C*-Algebras and W*-Algebras (Classics in Mathematics)

Ergebnisse der Mathematik und ihrer Grenzgebiete Band 60

Herausgegeben von

P. R. Halmos - P. J. Hilton - R. Remmert - B. Sz6kefalvi-Nagy Unter Mitwirkung von

L. V. Ahlfors • R. Baer • F. L. Bauer - R. Courant A. Do ld - J. L. Doob • S. Eilenberg • M. Kneser • G. H. Müller M. M. Postnikov • B. Segre • E. Sperner Geschdftsführender Herausgeber: P. J. Hilton

Shôichirô Sakai

C*-Algebras and W*-Algebras

Springer-Verlag Berlin Heidelberg New York 1971

Shôichirô Sakai Professor of Mathematics, University of Pennsylvania

AMS Subject Classifications (1970) : Primary 46 L 05, 46L 10; Secondary 81 A 17

ISBN 3-540-05347-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-05347-6 Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. by Springer-Verlag Berlin Heidelberg 1971. Library of Congress Catalog Card Number 75-149121. Printed in Germany. Printing: Zechnersche Buchdruckerei, Speyer. Binding: Konrad Tritsch, Graphischer Betrieb, Würzburg.

To Masato and Kiyoshi

Preface

The theory of operator algebras in a Hilbert space was initiated by von Neumann [126] in 1929. In the introduction to the paper [119] in 1936, Murray and von Neumann stated that the theory seems to be important for the formal calculus with operator-rings, the unitary representation theory of groups, a quantum mechanical formalism and abstract ring theory. These predictions have been completely verified. Furthermore the theory of operator algebras is now becoming a common tool in a number of fields of mathematics and theoretical physics beyond those mentioned by Murray and von Neumann. This is perhaps to be expected, since an operator algebra is an especially well-behaved infinite-dimensional generalization of a matrix algebra. Therefore one can confidently predict that the active involvement of operator algebra theory in various fields of mathematics and theoretical physics will continue for a long time. Another application of the theory is to the study of a single operator in a Hilbert space (see, for example, [99]). Nowadays we may add further the theory of singular integral operators and K-theory as other applications. Such diversifications of the theory of operator algebras have already made a unified text book concerning the theory virtually impossible. Therefore I have no intention of giving a complete coverage of the subject. I will rather take a somewhat personal stand on the selection of material —i. e., the selection is concentrated heavily on the topics with which I have been more or less concerned (needless to say, there are many other very important contributors to those topics. The reader will find the names of authors who have made remarkable contributions to those discussed in the concluding remarks of each section). Consequently parts of the book tend to be somewhat monographic in character. Let me explain briefly about the contents. There are essentially two different ways of studying the operator *-algebras in Hilbert spaces. The first alternative is to assume that the algebra is weakly closed (called a W*-algebra). These algebras are also called Rings of operators, and more recently, von Neumann algebras.

VIII

Preface

The earliest study along this line is due to von Neumann in 1929. In a series of five memoirs beginning with [119], Murray and von Neumann laid the foundation for the theory of W*-algebras. Virtually all of the later work on these algebras is based directly or indirectly on their pioneering work. We call a W*-algebra a factor if its center is just the complex numbers. Murray and von Neumann concentrated most of their attention on factors. However, von Neumann [132] obtained a reduction theory by which the study of a general W*-algebra may be to a large extent reduced to the case of a factor. At the same time a number of authors have pushed through the major portions of a global theory for general W*-algebras (the reader may find a long list of papers by many authors in the bibliography in [37]). The second alternative is to assume only that the algebra is uniformly closed (called a C*-algebra). The earliest study along this line is due to Gelfand and Naimark [55] in 1943. A notable advantage of the 0:algebra is the existence of an elegant system of intrinsic postulates, formulated by Gelfand and Naimark, which gives an abstract characterization of these algebras. Using this approach, Segal [180] in 1947 initiated a study of C-algebras. The subsequent development which contains many beautiful results (cf. Chapters 1, 3, & 4) has been carried out by a number of authors. The theory of C*-algebras fits naturally into the theory of Banach algebras, and in certain respects they are among the best behaved examples of infinite dimensional Banach algebras. In chapter 1, the characterization of W*-algebras obtained in [149] is used to define W*-algebras as abstract Banach algebras, like C*algebras, and to develop the abstract treatments of both W*- and C*algebras. Chapter 2 is concerned mainly with the classification and representation theory of W*-algebras which are developed along classical standard lines. Some proofs may be new. In chapter 3, the reduction theory is discussed. Here, a modern method, developed recently in [146], [159], [218] (i. e. the decomposition theory of states) is used. Also discussed are some recent results obtained by theoretical physicists. Chapter 4 consists of some special topics from the theories of W*algebras and C*-algebras. This chapter is the most personal in the book. All the topics covered are ones with which I have been more or less concerned. They are: Derivations and automorphisms on operator algebras; examples of factors; examples of non-trivial global W*algebras; type I C*-algebras and a Stone-Weierstrass theorem for C-algebras.

Preface

IX

I express sincere thanks to C. E. Rickart, S. Kakutani and I. E. Segal who made it possible for me to do research at Yale and M.I. T. during the various stages of the preparation of this book. I express thanks to B. Sz. Nagy who invited me to write this book for the Ergebnisse series. Also the encouragement given to me by P. Hilton for the completion of the manuscript is much appreciated. I thank R. Kal 'man and one of my students, Mrs. D. Laison, for their reading carefully the final manuscript. My deepest thanks go to Miss P. Fay who with great patience and skill typed most of the manuscript. I also wish to acknowledge the financial support which was given at various stages of the writing by the National Science Foundation (NSF—G-19041, NSFGP-5638, NSFGP-19845), John Simon Guggenheim Memorial Foundation and the University of Pennsylvania. Finally I wish to express my appreciation to Springer-Verlag (especially, Dr. Klaus Peters) for their most efficient and understanding role in bringing this book to its completion. Philadelphia, Pa. U.S.A., January 1971

S. Sakai

Contents

1.

General Theory 1 1.1. Definitions of C*-Algebras and W*-Algebras 1.2. Commutative C*-Algebras 3 6 1.3. Stonean Spaces 1.4. Positive Elements of a C*-Algebra 7 1.5. Positive Linear Functionals on a C*-Algebra 9 10 1.6. Extreme Points in the Unit Sphere of a C*-Algebra . 14 1.7. The Weak Topology on a W*-Algebra 19 1.8. Various Topologies on a W*-Algebra 22 1.9. Kaplansky's Density Theorem 24 1.10. Ideals in a YV*-Algebra 1.11. Spectral Resolution of Self-Adjoint Elements in a Pr-Algebra 26 1.12. The Polar Decomposition of Elements of a W*-Algebra . . 27 1.13. Linear Functionals on a W*-Algebra 28 1.14. Polar Decomposition of Linear Functionals on a W*-Algebra 31 1.15. Concrete C*-Algebras and W*-Algebras 33 1.16. The Representation Theorems for C*-Algebras and W*40 Algebras 42 1.17. The Second Dual of a C*-Algebra 45 1.18. Commutative W*-Algebras 1.19, The C*-Algebra CVO of all Compact Linear Operators on a 46 Hilbert Space cYf 48 1.20. The Commutation Theorem of von Neumann 50 1.21. *-Representations of C*-Algebras, 1 58 1.22. Tensor Products of C*-Algebras and W*-Algebras 1.23. The Inductive Limit and Infinite Tensor Product of C*70 Algebras 75 1.24. Radon-Nikodym Theorems in W*-Algebras 2.

Classification of W*-Algebras

2.1. 2.2.

Equivalence of Projections and the Comparability Theorem. 79 Classification of W*-Algebras 83

XII

Contents

2.3. Type I W*-Algebras 2.4. Finite W*-Algebras 2.5. Traces and Criterions of Types 2.6. Types of Tensor Products of W*-Algebras 2.7. *-Representations of C*-Algebras and W*-Algebras, 2 . 2.8. The Commutation Theorem of Tensor Products 2.9. Spatial Isomorphisms of W*-Algebras

87 89 95 98 102 108 111

3.

Decomposition Theory

3.1. 3.2. 3.3. 3.4. 3.5.

Decompositions of States (Non-Separable Cases) Reduction Theory (Space-Free) Direct Integral of Hilbert Spaces Decomposition of States (Separable Cases) Central Decomposition of States (Separable Cases)

4.

Special Topics

4.1.

Derivations and Automorphisms of C*-Algebras and W*Algebras 153 Examples of Factors, 1 (General Construction) 171 of Types II I , Examples of Factors, 2 (Uncountable Families II III 183 Examples of Factors, 3 (Other Results and Problems) . . • 202 Global W*-Algebras (Non-Factors) 216 Type I C*-Algebras 219 On a Stone-Weierstrass Theorem for C*-Algebras 236

4.2. 4.3. 4.4. 4.5. 4.6. 4.7.

.

121 131 137 140 146

Bibliography

243

Subject Index

251

List of Symbols

255

1. General Theory

1.1. Definitions of C*-Algebras and W*-Algebras Let si be a linear associative algebra over the complex numbers. The algebra d is called a normed algebra if there is associated to each element x a real number I Ix II, called the norm of x, with the properties: I 11)4 -_0 and 11x11 =0 if and only if x=0; II Ilx+Yll 11x11 + il.Y11; III IlAxIl =PI 11xII, A a complex number; IV lixyll -- 11x11 IIYII .

If si is complete with respect to the norm (i.e. if Jai is also a Banach space), then it is called a Banach algebra. A mapping x---4x* of sit into itself is called an involution if the following conditions are satisfied I (x*)* = x; II (x +y)* = x* + y*;

III (xy)* = y* x* ; IV (A x)* = I x* , A a complex number. An algebra with an involution * is called a *-algebra. 1.1.1. Definition. A Banach *-algebra al is called a C*-algebra if it satisfies Ilx * x11=114 2 for x e sit. 1.1.2. Definition. A C*-algebra di is called a W*-algebra if it is a dual space as a Banach space ( i.e., if there exists a Banach space A, such that (4 ) *=.11, where (Al)* is the dual Banach space of A :). We shall call such a Banach space di* the predual of dI. Remark. It is not true in general that a dual Banach space is the dual space of a unique Banach space. For example, c* and ct, are isometrically isomorphic to li , but c is not isometrically isomorphic to c o , where c is the Banach space of all convergent sequences, c o the Banach space of all sequences convergent to zero, and ll is the Banach space of all summable sequences. However, we shall show later that a W*-algebra is the dual space of a unique Banach space.

2

1. General Theory

1.1.3. Definition. The topology defined by the norm II II on a C*-algebra si is called the uniform topology. The weak *-topology H, 4Y) on a W*-algebra di is called the weak topology or the a-topology on dl. 1.1.4. Definition. A subset V of a C*-algebra sat is called self-adjoint if XE V implies X * E V. A self-adjoint, uniformly closed subalgebra of d is also a C*-algebra. It is called a C*-subalgebra of si. A self-adjoint, a-closed subalgebra ,Af of a W*-algebra di is also a W*-algebra, because (141,/{f °)*. , K, where S .° is the polar of Jr in dl* . Af is called a W*subalgebra of di. 1.1.5. Definition. Let tdahel be a family of C*-algebras. The direct sum E 0 ._5212 of I.slOEL EI is defined as follows: elements of E 0, ,saf„ are aell

ad

composed of all families (a2)„Eu such that aa ed,, and sup 1 c/a 11 < ± 00, a

and the operations: 2(a2)=(Aa2) ( 2 a complex number), (a, c)+(ba)=(aa+b„), (aj(k).(aŒ ba), (aŒ)* .(at) and II(a2)11= sup MaŒ ll. E 0 sic, is again a a ael C* -algebra. Let Idtalaa be a family of W*-algebras. The direct sum E $ d/ta is a dual space since

ozEI

E 0 dic, = ( E ® dice)* , where the norm of an 2211

element (fa)(LEA.) in hence it is a W*-algebra.

«elf

(

li

is defined by 2211

1'

11(fa)11 = E 1 fŒll ; 221[

1.1.6. Lemma. Let di be a C*-algebra; then li xll = II x* II for x E si.

Proof. IIxII 2 = ilx * xll -Ilx * II 114; hence 114 ilx* Ii and analogously

Ilx* I1 --11x ** 11=11x11.

q.e.d.

1.1.7. Proposition. Let si be a C*-algebra without identity, and let di the algebra obtained from si by adjoining the identity 1. For X E .4,be

A a complex number, define 112 1+ xII = sup HAI o C* algebra with norm II Il.

MAY±X.Y11

. Then al is a

MA

-

Proof. It is easy to see that d i is a Banach *-algebra. Suppose pc is an arbitrary positive number less than 1. Then there exists a yea' such that ly11=1 and iill 2 1+x11 < 11).Y+xyll .

Then,

11 2 11 21 +x11 2 0, consider the finite division d of the interval [0,11a11+1]:0=.1 0 O for heP heP

Define p(a+ ib). g(a)± i g(b) for a, be ./Ks . Then ço is a linear functional on d/. Moreover, since ,///s is closed, the *-operation is cr-continuous. Hence (p. is a cr-continuous positive linear functional such that q.e.d. (p(a)< O. The remainder of the proof is clear. 1.7.3. Definition. We call a directed set whenever oc>

fxOEI

in 4's increasing, if x,c _x

1.7.4. Lemma. Every uniformly bounded, increasing directed set converges to its least upper bound. Further, if x =1. u. b. xOE, then a* x a =1. u. b. a* x Πa. a

Proof. Let E be the set of all finite linear combinations of elements in T. It is clear that the topology (7(4', E) is weaker than OA jil*), and (7(4 , E) is a Hausdorff topology. Since S is cr(A, A)-compact, (7(4 , E) is equivalent to 04' , A) on bounded spheres. Therefore, to show that a uniformly bounded directed set {xc } is a Cauchy directed set in o-(/ll, 41* ), it is enough to show that for any (pe T and positive E> 0, there exists an index oco such that Icp(x,,— x p)1<e for a, fi ao If fxOEI is a uniformly bounded increasing directed set of self-adjoint elements of d/, then for every (p e T, {y9(x 2)} is a uniformly bounded increasing directed set of real numbers. Hence fxOEI is cr(4% A)-Cauchy so that by compactness of S, it converges to some element x. Moreover by 1.7.2, x = 1.u.b. { xOE}. If u is an invertible element in 41, then clearly 1.u.b. lu*x 2 ul=u*Il.u.b.x 2 lu=0X/4. a

a

1. General Theory

16

If a is an arbitrary element of 41, there exists a suitable number A > 0 such that Al+ a is invertible. Then cp((A1+ a)* x,c(A1+ a)). A 2 (p(x,c)+ A (p(a* x) + A ço(xa)+ (p(a* xo,a)--(p((A1+ a)* x(A1 + a)) for cp e T.

On the other hand,

I (p(a* (x, — x AI= icp(a* (x „— x fl)1 (x „— -. cp (a* (x Gt — x fl ) a)+ (p(x „— xfi )+

for a > fl

1

and cp ((x ,,, — x p)a)1= IT (a* (x ,,— x0))1. Hence, A 2 (p(x„)+ A (p(a* xa,)+ A (p(x„a)-- A 2 p(x)+ Acp(a* x)+ A (p(x a).

Therefore, ço(a* x „a)-- (p (a* x a) and so 1. u. b. a* x „a= a*x a. q. e. d. a 1.7.5. Lemma. If C is any maximal commutative C*-subalgebra of 4', then its spectrum space is Stonean. Proof. Let {a„} be a uniformly bounded, increasing directed set of positive elements in C and let a0 = 1. u.b. ac, in Jts. If u is any unitary a element in C, then 1. u. b. u* acc u = u* (1. u. b. a,c)u and 2 2 1.u.b. u*acc u=1.u.b. a= a0 . a a

Hence u* ao u = ao . Since every element of C is a finite linear combination of unitary elements in C, ao commutes with every element of C. Hence ao e C and the spectrum space is Stonean by 1.3.2. q. e.d. 1.7.6. Lemma. Let e be any projection of .1 f. Then the subalgebra e."11 e is o--closed and the mapping x—*e xe is o--continuous. Proof. e(Pn S)e consists of those elements of P n S which are <e. If {x,,} is a directed set in e(P nS)e which converges to an element xo 0, then e — x OE > 0, so that e — xo > O. Hence e(P nS)e is closed. Since e(4is n S)e = e(P n S)e — e(P n S)e, the compactness of e(P n S)e implies that e(dis n S)e is closed. Hence edif e is closed. We shall next show the continuity of the mapping x — exe. For this it is enough to show that the kernel (1 — e)4/ +.141 — e) of the mapping is closed, since 4' is the algebraic direct sum of edie and (1 — e)4' + jK(1 — e).

First, we shall show that if le a„(1 e)} (a,c e 4's n S) converges to a, —

eae=(1—e)a(1—e)=0.

1.7. The Weak Topology on a W*-Algebra

17

For any integer n and complex number c (Ici = 1), Ilea.( 1— e)+cne11= II fea.( 1— e)+ cne} {(1 —e)a„e-FTne}11+ = Ilea,(1—e)ci a,e+n2 e11 1 (1+n2 ). Now suppose that e ae 0 and that there exists a positive number A> 0 in the spectrum of

e ae + e a* e (otherwise, consider { —a2 }). Then 2

11eae+ne+ea(1—e)+(1—e)ae+(1—e)a(1 — 011 -11e(a+n 1 )ell

eae+ea*e + ne > A + n 2

Therefore, Ma+nell > (1 + n 2)+ for a large positive number n. This is a eae+ea*e = 0. contradiction. Hence 2 Analogously, suppose there exists a positive number A> 0 in the ie a* e—ieae (otherwise, consider { — a0). Then spectrum of 2 Ma -Enid

Ileae+niell

je a* e—ieae

2 >n+A, a contradiction.

+ in e

ieee—ieae = Henc 0 and so eae=0. 2 Similarly, suppose that (1 — e)a(1 — e)0 0; then Ilea2( 1— e)+ cn(1 — e)11= 11{ ( 1 — e)a,,e +? n(1 — e)} t7„(1 — e)+ cn(1 — e)}11 = 11(1 — e)a,c e a„(1 —e) +n2 (1 —e)11 1 -(1+n 2 )1 . We obtain a contradiction, and so a= e a(1 — e)+ (1 — e)ae. Therefore the closure of (1 — e)Se is contained in e(1 — e)+ (1 — e)dt e. By symmetry, the closure of eS(1—e) is also contained in e JIG — e)+ (1 — e)dt e. From the above remarks and the compactness of S, we can easily conclude that e S(1 — e)+ (1 —e)Se is closed, so that edi(1—e)+(1—e)Me is closed. Hence (1—e),11+.11(1—e).(1—e)die+edt(1—e)+(1—e)di(1—e) is closed.

q. e. d.

1.7.7. Lemma. If e is any projection of 4', then the mappings x—*ex and x—*xe are o--continuous.

1. General Theory

18

Proof. Suppose that le a„(1 -e)} (aŒ eS) converges to a and

(1 - e)ae00.

By the proof of 1.7.6, a=e a(1 - e)+ (1

el, a e Ila+n(1- e)aell = Ile a(1 - e)+(n+1)(1 = max{Ilea(1 (n+ 1) 11(1 - e)aell} for any positive number n. Hence, II a + n(1 - e) a ell = (n + 1)(1 - e) a ell for a large positive number n. On the other hand, -

-_,.max{1,n11(1-e)aell

Ilea2(1-e)+n(1

aeM

for a large positive number n. This contradicts the above equality. Hence x(1 - e), (1 - e)x e, e x, e # (1 e) is closed. Therefore, the mappings x and xe are a-continuous. q. e. d. -

1.7.8. Theorem. The mappings x - x*, ax, and x a are a- continuous for x, . Proof. di 4's +iJP, 4's ni,/lls = (0), and by 1.7.1, Jr is closed; hence the mapping x-*x* is a-continuous. Let C be a maximal commutative C*-algebra of _ ft containing a self-adjoint element h. Then the spectrum space of C is Stonean. Hence, by 1.3.1, for any positive number E (> 0), there exists a finite family {e i } of mutually orthogonal projections

belonging to C such that real numbers.

h-

E i=

<e, where

RI

is a family of

1

Let {xa } (11.x„Il< 1) be a directed set converging to 0; then for any a-continuous linear functional g, -

101 -Xce)1 = g

MOE+ Hence lim

E.

A i ei)x,c) i=

i=1

g((1 A i ei)x,c) i=

I Ail Ig(eix01

Therefore lim g(h x a). O. Hence a linear

functional 1(x)=g(hx) is a-continuous on S, so that by the Banach Smulian theorem, 1(x) is a-continuous on ,/# ; so the mapping x-41x and therefore also x--ax, is a-continuous. Finally the mapping =x a is continuous, q. e. d. 1.7.9. Corollary. Let H be a subset of ./1, si be the C*-subalgebra of 4' generated by H, and let 2 be the a(', 4')-closure of d. Then Jail- is a W*-subalgebra of si is called a W*-subalgebra of ,./K generated by H. Moreover, si is commutative if si is commutative. References. [149], [154].

1.8. Various Topologies on a W*-Algebra

19

1.8. Various Topologies on a W*-Algebra

From the theory of locally convex spaces, we can identify the Banach space with the Banach space of all o-(4', A)-continuous linear functionals. Now, let t(#, ./K) be the Mackey topology on div, that is, the topology of uniform convergence on all relatively 44, 4)-compact convex subsets in A. 1.8.1. Notation. Let si be a C*-algebra, d* the dual Banach space of d. For fed*, denote f(x)=<x, f> for XESi, <X, L a f> = , and <x, Ra f > = <xa, f> for x, a Ed. La f, R a f are in d*. 1.8.2. Definition. Let V be a linear subspace of d*. V is said to be invariant if fEV implies R a f, L a f eV for aed. V is said to be self-adjoint if f EV implies f* EV. the Let 4' be a W*-algebra, 41 * the dual Banach space of ', predual of ,/g. Then is a closed linear subspace of 41 * . By 1.7.8, is self-adjoint and invariant. 1.8.3. Proposition. The mappings continuous.

f*, L a f, R a f in J1* are

Proof. Let { f OE} be a directed set of elements in Then,

A, converging to O.

= = --03 , and <x, Ra f> = <xa, f„>-03 for xejK . q. e. d. 1.8.4. Proposition. The mappings a--q,a f, R a f of 4' with 04' ,A) into 4/4, with cr(.414„ /#) is continuous for each f

Proof. Suppose faal converges to zero in the OA 4,)-topology; then <x,L„f > = —* O. Hence L a f and analogously a—* R a f are continuous. q. e. d. 1.8.5. Proposition. The mappings

x*, ax, x a are t(4' , 4 ) -continuous.

Proof. Let fx OEI be a directed set in ,/ll converging to 0 in the T(///,,A) topology, and let G be a relatively o-(4,, 4)-compact convex subset in A. Then G*, L a G and R a G are also relatively 0- (4'* , dé)-compact. Hence , G> = a„ G*> 0 = <xa, L a G>-- 0 and <xa, a,G> 0 q. e. d.

(uniformly), (uniformly), (uniformly).

1. General Theory

20

1.8.6. Definition. Let T be the set of all cr(J1,A)-continuous positive linear functionals on dl. For 9e T, define a„(x)= 9(x* x)i for xe,/11 Then a, will define a semi-norm on /N. The family of semi-norms { a„! all 9e T} defines a locally convex topology. This topology is called the strong topology or the s-topology on ./ K, and is denoted by 4/ , A) or s. 1.8.7. Definition. For 9e T, define a,*(x)= 9(x x*)i for xe4'. Then a: is a semi-norm 41 . The locally convex topology on 41 defined by a family of semi-norms la,,a„*1 all 9 e TI is called the strong *-topology or the s*-topology on 4', and is denoted by s*(4', A) or s*. 1.8.8. Notation. Let Q be a set. Let t 1 , t2 be two topologies on Q. We shall write t 1 = <x,Lx: 9> .

Since L9 OE LsQ and Ls 9 is c)-(4', 4 ) -compact (cf. 1.8.4), oc,(x,)--4). Hence fxOEI converges to 0 in the s(//,,/# ) topology. Therefore, any s(dl, 4J-continuous linear functional on S is also 4/K, 4/0-continuous on S. Hence it is also 0-(W, dl)-continuous. Conversely, we shall show that any f e A, is s(il , A ) -continuous. It suffices to assume that f* =f. By the a-compactness of S, there exists a self-adjoint ac, in S such that 11f11 f(ao). Let ; = {Of (a) = I f II, ae Sn then g is a u-compact convex set. Let u be an extreme point in g; then it is extreme in 4's n S. In fact, let u = (a+ b)I2 (a, be jh's S). Then (f (a) + f (b)) f (u) = I f I • Hence f (a) = f (b)= f . Hence a, bee. 2

By 1.6.3, u is self adjoint and unitary. Also (L u f)(1)= f(u). 11f11 and so Lu f e T. Conversely, I (x) I = If (u2 I = I (L. f )(u f)(u 2 )4 (L u f)(x* Hence f is s(4', 41 ) -continuous. Therefore, the dual space of dl, with the s(,/d, A) topology, is dl. Hence by Mackey's theorem, -

o-(j11,A) Clearly s(4',A)<s*(A4 c ). The *-operation is z-continuous. Hence s* (di , (4' , q. e. d.

1.8. Various Topologies on a 14/*-Algebra

21

Remark. It is known that s*(/#, A) is equivalent to t(4', A) on S (cf. [1], [157]). Therefore, if a linear mapping 0 of a W*-algebra into another W*-algebra is a-continuous, then it is s*-continuous on bounded spheres, since 0 is -c-continuous. 1.8.10. Corollary. Let f be a linear functional on a W*-algebra 4'. The following eight conditions are equivalent. 1. f is continuous in the cr(di,,/14) topology (resp. 2. s(4',A), 3. s*(4',A) and 4. -c(j#, A)) ; 5. f is continuous on S in the cr(4',A) topology (resp. 6. s(4',A), 7. s*(41,A) and 8. -c()I,A)) 1.8.11. Corollary. Let C be a convex set in 4'. Then the following eight conditions are equivalent. 1. C is closed in the o-(4',A) topology (resp. 2. s(dt,41* ), 3. s*(W,A) and 4. ''K)) ; 5. for every A>0, CnAS is closed in the 4/#, A) topology (resp. 6. s(/#,A), 7. s*(idl, A) and 8. 4/11,41* )). 1.8.12. Proposition. The mappings x—*ax, xa are s(dt,,4'* )-continuous, and moreover the mapping (x,y)--xy is jointly s(jK,A)-continuous on ASx,/i/ (A>0) (a, x, ye4).

Proof. Suppose IxOEI converges to 0 in s(di',A). Then, for (pc T, ocv (axa)2 = cp(x: a* axa) ll a* all (P(x« x = a* all «,(x 0)2 0 and

oc,,,(x 2 a)2 = (p(a*

x,,a) = ocLa*R a (p(x 2)2 o. Moreover, suppose y fl —q in s(#, i#) and yfi e S. Then, for (peT, av (y x „ — y x) oc v (y fl (x „— x))+ cx,p((y 13 — y) .

q. e. d.

Remark. In general, the *-operation is not s(4', 4'4 )-continuous on bounded spheres (cf. 2.5). Concluding remarks on 1.8. 1. A is 0J/4,j/0-sequentially complete (cf. [1], [151]). 2. Let ,/ll be a type I-factor (cf. chapter 2) and let ((p,i)c T be a ',I)-Cauchy sequence. Then (cp,i) is a Cauchy sequence in the norm topology of A. (Cf. [1], [27]). (In this theorem, we can not replace T by A (cf. [151]).) Problem. Is the converse of the theorem true? 3. There are many C*-algebras, W*-algebras and their preduals (cf. [155]) which are not topologically contained in the classes of the socalled classical Banach spaces ((M), (m), (C), (c), (C(P))p_1, (LP)1 co) mentioned by Banach (cf. [8]); therefore it is very meaningful to examine whether or not many unsolved problems concerning Banach space are positive in these examples. Reference. [154].

1. General Theory

22

1.9. Kaplansky's Density Theorem Let be a W*-algebra and let di* be the predual of Let V be a self-adjoint, invariant linear subspace of (i. e. f eV implies f*, L ai, R afeV for ae4') which is norm-dense in 4'"* • Then we can consider the Hausdorff locally convex topology o- (4', V) on di'. Clearly o-(41, V)< #, ')• Using the methods of 1.8, we can easily show that the mappings x—ax, xa and x* on 4' are T(41, V)-continuous. 1.9.1. Theorem. Let si be a self-adjoint subalgebra of a W*-algebra J! which is 0(4' , V)-dense in J!. Then sly n S is -44 ', A c )-dense in S, where S is the unit sphere of 1 . Proof. It is enough to prove that sinS is /#,,./K* )-dense in S, since the T(4',A)-closure of si S is cr(4',4'* )-c1osed. Since V is norm-dense in J! d n S is uniformly bounded, it suffices to show that si n S is old , V)-dense in S. We can assume that d is uniformly closed. Let aell. Since si is a convex set and si is o-(jll, V)-dense in d is T(4', V)-dense in di Hence there exists a directed set {aa} in d such that the -44', V)-limit of the aa is a. Since 11(1+ a: a„)- 1II„ —

— I al , for some a l . On the other hand, a(1+ a* a) 1 a* aa(1+ aa* aŒ) 1 — a(1+ a* a) 1 a* a(1+ aa* aar = a(1 +a* a) 1 a* (a — a)(1+ aa* aŒ) .

1.9. Kaplansky's Density Theorem

23

Hence Ka(1 + a* a) - a* (a,, —a)(1+

f >I < e

for a > a2 , for some a 2 • Therefore,

1I

Ka(1 + a* a)_ 1 — a(1 + a„)- , f + Ka(1+ a aOE) - — a(1 + a aar 1 ,1> 1 —a(1 + a* 0 -1 a* a„(1 a: acc) - , f >I • Ka(1 +a(2a) 1 —aŒ(1 + acc) , f >I + Ka(1 +a* a) a* a(1 + a„)—a(1+ a* 0 -1 a* a(1 + a,)- , f >I a 3 > al , a2 , ao , for some a 3 . Hence, the limit of 2(a„(1 + a ay') in the o-(41, V)-topology is 2a(1 +a* a) 1 . Since 112(0 + a,,* aOE) - 1 11 < 1 and 2a„(1+ aac) - es/ (even if d does not contain 1), the 4 1/, A)-closure of d n S contains

12a(1+ a* a) - 1 1(2E41. Let u be an extreme point of S. Then

2u(1+u* u) -1 = 2u(1 + p)- = 2u(1/2p-1-(1 —p)), where p=u* u is a projection (cf. the proof of 1.6.4). Hence 2u(1 +13)-1 = up=u, so that the o-(, )-closure of csai n S contains all extreme points of q.e. d. S, and thus coincides with S. Concluding remarks on 1.9. The density theorem of Kaplansky is one of the most useful theorems in the theory of operator algebras. The problem of extending this theorem to more general algebras is quite important. 1. It is easily seen that we can not replace the algebra by a linear subspace. For example, let L"(0, pc) be the commutative W*-algebra of all essentially bounded measurable functions on a measure space 0 with a measure it (cf. 1.18). Let f be a maximal ideal of L"(0„1), and let E be a cr(Lœ ,V)-closed linear subspace of L(0,p) with deficiency one and 10; then L"(0,,u)=J+C1= E +C1, where C is the field of complex numbers. Therefore we can construct a linear isomorphism 0 of f onto E. Let E* be the predual of E, and let E* (resp. 5*) be the dual of E (resp. 5). Since E* is u(E*,E)-dense in E*, 0* (E *) is Of*, 5)dense in 5*, where 0* is the dual of 0. If the unit sphere of 0*(E*) is

1. General Theory

24

.(5*, 5) dense in the unit sphere of 5*, then the unit sphere of S is 0 5 *(E ))compact and so S is the dual of 0*(E*); hence ea for all °cell, then dip diea for all °cell. Hence p> el and q. e. d. so V eoc =ei . Analogously ea =e2 . a

Aa

Let a be an element in a W*-algebra 4'. Let Y = { x ix a= 0, x e 4/}. Then Y is a QV,/ ,d10 - closed left ideal. Hence Y =4'e for a unique projection e in di'. Then 1—e is the least projection of all the projections q in 41 such that qa=a. 1—e is called the left support of a and is denoted by 1(a). Similarly, let 3= fylay= 0, ye4'1. Then 4£) is a o(4',4') -closed right ideal. Hence IA=fdi for a unique projection f in ,ik'. 1—f is the least projection of all the projections p in /4' such that ap=a. 1—f is called the right support of a and is denoted by r(a). If a is self-adjoint, 1(a).--r(a) is called the support of a and is denoted by s(a).

1.10.3. Definition.

1.10.4. Proposition. Let h be a self-adjoint element of a W*-algebra 41, and let C be the W*-subalgebra generated by h. Then the support s(h) belongs to C. Proof. Let p be the identity of C. Then ph = h and therefore s(h)de/(A) + f

(.1.0 —.1.)de1 (.1..), and so A0-0

e((.1 0 1 — x) +)= e(.10) = Ç de' (.1.). e' (.1. 0 — 0). e'(2 0) .

q. e. d.

-

1.12. The Polar Decomposition of Elements of a W*-Algebra 1.12.1. Theorem. Let # be a W*-algebra, and let a be an element of A a can be decomposed as follows: a=ulai, where lai=(a* a)4 and u is a partial isometry of iI such that u*u=s(lal) and uu* =s(la*1). Such a decomposition is unique. This decomposition is called the polar decomposition of a. 1 )4 Proof. Put h(n)= ( a* a + 1 (n a positive integer) and —

1 a(n). a (a* a + —1

n

-4

. ,

1. General Theory

28

1 1 1 )- 4 then a(n)* a(n) =(a* a + — 1) a* a a* a + — 1 = n n

a* a

1 a* a + — 1 n

1 )4 a(n)11 1 and a(n) a* a + — 1 = a. Since h(n)--qa* a)4 (um-

hence

formly), for arbitrary g> 0 there exists an n o such that

Ilh(n)—(a*a)1 < g (n

0) .

By the weak compactness of the unit sphere S of ,/d, there exists an accumulation point b of {a(n)}. Since fa(n)(a* a)+1 c a+gS

(n_n o),

b(a* a)e a + eS.

is arbitrary, a= b(a* a)4 . Let p (resp. q) be the support of (a* a)4 (resp. (aa*)4). Then a= q a, since {(1 — q) a} {(1 — q) (1 — q)a a* (1 — q). (1 — q)a a* = 0. Hence a=qa=qb(a*a)1 . Since

E

a* a = (a* a)+ p b* q b p (a* a)4 and so (a* a)1 (p — p b* q b p) (a* a)4 = 0. 1, we conclude that p=pb*qbp. Define u= qbp. Then u is a partial isometry having the initial projection p. Moreover

Since

a a* = u(a* a)u* and so the final projection is up u* = q. Now suppose a= uial= u'ial is another polar decomposition of a. Then u'* 'al=u'* u 14 Hence (p — u'* u)lal =O. Let

= {xl(p — u'* u)x =0, xe 4}. Then M is a a-closed right ideal. Hence M.= e 4' for some projection e. Hence s(I al).p < e. Therefore (p — u'* u)p= O. On the other hand, pu'* up = u'* u. Hence p = Li* u and so q.e.d. 1.13. Linear Functionals on a W* Algebra -

is 1.13.1. Definition. A positive linear functional (p on a W*-algebra said to be normal if it satisfies (p(1.u.b. x OE ) = 1. u.b. cp(x) for every uniformly bounded increasing directed set tx a} of positive elements in 1.13.2. Theorem. Let (p be a positive linear functional on a W*-algebra Then the following conditions are equivalent. 1. (p is normal 2. cp ic (7(4 ',A)-continuous.

1.13. Linear Functionals on a W*-Algebra

29

Proof. 2.1. was proved (cf. 1.7.4). We prove the implication 1.2. Let {p} be an increasing directed set of projections such that x--cp(x p„)

is 0,71,4'0-continuous. Let p be the I. u. b. of the a projection (cf. 1.13.4). Therefore, 140 (x(P — P )) I

(P(x(P — P)x * )1

Then p is also

— Pa)/ (PM+ 49 (P — POI for Xe S.

(S is the unit sphere of 4'). Hence (p(xp) is a uniform limit of the directed set {(p(xp a,)} on S, so that q)(x p) is also 471, .40-continuous on S, and so it is continuous on N. Therefore, there exists a maximal projection po such that x--(p(x p o) is o-(4', 4)-continuous.

Suppose p(n2 +1)1 Iln w * +x 0 ( 1 —P)11

1.15. Concrete C*-Algebras and W*-Algebras

33

for a sufficiently large number n. Hence, g(x(1 — p))= 0 for xe,il. Similarly, g (x) = g(q x) for xe,4'. Hence, g(x) = g(x p) = g(x w w*) = R iog (x w). Hence g = R(p, where (p = Rg >-__ 0 and 11911 =11 R..911= 11(P11. Finally, we show the uniqueness of the decomposition. Let g = Rp = 12,„, (p' . Let w' w'* =;5, w'* w' =. Then g (x) = (p (x w)= (p' (x w'), and (p(x)= cp(x q) = (p(x w* w) = (p' (x w* w') = (p' ( x w* w'). Hence cp(1 - -4)=-- 0, and so q= s((p) -_ . V . Similarly 7q= s((p') q, and so q= . V . Put w'*w=-- h+ ik (h,k self adjoint). Then h,k e qjh'q since wi* we q,Wq. cp(w'* w) = (p' (w'* w') = (p' (q) = 1 = (p(h)+ i cp(k). Hence, cp(h) = 1 . Since I h I < 1, h = q, and since IIW* w I 0, there exists an integer n o such that

(am—an)2=

(rn,n

a

aci ), we have

For an arbitrary finite subset

E (am — ang.,11 2 <E (m,n_no).

i= 1

Hence

lim i=1 E I (ani an) m-.00 —

Therefore, we have

1 2 = E (ao an)«, 1

2

—

(n> no).

i= 1

E I (ao — an) acll

E

0), and so abeH and

an --a0 in H. Therefore H is a Hilbert space. ; hence a =-0 for all For aeH, a ccll 2
0. Then, i(n (p(x) ,l'1) — (n (p(x) a4,, b n(p)i -- i(n(p(xW — a.0, 11)I +

__ 11x1111--cin v Il 11n11 + 11x11 Ilarupll Iln—bri v 11-03

(n —* cc).

Hence, f (x) is a uniform limit of sequences {f,i(x)} on the unit sphere of j#, where f n(x)=- (ir,p (x) an ,„ kw). cp(b „* x an). f e A, since f n e A. Hence, the mapping x-œng,(x) of 4', with the 44',A)-topology, into B(Yf (p) with the u(B(Y(),B(Yf) *)-topology, is continuous on bounded spheres, and so it is continuous on 41 (i.e., {n,„ Yf v } is a W*-representation of 4' (say fnvw, Yf cp1)). 1.16.7. Theorem. Let ' be a W*-algebra. Then, it has a faithful W*-representation {n,/f} (i.e., n(a)=0 if and only if a=0). Therefore, is *-isomorphic to a weakly closed self-adjoint subalgebra of B($() on some Hilbert space Ye.

1. General Theory

42

Proof. Let gn, be the set of all normal states on 4', and consider the W*-representations [{7, Yf v } I cpe Y'n] of di. Set Yf =

and 7r(x)=

E

e

no,(x) (xe 4'). Let F be the set of all finite linear

qE

combinations of elements of U Yfv . F is dense in Yf. Let

=

E

Ec

and

i=1

ve.9'm

i = 1, 2, ... , n). Then f (x)=--(7r(x),n).

E

is f edif* , and so (7c(x)' ,,f)e 4 1* (' ,q' e ff). Therefore,Henc, {m, a W*-representation of di (say Suppose n(a)=O. Then n4,(a)=0 for all (pe,99 . So q(a)=O for all (peg,,',. Hence, a= 0 (1.7). q. e. d. Concluding remarks on 1.16. Theorem 1.16.6 is due to Gelfand-Naimark [55]. They defined a C*-algebra al as a Banach *-algebra satisfying the following conditions: 1. Ilx* xll = IlxII 2 and 2. 1 +x*x has an inverse (xe 41). Then they proved Theorem 1.16.6. They further asked whether axiom 2. was superfluous. This question was solved by Fukamiya, Kelley-Vaught and Kaplansky (cf. Concluding remarks on 1.4). Also they asked whether the axiom 1. could be replaced by lix* xll = Ilx* IIIIx. This question was solved by Glimm and Kadison [61] and Ono [136]. References. [55], [149], [154]. }

fe,J21).

1.17. The Second Dual of a C* Algebra -

Let si be a C*-algebra, d* the dual Banach space of si and let d** be the second dual of si. Let F be the set of all positive linear functionals ço on si such that 1. Then F is u(si*,d)-compact. Let C(F) be the Banach space of all continuous complex valued functions on the compact space F with the usual supremum norm. For ae sl with a*= a, let A be the commutative C*-subalgebra generated by a. Then there exists a character x on A such that x(a)=11all or — Let be an extension of x on si such that I l xii = 11211. 2 is a state. Hence, sup l(p(a)1=liall. Therefore, the self-adjoint portion sis of a/ may (per'

be isometrically embedded into the real Banach space Cr (F) of all continuous real-valued functions on F. Hence, si may be topologically embedded into C(F). 1.17.1. Proposition. Let g be a bounded linear functional on si. Then g is a finite linear combination of states.

1.17. The Second Dual of a C*-Algebra

43

Proof. Consider si c C(F). g can be extended to a bounded linear functional 4 on C(F). Then by Riesz's theorem, there exists a complex Radon measure p, on F such that g(a) = f (p(a)d,u((p)

(ae .4) .

F

Let 4u=p i --,u2 + ip 3 — iy 4 , where pi (i= 1,2,3,4) are positive Radon measures on F. Then g i (a) = f p(a)dm(p) for ae al (i= 1,2,3,4) are F

bounded positive linear functionals on si.

q.e. d.

1.17.2. Theorem. Let siv** be the second dual of the C*-algebra si. Then sl** is a W*-algebra in a natural manner. Moreover, sal is a C*-subalgebra of at" when it is canonically embedded into si. Proof. Let {U, K } be the universal *-representation of st Then the mapping x--U(x) of si onto U(si) is an isometric *-isomorphism. We shall identify si with U(sI). Let U(d) be the weak closure of si. U(d) is a W*-algebra. Let U(si) * be the predual of U(si). For feU(91) * ,

I sup I f (x)I 1 f I = sup 11x111 f (x)I = 11x11.1 xesi xEv(s4) by Kaplansky's density theorem. Therefore, the mapping fœq' id is isometric, where f Id is the restriction of f to d. Since {fIdIfe U(si)} contains all states on si, by 1.17.1, it is d*. Hence si* = U(si) * and (U(d) * )* = U(d). So U(d) is the second q.e. d. dual of a/.

1.17.3. Corollary. Let si be a C*-algebra and let 5 be a uniformly closed two-sided ideal in d. Then 5 is self-adjoint and the quotient algebra si If is also a C*-algebra. Proof. Letr ° be the bipolar off in si'. Theni" is the cr(._21**, si*)closure of f in si**. Hence, it is a u(s1**,s1*)-closed ideal of d**. Therefore, there exists a central projection z in si** such that I" = al** z. Since 5J00 n si = d**z n si, 5 is self-adjoint. .saf I5,'z:,' .91 + Af** z1.4** zz-', .91(1 — z).

q.e.d.

1.17.4. Corollary. Let si, .4 be C*-algebras and let 0 be a *-homomorphism of si into M. Then the image 0(21) of si is a C*-subalgebra of M. Therefore, the image of a *-representation of a C*-algebra is also a C*-algebra. Proof. Let f = {x10(x). 0, xe Jai } . Then 0 induces a *-isomorphism 0' of the C*-algebra si/f into M. Hence the image is closed, since 0' is isometric (cf. 1.2.6). q.e. d.

1. General Theory

44

1.17.5. Corollary. Let al be a C*-algebra, .1 a C*-subalgebra of d, and letfbe a uniformly closed two-sided ideal of st Then .4 +.54 is also a C*subalgebra of st

Proof. Consider the quotient C*-algebra s2115 and the canonical homomorphism 0 of d onto d/it. Then 0(.4)=.1+5 is uniformly closed. q. e. d. 1.17.6. Proposition. Let si be a C*-algebra, .4 a C*-subalgebra of sd, and c: . Put let Y be a uniformly closed left ideal satisfying inf x + y 1114 =-- yeY

(xe,1).

Then 11141= inf Ilx+ a+Y I or= a n „I as Banach weY spaces. Moreover .1 n .4 is a C*-algebra since Y n a is a two-sided ideal in .4. Proof. Consider d** and the bipolars, 400 and 2'°°, 2'°°, in d* * . 2200 is a cf(d**,d*)-closed left ideal. Hence, there exists a projection e in U./ where C is the field d** such that 2' °° =.4**e. Let M = 030 ± .4-1 of complex numbers and 1 is the identity of a**. Since YacY, y oo 400 27 00 and so y°°m„27 00. is a cf(d**,41 *)-closed selfadjoint algebra of d**. Let u be any unitary in M. Then oyoou c 20= u * u y00 u* U U * y00 u. ,

Hence, u*27 00 u.27 00 , and so u* eu=e for every unitary ueM. Hence, e commutes with M. Clearly the polar c.r ° of 2' is R (1 _ e) d*. So Illx111 = inf Ilx+YII = sup If(x)1 = sup f (x(1 — e))I = Ilx( 1 — e) ye.,r

f,290

fed*

11f11= 1

11f11= 1

In particular, the mapping w(1 — e) (w e) is a *-homomorphism, since (1 — e)commutes with M. Clearly inf Ix +y _inf., x +y ye-)or).z. yeY On the other hand, if wEM, w(1—e)-- 0, then wed** e. Hence, we IL' n a. Therefore, there exists a *-isomorphism of M/Mn into q. e. d. M (1 — e). So 11 w( 1 — e)11 = inf x + Y ye..r Now let 4' be a W*-algebra, if the dual of dl and let dl** be the second dual of 49. Let d4 be the predual of 4' and let be the polar of A: in 4'**. Then 41: is a cr(4**,j11*)-closed two-sided ideal, for A, is invariant under L a,R a (ae Let z be the central projection of di** such that 4=-41 ** z. Then e (4'0* =dl. Hence di" 4'**zaW**(1—z). Let aedi**. Then a there exists an element be ' such that al41* =-- b, and so a—be4=4'**z. Therefore ll**=4'+4'**z=dl**(1—z)+**z and j#*=--Rzj11*()R1_„/N*--Rzd/*041*. (XEsi).

,

1.18. Commutative W*-Algebras

45

1.17.7. Proposition. Let 41 be a W*-algebra, dl. predual of 41, and let d1* be the dual of J1. Then there exists a linear mapping R ( , _z) (z a central projection of J/**) of di* onto 4'* satisfying the following conditions: 1. R=R l _ z ; (fe jW * ); fI (fe,W*); if f 3. Ri f 4. ./#* is a closed subspace of 4'* invariant under L a, R a (a,be.W**). 2.

-

1.17.8. Proposition. Let 4' be a W*-algebra and let ./K** be the second dual of jll. Then there exists a W*-homomorphism of 4'** onto ,./1. Proof. di** =-- di**(1 — z)+ 41 ** z = +di** z and

z). Consider the mapping 0: x--x(1 —z) of 4' onto dl**(1 — z). Then 0 is a W*-isomorphism of 4' onto ,idl**(1—z). Let 01 (y).-- 0 -1 (y(1—z)) (ye,W**). Then clearly 01 is a W*-homomorphism of /.N** onto dl. q.e.d. di(1 — z). di**(1

—

Remark. A positive linear functional belonging to R„/&* is said to be singular. Then, it is known that a positive linear functional cp on di is singular if and only if for any non-zero projection pedl there exists a non-zero projection q in 4' such that cp(q). 0 (cf. [203]). Concluding remarks on 1.17. Theorem 1.17.2 is due to Sherman [188]. A complete proof was given by Takeda [200] and Grothendieck [66]. References. [66], [150], [181], [188], [200]. 1.18. Commutative W*-Algebras Let (0„ u) be a measure space with 12(0)< cc, let L"(0,12) be the C*-algebra of all essentially bounded ,u-measurable functions on Q, and let Ll (Q„ u) be the Banach space of all tt-integrable functions on Q. Then, by the Radon-Nikodym theorem, V(Q„u)*=-- U(0,12). Hence (Q„ u) is a commutative W*-algebra. More generally, let (T, y) be a localizable measure space (i.e., a direct sum of finite measure spaces (cf. [182])), and let if (F,y) be the C*-algebra of all essentially bounded locally y-measurable functions on Q, and let 12(F,v) be the Banach space of all y-integrable functions on F. Then (T, v )* = L"(F,y). Hence, I ( F, y) is a commutative W*-algebra. 1.18.1. Proposition. Let dl be a commutative W*-algebra. Then di is *-isomorphic to a W*-algebra Lœ(F,y) on some localizable measure space (F,y).

1. General Theory

46

Let Q be the spectrum space of 4'. Then jll = C(0). Let (p be a normal state on dl. By Riesz' representation theorem, cp defines a unique positive Radon measure ,u,p on Q such that cp(a)-- a(t)d,uv (t) (ae Proof.

Let

a — *a(t) be the mapping of 41 into the W*-algebra 12 )(Q„ ucp). is a *-homomorphism. Moreover, for be C(Q), a(t)b(t)d p(t)=. (p(ab) f2

and C(Q) is dense in L1 (Q ,pt,p). Therefore, Ov is a W*-homomorphism. Hence 0(4') is a W*-subalgebra of Loe(Q,,u,p), and so 4)(,a)=L` (52„u,p ), since &,( W ) C(Q) in /2° (Q„uv). = dlz for some projection z Let 5,0 be the kernel of Ov • Then in dl. On the other hand, cp(z)= z(t)dg q,(t)=0. Hence, z 1 — s((p). f2

The converse is also true. Therefore, z =1 —s((p). defines a *-isomorphism of s((p) onto If (Q„ uv). Let { (p a maximal family of normal states on ,W such that }be s(p 1 ) * s(49« 2 )= O (al a2 e fi, al Then,

À' =

(D

s((p„)= E

aell

(SCli v.)

Loe(s(cp c,), /Iv.)

L"(Us(u v„), a

2E11

E e Pa)

asp

is the support of pv.). Therefore, 41 is *-isomorphic to if(

U so-tv.), 2211

2E11

E $ py.).

on the localizable measure space ( 221

2611

Let K be a Stonean space. A positive Radon measure /2 on K is called normal if for every uniformly bounded increasing directed set u. b..1, {Lc} of continuous real valued functions on Q, 1. u. b. It( f at) = j fpfilfleo of normal hyper-Stonean if K has a faithful family K is called measures (i.e. for f 0)e C(K), j(f)= O for all 13e,..11 imply f = 0). Then, C(K) is a W*-algebra if and only if K is hyper-Stonean (cf. [31]). Reference. [182]. Remark.

1.19. The C*-Algebra CVO of all Compact Linear Operators on a Hilbert Space 1'

Let Yt° be a Hilbert space, C(Yf) the C*-algebra of all compact linear operators on lf , T(Yf) the Banach space of all trace-class operators, B ( r) the W*-algebra of all bounded linear operators on Yf. TC_Yel can be identified with the predual B(49)* of 13(A ) (cf. 1.15.3). For xeC(1() and a e T(4 ) , let O(x)= Tr(x a). Then Oa is a bounded linear functional on CVO.

1.19. The C*-Algebra C(") of all Compact Linear Operators on a Hilbert Spacer 47

1.19.1. Proposition. The mapping a--11Ja of T(Yf) into CVO* is an isometric linear mapping of T(Yt) onto CVO*. Therefore, under the mapping a--11Ja, T(Y() can be identified with C(Yf)*. Hence, we have C(Ye )* =T(i() and T(Ye )* =B(Yf).

Proof. Consider the second dual CVO** of C(rf). Let 1 be the identity of C(A 9)**. Let (pa) be a uniformly bounded increasing directed set of all projections in C(Yt ). Let p=1.u.b.pa in C(A )**. Then the , s(C(/f)**, CCYfr)-limit of the p„ is p. For aeC(if), IIPaaPa— all --0, since a is a compact operator. Hence C(Yt)cpC(Yelp and so p=1. Let (p be a state on C(Al. Then lim(p(p,,)=(p(1)=1. Hence, there

exists a sequence of indices (GO such that cp(1 —p„n)< 1/n. Then sup i(p(x)—(p(p a,n xpc,n)! iixii i accon sup I (f) 01 11x 11

—

p OE„Pc p ,„) + (p (19 Œri x (1 p„„)) + 0(l p .„) x(1 —p)1 —

—

xeC(0)

*- 3 0 1 — Pi — * 0 (n-- co).

Hence, lim li(p — LR p (1)11.0. Since L p.. R p.n (p is zero on ( 1 — Pan)C(Y0Pa„+Pan C(Y0( 1— Par,)+(1— Pa,)C(A )( 1— Pa„), Lia.RpŒn P can be considered as a positive linear functional on pan C(Yf)p an . Since pan C(Ye)p an is finite-dimensional, there exists a positive element pan anpan of pan C(Jr)pan which satisfies (p(pan xp an). Tr(pan anpanpan xp an)=- Tr(pan anpan x).

Moreover, IlLp„„Rp.„ (p — Lp,„.Rp.,.(P11 =- 11P0,„anPan — Parn amPam Ili .

Therefore {p o,n anpŒn} is a Cauchy sequence in T(r). So there exists a trace class operator ao such that limp(p an x p an)= cp(x)= Tr(ao x) for n

q. e. d. xeC(Jr), and IIPII = Ilaolli= Tr(iao l). Concluding remarks on 1.19. The problem of extending Theorem 1.19.1 to more general Banach spaces is interesting. Grothendieck [65] has solved this problem in some special cases. In general, the second dual of a Banach algebra is again a Banach algebra (cf. [7]). Hence, C(E)** (E, a Banach space) is a Banach algebra. References. [30], [174].

1. General Theory

48

1.20. The Commutation Theorem of von Neumann Let Yt9 be a Hilbert space, B(Yt) the set of all bounded linear operators, and let Y be a subset of B(Y(). We denote by the set of elements of B(/f) commuting with all elements of 2', and we call or' the commutant of Put (Y)'=.29" (the bicommutant of Y), ( 2" )' = , . It is clear that 2" is a subalgebra of B(Jr) containing the identity, Y" , and c:2''. Hence, Y' .291 c 2 '2 implies D 2' . Therefore 2")'=2". On the other hand, c(Y')" =Y" . Hence, 2" = 2" Y = Y (5) = • • • and Y" = Y (4) • . If Y is a self-adjoint set, 2" is a self-adjoint subalgebra. Let Yfl and Yta2 be two Hilbert spaces and let Y4 Yea2 be the algebraic tensor product of ,Y4 and A°2 . Then there exists a unique pre-Hilbert space structure on Y4 Yt°2 satisfying , 172 e ,Y(2 ). 2, 1710n2)=-, (1, 171 )( 2, 172) (1, 111eYti. and The Hilbert space (denoted by Y4 0,1(2 ) obtained by the completion of Yei /ta2 is called the tensor product of Y4 and Ye2 . Let ai ,bi eB(Y4) and a2 ,b2 eB(Y( 2). The algebraic tensor product 0 a2 defines a unique continuous linear operator on Y6 0 Yt92 such that (a1 0 a2) Q = a 0 a2 2 al a2 is bilinear in al and a2 , (a1 b1 ) ®(a2 b2) = (a1 a2)(b1 Q b 2 ), and (a1 a2 )* = a Q a. Let (n o,)„, ll be a complete orthonormal system of el(2 . The mapping 1 10 pia, is an isometry uo, of Y4 onto a closed subspace Yt" of The ft le"l aceli are mutually orthogonal, and Y4 0Ye2 = e u: is a linear mapping of Y4 492 onto Y4 such that Y(2

e Jr) = 0,

is the ortho-complement of ,Yr in Aol oyt°2 . It where .j Yt°2 e is also an isometry on Yr, and ua u: is the projection ea of Yfl 0 iii°2 onto 0". Let a e Bpri y€02 ). Then u: a up e B(Y4). Let aap = u: aup . a is perfectly determined by the matrix (aap). In fact, if aap = bap for a, 13611, u: aup =-4bup , and so ea aep = ea bep . Hence a=b. Moreover (.1 a)ap = aap (A a complex number), (a + Map = aap + bap , (e) p =(apa)* and (ab)api = u:(ab)upi = u:a( E uy u,1)bupi = aay b yell for e .*'. (a1 0 1 02)0 =

0 1 02)ufl =

(al 0 1 02)

no

rip) = 6 .fiai (ai eB(,Y4), e Y4, 1 0i (resp. 1 02) is the identity of B(Y4) (resp. B(A 92)). Hence (a1 0 1 02)0 = 5ap a1 . =

1.20.1. Lemma. If ae B(Yf l 01(2) commutes with ua u; (a, /3E4 a is of the form a l 0 1 02 (al e 40'0).

1.20. The Commutation Theorem of von Neumann

49

Proof. ao =u: aup =uy* uy u: aup = uy* auy u: up . Since u: up = 0 for oc013 and u Œ = l„ ao = 0 for oc0 )6' and a=uy* auy for yell. Hence ao = „p ai with al e B(Y4). q. e. d.

1.20.2. Lemma. For a subset D of B(A° 1 ) containing 0, let A) be the set of all elements a of BCYf 1 0 /f'2 ) such that aoeD. Then (D 1,02)' = iy and (DØ 1 02)" = D" 0102 . Moreover, if D contains 1, (A ))/ = D' 0 1/(2 and (d ID)" = Proof. Let al 0 1 02 (a 1 e D) and be (D 0 1 02 ) . Then

{(a 1 0 1 2)b} = a1 b = {b(a 0 1 02)} =b al • Hence (D® 1 0 2)' c j#D , • The converse is clear. Hence, (D® 1 0 2)' = Ay. Further, (uy ulc) 1 = u ay /4 u1/ =0 (ocOy or /306) or =1 (a =y and )6=0). Therefore, ti y 14 ; 4 e . Hence (D® 1 02 )" = (Ay)' B()®1 2 , and (D® 1x2) (D' 01 02 ) = / 11) ,, , so that (D 1 x 2) = D" 01 02 . Finally, suppose 1 01 e D . Then u y 4e,71/D , (4'D)' =B(A)O 1„02 , and A)D DO 10 2 . Hence, (.4'D)' =D'01 02 ; (WD)" = (Df 102r = q. e. d.

1.20.3. Theorem. Let 4' be a self-adjoint subalgebra of B(Yti on some Hilbert space and suppose that jll contains the identity of B(Ye). Then the following conditions are equivalent. 1. di is weakly closed;

2. dI"=dI. Proof. Clearly, 2. implies 1. Now suppose that ,// is weakly closed and dl dr . Then there exists a linear functional f on 4' continuous with respect to the weak operator topology, such that f(//)=0 and f(a)0 0 for some ae dr. Let f (x) =

(x

",,

(xe

E

(i = 1,2, ..., n)). Let

(n Ii= 1, 2, ..., n) be a complete orthonormal system in a n-dimensional Hilbert space Put Ye i = Y(0 ni , and let u • be an isometry Yf onto

0 th (e.Yt'). Then

Yf i such that f(x) =

E (x

= E (x ut ui

=

ui

=

= E (ui xut u1 ui )=((x 0 1 01 )

where

=

E

i=1 and Oa® 1 01 g,

(x

E ui e,Ye(1),Yf i . Therefore, 041 0 1 0i ) i=i O. Let g be the closed subspace of ,lf

=

0

50

1. General Theory

generated by the set {(4' 0 1) and let e be the projection of YfOXi onto A'. Since ,T is invariant under the *-algebra ./41 0 1, e belongs to (410 On the other hand, .K' ® 1 =(#® 1 0)", and 4'0 1 0, contains the identity. Therefore, e(a 101 ) = (a 1,r1 )e = (a 1.yei ) Hence, (a ® . Therefore, there exists a sequence (an) of di such that II(an ® 1,r1 ) — (a ® 1 .y(1 ) I —* 0 (n co). Hence, ((#® 1,ri g, implies ((a® q. e. d. = 0, a contradiction. 1.20.4. Corollary. Let JP and X be two weakly closed self-adjoint subalgebras of B(/f) containing the identity 1 0 . Then R(41,./1()=(//t where MAX) is the weakly closed self-adjoint subalgebra of B(0) generated by and X. 1.20.5. Proposition. Let X be a weakly closed self-adjoint algebra containing 10 on a Hilbert space Yt, =iVn X', tii (i,j = 1, 2, ..., n) elements in X, t; .i (i,j= 1, 2, ..., n) elements in X'. Then the following conditions are equivalent:

1. E tik t;ci = 0 (i,J= 1, 2, ..., n);

k=1 2. there exist elements z u (i,j =1, 2, ..., n) in le such that

E

tik zki =0,

k=1

E z ik ek; =(, (i,J=1, 2, ..., n). k=1

Proof. It is easy to prove that 2. 1. We shall prove 1.=2. Suppose

E tik t;,i =0 (i,J= 1, 2, ..., n). Let X. be an

n-dimensional Hilbert space.

k= 1

Let t = (tii)i,i= 1 ,2 , nE '117. B(X) and t' , (0i = 1,2,...,n 6 Then t t' = O. Put f'= fx' It x' = 0, x/e./1(' ® B(1)1. Then f' is a a-closed right ideal of ,Af'0 B(1). Hence there exists a projection z'=(zii) 1 ,...,„ such that 5' = z/(4( 1 ® B(X)). Since (X' 1)f' f', z u e and so

zii e,A(n

(i,j=1, 2, ..., n).

Since

tz'= 0,

E tik z ki =0 k=1

(0=

1, 2, ..., n). Since z' t' =t',

E zik eki =t;; (i,;= 1, 2, ..., n).

q.e. d.

k= 1

1.21. *-Representations of C*-Algebras, 1 1.21.1. Definition. Let {n,'} be a *-representation of a C*-algebra {n,} is said to be cyclic if there exists an element in ,rf such that [n(d) ]. ([7r(d) ] is the closed subspace of Yt generated by is called a cyclic vector of {it,*'}. {n(a)laesd}). Such an element An element ri in A' is called a separating vector of fn,Yel, if

1.21. *-Representations of C*-Algebras, 1

51

1.21.2. Definition. Let fn i ,Yf i l, {7r 2 ,'2 } be two *-representations of a C*-algebra d. {n i ,Yfi } is said to be equivalent to In2 ,Y(2 1 if there exists an unitary operator U of Yfi onto Y(2 such that Tc 2 (a)U =U ni (a) for ae,s21. We shall identify equivalent *-representations.

1.21.3. Definition. Let {n„,,Yfa} a be a family of *-representations of a C*-algebra d. The sum of the {n„,,Y4,1, {n,Y(} = E or„,, is a aelf

*-representation ITC, Yi° 1 of d defined as follows: Yt = n(a) = E fir(a) (ae d).

E 1CD ,Yfc,

and

fl

1.21.4. Definition. Let {it, .'} be a *-representation of d and let E' be a projection in 4524. Consider the *-representation {n 1 ,16} defined as follows.Y ( 1 =E' Yf and n1 (a)=7r(a)E' (aed). This representation, denoted by Or E', E' /0, is said to be a sub*-representation of Or, Yfl. We define analogously a cyclic W*-representation, the equivalence, a sum and a subrepresentation for W*-representations of a W*-algebra.

1.21.5. Proposition. Let {n,,YP} be a *-representation of a C*-algebra d.

(xed). Suppose that {7E,Y1 has a cyclic vector and put (p(x)=(n(x) (p is a bounded positive linear functional on cd, and fir, Yfl is equivalent to Proof. It is clear that cps is bounded positive. Consider the mapping 7r(a)--a (aed). Iln(a)a2 = (P(a* = a 11 2 . Hence, this mapping can be uniquely extended to a unitary operator U of Yt onto Yfq,. Further, 7r,p (b)U 7r(a) =(b a)9 = U n(b)Tc(a). Hence U 7r(b)= n,p (b)U (bed). q.e.d.

1.21.6. Proposition. Let (f) be a non-zero bounded positive linear functional on a C*-algebra si. The *-representation {n9 ,Yf q,} of d is cyclic. Proof. Let In,pw,Y(9 1 be the W*-representation of d** obtained from {n,Yf q,I. {7c,pw,Ye°,} is equivalent to the *-representation {7r,,,,Yt°9 } of d** constructed via the cy(d**,d*)-continuous positive linear functional cp. Hence 1,0 e It°,0 and [7 rq,(21 ) 1]= rcp (d * *) 1 q ]=Ye So 1,0 is a cyclic vector. q. e. d.

1.21.7. Proposition. Let {7r,} be a *-representation of a C*-algebra si =0 (all aed ) imply = 0 ( called a nowhere trivialsuchtai() *-representation). Then {ir, ,Y(} is equivalent to a sum of cyclic *-representations, constructed via bounded positive linear functionals. Proof. Let gOELED be a maximal family of non-zero vectors in Yt such that [n(d) c,] is orthogonal to [n(d) fl ] (oc,flen, cc0,6). Then E [n(d)M= Ye In fact, suppose there exists a non-zero vector

1. General Theory

52

in A' such that is orthogonal to [n(si),i] for all °cell. Then (7r(s1), 44)0= (, n(d) 0 = 0, and [n(d) fl 0 (0). This contradicts the maximality gal. Let E'„ be the orthogonal projection of onto [n(d) OE]. n(a)E',, Ye c [n(d) a ]. Hence, E'Œ n(a)E'ac = n(a)E'„ and n(a)E'„= E'„n(a) (if a* = a), and so n(a)E= E'n(a) (aeJai). Therefore, fnE',,E'Œ Yel is a sub*-representation of { ir, Ye}, and { n, Yf } =

E InE,E'„Yel .

q. e. d.

aEll

1.21.8. Definition. Let { n, Ye} be a *-representation of a C*-algebra si. {n,Ye} is said to be irreducible if every non-zero element in ,re is a cyclic vector of In, Yta 1.

1.21.9. Proposition. Let {it,} be a *-representation of the C*-algebra d. The following conditions are equivalent: 1. In, Al is irreducible; 2. n(d)=B ( r), where n(d) is the weak closure of n(d) in B(ff).

Proof. Suppose In,Yel is irreducible. Then n(d) contains l ye,. Consider 74.91)', a W*-algebra. If 7r(d)' contains a non-trivial projection E', then it(d)E'' g E' Ye 0 Ye, a contradiction. Hence n(d)' = (C1,r) (C, the field of complex numbers). Hence 74,4)=45aq =B(Ye). Conversely, suppose there exists a non-zero element 0 in Yf such that [n(si) ] A°, and let E' be the orthogonal projection of Yt° onto [74,s:4/g 0]. Then E' is in n(d)'. Hence, n(d)'0 (C 1 0) and so q. e. d. n(s1)" 13(Ye).

1.21.10. Theorem. Let (p be a state on the C*-algebra d. Then the following conditions are equivalent: 1. In,r,p 1 is irreducible; 2. (p is extreme in the state space Y of d. Proof. One may assume that si has an identity. Suppose that {n„,Yeo } is irreducible and (p=((pi + (p 2 )/2 ((p i , (p 2 e g). Since B(a49,b,p).(p 1 (b* a) defines a continuous positive definite conjugate bilinear functional on Yt(p . Hence, there exists a bounded positive linear operator H on /tag, such that B(aw bv). (H a,,,, b (p) (a, bed). One easily sees that H e n(sl)'. Hence, H=.1. 1, and so (p i. = (p 2 = (p. Conversely, suppose that 7r,p(d)' 0(C l ye). Take a non-trivial .(n`P(a)P' 149,1v) and projection P' in n(p(sai)', and set (p1(a) Ai (P2(a) =

(n (a)(1 0 —P')1 ,1 ) v

i'2

tP (P

Wed, Ai =03/ 1 r,, 1,0,

53

1.21. *-Representation of C*-Algebras, 1

and 2.2 =((1 0Q — P')1,„1,p )). Then cp 1 ,(p 2 eg, and A i cp

22 (p 2 = cp, and

Assume that cp=cpi . Then itcp(a)P'

1 , 1 V Ai

P'

=

V Ai

ir(a)1

=

P'

a

=

V Ai

P' Hence is an isometry on Yfv . This is a contradiction, so VAi Similarly, (p2 0 cp. q. e. d.

1.21.11. Definition. An extreme state on a C*-algebra is called a pure state. 1.21.12. Theorem. Let d be a C*-algebra. Then d has a faithful family of irreducible *-representations tnY4,1 (i.e., n(x)=O for all a a imply x = 0) .

a

Proof. Let be the set of all positive linear functionals cp on d such that II (PII < 1 . Then is a o- (d*,d)-compact convex set. Let Ro be the set of all extreme points in then q)=0 or I (P I = 1 ((pea 0). If (p00, cp is a pure state on d. Consider all irreducible *-representations {n,,,,Y4,,I vER0 ,,p 0 of d. Then nv (a) ,- 0 for all cp ( 0)e R o implies a=O. q. e. d.

a

a;

1.21.13. Proposition. Let d be a C*-algebra, and let 0 be a bounded linear mapping of d into the W*-algebra 13(rf). Then 0 can be uniquely extended to a continuous linear mapping 40 of d**, with the topology o- (21**,d*), into B(10, with the topology cr(B(Yel,B(YO *). In particular, let {m, } be a *-representation of d. Then, the extended mapping, denoted by Al, is a W*-representation {e,Y(} of d**. Conversely, let {nw,Y(} be a W*-representation of V** on the Hilbert space A'. The restriction of TC on d is a *-representation { 7r, Yf } of d, and its unique extension Vit, Yfl to the W*-representation of d** coincides with {it'', .$"}. Hence, there exists a one-to-one correspondence between *-representations of d and W*-representations of d**. Proof. Let 0* (resp. 0**) be the dual (resp. the second dual) of 0. 0** is a continuous linear mapping of V**, with the a(d**,sal*)topology, into B ( (f)**, with the o-(13(1()**, B(r)topology. By 1.17.8, B(Yf)** = WO+ B(Ye)** z, where WO** z= BGY(P* in B (. r)**, and WO** z rill(Yt)=(0). The canonical projection Wi of B(lf)** onto B(Yt) is a W*-homomorphism. Now set "(x). W 0**(x) (xed**). Then 413 is a continuous mapping of d**, with the o ( d**, d*)topology, into WO, with the o-(B(Jr), B())-topology.

54

1. General Theory

Moreover, .io(a) , W1 0** (a). W 1 0 (a) = 0(a) (a e .4). Hence, '0 is an extension of 0. By the continuity of '0' and the density of d, this extension is unique. Next, suppose 0 is a *-representation. For x, ye d**, there exist directed sets {x Œ }, fyfi l in d such that x,--x, y p -œy in the o-(d**, d*)topology. Hence, 4; (x„y) = lip. 0(x - ac y fl )=1ifin0(x„yfl )=1ip0(xat)0(y 13)=0(x„)4i;(y) and

'0(x y) = lim:i(xy) =litn0(x)4i(y)=4;(x)4 ^ i(y). ^ Analogously we « « have 4;(x*). io(x)*. Hence 43 is a W*-representation of d**. The remainder of the proof is clear, q. e. d. 1.21.14. Definition. Let Inw,Y1 be a W*-representation of a W*algebra 4'. The kernel 5 = lain(a)=0, ae 41 is a o-(11,A)-closed twosided ideal. Hence there exists a unique central projection z in 4' such that 5 = 4' z. Put 1— z=s(n). s(n) is called the support of { e,} The restriction of it to 4 ' s(n) is one-to-one. 1.21.15. Definition. Let {nlv, Yfi }, {702v, y(2 be two W*-representations of 41. If s(n i )=s(n2 ), we say that feiv, 491 } is quasi-equivalent to {71, y(2 . }

}

Clearly the quasi-equivalence is an equivalence relation, so that by this relation we can classify W*-representations of 4' into quasiequivalence classes. Let g (4') be the family of all quasi-equivalence classes of the W*representations of'. Then, to each element p e g (W), there corresponds a unique non-zero central projection c(p) of 4' such that c(p)=s(n) ({ 7r, Y}EP). Conversely, let z be a non-zero central projection of 4'. Then 4' z is a W*-algebra, so that it has a faithful W*-representation {it,'}. Then the mapping a-œn(az) of 4' into WO is a W*-representation of 4'. Hence, there exists a one-to-one correspondence between all elements of .g(4) and all non-zero central projections of 4'. For convenience, we shall add an imaginary 0-element to g (4), and we shall denote by g'(4') the set g(41)u (0). There is a one-to-one correspondence between g'(41) and the set YP of all central projections of di'. Since l''' P is a complete Boolean algebra, we can canonically introduce its Boolean algebra structure into g'(//1). Also, we can regard every element p of g(dt) as a W*-homomorphism of 4' onto 4' c(p). Therefore, within the quasi-equivalence, the W*-representation theory of 4' can be completely reduced to the structure theory (ideal theory) of the W*-algebra .11. Now let d be a C*-algebra, {7r,,Y(} a *-representation of d. Then there corresponds a unique W*-representation lit, Y(} of d**. Using

1.21. *-Representations of C*-Algebras, 1

55

we define the quasi-equivalence and the support of fg,Ifl (i. e., s(ic). s(71)). 1.21.16. Theorem. Let d be a C*-algebra, Ja? the C*-subalgebra of d** generated by d and 1, and let fn i ,/fi l (i=1, 2, ..., n) be a finite family of mutually inequivalent irreducible *-representations of d 1. Let Ti eB(/( 1 ), Tn eB(X), and let E 1 ,E2 ,...,En be finitedimensional projections of B(Ye i ), B(Y( 2), B ( 14,), and e> 0. Then there exists an element aed such that En(a)=ET, g(a)E=TE (1<j_n) and 114 — c< max (IIT 11). j5n

EB( 1 ), Te B() be self-adjoint elements, and E1 , ..., En finite dimensional projections of B(Y( 1 ), B(Y6), and let e > O. Then there exists a self-adjoint element h of si such that Ei gi(h)= ni(h)Ei = TE (I< 0 for xed 0 93. A C*-norm a on d 0 93 is defined as a norm satisfying oc(x* x)= cx(x) 2 and oc(x y)oc(x)oc(y) (x,y e 0 93). 1.22.1. Proposition. Let (p and Jj be positive elements of d* and 93* respectively. Then (p 0 is a positive linear functional on d 0 0. Proof. Let x =

bi ed

93. Then

E cp(at (bt i= 1 For any family of complex numbers l; \-1,-; 2, • • • /ln), (f) tp(x* x) =

E

(bt b

= tp (( E) b

i,j=1

*

n

(E

/li b))

> o.

j=1

Hence, the matrix (ip(bt bi )V 2 ,..., n is positive, and so it is a positive linear combination of one-dimensional projections. Since any onedimensional projection is of the form (7i = 1, 2, ..., n, we have

E

cp(at ai)tp(bt O. q. e. d. _-: 1 Now let Q. {L y*R y tfr lyes10 (pe,g tpeA3 1, where 91, (resp. go) is the state space of d (resp. 93). Q is a set of positive linear functionals on si 0 93. Since every element of d* (resp. 93*) is a linear combination of positive elements, every element of d*0 93* is a linear combination of elements of the form (p 0 (p e.9, /íe.9). Therefore, if a is a norm on dO 0 such that a*((p Q o is finite ((pe,99 f , tP e A ) , then a* is finite. i,j

1.22.2. Proposition. There exists a least C*-norm oc oamong all C*-norm a on d Ø3 such that a* is finite. oto is a cross norm and .1.

MY*X*XY)

1.22. Tensor Products of C*-Algebras and W*-Algebras

61

if 'p® ifr(y* y)0 O. On the other hand, consider the *-representation Inv ® kb , )Ko® kb} of d®3. (,9 .-.), y(y* x* x y) sup = 11;poo(x)I1 2 . 0, ® ip(y.y)* o (f) C) 114* y) y ,,210*3

Let ao (x) = sup II nv 0 ,p (x)II (xed 0 0). ao is a C*-norm on dØ 0, ve9L, Q E 9' 3 and a(x)a o (x) (x ed 0 0). We shall show next that a o is a cross norm and ao > A. Consider the *-representations {m v ,,Y4p } and {iro , cYtki,} of d and 0, respectively. One can easily see that the mapping av 0 bip -- (a 0 b)(p ® ip of Ye;p 0 Yfip into ,Y(490 , extends to a unitary operator U of .)(4(') 0 Yfo onto Jrcpoip . Moreover, U (7- c(p(a) 0 1,,e • 1,r, 0 no (b))U* = 7 r, 0 4, (a 0 b). Hence, ao (a 0 b)._ 11 all II bll. On the other hand, ao (a 0 b)2 = ao (a* a 0 b* b) sup Ka* a 0 b* b, p® '>I= II al1 2 11b11 2 . Hence, a o (a 0 b) , 11.4 II M. Next, for x ed 0 0,

(49, 0)C41 X 43

sup l(p 0 tp(x a 0 b)I -._ a o (x a 0 b)_-_ ao (x)ao (a 0 b)_ao (x).

Ilan .ç1. ilbli 1

(VME4/ x Ai

Consider the second duals d** and 0" • Then for fed**, g e0**, we have the polar decompositions f= R u If I and g = Rv Igi (U E d**, Ve0**). By the Kaplansky density theorem, there exist directed sets (aŒ) and (bp) such that (2,c —* U in the t(d**,d*)-topology, b fl --1/ in the -c(0**, 01-topology (a„ed, bfl e 0 and 114;11, II bfl II __ 1). Hence, A(x) =

sup

I (x, f 0 01 =

sup

I (x, R a cp 0 R b tp>i

(f, g)e At* x Q3*

Ilf11=11g11=1

11:'1)6 11111,11b1IxY:1-

(xed 0 0). Therefore, a o (x)> A(x) for xeJai 0 0. Moreover, a o is a cross norm; hence, a o < 7. q. e. d. 1.22.3. Proposition. Let d be a commutative C*-algebra, and let 0 be a C*-algebra. Then, the C*-norm go on d 0 0 coincides with A, and d 02 0 = C0 (2,0) (Q is the spectrum space of d). Proof. Let d = C o (Q). Under the mapping 0, d 02. 0 = C0 (Q,0). Moreover, 1)(al 0 b 1 a2 ® b2 )— 0(a 1 a2 ® b 1 b2 )— (a l a2 )(t)b 1 b2 — (a l (t)bi)(a2(t)b2) =0(a1 0 b 1 ) 0(a2 0 b2),

62

1. General Theory

and 0(a* 0 b*)=a(t)b* =0(a b)* (a1 ,a2 ,aed; b 1 ,b2 ,beQ3). Hence, 0 is a *-isomorphism of d Q 3 into Co (Q, 0). Hence, A is a C*-norm on dØ 0, and so )=z, for .1.* is finite. Moreover the C*-algebra q. e. d. ® 3 is identified with the C*-algebra C 0 (2,0) under 0. 1.22.4. Lemma. Let d, 3 be two commutative C*-algebras and suppose that Q3 has an identity. Then there exists only one C*-norm a on d 0 Q3, and a= A. Further, A/0,1 0 = Co (Q 1 x Q2), where 0 1 (resp. 02) is the spectrum space of d (resp. Q3). Proof. Let a be a C*-norm on do O. By the uniqueness of the C*-norm on a C*-algebra (cf. 1.2.6) we have a(a 1 )= aI (ae-524 Let x be a character of d 0,Q3. For h( .0)e3, set fh (a). x(a 0 h). Since f1 is a character on d, we have x(a h)= fl (a)g(h). g can be extended uniquely to a state "d on O. Let ao ed with Mao). 1; then f 1 (ao)(h 1 ) f 1 (a0)4(h 2 ). x(ao 0 h i )x(ao 0 h2 )= x(a,?) ®h 1 h 2 ) =f1 (a0)2 4(h i h2 ) (h i , h2 e Q3).

Hence every character on d 00,0 is a product of two characters on d and O. Let F be the spectrum space of d(DA3. By the above results, F is a subset of 0 1 X Q2. Moreover F is closed in 0 1 X Q2. Suppose that F Q1 X Q2. There exists an open set G 1 X G2 in 0 1 X 02 such that G 1 X G2 n T=(0), where G 1 (resp. G 2) is an open set in ‘21 (resp. Q2 ). Hence, there exists a non-zero continuous function a (resp. b) with a compact support on ‘21 (resp. Q 2 ) such that s(a)c G1 and s(b) c G2. Then x(a Ø b)= 0 for all xeT and so a () b = 0 in si 0„ a contradiction. Hence F=Sli X Q2. Therefore, sup iXi X2011 (x1oc2) , s 2 x(22 for yedo O. Hence, d(31„0 = C o (Q i x ‘22) under the canonical identification. On the other hand, I = sup I I )v)= sup cx(Y) =

hail 1 0115_ 1 OP. le (4, x ^

=

sup Hail cc o by 1.22.6, and so a > a o . Moreover

a(a0b)=110(a0b)II =11n( 1 ') 7r( 1 )11 (aed, be93). 11b11 Hence oco < q.e. d.

IITE( 1 ')11 11 70)11

1. General Theory

66

1.22.8. Definition. Let d, 0 be two C*-algebras. We shall call the C-algebra ,s210„.0 the C*-tensor product of d and 93, or more simply, the tensor product of the C*-algebras d and 93 ( denoted by .s410 93). 1.22.9. Proposition. Let tion of d (resp. 93), and set

(resp. {m 2 ,Ye2 }) be a *-representa-

be0). n1On2(a0b)=(n1(o)0 lif2 )(1/6 n2 (b)) (ae Then it 0 n 2 can be uniquely extended to a *-representation (denoted by Or1On2,Y60Y(21) of d093 on Yf i alf2 . Moreover, if tni , /fil and {n 2 , Y(2 } are faithful, then {ni n2, Y€ 91 0 Jr2 } is faithful. Proof. Consider the mapping 0:

at Obi

E (ai Ot ie,)(1 01 0b1)

of .9100 into B(Y6 0/(2). One can easily see that 0 is a *-homomorphism. For Vi e , EA; (i= 1, 2, ..., n), set (P(X)

= (

(

x)

i=

1

2

E

`2)

(x e

(p(x* x)

Then cp e „szi* Q3* and is positive. Hence, ao (x*x) >

2'

E Vi

i=

and so ao (x)> 11 (x)11• Hence 0 can be uniquely extended to a *-homomorphism of „510„0 into B(491 0 492 ). Next, suppose that fn i ,Yfi l and {n 2 ,Y(2 } are faithful. Let Vi = the convex span of {(pi (pi (x).(x Ye (i= 1,2) } . Then I7 1 =9:, and 172 =,99*3 , when fn i ,l and {m 2 ,} are faithful. -1 2) is the o-(91*,,s21)-closure (resp. o- (93*,93)-closure) of Vi (171 (resp. 7 2 )). (resp.V Put a(x)=11 0 (x)II (xesd0 93). Then ,9.1. x ,99923 c (d0Œ 93)*. Hence ao :5_ a. Hence, ao = a. q.e.d. We now define the tensor product of W*-algebras. Let ,./K and .4( be two W*-algebras, and let ,///* and Jr* be their respective preduals. Consider the C*-tensor product j1/0„0 .K. Let at be the dual norm of oco in di*Q ,Af* . Since c d/* and ../I(* Af*, we can consider the norm at on ,/#4, 0 ,4();. 0„0(); is a closed subspace of (4'020 ,47)*. Since 414,0./V; is invariant under R x , L x (xe4'0,A1), jil4,0 t ,A7; is also invariant under Rx, Lx (x e Q OE.,K). Hence, the polar •54-, of ,A1: 04 .iff* in the second dual of diato •Ar is a two-sided ideal, and (.//14,0„t A(* )*. (4' 0„,„, ,K)* *If. The canonical mapping -i# 02 0-47.—* (4' Oa° -4( 5)/5

is a *-isomorphism, and so is an isometry. Hence, the C*-algebra (1) 20 ,/lf can be considered as a C*-subalgebra of the W*-algebra (4/0„0,K)**/5.

1.22. Tensor Products of C*-Algebras and W*-Algebras

67

Hence (A, 04 ,./1%;)*=-- (4'0 0,0 ,4()**/5 is a W*-algebra and the C*-algebra di0Ar is cr-dense in it. 1.22.10. Definition. The W*-algebra (414,0iff* )*, more simply denoted by 4'0 A; is called the W*-tensor product or simply the tensor product of the W*-algebras 4' and Af. 1.22.11. Proposition. Let InT,Yf i l (resp. {n',}) be a W*-representation of the W*-algebra (resp. Af). Put ni 0 n2(x 0Y)= (ni (x)0 ide)( 1 dr, n2(Y)) ce Y A() • Then n i n2 can be uniquely extended to a W*-representation on Yfi 0 Y6 (denoted by {(n i Øn2 )'', /6 Ø}) of ,/h,c, X. Moreover, if fitT,Y41 and {702',,Y(2 } are faithful, {(n i 0 n 2r, /6 0 Y(2 1 is faithful.

Proof. First of all, consider the *-representation(5 0 -Y(2 } tni .)‹ n 2 0 Then of the C*-algebra (ni 0 n2(x)i 0 2 ni I/2)=A of2(x) fi (a).--(n i (a) i m i ) and f2 (b).--(n 2 (b) 2 ,q2 ) for aejll, be,iff). Consider the mappings n i 0 n 2 : 13(Y4 0 Y( 2 ) and (ni 0 n2) * (-WO --Af )* B(Yei 0 Y(2)* By the previous considerations, one can easily see that (n1 0 n2)* (B(Yti 0 elf2)*) Al:020 /.* Let (n 1 ® n 2 )t be the restriction of (n i 7r2 )* to B(Y6 ®)*' and let ((n i 0 n 2 ))(:'; )* be its dual. ((it s 0 n2))* is a continuous mapping of 00 ,4(* )* = 41,0 S. with the o-(4/(7),K, (41 g - ,K) * )-topology into Wei 0 ) , with the o- (B(Yf1 0 Y(2 ), 13(Y 0 *))-topology. Clearly ((n 1 0 n2))* = n1 0 n 2 on 4'0 At. Since n i 0 n 2 is a *-homomorphism, one can easily show that ((n i 0 n 2),)* is a W*-representation of ,W(7, 4f. Now suppose n i , n 2 are faithful, and let 5 be the kernel of {(n i 0 n2r, yea, 0 Yt2 1. If 50(0), there exists a positive element h (>0) in f. On the other hand, by 1.15.5, for feJI, g e ,if7,;, there exists se00

quences

(1/11) c cYfi, WA (1) E 1111;11 2 < + 00 i=

1 , 2),

°2

such that

f (a) =

i=

E g)11 2 < +

(n i(a)

and g(b) = E (7r2(b)V, ni2) (ae 4', be ,47). Then i= 1

C0

If

=

E

i, j = 1 cc

(n1oir2(h)ti

`2,oni2)

I.

E 1(ni n2(h) nti 07/12) ,j= 1 Hence, h=0, a contradiction, and so (n i 0 n 2 )14' is faithful.

q. e.d.

1. General Theory

68

1.22.12. Proposition. Let L'(2,y) be the commutative W*-algebra of all essentially bounded locally /1-measurable functions on a localizable measure space O. Let di be a W*-algebra. Then (0, y) (i) 0, A = L1 (0, y) 0 y= (Q,

where 141 (0,y,A) is the Banach space of all A-valued Bochner integrable functions on Q. 1.22.13. Theorem. Let di be a W*-algebra with the separable predual A, and let LŒ)(0„u,,A) be the Banach space of all di-valued essentially bounded weakly * pc-locally measurable functions on a localizable measure space O. Then, Lc° (Q, u, si') is a W*-algebra under the pointwise multiplication, and its predual is L'(0, y,A). Moreover the mapping f(Da—f(t)a (feL"(S2,y),ae4') can be uniquely extended to a *-isomorphism 0 of the W*-algebra L"(0,y)(54/ onto L' (0, y, 4 1).

Proof. Since A is separable, by the Dunford-Pettis theorem [43], for any xe(1,1 (Q,p)Oy A)*, there exists a unique ./K-valued essentially bounded weakly* locally /1-measurable function gx(t) on Q such that x(®)

y. Hence { x} — {x'a } e J. Therefore, we have ("Y fl). Clearly .2"fl1 c ..To , if fi i -_fl2 , and U YOE =- Y. Hence ael

j1;1

162 d

(fll #2),

and the uniform closure of U ja, is d. ace!'

1.23.1. Definition. The C*-algebra d, denoted by

lim {s/Œ ; 013 , o, I (13 , oc)elf x IL and f3 __ ac} , is called the inductive limit of {.4Œ l °cell} defined by the family of morphisms {013 , 2 }. Then we have 1.23.2. Proposition. Let szi.-- liln {sic, ; Ofi ,„ I (fl, a) e II x11, fl > a } . Then there Ç d; is equivalent to oc a and aa e JOE . Then

A p (aa).--n011-1 (aa). wg 0,- 1 (0„ 00E- 1 (aa))= n0,6-1 ■Tofl Op , a(0a-1 (aa)) =WŒ 0; 1 (aa). Hence A a = Afi on siOE . For be U JOE , there exists some a such that

besiOE . Define

°eel

A(b). A a(b); A(b)

is well-defined. A defines a *-isomorphism of Moreover, A is an isometry, since d'a and ga are

U si. onto U g,c . « « *..algebras . Hence, A can be uniquely extended to a *-isomorphism A of d onto g. q. e. d.

1.23.3. Corollary. Let d be a C*-algebra with identity, and let fda laefil be a directed set, by inclusion, of C*-subalgebras of si containing the identity. Suppose that d is the uniform closure of U da . For ael

a a). Consider a *-isomorphism W 2 A 2 0; 1 of

„szirc, onto i3Œ

(ael).

Then for aa esiOE ,

n As Oi 1 (a2)= wq A, oi 1 ( 02 02- 1 ( a2)) =

A, .T•,, 2 (0 2 1 ( a2)) =W6 w„,„ Aa Oa— 1 (aa)-- W aAa0;1(aa). wg

1.23. The Inductive Limit and Infinite Tensor Product of C*-Algebras

73

Hence, n A6013-1 =- tA 2 0„-1 on JOE (fi > a). Therefore, we can easily define a *-isomorphism A of d onto 0 such that A=W2 A Œ 0Œ-1 on ,94. q. e. d.

1.23.5. Proposition Let d= lirn {A; Oni ,„ I m_-_. n, m, n= 1, 2, ...} and n Q3 = lirn {O n ; W„,,„ I m>n, m, n= 1, 2, ...}, and each sin , Q3„ is *-isomorphic n

to B(Jrn)(dim(A)< + co). Then si is *-isomorphic to 0. Proof. Consider {,54in } and Ii3„1. By induction, we shall define a *-isomorphism A n of sin onto i3n . For n=1, we take an arbitrary *-isomorphism A 1 of iii onto ii i . Suppose A n (n <no) is defined. We can write: dn-o+ 1 =,szi:0>e) ,„0 + 1 = ii no 0 il n' 0 . Take a — -24:0 and 61 morphism z1 0+1 of 1 ()sin onto 1 0 i3.'„0 , and set A no±i = A no ® A no+ 1. Then A n = Am on ,i (n <m). Therefore, we can define a *-isomorphism A of d onto 0 such that A= A n on dn . q.e. d.

1.23.6. Definition. Let sl be a C*-algebra with identity. d is said to be uniformly hyperfinite if there exists a directed set, by inclusion, of C*subalgebras {Alcxell} containing the identity such that the uniform closure of U da is sal, and saiOE is *-isomorphic to B(Y1 92 ) ael

(dim(ft:)< + co, ocell).

1.23.7. Proposition. Let si be a separable, uniformly hyperfinite C*algebra. Then there exists an increasing sequence (i.e., dn c „Qv'. (n<m)) of C*-subalgebras sin containing the identity such that dn is *-isomorphic 00

to B(/(n)(dim(Y4,)< + co), and the uniform closure of U ,szin is si. n=1

Proof. Let si= the uniform closure of U A. We shall choose by ctel1

induction an increasing sequence lAn l. Let (an) be a sequence which is dense in si. There exists an A, and b 1 e 1 such that Ma l — b i 11 0) e , and let Rb R v IR b (P1 be the polar decomposition of Rb (f) (bed!). Then the absolute value IR b cpl of Rb cp is bounded by I R b (P1 11b11 P.

Proof. Since IRb

R o R b cp, IR b (pi(x)=- (p(x v* b), so that we have

icp(h v* b)I 11v * bll (P(h) Ilbll OM (he 4',

q. e. d.

1.24.3. Theorem. Let Ji be a normal positive linear functional on such that tk < (p. Then there exists a positive element t o of dl, with 0 _O, 11 , 1- .._ 1 h 'f) 11g' by 1.24.2, so that there exists a positive element to e s(cp)ll s((p), 0 ._ t 0 ._-.. 1, such that If ' 1(y) — (y' t o ,O. Then If ' 101= Rof '01= f ' (y ' v'*)= g' (y' v'* h'0 ), where R v ,If'l= f' is the polar decomposition of f '. Hence, (1/ e {s(49 ) j#s((P)}'). Since [{s(cp),4' s((p)}' fl= Yf o , we have to = v' * If° and so

v' to = I! e* h'0 On the other hand, (y' vi v'* h'0

) = if V 0= -- Rci i" 1(y/)

= f(y/)=-4' h'0 , 0 Hence, v' v'* lio = 14

0/ e {s(04's((P)I')-

and so vi t o = lio Therefore,

t' (x) = (x h'0 , lio )= (x y' to , y' to ) = (x v'* y' to , to )=- (x e* ko , to ) = (x to , to )-- (t o x to , 0 = elb (to x to)

(xes(cp) //1 s((p)).

Now we have tp(x)=-0(s((p)xs((p))=- (-I/ (s(cp)x s(T)) = Cp(to s((p)x s(cp)t o)= cp(t0 x to)

(x e 4').

q. e. d.

1.24.4. Proposition. Let ,/# be a W* -algebra, and let cp, 0 be two normal positive linear functionals on 4' such that Ilf < cp. Then there exists a positive element h o in 4', with 0 < h o < 1, such that 0(x)=---1(p(h o x+ xh o)

(xe,4').

Proof. Let A 1 =- Ihe4' 1 I h I .- 1 , h * =Ill. A1 is convex and o-(jil, MOcompact. Hence, under the mapping h-4(L h (p + Rh (p), the image of A 1 is convex and o-(/#40 ,71)-compact. Suppose tp 0 a; then tp° ao, where tii ° and are the polars of (II and respectively in the self-adjoint portion of A Hence, there exists an element xo in 4's such that If (x 0)I -. 1 (fe and ilk(x0)1> 1. On the other hand, let xo =4 —x(T,. Then

a

a

a°

a)

where e = s(x+). Hence, 0(4 + x 0- ) -_ 1. Similarly p(((1 — e)— e)x o +x0 ((1 — e)— e)) = —(p(4 + x 0- ) _._ — 0(4 + x (7,) .

1. General Theory

78

Hence, —0(4 +4)> —1. Therefore,

—1 —0(4 +4)_0(x 0)=-- 0(4)— 0(4)_.-ç.(p(4 +x (,):5_1. Hence 100415..1, a contradiction. Hence 0(x)=- 1(p(h 1 x+xh 1 ) (xeA and some h i e Moreover, since

>0,

P( 1 )=(P(hi — hn- (P — ( 1 — 14)= (P(h i(P — ( 1 — P))+(P—(1 — Ahi) + hfl, where p=s(hi'). Hence tli(h)=--0, so that

q. e. d. (xej11). 1.24.5. Proposition. Let 41 be a W*-algebra, 41 a W*-subalgebra of A. Let f be a 4'17;A/0-continuous linear functional on A/: Then f can be 0(x)=-4(p(14 x+xht)

extended to a 04;410-continuous linear functional f on 4' with 11111= II/IIProof. First of all, suppose (p be a normal positive linear functional on ,yr. Let Ar ° be the polar of Af in di* . Then .iff is the dual of Hence, there exists a self-adjoint element geAk such that g= (p on A: Put g=g+ — g - and (p 1 =--g + +g- . Then (!) - _ (p i on A: Hence by 1.24.3, there exists a positive element h, with 0 0 (ue llu), f (u). 0 for all ue ll", except for a finite number of points, and f 1.

E

Ueda"

For feC and xe 4', set f-x= have g • ( f • x) =

E

E

f(u)uxu*. Then for g,fe C, we

g (u)u( f • x)u* =- E g(u)u(Ef(v)vxv*) u*

.E g(u)f(v)(uv)x(uv)* = E(E g(u)f(v)) w x w* Eg(u) f (u - w)w xw* =g*f • x, W

U

where g *f is the convolution multiplication of g and f. Hence g *f e C. 2.1.13. Lemma. Let a be a positive element of di and E> O. Then there exist an feC and a central element c in 4' such that 11 f • a— ell < E.

Proof. For an arbitrary positive integer r, there exists a finite family {z i , r, z2 ,, Z n,} of mutually orthogonal central projections with the sum 1 and an Le C such that coa(f,.• a) 0, is well-defined and is a normal state on f 3 41 f 3. Since (f) is faithful on f 3 di' f 3, tfr is again faithful on f 341 f 3. By putting f 3 =po , = po , we have the required result. Now let {p i ii= 1,2, ..., ml be a maximal family of mutually orthogonal projections in 4' such that pi —p o (1 0, and (x*x)h =0 if and only if x= 0; x--.x4 is o-(./#,A) and s(./#,J!)-continuous. (xe4', zeZ).

Proof. Put x4 = Cx Z (x e dI). It is clear that the mapping x—*x4 is linear; Ilx4 1=1; (zx) 4 =zx4 . Since Cu*xa n Z = u* C ur'Z. Cx nZ for u (x* x)4 = (x x*)4 . Clearly (x*x)4 >O. Now let (ha)aa be a uniformly bounded, increasing directed set of positive elements in 4'. Clearly (1.u. b. h 2 )4 >1. u. b. 'ILI. a

a

Suppose that (1.u. b. h4 4 >1.u.b. 14. Then there exist a non-zero central a

a

projection z1 in di and positive number A. such that (1.U. b. h2) 4 a

z1 > (1.U. b. ilt)Zi. ± a

94

2. Classification of W*-Algebras

Consider a sequence {O n} of normal states on J/z1 such that On(Y *

, 1 + — )IP nlY Y * ) n

(Ye 4'z i , n= 1, 2, ...).

Then

O 1 ((l. u. b. 11,,)L' zi)

+ ) tP„ (( u. b. hOE )zi) =

1 2 ± —)

Hence O n (1. u. b. a

+ )1. u. b. OA« zi)

/

n

a

+

(1 +

(1. + — n

1 )2

(1. a

z0 •

u. b. h'clz) (n=1, 2, ...), a

a

contradiction. Hence (1. u. b. h4° =1. u. b. ht, and so for an arbitrary (x e J1) is again normal. normal state p on a state (p'' on : Hence the mapping x--xt' (x ') is cf(W, 41,)-continuous. (p(x L1 * (x* x) , (p((xh* 4) = * x) < (,6M (x 11 * Since T(X* x L') (x* x) = 9((x* ((p e 9'). Hence x'al x 0 (xe Then 11 Since 1 2 = 11 (XL1' )* (Xkl )11 1 (X* X) 11 x* x11 1 -- (x* so is uniformly x 1 (x* '• Hence 11(x* and x* 'I1 Card(1I Œ) fiela

0(64 and let

.Ytf

be a N-dimensional Hilbert space. Then

{n2,-e2} = {(n i + 2) E'2 , E'2 (14 lf2 )}

=E E {(iri+ 7r2)Ea43,E,43(Yei

Ye2 )

=E{( 7c10 1,y()F.X.Ye0it f l, aell

where F'„ is a projection in the commutant zai za2 = 0 (al, oc2 oc2), FOEi • F 2 = O. Hence

E {(7ci ®

aell

0

7r 1

1(4')'.

Since

= {( 7c 0 11 ) (E F;), (E F«) (Ye 0 YO} aell

aell

q. e. d. 2.7.5. Definition. A weakly closed self-adjoint subalgebra on a Hilbert space Yf is called a W*-algebra on a Hilbert space f. Let X be a W*-algebra, containing 1 0, on a Hilbert space ye and let e (resp. e') be a projection in X (resp. X'). We shall consider the W*-algebras e X e and A' e (resp. e' and e' X' e') as W*-algebras, containing l ey( (resp. l e ,o), on a Hilbert space e ,lf (resp. e' ,lf). By (e e)' (resp. (e' X' e')') we shall denote the commutant of e X e (resp. (e' X' e')) on the space e (resp. e' f). ,

2.7.6. Proposition. Let X be a W*-algebra, containing 1 0, on a Hilbert and let e ( resp. e') be a projection in X ( resp. X). Then space (e X e)' = X' e and (e' X' e')' =X e'. Proof. Clearly (X' e)' D(e X e). Let ae (X' e)' and let b be a bounded linear operator on ,rf such that b=- a on elf and b= 0 on (1 0 — e))r. Then for x'eX 1, bx'=ebex'=ax'e=x'eae=x'b. Hence beeXe, and so (X' e)' e. Similarly (X e')' =e' X' e'. q. e. d.

2.7.7. Notation. Let X be a W*-algebra, containing 1 0, on a Hilbert space ye For e Jr, put (Mx).--(x 0(x e X). p is a normal positive linear functional on X. If çt = 1 , p is a normal state on X (called a vector state). .

104

2. Classification of W*-Algebras

2.7.8. Lemma. Let .X be a W*-algebra, containing 1 0, on a Hilbert space Y e. UV/ is a normal state on X such that s(0)=P1,1A (some ee ) then there is a vector n in A' with tk --(p, and [X n ] = and tif (P,,,,, is the orthogonal projection of Ye onto pr Proof. Since ço<tp, there exists a positive element h in ."17' with x h) by 1.24.3. If e is the support 0 d(t)) Hence On = It 1 dim (.f(t))= n} (n= 1,2, ...) is locally it-measurable and = {t 1 dim (r(t))=No } is also locally measurable. Now let SO be the set of all elements in I/0 with J 11(t)11 2 (IMO< + cc. If e ,Y6 , then

(1 11

+

(t)11 2 di-t(t)) 4

(I 11 i (t) 11 2 dkt(t))/ +

1 2 (t) 11 2 dp(t))if < +00.

Therefore is a linear space. For e set ( , t1)=-- J ((t),n(t))dp,(t). is a Hilbert space. SO is called the Then one can easily see that associated Hilbert space of Vo . Clearly {L2 (Q,) ' 1 , 2 , • • -1

3.3.2. Proposition. The linear subspace generated by 1L2 (OA- e i li= 1,2,...} is dense in X. Proof. Let i( be the closed linear subspace of it'O generated by i()=-- 0, 1L2 (S2, it) • ei li= 1, 2, ...} . For e with f (t) (e i (t), (t))d,u(t) = O

( f e L2 (S2 u); i= 1,2,...).

Since J g(011 2 4(t)< + co, there is a sequence {Fn } of measurable subsets in S2 such that Fn c Fn .", ,u(Fn)< + co and (t). 0 for all

I. Therefore the above equalities imply =O in X.

te Q —

q. e. d.

n=1

Let ,Y4,' be an n-dimensional Hilbert space and let (e7,4,...,e) be a complete orthonormal system of Yfn . Let Y(09 be an No-dimensional Hilbert space and let (ei° , , .) be a complete orthonormal system of ,Ifcc . De fi ne an isometry U(t) of YP(t)(tet2„) onto Yfn by U(t)e1(t)=e7 (i= 1,2,...,n), and an isometry U(t) of )r(t) (teS2 0,) onto YtŒ, by U(t)ei (t)= eix) (i= 1,2, ...). Then for feL 2 (0,p), U(t)f(t)e i (t)=. f(t)e7, (i= 1,2, .. .,n if n< co; i= 1,2, ... if n= co). Hence the family {U(t)1 t e S2} of isometries defines an isometry of it/O onto 00

n= 1

(L2 (on,

o

e (L2 (Q., kt) cp Ye.)

•

Hence we have

3.3.3. Theorem. Let {Y((t)iterl}

be a measurable family of Hilbert spaces, and let V0 ,1/1 be two maximal measurable subspaces of and let X; be the associated Hilbert spaces of Vo , V1 respectively. Then there exists a unitary operator V(t) on lf(t) (te S2) as follows: for e, set 11(0= V(t)(t); then qe,Y4 and the mapping gives a unitary

3. Decomposition Theory

140

mapping of ,Y6 onto ifi . Therefore we can obtain an essentially unique Hilbert space Si) from a given measurable family tYf WI t e fl} of Hilbert spaces. This Hilbert space ,Y6 is called the direct integral of 1, Y f (t)i t e S21 and is denoted by 1 Yt (t)d,u(t). Since f Y f(t)d ft(t) is isometrically isomorphic to L2 (E2 o0,112.0)C)/(,0 1 L2(f2n,1in)artan, the reduction theory n=1

CO

L' (Q c„, pc,) 0 B(,10 0

E

/2° (Qn , jun) 0 B(Jrn)

can be carried

on

n=1

/19 (t)d4t) under this isomorphism, where p c), (resp. iin) is the measure obtained by restricting it to Q On). References. [37], [132], [179]. f

3.4. Decomposition of States (Separable Cases) Let d be a C*-algebra with identity. Throughout this section, we shall assume that d is uniformly separable. Let .9' be the state space of cd; then 99 is o-(d*„s21)-compact. Let {a n in--1,2,...} be a sequence of non-zero elements in cd which is uniformly dense in the self-adjoint portion ds of d. ' 0 —0)(a)1 (polleg, set d(cp, i11)— / For . Then d is a metric on n=1

2n Ilan 11

Y which is equivalent to the topology o-(99,,521). Hence 9' is considered as a compact metric space, and so it satisfies the second countability axiom, since a compact metric space is separable. Throughout this section we shall deal with the topology a (99,d) on 99.

3.4.1. (General lemma of Choquet). Let E be a locally convex space, C a metrizable compact convex subset of E and let OC be the set of all extreme points of C. Then a C is a Ga -set. Proof. For (cp, tk)e C x C, set p((cp,tp))-- (q) +"). Then p is a con-

2 tinuous mapping of C x C onto C. Let 4= {(q),01(pe C}. Then d is closed and so C x C— d is open. Furthermore p(C x C— d)=-- C—OC. Let d be the metric on C and Un (cp)= {01 d(q),IP)0). Hence (hy,h)' =u' u + u u' —v' v—vv', so that by Schwartz's inequality hn= 0

4.

1C-p(h(a+ib)h)1-
— •— . ' Fp(h(a+ib)h)1> (h ah) >I F-p 4x M ) 2 2 4 2 This contradicts the inequality 5. Hence a=O. Next suppose that b0 0 and there exists a positive number ii in the spectrum of b (otherwise, consider { —x„}). It suffices to assume that it= 1. Then there exists a positive element k (11k11= 1 ) in d with kbklic 2 . II(k Y 7,, k)' — k(a + i b)kll < i

for some ni •

4.1. Derivations and Automorphisms of C*-Algebras and W*-Algebras

155

Let C1 be the C*-subalgebra of V generated by ky„,k and 1. Then ky ni k >1 . there is a character (p 1 of C 1 with c,91 Let Fp i be an ex4 11x. i ll 2 tended state of (,9 1 to d. Then

IF-p i ((ky n1 k)')I=0 and so 1Fp i (k(a + i b) k)i < On the other hand, 175 1 (k(a+ib)01

IF-p 1 (kbk)1

1 Ffilkynik )>. 1 Tro 1 (-1 k2). — 2 .411x„ i II) 4 ' 2

a contradiction. Hence a+ib=0 and we have a contradiction.

q.e.d.

4.1.4. Lemma. Let d be a C*-algebra and let .5 be a derivation on d. Let {7r,/f} be a *-representation of d. Put i5(n(a))=74(5(a)) (aed). Then S is a derivation on 4521) and (.5'- can be extended to a derivation iion the weak closure n(d) of n(d), and (5 is a-continuous on Proof. Let d** be the second dual of d, and let 6** be the second dual of 6. Then one can easily see that 6** is a derivation on d**, since 6** be the W*-representation of is a-continuous on d**. Let {i 1 ', d** with n(a)=7rw(a) (aed). Let J be the kernel of nw. Then J is a a-closed ideal of d**, and so there exists a central projection z in d** with f =.5211**z. }

6**(z) = 6**(zz)=6**(z)z+z6**(z)=26**(z)z. Hence 6**(z)=0 and so 6**(szi** z) c dzi** z. Therefore 6** defines a derivation on d**(1—z). Since sl**(1—z) is *...isomorphic to nw(d**)=- n(d) by the mapping ev, 6** defines a derivation 6 on n(d) with 3..3 on n(d). q. e. d. 4.1.5. Lemma. Let 4' be a countably decomposable type III W*-algebra and let a be a non-zero element in A and let C(a) be the uniformly closed convex set generated by lu* aulue4'u} (41" is the group of all unitary elements in ,//I). Then, C(a)nZ contains a non-zero element (Z is the center of A1). Proof. C(a)n Z 0 (0) by 2.1.16. It suffices to assume that 114 no- 1 e (1 e). If e majorizes a non-zero central projection z, then a> no- 1 z — (1 — z). Hence C(a) n Z> no- 1 z — (1 — z) and so C(a)n Z contains a non-zero element. Now suppose that e does not majorize any non-zero central element. —

—

—

156

4. Special Topics

Let c(e) be the central support of e in 4'. Since

=

c(e)

(1 — c(e)),

it suffices to assume that c(e)= 1. Then c(1 —e)= 1, since e does not majorize any non-zero central projection. Hence e 1 —(1— e) by 2.2.14. eno+ 1 ) of mutually orthogonal There exists a finite family (e 1 , e2 , no+ 1

E ei = e and ei — e (i= 1,2, ..., no + 1). i= Put 1 —e=e0 . Then there is a finite family {u o , uno +1 } of unitary elements in 4' with ui ej ut = ent(i) , where ai is the no +2 cyclic permutation of 0, 1, ..., no + 1.

equivalent projections in 4' with

no+ 1

Then

E ui ei tit =1 (1=0, 1, 2, ..., no + 1).

i= o

no+ 1

Set b = (n o + 2) - 1

E

ui a ut . Then

i=0

no+ 1

(no + 2)b

E

ui (no- (e + e2 + • • • + en. +1)— e o)ur

i=o = (n o + 1)n o- 1 1 — 1 =n(T 1 .

Hence C(b) Z n o- (no + 2) - 1, but C(a) n Z C(b) n Z.

q. e. d.

4.1.6. Theorem. Let 6 be a derivation on a W*-algebra 4'. Then 6 is inner—namely there exists an element a in di such that 6(x)= [a, x] (xed/). Moreover we can choose such an element a as follows: 11(211 <MM.

Proof. Let 4'u be the group of all unitary elements in put Tu (x)=(ux + (5(u))u - (xe Then if u, ve4114,

For ue,./Ku,

Tu Tv (x) = {u(v x + 6 (v)) v 1 + 6(u)} = {(u v x + u 6(v)) v - + 6(u)} = uv x + u6(v)v- 1u 1 + 6(u)u= (u v x + 6(u v)) (u v) - = Tuv (x) Hence Tu T,=Tuy . Let d be the set of all non-void a-closed convex sets K in 4' satisfying the following conditions: 1. Tu (K) K; 2. sup 11x 11 _< MOM. xeK

Since 11 Tu(0)11 = 11 6(u)u - 11 < OM, d is not empty. Define an order in d by set inclusion. Let (K ii be a linearly ordered decreasing subset in A. Then K2 eA, because KΠell) is compact. Hence there exists

n

ael1

a minimal element Ko in d by Zorn's lemma.

4.1. Derivations and Automorphisms of C*-Algebras and W*-Algebras

157

If a,beK o , then for ue u(a — b)u - = uau - — u bu - = u au- + (5(u)u- 1 — (u bu -1 + (5(u)u=Tu (a)— Tu (b).

Hence K o —K 0 is invariant under the mapping 0": x ux (xe,4). is a countably decomposable finite W*-algebra; Suppose now that then 41 has a faithful normal tracial state T. Put Ilx 11 2 = T(X * 44. (xe 4). Suppose that K o — Ko contains a non-zero element c with c= a — b

Let

(a, be K o).

= sup pc11 2 . Then for an arbitrary positive number e, there xeK0

is a u in 4' with

Il 7 u(412,

.(b)11 2

T

a+ b) iu

2 )

> —e, since Ko is minimal. Since 2

,

8-(T u(a) — u(b))113 =


O; then 11 (n,,, (011 > lIn v,(d)11. But II 7r92 (d)11 = since g is simple, a contradiction. Analogously if A (n (pi(d))11 = IA, a contradiction. Hence A= 0—namely = 4,2( 4 Thus (p 2 (d) = (it 92 (d)1 4,2 , 1 (p2 ) (0(7r 9i (d)) 1q72 1 92 ). Take a directed set {7r pl (x„)} (x cc e Jai) with 2t471 (x a)-7r(pi (d) in the a-topology. Then (0(7r vi ((0)1 92 ,1 (P2) =- 1im (0(7c vi (x 2)) 1 92 , 1 q,) = lim(ir(pi (xOE) 1 90 1 v1 )

1 4,1)= 'Pi (d). = (nvi (d) This is a contradiction. q.e.d. Now let 6 be a derivation on a C*-algebra st Then 6 is norm-continuous on d, and so 6 belongs to B(si) (B(d) is the algebra of all

162

4. Special Topics

bounded linear operators on si). Put 0(t)=exp tS (- co 01, where C is the field of all complex numbers. Denote the principal determination of logarithms by Log. Then we can define y = Log a such that a=exp y. Put ia (x)= ax a- ' and (ady)(x)=yx—xy(xeg). Then we have -

4.1.17. Lemma. If C is a closed subspace of g which is invariant under ia, then it is again invariant under ad y. Proof. (expt ady)(x).(exp ty)x(exp

—

ty) for real t, and

(exp ady)(x)=ax a-1 . Put Ry x = xy and Ly x =y x (x,y eg). Then ady = Ly - R y . Since y -4?), (resp. L y) is an isometric anti-isomorphism (resp. isomorphism) and g has an identity, Sp(y)=Sp(R y)=Sp(Ly) and so Sp(ady)c {A— ii IA, ii e sPCY)},

where Sp(R y), Sp(L) and Sp(ady) are considered in the Banach algebra B(g). Since Sp(y)c tileCIIImAleg,„

ioye,)), it is also invariant under the mapping

ad Log I TI -2 by 4.1.17.

(exp t ad Logi Ti - 2 )(a). (exp t Log I Ti -2) a(exp — t Log I Ti -2 ) = (exp 2 t Log I TI -1 ) a(exp — 2 t Log I Ti -1 ) (ae,e1). Set t=-1-; then we have that a--±ITI -1 alTI is an automorphism on .541, because ad Log' TI -2 is a derivation on d. Thus, udu* =- TITI - 1 si ITI T - 1 =O. Hence the mapping a--u au* of si onto 0 is a *-isomorphism. Hence we have 4.1.20. Theorem. Let d and 0 be two C*-algebras and suppose that d and 0 are isomorphic. Then they are *-isomorphic.

4. Special Topics

170

4.1.21. Corollary. Let si and 0 be two C*-algebras and let 0 be an isomorphism of .szi onto-$. Then 0 can be uniquely written as follows: 0=01 02 , where 0 1 is a *-isomorphism of si onto $ and 02 is an automorphism on si with 0 2 = exp (5 ((5 is a derivation on ,21). Proof. By the previous consideration,

0(a)= Ta T- 1 =u1T1 alTi - 1 u*

(ae Jai).

and 0 1 (a)=uau* (aed). Then 02 is an automorphism on .szi and 0 1 is a *-isomorphism of d onto $. Moreover, I TI al TI -1 =(expad Log TI)a. Since ad Log I Ti -2 = —2 ad Log I Ti is a derivation on V, a dLog I TI is again a derivation on d. Put 6 =ad Logl Ti; then we have exp6 = 02. The uniqueness of such a decomposition is not so difficult, because the polar decomposition of T is unique. q.e. d. Put

02 (a)=ITI al Ti -1

4.1.22. Corollary. Let d (resp. 0) be a C*-algebra on a Hilbert space Yf (resp. JO such that its weak closure „si (resp. 0) contains 1 0 (resp. l i ). Let 0 be an isomorphism of si onto $. Then there exists an invertible positive element h in si such that 0(a)=01 (hah -1 )(aes1), where 01 is a *-isomorphism of si onto $. In particular if si is a W*-algebra or a simple C*-algebra with identity, then h belongs to si. 4.1.23. Corollary. Let di and s- be two W*-algebras and let 0 be an isomorphism of di onto S .. Then 0 is a- and s-continuous. Remark. A homomorphism (even a *-homomorphism) of a W*algebra onto another W*-algebra is not, in general, a-continuous. For example, let l' be the W*-algebra of all bounded sequences, and let l' = C(K). Then K is the Cech compactification of all positive integers. Let t be a limit point in K and let 0, p, o- rationals): Put = {oc i (p,o-)}. Then W is a countable group and ,u is quasi-invariant under W. Moreover W is free. p(Eoc i (p,o-)). p p(E) for a measurable set E and so Since W o is ergodic, W is nonWo= 1 tiŒicp,o measurable by 4.2.8, and so ,i1( is a type III-factor. -

4.2.12. Example (a III factor). Let Q be the one-dimensional torus group and we shall consider Q as the set of all complex numbers z with 1z1= 1. Let ,u be the Haar measure of Q with p(Q). 1. z+u Consider the following homeomorphisms on Q: Œ2 (0,u): z 1 +u z (101= 1,1u10 and f(x) is bounded on F, and let

f ' () = f () on Fu(C)F7k) and f V) = 0, otherwise. k=1

Then

JI

f ' y 0 — f ' ()1 2 dpw_j . If (71 ) —f ()1 2

40

F

fl

f dttykoi - 1

I dit

1

f

2

I f ()I 2 d;4)

F

2) f

2

> min(

I f ()1 2 d 10

F

2 5\2 (

1 — (5 )

iv.

2 d,L(,)•

F

On the other hand, for a e , if k is sufficiently large, wc,( TO = wa(). Hence iim f I wa( yk)— wa()1 2 d 140= 0 . k-

+ co

Since {Wa} is a complete orthonormal system in L2 (Q,p), this implies lim f ig( y k) — g()1 2 cl,u() = 0 (g e L2 (.0, ,u)), and so f I f ()1 2 d ,u(0 =- 0 .

k -> co

F

Hence v(F)= f f () 4() = 0, a contradiction.

q. e. d.

F

Now let Yt° = Yfo = C be the one-dimensional Hilbert space, e G) be the /dr=-- C1ito , S -- G a discrete group, and let a-÷E./a = 1 unitary representation of G. Then the corresponding W*-algebra ,yr is *-isomorphic to the W*-algebra generated by the left regular representation of G.

4. Special Topics

182

We shall explain this special case in a different form. Let G be a discrete group and let 12 (G) be the set of all complex valued square summable functions on G. For f 1 , f2 e 12 (G), the convolution f 1 *f2 is defined as follows: (f1*f2)(a) = / f 1(b) f 2(b - 1 a). f1 *12 beG

is a bounded function on G (does not necessarily belong to 12 (G)), and 11f1 *f 111211121125 where 1111 is the norm of l"(G) and 1 112 is the norm of 12 (G). For aeG and hel2 (G), define (U(a)h)(b)=h(a-1 b) and (V (a)h)(b).. h(b a) (be G). Then U (a) and V(a) are unitary on 12 (G) and the mapping a--+ U (a) (resp. V(a)) is the left (resp. right) regular representation of G. Clearly U (ai )V(a 2 ). V(a 2)U (al ) (a1 , a2 e G). Let U(G) be the W*-subalgebra of B(12 (G)) generated by {U (a)iae G}. Denote by ea the function on G with Ea (b)=-- 0 if b 0 a and e a(a) = 1. Consider a mapping (P T -÷TEe of U(G) into 12 (G) (e, the unit of G). If Ti ee = T2 ee (T 1 , T2 e U(G)), :

V (a) T1 ee = T1 E a - 1 = V (a) T2 ee = T2 ea - 1

(a e G) .

Since the set g of all linear combinations of {ea 1 ae G} is dense in 12 (G), T1 = T2. Hence 0 is one-to-one. Moreover V(a -1 )Tee =T V(a - 1 )6e = T a and V(a - 1 ) T ee =(T ge )* Ea . Hence (T e e )* Ea = T ea (ae G), and so Tk=(Te e)*k for keg. For hel2 (G), take a sequence {k„} in g with 1 kn —h11 2 -+0 (n--3co). Then T k„--±T h in 12 (G) and (T e e)*k„--4T e e)*h in 1"(G). Since /2 (G) = 1"(G) and T k„--T h in the topology QV' (G), /1 (G)), T h=(T Ee)* h. By using the mapping 0, we shall identify U(G) with a subspace of 12 (G). Then by the above result, if f e U (G) c 12 (G) and hel2 (G), the corresponding operator 0 -1 (f) is as follows: 0 -1 (f)h= f*h. We shall denote 0 -1 ( f ) by U1 . Put T(f )=f (e)(f e U(G)). Then one can easily see that t is a faithful normal tracial state on U(G). Hence U(G) is a finite algebra. 4.2.17. Definition. A discrete group G is said to be an infinite conjugacy group if every conjugacy class {b ab I be Gl (ac G) is infinite except for a . e. 4.2.18. Lemma. If G is an infinite conjugacy group, U(G) is a type 11 r factor.

Proof. Let f e U (G) be a central element in U(G). Then (U (a) f U (a - 1 ))(b)= (cc, f Ca - 1)(b)= f (a - 1 b a). f(b)

(a, be G) .

Hence f is a constant on each conjugacy class. On the other hand, f belongs to 12 (G). Hence f(b)= O for b 0 e, and so U(G) is a finite factor. Since U(G) is infinite-dimensional, it must be a type 11 1 -factor. q. e. d. References. [121], [131], [139].

4.3. Examples of Factors, 2. (Uncountable Families of Types II I , II,,, and III)

183

4.3. Examples of Factors, 2. (Uncountable Families of Types Il ' , H OE, and III)

In this section, we shall show the existence of uncountably many examples of type II, (resp. II„„ 111)-factors on a separable Hilbert space. Let G be a discrete group and let U(G) be the W*-algebra generated by the left regular representation of G.

4.3.1. Definition. A uniformly bounded sequence (Ta) of elements in U(G) is called a central sequence if for all xe U(G), [x, T] 2 —>0 (n-- co), where 11x11 2 (xe U(G)) is the 12 (G)-norm of x, when U(G) is embedded into 12 (G) canonically. Two central sequences (TO, (T'a) in U(G) are said to be equivalent, if 11T.— T.112 -40 (n—> co).

4.3.2. Definition. For a W*-algebra A its unit sphere is denoted by (/#) i . If 4' and ,/lf are W*-subalgebras of a W*-algebra U(G) and 6>0, then (5

we shall write ,ArC di' to mean that given any Te some Se(4') 1 with 11T —

, there exists

4.3.3. Lemma. Let G be a discrete group and let E be a subset of G. Suppose that there exist a subset F c E and two elements a 1 ,a2 e G such that I. F ai F =E;II. aFa2 ,F,a2 Fa 1 OE E and they are mutually disjoint. Let f (a) be a complex valued function on G such that I f (a)I 2 < + oo aeG

and( E

if(ai aai- 1)—f(a)12)

< E (i = 1,2). Then(

E if(a)12) < 14e. aeE

aeG

Proof. Put v(M)

=

f(a)2 for every subset M in G. Then aeM

e>(

If(ai a

f(a)I 2)

Faj 1 )4 — v(F) 4 1 •

aeG

Putting v(E)4 =s, then iv(a i F a 1- 1 ) — v(F)I .1v(a 1

v(F)4 I iv(a F a1- 1 )-1 — v(F) I < 2 s e

and so v(a i F al- 1 ) < v(F) + 2 se; hence s2 < v(ai Faj 1 ) + v(F) s2 12— s e. Since

E If(a2 a a2- 1, —f (all, , 2) 1 = E if(a2 a2 a a2 a27 1 )—f (a2 1 aa2 )1 2)+ ,

(

aEG

we have Iv(a2 F

aeG

v(F)i< 2 S £ and Iv(a2- 1 F a2 )— v(F)i< 2 s e. Hence, s

2

S

2

— 3SE and v(a2- 1 F a2 )> — — 3SE. Therefore, 2 1 s2 = v(E) ._v(F)+ v(a2- F a2) + v(a2 F a2- 1 )> s2 —7 s e, that is s< 14e. q.e. d. v(a2 F a2-1 )> v(F)— 2 s e>

— 2

4. Special Topics

184

4.3.4. Definition. Let G be a discrete group and let H be a subgroup of G. H is called residual in G if there exists a subset S of G\H (the complement of H in G) and elements a l , a2 of G such that G\H =Su aj 1 Sa 1 , and S, a2-1 Sa, a2 Sa2-1 are mutually disjoint subsets in G\H. 4.3.5. Definition. A sequence (Ta) in { U(G)} 1 is called an c-central sequence if lim sup 1 [T„,X]11 2 no . Since U U(11) is a-dense in U(G), we have 1 [T„, X]11 2 -40 n=i

(n--* co) for Xe U(G). Hence (Tn) is central. q.e. d. Now let G.(G 1 ,G2 ,..., H 1 ,112 ,..) with Gi = Z (i= 1,2,...). Put 00

Q(G,n) =

E

1011

(n= 1,2,...)

i=n n

and Q(G,m,n)=

E 0 H (m • " > mt and m 1 >m 3 ... > 11'4_ 1 , with ;lit > n„ and —

—

U{Q(G [I[ 2],n„ Mt)}

50 C U {Q(G[ 11 1],nt-1,111t-1)} 50 C••• 50

C U{Q(G[ 11 1],n1,m1)}• Since Q(G[I11],h,k)(i=1,2) is a finite sum of the form

(G i , G2, —; H1,H2, • •') (j= 1,2, ...), it has the canonical, strongly residual sequence [Q { Q(G [11i ], h,k), nl]. For simplicity, we shall denote Q(G[Ili],h,k) (resp. Q {Q(G [11 ], h, k), n1) by Qi(h,k) (resp. Q7[(h,k),n]).

with G

4.3. Examples of Factors, 2. (Uncountable Families of Types II I , II 56

III)

189

56

4.3.16. Lemma. If U {Q i (h,k)}CU 1•2 2 (i,j)1CU {Q i (p,q)} with h> p and q> k, then for arbitrary positive integers r and w there exists a (lo)3s positive integer s such that U {42i[(h, k), s]l C U {42i[(i,j), r]l with s> w.

Proof. Suppose this is not true. Then for each n with n> w there is a Tne(U { 42[(11 ,4n]l)1 such that IlTn—S112(10)3 6 for all

sc(uti2i[(iAr]l)i. Since (42?[(h,k),n]) is a strongly residual sequence in Q i (h,k), (Te) is a central sequence in U[121 (h,k)]. On the other hand, Q i (p, q)---- Q i (h,k)C ■ C (C, some subgroup of 12 1 (p,q)), Hence (Ta) is a central sequence in U[Q1 (p,q)]. Take Le {U [Q 2 (i,j)]} with 1lTn — Ta se . On the other hand, 545

56

U {Qi(nt- 1 ,mt- 1)} CU{Q2(nt-2,mt-2)} CU{Qi(nt-3,mt-3)}• Hence by the reasoning similar to the proof of 4.3.16, for each r there exists a k with k — 1 > re - 1 such that (1o)36 U {I2 [(lt _ 1 ,m, _ 1 ),k]} C U

{Qi [(n„me),r]C)111 .

For Te(U IR 423 [(rie , m e), (rt , s )]l) i (U {123 Rne,mt), r t]l) 1 there exists a re(Utinnt i,mt i),rt i]ili such that MT r M 2 n1 and U{Q(G[11 1 ],r)} U {Q(G[11 2],m2 + 1)1. For X' e[U {Q(G [11 1],r)}] 1, take Xe[U{Q(G[11 2],m2 +1)3. such that 111X' — X111 2 < 6. Then

HIP(' ,

III PC, 711112 =

because [X, T] = O.

— 711112 +111[X , 1 ] 1112 +111[X —7 11112+111[X

—

7 ] 1112,

—

7

] 11i2

200

4. Special Topics

(pG[ii](Y X) for YeU(G [111 ]) and

Since (pG[lin(X

X elP 0 U (GPIJ), — 7 ]1112 -111X'(T' — T)111 2 + 111(T — T)X1112 — T 1112+ (PG[L]PC * (T' — T)*(T'— T)X') 4 6+ (pGui,i (X' X'*(T' — T)* (T' — T)) 4 6 + (PG[LPC (T — T) * (T — T)X' X' *)1 (PGDEAT — Tr (T T)) 1

111[X'

< (5 +41 (54 = = (n m). Since {Pn(x)} is uniformly bounded and V is norm-dense in ner,(d), this implies that limPn(x)=x in the a-topology. Hence Pn (X). /1 n 1 if z in the o--topology, so that 7r4,(541)" is a factor. q. e. d. CO

4.4.3. Proposition. Let Jai = C) O n be the C*-infinite tensor product n= 1

of C* -algebras (9%1 with identities, and let (I n be a pure state on (n = 1,2, .. .).

Then the infinite product state ( p =

again pure.

n=1

n on d

is

Proof. One can easily see that 0 p n is pure on 0 93n . Suppose Ji n=1 si with

is a positive linear functional on n=1

93n 010-• (m= 1,2, .. .). Hence

n=1

. Then tP=1//(1)(p on

= tP(1)(p on 0

n=1

q.e.d.

(X)

Now let d =

93 n be the C* infinite tensor product of W* algebras -

=1

f93n}, and let (pn be a normal state on 93n (n= 1 ,2,...). Let

-

=

n= 1

(Pn

be the infinite product state on si. For each n, 1 0 - • 0 nvn (sa)' 010. • • (say ..i1(n ) is a W*-subalgebra of B(Yf l,) and

204

4. Special Topics

4.4.4. Proposition. nv(d)' =R(Afn in=1,2,...), where R(4In= 1 , 2, • • .) is the W*-subalgebra of B(/4) generated by {.Arn in=1,2,...}. Proof. Write 9,i (x).(xl,pn , 1 (/, .) (x e3) and set (yeB(Y4.)).

Then On is a pure state on B (

On be the

,n). Let ep = 0 n=1

00

infinite product state on 0 B(Ye; pn). Then by 4.4.3, Cp is a pure state n= 1 CO

on 0 B(Yf„). n= 1

Since each 1

) Q =.).

a cyclic vector for nvh(O n), we can consider Il

00

Then 7r,p (szl)" c no (0 B()) --- B(4).). n=1

Clearly R(../1/„. In=1,2,...)' D Consider the conditional expectation Pm considered in the proof of If

4.4.2. It is a mapping of nv

CO

fl

M

0 4)(4)) onto Tcv ( 0 B(Yfvn)010...) . n

n= 1 (n = 1 For xeR(Kn in=1,2,...)' and yeR(Arn'In=1,2,...,m), Pm (x y)= Pm (x)y= Pm (y x)= y Pm (x). Hence

Pm (x)e R(SIn = 1,2,...,m)' n no n

6i B(Yf yn ) 0 1 0 - • •

we have p, .

(

By the commutation theorem of tensor products (cf. 2.8.1), Pm (x)en g,(6 93n 0 1 0 • • •)" . 11

Since Pm (x) -œx in the o--weak topology, xen 9 (4)" . q.e. d. Next we shall state results about a special class of factors (called hyperfinite factors) which is fairly manageable and an extensive study has been done for those factors. Those factors are also important, because they are appearing in mathematical physics and Ergodic theory. We will not give proofs to most of the results, because it requires too much space. 4.4.5. Definition. A factor, with 1, on a separable Hilbert space Yt' is said to be hyperfinite, if there exists an increasing sequence of type Inp-subfactors {,i/p } with np < + co (p= 1,2,...), containing the identity of 41, such that the cf-closure of

U C/dp P=1

is

41 •

Let d be a separable uniformly hyperfinite C*-algebra and let cp be a factorial state on d Then the W*-algebra n v(21)" is a hyperfinite factor.

4.4. Examples of Factors, 3 (Other Results and Problems)

205

Conversely let ,// be a hyperfinite factor on a separable Hilbert space 4' and let

be the (7-closure of U

di'

tWp

with

c 4;) +1 and

P=1

,/#p

a type Lip-factor (n p < + co). Let (p be a faithful normal state on

, and let d be the uniform closure of U 4fp in 41. Then d is a P=1 separable uniformly hyperfinite C*-algebra and the restriction Cp of (p to d is a factorial state and furthermore n ip(s21)" is *-isomorphic to 4'.

4.4.6. Theorem. (cf. [120. Let 41 be a type 11 1 -factor on a separable Hilbert space 4'. Then the following conditions are equivalent. 1. 4' is hyperfinite, 2. there exists an increasing sequence of finite-dimensional *-sub00

algebras (J11) in 4' such that the o--closure of U .A1 is 4°. i= 3. (2)41 is the W*-algebra generated by a family of countable elements, A ni e di, there exist a finite-dimensional (fl) for e >0 and A 1 ,A 2 , * -subalgebra ,Ar of ,/ll and B 1 , B 2 , , A n e S. such that 11,4 1 — B 112 < E for (11 . 112, thee trace norm on jil). 4. there exists an increasing sequence of type1,,i-subfactors (A) of 4' oo

such that the o--closure of U

/gin

is

jd.

n=1

This famous theorem of Murray-von Neumann implies that there exists one and only one hyperfinite type 11 1 -factor in a separable Hilbert space. Now let

= 0 93„ be the C*-infinite tensor product of {93„} with n= 1

a type 4n-factor (yn < + Go), and let (Pn be a state on 93„. Then

On,

=

n=

(pn is a factorial state on „a, because (pn is factorial on 93„ for

each n. Suppose that 93 n =93 with a type 1 2-factor 93 (n= 1,2,...), and let (pn in=- 1,2,...) be a sequence of positive numbers with 00 and 00

ki ( Q — y Q)

00

then 4'.

n=1

eL"(Q,„ /2„), where ,un is the ren=1

striction of it to Q. Therefore to assure that the reduction theory of W*-algebras is non-trivial, the following question should be answered affirmatively: is there a global W*-algebra 4', with a separable predual, whose reduction theory 4'. f 4'(t) OW satisfies the following condition: there f2

exists a measurable subset 0 0 in Q such that /2(Q 0)./.2(Q) and for arbitrary two distinct t 1 , t2 in 00 , .11(0 is not *-isomorphic to 4(t 2 ). Obviously, to answer this question affirmatively, there must exist uncountably many examples of factors in a separable Hilbert space. In this section, by using the examples in 4.3, we shall construct examples of global types II ' , Hop and III W*-algebras satisfying the condition.

4.5. Global W*-Algebras (Non Factors)

217

Now let Q be the set of all infinite sequences (pi) of positive integers with pi =1 or i (i= 1,2, ...), and let 5= {OM be the compact group of order 2. Let S=S for n=2, 3, ... and let F =

Sn be the compact n=2

group obtained by the weakly infinite direct product of {S„}. Let 11 1 = (pi) be an element of Q. We shall identify 1I with the group G[1[1 ] in 4.3. Define (lii )„. 1 if pn =n and (II 1 )n = 0 if pn = 1 (n=2, 3, ...). Then (0102,(111)3, • • .) will define an element y in F. Set p(II)=y. Then p is a one-to-one mapping of Q onto F. By using the p, we shall identify Q with F. Since F is a compact group, we have the corresponding compact group structure in Q by this identification. 4.5.1. Lemma. Let /11 = (G1 , G2,...;H1,H2,...) and z1 2 =(G1 ,G2 ,...;J1 ,J2 ,...) (cf. the group construction of 4.3). Suppose that there exists a homomorphism of Hi onto Ji for all i. Then we can define a homomorphism of z1 1 onto Z12 such that = the identity on G. and on Hi for all i, where G. and Hi are identified with the corresponding subgroups of (G1 ,G2 , ...; H 1 ,H 2 , ...). This lemma will be almost trivial. Now let Foo be the free group of denumberable generators and let A=(G 1 ,G2 ,...; Hi,H2, with Gi = Z and Hi = Foe . Let (r1 ,r2 , ...,rn) be a finite sequence of positive integers such that ri = 1 or i (1= 1,2,...,n).

Consider the group F

E

C)L,. i . Then there exists a homomorphism of i= We shall pick up one homomorphism and fix it;

r2 , r) and so is a function of (ri , r2 , ...,rn). we denote this by Let Ili e Q with 1[1 .(pi ,p2 ,...). Then = (Gi, G2, • • .; Mi(111), M2(110, •••) with Gi = Z (i= 1,2, ...). By 4.5.1, we can define a homomorphism of A onto G[II1 ] such that = the identity on G. and (111 ) =(Pl,P2, • • • , p„) on Hn (n= 1, 2, ...). Let y be the Haar measure on A such that v(e)=1 (e, the identity of A), and let 1 1(A) be the group algebra of A composed of all y-integrable functions on A. Let Itfrfi lfie j be the set of all positive definite functions on A with tlip (e)=1. Define 11 x 2 = sup (x* *x)(g)111 fi (g)dv(g) for x e 11 (A) (x*(g)= x(g 13 E'; A and x**x is the convolution of x* and x). Then ' will define a C*-norm on 11 (A). The completion of 11 (A) under this C*-norm II' II is called the group C*-algebra of A and is denoted by R(A).

4. Special Topics

218

Q, we shall define a trace T A on R(A) as follows: take the homomorphism (/1.) of A onto G[A] and define T,i (g)=6 (,1)(g) (eA ), where (5 (A)(n) is the function on GP] such that 6 (.)(g) (/). 1 if (.1)(g). l and ( W.)(g)(1) = 0 if ()) (g) l (l e GPI), and e is the identity of G [A ]. Then t2 is a central positive definite function on A with T2(e). 1; therefore it will define a unique tracial state (denoted by the same noFor /lei'

tation TA) on R(A). Let fir2 „Ye°,1 1 be the *-representation of R(A) on a separable Hilbert space Yfj constructed via TA . Then cA (R(A))" is *-isomorphic to U(G[]). Now suppose that a sequence (.1n) of elements in F with An — (1) n,1919 n,29 • • •) converges to .1.0 in F with .1,0 = (n 0,14) 0,29 •••) in the topology of Q F). Then for an arbitrary geA, there exists a positive integer n o such G 0 and that g belongs to a subgroup of A generated by G1 , G2 , Hi , H2 , , H = (Gi , G2, ...; H1,1-1 2 , ...) with Gi =Z and Hi =FG, for i= 1,2, ...). Since A n -- /3, 0 , there exists an n 1 such that pn i =p0 , i for i = 1,2, ..., n o . Hence lirn T An (g) = A .(g) (g e A). This implies that the sequence (TA.,) of traces on R(A) converges to the trace TA0 on R(A) in the .o-(R(A)*, R(A))-topology (R(A)* the dual Banach space of R(A)). Therefore the mapping 11: /3.--T 1 of F into the state space SI of R(A) is continuous; it is one-to-one, because rcA (R(A))" is not *-isomorphic to n,y (R(A))" if AO 2'. The compactness of F implies that the n is homeomorphic. Let cbl. S2) with the total mass 1; by using the n, be the Haar measure on F we can introduce a Radon measure ,u on 5 9 such that dpi(r2)=d2.. Define tfr(x) = f(p(x)d,u((p) (xeR(A)). Now we shall show. 4.5.2. Theorem. tt is the central measure of II'.

Proof. This is obtained immediately from 3.1.18. 4.5.3. Corollary. There exists a type H i W*-algebra 4' on a separable Hilbert space satisfying the following properties: 1. The center of ./W is isomorphic to Lœ(F,dA), where F is an infinite compact group and dA. is the Haar measure on F. 2. For its reduction theory ./# = 1,/#(2)d.1,,,/#(.11 ) is not *-isomorphic to ./#(2.2)

for arbitrary two distinct

)1 2

in F.

4.5.4. Corollary. There exist types IL, and III W*-algebras on a separable Hilbert space satisfying the similar properties to 4.5.3.

4.6. Type I C*-Algebras

Proof. Consider ,./N g - B(r) with dim(Ye)= ( )B(Yf) =

°(,11 ) 5 B(cif)

219

0 . Then

di()) (7) B(Yt)chl.

is not *-isomorphic to ,W(.12 )(7) B(Jr) if /11 0.1.2

(cf. 4.3). Analogously consider .7d' C- D. IP, where IP is the III-factor of Pukanszky (cf. 4.3). q. e. d. Remark. By using Powers' factors, we can construct another example of a non-trivial global type III-factor (cf. [169]). Concluding remarks on 4.5. Now the following problem would be very interesting. Let ./d be a global W*-algebra on a separable Hilbert space. Find nice conditions under which # can be written as s-c5 Z (Z, the center of .7#; Jr, a factor). By using these conditions, can we find directly a global W*-algebra such that the reduction ./#= 4'((p)dp((p) satisfies that /I is a con-

tinuous measure and there exists a pt-measurable subset Yo in Y with 1,45'2 — =0 such that for any two distinct (p i , (p 2 in Yo , .. #(pi) is not *-isomorphic to ,/#((p 2 )? References. [169], [173].

4.6. Type I C*-Algebras In this section, we shall give some considerations to type I C*-algebras which are important in the group representation theory, and in the theory of singular integral operators (cf. [24]). Let G be a locally compact group, tt a left invariant Haar measure on G and let Ll (G) be the Banach space of all complex valuedg-integrable functions on G. For f2 eL 1 (G), define a multiplication * and a *-operation as follows: f I * f2 (x) = fi (y) (y -1 x)d ,u(y) and fi*(x)= p(x) f (x 1 ), where p is the modular function on G. By a unitary representation u,Y(} of G on a Hilbert space it; we mean a mapping x--ti(x) of G into the group of unitary operators in Ye satisfying the following conditions: {

1. u(x i x2 )=u(x i )u(x2) (xi ,x2 e G); 2. a function x—qu(x),n) is continuous on G

220

4. Special Topics

For feLl (G), put n(f)=Su(x)f(x)4(x), where the integral is taken by using the strong operator topology. Then a mapping f satisfies the following conditions: 1. n(f1*f2)=n(f1)n(f2)(f1,f2e (G)); 2. ir(f *)= n( f)* ( fe (G)); 3. Ilit(f)ll I f 111 (fe (G)); 4. [n( f ) fe .12 (G)] = ye e . nowhere trivial, where M M1 is the norm on Ll (G). Such a mapping {n, A' } of Ll (G) into B(Ye) is called a nowhere trivial *-representation of 1,1 (G). Conversely we may show that given a nowhere trivial *-representation Al of Li (G), we can construct a unitary representation {u, } of G with Tr( f ) = u(x) f (x)d,u(x)(fe L l (G)). Therefore the unitary representation theory of G can be reduced to the *-representation theory of Li (G). Now let {11 . 11,1 0ce be the set of all C*-seminorms on Li (G), satisfying II f 11. I f 111 fe (G)), and define f I = supll f Le . One can efl

easily see that I is a C*-norm. Let R(G) be the completion of 1,1 (G) under the C*-norm 11.11. Then R(G) is a C*-algebra which is called the group C*-algebra of G. Then the unitary representation theory may be reduced to the *-representation theory of R(G). A basic principle in the representation theory is to pick up basic representations as building blocks, to determine all such basic representations and to construct an arbitrary representation by integrating basic ones. Therefore we have first to decide what kind of representations are suitable to basic ones. The representation theories of commutative groups and compact groups may suggest irreducible *-representations as the basic ones. But, from the reduction theory of W*-algebras, we may guess that this choice is not reasonable unless a corresponding W-algebra n(R(G)) is of type I. For the general theory, we should choose factorial *-representations as the building blocks. On the other hand, if n(R(G)) is of type I for every *-representation {n,Ye}, we may follow the classical line and expect a fairly nice representation theory. Thus it is a very important problem "which groups have type I representations only ? " A locally compact group G is called a type I group if u(G)" is of type I for every unitary representation It is known that semi-simple Lie groups, nilpotent Lie groups are type I groups (cf. [34], [74], [102]). But there exists a solvable Lie group which is not of type I (cf. [114]). Now we shall define.

4.6. Type I C*-Algebras

221

4.6.1. Definition. Let d be a C*-algebra. si is said to be a type 1 C*-algebra if for every *-representation {71,/f} of d, the weak closure n(szl) of 44) on A' is a type I W*-algebra. (a *-representation {n,.*'} of a C*-algebra sal is said to be of type I (resp. II, III) if the weak closure 491) of n(d) on A' is of type 1 (resp. II, III)). One can easily see that the study of type I C*-algebras is important for the group representation theory. On the other hand, we can not expect type I C*-algebras in mathematical physics in fact, the weak closures of the representations which appear in mathematical physics are of type III in many cases (cf. [3]). In this section, we shall give characterizations of type I C*-algebras. 4.6.2. Definition. Let si be a C*-algebra. si is said to satisfy the condition of Glimm if for every non-zero positive element h in ,sa f, there exists an irreducible *-representation {n,Y(} of si such that the dimension of the range space of Tc(h) 2. 4.6.3. Definition. Let ,s2I be a C*-algebra. ,s21 is said to be smooth if for every non-zero irreducible *-representation {n,clf} of sal, n(d) contains a non-zero compact operator on Yf. Then we shall show 4.6.4. Theorem. Let si be a C*-algebra. The following conditions are equivalent: 1. 2. 3. 4.

,s2I is a typeI C*-algebra; si is smooth; si has no type III factorial *-representation; any quotient C*-algebra of sal does not satisfy the condition of

Glimm— i. e. for every non-zero quotient C*-algebra 0 of si, there exists a positive element h (> 0) in 3 such that the rank of n(h) =(F' l, l) and 1 Q e E' ffQ, 0* (F') O. Since F'e1t,2(0 2 )" c 7r,2 (d2 )" and 0* is a-continuous, there exists a non-zero finite projection e in ,iff with e < F' and 0* (e) 0, and so there is a non-zero projection p with Ap O, in Q such that tfr(t)(a0 (t))0111(t)(0(a 0)(t)) for all t e G.

4. Special Topics

240

Clearly G n (0— Q)0 (0). Take a to e G n (Q— Q) and define a linear functional tk i on sd as follows: 1 (c)=4/(t0)(n(c)(t 0)) (ced). Then 0, is an atomic state on d , since n(d)(t 0)=Bi for some i. Put ao =n(co) for some co in d, and define a linear functional ty2 on 93 + C co as follows: 04 (d + c0). (t 0)(7r(d)(t 0) + 0(a0)(t 0)) for d e and a complex number .1. Then IV2(d + c 0)I

110 (011 11 7r(d)(t 0 ) + AO (a o)(t • 110 (t 011 II 0 (n(d) + A. 7* (t 011 • Illgt 011 II 4) (7r(d) ± 7* o))11 110(011 II n(d) + Tc(c 011 • Ilt/i(to) l II d+A.c0 M.

Hence 1/6 is well defined and is bounded. Let tp 2 be a linear functional on cd with 11021I = 1W2 and 02 =0'2 on 93+Cco . Since tP2( 1 )=V2( 1 )= 110(t0)115 02 is positive, and clearly tp 1 = on 93. Hence by 4.7.5, 0 1 =02 on d, so that tk I (co) = IP(to)(n(co)(to))=IP(to)(ao(to)) = 02

a contradiction.

(C

O = 0 (t0)( 4)

(C

O) (to))

q. e. d.

4.7.7. Definition. Let d be a C*-algebra. d is said to be amenable if there exists an increasing directed set ts41,„1 of type I C*-subalgebras in d such that the uniform closure of Usic, is d. Remark. Every type I C*-algebra is obviously amenable, and every uniformly hyperfinite C*-algebra is also amenable. 4.7.8. Corollary. Let d be a separable C*-algebra, 93 an amenable C*subalgebra of si. If 0 separates e u (0), then 93= Proof. It suffices to assume that d has an identity and 93 contains it. Suppose that 93 d. Take a self-adjoint element f, with 11 f 11= 1, in the polar 93 0 of 93 in d*, and let f=f+=f- be the orthogonal decomposition of f. Put cp. f+ + f . Then clearly n v(93)" Put do =n,p (si) and 93 0 = n,,,(93). Since 93 is amenable, there exists an increasing directed set of type I C*-algebras in 93 such that the uniform closure of U 93„ is 93. 2E11 Without loss of generality, we may assume that 1e93„ for all a el Since n,p (93 2)" is a type I W*-algebra, 7r,p (93 c )' is a type I W*-algebra. Hence by 4.4.19, there exists a conditional expectation Pa of B(Yf q',) onto

4.7. On a Stone-Weierstrass Theorem for C*-Algebras

241

Let B(B(Ye e )) be the algebra of all bounded linear operators on B(Yeço). Then B(B(1 9,0)).(B(19,„)07 414p)*)*. Take an accumulation point 0 of IPŒ LEil in the a-topology of B(13(Y‹,)). Then P(b)=b (ben,(9313)) for a f3, and so 0(b)= b (ben( (23 fl )) for fi el Therefore 0 is the identity operator on U 7000). /3E II

Since 0 is uniformly continuous, 0 is again the identity operator on the uniform closure of U ir„,(0 fl) ( = /3E11

On the other hand, 0(B())E- the a-closure of U POE(B(Y4 p)) and ael1 SO

0(B(Yf er,))c the a-closure of U ner,(93 0,)" ( =-_ the a-closure of

7r(p (0)).

°cell

Hence by 4.7.6, 0(a)=a for aen,(d), and so nv (d)cn,(0)", a contradiction. q. e. d.

Remark 1. If d is a type I C*-algebra, then any C*-subalgebra 0 is again a type I C*-subalgebra (cf. 4.6). Therefore this theorem implies that the Stone-Weierstrass theorem is true for all separable type I C*-algebras. Kaplansky [96] proved this theorem without the assumption of separability, using a different method.

Remark 2. Let A (resp. di ) be a separable uniformly hyperfinite C*-algebra with type (p 1 ,p2 ,...) (resp. (q 1 , qi , ...))—i.e. there exists an increasing sequence{-9/11 , n (resp. -2/2,n) of type 4. (resp. I gnyfactors with pn < + oo (resp. qn n such that pn is a divisor of q qn is a divisor of pm . Moreover he proved that if cl2 is a C*-subalgebra, containing the identity of A, of A, then for each positive integer n there exists an integer m > n such that qn is a divisor of pm . One can easily see that conversely if the sequence (qn) satisfies the above condition, we can find a uniformly hyperfinite C*-subalgebra 0 of s14 with type (q 1 , q i , ...) such that 0„ c A . (n=1, 2, ...), where {0„} is an increasing sequence of type 4-factor's in 0 with the uniform 00

closure of n =i

The following problem would be interesting: let 0 be a uniformly hyperfinite C*-subalgebra, with type (q 1 , qi , ...), of the C*-algebra A .

242

4. Special Topics

Then can we find an increasing sequence Ig n} of type Ipn-factors in d1 such that 23 n gn and the uniform closure of U gn is 02/1 . n=1

The above Stone-Weierstrass theorem might be useful to attack this problem. 4.7.9. Corollary. Let d be a separable C*-algebra, 0 a C*-subalgebra. Suppose that there exists a *-representation tn,Yel of si such that Tc(si)" is a finite W*-algebra and 740)" ir(szl)". Then 93 can not separate 61,u(0). Proof. It suffices to assume that dim(Ye), 19 La f , 19 Ra f , 19 -411,14), 19 Œ 0, 20 sy(, .i/14( ), 20 s, 20 ce(o , 20 s*(11, 1(*), 20 s*, 20 C, 23 de, 24 1(a), 25 r (a), 25 s(h), 25 c(p), 25 y ea, 30 ca s(T), 31 f ± , 31

f - , 31 WI, 32 B(Ye), 33 H, 35 T( A ), 36 (P., 37 Tr (h), 36 dim(), 38 In,,Yel, 40 {n o, Jrg,}, 40 flEw, feb 40 x P' 40 Y, 41 { U, K}, 41 d**, 42 Li (Q,p), 45 L' (C2, p), 45 C(X9), 46 fei 0 're2 , 48 48 2", 48 48 [n(d) 4 50 R (d(, X), 50 { ir. , Jr.}, 51 (

E

aell

In E', E' API , 51 0*, 53 0**, 53 s(n), 54 E 0 F, 58 E®fl F, 58 Co (Q, F), 59 Co (Q) OA F, 59 Ll (Q, p, F), 59 1)(0,p) 07 F, 59 cp 0 vi, 60 si 0a 0, 60

d®Œ , 66 d 0 0, 66 ,9, 66 oe4', 66 di g s., 67 Card (I1), 69 0 da, 75 ad

0 Aa , 75

aell

0 (P., 75

ctell

p — q, 79 pq, 79 p-