) = Z©- • -®Z, m copies of Z. Let e\, ...,em be nonzero minimal projections in A\ so that [e^] is one in each copy of Z. Since A is simple and TR(A) = 0, it follows from 3.5.12 that there are (for each i) two mutually equivalent and mutually orthogonal projections (ft,i, (ft,2 such that qiti,qi,2 ^ 0, qi:i + qi2 < e*, i = l,...,m and m{i)

^2x*js1i,sxijs=l,

i = l,...,m,s

= 1,2.

(e7.8)

By the perturbation lemmas in section 2.5, without loss of generality, we may assume that qi,s,xijs are in hn 2, i = l,...,m and s = 1,2. Write An = M fc(1) (C(Xi)) © • • • © Mk(n){C(Xk(n))), where X,- is a compact connected subset of S1, j = l,...,k(n). Let itj : An —> M f c ( i ) ( C ( ^ ) ) . By (e7.8) fa )*o ([««]) ^ 0. Therefore faMfe;]) > 2 (in Z). Thus, by passing to a subsequence, the lemma follows. • Lemma 4.7.3 Let (G,G+) = \imn^>.00(Gn, (G n )+^")> w/iere eac/t G„ is a finitely direct sum of Z, suc/i that each partial map of (f>n has positive multiplicity at least 2. Let n\ < n2 < • • • be an increasing sequence of integers. Then (G, G+) = \im.n^00(Sk, (Sk)+,Tk), where each Sk is a finitely direct sum of m(k) copies of Z with m{k) > nk and each partial map of rk has positive multiplicity at least 2. Proof. Without loss of generality, by passing to a subsequence, we may assume that each partial map of (f>k has multiplicity at least nk. Let Gk = ©j=i Gk,j, where each Gkj is Z, 4>k : Gk>i ->• Gk+ij be the partial map of cj>k and 4>k : Gk —$• Gk+\ —• Gk+iij is the composition. Let K(k, 1) be the minimum multiplicity of (j>k 'J' (j = 1,..., m(fc + l)). Note that K(k, 1) > nk. Let Fk be a direct sum of K(k, 1) copies of Z. Define a homomorphism fk • GkA -> Fk by sending n ->• (n,n,...,n) (n 6 Gk,i), nkj) : Fk ->•

198

Classification

of Simple

AT-algebras

Gk+i,j by sending {zi,...,zK(ktl)) to Y^^l'l) lszs (in Gk+i,j), where h = <j>(k'j)(l) - K(k, 1) + 1 and ls = 1 for s > 1 and define irk = e^ = ( i +1) 7r^' ) . Note that Z: > 0. We have n{kj) o / f c (l) = ^ 1 J ) ( 1 ) . Thus we have the following commutative diagram

//*

,u>

Gfc,!

>•

Gk+\j.

Now define 5i = Fi © f f i ^ d . i and S 2 = F3©ffi™(23)G3,i, ..., 5fc = F2fc+i © ©™22fc+ G2fc+i,i, •••• We also define r M , i = / 2 fc +3 ° (02fc+2 ° ^fe+i) which induces the following commutative diagram •c v 2fc+l

> 'P2k + 2

n

^2k+2

•

J^2k+3

(

n

-^2fc+3,l

Define T^IJ = 4>2k+2 ° n2k+i (j ^ !)• So the following commutes F2k+1

—^

G2fc+3,j

'T 2 fc+1

/

,*W) 2fc + 2

w

G 2 fc+2

Define r fc)iii = ^ + 2 ° 2k+i\G2k+i,i, i ^ hj ¥" 1 and r fc]iil = / 2 f c + 3 o (^1+2) 000(5fc,Tfc). Note that each partial map of Tk has multiplicity at least nk. Define \tfc : Sk ->• G2fc by (*fc)|.F2jt+1 = 7T2fc+i and (*fc)|ei>2c?2it+i,i = 2G2)o+i,i- Define j f c : G2k+i ->• Sfc by 0'*)|c?2fc+i,i = /2fc+i and (ifc)|ei>2G2fc+i,i = i<W 2 G 2fc+lii - Define Afc : G2fc -> -S'fc+i by Afc = jfc o <^2fc- Then we have the following commutative diagram £>k

r \*fc ~

Lr2fc

"Sfc+l /Vt

• \**+i

4'2k,2k + 2

>

'Jfc+2 / Ak

s~t

<J2fc+2

for all k. It follows from 1.10.2 that there exists an isomorphism r : S1 —> G. Since each Tfc, ^fc, Afc and <j>2k,2.k+2 are positive, r and r _ 1 are positive. •

Invariants

of simple

AT-algebras

199

Proposition 4.7.4 Let F be a countable torsion free abelian group. Then F — lim„_ >00 (F n ,a 71 ), where each Fn is a finitely generated abelian free group. This easy result is left to the reader (see 4.8.5). Theorem 4.7.5 Let G = lim„_>0o(GrTM n), where Gn is a finitely generated abelian free group and each partial map of 4>n is positive and has multiplicity at least 2, and let F be a countable torsion free abelian group. Then there is a unital simple AT-algebra A such that (Ko{A), Ko(A)+) = {G, G+) andK1(A) = F. Proof. Suppose that F = \xm.k-i<x{.Fk,o.k), where Fk is nk copies of Z. Let ak,i,j be the multiplicity of the partial map from i-th copy of Z to j - t h copy of Z. Set (3k = Y,itj <*fc,i,j> Ik = I l i U A a n d s(k) = fc7fc- By 4.7.4, we may assume that each Gn is mk copies of Z with mk > rik- By passing to a subsequence if necessary, we may assume that each partial map of k has (positive) multiplicity at least s(k) + 1. It follows from 3.4.9 that there is a unital AF-algebra B = \imk^oo{Bk,tpk), where Bk = M ; ( M ) ®---Ml(k,mk), (K0(Bk),K0(Bk)+) = (Gfc,(Gfc)+) and (tpk)*o = ctk, k — 1,.... Denote by ipk the partial map of ipk from i-th of summand of Bk to the j - t h summand of Bk+i- Define Afc =

©^ 1 M ; ( f c i l ) (C(5 1 ))©---M ; ( f c i n t ) (C(5 1 ))©M i ( f c i n f c + 1 ) ©---©M z ( f c , m f c ) .

Define 7rfc;i : Ak ->• M,(fc>i)(C(5'1)) for i < nk and 7rfcji : Ak ->• M^k,i) for i > rik be projections. Let £,k,i,£,k,2,..., be a sequence of points on S1 so that {^fc.ii •••i^fcs(fc)} is 27r/s(fc)-dense in S1. Define (for i < nk and j < nk+i) hk,ij : M^CiS1)) -> Ml{k+1J)(C(S1)) by h

k,i,j(f)

= &&g(f °tk,i,j, f{tk,l), -> f(Zk,r(k,i,j))),

(e7.9)

where t^ij : Sl —>• S1 is defined so that tk,ij(z) — zak-i-^ for z = e%e G S1 and r(k,i,j) + 1 is the multiplicity of the partial map of 4>{k'j)- A l s o / € Mi(kti)(C(S1)) is identified with matrix valued continuous functions in C(5 1 ,M/( fc]i )) in (e7.9). If i > nk and j < rik+i, we define hk,i,j{f) = diag(f(tk,i),

- , f(€k,r{k,i,j)+i)),

(e7.10)

where / is in the i-th summand of Ak and r(k,i,j) + 1 is the multiplicity of the partial map of k' • If i > nk and j < nk+i, define h^ij to be the composition of M^k+ij) —> M^^k+1j)(S1) and ipk • li i > nk

200

Classification

of Simple

AT-algebras

and j > n fc+1 , define hk,ij = ipk'3) • Set hk = ®i:jhktij : Ak ->• Ak+1, A = lim„_>.00(Afc,/ifc) and hktU = hn-i o • • • o hk (n > k). Therefore (K0{Ak),K0(Ak)+) = (Gk,(Gk)+), K^Ak) = Fk, (fcfc)*o = A and (/ifc)*i = ak. So A is a unital ^T-algebra and (K0(A),K0(A)+) = {G,G+) and K^A) = F. To see that A is simple, it suffices (by 3.5.10) to show that for any ideal I oi A with I ^ A, hk,oo(Ak) n I = {0} for all k. Let / G .4fc be a nonzero element, without loss of generality, we may assume that / > 0. Case (i): irk>i{f) ^ 0 for some i > nk. It is clear from the construction that the ideal generated by hk(TTkti(f)) is Ak+i- Case (ii): 7rk,i(f) ^ 0 for some i < nk. There exists n > k such that for any 27r/n-dense subset X of S1, Tfc,i(/)(C) ¥= 0 for some ( e l . Note that {^,1.^,2, -^n,r(fc,ij)> i s 2 7 r / n ' dense in S 1 for any s < 7„. Therefore, for each j , 7r n j O hktn(jrkii(f)) > f(((i,j))e(i,j), where dj £ S1 with f(((i,j)) > 0 and e(i,j) is a nonzero projection in nnj(An). Therefore the ideal generated by f{C,{i,j))e(i,j) in irn,j{An) is irnj(An) itself. Since this holds for each j , the ideal generated by hk>n(f) in An is An itself. Thus, in both cases, if hk>00(f) £ 7, then I D hn,oo(An) D hk>00{Ak). This implies that 7 = A. Thus 7 n hki00(Ak) = {0} for all k. It follows from 3.5.10 that 7 = {0}. Therefore A is simple. It remains to show that RR(A) = 0. We will show that TR(A) = 0. Since A is an ^4T-algebra, TR(A) < 1. It follows from 3.6.6 that A satisfies (SP). In particular, for any nonzero positive element a £ A, there is a nonzero projection e G Her(a). So [e] < [a]. Therefore, by applying 3.7.2, to show that TR(A) = 0, it suffices to show the following: Given any e > 0, any finite subset T C A and any nonzero projection e £ A, there is a finite dimensional C*-subalgebra B of A with ljg = p such that (1) \\px - xp\\ < e, (2) pxp £e B for all x £ T and (3) r ( l -p) < T(e) for all r G T ( 4 ) . Without loss of generality, we may assume that p C hit00(Ai). For any k > 1, there is a projection £>*; € j4fc such that (see (e7.9)) h\,k{x) = (1 - pk)hltk{x)(l

- Pk) ® Pkhi,k(x)pk

for all x £ Ai, pk(hitk(Ai))pk is a finite dimensional C*-subalgebra and r ( l -pfc) < 1/A; for all r £ T{Ak). Since for every trace r £ T(A), r o hkt00 is a normalized trace on Ak, T(hk,oo(pk)) < 1/k. On the other hand, since

Invariants

of simple

201

AT-algebras

A is simple, inf{r(e) : r G T(A)} > 0. Therefore, for large k, T{hkt00{Pk))

< l/2r(e)

for all r G T(A).

Choose p = hk>00(pk).

•

We will now present a criterion for simple AT-algebra to have real rank zero. Lemma 4.7.6 Let a,b,c G C(X,Mn)+ such that 0 < a,b, c < 1, 1 cb = c,ab = b, where X = S or a compact connected subset of S1, and rank(c(i)) > k for every t G S1. Then there is a projection p G C{X,Mn) such that pa—p and rankp(i) > k for every t G S1. Proof. By the end of the proof, it will become clear that we only need to consider the case in which X = S1. For each t G S1, there is a neighborhood vt such that dist(0,sp(c(s)) \ {0}) > dt > 0 for all s e vt and for some dt > 0. Therefore there is a ft G Co((0,1]) + such that ft(c)(x) = qt(x) is a projection of rank at least k for all x G vt and ft(c)(x)c(x) = c(x). The same is true for b. Note cb = c. Therefore, by a standard compactness argument, we obtain 0 = h < t2 < • • • < t2m =

2TT

and qi,...,q2m, e i , e 2 , . . . , e m , where qi G C(Ii,Mn) are projections, where Ii = {eis :se [ti_i,* i + i]} (t = 2 , . . . , 2 m - l ) with h = {els : s G [ti,t2] U fom-iMm)}; ej G C(Jj,Mn) are projections, where Jj = [tj,tj+2], i = l,-.,2m, j = l,...,m, such that qi G Her(7Tj(c)), ej G Her(^(6)), where m : C{S\Mn) ->• C(Ii,Mn) and ,• : CCSSMn) -»• C(Jj, Mn) are the quotient maps, and gi(ti) = gi(i2m), \\qi(s) - qi+i(s)\\ < 1/8 and q2j{t)ej{t)

= qij{t),ej{t)q2j+i{t)

= q2j+i(t)

for s £ It n / i + i , t e C j = {e ie : 0 G [%,t 2 j+i]}, i = l,...,2m, j = 1, ...,m. Moreover, rankgj(a;) > fc for all a; G X. Since e^ is equivalent to a constant projection in C(Cj,Mn), ejC(Cj,Mn)ej = C(Cj,Mi) for some integer I < n (but I > k). By 2.5.1, q2j(s) and q2j+i are equivalent in

202

Classification

ejC(Cj,Mn)ej. t h a t pj(elt2i)

of Simple

AT-algebras

Therefore, there is a projection pj G ejC{Cj,Mn)ej = q2j and pj(e'lt2'+1) = q2j+\. Now define p(s)

=qi(s)

for s G {eie : 0 G [0,i 2 ]},

p(s)

= pj(s)

for s G {eie : 0 G [t 2 j,*2j+i]} and

p(s)

= q2j+i(s)

1

T h u s p G C(S , Mn).

for s G {e i e : 61 G

such

[t2j+i,t2j+2]}.

Since a6 = b, from the construction of p, pa = p.

•

L e m m a 4.7.7 Assume that C(X,Mn), where X is a compact connected subset of S1. If f,g G C(X, M n ) + and /c,fc' G N 6e given so that f < g, r a n k / ( i ) < fc' and r a n k g ( i ) > k for all t G X. If k — k' > 1 and there are fo,fi,go,9i G C(X,Mn)+ such that / 0 / i = f0, ffi = A , 505 — 9 and 9i9o = k for all t G X. Since (1 — / i ) ( l ~ a) = (1 — a)> (1 — a ) ( l ~~ b) = 1 — 6, by 4.7.6 again, there is a projection q' G C(X,Mn) such t h a t q'(l — b) = q' and r a n k g ' ( s ) > n — k' for all s G -X". Let 92 = 1 —
Then the

following

Proof. We have shown in (4.3.5) the equivalence of (1) and (2). If (1) holds, t h e n every self-adjoint element is in the closed real span of projections. T h u s the span of projections is dense in A. Hence the projections separate traces.

Invariants

of simple

203

AT-algebras

We now show that (3) implies (1). We write A = lim.n->.00(An,(j>n), where

and where Xnj is a compact connected subset of S1. Claim: If x G (An)+ and e > 0, there exists an integer mo > n so that sup (l/n(j))\Tr(^m(x)(s))

- Tr(^m(x)(t))\

<e

s,t€Xnj

for all j and m > mo. Suppose, to the contrary, that there is x G (An)+ and EQ > 0 so that for every m > n, there is a summand C(Xmj, M m (j)) of Am and points sm and tm in Xmj such that {l/m{j))\Tr{<j>^m{x){sm))

- Tr{^m{x)(tm))\

> e0.

Let am be any state of A extending the trace am(a) = (1 /m(j))Tr(a(sm)) on Am; and let Tm be a state extending the trace Tm(a) = (l/m(j))Tr(a(tm)). Choose a subset S C N such that a = lim,s<7m and r = lims r m exist as weak-* limits. These are easily seen to be traces on A such that | s0. Hence a ^ r. Now suppose that p £ A is a projection. By 2.5.1 (see also 2.7.2), p is the limit of projections (f>n,oo{Pn) € An. For any rn > n, 4,n,m(Pn) is a projection in Am, and thus has a constant rank on each summand. As the trace of projections in Mk depends only on rank, it follows that crm(n,m(Pn)) = Tm(n,m(Pn))- Therefore a and r agree on each 0„ )OO (p„), and hence on every projection in A. This contradicts the hypothesis that projections separate the traces. We now apply (3.8.20). Let Oj € (An)+ (i = 1, ...,6) such that a,iai+i = cii, i = 1,...,5. If 6 € (0,1) is not in sp(a4), then p = X[S,oo)(ai) is a projection in An such that a\p = a\ and asp = p. If (0,1) n sp(a4) ^ 0 and f(t) = max{i(l — t),0}, b = / ( a 4 ) is nonzero, bai = a^b = 0 and 02(05 ~ b) = a.2- Let <5 > 0 and mo > n be as in Proposition 4.1.8 for b G (-An)+- Thus, if m 0 > n,

(l/m(i))rr(^) m (6)( S )) > 6.

204

Classification

of Simple

AT-algebras

for all s € Xnj, all j and for all m > mo. By applying the claim and by choosing larger mo if necessary, we may assume that sup

(l/m(j))|Tr(<^> m (a 5 - b)(t) - Tr(4>^m(a5 - b)(s)))\ < 5/2

S,t£Xm,j

for all j and m > mo- Hence for any s, t G rank(^) m (a 2 )( S ))

<

Xmj,

Tr(^> T O (a 5 - 6)(s))

<

Tr(^m(a5-b)(t))+m(j)S/2

<

Tr(^m(a5

<

Tr(^m(a5)(t))

- b)(t)) +

Tr(^m(b)(t))

< rank(^m(o5)(t)).

(e7.11)

Let k' = maxs{rank(n,m,(a2)(s))} and A; = max s {rank(^7 m (a5)(s))}. Then, by (e7.11), k — k! > 1. It follows from 4.7.7 that there is a projection p G Am such that a^p = 03, pa6 = p. It follows from (3.8.20) that TR{A) = 0 . • Corollary 4.7.10 Let A be a unital simple AT'-algebra with the unique normalized trace. Then TR(A) = 0. 4.8

Exercises

4.8.1 Let A = l i m ^ o o ^ n , n), where An = © ^ 1 ) C ( X n , i , M n W ) , where Xnj are compact metric spaces. Suppose that A is a unital simple C*-algebra. Show that n(j) —> 00

as n -4 00 for all j .

4.8.2 Prove 4.1.1. 4.8.3 Complete the proof of 4.1.2. 4.8.4 Let A be a unital C*-algebra. Suppose that every unitary u G U(A)o is a (norm) limit of unitaries in A with finite spectrum. Show that RR(A) = 0. 4.8.5 Prove 4.7.4. 4.8.6 Every unital hereditary C*-subalgebra of an ^IT-algebra is an AT-algebra.

205

Exercises

4.8.7 Let An be unital AT-algebras. Show that A = lim n _ >00 (A n , 4>n) is again an AT-algebra, where <j>n : An -> Ai+i is a homomorphism. 4.8.8 Define a homomorphism <j)n : C{S1,M4n)

-*• C ( 5 1 , M4n+i) by

/0 1 0 0\ 0 0 1 0 0 0 0 1 \z 0 0 0/ where z H-» Z is the canonical unitary of C(S1). Let A = 1 \imn^00(C(S ,Min),(l)n). Show that A is simple and RR(A) = 0. This C*-algebra is called Bunce-Deddens algebra. 4.8.9 Let A be a unital C*-algebra and u,v £ U(A) such that [u] = [v] in U(A)/U(A)0. Show that, for any e > 0, there is a unitary W G MJV(A) with finite spectrum for some positive integer N > 0 and a unitary U € MN+i{A) such that ||diag(u, W) - [Tdiag(v, W)l/|| < e

4.8.10 If A is a unital C*-algebra with RR{A) = 0, then conclusion of 4.8.9 holds with N depending only on s (independent of u, v and A). 4.8.11 Let A be a unital AT-algebra with Ki(A) = 0. Show that A is an AF-algebra. 4.8.12 Let A be a unital stably finite C*-algebra and T(A) be the space of normalized traces on A (with weak-* topology). Show that the map p^dp]) = T(JP) for projections in A induces a homomorphism pA : KQ(A) —J- Aff(T(A)), the set of all (real) affine (continuous) functions on T(A). 4.8.13 Let A be a unital simple AT-algebra. Show that RR(A) = 0 if and only if pA(K0(A)) is dense in Aff (T(A)). 4.8.14 Let A and B be two unital simple AT-algebras. Suppose that A has real rank zero and there is an order homomorphism a : (K0{A),K0{A)+,K1(A)) -> {K0{B),K0(B)+,K1(B)) such that ^ ( [ I A ] ) < [lfi]- Show that there is a homomorphism h : A —> B such that /i*o = (X\K0(A) a n d Ki = a\Kl(Ay

206

Classification

of Simple

AT-algebras

4.8.15 Let A be a unital AT-algebra. Fix a finite subset of projections S and the corresponding finite subset V in KQ(A)+. Let G(V) be the subgroup generated by V. Show that there exist an e > 0 and a finite subset Q C A such that for any C?-£-multiplicative contractive completely positive linear map L : A —»• B (where B is a unital stably finite C*-algebra), [£]|G(T>) i s well defined and positive. In the following two problems X will be a connected compact metric space with dimension d. A projection q £ C(X, Mn) is said to be trivial if q is equivalent to constant projection q' (q'(x) = q'(y) for all x,y G X). 4.8.16 Let q\,qi be two projections in C(X,Mn). Show that (see [88]) (1) there is a trivial pi G C{X,Mn) such that p\ < q\ and rankpi > rankgi — (d/2), (2) there is a trivial projection p 2 £ C(X, MN) such that q2 < pi and rankp 2 < rankg 2 + (d/2), where N ^, then q2 is equivalent to a subprojection of 4.8.17 Suppose that / , g G C(X, M„)+ and fc,A;' £ N are given so that there is a £ C(X, M n )+ such that fa = f,ag = a and rank/(a;) > fc and rank (7(0:) < k' for all x € X. Use (4.8.16) to show that there are projections p,q £ C(X,Mn) so that (1) pg = p and rankp(z) > k - ( l / 2 ) ( d + 1); (2) fq = f and rankg(z) > k' + (l/2)(d + 1). Let gi, #2 be two projections in C(X, Mn). Show that (see [88]) (3) there is a trivial pi £ C(X, Mn) such that p\ < q\ and rankpi > rank31 — (d/2), (4) there is a trivial projection p2 £ C(X,M^) such that q2 < p2 and rankp 2 < rankg 2 + (d/2), where N d, then q2 is equivalent to a subprojection of Qi-

4.8.18 Assume that C(X,Mn) is a unital subalgebra of a C*-algebra B with tsr(B) = 1. Let f,g G C(X,Mn)+ and k,k! G N be given so that / < rank f(x) < k' and rankg(a:) > k for all a; G X. If fc — k' > 2d + 1, there are a,b,c,d G C(X, Mn)+ such that a& = a, bf = b, gc = g and ccf = c. Show that there is a projection p G C(X, Mn) such that bp = b and n) be a simple C*-algebras, where An = ®rjL\Pn,jC(Xnj,Mn(j))Pn,j, where Xnj is a connected finite CW complex, and Pnj C C(Xnjj,Mn(j)) is a projection. We say A has slow dimension

207

Exercises

growth if dim-Xn^ . lim maxj{ ———-} = 0. n->oo rank(P„,j) Use (4.8.17) and (4.8.18) to show (1) if p, q G A with r{p) < r(q) for all tracial states on A, then p is equivalent to a projection in qAq, (2) A has stable rank one, (3) A has real rank zero if and only if projections in A separates the traces. 4.8.20 Let A be a unital simple C*-algebra with RR(A) = 0, tsr(A) = 1 and with unique normalized trace r and let j : C(S1) —>• A be two monomorphisms (j = 1,2). Suppose that (i)»i = {4>i)*i (i = 0,1) and T O 4>\ = T O (f>2-

Show that there exists a sequence of unitaries Un £ A such that lim UX<j>i{x)Un = <j>2{x) for all x G C(5 X ). n—»oo

4.8.21 Let i be a unital C*-algebra define cel(^4) = inf{L(w) : u G U(A)o}. If RR(A) = 0, show that eel(4) < IT. 4.8.22 Let A be a unital C*-algebra and u G U(A)o with cel(u) < b for some b > 0. Then there is a norm continuous path {«(£) : t £ [0,1]} in C/(A)o such that \\u{t) - u{s)\\ < (b + 5)\t-s\

for s,t £ [0,1].

4.8.23 Let C be a stable C*-algebra. Show that cel(A) < 2n. 4.8.24 Let Cn be a sequence of stable C*-algebras. Show that

Ko{Y[nCn) = Y\nK0{Cn). 4.8.25 Using 4.8.23, show that ffifll„ Cn) = Un

Ki(Cn).

4.8.26 Let A be a unital C*-algebra and p, q G A be equivalent. There is a v G U(M2(A))o with cel(v) < 7r such that v*pv = q. If pq = 0, then there also exists a t i £ U(A)o such that u*pu = q and cel(u) < 7r.

208

Classification

of Simple

AT-algebras

4.8.27 Let A and B be C*-algebras and t : A —¥ B be a homotopic path (t £ [0,1]) contractive completely positive linear maps. Suppose that for each t £ [0,1], there is ipt '• A -»• B K. such that ipt © t can be approximated by homomorphisms from A to B®K with finite dimensional range. Prove that for any finite subset T C A and e > 0, there is an integer n > 0 and there are homomorphisms hi,h2 : A —> B ® Mn{B) with finite dimensional range such that ||diag(<£0(a),/ii(a)) - diag(0i(a),/i 2 (a))|| < e for all a £ J. The following is a characterization of ordered K§ for a simple ATalgebra. 4.8.28 Use 3.9.14 and section 4.7 to show the following: Let (G, G+) be a unperforated simple ordered group with the Riesz interpolation property. Then (G,G+)=

lim(G n ,(G„) + ,4>„), n—+oo

where G n is a finite direct sum of Z with the usual order and where 4>n is a positive homomorphism so that each partial map has multiplicity at least 2. 4.9

Addenda

Definition 4.9.1 Let A be a unital G*-algebra with stable rank one. Let (K0(A) ®Ki(A))+ be the set of those elements [p©v], where [p] G Ko(A), [v] £ Ki(A) so that there is a projection p £ Mn(A) (for some n > 0) and a unitary v £ pMn(A)p such that [v] = [v + (1 — p)]. Define an order on graded group K0(A) ® Ki(A), by the cone (K0(A) © Ki(A))+. The following s another theorem of G. A. Elliott. Theorem 4.9.2 Let A and B be two unital AT-algebras. Then A = B if and only if there is an (graded ordered) isomorphism a : (K0(A) © K^A),

(K0(A) © K^A))^

[1A])

-»• (K0(B) © Ki(B), (K0(B) © ^ ( 5 ) ) + , [1 B ]). Theorem 4.9.3 (Elliott-Evans) For any irrational 8, the irrational rotation algebra, Ag, is a unital simple AT'-algebra of real rank zero.

Addenda

209

Notes Theorem 4.9.2 was proved by George A. Elliott ([56]) which initiated the program of classification of simple amenable C*-algebras. Since we are primarily interested in simple C*-algebras, it is perhaps appropriate to present here a proof of the theorem for simple AT-algebras of real rank zero. Theorem 4.2.8 was proved by the author ([104]) which is used in the proof of 4.9.2 presented here. It has other roles in the study of C*-algebras of real rank zero. In the second part of these notes, a generalization of this proof will be presented for a broad class of amenable simple C* -algebras with tracial topological rank zero. Lemma 4.1.2 and 4.1.4 are taken from [56]. Lemma 4.2.3, Lemma 4.2.4 and Lemma 4.2.5 can be found in [151]. The rest of 4.2 is taken from [104]. A similar version of Lemma 4.4.2 appeared in [69]. Most statements in section 4.5 appear the first time. Theorem 4.7.5 is due to Elliott. The last part of section 7 is taken from [6] and [5]. From the proof of 4.6.4, one should realize that by using an argument (the approximate intertwining-1.10.16) of Elliott, to prove a classification theorem, we need an existence theorem (4.6.2) and a uniqueness theorem (4.5.5). Much of the later chapters are devoted to establish a general uniqueness (see 6.3.3) and a few versions of existence theorem (see 6.1.11 and 6.2.9).

1-^tJlt

Chapter 5

C*-algebra Extensions

The theory of C*-algebra extensions began with the work of Brown, Douglas and Fillmore ([24], [25]) on classifying essentially normal operators (see 5.10.21). But soon it developed into a theory related to many fields of mathematics. Our presentation here is very limited. The main goals of this chapter are Theorem 5.6.4, Theorem 5.9.9 and other closely related results.

5.1

Multiplier algebras

Definition 5.1.1 Let A be a C*-algebra. Denote by M(A) ( called the multiplier algebra of ^4) the idealizer of A in A", i.e., M(A) = {x £ A" : xa, ax £ A for all a £ A}. Proposition 5.1.2 For any C* -algebra A, M(A) is a C* -algebra. Moreover, A is an ideal of M(A). Proof. It is easy to verify from the definition that M(A) is a *-subalgebra of A". To see it is closed, let y £ A" such that there are xn & M(A) with ll^n - y\\ ->• 0- Then \\ya — xna\\ —> 0 and ||ay — axn\\ —>• 0 as n - > o o for all a £ A. Since xna and axn £ A, we conclude that ay, ya £ A for all a £ A. Therefore y £ M(A). This implies that M(A) is a C*-subalgebra of A" containing A. It follows from the definition that A is an ideal of M(A).

• 211

212

C-algebra

Extensions

Example 5.1.3 Let A = C0(X), where X is a locally compact Hausdorff space. Then M(A) = Cb(X), the set of all bounded continuous functions on X (with supremum norm). It is known that Cb(X) = C(j3(X)), where (3(X) is the Stone-Cech compactification. Therefore one sometimes calls M(A) a non-commutative Stone-(7ech compactification of the (non-unital) C*-algebra A. Here is a non-commutative example: M(K.) = B(l2). Definition 5.1.4 Let A be a C*-algebra. A (closed two sided) ideal I
Proposition 5.1.5 Let A be a C*-algebra and B be an ideal of A. Then there is a homomorphism a : A —> M(B) such that O~\B = id^ and kercr = {a £ A : ba = ab = 0 for all be

B}.

In particular, if B is an essential ideal of A, then a is injective. Proof. By 1.11.54, we may identify B" with the strong closure of B in A". Let p be the identity of B". Then p is a projection in A". Let {ea} be an approximate identity for B. Then ea —• p in B" (weakly). For any a £ A, aea G B. So paea = aea. Therefore paea —¥ pap and paea = aea = apea -> ap (weakly). Similarly, we have pap = pa. We conclude that pa — ap and pa G B" for all a £ A. Thus the map a : A —> B" given by a {a) = pa is a homomorphism and takes A into M(B), since for each b £ B and o 6 A, (j(a)6 = apb = ba £ B. The kernel of a is {a £ A : pa = ap = 0}. If B is essential, the kernel is zero. Thus a is an isomorphism. • Definition 5.1.6 Let A be a cr-unital C*-algebra. Define a topology on M{A) by the semi-norms (of x): \\ax\\, \\xa\\ (a £ A). This topology is called the strict topology. If {xa} C M(A), {xa} converges in the strict topology if and only if {xaa} converges for each a £ A (in norm).

Multiplier

algebras

213

We leave the following as an exercise (see 5.10.1). Theorem 5.1.7 Let A be a cr-unital C*-algebra. Then completion of A in the strict topology is M(A). E x a m p l e 5.1.8 Let B be a cr-unital C*-algebra. Let us describe M(SB ® K). Denote by C b ((0,1), M(B K)a) the algebra of all bounded continuous mappings from (0,1) into M(B(g>/C) with the strict topology on M{B®K). Note that i f / G C 6 ( ( 0 , 1 ) , M ( 5 ® / C ) £ r ) and c; G SB®K (which we identify with continuous mappings from [0,1] into B ® K, (with norm topology) vanishing at 0 and 1. Then by definition of C 6 ((0,1), M(B®K,)a), fg eSB®)C. Let / G C*((0,1), M ( f l ® £)„). Define ||/|| = sup{||/(t)|| : t G (0,1)}. It is easy to check that M(SB K.) S Cb((0,1),M(B K)a) which maps SB K. to C 0 ((0,1), B®IC). Let X be a locally compact and cr-compact Hausdorff space and B — C0(X, A), where A is a cr-unital C*-algebra. Then M(B) = Cb(X, M{A)a), the space of all bounded continuous mappings from X into M(A) (with the strict topology) (see 5.10.2). 5.1.9 Let B be a unital C*-algebra. One can use certain infinite matrices to identify elements in the multiplier algebra M(B K). Let {ey} be a system of matrix units for K. and e„ = 52™=1 en. Let x = (ay) be a formal infinite matrix with ay G B. If, for any e > 0 and Z > 0, there is an integer N > 0 such that ||(e„+ m - e„)(ay)ez|| < e and ||e;(ay)(e n + r n - e„)|| < e for all n > N and for all m, then a; represents an element in M(B ® K,). Conversely, let x G M(B ® /C) and ay = enxejj G B. One shows that the matrices (ay) must satisfy the above condition. In particular, if we identify those infinite matrix in M{B®K) with scalar entries with elements in B(l2),

then B(l2)

CM{B®K).

We leave the details as an exercise (see 5.10.3). Lemma 5.1.10 Let A be a a-unital C*-algebra and {en} be an approximate identity for A. Suppose that {an} is a bounded sequence in A. Then oo

X ] ( e « + i ~ en)1/2an(en+i

- en)1/2

n=l

converges in the strict topology to an element in M(A).

214

C*-algebra

Extensions

Proof. Let M = sup{||a n || : n = 1,2,...}. It suffices to consider the case in which 0 < an < M. Set xk = £ * = 1 ( e n + i - en)1/2an(en+1 en)1/2, 1 2 1/2 k = 1,2,.... Since (en+1 - e n ) / a n ( e „ + i - en) < M(en+X - e„), we conclude that ||a;fc|| < M\\ J2n=i(en+i ~ en)\\ < M. We need to show that \\a(xk+m - Xk)\\ -> 0 and ||(a;fc+m - a;fc)a|| ->• 0 as A; -» oo for all a € A Fix a € A and £ > 0. There is K > 0 such that ||o(l - efc)|| < s2/\\a\\(M + 1) and ||(1 - efc)o|| < e2/\\a\\(M + 1) for all k>K.

Therefore

\\a(xk+m - xk)\\2

<

||a(a;fc+m - xk)2a\\ < M\\a(xk+m

- xk)a\\

ro+fc

<

M\\a{ ^

<

e2

M(en+1 for

- en))a\\ < M 2 | | a ( l - ek)a\\

k>K.

Similarly, ||(a;fc+m - xk)a\\ < e

for for k>

K.

Lemma 5.1.11 Let A be a a-unital C*-algebra and {fn} be an approximate identity for A. Let an 6 (fn+i — fn)A(fn+i — fn) be a bounded sequence of elements in A. Put oo

Vk = ^

oo

an enk and V = y^Vfc-

n=l

fc=l

Then both sums converge in the strict topology as elements in M(A (g> K). In particular, suppose that fn+ifn — fn (for all n) and an = (fn — / n - i ) 1 ^ 2 (with /o = 0 / Then V is an isometry in M(A ® K) such that (EZi fnennW = V and P ( £ r = i /„ ® enn) = P, where P = VV*. Proof. Note that a2no-2k = o-2k0.2n = 0 and a2n+ifl2fc+i = «2fc+ia2n+i = 0 (k ^ n). Therefore Y^n=i a2n and Yln°=ia2n+1 converge in the strict topology. So 2^°=i an and ^ ^ L j anan converge in the strict topology. For any m, m

m

m

||(^]a Tl e n fc)*(^a rl (g)e n fc)|| = | | ^ a * a n | | . n=l

n=l

n=l

Multiplier

215

algebras

Therefore there is M > 0 such that || £)™=1 a « ® e"-k\\ < M. Fix a € A /C. Then m

CXI

| | a ( ^ a n ® e n f c ) | | < sup{||a„||}||a(l - ] T e„ n )|| -» 0 as m —> co. We also have I

oo

(y~] an ® e„fe)(l - ^ n=l

ejj) = 0

i=l

if I > k. Write a^ = euaejj, i,j = 1,2,.... Then there is lo > 0 such that if Z > Zo ||(1 - /,)ay|| < e/(fc + l ) 2 , i , j = 1,2,...,fc+ 1. Then, if I > max{/c, Z0}, OO

||(^an®enfc)a||

OO

<

n=Z

n=£ oo

+

fc+1

| | ( ^ a „ e „ f c ) ( ^ e i i ) a | | i=l fc+1

||(y^anOerafc)(l - y ^ e j i ) a | |

= M|| 5^(1 -/,)a«|| <e. ».j=i

This implies that Vj. converges in the strict topology. So Vfc € M(A/C). A similar argument (5.10.4 ) also shows that X]fc=i ^fc converges to V in the strict topology. So V <5 M(A®/C). If / „ + 1 / „ = / „ and a n = (/„ - / „ - i ) 1 / 2 , we compute that ^ * V = 1. This implies that P = VV* must be a projection in M(A /C). We also have Q T ^ / „ e „ n ) F = V. D Lemma 5.1.12 LetB be aC*-algebra andp e M{B®K) be a projection. Suppose that there are two projections qi < p and qi < (1 — p) such that q\ and qi a r e Z>o£/i equivalent to 1. TTien p and 1 — p are both equivalent to 1. Consequently K0(M(B ® /C)) = {0}. Proof. It suffices to show that, for any projection / € M(B /C), / © 1 is equivalent to 1. Let B(l2) C M(B /C) be as in 5.0. Then, we obtain a sequence of mutually orthogonal projections {/„} so that the sum X^^=i fn converges to 1 in the strict topology and each fn is equivalent to 1. Thus,

216

C * -algebra

Extensions

for each n, there is a projection pn < fn such that pn is equivalent to / „ . By using a shift one can easily shows that oo

oo

n=pi

PI ® Yl f

® Y;(pn

n=2

+

(fn ~pn^

n=2

is equivalent to Y^n°=2 fn which is equivalent to 1. Consequently / © 1 is equivalent to 1. • Lemma 5.1.13 Let A be a a-unital C*-algebra and P = E^Li e 2n2nSuppose that Q £ M(A ® /C) is a projection so that both Q and 1 — Q are equivalent to 1. Then, for any 1 > S > 0, there are projections P' < P and Q' < Q such that P — P', P', Q' and Q — Q' are all equivalent to 1 and

WP'Q'W < s. Proof. There is a unitary W € M(A K.) such that Q = WPW*. Let {/„} be an approximate identity for A such that fn+ifn = fn (n — 1,2,...) and an = (fn - / n - i ) 1 / 2 (with /o = 0). Put gn = an®einin and hn = WgnW*. For any n and m, gnWgm = gnWPgm and gnWP € A K.. Therefore (dnWP)gm —>• 0 for any fixed n a s m - t oo. Thus lim gnhm = lim (gnWPgm)W* m—>oo

=0

m—foo

for any fixed n. Similarly, linin^oo gnhm = 0 for any fixed m. Therefore, for any S > 0, we obtain two subsequences {gn(k)} a n d (^m(fc)} s u c n that ||ffn(i)^m(j)ll < S/2l+J, i, j = 1, 2,.... As in 5.1.11, we obtain two projections P> ^ (E» a «(i) ® e4n(i)4n(i)) < -P and Q' < W(J2j am(j-) (gi e4m(:7-)4m(i))W* < Q which are both equivalent to 1. Since P = Yln°=i e2n2n, we see that P—P' is also equivalent to 1 (see 5.1.12). Similarly Q — Q' is also equivalent to 1. Moreover, \\P'Q'\\ = i

\\nYt9n(i))'Zhm(j))Q'\\<5. 3

•

Lemma 5.1.14 Let u e U(M(B AC)). Suppose that p <E M ( S ® /C) is a projection so that both p and 1—p are equivalent to 1. Suppose also that up = pu — p. Then u G U(M(B <S> £))o a i d cel(u) < 27r. Proof. We may write u = diag(v, 1), where v is a unitary. By writing 1 = diag(l, 1,...) (see the proof of 5.1.12), we define w = diag(v*,v,v*,v,....). It follows from 4.2.5 that diag(v*,v) is connected to

Extensions

217

of C* -algebras

diag(l, 1) in U-2(M(B®K)) with a path z(t) of length no more than ir. Put Z(t) = dia,g{v, z(t),z(t),...). Then Z(0) = u and Z(0) = diag(v,w) such that the length of Z(t) is no more than IT. The same argument also shows that dia,g(v,w) is connected to 1 with a path which has length no more than 7r. • Theorem 5.1.15 Let A be a a-unital C*-algebra. Then U{M{A®K)) connected. Moreover

is

cel((M(A ® K.)) < 6TT (see 4.8.21) Proof. Fix U G U(M(A K)) and 5 > 0. Let P be as in the proof of 5.1.13. Let Q = U*PU. It follows from 5.1.13 that there are projections P'

Q"(l

- P')

=

Q".

Since 1 — P is orthogonal and equivalent to P and P > P', by 4.8.26, there is a unitary Vl G £/((.A ® /C))0 with cel(yi) < TT such that ^ P ^ = P ' . By 4.8.26, there is a unitary V2 £ J7((vl ® /C))0 with cel(F2) < 7r such that VJP'Vz = (1 — P ) . Since P ' is orthogonal and equivalent to (1 — P ) and Q", there is a unitary V3 G £^((-<4 ® /C))o with cel(Va) < n such that V3*(l - P)V 3 = Q". Since \\Q" - Q'\\ < 5/2TT, there is a unitary V4 £ U((A ® X;))0 with c e l ^ ) < S such that V£Q"V4 = Q'. Since Q' < Q and (1 — Q) is orthogonal and equivalent to Q, there is a unitary V5 G f/((A ® /C))0 with cel(V5) < TT such that V^Q'Vs = Q. Thus we obtain a unitary W G f7((A®£))o with ce\(W) < 4TT + J such that W*UPUW = P. It follows from 5.1.14 that UW G U((A ® K))0 with cel(C/W) < 2TT. The lemma follows. D

5.2

Extensions of C*-algebras

Definition 5.2.1 Let A and P be C*-algebras. An extension of A by P is a short exact sequence 0 -> B^-E-^A of C*-algebras.

-> 0

218

C -algebra

Extensions

Example 5.2.2 Let B = C 0 ((0,1)) and A = C. Then Ex = C^S 1 ) is an extension of A by .B. It is also clear that E2 = Co((0,1]) and E3 = C © Co((0,1)) are also extensions of A by B. They are three quite different extensions. Example 5.2.3 Let S be the unilateral shift defined in 1.11.4. Let T be the C*-algebra generated by S. An exercise (5.10.5) shows that T is an extension of C(S1) by /C. Definition 5.2.4 Let A and B be C*-algebras and <j> : A —> B be a homomorphism. Define the mapping cone of , denoted by 0$, to be {(a,/):(«) = / ( l ) } C .4 ©C 0 ((0,1],B). There are maps s : C$ -» A and j : S B —» C,/, given by s((a,f)) j ( / ) = (0> / ) • Therefore C^ is an extension of A by SB :

= a and

0 -> S J 5 4 C 0 A A -> 0.

One of the differences between E2 and E3 is that £3 contains elements that are orthogonal to the ideal Co((0,1)). Let C = {x £ E3 : xa = ax = 0 for all a £ CQ((0, 1))}. Then C — C. Such extensions do not seem too interesting. This leads to the following definition: Definition 5.2.5 An extension E of A by B is essential if A is an essential ideal of E. Thus E3 in 5.2.2 is not essential. On the other hand, E\ and E2 are essential extensions of A by B. Moreover T in 5.2.3 is also essential. Definition 5.2.6 Given an extension 0->B^E->A->0. Then B sits as an ideal of E. Let a : E -> M(B) be as in 5.1.5. The map cr is injective if and only if the extension is essential. Let IT : M(B) —> M(B)/B be the quotient map. Define r : A = i?/B —^ M{B)/B by the map 7r o B —> E —> A—> Q. For extensions, we will study their Busby invariant. Definition 5.2.7 Let A and B be two C*-algebras. Consider extensions of A by B /C. To save notation, we denote by Q(B) the quotient C*algebra M(B <8> /C)/B ® /C. An extension T : A -> Q(B) is called trivial if there exists a : A —> M(B K) such that IT O a = r. Let r i , T2 : A -> Q(B) be two extensions. We say T\ and T 2 are unitarily equivalent if there is a unitary u € Q(B) such that adu o n = r^-

Extensions

of C -algebras

219

Denote by Ext (A, B) the set of all equivalence classes of extensions of A by B ® K. Let TI,T 2 : A —> Q(B) be two extensions. By identifying M2(Q{B)) with Q(B) via an isometry v G M2{Q{B)) with v*v = 1 2 and vv* = 1, we may add ri and T 2 by defining T\ © r 2 to be V(TI © r 2 )u*. By 5.10.6, if A is separable, every element in Ext(A,B) is represented by an essential extension. Let T\ and r 2 be two trivial extensions. It is easy to see that T\ © T 2 is also a trivial extension. Let Exto(A, B) be the set of unitary equivalence classes of trivial extensions. Denote by Ext(A,B) the semigroup of Ext(A,B)/Ext0(A,B). An extension To is said to be absorbing if for any trivial extension r, r © To is unitarily equivalent to To. Proposition 5.2.8 Ext(A,B) = {0}.

Let A = C o ((0,1]). Then for any C*-algebra B,

Proof. Let T : A —• Q(-B) be an extension. Let x E Abe the self-adjoint generator with sp{x) = [0,1]. There is a self-adjoint element y G M(£? ® JC) with ||y|| = 1 such that 7r(y) = T(X), where 7r : M ( B ® /C) ->• Q(-B) is the quotient map (see 1.11.10). Define cr : A -» M(B ® /C) by cr(f(x)) = f(y) for / G C o ((0,1]). We see that TT O a = r. Therefore £a;t(A, B) = {0}. D Proposition 5.2.9 Every element z £ Ko(Q(B)) can be represented by a projection e G Q(B) for any a-united B. Moreover, [1] = 0 in KQ(Q(B)). Proof. Suppose that z = \p] — [q], wherep,g G Mn(Q(B)) are projections. By 5.1.12, [ l n + i © p © 1] = [pffi 1]. Therefore / = (p© 1) © ( l n + 1 - g© 1) is a projection and [/] = z. There is v G Mn(Q{B)) such that v*v = ln and wv* = 1. So we may assume that / G Q(B). • Proposition 5.2.10 Lei A = C([0,1]) and B be a a-unital C*-algebra. Then there is an isomorphism 7 : Ext{A,B) —¥ KQ{Q{B)). In particular, Ext{A,B) is a group. Proof. Given z G KQ(Q(B)), by 5.2.9, there is a projection p G Q{B) such that \p] = z. It is easy to find a self-adjoint element x G n(eB(l2)e) with sp(x) = [0,1], where TT : M(B®K) -> Q(B) and e G B(l2) is an infinite projection. Here we again identify B(l2) with a unital subalgebra of M ( B ® /C). By 5.2.9, we may assume that p > e. Therefore the C*-subalgebra D generated by p and x is isomorphic to C([0,1]). Thus we obtain an injective homomorphism r : A -> Q(B). Then r»o gives a homomorphism

220

C*-algebra

Extensions

from Z = Ko(A) to KQ(Q(B)). This defines a (semi-group) homomorphism 7 : Ext(A,B) -» Hom(Z,K0(Q(B))) S ir 0 (Q(.B)). The above also shows that 7 is surjective. To see 7 is injective, we assume that [T(1A)] = 0 in K0{Q(B)). Thus r (l^i) © In is equivalent to 1„ for some integer n. Thus we may assume that T{1A) = 1. Let a; € r(^4) with sp(x) = [0,1] such that T{X) and 1 generate T(A). There is y € M{B®K) such that sp(j/) = [0,1] and ir{y) = x. Define ®i=o,iKi(Q(B)). See (5.10.7). We leave the following to the reader as an exercise (5.10.8). Proposition 5.2.12 Let that [T\] = [T2] in Ext(A,B). such that

: A —• Q(B) be two extensions. Suppose Then there exists a unitary u € M(B (g> K)

T\,T-I

7r(u)*(Ti © TO)TT(M) = T2 © TO

/or some trivial extension TQ. Definition 5.2.13 Let A be an amenable C*-algebra. Define by Ext{A,SB). So KK(C0{{0,1],B) = {0}, tfA-(C([0,l],fl) = and Jftr/sT(C(51)) = (Bi=o,iKi(Q(B)).

KK(A,B) K0{Q{B))

Definition 5.2.14 (Mapping torus) Let A be an amenable C*-subalgebra of B and B be a o"-unital C*-algebra. Suppose that a : A —> B is a homomorphism. Then a gives an element in KK(A, B) as follows. Define the mapping torus by Ma = {/ G C([0,1],B) : /(0) G A and / ( l ) = a(/(0))}. This gives a short exact sequence: 0 - • SB -> M a -> A -> 0. By identifying C([0,1], 5 ) with a C*-subalgebra of euM(SB K)eu, we obtain an extension a : A -» Q(SB). We will write [a] for the corresponding element in Ext{A, SB) = KK{A, B).

Completely positive maps to M n ( C )

5.3

Completely positive maps to

221

Mn(C)

Lemma 5.3.1 Let A be a separable C-algebra and L be a closed left ideal of A. Then there is a sequence of positive element {en} c L such that e n e n +i = e„, ||e n || < 1 and \\xen - x\\ -> 0 and \\eny - y\\ -» 0 as n -4 oo for all x € L and y £ L*. Proof. Let B = L f) L* be the hereditary C*-subalgebra (see 1.11.51). Since B is separable, B = aAa for some positive element a £ A (1.5.10). Let {e n } be an approximate identity for B such that enen+i = en (n = 1,2,...). If x £ L, then a;*a; G L n L* = JB. So ||x(l - e n )|| 2 = ||(1 - en)x*x(l

- en)\\ -+ 0

as n —>• oo. So ||a;(l — e n )|| —>• 0. Consequently, ||(1 — en)y\\ —> 0 as n —>• oo for all y G L*. D Lemma 5.3.2 Let A be a unital separable C*-algebra and <j> : A—> C be a pure state. Then for any a £ A there are zn G A+ with \\zn\\ = 1 such that znzn+i = zn+i and ||z„(<£(a) -a)zn\\

-»0

as n —>• oo. Proof.

Let

Z^ = {a G A : <j){a*a) = 0} and Nj, = {a G A : <j>{a) = 0}. Note that L$ is a closed left ideal. Let {en} C L with e„ > 0 and ||e n || = 1 be as in 5.3.1. It follows from 1.11.35 that Nj, = 1$ + LJ. For any x G A, <j>{x) • 1^ — x G N^. So there are a £ L and b £ L* such that 0(x) - a; = a + b. By 5.3.1, ||(1 - e„)(0(:c) - z)(l - e „)|| = ||(1 - e„)(a + 6)(1 - e„)|| < ||(1 - e„)a(l - e„)|| + ||(1 - e„)fo(l - e„)|| -»• 0 as n —• oo. Set zn = 1 — en. We note that z n G A+, \\zn\\ = 1 and znzn+\ = (1 - e n )(l - e n + i ) = 1 - e n + 1 = 2 n + 1 . D

222

C* -algebra

Extensions

Definition 5.3.3 Let A and B be two C*-algebras and <j> '• A —• B be a linear map. The map is said to be full if the (closed) ideal generated by 4>(A) is B. Definition 5.3.4 Let A be a C*-subalgebra of a C*-algebra B. Define dn{o) = diag(a, a, ...,a) : yl —• Mn(B), the diagonal map (where a is repeated n times. We also use d^ : A -) M(B ® K) for the diagonal map d<x>(a) = diag(a, ...,a,...). Lemma 5.3.5 Let A be a unital separable C*-algebra and let j : A —¥ B be a unital full embedding. Then, for any state : A —>• C I 5 C B, any finite subset J- C A, and any e > 0, there is a partial isometry V G Mn(B) (for some integer n > 0) such that \\(a) - V*dn{a)V\\ <e

for a e f and V*V = lB.

Moreover, if 4> is only assumed to be a nonzero positive linear functional with \\4>\\ < 1, then the above still holds when V is merely a contraction. Proof. First assume that <j> is pure. By Lemma 5.3.2, there are zi,z2 A+ with Z1Z2 = z\ and ||ZJ|| = 1 (i = 1,2) such that \\4>{a)zf -

Ziazi\\

<e

a 6 f ( j = l,2).

£

(e3.1)

Since j : A —• B is full, the ideal generated by A is B. Therefore, by 3.3.6, there are x\,..., xk £ B such that k

^2x*zfxi = 1 B . Define V* = (x\z\,x\z\, ...,x*kzi) (so V is a column). Note that V*V = 1BFurthermore, dk(z2)V — V since z2z\ = z\. We estimate (by (e3.1)) U(a)-V*dk(a)V\\

=

||^(a)F*dfc(z22)^-^*(4^2az2))^||

<

\\cj>(a)dk{zl) - dk(z2az2)\\

<£

for all a G T. For a general state , by the Krein-Milman theorem, we have positive numbers a 1 ; ..., am with X)i=i a j = 1 a n d pure states i,..., 4>m such that m

||^(a) -J2aiMa)\\

<e/2,

a£f.

223

Completely positive maps to M n ( C )

Let k(i) be the integer k in the first part of the proof corresponding to i. Set n(i) = Tlj=i k(J) w i t n n ( ° ) = °- L e t Vi e Mn(m){B) be given as in the first part of the proof such that n(i-l)

V*Vi = 1B,W;

< (0^0',dKi)(lB),0,...,0)

and

||^»(o)-V;*d f e (i)(o)Vi||<e/2, Set V = E H i i / W ^ j G Mn{m)(B). Moreover

a£F.

We see that ^ * V = £ ™ i "ilfl = I s m

||0(a) - F*d„(a)V||

=

||
m

m

<

||
<

e/2 + s/2 = e,

i=l

i=l

a e F.

(e3.2)

To see that the last statement of the theorem holds, we note that there is 1 > A > 0 such that <j>(a) = A • ip(a) for some state ip and for all a G A. D We leave the proof of the following lemma to the reader (see 5.10.9). Lemma 5.3.6 Let pi,...,pk be infinite dimensional projections and ai, 0-2, •••,am in B(l2). Then, for any e > 0, there are infinite dimensional projections qi
i ± j,l = l,2,...,m.

Lemma 5.3.7 Let A be a unital separable C*-subalgebra of B(l2) with 1A = 1B(/2) and <j>: A —> C be a nonzero positive linear map with \\<j>\\ < 1. Suppose that 4>\AHK = 0. Then, for any e > 0 and any finite subset J- C A, there exists a contraction V G B(l2) such that \\{a)-V*aV\\<e

for a G T,

where we identify C with the center of B(l2). Proof. The proof of this lemma is a modification of that of 5.3.5. Let 7T : B(l2) —> B{l2)/K be the quotient map and let e„ be as in the proof of 5.3.2. Since 0 vanishes on A n /C, gives a positive linear functional <> / on ir(A). Let L^ = {a G A : <jf>(a*a) = 0}. If 0 £ sp(ir(en)), then 7r(en) is

224

C* -algebra Extensions

invertible. This would imply that n(L^) = ir(A). Hence 4> = 0. But then = 0. In other words, z\, z<± in the proof of 5.3.5 may be assumed to have the property that 1 G sp(n(zi)), i = 1,2. Let p be the spectral projection of z\ corresponding to the set {1}. Then p £ K. Let q < p be any infinite dimensional projection. There is a V € B(l2) such that VV* < q and V*qV = 1B- Since qz\ = zxq = q, V*zxV = V*qV = 1B, we conclude that V*z{V = 1B- Thus, as in the proof of 5.3.5, if 0 is a pure state, then \\(a) - V*aV\\ <e

a € T.

Let (f>i,...,(j)k be pure states, Zj,Zj,...,Zj

be positive elements with

z

\\ j II = 1; 3 — 1.2, i = l,2,...,k with z\ z[ = z[1' and let pi,...,pk be infinite dimensional projections corresponding to these pure states as above. In particular,

wuo)(4])2 - zj°a*j°ii < £/4fc2

(e3-3)

i = 1,2,..., k and j = 1,2. Set g = { 1 B , (^ ( 0 ) 2 , o : a e f , l < ! < M = l , 2 } . There are infinite dimensional projections q\,...,qk such that <7J < pi and Hgi^jH < e/4/e2,

i ^ j , i,j =

l,2,...,k

for all a; £ 0, by applying Lemma 5.3.6. So in particular H^ffr'H < s/4k2. Therefore, by what we have proved, we obtain Vi, i = 1,2,..., k such that V*Vi = 1B, ViV* < qi and ||^(o) - V*aV4 < e/4 for all a £ !F. Suppose that 4> is a state and fc ||0(o)-53o^i(o)||<e/2

fl£f,

i=l

where 0 < en < 1 are positive numbers so that Y2i=i ai 12i=i y/^iVi- We estimate that k

\\(a)-W*aW\\

<

\\4>{a)-YJ^iV;aVi\\ »=1

=

1- Set W

k

+

\\Y/^OijV:qiaqjVj\\ iy^j

=

Completely positive maps to M n ( C ) k

<

225

k

U{a) - £ ) a i & ( a ) | | + ^ a ; | | < £ i ( a ) - V*aVi\\ i=l

i=l

k

W^aia^e/Ak2^

+

< e/4 + e/4 + e/4 = 3e/4

for all a £ J7. Note that fe

liw^ll = ii^^H + E v ^ l l ^ H i=l

<

i^j

l + k2{e/Ak2) =

^ .

Set V = (^/4/(4 + £))Wr. Then ||V|| < 1 and for all a G T U(a) - VaV\\

= ^ i - + ^

110(a) - W W | | < e/4 + (3e/4) = e.

• L e m m a 5.3.8 Lei A be a unital separable C*-algebra and let j : A —»• £? 6e a unital full embedding. Then, for any nonzero contractive completely positive linear map <j> : A —>• M„(C) C Mn(B), any finite subset !F C A, and any e > 0, £/iere is a contraction V G MK(B) (for some integer K > 0 ) SMC/I t/iat (see 5.3.^ for the definition of dx ) ||0(a) - V*dtf(a)V|| < e

(a e F) and V*V = 1B.

Proof. Write 4>(a) = Y^=i ij{a)®eij (a & A), where {e^} is a system of matrix units for Mn and fcj : A —• C is linear. Define ip : Mn(A) - > C c B by ip((aij)nXn) = Y,i,j=i
6 € 0.

Note that we may also assume that W*W < diag(l^,0,..., 0). So we may write W as an nK x 1 column. Let Vi = (0,..., 0,1,0,..., 0) (with 1 in the ith place and zero in the rest) and Vi = dx(vi). Note that, for o e i , v*dK{a)vj = djr((6 s t)nxn), where bij = a and bst = 0 if (s,t) ^ (i, j). Therefore

Uijia) - W'iJjdK-Wi'iWII < s/n2.

226

C*-algebra

Extensions

Define V = {viW, V2W,..., vnW). Then we have \\(a)-V*dK(a)V\\<e

a£?.

Corollary 5.3.9 Let A be a separable unital C*-algebra so that A C B{12) with 1A = l B (/2), and let 0 : A ->• M n (C) C Mn{B{l2)) be a contractive completely positive linear map which vanishes on A D /C. TTien £/iere exists a sequence of contractions Vk € M n (B(Z 2 )) such that ||0(ci) - Vfc*aVfc|| -> 0

as fc -> 00

/or o// a € A Proof. We use the same proof of 5.3.8 with obvious modification (by applying 5.3.7). • B(l2) is a very special C*-algebra and B{l2)/K. is a purely infinite simple C*-algebra. Every element in B{l2)/K looks like an infinite diagonal element. This is why Lemma 5.3.7 and 5.3.9 work without using aV (see also 5.10.11). The following corollary illustrates this point and goes a step further. Corollary 5.3.10 Let A C B(l2) be a separable unital C*-algebra such that \A = I s , and let 0 '• A —» Mn(C) be a nonzero contractive completely positive linear map with 0|An/c = 0. Identify M n (C) with B(HQ), where Ho C I2 is an n-dimensional subspace. Then there exists a sequence of contractions Vk £ B(Ho,l2) such that (1) 0(a) - Vk*aVk £ K, (2) limfe ||0(a) - Vk*aVk\\ = 0 for all a £ A and (3) for any finite rank projection p, any e > 0 and any finite subset J- C A, we can choose Vk*p = 0, ||Vfc*ap|| < e and ||paVfc|| < £ for all a £ T. Proof. Fix a finite subset T and let T\ — J- C F2, • •• be an increasing sequence of finite subsets so that UnJT„ is dense in A. Set 0 = 0(a) ® 1B(I2) '• A -> Mn(B(l2)). Applying 5.3.9 to 0, we obtain Wk £ Mn(B{l2)) such that ||0(a) — W£aWfc|| = g(k,a) where g{k,a) > 0 and g(k,a) —>• 0 as k —> 00 for all a £ A. Let {e„} be a sequence of mutually orthogonal rank one projections such that 1B(I2) = Y^=i e «- Set ^k = diag(efc,efc, ...,ek), where ek repeats k times. Then Ek commutes with the range of 0. We may identify 0(a) with 4>(a)Ei for all a £ A. Thus ||0(a) - EiWZaW&W

< g(k,a),

a £ A.

Amenable

completely positive

227

maps

Let Vfc = WkEx. Note that <j>{a) - Vk*aVk G K. for all a G A. This proves (1) and (2). To see (3) holds, let p\ be a finite rank projection. Fix k. There is m(2) > 1 such that \\Emi2)WZPl\\

< e/4, ||^ m( 2)W?oPill < e/4 and || P l aW 2 J B m ( 2 ) || < e/4

for a e f . Set Vfc' = (1 - p i ) W f c E m ( 2 ) . Then H0>)£ m (2) " {VlYaVlW < g(k,a) + + \\Em(2)Wk*(l

\\Em(2)W^PlaWkEm{2)\\

-Pl)aPlWkEm{2)\\

<

g(k,a)+e/2

for a^T. There is a unitary Ux G Mn(B{l2)) such that U*4>(a)Emi2)U = 0(a) for all o G A Set Vfc = V^Ui = (1 - Pi)Wfc£?m(2)I/'i. Clearly (i) 0(a) - Vfc*aVfe G £ for all a G A We also have (ii) ||0(a) - Vfc*aVfc|| < g(k,a)+e/2 for a e f . Moreover, (hi) Vfc*p! = {7j.Em(2)Wfc(l - p i ) p i = 0, ||Vfc*api|| < £ / 4 and ||piaVfe|| < e/4 for all a e f . •

5.4

Amenable completely positive maps

Definition 5.4.1 Let A and B be two C*-algebras and 0 : A —>• B be a contractive completely positive linear map. We say 0 is amenable if for any £ > 0 and any finite subset J- C A, there are contractive completely positive linear maps a : A —> Mn (for some n > 0) and ip : M n —»• B such that ||0(a) - V> o
for a e f.

If either vl or B is amenable, then 0 is always amenable. Lemma 5.4.2 Let A be a unital separable C*-subalgebra of B such that the embedding j : A —¥ B is full and unital, and : A —>• B is amenable. Then, for any e > 0 and finite subset T C A, there is a contraction V G Mn(B) for some integer n such that ||
<e

foraGf.

228

C* -algebra

Extensions

If we write V* = (x\, x\,..., x*n) with x{ G B and || X)"=i x*xi\\ £/ien

< 1,

n

||^(a) - ^ a ; * a ^ i | | < £ for a G J7. i=i

Proof. Fix a finite subset J 7 and e > 0. Since 0 is amenable, by definition 5.4.1, to save notation and without loss of generality, we may assume that = tp o a, where a : A —> Mn and ip : Mn —>• B are contractive completely positive linear maps. Identify Mn with Mn • 1^ C Mn(A). Write

a(a) = ^2 a^ (a) ® e^,

aei.

Note that CI A by H(aij))

=

^VijiVij)

for (aij) G Mn(A). By 5.3.8, there exists a contraction W G MK{A) some integer K > 0) such that ||cr(6) - W*aV(&)Wi < £/2n 3

(for

6 G 5,

where Q = {(aij) : a^ G J-}. We write

^(a) = 'Y2crij(a)'lP(eij)

a

£ A.

Since the matrix ( e y ) n x „ is positive and ip is completely positive, (ip(eij))nXn is positive. Let (rjj) nX n 6 Mn(B) be the square root of (e-yOnxn- Then ^ ( e ^ ) = YTS=\ rkirkj- As in the proof of 5.3.8 with v[s in 5.3.8, u*ai>j = (cst)nxn,

Cij = a,cst = 0 for (s,t) ±

So \\aij(a) - W*v*dK(a)vjW\\

< e/2n3

where Vi — dx(vi), i = 1,2, ...,n. We have

i,3

ij,s

a G F,

(i,j).

Amenable

=

completely positive

229

maps

y~Vsj0ij(a)7Y,- (recall Cy(a) e C - l ^ t h e center of A). i,3,s

We then estimate that H(a)-Y,r:iW'vidK(a)vjWraj\\ ij>s

< II *52r*i(°ij(a)

- W*vidK{a)vjW)rsj\\

< n3e/n3

=e

for all a iWrSi, i,s = 1, 2, ...,n. Then \\4>(a)-V*dKn2(a)V\\<e

&T.

n

Lemma 5.4.3 Let A be a separable C* -algebra, B be a C* -algebra, I be an ideal of B and : A —• B/I be a contractive completely positive linear map. Suppose that there is a sequence of contractive completely positive linear maps ipn : A —> B such that lim |-7T o tpn(a) — (a)|| = 0 for all a & A, n

where IT : B —> B/I is the quotient map. Then there is a contractive completely positive linear map ip '• A ~• B such that ir o tp — <j>. Proof. Since A is separable, there is a separable C*-subalgebra C of B such that i>n(A) c C and <j){A) C C. Let I0 = C nl.By replacing B by C and I by IQ, we may assume that B and / are also separable. Let T\ C J~2,--- be an increasing sequence of finite subsets of A such that UnJ7,, is dense in A. Let {ea} be a sequence which is a quasi-central approximate identity for I (and B) such that eaep — ea if a < (3 (see 1.11.53). We may assume that

|KoV„(a)-
£ I such that

HV'n(a) - ^i(o) ~ c{n, i, a)\\ < l / 2 i + 2

a e Tu

i = 1,2,..., n — 1. For r)\ > 0, there is e\ C {ea} such that ||(1 - ei)c(l,i,a)|| < 1/4,

and

||eiV>i(a) - Vi(a)ei|| < 771/2 i = 1,2

230

C*-algebra

Extensions

and all a £ T\. We choose 771 so small that \\z1'2^i{a)-iji{a)zll2\\
\\zl/2Ma)-Ma>1/2\\ e\ such that ||(1 - e 2 )ei|| < 1/8, ||(1 - e2)c(n,i,a)\\

< 1/8

(n,t = 1,2) and

HeaV't(o) - Tl>i{p)e2\\ < %/2

(i = 1,2)

for all « e f 2 - Let Ex = e{/2 and E2 = (e 2 - ei) 1 / 2 . Thus \\E2^(a)E2

< l/2i+1

- E2^(a)E2\\

(a e ^ )

for i = 1,2. Moreover 11-^2^(0) - ipi(a)E2\\ < 1/23 for a £ j j

and i = 1,2,3.

Continuing this fashion, we obtain an increasing subsequence {en} C {e a } such that (with En = (en — e n - i ) 1 / 2 for n > 1) (1) HKV'i(a) - Ma)En\\ < l / 2 n + 2 oe7f(i 2) (3) \\EniPn{a)En - En^i{a)En\\ < l / 2 i + 2 for a e ^ (i = 1, ...,n - 1). Set ^(o) = Y^=i Eni'nio^En (a g A). This sum converges in the strict topology (see 5.1.10). Furthermore, since each ipn is contractive, ||^>(a)|| < 1 if ||a|| < 1. So V maps A into M{I). We shall also use n to denote the quotient map from M(I) to M(I)/I. We have 00

00

^2 Enipn(a)En

- (1 - eN)ipN(a)

= ^

n=N

(Enipn(a)En

-

EnipN(a)En)

n=N 00

+ £

(En^N(a)En

- E2nil>N{a))

( £ 4.4)

for all a e f i v - Note that for n > N \\Enipn(a)En

- Eni>N{a)En)\\

< 1/2 W+2 , a G ^jv-

(e4.5)

Amenable

completely positive

Since eaep = ea if a < /?, EnEm

231

maps

— 0 if \n - m\ > 2. Therefore

oo

||

Yl

(EnMa)En

- En-4)N(a)En)\\


2

and

n>N)even oo

||

^

{Eni>n(a)En - E n Vjv(a)J5 n )|| < 1 / 2 ^

for a G . 7 ^ .

n>JV,odd

Thus CO

|| 5 ] (EnMa)En for all a G

JFJV-

- £ n VAr(a)£„)|| < 2(1/2 W + 2 ) = l / 2 A r + 1

Also for a G

(e4.6)

.T-JV,

OO

OO

|| 5 2 (En^N(a)En

- E2ni>N{a))\\ < ^

n=JV

2 i + 2 = 1/2 JV+1 .

(e4.7)

i=JV

Hence, by (e4.4)-(e4.7), we estimate that CO

|| 5 2 En^n{a)En

- (1 - eN)^N{a)\\

< \/2N

for a & TN

Since oo

n{ip(a)) = 7r(52 Entpn(a)En)

and ir(ipN(a)) = 7r((l -

eN)ipN(a)),

n=N

we conclude that ||7r(V(a)) - ir{iPK{a))\\ < 1/2* -+ 0 (as K -> oo) for all aeFN{K>N).

Therefore

lim7r o V'n(a) = 7r o ijj{a) for all a G A. n

This implies that TT O ip(a) = 4>(a) for all a € A. Since 0(a) G B/I for all a G A, we conclude that ip(a) G -B for all a £ A. Since each an is completely positive, it is easy to check that the sum tp is also completely positive. • T h e o r e m 5.4.4 (Choi & Effros) Let A be a separable C*-algebra, B a C*-algebra and I an ideal of B. Suppose that <j> : A —> B/I is an

232

C* -algebra

Extensions

amenable contractive completely positive linear map. Then there is a contractive completely positive linear map ip : A —• B such that n o tp = , where -ip : B —>• B/I is the quotient map. Proof. Since is amenable, there are sequences of contractive completely positive linear maps an : A —• My^ and r n : Mfc(„) —>• 5 / 7 such that lim(<£(a) - r n o ern(a)) = 0 n

for all a G A By 5.4.3, it suffices to show that for each n there is a sequence of contractive completely positive linear maps ipm : A -+ B such that lim(r n o crn(a) - 7T o ^\a))

=0

k

for all a € A To show this, by applying 5.4.3 again, it suffices to show that, for each n, there is a sequence of contractive completely positive linear maps *; (ri) : Mk{n) -> B such that lim(rn(y)-7rott,(n)(|,))=0 for ally e M fc(n) . Thus, to save the notation, we may assume that A = Mn. By replacing B by B, we may assume that B is unital. We identify Mn with scalar matrices in Mn(B/I). So j : M n —>• Mn(B/I) is a full unital embedding. Thus there are contractions V^ e M^(„) (B/I) such that limVn*djr(n)(a)K = ^(a)

«eM„.

There are contractions Wn e Mx(n)(B) such that ir(Wn) = Vn, where ir also denotes the quotient map from MK^n)(B) to MK(n){B/I). We will also identify Mn with the scalar matrices in Mn(B). Define ipn : Mn —> B by i/jn(a) = W*dK{n)(a)Wn. Then n o tpn = V*dK{n)(a)Vn. So we can apply 5.4.3. • Corollary 5.4.5 (Choi & Effros) Let A be a separable amenable C*algebra, B a C* -algebra and I an ideal of B. Suppose that (j)'•A —• B/I is a contractive completely positive linear map. Then there is a contractive completely positive linear map ip : A —> B such that TT o ip = , where IT : B —>• B/I is the quotient map.

Absorbing

5.5

233

extensions

Absorbing extensions

Lemma 5.5.1 Let A be a unital separable C*-subalgebra of B so that the embedding j : A —> B is unital and full. Identify the constant functions in C([0,1],B) with B. Suppose that an : A —» Mk(n){SB) are amenable contractive completely positive linear maps and a : A -> M(SB (g> K.) is defined by a{a) = diag(<Ti(a),..., c n (a),...) for a £ A, where the convergence is in the strict topology. Then there is a sequence of contractions Vn £ M(SB ® K) such that (j(a) - K*doo(a)K £SB®K

and lim \\a{a) - V^d^a^W

=0

n

for all a € A. Proof. Fix a finite subset J- and e > 0. Let T\ = J-,...,!Fn C J-n+\, n = 1,2,..., such that U„f n is dense in A. By 5.4.2, there is a sequence of contractions Wn £ M K ( n ) (C([0, l],B) such that (JK"(n) > k(n)) W*Wn £ M f c ( n ) (C([0,l],B))and

K ( a ) - Wn*dK(n)(a)ttg| < £/2™+2 for a e .F n , n = 1, 2,.... There is gn £ M fc („)(55) with 0 < gn < 1 and ||ffnff(a)5n - crn(a)\\ < e/2n+2,

a £ Tn,

n = 1,2,.... Set Vn = Wngn, n = 1,2,.... So Vn 6 MK(n)(SB). K ( a ) - K T ^ ( n ) ( a ) K | | < e/2n+1 and crn(a) - V n *d K(n) (a)V n e MK{n)(SB),

Then

a £ jfn,

n = 1,2.... Write

doo(a) =diag(d i C ( 1 ) (a),d K ( 2 ) (a),...,d i c ( n ) (a),...) for s e A Define also V = diag(Vi, V2, •••, Vn,...). The convergence is in the strict topology. So we may view V £ M(SB K). For any fixed N, N

V*dia.g(dKW(a),

...,dK(N)(a),0,

...0, ...)V - Yi°n(a)

£ ML(N)(SB)

and

n=l JV

|| V*diag(d K (i)(a),..., dK(N) (a), 0, ...0, ...)V - £

JV


£/2

n+1

234

C* -algebra

Extensions

for all a £ J7^, where L(N) = J2n=i M n ))- Therefore n

V*dia,g(dKW(a),

...,dK In)

(a), 0, ...0, ...)V - J2 CT» ( G MHn){SB)) converges in norm to V*d00(a)V — a(a) for every a £ UnJ~n- In particular, V*d00(a)V - a (a) £ SB K (for a £ U n .F„). Since U„f n is dense in A, this implies that V*d00(a)V — a {a) £ SB ® K for all a £ A. We also have WVdnWV-o^W

<e

for all a£T.

•

Corollary 5.5.2 .Le£ A be a unital separable C* -subalgebra of B so that the embedding j : A —> B is unital and full. Suppose that o~n : A —>• -Wfc(n)(2?) are amenable contractive completely positive linear maps, and a : A —• M(B®K) is defined by a(a) = diag(<7i(a), ...,an(a),...) /or n e i , where the convergence is in the strict topology. Then there is a sequence of contractions Vn £ M(B <8> K) such that a{a) - V'd^Vn

£ 5 ® / C a n d lim ||
= 0

n

for all a £ A. Proof. The proof of this corollary is exactly the same as that of of 5.5.1 except for one thing: it is easier. One does not need to introduce the elements gn in the proof. • Lemma 5.5.3 Let A be a unital separable C*-subalgebra of B so that the embedding j : A —> B is unital and full. Identify the constant functions in C([0, l],B) with B. Suppose that 4> '• A —• M(SB <8> JC) is an amenable contractive completely positive linear map. Then there is a sequence of contractions Vn £ M(SB K.) such that (a) - V*d^{a)Vn £SB®K

and lim \\(a) - T O M a J K U

=

°

n

for all a £ A. Proof. Fix a finite subset f c i and e > 0. Let T — T\ C Ti,... be a sequence of finite subsets of A such that U„f n is dense in A. It is easy to find an approximate identity {en} for SB K, such that en £ Mi^(SB).

Absorbing

235

extensions

As in (1.11.52) a subsequence of convex combination of {e„} forms a quasicentral approximate identity for SB ® /C. To save notation, we may assume that {e n } is quasi-central and en G Mfc(n)(S'-B), n = 1, 2,.... By passing to a subsequence, if necessary, we may assume (eo = 0) ||(e„ - e n _i)0(a) - cf>(a)(en - e n _i)|| <

e/2n+2,

||(e n - e n - i ) 1 ' 2 ^ ) - # a ) ( e n - en^f'2\\

< e/2n+2

2

||(l-e„+1)e„||<£ /2

2(n+4)

and

.

for all a G ^"n, n — 1,2,.... Let ^ = ei and 2£n = (e„ + i - en)1^2 and set o-„(a) = En(a)En (0 < n and a e A). Note that each <jn : A ->• Mk(n)(SB) is also amenable. One verifies that V7 = 2^°=i °n(a) converges in the strict topology for every a £ A (see 5.1.10). Furthermore, oo

U(a) ~ ip(a)\\

=

oo

\\Y,{en-en^)<j>{a)-Y,Enct>{a)En\\ oo

<

oo

Yl WEn(En / 2 " + 2 < e/4 n=l

n=l

for all a G T. Since ] T [ ( e n - e„_i)(a).En] e

Mk{N){SB)

n=l

for all a G A and oo

|| Yl

oo

(en-en-iMa)-

n=JV+l

£

oo

a„(a)|| < £

n=N+l

£ /2"+

2

-• 0

JV+1

as AT —> oo for all a £ Fk (any fc), we conclude that (a) — ip(a) G 5 5 /C for all a G ^ (any fc). Therefore ( /C for all a € A To save notation, without loss of generality, we may now assume that <j>(a) = Yl^Li
£2/22(n+6) +

£2/22(„+3)

<

£2/22(n+2)_

236

C* -algebra

Extensions

Hence ||(e n + 2 - e n _ i ) ( e n + i - enf'2 - (en+1 - e„) 1 /2||

<

e j 2

^\

Also (with e_i = 0 ) n

W*di&g(a1(a),a2(a),...,o-n(a),0,

...)W = ^ ( e i + 2 - e i _ 1 )CT i +i(a)(e i+2 - e ^ i ) i=0

for all a £ A. These imply that for all x G A oo

oo

W*a(x)W-Yj°n{x)eSB®K.

and \\W*a{a)W - ^an(a)\\

n=l

< e/2

n=l

for all a G T (the sums converge in the strict topology). By applying 5.5.1, we obtain Z G M(SB ® K) such that, for x £ A, a(x) - Z*doc(x)Z

£SB®K

and ||cr(a) - Z*doa(a)Z\\

< e/4

for a G T. Set V = ZW. Then T ^ d ^ a ) ! / - 0(a) = (W*(Z*d00(a)Z OO

-

a(a))W)

OO

+(W*a(a)W - J2(a)) + (Yl a^a) ~ ^ a )) G SB ® ^ We also have UVdocCaJV-^o)!! < £ for a e f. Theorem 5.5.4 Let A be a unital separable C*-suhalgehra of B so that the embedding j : A —»• B is unital and full. Identify the constant functions in C([0,1], B) with B. Suppose that <j> : A —>• M{SB K) is an amenable homomorphism. Then there is a sequence of isometries Vn G M(SB ® K) such that 4>{a) - V^doofaJK &SB®K

and lim \\cj>(a) - !£<*«, (a) Vn|| = 0 n

/or a/Z a G A. Proof. This follows from 5.5.3. Let bn = \\4>(lA) - V*Vn\\, n = 1,2,.... We may assume that bn < 1/2 for all n. Since p = (J>(1A) is a projection, we may assume that Vn = Vnp, n = 1,2,.... From

\\P-pv:vnP\\
1/2,

Absorbing

237

extensions

we know that cn = pV*Vnp is invertible in pM(SB V^lpra(c7l)1/2 is a partial isometry and

JC)p. Then Wn =

I I K - W n l l - + 0 (since bn -> 0). We also have p - pVn*Kp £SB®K,. Let TT : M ( 5 B /C) -> Q(5J3) be the quotient map. Then Tt(cn) = ir(p). Thus n(Wn) = n(Vn). In other words, n((a) - W*nd00{a)Wn = n{(a) -

Vy^a^

for all a G A. So 4>(a) - W*d00(a)Wn eSB®)C

for a 6 A.

We now replace Vn by Wn.

O

Corollary 5.5.5 Let A be a unital separable C*-subalgebra of B so that the embedding j : A —$• B is unital and full. Identify the constant functions in C([0,1],B) with B. Then d^ : A —> M(SB K.) gives an amenable absorbing trivial extension. Proof. Let r : A —> M(SB®JC)/SB®1C be an amenable trivial extension. It follows from 5.4.4 that there is a contractive completely positive linear map <j> : A —> M(SB K) such that 7r o <j> = r. It follows from 5.5.4 that r © -K o cfoo = n(V)*ir o doo^C^) for some isometry V G M(B /C). D Corollary 5.5.6 Lemma 5.5.3, Theorem 5.5.4 and Corollary 5.5.5 hold by replacing SB by B. Proof. The proof of each is exactly the same as that of 5.5.3, 5.5.4 and 5.5.5 (applying 5.5.2). • Lemma 5.5.7 Let A C B(l2) such that 1^ = 1B(I2) and an : A —>• Mk(n) be a sequence of contractive completely positive linear maps such that o-n(a) — 0 if a G A(~\)C. We identify Mfc(n) with a C* -subalgebra ofK,. Suppose that the ranges of an are mutually orthogonal and a : A —• B(l2) is defined to be a(a) = diag(ai(a),a2(a), ...,crn(a),...) for all a G A. Then there exists a sequence of contractions Vn G B(l2) such that a(a) - V*aVn e K and lim \\a(a) - V*aVn\\ = 0 n

for all a €: A.

238

C* -algebra

Extensions

Proof. The proof is very much the same as that of 5.5.1. Let J- = {xi, ...,xm} be a fixed finite subset of A and e > 0. It suffices to show that there is a contraction V G B(l2) such that a(a) - V*aV G K (a G A) and \\
G K for all a G A and ||<7i(z) - W^zW^H <

e/21+1

for a; G T\. We may assume that Wipi = W\. Let q\ be the range projection of W\pi. Then q\ is finite dimensional. We claim that there are contractions Wn G B(l2) with W„p n = Wn, (i) cr„(o) - W*aWn G /C for a G A, (ii) ||<7n(a) - W*aWn\\ < e/2n+1 for all a G .F„, (iii) WJ-Wj- = 0 (i ^ j) and HW^aWjH < £/2 1 + i + ^ for all a G ^ , 2 = 1,2,....

Suppose that we have obtained such W\,...,Wn. Let qi be the range projection of W^pi (i = 1,2, ...,n). Note by (iii) g^- = 0 if i ^ j and J, j < ra. Then <5n = Y^7=i 9* ^s a n n i t e rank projection. By applying 5.3.10, we obtain a contraction Wn+\ G B{12) with W^+iPn+i = Wn+i such that cr n+1 (a) - W*+1aWn+i

G /C for all a £ A,

||crn+i(a;) - W^ +1 a;W n+1 || < e / 2 n + 1 for x G Tn+i and W;+iQn = 0,

< e/2n+n

\\W:+1aQn\\

for a G ^ n + 1 .

This ends the proof of the claim. Set V = E r = i Wn- This is well-defined since W*Wj = 0 for i ± j . Furthermore, V is a contraction. Note that oo

\\V*aV-Y,W:aWn\\

<

Wj^W^aWjW

< ^2s/2i+j+1

<e/2

Absorbing

239

extensions

for a G J-. Therefore oo

\\a(a)-V*aV\\

< Ha) - £

W*aWn||

n=l oo

+

\\V*aV~Y,W:aWn\\<e/2

+ s/2 = s

n=l

for all a G F. We also have, for any fixed K, oo

oo

| | ( l - Q t f ) ( V * a y - £ w ' n * a W n ) | | = || X ) n=l

= ||

>

y

n=K+l

W W

J!

oo

W

> iW <

n>if+l,jyn

H

2 +J+1

"

-

E

W^nl

71=^+1

< 2^+1

n>K+l,j^n

for all a G TL with L < K. Let if -> oo, we conclude that V*aV E ^ = i ^ > W n e K for all a G f j (any L). Therefore OO

V*aV - J2 w>wn

G /C for a G A

n=l

We also have oo

a(a) - ^

W > W „ G /C for a € A

n=l

g

Lemma 5.5.8 (Voiculescu) Lei A C B(l2) be a separable unital C*algebra such that 1^ = 1B(I2) and let <j> '• A —i B(l2) be a contractive completely positive linear map such that • M(B® /C) be two homomorphisms (i = 1,2). We write 4>i — 4>2 if there is a

240

C* -algebra

Extensions

sequence of isometries Un G M(B ® K.) such that U'M^Wn

~ h(a)

G B ® /C and lim ||t/*^(a)C/„ - 4>2(a)|| = 0

for all a £ A. Theorem 5.5.10 (Voiculescu) Let A be a separable C* -subalgebra of B(l2) so that the embedding is non-degenerate and IT : A —» B{12) be a representation such that IT vanishes on An K,. Then id © 7T ^ id. Proof. To be non-trivial, we may assume that A <£ /C. By adding an identity to A and assuming 7r(l) = p, where p is the range projection of ir(A)(l2), without loss of generality, we may assume that A is unital and 1A = ls(/ 2 )- Put ip = 7r © id By 5.5.8, there is a sequence of contractions Vn G B{12) such that V^aK - 4>(a) G /C and lim \\V*aVn - <j>(a)\\ = 0 n

for all a s A Now using the same argument as in 5.5.4, we can replace Vn by isometries. • Corollary 5.5.11 Let A be a separable C*-algebra. Then every trivial essential extension r : A —t B{l2)/K, is absorbing. Let j : A —• B(l2) be a non-degenerate faithful representation. Let doo : A —»• -B(/2) by d^a) = diag(j(a), j(a),...), where we identify I2 with the Hilbert space which is the countable direct sum of I2. Then by 5.5.11 dx gives a trivial essential absorbing extension. We leave the following theorem of Kasparov as an exercise. Note every essential trivial extension of A by K. is unitarily equivalent to d^. So, using (the proof of) 5.5.3, we have the following: Theorem 5.5.12 Let (f> : A —• M(B ® K) be an embedding, where B is a a-unital C* -algebra. Suppose that cf> factors through M()C) and Tr(A) ("I K. = {0}. Then for any amenable contractive completely positive linear map tp : A —> M{B /C), there are contractions (or partial isometries if IP(1A) is a projection) Vn G M(B (g> K) such that xjj{a) - V*cj>{a)Vn G B ® K and lim \\^{a) - V*<j>{a)Vn\\ = 0 for all a € A.

Absorbing

241

extensions

In particular, <j> gives an amenable absorbing extension of A by B ® /C. The following is a result of M. Dadarlat which is an approximate version of the next theorem. Theorem 5.5.13 Let A be separable unital RFD C*-algebra (see 3.7.6) and let h : A —» B be an amenable homomorphism to a unital C* -algebra B. Then there is a sequence 4>n : A —>• Mr(n)(B) of contractive completely positive linear maps and there is a sequence of homomorphisms ipn : A —• Mrtn\+i(B) with finite dimensional range such that lim \\diag(h(a),(j)n(a)) — ipn(a)\\ —0 for a G A. n—*oo

If h(lA) = 1B> then we may arrange for (/>„ and ipn to be unital. Proof. It is clear that, without loss of generality, we may assume that /i(l/i) = I s - Let 7r„ : A —>• B(Hn) be a separating sequence of unital finite dimensional representations such that each nn repeats infinitely many times in the sequence. Let Kn = Hi®- • -®H„ and H = ®^L1Hn. Put p = ©~=17rn and pn = ©£=17rfc. So p maps A into B(H) C M(B ® K). Letfc(n)be the dimension of Kn. We identify B with e n M ( B ® K)eu (and 1 B ® e u with e n ) . Let Vn e M{B®KL) be partial isometries given by 5.5.12 applied to h and p. So lim ||fc(a) - V*p(a)Vn\\ = 0 for all a G A n—>oo

Since ^ ( l ^ ) = e u and replacing Vn by V^en, we see that, without loss of generality, we may further assume that V*Vn — eu. Put / „ = VnV*. Since now Vn e B ® K., there is a subsequence {L(n)} such that lim ||/i(o) - V:PL{n){a)Vn\\ = 0 for all a G A Without loss of generality and saving notation, we may also assume that L(n) = n. We may further assume that / „ < J2i=i ea = PU(1A)- From (Vnh(a) - pn{a)Vn)*(Vnh(a) = (V:Pn(a*a)Vn

-

pn(a)Vn)

- hn(a*a))

+(h(a*) - V:Pn(a)Vn)h(a)

+ h(a*)(h(a) -

V:Pn(a)Vn)

we see that ||V„/i(a) - pn(a)Vn\\ -> 0 for a G A Moreover [Pn(a),/n] - (Pn(a)Vn - Vnh(a))V: + Vn(Vnh(a*) -

pn(a*)Vn)*,

242

C* -algebra

Extensions

we see that ||[/0„(a), / n ] | | -> 0 for all a £ A. Set r(n) = 2k(n) and let un be the unitary of M r ( n ) (B) defined by Vn

J-r-(n)

. l r (n) - e n

in

V*

Define the maps <j>n, ipn and hn by ^(a)

=

< f ( l r ^ - MPn{a){lr{n)

lM«)

=

u n (^

h

_

(n)

0

^(a)jU„and

Q

*(fnPn{a)fn

n„ w - « ^

-fn)

0

0\

oj^-y

_/KV„(a)F„

o

0

o

Then \\hn(a) + n{a) - xpn(a)\\ < ||(1 - /„)p„(a)/ n || + ||/„p„(a)(l - /„)|| -»• 0, since ||[p n (a), /„]|| ->• 0. Also ||/i„(a) - h(a)\\ -> 0. Therefore ||/i(a) + <£n(a) - V'nWH -4 0 for a e A. Finally we note that ipn is a homomorphism with finite dimensional range.

• To end this section, we offer the following theorem which was proved by Arveson in the case in which B = K, : Theorem 5.5.14 Let A be a separable amenable C*-algebra and B be a a-unital C*-algebra. Then Ext(A,B) forms a group. Proof. We may assume that A has a unit. Let r : A —¥ Q(B) be an extension. Since A is amenable, there exists a contractive completely positive linear map L : A -> M(B (g> /C) such that o L = r, where w : M(B ® /C) —> Q(B) is the quotient map. It follows from 5.5.12 that there is an absorbing trivial extension given by an injective map ip : A —> M(B ® K-). Thus, by 5.5.12, there is a partial isometry V € M(B ® JC) such that L{a) - V*iP(a)V eB®K,

for a e A.

Let 7T : M(B K) ->• Q{B) be the quotient map. Note r(a) = Tr(V*ip(a)V) (a £ A). Since we assumed that A is unital, ir(V) is a partial isometry. Let

A stable uniqueness

243

theorem

p = TT(VV*). Write IT o ijj in 2 x 2 matrix form using t h e decomposition p + (1 — p) = 1 t o obtain / T(O) \P2i(a)

pi2(a)\ P22{a)j

Note, if a G Asa, then /012(a) = p2i(a)- For any a G A , a , r ( a 2 ) = r ( a ) r ( a ) = pir o ip(a)ir o ip(a)PT h u s , we compute t h a t T(a

2

) = T(a)r(a)

+ pi 2 (a)P2i(a) = r(a2) +

pii{a)2

for all a G Asa. Hence p\2 = 0. Therefore p 2 i ( a ) = 0 for all a G Asa. T h u s P21 = 0. Therefore p2i : A ->• Q(-B) is a homomorphism. Let T _ 1 = P22 © 7T o •;/). T h e n T _ 1 : A —• Q(B) is a monomorphism. T h e n we see t h a t [ T © T _ 1 ] = [ T T O ( I / ; © V)]

Since tjj®ij; is trivial, Ext(A,

5.6

inEa;t(A,B).

B) forms a group.

•

A stable uniqueness theorem

5 . 6 . 1 Let A C B, where A a n d B are C""-algebras. Recall t h a t doa(a) = d i a g ( a , a , ...,a,...) is a diagonal element in M(B ® K). Let { e ^ } b e t h e m a t r i x units for K. Let em = Y^iLieaFor convenience, in what follows, we use doo-m(a) for (1 — em)d00(a). In other words, rf00_m(a) = diag(0, . . . , 0 , a , a , . . . ) with m zeros. We will use notation e m a n d doo_ m in this section without further notice. L e m m a 5 . 6 . 2 Let C be a unital C*-algebra, and let A be a unital C*subalgebra of C. Suppose that U G M(C ®K) is a unitary such that C/diag(6(a), C ; oo (a))C/* - (c{a), d^a))

G C ® K,

where a G A, and for some b(a), c(a) G C (depending e > 0 and any finite subset J- (Z A, (1) there is an integer N > 0 such that

on a). Then, for any

||(1 - e t f ) t f d i a g ( d 0 0 _ i ( a ) ) £ T ( l - eN) - (1 - e J V )(d 0 O _ 1 (a))(l where ejy = 5Z i = : 1 e^, for a £ J7.

ejv)||

< e/32,

244

C*-algebra

Extensions

(2) Let p = U*eNU. If Ni > N such that \\eNlp - p\\ < e/4, then Hdiag^.djVjCa)) - [pd00-i(a)p+

(eNl - p)d00_i(a)(eNl

- p ) ] | | < y/e + e

for all a £ J7. Proof. To simplify notation, we may assume that T is in the unit ball of Asa. Let g = FU{ab:a,beJr}. Let M = max{||6(a)||||c(a)|| : a G £ } . Note the matrix units {e^} are in C ® /C. Therefore, Ue\\ G C K.. Since {e„} is an approximate identity for C®K, we obtain an integer N > 1 such that ||(1 - eN)Uen\\ < e/64(M + 1) and (1 - e w )t/diag(6(a),d 00 (a))C/*(l - eN) w e / 6 4 (1 - e w )diag(c(o),rfoo(a))(l - eN) for all b, c G M(C) and a G £. Note that (l-eAr)diag(c(a),rf tx) (a))(l-ejv) = (1 — eN)dca(a)(l — ejv) = d00-N(a) if TV > 2. Therefore ||(1 -

ej v)tfdoo-i(a)tf*(l

-

ejv)

- ^oo-iv(a)|| <

Me 6 4 ( M + 1)

+ £/64 = £/32

for all a € Q. This proves (1). It is clear that for some large N\ > N, ||ejViP — p\\ < £/8. We apply an argument similar to 5.5.14 to show that

Updoo-ifaXl-pJKV^ for all a £ !F. From above, we have, if a G J-, ||(1 - e j y j l / d . x , . ! ^ 2 ) ^ ^ ! - eN) - doo-Nia2)]] < £ / 3 2 and ||(1 - e JV )f/d 00 _ 1 (a)(l - p)d00_1(a)CA*(l - ejv) - doo-iv(a 2 )|| < e/16 Therefore, for a E !F, (1 -

e i v )t/doc-i(a)(l

- p ^ - ^ c O C / ^ l - ejv)

~3E/32 (1 - e j v ^ e k o - i t a 2 ) ? / * ^ - ejv). Consequently (since (1 — p) = i/*(l — es)U), ||(1 - p ) d o o - i ( a ) ( l - p ) d 0 0 - i ( o ) ( l - p ) - (1 - p ) d o o - i ( a 2 ) ( l - p ) | | < 3 £ /32. But (l-p)doo-i(a2)(l-p)

-

(l-p)doo-i(o)(l-p)doo-i(a)(l-p) +(1 -p)d 0 0 _i(a)pd 0 0 _i(a)(l - p ) .

A stable uniqueness

245

theorem

Thus ||(l-p)d00_i(a)pd00_i(o)(l-p)||<3e/32 for all a € T. Since a = a* (as we conveniently assumed), we have

\\pd00_l{a){l-p)\\<^/2 for all a e T. It follows that Noo-i(a) - [ M x , ( a - l ) p + ( l - p ) d 0 0 _ 1 ( a ) ( l - p ) ] | | < \fi for all a £ !F. Finally, for all a G !F, Hdiag^.djv^a)) - \pd(X)-i(a)p + (eNl -p)doc-i(a)(eNl

-p)}\\ < y/e + e.

a

Lemma 5.6.3 Let A be a unital C*-subalgebra of a unital C*-algebra B. Suppose that there is a unitary W G M(B K) such that W*d00(x)W

- doc (a:) G B ® /C for i £ i

and

/or a in a finite subset J-'. Then there is a norm-continuous path of unitaries V € M(B ® £ ) sucfc that V(0) = 1, V(l) = diag(W, W*), y(t)*diag(d 00 (2;),d 00 (a;))y(i) - diag(d 00 (a;),d 00 (a;)) e f l « K or a;£ A and ||V(t)*diag(d 00 (a),d 00 (a))V'(f) - diag(d 00 (a),d 00 (o))|| <e for all a € Proof.

Define 7

= 1

Z3

(W V°

- (, 0

°\ V 1J

/ m J H ^ ) V8111^/2) and

Z4(f)

-sin(7ri/2)\ COs(lTt/2) J

- V - sin(7ri/2)

Set

v(t) = z1z2(t)z3zi{t)

te[o,i].

Note that V(0) - 1 and 7(1) = diag(W, W ) .

cos(7rf/2) J '

246

C-algebra

Extensions

Moreover, since for each t, Z2(t) and Z\{t) are scalar matrices, they both commute with the diagonal matrix diag(d00(a),d00(a)) for all a G A. Let TT : M2{M(B ® £)) -» M2(M(B ® K))/M2{B ® /C) be the quotient. Then TT(Z\) and 7r(Z3) commute with 7r(diag(doo(a),doo(a)) by the assumption given. Therefore 7r(V(£)) commute with 7r(diag(rf00(a),cfoo(a))) for all a G A On the other hand we have, for each t G [0,1], y(i)diag(d 0 0 (a),d 0 0 (a))

=

Z1Z2{t)Z3dia.g(d00(a),d00(a))Z4(t)

«e/2

^iZ2(t)diag(doo(a),(ioo(a))^3^4(i)

= w£/2

Xidiag(c? 00 (a),d 00 (a))Z 2 (i)^3^4(i) diag(d 0 0 (a),d 0 0 (a))y(i)

for all a&T.

•

Theorem 5.6.4 Lei B be a unital C*-algebra and let A be a separable amenable full unital C* -subalgebra of B. Let a : A —> B and (3 : A —• B 6e iwo unital homomorphisms. If [a] = [/?] in KK(A,B), i/ien, for any e > 0 ami any finite subset !F C A, there is an integer L > 0 and a unitary U G Mx,+i(5) SMC/I i/iai \\U*dmg(a(a),dL(a))U

- dmg(f3(a),dL(a))\\

< e

for all a G J-. Proof. Fix £ > 0 and a finite subset T C A. Without loss of generality, we may assume that J- is in the unit ball of Asa which contains 1^. Let G = Fl){ab: a,b&T}. Let T\ and r2 be two unital essential extensions of A by SB via maps a and j3 as mapping tori: Ma = {/ G C([0,1],B) : /(0) = a and / ( l ) = a(a), a £ A} and Mp = {fe

C([0,1],B) : /(0) = a and / ( l ) = /3(a), a G A}.

Since [a] = [/?] in iTif (A, B), there is a unitary W in M(SB K.) ( which is identified with C 6 ((0, l),M(B ® /C)a)—see 5.10.2) such that 7r(W / )*dia5(ri,r 00 )7r(W) = diag(T2,Too),

(e6.8)

where 7r : M{SB®K) -» M(SB®K.)/SB®K is the quotient map. For any a G A let / 0 G M a ( A B) and 5 a G M ^ A B) with / o (0) = a, / a ( l ) = a(a),

A stable uniqueness

247

theorem

ga(0) = o and ga(l) = (3(a). Set c(a,fa,9a)(t) =

W^Qdiagifait^dooiaWW-diagfaitldnia))

(t G (0,1)). Then, by (e6.8), we have c(a,fa,ga) G SB®K. In particular, \\c{a,fa,9a){t)\\ ->• 0 as t -> 1 or i -> 0. Fix 0 < h < 1 so that | | c ( a , / 0 ) 5 o ) | | < e/64, ||/ a (t) - a(a)|| < e/64 and || flo (t) - (3(a)\\ < e/64 for all a G 5 and 1 > t > ti. Fix 0 < t0 < h so that l|c(a,/ ai
(e6.10)

for all a G Q. Set V(t) = diag(W(i), W(t0)*) for t G [t 0 ,*i]. We have V(*i)*diag(a(a),doo(a),doo(a))V(*i) « 3e /64 diag( / 9(a),d 00 (a),d 00 (o))

(e6.11)

for all a £ Q. By applying 5.6.3, we obtain a continuous path {V(t) G M2(M(B ® /C)) : [0,i 0 ]} such that V(0) = 1, V(t0) = V(t0) (notation already defined) and V(tydiag(d00(x),d00(x))V(t)

- (d00(x),d00(x))

&B®K

(e6.12)

for all x G A and V(t)*diag(doo(a),doo(o))V(t) «3E/32 diag(d 00 (a),d 00 (a))

(e6.13)

for all o e § . We now identify dOQ(a) with diag(d00(0'),rfoo(a)) and M2(M{B ® £ ) ) with M ( B ® /C). Set C = C([0,i 1 ],5). Then C is unital. We identify B with the constant functions in C. We also have (denote also by V, fa and ga, the restriction of V, fa and
GC®K.

248

C*-algebra

Extensions

for all a G A. For any e/256 > 5 > 0, it follows from 5.6.2 that there is an integer N > 0 such that ||ejv^(t)*diag(/ 0 ,d 0 0 (a))V(t)-F(t)*diag(/ 0 ) d 0 0 (a))V(t)e J V || < 6 and ||(1 - e J V )V(t)*(d 0 0 _i(a))y(t)(l - eN) - d^N{a)\\

< 6 for a G Q

and for all t G [0,*i]. It follows from (e6.9) and (e6.10) that for all a G Q, eiv^(ii)*diag(a(a),doo(a))^(ti)e A r « 6 £ / 6 4 diag(/3(a),djv_i(a)) (e6.14) for a £ Q. Set p(i) = V(t)*eNV(t) for i G [0,*i]. Since W\(0tl) is in b C ((0,l),M(B ® /C)a) and V(i) is norm-continuous on [0,t0], p(t) is a norm-continuous function on [0, N such that HejVjP^) - p(t)\\ < 6/16 for all * G [0,*i]. It follows from 5.6.2 (with C = C([0,ii],5)) that (for all t G [0,*i]) dia,g(0,dNl-i(x))

«e/512

\p{t)doo-i(x)p(t)

+ (eNl - p(t))doo-i(x)(eNl

- p(t))]

(e6.15)

on x G T, if <S is chosen small. With a = 1, from (e6.15), we have p(i)diag(0,ejv!-i) «£/256 d i a g ^ e ^ - i M * ) for all t G [0,*i]. There are 0 = tm < ... < £2 < t\ such that ||p(£$+i) — p(ii)|| < e/256, i = l , 2 , . . . , m - l . There are q[ < eNl such that ||g--p(ti)|| < e/128 and ^diag(0,ejVi-i) = diag(0, e ^ - i ) ? * , * = 1,2, ...,ra. Note pm = eN. Set qt = Q-diag(0,e iVl -i). Then qm = ejvdiag(0, e W l - i ) = diag(0,ejv_i). We have \\qi-qi+i\\<e/64

i = 1, 2, ...,m - 1.

(e6.16)

Set m(a) cr^a)

= =

p(*i) d °o-i( a M*i), »7t(a) =gidiag(0,div 1 -i)(a))gi, (eNl -pi)dia,g(0,dNl-i(a))(eNl -p{) and

<7;(a)

=

( e ^ -gi)diag(0,dATi-i(a))(ejVi ~ 9 i ) , i = 1,2, ...,m.

We have oi(a) « e / i 2 8 (ejvx - q , i)diag(0,djv 1 -i(a))(ej Vl - qi) for all a e f . Note that <7m(a) = ( e ^ - eN)dNl(a), gidoo-i(a))9i « £ /64 »7i(a), g-diag(0,djv 1 -i(a))g- = 7ft (a) and rjm(a) = diag(0,djv_i(a))

o€A

A stable uniqueness

249

theorem

Furthermore by (e6.15), diag(0,djv_i(a)) « 3 < r / 6 4 rn(a) + a^a)

(e6.17)

for all a G T and 1 < i < m. We also have, by (e6.16), 11^(0) - »7i+i(a)|| < e/32 and \\ai(a) - ai+1{a)\\ < e/32

(e6.18)

for all a G J- and i = 1,2, ...,m — 1. Set d

m(JVi-i)( a ) = (eNi - eN)dNl(a)

©d m _i(diag(0, dNl-i(a))

a

C(Ar!-i)+(iv-i)( ) = dm(d\ag(0,dNl-i(a)))

®

®dN_i(a),

dN-i(a),

Ai(a)

=

diag(cr 1 (a),772(a),a 2 (a),...,77 m (a),cr m (a),djv_i(a)),

A 2 (a)

=

diag(cr m (a),77 m (a),(7 m _i(a),...,772(a),(Ti(a),djv-i(a)) and

A 3 (a)

=

diag(cr m (a),77 m _i(a),(T m _i(a),...,?7i(a),cri(a),div-i(a))

for a G A. Then by (e6.17) and (e6.18) ||7?1(a)©A1(a)-^(JVi_1)+(JV_1)(a)|| <3£/64

(e6.19)

||A 3 (x) - A 2 (a)|| < e/32 and

(e6.20)

l|A3(a)-^(Jv1_1)(a)||<3£/64

(e6.21)

for all a G J-. Note that u\(l) = (e^1 — pi)diag(0,a'jv 1 -i(l)) and
ejv

©d^(Ni_1}(l).

Then, for any a G F, by (e6.19), (e6.15), (e6.14), (e6.20) and (e6.21), Z*dmg{a{a),d'Jn{Ni_1)+{N_1)(a))Z «£/5i2+3e/64

ejv^(ii)*diag(a(a),d 0 0 (a))V(i 1 )e i v

®z*Ai(a)z

«6E/64 diag(/3(a),djv-i(a)) © A 2 (a) « 3 e / 6 4 diag(/3(a),djv_i(a)) © A 3 (a) «3e/64 diag(/3(a),div-i(a)) © d' m(JVl _i)(a) for all a £ T. Therefore there is a partial isometry Z\ G Li > (m + l)Nx such that

ML1(B)

with

Z 1 *diag(a(a),d^ ( J V i _ 1 ) + ( J V _ 1 ) (a))Z« £ diag(^(a),d m ( W l _i ) + ( J V _ 1 ) (a))

250

C* -algebra

Extensions

for all a £ T. Thus we conclude that there is a unitary U £ L > L\) such that \\U*dmg(a(a),dL(a))U

- diag(P)(a),dL(a))\\

ML+I(B)

<e

for all a € T.

5.7

(with

•

if-theory and the universal coefficient theorem

In this section and the next, we will present several important results without proof. Definition 5.7.1 Let A be a unital C*-algebra and u G Un(A). Let v e U(C([0,l],M2n(A)))0 such that v(0) = l 2 n and v(l) = diag(u,u*). Put p = v

{o o)v-

Then p is a projection in M 2n (S'yl). Let n : Mzn{SA) —>• M 2 7 l (SA)/M 2 n (SA) = M2n be the quotient map. Then w(p) = diag(l„,0). In particular \p] — [1„] € K0(SA). Define OA([U})

= [P] -

[ln].

Then 6A gives a homomorphism from K\{A) to Ko(SA). If yl is not unital, then 9£ gives a homomorphism from Ki(A) to AToCS'.A). One can verify that 6£ gives a homomorphism from -ftTi(j4) to KQ(SA). Definition 5.7.2 Let A be a unital C*-algebra and p a projection in Mn{A). Define a projection loop fp : S1 —> Un(A) by fp(z) = e^tp+(ln-p),

iG[0,l].

Since / p (0) = / p ( l ) = l n , we see that fp e J7„(5A). One also has (1) fp(Bq = fp® fq for any projections in M M (A), (2) /o = 1 and (3) If p, q G M n (A) are two projections with p ~h Q m Proj(M„(A)), then / p ~ h fq in !7„(Si4). Thus we obtain a homomorphism /3^ : KQ(A) —• K\{SA).

K-theory

and the universal coefficient

theorem

251

Definition 5.7.3 Now suppose that A is a non-unital C*-algebra. We have the following diagram with split exact rows: 0

->

K0(A)

->

K0{A)

^

K0(C)

0

->

Ki(SA)

-+

K^SA)

^

Ko(SC)

->

0

-»• 0

It is easy to check that the square in the above diagram is commutative. Therefore there is a unique homomorphism /3A : Ko(A) —• i^i(S^4) making the following diagram commute: 0

->

ff0(4)

0

-•

^(SA)

->

Ko(i)

^

K0{C)

-•

4-0X #i(Si)

^

4-ft ^(SC)

4-/3A

-»• 0 ->

0

The map /3A : KQ(A) —>• Ki(SA) is called the Bott map. Note also that 6 A can also be defined for the non-unital case using a corresponding diagram. Let A and B be two C*-algebras and (j) : A —»• B. We denote by (f>s the induced homomorphism from SA to SB. Theorem 5.7.4 Let A and B be two C*-algebras and 4> '• A —»• B be a homomorphism. Then we have the following commutative diagrams: Ki(A) Ko(SA) Moreover

*-$ ^

K^B)

K0(A)

Ko(SB)

K^SA)

9A, 6Bi PA,PB

are

0

A°

K0(B)

^

K^SB).

isomorphisms.

Definition 5.7.5 Let A be a unital C*-algebra and u € Un(A/I). Choose a unitary w £ M 2 „(A) so that ir(w) = diag(w, u*) (see 4.2.5 and 2.8.21), where 7r : M2n(A) -»• M2n{A/I) is the quotient map. Define Si : Ki(A/I) -> K0(I) by d^u]) = [wlnw*] - [ln]. We note since ir(wlnw*) = 1„, [wlnu;*] - [l n ] € K0(I). Also, if there is U € Un(A) such that n(U) = u, then diag(U,U*)ln = l n . So <5i([w]) = 0. In particular, if u € Un(A/I)0, <5i([u]) = 0. Moreover, we note that even though wlnw* ~h l n in M-zn{A), wlniv* may not be connected to l n in MN(I) for any N > 0. To see that Ji is well-defined on Ki(A/I), we check the following:

252

C*-algebra

Extensions

(1) Si does not depend on the choice of w. (2) If v G Un+m(A/I) such that v*di&g(u, l m ) G t/'„ + m (A//) 0 , then S1([u}) = 61([v}). For (1), let » ! G £727l(^4) such that 7r(wi) = diag(u,u*). Put z = w\w*. Then 7r(z) = 12„. So z £ U2n{I)- In particular, [zgz*] = [g] in K0(I) for any projection g G M 2 n ( / ) . But z(wl n w*)2* = iwilnu>J, so K l n W ] " [In] = [wilnWl]

~ [ln]

(in # , , ( / ) ) .

For (2), let m > 1. Set vi — diag(u, l m ) . Then there is a scalar unitary matrix U G M2(n+m) such that f/diag(w, l 2m )C/* is a lift of &\&g{v\, v\). Put Wi = Udmg(w,l2m)U*. Then, since U G Af 2(n+m) , [Wlln+mW{] = [wlnw* © l m ] . This implies that <*i([vi]) = *i([«])• It remains to show that if u, v G Un(A/I) such that u = VVQ with v0 G Un(A/I)Q. Then <5i([u]) = <*i([u]). Let w G C/2„(A) such that ir(w) = diag(v,v*) and Z G Un(A)0 such that 7r(Z) = v0 Set W = wdiag(Z, Z*). Then TT(W) = diag(u,u*)- But Wln+mW* = w(l n + m )u;*. Therefore S1([u]) = S1([v]). So <5i is well-defined. It is easy to check that Si is a group homomorphism. If A is not unital and u G Un(A/I), we also define <Si([u]) = [iwlnio*] — [1„], where w G C/2n(A) is a lift of diag(w, u*). One needs to check that [wlnu)*j — [1„] G KQ(I). This is immediate, since Tr(wlnw*) = ln and J*O([W1„K;*] — [l n ]) = 0, where j : I —> A is the embedding. Remark 5.7.6

If there is a partial isometry v G Mn(A) such that n(v) =

u, then n(w) = diag(w,u*), where w = I ,

,

\ln-vv

,

v*

I . Therefore

J

Si{[u}) = [1 - vv*} - [1 - v*v]. In the case in which A = B(l2) and I — IC, this index map is precisely the Fredholm index. Definition 5.7.7 Let A be a C*-algebra and / be an ideal. Denote by S{ : Ki(S(A/I)) —>• KQ(SI) the index map for the suspensions. We define the exponential map do : K0{A/I) ->• Kx{I) to be 6J1 o <5f o / 3 A / / . Let p G M ^ A / / ) be a projection. Take a self-adjoint element a G A such that 7r(a) = p, where 7r : A ->• A / / is the quotient map, and put u — exp(27rio). Note that a 0 / in general. Then u G £/(/)• It can be shown

K-theory

and the universal coefficient

theorem

253

that *o([p]) = M = [exp(27rm)] (see 5.10.13). T h e o r e m 5.7.8

Let 0 -)• 1 - 4 A ^ B -)• 0

6e a s/iori exact sequence of C*-algebras. Then one has the following sixterm exact sequence. K0(I)

H

K0(A)

^

K0(B)

X1(A//)

^

Kl(A)

£i

tfl(J)

Both 5.7.4 and 5.7.8 are very important in -ftT-theory. Exercise 5.10.14 and 5.10.15 show that they can be quite useful for calculating .RT-theory. Corollary 5.7.9 Ki-!(A),i = 0,1. Proof.

Let A be a a-unital C*-algebra. Then Ki(Q(A))

=

Consider the short exact sequence: 0 -¥ A ® K ->• M(A /C) -> Q(A) -> 0.

By 5.1.12 and 5.1.15, Ki(M{A ® /C)) = {0}. It follows from 5.7.8 that K0(A) * ^ ( Q ^ ) ) and ^ ( A ) = K0(Q(A)).

Q

Definition 5.7.10 Let A be an amenable C*-algebra and let r € Ext(A,SB). Then r gives a homomorphism from A to Q(SB). Therefore we have TH : Ki{A) ->• Ki{Q{SB)). By 5.7.9, Ki(Q(SB)) = K^SB). Then by 5.7.4 Ki-i(SB) = Ki(B). Thus r gives two homomorphisms from ifi(i4) to ^ ( 5 ) (z = 0,1). We then obtain a map 7 : KK(A,B) -» Example 5.7.11 Let A = C([0,1]) and B be a cr-unital C*-algebra. It follows from 5.2.10 that 7 is a surjective homomorphism. Definition 5.7.12

Consider the short exact sequence

254

C*-algebra

Extensions

and denote by r the element in Ext(A, SB) representing this extension. Suppose that 7(7-) = 0. Then, by 5.7.8, we obtain two short exact sequences: 0-> Ki(SB)-+Ki(E)

^ Ki(A)->

0 i = 0,l.

When Ki{A) are free groups, the above short exact sequence splits. However, when Ki(A) is not free (or Ki(SB) is not divisible), the short exact sequence may not split. In that case, the corresponding extension is not trivial. Therefore there is 77 : ker7 —>• (Bi=o,iExtz(Ki(A),Ki(SB)). By applying 5.7.4, we obtain a group homomorphism 77: ker7 ->•

@i=0ilExtz(Ki(A),Ki-i(B)).

It is natural to ask if 7 is surjective and if 77 is an isomorphism. Let A and B be C*-algebras. We define KKa{A,B) = KK{A,B) 1 KK (A,B) = Ext(A,B). Definition 5.7.13

and

Let 0 ->• Gi -4- G2AG3 -> 0

be an exact sequence of countable abelian groups. G2 is an extension of G\ by G3. If, for any g £ G3 with finite order, there is g' € G2 with the same order such that s(g') = g, then we say the extension is pure. The set of all pure extensions of G\ by G3 is denoted by Pext(G\, G3). Note that it G2 is a pure extension, then every finitely generated subgroup of G3 can be lifted. Definition 5.7.14 Let 7o be the subgroup of those elements in KK(A, B) which correspond to the pure extensions: 0 -> Ki(SB) Denote by KL(A,B)

-> Ki{E) -»• Ki(A) -> 0 * = 1,2.

the quotient

KK{A,B)/%.

Definition 5.7.15 Let A be a G*-algebra. A is said to satisfy the Universal Coefficient Theorem if for any cr-unital C*-algebra B, the map 7 is surjective and the map 77 is an isomorphism. If A satisfies the Universal Coefficient Theorem (UCT), with S = r]^1, we have the following short exact sequence: 0 -> Extz{K,(A),K,{B))\KK*{A,B)-^Hom{K»(A),K,{B))

->• 0, (e7.22)

Characterization

of KK-theory

and a universal raulti-coefficient

theorem

255

where 7 has degree 0 and S has degree 1. Definition 5.7.16 Let M be the smallest class of separable amenable C*-algebras with the following properties: (1) M contains all separable commutative C*-algebras; (2) N is closed under inductive limits and (3) if 0—> A ^ D —> B —>• 0 is an exact sequence, and two of the terms are in M, then so is the third. In particular, J\f contains all type I C*-algebras and their direct limits. Theorem 5.7.17 (Rosenberg & Schochet) Every C*-algebra A G M satisfies the Universal Coefficient Theorem.

5.8

Characterization of KK-theory coefficient theorem

and a universal multi-

From the definition, one certainly has Ext(A,B /C) = Ext(A,B). Let e : A —>• A /C be an embedding (which maps A to e\\A <E> JCen). It is not difficult to prove the following: Theorem 5.8.1 Ext{A,B).

Let A and B be two C*-algebras. Then Ext(A®K., B) ^

We also have the following: Theorem 5.8.2 Let A and B be a C*-algebras. Suppose that A\ and Bi are two other C*-algebras such that A ~h A\ and B ~h B\. Then Ext{A,B) Theorem 5.8.3

^Ext(AuBi).

Let 0 -» IAE^IE/I

->• 0

be a split exact sequence of C*-algebras. Then one has the following two split exact sequences: 0 ->• Ext(A, I)^Ext(A,

E)4*;, Ext(A, E/I) ->• 0

0 ->• Ext{E/I,B)A\,Ext(E,B)hExt(I,B)

and

->• 0

5.8.4 Let A = \im.n^>.00(An, hn) be an inductive limit. For each n, (/in)» gives a homomorphism from Ext(An+i,B) -> Ext(An,B). Then there is a

256

C* -algebra

Extensions

homomorphism s : Ext(A,B) —¥ lim*- Ext(An,B). The following theorem says that s is always surjective. However, the kernel of s may not be zero. Theorem 5.8.5 Let B be a a-unital C*-algebra and let A = limT!,_).00 An. Then there is a natural Milnor lim -sequence 0 -s- \\mxnExt(An,B)

-> Ext(A,B)

-> \imExt(An,B)

We now present a characterization of KK-theory

-> 0.

due to Higson.

5.8.6 Consider covariant functor F from separable C*-algebras to abelian groups which are assumed to be stable, homotopy invariant and split exact. In other words: (i) F is a homotopy functor; (ii) for every separable C*-algebra B the homomorphism e* : F{B) —> F(B (g) JC) is an isomorphism; and (iii) if 0 —• / —>• E ^ E/I —• 0 is a split exact sequence of separable C*algebras then the sequence 0 —> F(J) —>• F(E) ^ F(E/I) —>• 0 is also split exact. Fix a separable amenable C*-algebra A, then Ext(A, —) is one of these functors. Define a natural transformation between two such functors F± and F2 to be a collection of functions a A : F\(A) —> F2(A), one for each separable C*-algebra, such that if h : A —• B is a homomorphism then the diagram Fi(yl)

^

F^B)

F2(A)

h

F2(B)

commutes. The functions a A : i*i(A) -»• -?2(-<4) a r e g r o u P homomorphisms. We are particularly interested in the relation between the functor Ext(A, —) and other functors. Theorem 5.8.7 Let F be a functor from separable C* -algebras to abelian groups which is homotopy invariant, stable and split exact. Let A be a separable amenable C* -algebra. Then, for any x £ F(A), there is a unique natural transformation a : Ext(A, -) -> F such that (^([id^]) = x. Here [id^j is the element in Ext(A,A) map (see 5.2.14).

which represents the identity

Characterization

of KK-theory

and a universal multi-coefficient

theorem

257

Definition 5.8.8 Let 6n : C(Sl) ->• C(Sl) be the degree n map, namely, 6n is denned by z -> z n for z on the unit circle. By identifying C 0 ((0,1)) with the C*-subalgebra of C(S1) (vanishing at 1 ), we obtain a degree n homomorphism from C 0 ((0,1)) ->• C 0 ((0,1)). We may identify Co(R) with Co((0,1)). Denote by C n the mapping cone (see 5.2.4) Cgn of 6n. This is a (non-unital) commutative C*-algebra. Note that SCQ(W) = C0(]R2). There is a short exact sequence

0 -> CoC^-Vc^-VcbW -»• 0, where j ( / ) = ( 0 , / ) and s((a, / ) ) = a (but 0 n (a) = f(l)). By applying the six-term exact sequence (in 5.7.8), one computes that Ko(Cn) = J*/n2* and tfi(Cn)=0. Definition 5.8.9 Let A be a C*-algebra. Using degree n map 0 n and the mapping cone Cn, we obtain the following short exact sequence 0 _>
-> 0

Define ^ ( A , Z/nZ) by if;(A C„), 2 = 0,1. Definition 5.8.10 By 5.7.4, let (3A : K0{A) -fTi(A) -» X O ( 5 J 4 ) be the isomorphisms. Put

->

tfi(SA)

and 6^ :

^ ^ ( A J - ^ ^ Z / n Z ) by pW = (i ® j(en)U o 0 S A o fo and PM = (l ® j ( 0 n ) ) t l o /3 S A o 0 A , P® : Ki{A,Z/nZ) 0)

/3i = ^

^ Ki-^A)

by

° (1 ® *(»n))*o and /#> = ^

o (1 ® s ( 0 n ) ) , !

Then we have the following short exact sequence K0(A) Txn

"-=•

K0{A,Z/nZ)

^

^(A) 4-Xra

Ko(A)

&

Xi(.4,Z/nZ)

£

#i(A)

The up and down maps are multiplication by n.

258

C*-algebra

Definition 5.8.11

Extensions

Let 8ntTn : Cm —>• Cn be a map making the diagram

0->C 0 (R 2 )

i (

V

Cm

s(

V

C0{R)

^

Co(M)

0~>C 0 (R 2 )

i(

-V

C„

' ( -V

C 0 (R)

^

C 0 (R)

commute. Define Kn.m* = (id ® e n>m )»i : iCi(i4, Z/mZ) -> ^ ( A , Z/nZ), i = 0,1. We obtain an exact sequence: Ki(A, Z/mZfh,nKi(A,

Z/mnzfn-^nKi(A,

Z/nZ).

To obtain a long exact sequence, we define (3lm n = pm

° Pn, i = 0,1,

77i, n = 1 , 2 , . . . .

Proposition 5.8.12 K0(A,Z/mnZ)

There is the following exact sequence: ""-S""

#0(AZ/nZ)

*V

Jifi(4,Z/mZ)

I ™mn,m

4- 'Snn,m

if 0 (A,Z/mZ) Definition 5.8.13

V"

tfi^.Z/nZ)

V""

K^Z/mnZ)

Define £ ( 4 ) = ©<=o,i ®n=i ^ ( A Z / n Z ) .

Denote by Hom\(K_(A),K_(B)) (<^°>^n).

the set of all homomorphisms <j>n =

where

^W : Jfi(i4,Z/nZ) ->•

K0(B,Z/nZ)

are homomorphisms which preserve the exact sequence in 5.8.12 and the exact sequence in 5.8.10. Given an element z € KK(A,B), we obtain an element T(z) in Hom\{K_{A),K{B)). The map T is a homomorphism. Proposition 5.8.14

Let A = limn_yoo An.

HomA(K(A),K(B))=

lim

Then

HomA(K{An),K{B)).

We end the section with the following universal multi-coefficient theorem

Approximately

trivial

259

extensions

Theorem 5.8.15 (Dadarlat & Loring) Let A € W and B be a a-unital C*-algebra. Then there is a short exact sequence 0 -»• Pea;t(^(A),^( J B))AK J F£:(A,5)^ J H'omA(K(A), ; K( J B)) -> 0 which KL(A,B)

5.9

is =

natural in HomA{K{A),K{B)).

each

variable.

Therefore

Approximately trivial extensions

In this section, we will show that if A satisfies the UCT, then Pext(K*(A),K*(SB)) corresponds to the subgroup of so-called (stable) approximately trivial extensions. We also give another version of 5.6.4 (see also 5.10.20). Let {Bn} be a sequence of C*-algebras. We would like to compute K i(UnBn) (* = M ) ' B y 4.8.24 and 4.8.25, if each Bn is stable, then = ^ t ( r i n - ^ n ) nnKi{Bn)I n general, the situation is rather different. Definition 5.9.1 Let r : N ->• N and T : N 2 -> N be two maps, and A be a C*-algebra. (1) We say A has ifo-r-cancellation if p® iMr(n)(A) ~ 1® ^Mr,n)(A) f° r an Y two projections p,q € Mn(A) with [p] = [q] in K0(A). (2) We say A has Xi-r-cancellation if u © lMr(n)(A) a n d v © lMr(n)(A) a r e in the same path-connected component of U(Mn+r^(A)) for any pair of unitaries u,v € M n (A) with [it] = [v] in K\(A). (3) Let s > 1. We say A has i n s t a b l e rank s if J7(MS(A)) is mapped surjectively to K\{A). (4) We say A has stable exponential length b : N -> R if cel(M m (A)) < 6(m) for all m. (5) We say A has K0-divisible rank T if - n [ l A ] < fcz < n[U] implies - T(n, k)[lA] <x<

T(n, k)[lA]

for anyfc,n € N . There is a longer list of those conditions. But we only use these here. 5.9.2 We have the following easy facts: (i) If A has ifi-stable rank s for some integer s > 0, then A has K\-rcancellation with r(n) = s.

260

C* -algebra

Extensions

(ii) If every z & KQ (A) can be represented by two projections p, q G A with z = [p] — [q], then A has if 0 -0-cancellation as well as if 0 -divisible rank T with. T(n,k) = 1. (hi) It follows from 4.8.21 and 3.1.14 that every C*-algebra A with RR(A) = 0 and tsr(A) = 1 has stable exponential length 7r and has K0-0cancellation and .fiTi-l-cancellation (3.1.10). (iv) Let B be a cr-unital C*-algebra. It was shown (5.1.15) that M{B®KL) has stable exponential length 67r. Furthermore, Ki(M(B ® K)) has Ki-0cancellation (i = 0,1). (v) Let Q(B) = M(B ® K)/B ® K. Then Q(B) has stable exponential length Q-K and Ki(M(B /C)) has ifj-O-cancellation (i = 0,1) (see 5.1.12). (vi) If KQ{B) is a weakly unperforated ordered group, then K0(B) has ifo-divisible rank T with T(m,n) = m + 1. Definition 5.9.3 Let {£«} be a sequence of unital C*-algebras. Let Ylb Ko{Bn) be the set of all sequences {xn}, where xn G Ko(Bn) and x„ can be represented by [pn] — [qn] for projections pn and qn in M/,(5 n ) for some integer L > 0 (L, of course, depends on the sequence {a;n} but not on n), and let Y\bKi(Bn) be the set of all sequences {zn}, where zn £ K\(Bn) and zn can be represented by unitaries in ML(BH) for some integer L > 0 (L depends on the sequence {zn} but not on n). There are natural homomorphisms 9i : if j ( J | n Bn) —¥ J] 6 (ifj(B n )) (i = 0,1). It is obvious that the 6i are surjective. We leave the following to an exercise (see 5.10.17). Proposition 5.9.4 Suppose that Bn are C*-algebras and r : N —>• N is a map. (1) / / each Bn has Ko-r-cancellation, then ker#o = 0. (2) Let b : N —• N. If each Bn has K\-r-cancellation and ce\(Mm(Bn)) < b(m), then ker#i = 0. (3) if Bn has Ki-stable rank s (s > 1), then Kx{Y[nBn) = Y[nKi(Bn). (4) If Bn has Ko-r-cancellation, then Y[K0(Bn)

= {{xn} : -L[1BJ

<xn<

L[lBn)

for some L > 0}.

6

(5) Suppose that Bn has Ko-r-cancellation and j : Y[n Bn —>• Yin B ® K. is the embedding. Then j*o is infective.

Approximately

trivial

261

extensions

5.9.5 It follows from 4.8.24 and 4.8.25 that if Cn is stable, then KidlnCn) = UnKi(Cn), K^n Cn, Z/fcZ) = I L W n , Z/fcZ), KiiQ) = IlnKi(Cn)/®nKi(Cn) &ndKi(Q,Z/kZ) = YlnKi(Cn,Z/kZ)/(Bn Ki(Cn,Z/kZ), where Q = Yln Cn/ ©„ C„. Let j : ]}„ Bn -» fin 5 ® * b e the embedding. Then there are homomorphisms j \ ' : Ki(Yln Bn, Z/kZ) —¥ UnKi(Bn,Z/kZ), i = 0,1 irnd k = 2,3,...,. T h e o r e m 5.9.6 Let r : N -s- N, T : N2 -> N, Z : N -> R and s > 1. / / eac/i 5 „ /ias Ko-r-cancellation, Ko-divisible rank T, stable exponential length I, and K\ stable rank s, then both JQ ' and j[ are injective. Proof. By 5.9.4, K0(UnBn) = UbK0(Bn) and K^^Bn) = Y[n Ki(Bn). It follows from 5.8.10 we have the following commutative diagram: n 6 ^ o ( B n ) ^ nbK0(Bn) +j.o

-» K0(UnBn,Z/kZ)

-M»O

-»• tfi(II„£») ^

-!"(*)

tfi(II„fl™)

^i.i

n n #o(B„) ^ n n Ko(Bn) -+ Y\n K0(Bn°Z/kZ)

-w.i

-> ft, *i(B„) ^

Both horizontal sequences are exact. To see that jg

Il„ ^ i ( B - )

is injective, we let

(k)

x G kerj 0 . Note that j*i is an isomorphism. By the third commutative square from the left, we see that there is {zn} G Y\b^°(Bn) such that x is the image of {zn}. Since j*o is injective, by the second commutative square from the left, there must be {yn} G ]T n K0(Bn) such that k{yn} = kj*o{{zn}). By (3) in 5.9.4, there exists L > 0 such that -L[lBn\

< kyn < L[1 B „], n = l,2,....

Therefore, since Bn has Ko-divisible rank T, -T(L,k)[lBn]

< yn


By (4) in 5.9.4, {
tfi(n„fl»)A*i(n„5»)-»tfi(n„B».z/fcz)-> r u - M ^ A ru-MSn) vjii 4-j,i n n ^ i ( B n ) ^ UnKi{Bn)

"M*' -• UnKi{Bn'z/kZ)

4-J«O

-• TlnK0(Bn)

4-J.O

A

UnKo{Bn)

Since j*i is an isomorphism and j*o is injective, this time the injectivity (k)

of j \

follows from a standard diagram chase (the Five Lemma).

•

262

C*-algebra Extensions

Remark 5.9.7 The reader may notice that the theorem is not stated symmetrically. A more general version can be found in [84]. We only need a special case here. The following is left for an exercise (5.10.18). Proposition 5.9.8 (1) The map Ki(Un Bn,Z/kZ) -> Ki(Un Bn/ ®„ Bn, Z/kZ)) for any C* -algebras Bn (unital or not). (2) There is a natural identification

is surjective

KiiJJBn ® /C/ ©„ Bn ® K,Z/kZ) = Y[(Ki(Bn,Z/kZ)/ ® Ki{Bn,Z/kZ)). n

n

(3) If jl

is injective, then the map

Ki([[Bn/(BnBn,Z/kZ)) n

-> Y[Ki(Bn,Z/kZ)/

®nKi(Bn,Z/kZ)

n

is also injective. Theorem 5.9.9 Let r : N -> N, and I : N -> R and s > 1 with r(n) > 1 and l(n) > TT. Let B be a unital C* -algebra with K^-r-cancellation, stable exponential length I and K\-stable rank s. Let A be a unital amenable separable C* -algebra in Af and j : A —* B be a unital full embedding. Suppose that a : A —> B and (3 : A —>• B are two homomorphisms such that [a] = [/?] in KL(A, B). Then, for any e > 0 and any finite subset J- C A, there is a unitary u € MK+I{B) (for some K > 0) such that ll«*(a(o), j(a), ...,j{a))u - (/3(a), j(a), ...,j(a))\\

<e

for all a £ J-, where j{a) repeats K times. Proof. Let a : A ->• M(SB ® K)/(SB ® K) such that [a] = -\/3] in KK(A, B) (= Ext(A, SB)) and 7 = a © a. In KK(A, B), [7] gives an element in ®i=otiPext(Ki(A), Ki+i(B)) which is represented by the following pure extensions: 0-* Ki+1(B)

^ Hi •$ Ki(A)-*

0

(:GZ/2Z).

Let Ki{A) be generated by pi,52, ••••,9n, ••• and let G4 be the subgroup generated by gi,..., gn. There is an injective homomorphism j n : G„ —• Hi such that pojn = id^ ( i). Note that jn{gk) -ji{9k) & K0(B) for any n,i>k. n

Suppose that jn{gk)—jk(gk)

is represented by a difference of two projections

Approximately

trivial

263

extensions

in Mi(fc,„)(B). Let L(n) = max{Z(fc,n) : k < n}. Let BL = n ^ L ( n ) ( S ) . Since B has iC0-r-cancellation, by 5.9.4, K0(BL) == n& ^o(-B). We have for any gG G ^ , (0,...,0,jk+1(g)

- jk(g),...,jn(g)

- jk(g),-)

£ K0(BL).

(e9.23)

Let Qoo = {a}, Ax, = {/3} : A -^ BL (where {a} = {a, a,...} and {(3} = {(3,(3,...}). Note a.,*, and Ax> give essential extensions of A by S(BL) by the "mapping tori" (see 5.2.14). Let 0 -» 5J5 /C -» £CT ->• A ->• 0 be an extension given by cr. Define i?£. = {{e n } : e n = e, e G £? /C C S , 5 L ® £ if each en = e. Let Ea = E'a + (SBL) /C. Then £CT is an extension of A by S(BL) /C. This extension is also denoted by <7oo- Let ,E7 be defined the same way as Ea by replacing a by 7. Since A is amenable, let a' : A —» M(SB /C) be a contractive completely positive linear map such that IT o a' = cr, where IT : M(SB ® /C) —»• Q(SB) is the quotient map. Since [a] — —[7], there is a unitary it G M(SB /C) such that u*((3(a) © cr'(a) © doo(a))u - doo(a) = c(a) G

SB^K

for all a G A, where rfoo(a) — diag(j(a), j(a),...) is an absorbing trivial extension. Let UQQ = {u,u,...}. Note that Uoo G M(S(BL) £)!) Therefore «^o(/3oo(a) © a'ooia^Uoo - doo(^oo(a)) = {c(a)} G S'(B L ) ® /C. This shows that [Ax>] = -[ax,]- Consequently, [7^] = [a^] - [Ax>] in KK{A,BL). In fact [700] gives the following pure extension 0 -> K i + i ( 5 L ) -> Ft 4 Ki(A) ->• 0, where F* = {{ 5 } + {gn} : g G #i,{c/ n } G K i + i ( S L ) } . Let K„ : KX{A) -» i?i be a map (not necessarily additive) such that (nn)\cw = jn- By the construction above, { K „ } maps ^ i ( A ) into F\. Let II : BL -> BL/®nML{n)(B) be the quotient map. Then TIoaio^-TIo/^] = [ n o 7 o o ] G ©i=o,i-Pea;i(i ; i:i(A),i ; fi + i(B L /©„M L („ ) (B)). It is easy to check that Ki(BL/ © ML{n)(B)) = Ki{BL)/ © K;(B), i = 0,1. We also have that (II)*(-F\) = Fi/ ®KQ(B). Moreover the corresponding short exact sequence

264

C* -algebra

Extensions

not only is pure but actually splits: 0 -> K0(BL/

®ML(n){B))

- • (n),(F!) -> ^i(,4) -> 0.

In fact by (e 9.23), (II), o ({/«„}) is an injective homomorphism from K\ (A) into F\l © KQ{B). This implies that [II o 700] gives a trivial element in Pext(K1(A),K0(BL/ © n ML{n){B))). On the other hand, since B has stable exponential rank b and ii^-stable rank s, by 5.9.4,

Thus a similar but easier argument shows that [IT o 7^] gives also a trivial element in Pext(K0(A),Ki(BL/ ©„ ML{n)(B))) ( we do not need to worry about the size of jn(g)—9k{g) as in the KQ case, since Ki(BL) = Y[Ki(B)). This implies that [IT o Q o o ]

= [n o /3oo]

KK(A,BL/®nML(n)(B)). Note also that if j : A —> B is a unital full embedding, then En : A —> Mi,{n){B) defined by sending a to diag(E(a), ...,E(a)) gives a unital full embedding E : A -¥ BL and a unital full embedding E = IT o E : A —)• BLI © ML(n}(B). From 5.6.4, for any e > 0 and any finite subset T C A, we obtain a unitary U G BL/ © ML^n){B) and an integer K > 0 such that in

\\U*diag(Uoaoo(a),

E(a),..., £J(a))t/-dia 5 (no/3 0 0 (a), E(a),..., E(a))\\ < e/2

for all a £ J7 (where E(a) repeats K times). It is easy to see (see 2.8.9) that there are unitaries Un G MK{ML(U){B)) such that U({Un}) = U. We have, for sufficiently large n, \\U*diag{a(a),E(a),....

£(a))t/„ - diag((3(a), E(a),..., £(a))|| < e

for all a £ T (where E{a) repeats K(L(n))

times).

•

Definition 5.9.10 We now turn to extensions of A by B, where B is u-unital. An extension r : A -» M(B)/B is said to be approximately trivial if there is a sequence of trivial extensions r„ : A —> M(B) such that lim„ r n (a) = r(a) for all a G A. An extension is stably approximately trivial, if there is a trivial extension To : A —> Q(i?) such that r © To is an approximately trivial extension of A by B /C. Denote by 7o(A -B) the set

265

Exercises

of stably approximately trivial extensions. One calls To(A, B) the closure of stably trivial extensions. Theorem 5.9.11 Let A be a separable C*-algebra in TV and B be a aunital C*-algebra. Then T0{A,B) = ®i=0liPext(Ki(A),KH1(B)). Proof. We will show that ®i=0tlPext(Ki{A), Ki+1(B)) C T0(A,B). For the other inclusion, see 5.10.20. We will consider the case that A is unital only. Let T : A —• Q(B) in %(A, B). Let r 0 be an absorbing unital trivial extension. It is easy to check (see 5.10.19) that r 0 : A —>• Q(B) has to be full. Note that Q(B) has .Ko-O-cancellation, Xi-stable rank 1, and stable exponential length 67r (see also 5.1.15). Therefore, by 5.9.9, there are unitaries un G Q{B) such that lim unT0(a)un

= r © T0(a)

n—+00

for all a G A. 5.10

O

Exercises

5.10.1 Prove 5.1.7 5.10.2 Let X be a locally compact and cr-compact Hausdorff space and A a cr-unital C*-algebra. Show that M{C0{X,A)) = Cb{X,M{A)„). 5.10.3 Let B be a unital C*-algebra. Show that an infinite matrix (a^) with a^ G B represents an element in M(B®K) if and only if the following holds: for any e > 0 and integer I > 0, there exists N > 0 such that ||(e„+ m - en)(a,ij)ei\\ < e and \\{ei(a.ij)(en+m

- en)\\ < e

for all n > N (and for any m). 5.10.4 Show that {Vfc} converges to V in the strict topology in 5.1.11. 5.10.5 Let S be the unilateral shift and T be the C*-algebra generated by S. Show that T is an essential extension of C(S1) by /C. 5.10.6 Let A be a separable C*-algebra and r : A —>• Q{B) be an extension. Then there is a trivial extension T0 : A —¥ Q(B) such that T®TQ is essential. 5.10.7 Prove 5.2.11. 5.10.8 Prove 5.2.12.

266

C* -algebra

Extensions

5.10.9 Prove the case that k — 2 for 5.3.6. Then prove the general case by induction. 5.10.10 Let B be an amenable C* -algebra and T C B be a finite subset in the unit ball. For any e > 0, there is a finite subset Q in a finite subset K c C and 6 > 0 such that if L : B —> A is an J-^-multiplicative map such that ||L(a)|| < 1 for all o f f , \\(aL(a) + /3L(b)) - L(aa + (3b)\\ < S and \\L(a*)-L(a)*\\ < 6 for all a,b eQ and a, (3 G K, where ^4 is a C*-algebra, then there is a contractive completely positive linear map : B —• A such that \\L(a) - 4>(a)\\ <s for a G B.

5.10.11 (Kirchberg) Let A be a unital separable amenable C*-algebra and B be a unital purely infinite simple C*-algebra. Using the methods in section 5 of this chapter, show that any trivial extension r : A —> Q(B) is absorbing (see the remark after 5.3.9). 5.10.12 Let A be an amenable C*-algebra and I be an ideal. Show that A/1 is amenable. 5.10.13 Let A be a C*-algebra, / C A be an ideal and let p G A/I be a projection. Suppose that a G Asa such that 7r(a) = p, where 7r : A —> vl/7 is the quotient map. Show that <Jo([p]) = [exp(2Tria)]. 5.10.14 Let S be the unilateral shift on I2. Let T be the C*-subalgebra generated by S. Compute the -ff-theory of T. If 7fc is the C*-subalgebra generated by Sk and K,, what is its if-theory? 5.10.15 Let n > 0 be an integer. Compute the if-theory of Cn. 5.10.16 Prove the statements in 5.0. 5.10.17 Prove 5.9.4. 5.10.18 Prove 5.9.8. 5.10.19 Let A and B are unital C*-algebras. Show that every absorbing unital extension r : A —>• M(B)/B is full. 5.10.20 Show the converse of 5.9.9. Also complete the. proof of 5.9.11. 5.10.21 (Brown, Douglass and Filmore) An operator T G B(l2) is called essentially normal if 7r(T) is normal in B(l2)/IC. Denote by spess(T) the spectrum of 7r(T). There is an injective homomorphism hf : C(spess(T)) —>•

Exercises

267

B(l2)/IC. Set Ind T = (hT)*i : A"i(C(sp e «(T))) -> Z. Let 7\ and T 2 be two essentially normal operators. Use the UCT and Voiculescu's theorem to show that T\ and T2 are approximately unitarily equivalent if and only if spess(T\) — spess(T2) and Ind^ = IndT2. In particular, T = N + K, for some normal operator N and compact operator K if and only if Indr = 0.

Notes Theorem 5.4.2 provides a proof of 5.4.4 which is different from and much shorter than that of Effros and Choi. We avoid the topic of tensor products entirely. Moreover, the proof of 5.4.2 that we present here fits well in the chapter. Theorem 5.5.8 was known as Voiculescu's non-commutative Weylvon Neumann theorem. Theorem 5.5.12 was obtained by Kasparov. The renewed interest in absorbing extensions began with Kirchberg's (5.10.11) work on classification of amenable purely infinite simple C*-algebras (a fascinating subject which, unfortunately we could not cover). The important fact that the trivial extension d^ in 5.5.4 is absorbing was originally proved by the author ([126]) which led to a version of Theorem 5.6.4. It was Dadarlat and Eilers who observed that a simplicity condition in original version in [126] could be replaced by the condition that the embedding is full. Author's original proof of 5.6.4 also used a theorem of G. Pedersen on derivations (8.6.12 in [147]) and Kadison-Ringrose's derivable automorphisms ([9l]). The proof given here is more elementary but analytical. Theorem 5.6.4 plays a very important role in the classification of amenable C*-algebras. It can be applied to the case in which C*-algebras are finite as well as the case in which C*-algebras are purely infinite. Theorem 5.9.11 has the roots in the BDF-theorem, and is a more general version obtained by Salinas ([169]). See also [172]. The proof presented here uses 5.9.9.

— 1076 — & ft, (10S6 - 1101)

Chapter 6

Classification of Simple A m e n a b l e C*-algebras

In this chapter, we will present some classification theorems for certain simple amenable C*-algebras. The main results include Elliott-Gong's theorem for simple AH-algebras of real rank zero. 6.1

A n existence t h e o r e m

6.1.1 Let A and B be two C*-algebras and L : A —> B be a contractive completely positive linear map. Let C n be as in 5.8.8. Then Lid : A®Cn —> B®Cn is a contractive completely positive linear map. In order to simplify notation we will use L for L id. We will also use L for the induced map L ® id : A (g) SCn —> B (g) SCn. Given an integer n, a positive number e > 0 and a finite subset T C A ® C n , there is r? > 0 and a finite subset Q C A such that if L is ^-77-multiplicative, then L is JT-e-multiplicative on A Cfc and A ® SC^ for all k < n. Given a finite subset V of projections in ©fc=iMoo(A 0 Cfc) © ©fc=iM00(A SCfe), there is rj > 0 and a finite subset Q C A such that if L is ^-^-multiplicative then [L]\-p is well-defined (4.5.1). Suppose that k[p] = [p] in K0(A® Cn). With sufficiently large J 7 and sufficiently small 77, as in 4.5.1, /c[L]([p]) = [L]([p]). Therefore, by 4.5.1, if G C K_(A) is a finitely generated subgroup, with sufficiently large T and sufficiently small 77, [L]\a is well-defined. In what follows, we denote by P(A) the set of projections in ®k=iM00(A ® Cfc) © ©fc=iMoo(yl SCk). Let P be a finite subset of P(A) and G be the subgroup oi K_(A) generated by V. Whenever we write [L]\p and [L]\a we imply that they are well defined. Note that we will write [L]\-p instead of [£]|-p, where V is the image of V in K_(A). We say [L]\G is positive if [L](a;) > 0 whenever x G G H KQ(A)+ is positive. 269

270

Classification

of Simple Amenable C -algebras

If L : A —>• B is a homomorphism then [L]\G is always well-defined and positive. Let C C A be a C*-subalgebra. Suppose that P C P ^ ) and the subgroup G' of Xo(C) generated by V is isomorphic to G (one always has a homomorphism from G' onto G) and /i : C —• 5 is a F-e/2multiplicative contractive completely positive linear map so that [h]\c is well defined and positive. Suppose that the finite subset T required above is also in C. Suppose further that either A or B is amenable. Then there is sequence of completely positive linear contractions Ln : A —>• C such that lim„ £ „ (a) = a for all a £ T (see 2.3.6). One sees easily that, for large n, L = h o Ln gives an .F-e-multiplicative completely positive linear contractions such that [L]\G is well-defined and positive. This will be used later. Definition 6.1.2 Let A be a residually finite dimensional C*-algebra (RFD C*-algebra) (see 3.7.6). Recall that A has a separating family of finite dimensional irreducible representations. Let B be a C*-algebra. We use the notation B+ for the C*-algebra obtained by adding a unit to B (even if B is already unital). An asymptotic sequential morphism <j> = {n} from A to B is a sequence of completely positive linear contractions {(f>n} '• A —¥ B+ ®K, such that (1) \\4>n(ab) — (f>n(a)(f>(b)\\ —> 0 as n —> oo, for a, b G A,

(2) there exist two sequences of homomorphisms hn, h'n : A -> K and a unitary un £ K. such that \\u^diag(TT o n(a), hn(a))u

— h'n(a)\\ —> 0 for all a £ A,

where IT : B+ ® K. —>• K. is the quotient map, (3) there exist two sequences of homomorphisms hn, h'n : A —>• C • 1^+ ® /C with finite dimensional range satisfying the following: for any finite subset V C P(^4), there is n > 0 such that, [fc]|p is well defined for all k > n, [4>k © /lfc]|7> - [/ifelk = fan © fcn]|p -

\K)\v

and {[„ © /in] - [h'n]} defines an element in Hom^{K_{A),K_{B+ ®K)), + i.e., there exists a € i7omA(^(^l),^(jB <8> K)) such that, for each finite subset V C P(A), ([4>n © /in] - [/i„])|p = a\v for all large n. An asymptotic sequential morphism (f> — {fin} is said to have finite dimensional range if the range of each n is contained in a finite dimensional C*-subalgebra.

An existence

271

theorem

Let

~ V if there are two sequences of homomorphisms hn, h'n : A -» Cl B + ®/C with finite dimensional range and unitaries u n € (B+ ® /C) such that \\u*ndiag{(pn{a),hn(a))un

- diag(tpn(a),h'n(a))\\

->• 0 as n -s- oo.

We denote by (0) the equivalence class of asymptotic sequential morphisms represented by

and tp be two asymptotic sequential morphisms from A to B. We define

+ tp)(a) = diag(4>(a),ip(a)). This clearly defines an addition on £(A,B). In the rest of this section A is a separable amenable RFD C*-algebra unless otherwise stated. Lemma 6.1.3 Let (pn : A -4 B+ ® K be an asymptotic sequential morphism with finite dimensional range. Then there is a unitary Un G + UQ(B+ K,) and an asymptotic sequential morphism tpn : A —> B /C with finite dimensional range such that the range of a,dU o (cpn + tpn) is contained in Clg+ Cg) K. Moreover, if /C such that qni = Pn. Let En = Yli=i 1m- Suppose that e\ < qni (i > 1) be a projection with e\ ~ et. Set pnl - (Pn - ex) + Ylt^a e'i a n d Vm = (qin ~ e 'i) + ei (* > !)• So p„i are mutually orthogonal and mutually equivalent. Thus we obtain an isomorphism hn : Fn —> En(B+ <8) K)En such that

272

Classification

of Simple Amenable C* -algebras

ei © hn(ei) e En(C • lB+ ® )C)En. Set tpn = hno (pn. Thus, by the proof of 3.4.6, there is a unitary U' £ En(C- 1B+ ® JC)En such that imad/7' o (n © ipn) £ En(C-lB+ ®K)En. £„£/ = UEn = U'.

There is a unitary U £ C/0(B+ K) such that D

Proposition 6.1.4 (1) £ (A, B) is an abelian group. (2) For fixed A, £(A, —) = £A(~) *S a covariant functor from separable C* -algebras to abelian groups which is homotopy invariant and stable. (3) If 0 —• J —» Z) —>• I ? / J -4- 0 is a s/iori exaci sequence, then £A(J)^£A(D)^£A(D/J) is exact in the middle. Proof. To obtain (1) we apply 5.5.13 which asserts that there is a sequence of completely positive contractions r„ : A —• Mun\{A) and a sequence hn : A —>• Mfc(„)+1(j4) of homomorphisms with finite dimensional range such that lim \\diag(a,Tn(a)) - /i„(a)|| = 0 71—>00

for all a £ A. Fix a homomorphism h' with finite dimensional range. Then we have

( M + [h'])\v ~ [K]\v = [ti] - [idA\\v

(el.l)

for any finite subset V C P(A) and for all large n. Let .Fn be the range of hn. Suppose that {n} is an asymptotic sequential morphism from A to B+ ® fC. Let {.Fn} be an increasing sequence of finite subsets of A such that the union is dense in the unit ball of A. Since dimFk < oo, there exists n(k) (n(k) > n(k — 1)) such that ||0 n (a) -gW{a)\\
+ k)

for all a £ hk{J-k) and n > n(k), where g„ : Ffc —>• 5 + /C is a homomorphism (see 2.5.9). Therefore, by letting r^ = Tfc, /i^ = /ifc and tpn = ffn for n(fc) < n < n(fc + 1) (and noting that <j>n are contractive), that lim \\diag(4>n{a),(j)noT'n{a)) - i/jn o h'n(a)\\ = 0

(el-2)

n—>oo

for all a € A Note that tj)n o hn is a homomorphism with finite dimensional range. It follows from 6.1.3 that we may assume that ifjn o h'n has image in

An existence

theorem

273

C- 1 B + ®/C- To see 4>n°Tn is a n asymptotic sequential morphism, one notices that condition (3) in 6.1.2 follows from (e 1.1). Moreover, the condition (2) follows from the fact that n is an asymptotic sequential morphism and (el.2) above. This implies that £A(B) is a group. This proves (1). It follows that £A(—) i s a covariant functor from separable C*-algebras to abelian groups. It is obvious that it is stable. The homotopy invariance follows from 4.8.27 (and 6.1.3). Now we show (3) holds. Let ((j>) G £A{J) and be represented by an asymptotic sequential morphism {<j>n} '• A —> J+ /C. It immediately follows from condition (2) in 6.1.2 that 7r» o j„ = 0. Next assume that (0) € £A{D) and is represented by <j>n. It is easy to see that we may assume that xm<j)n C Mfc( n )(D + ). Suppose that there are asymptotic sequential morphisms hn, h'n : A —> C • 1(D/J)+ ® )C with finite dimensional range and unitaries un £ ((D/J)+ ® K.) such that lim n ^oo ||u*diag(7r o n(a), hn(a))un - h'n(a)\\ = 0 for all a e A. A standard argument shows that we may assume, without loss of generality, that imhn C Mn(n), im/i^ C MKI^ for some large K(n) and K'(n). We may further assume that K{n) = k(n) and K'(n) = 2k(n). It is then easy to see that we may replace un by vn e t/(M 2 fc(„)((.D/J) + )). By replacing un by vn © i£, we may assume that un £ UQ{{D/ J)+ ® /C). Therefore there are zn € U(D+ ® K) such that ir{zn) = un. By identifying C • 1(D/J)+ ® £ with C • 1D+ ® £ and C • lj+ ® /C, we may view hn, h'n : A —¥ C • 1^+ ® /C. Let n : r I n ( - D + ® ^ ) -> F L C 0 * ® / C ) / ©" (-° + ® /C ) b e t h e quotient map. Let U = {zn} and w = U(U). Let also $ = {„}, H = {hn} and tf' = {tin}. Then, 17*(II o $(o) © ff(a))tf - n o # ' ( a ) e n „ ( J ® £ ) / ©« ( J ® *Q f o r all a e A. Therefore, iim7*(II o $ © # ) [ / e I L ( J + ® £ ) / ©« ( J + ® £)• Since A is amenable, there is a contractive completely positive linear map L = {ln} : A -> n „ J + ® ^ s u c h that n o L = U*(Il o $ © # ) [ / , where Zn : A —>• J + /C is a contractive completely positive linear map. Then limn^oo \\l„(a) - z*(<£„(a) + hn(a))zn\\ = 0 for a £ A. Thus ({/„}) is a sequential asymptotic morphism in £A(J) and

<{U) = Wn} + {M>This implies that () is in the image £A(J+)Corollary 6.1.5

£A{—) *s split exact.

•

274

Classification

of Simple Amenable C* -algebras

Proof. We first show that if D/J is contractible, then the embedding i : J —• D gives an isomorphism from £A{J) to £A(D) (without assuming the exact sequence 0 —> J —> D —• D/J —>• 0 splits). Since 5A{—) is homotopy invariant, £A{D/ J) = 0. By the exact sequence 0 ->• J ->• D -» D / J ->• 0 and 6.1.4, i» is surjective. We also have £A(D/J) = £A{S{D/J)) = £A(C0((0,1], J)) = 0. Set S(D, D/J) = {{a, 6) e D © C 0 ([0,1), D / J ) : TT(O) = 6(0)} and Z{J,D) = {x€ C([0,1],D) : x(0) G J } . From the exact sequences: 0 = £A(S(D/J))

-> £A(S(D,D/J))

0 = 5 A (C 0 ((0,1], J ) -> £A(Z(J,D))

-> £4(2?) and ->

£A(S(D,D/J))

it follows that £A(Z{J,D)) -» £A{S(D,D/J)) -» 5 A (D) are monomorphisms However, Z(J, D) is homotopically equivalent to J, and the composition J —• Z(J,D) —> S(D,D/J) —> D coincides with i. Therefore i* is injective. In general, let z : J -» S(D,D/J) be defined by i(6) = (6,0). Then S(D,D/J)/i(J) = C 0 ([0,1),D/J) is contractible. So from the above, i* is an isomorphism. To see that £A is split exact, let 0 -> JA-D^SD/J

-» 0

be a split exact sequence of separable C*-algebras. By 6.1.4, we know that £A{J) —> £A{D) —>• £A{D/J) is exact in the middle. From the map s*, we see that 7r* is surjective and 7r» o s» = i d ^ / j . It remains to show that j„ is injective. Using the exactness of eA{S{D/J))

-> £A(S(D,D/J))

and identifying £A{J) with £A(S(D,D/J)), where n : 5 ( D / J ) -> S(D,D/J). Let

-> 5A(Z?) we see that kerj, C im(ii)*,

/ = {(s(6(0)), 6) e 5(2?, D / J ) : 6 e C([0,1), D / J ) } . Then I = C([0,1),D/J), which is contractible. However, iirni C 7. Therefore (ii)» = 0. Thus kerj, = 0. In other words, j * is injective. •

An existence

275

theorem

6.1.6 There is a map Y from KK(A,B) to HomA(K_(A),K{B)) (see 5.8.15). It is known that KK(A,-) is a covariant functor from separable C*-algebras to abelian groups which is homotopy invariant, stable and split exact (see section 5.8). HomA{K_{A),K_(—)) is also a covariant functor from separable C*-algebras to abelian groups. From the definition, we see immediately that it is homotopy invariant and stable. Suppose that 0 -> lAB^sC

-» 0

is a split exact sequence, where j : / —• B is an embedding, 7r : A —> C is surjective and it o s = idc- We obtain the split exact sequences 0 ->

Hom{K*{A,Z/nZ),K*{I,Z/nZ))hHom{Kat{A,ZlnZ),Kif(B,Z/nZ)) ^s„Hom(K*(A,Z/nZ),K*(C,Z/nZ))

->• 0

(n = 0,1,...). Since j , IT and s are homomorphisms, j * , 7r* and s* respect the two exact sequences

Kii-^Kii-^Kii-WnZfiKi+ti-) and Ki+1(-, Z/nzfh"Ki{-,

Z/mzf^^Kii-^/mnzf^Kii-^/nZ)

for all m, n £ N , i G Z/2Z. Therefore we have the split exact sequence 0 -»• HomA(K(A),K(I))l4HomA(K(A),K(B))U[s]HomA(K(A),K(C))

-> 0.

We define a subset F o{HomA{K_{A),K_{B+®K)) as follows. We say a € F, if there exists a sequence {hn} of homomorphisms from A to C • 1B+ 0 such that a

lp = [K}\v-

We write x ~ y for a;, 1/ e HomA{K{A),K_(B+ /C)) if there exist two sequences [/i|J, [h'^\ e F such that, for any finite subset "P C P(-A), there exists n > 0 satisfying

z+[MT>=y+[>4]l*>

276

Classification

of Simple Amenable

C*-algebras

for all m> n. Clearly if x ~ y then x+z ~ y+z. Since HomA(K_(A),K_(—)) is split exact, we may view Hom\(K(A),K(B)) as a subgroup of HomA(K(A),K(B+ ®/C)). Denote by KX F (A,.B) the quotient of HomA{K_(A),K_(B))+F by the equivalence relation "~" defined above. So iiT.Lir(A, 5 ) is a quotient of Hom\{K{A), K(B)). We use q for the quotient map. In particular, if x £ F, then q(x) = 0. We omit the proof of the following. Theorem 6.1.7 Let A be as above. Then KLp(A, —) is a covariant functor from separable C*-algebras to abelian groups which is stable, homotopy invariant and split exact. Definition 6.1.8 There is a homomorphism (3% from £A(B) to KLF{A,B) defined as follows. Let = ({>„}} G SA(B). By the definition, there are homomorphisms hn,h'n : A —t B+ ® K. with finite dimensional range such that x = {[(f>n © hn}] — {[h'n]} defines an element in HomA(K_(A), K_(B+ ® K)). Note that we have a split exact sequence 0 -»• HomA{K(A),K(B))

->• HomA(K(A),K{B+

[

^[s]HomA(K{A),K(C

® K.))

• 1 B + ®/C)) -»• 0,

hence x— \SOTT]{X) defines an element z in HomA(K_(A),K_(B)). We define (3^{4>) = g(z). To see that it is well-defined suppose that f'n,f!l[:A-+B+® K. are two sequences of homomorphisms with finite dimensional range such that y = {[<j>n © f'n] - [/;']} defines an element in HomA (K(A), K_(B+ <8> K.)). Then, letting {Pn} be an increasing sequence of finite subsets of P(A) with U„ = i'P r i = P(A), we have (by passing to a subsequence if necessary) {x-y

+ [tin] + [a)\vn

= ([hn] + [fn])\vn

So q(x) = q{y). Suppose that cj>' = {4>'n} is an asymptotic sequential morphism which is equivalent to . Then there are two sequences of homomorphisms gn, g'n : A —>• C1 B + ® fC with finite dimensional range and a sequence of unitaries un G B+ ® /C such that \\u*ndiag{<j)n(a) ® gn{a))un - diag{<j>'n{a) © g'n(a))\\ -> 0, a s n - > o o for all a £ A. Therefore, we may assume that \4>n®gn\\vn

=

['n®9'n\\vn-

An existence theorem

277

A similar argument above shows that <j> and <j>' define the same element in KLF(A, B). So (3% is well defined. It is clear that (3 : SA(~) -> KLF{A, -) is a natural transformation. We denote by TLF(A, B) the image of T(KK{A, B)) in KLF(A, B) (see 5.8.13 for the definition of T). L e m m a 6.1.9 The transformation j3A maps £A(B) each separable C-algebra B.

onto TLF(A,B)

for

Proof. By 5.8.7, there is a unique natural transformation a : KK{A,-) -> £A{~) such that (^([id^]) = (id^), Let 7 : KK(A, —) -4- KLF(A, - ) be the natural transformation induced by the map T : KK(A,B) -> HomA(K_(A),K_(B)) and the quotient map from HomA(K(A),K(B)) to KLF(A,B). We have PAoaA(idA)

= q{[idA]).

Since 7(id^) = (/([id^]), by the uniqueness (5.8.7), /3 o a = 7. Since 7 maps KK{A,B) surjective for each 5 .

onto r i F ( A , B ) , j3A : £A(B)

->• r L F ( A , 5 ) is •

Definition 6.1.10 Let G be an ordered group. We say G+ is locally finitely generated if for any finitely generated subgroup Go C G there are 9i,-,9k G (G 0 )+ such that {Y^i=imi9i • ™i G 1+} = (G 0 )+. T h e o r e m 6.1.11 Let A be a separable C* -algebra satisfying the UCT such that A is the closure of an increasing sequence {An} of amenable RFD C* -algebras. Let B be a unital separable C* -algebra. Then, for any a £ Hom\(K(A), K(B)), there exist two sequences of completely positive contractions <$ : A —> B ® K, (i = 1,2) satisfying the following: (1) \\n are contained in a finite dimensional C* subalgebra ofB®K for eachn, and for any finite subset V c P(^4), [^>n \\v and [<j>n ']\a are well defined for all large n, where G is the subgroup generated by V; (3) for each finite subset ofVc P(A), there exists m > 0 such that

[«]|„ = a +[<£(?)] | P for all n> m;

278

Classification

of Simple Amenable

(4) for each n, we may assume that n

C*-algebras

is a homomorphism

on An;

and (5) if KQ(A)+ is locally finitely generated and O\K0(A) is positive, we may assume that both 4>h, and 0„ are positive on G PI KQ (A), where G is the subgroup generated V. Proof. We first prove this for all RFD C*-algebras and every a £ Y(KK(A,B)). So assume that A is a RFD C*-algebra. It follows from 6.1.9 that all conclusions remain valid if we do not demand that the images of maps are contained in B ® K (they are in B+ ® K.). Furthermore, „ can be taken to be homomorphisms. Therefore, with the same notation, we may assume that we have obtained <j>n satisfying (l)-(4) but with images in B+ ®K. Since B is unital, we have B+ <S) /C = (B®/C) ©K. Therefore, we may write 4>\ — 4>$ © $ , where „ : A -> B ® K and $ : A -> /C. It immediately follows that

We now consider the general case. We write A = lim„ An where each An is a RFD C*-algebra. So for a finite subset V C P(-A), we may assume that there is a RFD C*-algebra Am C A such that V C P(.4 m ). By 5.8.14 and 5.8.5, we have the following commutative diagram -*KK{A,B)

-»•

UmKK(An,B)^0

r 4. HomA(K(A),K(B))

I hm r„ 4

limHom A {K(A n ),K(B)).

When A satisfies the UCT T is surjective by 5.8.15. So, from the commutative diagram, the map lim„ T„ is also surjective. We may assume that there is /? e T(KK(Am,B)) C HomA(K{Am),K(B)) such that do[jm]v

= f3\v.

Fix e > 0 and a finite subset T C A. By choosing an even larger m, we may assume that T C Am. By applying the first part of the proof to (3 and Am, we obtain an JF-e/2-multiplicative completely positive contraction <j> : Am —>• -6 ® /C and a homomorphism /i : Am -* B ®K such that

Since Am are amenable, by 2.3.13, we may extend <j> to A.

Simple AH -algebras

279

To see (5) holds, let g\, ...,gk be positive elements in G n KQ(A) such that g\,...,gk generate Gn K0(A)+. Then there are fi,...,fk € K0(An)+ such that (j n )*(/i) = 9i f° r some integer n > 0. Since we may assume that 4>n are homomorphisms on An, [ 0 for sufficiently large n. D

6.2

Simple

AH-algebras

Definition 6.2.1 Let In be the n-dimensional unit cube. Fix k > 0. A 1/fc-frame Q. of In is a closed subset In which is a union of hyperplanes (intersecting with In) satisfying the following: Q, = Xi x J " " 1 U I x X2 x 7 n " 2 U • • • U J™"1 x X„, where Xi = {*0'*ii *2> —'*JU-2} w i t n *o = 0 and tj. + 2 = 1 such that l/2fc < \t} - t)+1\ < 1/k, j = 0,1, ...k + 1 Let d > 0. Define ftd = {a; G J n : dist(a;, 0) < d}. A 1/fc-frame is called standard if each Xi = {0, l/(fc+l), 2/(fc+l),..., 1}. The following is an easy fact. Proposition 6.2.2 Fix k > 0, 1/8/c > d > 0 and a 1/k-frame Q, of In. Let N > max{(8fc) 2 ,8/d} be a positive integer. There is r\ > 0 and finitely many (l/k)-frames {f2i, ...,Q,i} satisfying the following:

( i ) ^ C ( l d / 2 , i = l,2,..,I. (ii) for any l/k-frame such that

QQ with Cl0'

C ftd/2, there is fij with 1 < j < L

n? c nl/N. Lemma 6.2.3 Let k > 0, 1/8A; > d > 0 and ft a l/k-frame of In. For any a > 0, there ezisis an integer N > (8fc)2 swc/i i/iat, /or any normalized Borel measure fi on In, there is a (l/k)-frame fto satisfying: (a) n20/N

(b) M (Oj

C Sld<2 and /JV

) < a/2.

Proof. Let J = min{d, o~). To save notation, without loss of generality, we may assume that k > 2 and Xi = {0, l/(fc + 2),2/(fc + 2),...,!} for

280

Classification

of Simple Amenable C -algebras

each i. Choose an integer L > 2n(k + 2)/5. Consider L disjoint closed subintervals j[iJ),..., J{L,J) of [(j/k+2)-S/8, (j/k+2) + 6/8],e&ch of which has length 6/16L. Put F(l,j,m) = j£J) x In~\ ...,F(n,j,m) = In~l x J^'j) (m — 1,...,L). For each i and j , there is at least one of F(i,j,m) has measure no more than 1/L. We may assume that [i(F(i,j, 1)) < 1/L. Set F = U? =1 \j)±l F(i,j, 1). Then /x(F) < n(k + 2)/L < 5/2 < a/2. Choose N > 64L/S. We note that there is a 1/fc frame fl0 C F such that fiJ/JV C F. It is also clear that (a) holds. • Theorem 6.2.4 Let A be a simple AH -algebra with real rank zero. Suppose that A satisfies the property that p is equivalent to a subprojection of q for any p, q £ Mn(A) (for any n) if r(p) < r(q) for all r € T(A). Then TR{A) = 0. Proof. Write A = ]han(An,„). Fix a finite subset T C A, e > 0 and a € A+ \ {0}. Without loss of generality, we may assume that there is a finite subset Q C A\ such that 4>il00{Q) = J~- To save notation, we may further assume that J- and Q are in the unit balls of A and A\, respectively. We will show that, for any nonzero a e A+, there is a finite dimensional C*-subalgebra C C A with lc = p such that (1) ||pa; — xp\\ < e, (2) pxp Ee C for all x € T and (3') 1 — p is equivalent to a projection in Her(a). Since RR(A) = 0, there is a nonzero projection q £ Her(a). Thus, we may replace a by the projection q, and so we may replace (3') by (3) r ( l — p) < a for all r e T(A), where a > 0 is previously given. Let pi, ...,pm be central projections of A\. By considering each ^>i,oo{Pi)A(f)it00(pi), we can reduce the general case to the case in which A\ = PMi(C(X))P, where X is a connected finite CW complex. We first consider the case that Ai = Mi(C(X)). If (1), (2) and (3), can be established for the case that A\ = C(X), then it is clear that (1), (2) and (3) can be established for the case that A\ = Mi(C(X)). Therefore we reduce to the case in which A\ = C(X). So let Ai = C(X). There is an integer n such that X C In. There is a surjective map ip : C(Jn) —¥ C(X). We may further assume that Q C C(In). Now we choose 6 in 4.3.3 for e/2 and the finite subset Q with X = In. We will apply Lemma 6.2.3 and Proposition 6.2.2. Fix k > 0 so that 1/fc < 6/4, d = l/(8fc + 1) and a standard 1/fc-frame fi of In. Let N be as in 6.2.3 for the above fc, d and a. Let 77 > 0 be as in 6.2.2 and fii,..., flz, be finitely many 1/fc-frames of In satisfying (i) and (ii) in 6.2.2.

Simple

AH-algebras

281

Let 1 > fi > 0 be in C(In) such that fi{t) = 1 on fi?/J and fi(t) = 0 o n X \ ftf, i = 1,2,..., L. Since RR(A) = 0, there are mutually orthogonal projections {pi,i, ••-,Pi,z(i)} C A such that ||^i,co(/i)-^A(i,jKJ||
-^HhJWrn.ooiqi^W

< ^/8

J'=l

for all 1 < i < L. It follows from 1.10.6 that, for even larger m, we may assume that i(i)

ll0i,m(/O-X)A(*.J')gijll<^/4,

l<»
(e2.3)

We may assume that Am = PMK{C(Y))P, where P <E MK(C{Y)) is a projection and V is a disjoint union of Yi,Y2, ...,Yj, where each Yj is a connected finite CW complex (1 < j < J). Let yj € Y^ and let Tj = tr ° Pyj ° ^l.m, J — 1,2,..., J, where tr is the normalized trace on Ms^, Pj = P\YJ, s(j) is the rank of Pj, and pVj is the point-evaluation at yj. Let Hj be the normalized measure induced by Tj. We now fix j . Let Bj = m,oo(Pj)A<j)m!oo(Pj)- By applying 6.2.3 and 6.2.2, there is a 1/fc-frame such that fto,j such that Vjitfjf)

< o/2 and ^1%N C fid/2.

Therefore by 6.2.2, there is an i(j) such that

It follows that Tj(/i(j)) < ijm, where £ is any other point on Yj. Since 1} is connected, each T^(qij) = Tj(qij). By (e2.3)) this implies that ^ ( / ^ j ) ) <
282

Classification

of Simple Amenable

C*-algebras

Let In \ O^/ 16 = uf=1Ot, where Ot are mutually disjoint open subsets with diameter < 6. Let 0 < h^h'i < 1 be in C0(Oi) C C(In) such that hi(t) = 1 on O i n(/™\0''/ 2 ) and hi(t) = 0 in n „ / 4 ; ti^t) = 1 if O i n(7 r i \0 7 '/ 4 ) and h'iit) = 0 in FLv/&. Note that hih'i — hi. It follows from 3.8.23 that, since RR(Bi) = 0, there is a projection e* G Her(0iiOO(/iJ)) such that ei(f>ii00(hi) = i,oo(hi). Let Qj = £ \ = 1 e*. Then lBj - Qj < 4>i,oo(fi(j))It follows from 4.3.3 that (i) \\QjX - xQj\\ < e, for all x G i)00(£), (ii) QJXQJ G£ Cj, where Cj is the finite dimensional C*-algebras generated by ei,...,eK, (hi) T(1BJ — Qj) < crr(ls i ) for all tracial states r on A Applying this to each j , we obtain a finite dimensional C*-subalgebra C c A with l c = p such that (1) ||pz - zp\\ < e, (2) pzp G£ C for all z G 4>i,oo(G) and (3) T ( 1 - p) < (T for all r G T(A). Now we consider the case in which Ai = PMi{C{X))P. By 4.8.16, there is K and a projection Q G MK(PMi{C(X))P) such that QMK(PMi(C(X))P)Q ^ M ^ C p f ) ) for some i . Let e = l A l be identified with a projection in ML{C(X)). Let T\ = {e} U T. If (1), (2) and (3) can be established for the case that A\ = ML(C(X)), then (1),(2) and (3) can be established in i)tx> (Q)MK(A)^it00{Q) for Tx and e/32 < 1/64. In particular, ||pe — ep|| < £/32 and pep G£/32 C. Thus there is a projection p' G e<j>it00(Q)MK(A)<j>ii00(Q)e = A such that ||p'-pep|| < e/16 (see 2.5.4). There is a projection q G C such that ||g — p'|| < e/8 (see 2.5.4). There is a unitary u £ A such that ||u — 1|| < e/4 with u*gu = p' (see 2.5.1). Set C\ = u*(qCq)u. Then Ci is a finite dimensional C*-subalgebra and lc1 = p'• Moreover, since ||(e - p') - (e - pep)\\ < e/16, by 2.5.4, e - p' is equivalent to a projection in 1 — p. Now we have (1) \\p'x-xp'\\<e/4, (2) p'xp' GE/2 d (3) r(e -p')
This follows from 6.2.4 and 4.8.19.

Lemma 6.2.6

Let A be a unital C* -algebra with RR(A)

• =0,/:i->C

Simple

283

AH-algebras

be a state and X be a compact metric space. Suppose that h : C(X) —¥ A is a unital homomorphism. Then, for any e > 0, any a > 0, any finite subset T C C(X), there is a projection p £ A and a homomorphism h\ : C(X) —• pAp with finite dimensional range such that \\h(x) - ( l - p ) / i ( a ; ) ( l - p ) - / i i ( x ) | | < e for all x € T and

/ ( 1 - p ) < a.

Proof. Set T — / o h. Denote by B the C*-subalgebra generated by 4>(C(X)). Then B = C(Y) for some compact subset of Y. It is clear, without loss of generality, we may assume that Y = X. Fix an e > 0 and a finite subset !F, and let S > 0 be as in 4.3.3. The restriction o f r o n B gives a state on B. By the Riesz representation theorem, the state defines a normalized Borel measure fj,T on X. For any open subset O C X, set g G B{= C{X)) such that 1 > g(t) > 0 for all t G O and g(t) = 0 for all t e X \ O. Let b = h(g) and Bo be the hereditary C*-subalgebra of A generated by 6. Then {6 1 /"} forms an approximate identity for Bo- It is clear that fiT(0)

= limn^0OT(b1/n)

= \\T\Bo\\.

Let {e° } be any approximate identity for Bo consisting of projections. Then »T(0) =

\\T\Bo\\=sup{T(e°n)}.

For any e > 0, there is a finite subsets {Ci,C2, —Xm} of X such that for any £ £ X, there is an integer i such that dist(Ci,C)<<&/32 and for any i, there is j / i such that dist(Ci,G)<<*/16. For each i set Xt = {C : S/32 < dist(Ci, C) < 5/16}. Fix i, for each 5/32 < r < J/16, set

C r = {C:dist(C,<0=r}. Since /xT(X») < 1 and C r n Cr> = 0, if r ^ r', there are only countably many r in (5/32, 6/16) such that Mr(C r ) > 0.

284

Classification

of Simple Amenable C* -algebras

We conclude that for each i, there is rt e (6/32, <^/16) such that fiT(Cri)=0. Now X \ UC ri is a disjoint union of finitely many open sets 0\, O2,..., OJV such that the diameter of each Oj is less than <5/4 and /i T (UC r J = 0. Let {e„ } be an approximate identity for S o i consisting of projections, where Bo{ is the hereditary C*-subalgebra corresponding to the open subset Oi (notice that Boi has real rank zero, whence such an approximate identity exists (see 3.2.7)). Then r(eW) /

Hr{Oi)

n = 1, 2,... and i = 1,2,..., JV. Since /j.T(X \ uf=1Oi) = 0, N i=l

So T(1

N - y ^ e ^ ) ->• 0

as n -» 00.

»=i

Let pj — e„ for sufficiently large n so that r ( l — ^Zi=i e « ) < a- Define h\(x) = Z)i=i x(&)Pi and p = X)i=iP»- The lemma now follows from 4.3.3.

• Lemma 6.2.7 Let X be a compact metric space. For any finite subset F C C(X), any a > 0 and any e > 0, there is 5 > 0 and /mite subset G C C'(X) satisfying the following: If for any integer n > 0, L : C(X) —> Mn is a G!-<5-multiplicative contractive completely positive linear map, then there exists a homomorphism h : C(X) —> Mn with h(lc(x)) = P such that tr(l — p) < a and \\L{x)-{l-p)L(x)(l-p)-h(x)\\<e for all x € J-, where tr is the normalized trace. Proof. Suppose that the lemma is false. There is £0 > 0, o~o > 0, a finite subset J7, a sequence of contractive completely positive linear maps

Simple

285

AH-algebras

Ln : C{X) -> Mfc(„) with \\Ln(ab) - Ln(a)Ln(b)\\ -» 0 as n -> oo for all a, 6 G C(^Q, such that for any projections pn G Mj.(n) infsup{||L„(:r) - (1 -p n )L(a;)(l - p n ) - hn(x)\\ : x G J 7 } > e 0 , where the infimum is taken over all unital homomorphisms hn : C(X) —• PnMk^pn with i r ( l - p n ) < cr0. Define $ : C(X) -> n „ M fe(n) by $(a;) = {L„(i)}. Since C(X) is separable, there is a sequence n(m) such that tr(L n ( m )(a)) converges for all a G C'(X). Let r be a weak*-limit of tracial states £n(m)({afc}) = tr(an(m)). By passing to a subsequence if necessary we may assume that, for any a G C{X), tr(Ln(a)) converges. If x = {an} C ©nMfc(n), t n e n ll a n|| ~> °Therefore tr(a n ( m )) —> 0. In other words, T(X) = 0. Thus T gives a normalized trace f on C = YlnMk(n)/ © -Mfc(n)- Let 7r : rj„ ^4(n) -» C1 be the quotient map. Then $ = 7r o ^ : C(X) —> C is a homomorphism. Note that RR(C) = 0. It follows from 6.2.6 that there is a projection p G C and a homomorphism H\ : C(X) —> pCp such that | | $ ( a ; ) - ( l - p ) a ; ( l - p ) - # i ( : r ) | | < e/2 and f ( l - p ) <

(e2.4)

1/2
(cc G J 7 ). We may assume that H\(x) = J^ i = 1 a;(^)cfj (a; G C(X)), where £ii —,£N S -^ a n d di,d,2,...,dN are mutually orthogonal projections in C. There are projections P = {pn}, Di = {D%'} G FJn ^fc(n) (where p n , .D£ I) G Mfc(n) are projections) such that n(P) = p and -K(Di) = di. Furthermore, are mutually orthogonal projections and J2t=i^i follows from (e 2.4) that, for sufficiently large n, DI,D2,.-.,DN

=

P- ^

N

\\Ln{x) - (1 -pn)x{\

-pn)

- ] T £(&)£>£> || < e and tr{\ - pn) < a0 i=i

(for x G J-). We reach a contradiction.

D

L e m m a 6.2.8 Let X be finite connected CW complex, C = C{X) and A be a unital simple C*-algebra with TR(A) = 0. Let V C P(A) be a finite subset and a G KL(C,A) such that CX\K0(A) *S positive. Then there exists a sequence of contractive completely positive linear maps Ln : C —>• A such that [Ln]|-p = oc\v and \\Ln(a)Ln(b) for all a,be

C{X).

— Ln(ab)\\ —¥ 0 as n —>• oo

286

Classification

of Simple Amenable

C*-algebras

Proof. By replacing A by one of its corner, we may assume that Q([1C]) = [IA]- Fix e > 0 and a finite subset T c A. It follows from 6.1.11 that there are a sequence of contractive completely positive linear maps ij)n : C{X) —>• A®Mk(n) and a sequence of homomorphisms Hn : C(X) —• .4 0 (n ->• oo) and [V>n]|7> = a|p + [Hn]\v for all n, where l{n) < k(n) are integers. Let G be the subgroup generated by V. Suppose that k is an integer such that G D Ki(C(.X"), Z/mZ) = {0} for all m > k. By replacing ipn by VVi © i7 n and Hn by i/„ © Hn, where Hn is a direct sum of fc! — 1 copies of Hn, we may assume that [Hn]\GnKi(C(X),Z/mZ)

= °

a n

d

[V'n]|Gn.K'i(C(X),Z/mZ) =

for all m < k. Moreover, [ipn(lc)] = a ( [ M ) + fc![4n)]

°'\GnKi(C(X),Z/mZ) for

some projection eo G Mfc(„)(A). Fix an integer n. We may assume that tpn(lc) is a projection. Put D„ = tpn(lc)Mk^n)(A)ipn(lc). So we write V>n(lc) = 1D„Since TR(Dn) — 0, there is a projection qn G £)„ and a finite dimensional C*-subalgebra Bn C Dn with 1B„ = g n such that (1) \\qnx — xqn\\ —>• 0 for all a; G V'(^r) ( a s n ~* °°)> (2) dist(g n xg n , B„) —>• 0 (as n —>• oo) for all x G VK-?7) and (3) T{1D„ — Qn) —>• 0 uniformly on T(^4) as n —> oo. Let Qn be a finite subset of Bn such that dist(qnxqn, Qn) —>• 0 for all zGJF. There is a contractive completely positive linear map $ „ : Dn —• ( l n n — 9n)^n(l£» n ~9n) defined by $„(z) = (1 D „ -9n)^(l£»„-9n) (for z G D„) and there is another contractive completely positive linear map A„ : Dn —> Bn such that ||An(6) - b\\ < 1/2" for all b G 0„. We may assume that r(l fc ( n ) - qn) < T(1A)/2 for all r G T ( J 4 ) . By applying 6.2.7, for sufficiently large n, we may without loss of generality write A„ o ipn = L'n © /i n , where X^ : C(X) -> ( dnBndn is a homomorphism, d n G £ „ is a projection and rfg n — d n ) < T ( 1 A ) / 4 for all r G T(A). Suppose that /i„( a ) = £i!Li °(£i' l) ) c £ ) > where c« , c„ ,.-->cn are mutually orthogonal projections. If follows from 3.8.24, for each n, there arefc!mutually equivalent and mutually orthogonal

287

Simple AH -algebras

projections e P , . . . , e ^ 0 such that T(<4° - E j L i ^ i , j ) ) < r(l A )/4AT(n) for all r G T(A). Set e ^ = £ " = i e ^ J C )

and e n = X ^ e ^ .

Define

) e )

/»n(o) = E l I r « ( d " ) " for a e C{X) and h'^ = hn - h'n. Define ^ 0 ) ( a ) = £ £ ( 1 n ) a ( ^ n ) ) e £ , 1 ) for a £ C(X). Note that / # : C(A") -»• (d„ — en)Bn(dn — en) is a homomorphism. To save notation (by conjugating by a unitary if necessary), we may assume that (ljo n — qn) © (
(fc!)(Ef=(r)[^1)])-(fc!)[4n)iDefine L„(a) = $ „ o */>n(a) © -^4 © ^n( a ) © ^n° 0 \ where /il° 0) (a) = ( 0 / n a n d £ € X for a £ C{X). We now verify that Ln satisfies the requirements. First [Ln(lC(x))] = a (lc(X))- Write K0(C(X)) = Z®kerdx, where keidx is finitely generated. Note that for any homomorphisni with finite dimensional range, [](kerdx) = 0. Since Hn,hn,h'n,h'^ and /i°° are homomorphisms with finite dimensional range, we also have [V'nJIkerdxnG = Oi\kerdxnG and [Ln]|kerdxnG = a|kerdxnG- We have a

[Ln]\K0(C(x))nG = a>\Ko(c(X))nGSince [h'n] = k\[h^} and [/„] =

fcKE^eJj'1']

[K}\GnKi(c(X),z/mZ) = [hn00)}\GnKi(c(x),z/mZ)

- [e^]), =0

we have

for all m < k.

Since [„]|G = ([*„OV>„] + [ I 4 ] + [/I„])IG and [hn] = [K] + [h'n], we conclude that [Ln]\GnKi(C(X),Z/mZ)

= ['>Pn]\GnKi(C(X),Z/mZ)

=

<x\GnKi(C(X),Z/mZ)

for all m < k. Since Hn, hn, h'n, h'^ all have finite dimensional range, we also have [in]ki(C(X))nG = a|x 1 (C(X))nG-

„

Theorem 6.2.9 Let A be a unital AH -algebra and B be a unital simple separable amenable C* -algebra with TR(A) = 0. Suppose that a € KL(A,B) such that O\K0(A) *S positive and V C P(A) is a finite subset. Then there is a sequence of contractive completely positive linear maps Ln : A —• B such that [Ln\\-p = a\v and ||L n (a6) — Ln(a)Ln(b)\\ for all a, b e A.

—> 0 as n —>• oo

288

Classification

of Simple Amenable C -algebras

Proof. We first show that Lemma 6.2.8 holds for C = PMn{C{X))P, where X is a connected finite CW complex and P G Mn{C{X)) is a projection. It follows from 4.8.17 (see also 4.3.2) that C K, ^ C{X) K. Since KL(C,B) = KL(C(X),B), we may view a e KL(C(X),B). Thus, by 6.2.8, there is a sequence of contractive completely positive linear maps A n : C(X) -»• B such that ||A„(a6) — A n (a)A„(6)|| —>• 0 as n -> oo and a\v = [A„]|p. Let 14 = A„ id n and Ln = {L'n)\PMn(c(X))P- T n e n it i s clear that [£„]|-p = a\-p for all large n. Now let C = PM n (C(Y))P, where 7 is a finite CW complex (not necessarily connected). By considering each summand separately, we see that Lemma 6.2.8 holds for C = PMn(C{Y))P. Suppose that A = limn_).oo(^4n,n), where An = PnMk(n)(C(Yn))Pn, where Yn is a finite CW complex and Pn £ M^n){C{Yn)) is a projection. Fix a finite subset J- C A and e > 0. Since K_(A) = linw,-^ K(A„\ we may assume that V C [n] {Pn), where Vn is a finite subset of P(An) for some n. We may also assume that T C 4>n(G), where Q is a finite subset of An. Let f3 = ao [n]. Then /3 € KL(An, B). By what we have just proved, there is a sequence ^ n : An —• P such that | | * n ( a 6 ) - * n ( a ) * „ ( f e ) | | <e/2

for a , 6 e g „ and [*„]| Prl = {3\v.

Since both A and An are amenable, there is a contractive completely positive linear map $ „ : A —> An such that | | $ ( a ) - a | | <e/2

for all a&Q.

Define Ln — ^n o $ „ . It is then clear that Ln is J'-e-multiplicative and [Ln]\v =Oi\v-

6.3

_

The classification theorems

Theorem 6.3.1 Let A be a separable unital simple C*-algebra. For any £ > 0 and any finite subset T C A, there exist 5 > 0, a finite subset Q C A, a finite subset V C P(^4) and an integer n > 0 satisfying the following: For any simple unital C*-algebra B with RR{B) — 0, tsr(P) = 1 and

The classification

289

theorems

weakly imperforated KQ{B), if , tp, a : A —$• B are three Q-6-multiplicative contractive completely positive linear maps such that

and a is unital, then there is a unitary u € Mn+i(B)

such that

\\u*diag((f>(a),a(a), • • • ,a(a))u — diag(ip(a),a(a), • • • ,o~(a))\\ < e for all a £ J-, where a(a) is repeated n times. Proof. Suppose that the theorem is false. Then there are SQ > 0 and a finite subset T C A such that there are a sequence of positive numbers {Sn} with Sn \ 0, an increasing sequence of finite subsets {Gn} which is dense in the unit ball of A, a sequence of finite subsets {Vn} of P(A) with W^=1Vn = P(A), a sequence {k(n)} of integers and sequences {4>n}, {4>n} and {o~n} of n]|p„ = \^\\vn satisfying the following: ini{sup{\\u*diag(^n(a),an(a), -diag(tpn(a),an(a),

• • • ,o-n(a))u • • • ,an(a))\\ : a £ F} > e0

where an{a) is repeated k(n) times and the infimum is taken over all unitaries in M fc ( n ) +1 (B„). Set C 0 = ®™=xBn and C = H™=lBn. Define $, \f, E : A -» C by $(a) = {(j>n(a)}, *(a) = {t/>„(a)} and S(a) = {(Tn(a)} for a G A. Let IT : C —* C/CQ be the quotient map and set = 7ro, 'I' = 7ro\I> and S = 7r o S. Note that $ , ^ and S are monomorphisms. Given an element p E Vk for some k, we have [^n(p)] = [V'nO')] f° r a u large n. Note that since RR(Bn) = 0, tsr(B n ) = 1 and Ko(Bn) is weakly unperforated. As in 5.9.4, 5.9.6 and 5.9.8 (see also (hi) of 5.0), we have

K0([[Bn)=n#o(£n),#i(nB™)=n^i(s«)> b

Ki([[Bn,Z/kZ)

c Y[Ki{Bn,Z/kZ),

k > 0,t = 0,1.

We also have Ko(C/C0)

= ([[Ko(Bn))/(BKo(Bn), b

K^C/Co)

= (n^i(-B„))/©i^i(B„)

290

Classification

of Simple Amenable

C*-algebras

and Ki(C/C0,

Z/kZ) c ( J ] Ki{Bn, Z/kZ))/

© Ki(Bn, Z/kZ),

i = 0,1.

Therefore we conclude that [$(p)] = [*(p)] for all p G Vk- Thus [$] = [\&] in KL(A,C/C0). Since, for each n, tsr(B„) = 1, tsr(C) = 1. It then follows that tsr(C/C 0 ) = 1 (3.1.2). Similarly, RR(C/CQ) = 0. By applying 5.9.9, we obtain an integer N and a unitary u G MN+I(C/C0) such that \\u*diag($(a), £(a). • • •, E(a))u - diag($(a), S(a). • • •, 2(a))|| < e 0 /3 for all a G J7, where E(a) is repeated N times. As in 2.8.9, there is a unitary U G MN+I(C) such that TT(U) = u, and for each a £ J- there exists c a G MJV+I(CO) such that \\U*diag(${a), E(o). • • •, £(a))[/ - dta<7(*(o), E(a), • • •, 12(a)) + ca\\ < e0/3 where E(a) is repeated N times. Write U = {un}, where un G MJV+I(-B„) are unitaries. Since ca G M/v + i(Co) and T is finite, there is A?o > 0 such that, for any n > No, \\u^diag(
• • • ,an(a))\\ < eQ/2

for all a £ J7, where crn is repeated N times. The above inequality contradicts our earlier assumption that the theorem is false. • Lemma 6.3.2 Let A be an amenable simple unital C*-algebra with TR(A) = 0. For any e > 0, any finite subset T C A and any positive integer n > 0, there exist projections pi and p2 such that (1 — pi)A(l — P\) = Mn(p2Ap2) with pi ^ P2, and there is a unital T-e-multiplicative contractive completely positive linear map : A —> F\, where F\ is a finite dimensional C*-subalgebra of P2AP2, such that (a) \\xpi — Pi'x]\\ < e for all x G J-; (b) ||ir — (p\xpi ® diag((f)(x),(x), ...,(x)))\\ < e for all x G J- (where (j)(x) is repeated n times). Proof. To simplify notation, we may assume without loss of generality that T is in the unit ball of A. Since TR(A) = 0, there is a finite dimensional C*-subalgebra F C A with lp = p such that

The classification

theorems

291

(1) \\px — xp\\ < e for all x £ T, (2) pxp Ge/2 F for all x £ T and

(3)(n + 2)[l-p]]. Write F = M n i © M„ 2 © • • • © M „ t . Let di be a (nonzero) minimal projection of Mni, i = 1,2,..., k. By applying 3.8.24, we obtain

d^®df)®---®dfn)+ri,

di =

where d ^ are mutually equivalent projections and rj £ diAdi is a projection with [n] < [d\l)]. Let a = d\1] + df\

fi = di- n and

k

k

P2 = SjTldiag(ei,ei,...,ei)

and E = y]diag(fi,...,

i=l

fj),

i=l

where e^ and /j repeat n^ times on the diagonal. So EAE = Mn{p2Ap2). Let L : A -4 F be a contractive completely positive linear map which is an extension of the identity map on F (see 2.3.5). Then L is .F-e-multiplicative. Define 4>{x) = piL{x)pi. Note that p^y — yp2 for all y £ F. So is also .F£-multiplicative. Define L\{x) = EL(x)E for x £ A. We may write L\(x) = diag((x),(x), ..., is repeated n times on the diagonal. Let Fi — EFE. Since Ey = yE for all y £ F, F\ is a finite dimensional C*subalgebra. Let r = Y^i=i ri- We compute that r ( l - p ) + r(r)

<

^2-r(p) + ^ T ( ^ )

" ^ r l T ( P 2 ) + 2^ T(P2) < T{P2) for all r G T(A). Set pi = (1 - p ) + r. Since TR(A) 3.7.2 that {pi} =

Now we see that (a) and (b) hold.

= 0, it follows from

[(l-p)+r]<[p2}.

•

Theorem 6.3.3 (Uniqueness) Let Abe a separable unital amenable simple C* -algebra with TR{A) = 0 satisfying the UCT. Then, for any e > 0 and any finite subset T C A, there exist S > 0, a finite subset V C P(A) and a finite subset Q C A satisfying the following: for any unital C* -algebra B

292

Classification

of Simple Amenable C* -algebras

of real rank zero and stable rank one and with weakly imperforated and any two Q-8-multiplicative morphisms L\,L2 : A —• B with [L1]\V

=

KQ(B),

[L2]\P,

there exists a unitary U S B such that ad(U) o L\ R3e L2 on T. Proof. Let Q\ = 0(^,6), Si = 5(!F,e) and V\ = V{T,e) be as required in Theorem 6.3.1, and let n be the integer in 6.3.1. Choose a finite subset Q2 and a positive number rj < 5\/2 satisfying the following: if Hi : A —> B are both ^2-^-multiplicative with H\ ttv H2 on Q2 then [H{\\Vl = [H2}\Vl. Let S2 = min(<5i/2,77/4). Since A is amenable, from 6.3.2, there exist a finite dimensional C*subalgebra F of A with p2 = lp and (^-^-multiplicative morphism • F such that id A «« 2 Pixpi © diag{4>{x), <j>{x),..., <j>(x)) on Q2, where is repeated n times, and (1 — pi)A(l — p\) = Mn(p2Ap2). By the choice of 77, [ip]\vi a n d {^WV-L a r e w e u defined and [ip]\vi © [4>}\vi — [^A]\V> where ip(x) = PixPi f° r ah xFix a finite dimensional C*-subalgebra F. Choose a large Qs, a small (53 > 0 and a finite subset Q C P(F) so that 4.5.3 applies, i.e., if Aj : F -> B, Aj are £3-^3- multiplicative morphisms and [Aj] | Q are well-defined and are the same, then Ai ~ , / 4 A2 on (Gi). Now let <5 = min(<Ji/2,5 2 ,5 3 /2), let Q D 0(&)UV'(0i)lW2)UV'(&)U& and choose V = V\ U Q (note that P ( F ) C P ( i ) ) . We also assume that 1A, 1A — P, a n d diag(0,0, ...,p 2 ,0, ...,0) (with p2 in the ith place) are in Q. Suppose that Li : A —• p2Bp2 satisfy the hypothesis of the theorem for the above 6, Q and V. Put L'iix) = Li{ip(x)), i = 1,2. Then Li « diag(L^, L^ o , L; o ,..., L; 0 , £,< o )

The classification theorems

293

on Q. Note that Li\p are ^-^-multiplicative. We also have [ii]|-Pi = [diag(L-,LjO^,Lj o<j>,...,Li o<j>,Li o 4>))Vl. Furthermore, we have [Li\F]\Q = [L2\F]\Q. By applying 4.5.3, we obtain a unitary V £ B such that ad(V) o Li w^/4 L2 on <^>(<7i). Note that diag(0,0, ...,p 2 ,0, ...,0) are in Q. So we have L 2 -v/2 diag(L2,^i °,Li o,...,Li °,Li o<j>) on Q\. Therefore by our choice of 77, [^211-Pi = [diag(L 2 ,Lio^,L 1 o<j>,...,L1o,Li °
[LSIIPD

[L\}\Vl

we obtain that

= [L'2]\Vl-

Now 6.3.1 provides a unitary U £ B such that ad(U) o Lo ~e £1

on

•?r-

• Theorem 6.3.4 Let A be a unital amenable simple C*-algebra with TR(A) = 0 which satisfies the UCT. Then an automorphism a : A —>• A is approximately inner if and only if [a] = [id^] in KL(A,A). Proof. We only prove the "if " part. The "only if part is left to the reader (see 5.10.20). Now assume that [a] = [idA] in KL(A,A). Then, for any finite subset V C P(A), we have [a]\v = [idA]\vIt follows from 6.3.3, that for any finite subset J- C A and any e > 0 there exists a unitary U G A such that ad(U) o a « e idyi on J-'. This implies that a is approximately inner.

D

294

Classification

of Simple Amenable C* -algebras

Theorem 6.3.5 Let A and B be two unital separable simple C* -algebras with TR(A) = TR(B) = 0 and satisfying the UCT such that (Ko(A),K0(A)+,

[1A], WA))

s (K0(B),K0(B)+,

[lg^K^B)).

Let a 6 KL(A,B) and (3 £ KL(B,A) be two elements which carry the isomorphism. If, for any finite subsets V C P(A) and Q C P(-B), there are sequences of contractive completely positive linear maps Ln : A —>• B and A„ : B —>• A such that \\Ln(ab) - Ln(a)Ln(b)\\

-> 0, ||A„(o6) - A„(o)A n (6)|| ->• 0,

[Ln]\v = OL\V and [\n]\v = P\vThen

A^B.

Proof. Since both A and B satisfy the UCT, there are a G KL(A, B) and (3 e KL(B, A) such that ao/3 = [ids] and /3oa = [idA]. Let Tn and T'n be two increasing sequences of finite subsets of the unit ball of A and B, respectively. Let {en} be a decreasing sequence of positive numbers such that Y^=i£n < oo. Let 5i = 5(ei,!Fi), Q\ = Q{ei,J:i) and V\ = V(£\,!Fi) be as required by 6.3.3. We may assume that 5\ < e\j2 and Qi D T\. By assumption, there is a <7i-<5i-multiplicative contractive completely positive linear map L\ : A —>• B such that [LIJIPJ = a\vi- Let rji = 6(5\/2,J:'l ULi(5i)), 0i = 5 ( < J i / 2 , ^ ( U L i ^ i ) ) and V[ = V(51/2,F!iUL1{g1)) be as required by 6.3.3. By assumption we obtain a Q[-r)i- multiplicative contractive completely positive linear map A^ : B —> A such that [ A J J I ^ u ^ j ] ^ ) = Pl^ulLMVi)- Therefore [Ai o-kitautLiKPi) = lidA}\v2u[Li](Vi)It follows from 6.3.3 that there is a unitary u\ £ A such that idyi « £ l ad(ui) o A^ o L\

on T\.

Define A„ = ad(ui) o A^. We may assume that r)\ < e\/2. To simplify notation we may assume without loss of generality that T2 D A„(£ 2 ) a n d e 2 < min{£i/2,r ? 1 /2}. Let S2 = S2{e2/2,F2), Q2 = Q{e2/2,J:2) and V2 = V(e2/2, T2) be as required by 6.3.3. We may further assume that [Ai]CP{) C V2. By assumption we obtain a ^-^-multiplicative contractive completely

Invariants

and some isomorphism

295

theorems

positive linear map L'2 : A —> B such that [ i ^ t a

= a

l^2- ^°

[L'2oAi]|7>2 = [id B ]|p 2 . It follows from 6.3.3 that there is a unitary v2 G B such that ids « £ 2 ad(v2) o L'2 o Ai

on T[.

Therefore we obtain the following (not necessarily commutative) diagram: A

^

A

\-L\

/Ai

4-Z/2

B

^

B

so that the upper triangle is .Fi-£i-commutative and the lower triangle is ^"{-Si-commutative. Continuing in this fashion, we obtain the following approximately intertwining diagram: A

**4

A

4-Li

/ Ai

4-L2

^ / A2

A

^-.-

4-1/3

/ A3

A

Furthermore, since Ln and A n are ^"n-£:n-multiplicative and ^F^-Snmultiplicative, respectively, it follows from 1.10.14 that there is an isomorphism hi : A —• B and h? : B —• A such that /1J1 = hi. D Theorem 6.3.6 (Elliott &; Gong) Let A and B be two unital simple AHalgebras with slow dimension growth and with real rank zero. Then A = B if and only if (Ko(A),K0(A)+,[lA],Ki(A))

9*

(K0(B),K0(B)+,[lB],Ki(B)).

Proof. It follows from 6.2.4 that TR(A) = TR(B) = 0. The proof follows immediately from 6.3.5 and 6.2.9. •

6.4

Invariants and some isomorphism theorems

Definition 6.4.1 Let k > 1 be an integer and let Cfc be the mapping cone defined in 5.8.8. Denote by Yjt the spectrum of Cfc. In this section, we

296

Classification

of Simple Amenable C* -algebras

denote (Z © F)+ ^ { ( n , z) : n > 0, z G F } U {(0,0)} for a finite group F. We note that K0(Ck) = Z © Z/fcZ and (see 4.8.17) {(n, z) : n > 3, 2} U {(0,0)} C KQ{Ck)+ C (Z © Z/JfeZ)+. Let Tk = SCk. Then ir0(T/b) = Z and ^(T*.) = Z/JfeZ. Denote by Zfe the spectrum of Tk. Lemma 6.4.2 Let p be a prime number, and let n and m be positive integers. (1) For any homomorphism a : Z/rnZ —> Z/mZ there is a homomorphism h : Cm —> Cm such that (h*o)\ic0(cn) = a and there is a homomorphism h! : Tm —> Tm such that h'ti = a; (2) Let (3 : Z/pn+mZ —> Z/p n Z be the canonical quotient map. Then there is a homomorphism g : Cpn+m —> Cpn such that g*o = (3 and there is a homomorphism g' : Tpn+m —> Tpn such that g'^ = (3. Proof. Suppose that a(I) = k. Define a homomorphism h from CoiS1 \ {1}) © C0({0,l),Co(Sl \ {1})) to itself by h{(a,b))(z,t,s) = (a(zk),b(t,sk)), where z,s G S1 and t G (0,1]. From the definition, one checks that h defines a homomorphism from Cm into Cm. Consider the short exact sequence: 0 -> C 0 (E 2 ) ->• Cm -»• (^(S1) -> 0. Let h : C(S1) —>• C(S1) be the homomorphism induced by /i. Then /i*i : Ki(C(S1)) -> Ki(C(S1)) is multiplication by k. From (a part of) the sixterm exact sequence in ii'-theory, we obtain the following 0^ 0^

K^CiS1))

-> K0{C0(R2)

-i-k

4-/i*o

Aries'1))

-»• ifo(Co(IR2)

->Z/mZ->0 Y/I.O

-^Z/mZ->0

Therefore /i*o is a multiple of k and /i*o = ct. We then obtain h! as the suspension of h. To see (2) holds, define g((a,b))(z,t,s) = (a(zP m ), 6(i, s^"1)). One verifies that g gives a homomorphism from Cpn+m to Cpn. A similar computation as above shows that g*o = /?• Again, one obtains g' by suspension. • Definition 6.4.3 Let F0 = Z/k{l © • • • Z/A;mZ and Fl=Gx® Z/rxZ © • • • Z/r/Z, where ki and r^ have the form pm for some prime number p

Invariants

and some isomorphism

theorems

297

and positive integer m, and G\ is a direct sum of s copies of Z. Define l{F0,Fi) = m + s + l. Let X and Y be two connected finite CW complexes. Recall that X\/Y is the space obtained by identifying a base point of X with a base point of Y. Let y ( F 0 , F ! ) = Ykl V • • • V Ykm V S1 V • • • V S1 V Zri V • • • V Zn (where S1 is repeated s times). Let CFotFl = C(Y(F0,Fi)). Then K0(CFoiFl) = Z © F0 and K1{C(F0,F1)) = F1. There is a quotient map 7Tj : CFo Cki such that (7Tj)*o : Z © Fo —» Z © Z/fcjZ is a projection, and there is a quotient map 71^ : C F 0 , F I —>• T^ such that (7r')»i : Fi —> Z/r^-Z is a projection. There is also a quotient map ir't : Cp0}Fl —> C(S1) which induces the projection from F\ to the tth copy of Z (t < s). There is an obvious embedding e^ : Cki —>• C F 0 , F I such that (ei),o : Z © Z/fciZ —• Z © Fo is the embedding, and there is an obvious embedding e'j : Trj —> CFotFl such that (e^)*i : Z/r^Z —• Fi is the embedding. Furthermore, there is an obvious embedding from C(S1) to C F 0 , F I which induces the embedding from Z to the tth copy of Z in F\. There is u G U(M3{Tk)) such that [u] generates Ki(Tk). If 77 : Z -» Z/rjZ is the quotient map, then h^k\z) = u can be extended to a homomorphism h : C(Sl) -> M3(Tfc) such that / ^ ( l ) = 1. Let £ G Y(F0,Fi) be the common point which connects Ykl, ...,Ykm, Sl, ...,S\ Zri,...,Zri. Define C g F i = CoQOFb.-Pi) \ {£}). From above discussion and applying 6.4.2, if (a,/3) : (Fo,Fi) ->• ( F Q , F { ) is a homomorphism, where F 0 , FQ are finite groups, L = 3Z(F0, Fi) and Fi, F{ are finitely generated groups, then there is a homomorphism h : Cy F —• M L (C) r 7 F ,) such that /i,o| = a. and /i*i = /3. We also note that {(n, z) E (Z © F 0 )+ : n > 3} U {(0,0)} C ^ 0 ^ , ^ ) + C (Z © F 0 )+. Now let (a',/?) : (Z © F 0 ,Fi) -+ (Z © F^,F{). Let a ' ( l © 0) = m © x, where x £ FQ and m G Z + . Denote m by m(a'). Suppose that Tn(a') = 3Z(F 0 ,Fi) + N + 3. There is a projection p G M6(C(F(),F1)) such that [p] = 3 © x (see 4.8.17). Let £i,...,&v be fixed points in Y ( F 0 , F i ) . Define the point evaluation hp : C(F 0 ,Fi) -> MN+6(C(F^,F{)) by hp{f) = diag(/(6),-.-,/(6v-i))©/0fiv)pfor / G C(Fo,.Fi). Then (h®hp)U0=a' and (/iffi /ip)*i = (3.

298

Classification

of Simple Amenable C* -algebras

Thus we have the following: Proposition 6.4.4 Let Fo, FQ be finite abelian groups and Fi and F[ be finitely generated abelian groups. Then for any positive homomorphism (a, /3) : (Z © F 0 , Fi) ->• (Z © F&, F{) with m(a) = 3l(F0, F^+N + 3 there is a homomorphism h = h^ © ftW : Cir0,Fi —> ^m(a)+3(GF0,.Fi) such that h*o = a, /i*i = f3 and h(°> is a point evaluation using N points. The following results was obtained by Elliott and Gong. The proof presented here is a routine variation of that of 4.7.5. Theorem 6.4.5 Let (G, G+,g) be a countable weakly unperforated simple ordered group with the Riesz property and with an order unit g, and let F be a countable abelian group. Then there is a simple AH-algebra A = lim„j4 n , where An = PnMk(n)(C(Xn))Pn and X is a finite CW complex with dimension no more than 3 such that TR(A) = 0 and (K0(A),K0(A)+,[1A],K1(A)) = (G,G+,g,F). Proof. Let Go be the torsion subgroup of G and Gy = G/GQ. Define (Gi)+ = 7r(G + ), where IT : G —>• G\ is the quotient map. It is easy to see that Gi is an unperforated simple ordered group with the Riesz property. It follows from 4.8.28 that Gl = limn_>oo(Z © G^,(pn), where G is a finite direct sum of Z and Z © G^ with the usual order and each partial map of „ has multiplicity at least 2. Since each Z®(?W is free, there is j„ : Z©G(") ->• G such that nojn = id z e G (n). Thus we obtain homomorphisms 7^ : G 0 © Z © G^ —¥ Go © Z © G(™+!) with (7^) I Go = id G 0 and (7n)lzec(") = (a'n, M, and where a'n : Z© Q(n) —> G 0 is a homomorphism. We may also write Go = lim r n. 00 (Gon ! VVi)> where each Gon is a finite group and tpn is injective. Therefore we have G = lim„^ o o (G 0 „ © (Z © GW), $ „ ) , where ( $ „ ) | G 0 „ = Vv», ($n)lz©G("> = (Vn, n), where r)n : Z © G^ ->• G0n+i is a homomorphism. Put r „ = {(go, n, m) e G0n © (Z © G 3, m > 0} U {(0,0,0)}. Then Tn c (G0n © (Z © G< n )) + (with the usual order). Suppose that G^ is l(n) copies of Z and F = lim„(F n ,/3„). Set A\ — By © Gi, where B\ — CG01,F1 and C\ is a direct sum of 1(1) copies of C. So KoiAy) = G01 © Z © G ^ , K^Ay) = Fx and 1^ C #0(^1)+ C (GoieZeG^^. Now we begin to construct A.

Invariants

and some isomorphism

299

theorems

(Step I): Let {.T^} be an increasing dense sequence of finite subsets in the unit ball of Cc^^i- Let {£A } be a dense sequence of Y(G0i,Fi). We may assume that, for any / 6 J-[ , there is j < s\ such that /(£• ) ^ 0. Since every partial map of 4>n has multiplicity at least 2, there is n\ > 1 such that every partial map of <j>ni o ni-\ ° • • • 4>\ has multiplicity greater than 9Z(G0i, F\) + si + 3. By passing to a subsequence if necessary, we may assume that n\ = 2. By 6.4.4, there is a homomorphism h\ 2 • Cc01tFi -> MN0(CC02,F2) with No > 9 Z ( G 0 i , F i ) + s i + 3 such that (hf^)*o

=

PO2°(I)*O\G01,

where

0)

P02 • G02 © (Z © GW) -> G02 © Z is a projection, and (/ii°2 )*i = /?iMoreover, h^ 2

= ^1 2

© ^1 2

> where /ij 2

is a point evaluation

using points which include {£\ , ...,^si }• Therefore the closed ideal generated by hQ0 (/) isQ2MjVo(C,Go2,F2)q,2, where g2 = h{°2°'(lBl).

Furthermore,

[h^°\lBl)]>2[h^\lBl)]. By using Z(2) point evaluations each of which uses the points {£1 > •••>^si }i we also obtain a unital homomorphism ^i 2 • ^o0\,Fi —> £21 © • • • © £2/(2), where E^i is a finite dimensional simple G*-algebra, such that {h^20) © hf^)^ = (^>I)*O|G0I©ZIt follows from 3.4.6 that there is a unital monomorphism h\ 2 . C\ —• E21 © • • -®E2l,2\, where E'2i is a finite dimensional simple C*-algebra, such that ( f c ^ J . o = P12 o (0i)»o|o(i), w here P 1 2 : G 02 © Z © G<2) -> G<2) is the projection. Since each partial map of <j>\ has multiplicity greater than > 9Z(Goi,Fi), there is iVi > 0 and a unital monomorphism h\ 2 ' : Ci ->• q2MN1(CG02,F2)q'2, where q'2 e MNI(CGO2,F2) is a projection, such that (/i^20))*o = PO2O(($I),O)|G where i? 2i i s a finite dimensional simple C*-algebra with dim-E2i = dimE 2 i + dimE'2i. Let A2 = B2 © G 2 . Note that if 0 (A 2 ) = G 02 © Z © G< 2 \ Ki(A 2 ) = F2 and T 2 C K 0 ( A 2 ) + c (G02©Z©G(2))+. One sees now that we obtain a unital homomorphism h\t2 '• B\@C\ —> B2®C2 such that (/ii)2)*o = $1 and {h^2)*i = (3\. Moreover, /i 1)2 = /4°2 © /ij 2, where /i^ 2 has finite dimensional range and [/ij 2(1-Ai)] > 2[/4 2 ( 1 A J ] in Ko(A2). Since each E2i is simple, the closed ideal generated by /ii j 2 (/) is B2 © G2 for every f £ F\. (Step II): We now repeat the above construction. Let A2 = B2 ® C2.

300

Classification

of Simple Amenable

C*-algebras

Note that K0(A2) = G 02 © Z © G^\ K1(A2) = F2 and T2 c K0(A2)+ c (G02©Z©G(2))+. Let {A2)} be an increasing dense sequence of finite subsets in the unit ball of B2 and let {t;k '} be a dense sequence of Y(GQ2,F2). We may assume that TT(2) O hitiFz) C F[2), where n^ :B2®C2-*B2 is the projection. For any / £ T2 , we may assume that there is j < s2 such that f(Q ') ^ 0. As in (Step I), we may assume that each partial map of 4>2 has multiplicity greater than 3(1 + 22)l(G02, F2)) + s2 + 3. By 6.4.4, there is a homomorphism h23 ' : CC02,F2 —^ MN/(CG03,F3) with NQ > 1 such that (h2°f')*0 = P03 o {2)*o\G02, where -Po3 : G03 © (Z © G<3)) ->• G03 © Z is a projection, and C4,3 )*i = Pi- Moreover h23 ' = h2 3 ' ' ® h23 ' \ where h23 ' ' is a point evaluation using points which include {£{ , ...,£i 2 }. Moreover 0) [/4° 3 °' 0 ) (1C G O 2 ,.F 2 )] > 4 [ 4 ° 3 ° ' 1 ) ( 1 C G O 2 , F 2 ) ] - The map h2°'3 extends to a map from MNo+Nl(CGo2:F2) to MN^NO+NI)(CGQ3,F3)Thus we may assume that h23 is a unital map from B2 to q'3M^i(N0+Ni)(GGo3,F3)q'3, where q3 € •^iV'(iVo+JVi)(CGo3,i;3) is a projection. Moreover the ideal generated by hffHf) i s l3MK(No+N1)(CGo3,F3)q3 for any / € J^2). As in (Step I), we obtain a homomorphism h2

3

' : B2 —¥ E3\@- • • .63/(3),

where E3i is a simple finite dimensional C*-algebra, such that (h2 ,

h23 )*o —

>

(4 2)*O\G02®2

and the closed ideal generated by h23 \f)

3

' ©

is E3X ©

•••£3*(3)forall/G^2). We obtain a unital monomorphism h23

• C\ -» E'31 © • • • © E'3l,3-.,

where E3i is a simple finite dimensional C*-algebra such that (h2 3 )*o = P13 ° (<^2)*O|G<2) • We also obtain, for some iVi > 1, a unital homomorphism h{l20) : C2 -> q3MNi(CGo3tF3)q3\ (where q'3 € MN[{CGo3tF3) is a projec2 tion) such that ( ^ 3 )*o = -PO3°(($2)*O)IG( )- Let 5 3 = P3MN2(CGO3}F3)P3, where N2 = N^(N0 + iVj) + JV{ and p3 = q3 © 93 a n d ^3 = ®E3\ © • • • £ 3 ' /(3) , where E3i is a finite dimensional simple C*-algebra with dimi?^ = d i m ^ i + dim£ 3 i . Let A3 = B3 © C3. Note that #0(^3) = G03 © Z © G<3), ifi04 3 ) = F 3 and T 3 C #0(^3)+ C (G 03 © Z © G( 3 ))+. One sees now that we obtain a unital homomorphism h2>3 : B2 © C2 —>• B3 © C3 such that (^2,3)*o = $2 and (/i2,3)*i = #2- Moreover /i 2 ,3|s 2 = /i2°3 ®hz,3, where h2°/3 is a point evaluation and [h2^3(lA2)} > 22[h23((lA2)] in K0(B3). Further(2)

more, the closed ideal generated by h2j3(/)

is 5 3 ffi C 3 for every f e F2 '.

Invariants

and some isomorphism

theorems

301

By continuing in this fashion, we obtain a sequence of C* -algebras An = Bn® Cn, where Bn = pnML„(CGon,Fn)Pn, Pn is a projection, C„ is a finite dimensional C*-algebra, and a sequence of unital homomorphisms hn : An -> An+l such that ( ^ o ( A „ ) , ^ i ( ^ „ ) = (G0n © (Z © C?(")),F n ), (/in)*o = $n and (/in)*i = Ai- Since each partial map on Z © G^ has multiplicity at least 9, we have (KUiGon

© (Z © G ( n ) )+ C r n + i .

Therefore G+ = lim n K0(An)+. Furthermore, hn = hh © /i„ , where hnl> is a point evaluation and [hn\lAn)\ > 2n[h{n\lAn)} in K0(An), and for each n, the closed ideal generated by hn>n^.k (/) is An+i for every / G .FA™ , where {T[n'} is dense in Bn, Tx^ohn-l{TnvL[1)) C F x ( n ) and 7r(n) : S „ © C „ -»• Bn is the projection. Thus B = lim 7l _ >00 (A n ,/i n ) is a unital simple C*-algebra with stable rank one (by 4.8.19). The condition [hHt\lAn)} > 3"[/il0)] implies that TR(A) = 0 (see the proof of 4.7.5). Moreover, {K0{A),K0(A)+,K1 {A)) = (G,G+,F). Since g is an order unit, there is a projection e 6 MJV(-B) for some n such that [e] = g in K0(B). Let A = eMN(B)e. D In the rest of this section, we present three classification results. These are classification theorems for C*-algebras which we do not assume to be inductive limits of the usual building blocks. We will give proofs for the first two (6.4.8 and 6.4.10). 6.4.6 If G is a subgroup of Q, then G+ is locally finitely generated. One can easily show that positive cone of a dense subgroup of R is finitely generated if and only if it is a subgroup of Q. Furthermore, if 1 € G, then any positive homomorphism h : G —> G has the form h(g) = rg for some r € G. Lemma 6.4.7 Let A be a unital amenable separable simple C* -algebra with TR(A) = 0 and with unique normalized trace r, where K* (A) is torsion free and such that Ko(A) is a dense subring o/R with [1A] = 1 with KQ(A)+ locally finitely generated, and let B be a unital simple AT-algebra with real rank zero and with the same (ordered) K-theory, and let a € KL(A,B)+ be such that a|x ; (A) = id for i = 0,1. Then for any finite subset V C P(A), finite subset J- C A and e > 0, there exists an T-e-multiplicative completely positive contraction $ : A —»• B such that

[$}\v = o\v-

302

Classification

of Simple Amenable C* -algebras

Proof. It follows from 3.9.17 that A is the closure of the union of an increasing sequence, {A„}, of amenable RFD C*-algebras. We will apply 6.1.11. We assume that A is not elementary, otherwise the theorem follows from 3.4.6. Let V be a finite subset of P(A) which contains 1^. Note that here KL(A,B) = Hom(K*(A),Kt(B)). Let G be the subgroup generated by V and G0 = G C\KQ{A). Let D = KQ{A) which we identify with a dense subring of Q (see 6.0). Let 4>n = 4>n and <j>n be as in 6.1.11 (corresponding to a here). We may assume without loss of generality that [„ ]|G(P) i s well-defined. We may also assume that (G 0 )+ = Go C\KQ(A)+ is finitely generated. Therefore, we may assume that both [0„]|G O a n d [n ]\G0 a r e positive (6.1.11). We may also assume that s = [„] (1^) G D and s = 1 + t, where t = [„ ](1^) G D. Furthermore, n(lyi) is a projection. Since D is dense in Q, for any integer J, there is p/q G D such that p and q are relatively prime and q > J. There are ni,ri2 G 7L such that nip + ri2q = 1. Therefore 1/q = n\{p/q) +ri2 G D. Thus there exists a sufficiently large integer q G Z + such that 1/q G D and q = Nt + I, where N > 0 and I > 0 are integers such that l/q < l/s. Let ki = (q/t)(l - (l/q)s). Then h = (l/t){q -l-lt)

= {l/t){Nt

-lt)

=

N-l

is in Z+. Define (3{r) = (l/q)r for all r G D. Then /3 : Xo(fi) ->• K0(B) is a positive homomorphism. It follows from 4.8.14 that there is a homomorphism h : B -> B such that [/i]| je0(B) = /? and [ / i ] ^ ^ ) = idtfi(.B) (here we identify tfo(-B) with K0{A) = D). If 1/q = l/s, we let $ = /i o 0 n . Otherwise, we let /3i(r) = (ki/q)r for r G D and h\ : B -> B such that [/ii]|/f0(B) = /?i and [AJIifjfs) = i d x ^ s ) . Set * = hi o i'. Note that (h/q)t

= (q/t)(l - (l/q)s)(t/q)

= 1-

So

(l/q)s + (h/q)t = 1.

(l/q)s.

Invariants

and some isomorphism

303

theorems

Therefore [hocf>n}{[lA}) + m{[lA})

= l and [hon] +

[*]\Kl(A)=a\Kl{A).

Set $ = h o (f>n © \t. Since [$] is positive on G 0 C D, [$}\v = a\v as desired.

•

T h e o r e m 6.4.8 Let A and B be two united separable simple amenable C* -algebras satisfying the UCT, with TR(A) = 0 and with torsion free K* such that KQ(A) is a dense subring of Q. Suppose that (K0(A),K0(A)+,[1A],K1(A))

Then

- (tfo(B), #«>(£)+, [1B], tfi(B))-

A^B.

Proof. It suffices to show that A is isomorphic to an AT-algebra. This follows immediately from 6.3.5 and the above lemma. D Example 6.4.9 Let D = Z[l/2]. Then D is a dense subring of Q. One sees that there are many dense subrings of Q. If A is a separable unital simple amenable C*-algebra with TR(A) = 0 satisfying the UCT, and the if-theory of A is the same as that of the Bunce-Deddens algebra B (see 4.8.8 and 3.7.11), then 6.4.8 says that A £* B. T h e o r e m 6.4.10 Let A and B be two unital separable simple amenable C*-algebras satisfying the UCT with TR{A) = 0 and K0{A) = Q © tor(Ko(A)). Suppose that (Ko(A),K0(A)+,[lA],K1(A))

Then

* (tf 0 (£),ffo(B) + ) [1 B ], # ! ( £ ) ) .

A^B.

Proof. We may assume that [Is] = 1 in Q. By 6.4.5, there is a unital simple AH-algebra C with no dimension growth such that TR(C) = 0 and (Xo(C),Xo(C) + ,[lc],ifi(C)) = (Ko(A),K0(A)+,[lA],K1(A)). It suffices to show that A = C. It follows from 6.3.5 and 6.2.9 that it suffices to show that, for any finite subset V c P(A), there is a sequence of contractive completely positive linear maps Ln : A —» C such that ||L„(afe) — Ln(a)Ln(b)\\

—> 0 for all a, b € A and [ i n ] | p = id-p.

304

Classification

of Simple Amenable

C*-algebras

By 3.9.17, A is an inductive limit of residually finite dimensional C*algebras. Let n and n' be as in 6.1.11 for a with C^K^A) = i&KdA) (i = 0,1) and B = C. Let G be the subgroup generated by V. Suppose that G n Ki{A,Z/mZ) = {0} for all m > k. Set ipn1] = <$ + $1 2 ) , where $£ 2) is a direct sum of k\ — 1 copies of cj)n' • Therefore bPn ]\GnKi(A,Z/mZ)

=

for all m < k. Since the image of $ „ C*-subalgebra, we also have [>Pn ]\Gntor(K0(A))

= ^Gntor(K0(A))

^GnKi(A,Z/m2)

is contained in finite dimensional

a n d [ip^] | G n K i ( A ) = i d G n K ! ( A ) -

We may assume that ^4 maps A to Mi^(C) for some integer l(n) > 0. Now let [ipnn'(lA)] = rn G Q. We may assume that r n > 1. Let (3n G KL{C,C)+ such that /3 n (r) = r/rn, (3n\tor(K0(C)) = ^tor(K0(C)), Pn\Ki(c,z/ki) = id^c.z/fcz) and (3n\Kl(c) = id|/Ci(C)- It follows from 6.2.9 that, for any -qn > 0 and finite subset ,F n C M^n){C) and any finite subset Qn C P(C), there exists an ^-.T^-multiplicative contractive completely positive linear map r „ : M^n)(C) ->• M^n)(A) such that [r„]| Q „ = /3n\Qn. For sufficiently large n, sufficiently large Tn and small r\n, we have [Tno-ipn}\-p = [d\-p and ||r n o V»„(a6) - T n o ^ „ ( a ) r „ o ^ n (b)||->• 0 for all a, b G A Set L„ = T„ o ?/;„.

•

Finally, we have the following theorem: Theorem 6.4.11 Let A and B be unital separable simple amenable C*algebras with TR(A) = TR{B) = 0 and satisfying the UCT. Then A^B if and only if (K0(A),Ko(A)+,

[ U ] , # i 0 4 ) ) * (K0(B),K0(B)+,

[1B],

K^B)).

Notes Theorem 6.1.11 appeared in [130]. A more general version was also obtained in [47]. A version of 6.2.5 was implicitly contained in [65]. Theorem 6.2.4 that we presented here did not require that A/f-algebras have bounded dimensions. The proof is shorter than that in [65]. Theorem 6.2.5 also saves

Invariants

and some isomorphism

theorems

305

a reduction step ([36] and [78]). Theorem 6.3.6 was obtained by Elliott and Gong ([65]). The proof we present here is different (applying 6.3.5) from theirs and can be used to prove 6.4.11. Classification of separable amenable simple C*-algebras with tracial topological rank zero was obtained by the author.

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316

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[178] S. Zhang, A property of purely infinite simple C*-algebras, Proc. Amer. Math. Soc, 109 (1990), 717-720. [179] S. Zhang, A Riesz decomposition property and ideal structure of multiplier algebras, J. Operator Theory 24 (1990), 209-225. [180] S. Zhang, C* -algebras with real rank zero and the internal structure of their corona and multiplier algebras, III, Canad. J. Math. 42 (1990), 159-190. [181] S. Zhang, Matricial structure and homotopy type of simple C -algebras with real rank zero, J. Operator Theory, 26 (1991), 283-312. [182] S. Zhang, C* -algebras with real rank zero and the internal structure of their corona and multiplier algebras, Part I, Pacific J. Math., 155(1992), 169-197. [183] S. Zhang, Certain C* -algebras with real rank zero and their corona and multiplier algebras, IIK-Theory 6 (1992), 1-27. [184] S. Zhang, C* -algebras with real rank zero and their corona and multiplier algebras, IV, Internat. J. Math. 3 (1992), 309-330. [185] S. Zhang, On the exponential rank and exponential length of C*-algebras, J. Operator Theory 28 (1992), 337-355. [186] S. Zhang, Exponential rank and exponential length of operators on Hilbert C -modules, Ann. of Math. 137 (1993), 129-144.

Index

W-algebra, 65 cr-unital, 22 cr-weak topology, 38 d„(a), 222 doo(a), 222 E x t ( A , B ) , 219 cel(A), 207 cel(u), 171 sp(a), 2

•-isomorphism, 15 AF-algebra, 55 ,4T-algebra, 81 C*-algebra, 9 purely infinite, 145 residually finite dimensional, amenable, 76 cancellation, 119 Cuntz, 164 elementary, 64 finite, 127 finite dimensional, 43, 46, 77 group, 50, 65 inductive limit, 53 liminal, 46 nuclear, 76 postliminal, 46 reduced group, 50 RFD, 271 simple, 43 stably finite, 127 type I, 46, 66 unital, 10 C*-norm, 9 C-subalgebra, 9 Ext(A,B), 219 K0(A), 84 K^A), 85 M(A), 211 Q(B), 218

positive homomorphism, 129 adjoint, 9 AF-algebra, 133 algebra, 1 AT-algebra, 81 AF, 55, 133 Banach commutative, 2 unital, 2 irrational rotation, 48 multiplier, 211 normed, 1 von Neumann, 36 enveloping, 36 amenable group, 50 amplification, 34 analytic function, 3 approximate identity, 20 quasi-central, 65, 229, 235 317

318

Index

approximately commutative, 59 approximately commutes, 56 approximately intertwining, 56, 59, 196 approximately the same, 55 approximately unitarily equivalent, 55 asymptotic sequential morphism, 270 automorphism, 46 inner, 64

unitary, 10 Elliott, 136, 295 enveloping group, 84 exponential map, 252 extension, 217 absorbing, 219 essential, 218 pure, 254 trivial, 218

Borel functional calculus, 40 Busby invariant, 218

finite dimensional range, 270 frame, 279 standard, 279 full map, 222

cancellation of projections, 119 canonical generator, 134 canonical generators, 44 canonical map, 52, 134 Cauchy-Schwarz inequality, 27 commutant, 35 compact operator, 16 completely positive map, 67, 72, 78, 227 amenable, 227 completely positive maps, 221 composition series, 65 continuous functional calculus, 15 covariant functor, 256 Cuntz algebra, 164 direct product, 50 direct sum, 50 double commutant, 35 dual group, 63 dynamical system, 46 classical, 46 ergodic, 48, 63 minimal, 48, 63, 65 element finite, 109 invertible, 2 normal, 10 positive, 16 self-adjoint, 10

Gelfand transform, 13 Gelfand-Naimark, 32 Gong, 295 Grothendieck group, 84 group C* -algebra, 50 hereditary C*-subalgebra, 22 homotopic, 98 ideal, 24 inductive limit, 51 intertwining, 57 invariant subspace, 45 irrational rotation algebra, 48 isomorphism, 25 Jordan decomposition, 30 Kirchberg, 266 left Haar measure, 50 left kernel, 31 length of a map, 171 linear map, 25 n- positive, 67 completely bounded, 67 completely contractive, 67 completely positive, 67 positive, 25

319

Index self-adjoint, 25 mapping cone, 218 mapping torus, 220 Markov-Kakutani, 46 matrix units, 44 maximal ideal, 12 maximal ideal space, 12 multiplicative linear functional, 12 multiplier algebra, 211 natural transformation, 256 operator essentially normal, 266 operator topology, 32 strong, 32 weak, 32 order ideal, 127 order isomorphism, 129 order unit, 127 ordered group, 127 unperforated, 130 weakly unperforated, 130 scaled, 127 simple, 127 orthogonal elements, 64 partial isometry, 41 positive linear functional, 25 positive operator, 16 projection, 10 equivalent, 83 range, 41 stably equivalent, 83 trivial, 206

covariant, 46 cyclic, 31 faithful, 31 finite dimensional, 43 irreducible, 46, 50 left regular, 50 non-degenerate, 36 non-degenerated, 240 unitary, 49 universal, 32 resolvent, 2 RFD C*-algebra, 157 Riesz interpolation, 130 Riesz property, 130 S. Zhang, 132, 145, 146, 162 scaled ordered group, 127 slow dimension growth, 207 SP, 142, 143, 149 spectral mapping theorem, 15 spectral measure, 40 spectral radius, 5 spectrum, 2, 11 stable rank, 124 stable rank one, 113 state, 25 pure, 63 tracial, 26 Stinespring, 68 Stone-Cech compactiflcation, 45, 212 strict topology, 212 strongly continuous, 34

quasitrace, 162

topological group, 49 trace, 26 tracial topological rank, 147 tracial topological rank zero, 148 translation invariant measure, 46

range projection, 41 real rank, 124 zero, 116 real rank zero, 116 representation, 30

UCT, 303 unitarily equivalent, 55 unitary, 10 unitary representation, 49 unitization, 10

320 universal weak closure, 36 von Neumann, 36 weakly continuous, 34

Index

The theory and applications of C*-algebras are related to fields ranging from operator theory, group representations and quantum mechanics, to non-commutative geometry and dynamical systems. By Gelfand transformation, the theory of C*-algebras is also regarded as non-commutative topology. About a decade ago, George A. Elliott initiated the program of classification of C*-algebras (up to isomorphism) by their K-theoretical data. It started with the classification of AT-algeras with real rank zero. Since then great efforts have been made to classify amenable C*-algebras, a class of C*-algebras that arises most naturally. For example, a large class of simple amenable C*-algebras is discovered to be classifiable. The application of these results to dynamical systems has been established. This book introduces the recent development of the theory of the classification of amenable C*-algebras — the first such attempt. The first three chapters present the basics of the theory of C*algebras which are particularly important to the theory of the classification of amenable C*-algebras. Chapter 4 ofters the classification of the so-called AT-algebras of real rank zero. The first four chapters are self-contained, and can serve as a text for a graduate course on C*-algebras. The last two chapters contain more advanced material. In particular, they deal with the classification theorem for simple AH-algebras with real rank zero, the work of Elliott and Gong. The book contains many new proofs and some original results related to the classification of amenable C*-algebras. Besides being as an introduction to the theory of the classification of amenable C*-algebras, it is a comprehensive reference for those more familiar with the subject.

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