Introduction to Operator Algebras

INTRODUCTION TO LI BING-HEN Academia Sinica, Beijing ' • Ib World Scientific Singapore • NewJersey- London • Hong K...

Author: Li Bing-Ren

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INTRODUCTION TO

LI BING-HEN Academia Sinica, Beijing

'

•

Ib World Scientific

Singapore • NewJersey- London • Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd. POBox 128. Farrer Road. Singapore 9128 USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

INTRODUCTION TO OPERATOR ALGEBRAS Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orbyany means, electronic or mechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-0941-X

Printed in Singapore by Utopia Press.

Introduction

An operator algebra here, precisely speaking, is a * algebra consisting of bounded linear operators on some Hilbert space, i.e., a * subalgebra of B(H), where B(H) is the collection of all bounded linear operators on some Hilbert space H. Since it is an infinite dimensional object (generally, H is infinite dimensional], so for studying it we must ask that it is closed under some topology. Under usual linear topologies in B(H), we find that the closures of operator algebras are just two classes: weak closure and uniform closure. Hence, we need mainly to study the weakly closed operator algebras (Von Neumann algebras or W·-algebras) and the uniformly closed operator algebras (C·-algebras). When J. Von Neumann studied the spectral theory of operators and the quantum physics, he recognized that we need new mathematical tools for studying the infinite dimensional systems in the physical world. In 1929, he introduced the concept of weakly closed rings, renamed Von Neumann algebras by J. Dixmier latter. He pointed out two essential characteristics: 1) such rings must be self-adjoint; 2) such rings are weakly closed. So that each spectral projection of any self-adjoint operator in such ring belongs still to this ring. J. Von Neumann and his collaborator F. Murray laid down the foundation for this new field of Mathematics during the period of the 1930's and early in the 1940's. Their results are still important up to now. Then, many important results and powerful techniques were added to this theory, for example, the structure theory of factors, the general theory of weights and traces, Tomita-Takesaki theory, the Connes classification of type (III) factors, Jones index theory, and etc. The theory of Von Neumann algebras got great progress and many applications. In 1943, I.M. Gelfand and M.A. Naimark laid down the foundation for the theory of C·-algebras. They showed that a Banach * algebra can be isometrically * isomorphic to a uniformly closed operator algebra on some Hilbert space under few conditions. Then, I.E. Segal gave its perfected form, i.e., the famous GNS construction. The theory of C·-algebras can be divided

VI

into two parts: the essential structure of the algebras, and the theory of representations. Of course, these two parts have very closed connections. As well as the theory of Von Neumann algebras, the developments of the theory of C·-algebras are also very great. Now the theory of operator algebras becomes one of the most important fields of mathematics. This book consists of 17 Chapters and an appendix. Chapter 1 is the fundamentals of Von Neumann algebras. It contains the definition of Von Neumann algebras; Von Neumann's double commutation theorem; the commutation theorem of tensor products of Von Neumann algebras; Kaplansky's density theorem; the comparison of projections; the normalness, polar decomposition and the orthogonal decomposition of linear functionals; the Radon-Nikodym theorem; the structure of normal * homomorphisms; the comparison of cyclic projections; the spatial theory; a-finite Von Neumann algebras, and etc. In particular, the analysis of topologies is given in detail. Moreover, by the Van Daele-Rieffel approach, we can put the commutation theorem of tensor products in this chapter, and it is convenient for the latter. Chapter 2 is the fundamentals of C·-algebras. It contains the definition of C·-algebras; the Gelfand-Naimark-Segal construction; the characterization of the extreme points in unit ball; Kadison's transitivity theorem; the theory of representations; the enveloping Von Neumann algebras; the multiplier algebras; the axioms of complex and real C·-algebras, and etc. From the GNS construction, every (abstract) C* -algebra can be isometrically * isomorphic to a uniformly closed operator algebra on some Hilbert space (concrete C*algebra). Hence, the definition of C·-algebras is independent of the Hilbert spaces. Moreover, last section of this chapter is devoted to the famous GelfandNaimark conjecture: the axioms for C·-algebras. This conjecture is very interesting and important, and in a long period of time many mathematicians studied it. In Chapter 3, we discuss the tensor products of C* -algebras. The tensor products are a useful method to construct new C·-algebras from given C*algebras. On the algebraic tensor product of C·-algebras, the first natural C·-norm is the spatial C*-norm. But M. Takesaki discovered that there exist more than one C·-norm possibly. So we must study the properties of general C·-norm. Here, the important result is that each C·-norm on the algebraic tensor product of C·-algebras is a cross-norm. We also discuss the completely positive maps on C·-algebras, inductive limit of C·-algebras, infinite tensor products of C*-algebras. In last section of this chapter, we give an introduction to an important class of C·-algebras: nuclear C·-algebras. W*-algebras are the "abstract" Von Neumann algebras. In Chapter 4, using Tomiyama's projections of norm one, the Sakai's theorem (any W*-algebra can be * isomorphic to some Von Neumann algebra) is proved. We also discuss the

Vll

normal part and the singular part of a linear functional, the characterizations of weakly compact subsets in the predual, and etc. In some sense, this chapter is the continuance of Chapter 1. In Chapter 5, we study the abelian operator algebras. It contains the general forms of abelian W·-algebrasj the characterizations of the spectral spaces of abelian W·-algebras; the measure description of an important class of representations for an abelian C·-algebra, and etc. Moreover, the section 5.1 can be seen as preliminaries of this chapter. Chapter 6 is the traditional program: the classification of Von Neumann algebras. First, we give the Murray-Von Neumann decomposition of a Von Neumann algebra. Then we discuss the properties and characterizations of the finite, the semi-finite, the pure infinite, the discrete (or type (I)), the continuous and type (II) Von Neumann algebras. Finally, we study the classification of the tensor products of Von Neumann algebras. The theory of factors occupies the central place in the theory of Von Neumann algebras. In Chapter 7, first we describe the classification of factors with dimension theory. Then we point out the uniqueness of the hyperfinite type (Ill) factor up to * isomorphism; and the existences of type (II) and (III) factors by the standard method of group measure spaces. Moreover, the examples of non-hyperfinite type (lid factors and non-nuclear C·-algebra are given. Chapter 8 is the Tomita-Takesaki theory. It is an important part of modern theory of operator algebras. Following Van Daele-Rieffel approach, we discuss the KMS condition, and the modular automorphism group of a W· -algebra, It also contains the innerness of the modular automorphism group in semi-finite case; the Connes unitary cocycle theorem; and etc. Since we don't study the general theory of weights and generalized Hilbert algebras, so the results are restricted to cr-finite case. In 1930's, Murray-von Neumann divided the factors into five classes. Later, we know that such classification is not complete. Even there exist uncountably many non-isomorphic type (lId, (1100 ) ad (III) factors on a separable Hilbert space. 1973, A. Connes got a great progress on the classification of type (III) factors. Chapter 9 is devoted to this theory. It contains the Arveson spectrum; the Connes spectrum; the type (III),,) (0 < .x < 1) classification of type (III) factors in er-finite case; and the examples of type (III),,) factors. We discuss the Borel structure in Chapter 10. This chapter can be seen as preliminaries for some later parts. E. Effros introduced a standard Borel structure into the collection of all Von Neumann algebras on a separable Hilbert space. Chapter 11 is devoted to this theory. Chapter 12 is the important traditional program: Reduction theory. The aim of this theory is to simplify the study of general Von Neumann algebras.

Vlll

By the view of Borel maps, we discuss the measurable fields of Hilbert spaces, operators and Von Neumann algebras. Then we discuss the decompositions of Hilbert spaces and Von Neumann algebras. Moreover, the Borel subsets of the Von Neumann algebras space and the state spaces on separable C·-algebras are also studied in this chapter. In Chapter 13, we discuss the spectrum of a C·-algebra, CCR, GCR, NGCR algebras. The main result is that GCR¢::::> type I for a C·-algebra (KaplanskyGlimm-Sakai) . Chapter 14 is the decomposition theory for the states on C·-algebras. By Choquet theory and the Sakai-Ruelle construction, any state on a C·-algebra can be represented as an integral of pure states or factorial states (roughly speaking). This theory is also very useful for the mathematical physics. In 1960, J. Glimm introduced a class of C·-algebras: uniformly hyperfinite (UHF) algebras. Then in 1972, O. Bratteli introduced approximately finite (AF) algebras, which are the generalization of (UHF) algebras. Chapter 15 is devoted to this theory. It contains the equivalent definitions; isomorphism theorem; Bratteli diagrams; dimension groups (Ko-groups), stable isomorphism theorem, and etc. Now the K-theory is very important in the study of C·-algebras (see [11]), and the dimension group is an excellent example. Chapter 16 is the theory of crossed products. It contains W· -crossed products, Takesaki duality theorem; C·-crossed products, Takai duality theorem; and some examples. Crossed products have long been used to construct interesting C·-and W·-algebras. But it is difficult to obtain some good information about the internal structure of crossed products. Hence, we should pay more attention to this field, and study it furthermore. Chapter 17 is an introduction to the Jones index theory. 1930's Von Neumann and Murray introduced the concept of coupling constant. In 1983, V. Jones gave a new form to this concept. Then he defined the index for subfactors, and proved a surprising theorem on the values of index for subfactors. And also he introduced a very important technique in the proof of this theorem: the towers of algebras. Now this theory becomes a focus of many fields in mathematics and physics. The appendix is a survey on weak topology and weak * topology in a Banach space and its conjugate space. Maybe, it is helpful to understand the topologies in a Von Neumann algebra. This book is an introduction, written in a self-contained manner (for reading it, just need the general knowledge of functional analysis and measure theory, [178], [67], and some foundation of abstract harmonic analysis). We hope that it can offer the basic concepts, techniques, structures and important results of the theory of operator algebras. And we also expect that it can serve as a text for graduate students. The author would like to express here his sincere gratitude to Professor K.

IX

Shiraiwa, Professor J. Tomiyama and Professor Xia Daoxing for their recommendation and the moral support. He acknowledges gratefully the support of the NSF of China. And the author is also very grateful to his home Institute for its support and typing the manuscript.

Contents

Introduction Chapter 1 Fundamentals of Von Neumann Algebras

1

Banach spaces of opertaors on a Hilbert space Locally convex topologies in B (H) Von Neumann's double commutation theorem Tensor products of Von Neumann algebras Comparison of projections and central cover Kaplansky's density theorem Ideals in Von Neumann algebras Normal positive linear functionals Polar decomposition and orthogonal decomposition Radon-Nikodym theorems The equivalence of the topologies s" and r in a bounded ball Normal * homomorphisms Comparison of cyclic projections and spatially * isomorphic theorm 1.14. a-Finite Von Neumann algebras

63 66

Chapter 2 Fundamentals of C*-Algebras

71

1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 1.9. 1.10. 1.11. 1.12. 1.13.

2.1. 2.2. 2.3. 2.4. 2.5. 2.6.

Definition and basic properties of C*-algebras Positive cones of C"-algebras States and the Gelfand-Naimark-Segal construction Approximate identities and quotient C"-algebras Extreme points of the unit. ball and the existence of an identity Transitivity theorem and irreducible * representations

1 8

15 20 30 35 38 40 46 50

55 60

71 75 79 89 94 98

Xll

2.7. Pure states and regular maximal left ideals 2.8. Ideals and quotient C*-algebras 2.9. Hereditary C*~subalgebras 2.10. Comparison, disjunction and quasi-equivalence of representations 2.11. The enveloping Von Neumann algebras 2.12. The multiplier algebra 2.13. Finite dimensional C*-algebras 2.14. The axioms for C·-algebras 2.15. Real C*-algebras

103 107 112

*

Chapter 3 Tensor Products of C*-Algebras 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 3.9.

Tensor products fo Banach spaces and cross-norms Tensor products of C*-algebras and the spatial C*-norm The maximal C·-norm States on algebraic tensor products The inequality >.(.) < ao(') < a(.) < 1(') Completely positive maps The inductive limit of C·-algebras Infinite tensor products of C·-algebras Nuclear C·-algebras

Chapter 4 W·-Algebras 4.1. 4.2. 4.3. 4.4. 4.5.

Projections of norm one W·-algebras and their * representations Tensor products of W·-algebras Completely additive functionals and singular functionals The characterizations of weakly compact subsets in the predual

Chapter 5 Abelian Operator Algebras 5.1. 5.2. 5.3. 5.4.

Measure theory on locally compact Hausdorff spaces Stonean spaces Abelian W·-algebras * Representations of abelian C·-algebras

Chapter 6

116 119 123 129 130 148

163 163 165 171 175 179 187 196 200 205

215 215 219 223 226 233

237 237 242 248 257

Xlll

The Classification of Von Neumann Algebras

267

6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8.

267 269 275 285 288 298 302

6.9.

The classification of Von Neumann algebras An ergodic type theorem for Von Neumann algebras Finite Von Neumann algebras Proprly infinite Von Neumann algebras Semi-finite Von Neumann algebras Purely infinite Von Neumann algebras Discrete (type (I)) Von Neumann algebras Continuous Von Neumann algebras and type (II) Von Neumann algebras The type of tensor products of Von Neumann algebras

Chpater 7 The Theory of Factors 7.1. 7.2. 7.3. 7.4.

Dimension functions Hyperfinite type (lId factors Construction of factors of type (II) and type (III) The existences of non-hyperfinite type (lId factors and non-nuclear C·-algebras

306 308

313 313 317 327 338

Chpater 8 Tomita-Takesaki Theory

347

8.1. 8.2. 8.3.

347 354 360

The KMS condition Tomita-Takesaki theory The modular automorphism group of a a-finite W"'-algebra

Chapter 9 The Connes Classification of Type (III) Factors

369

9.1. 9.2. 9.3. 9.4. 9.5.

369 374 385 389 393

Preliminaries The Arveson spectrum The Connes spectrum The Connes classification of type (III) factos ( a-finite case) Examples of type (IlIA) factors

Chapter 10 Borel Structure

405

10.1.

405

Polish spaces

XIV

10.2. 10.3. lOA.

Borel subsets and Sousline subsets Borel maps and standard Borel spaces Borel cross sections

Chapter 11 The Borel Spaces of Von Neumann Algebras 11.1. 11.2. 11.3. 11.4.

The standard Borel structure of W(X*) Sequences of Borel choice functions The Borel spaces of Von Neumann algebras Borel subsets of factoial Borel space

Chapter 12 Reduction Theory 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. 12.7. 12.8.

Meausrable fields of Hilbert spaces Measurable fields of operators Measurable fields of Von Neumann algebras Decomposition of a Hilbert space into a direct integral The relations between a decomposable Von Neumann algebra and its components The constant fields of operators and Von Neumann algebras Borel subsets of the Borel space of Von Neumann algebras Borel subsets of the state space of a separable C·-algebra

Chapter 13 Type I C·-Algebras The spectrum of a C·-algebra Elementary C·-algebras and CCR ( liminary ) algebras GCR ( postliminary) algebras and NGCR ( antiliminary ) algebras 13.4. The existence of type (III) factorial * representations of a NGCR algebra 13.5. Type I C·-algebras 13.6. Separable type I C·-algebras

13.1. 13.2. 13.3.

Chapter 14 Decomposition Theory 14.1. 14.2.

410 414 420

426 426 430 435 438

448 448 455 459 464 470 474 477 487

490 490 501 506 512 528 533

538

Choquet theory of boundary integrals on compact convex subsets 538 The C-measure and C-isomorphism of a state 542

xv

14.3. 14.4.

Extremal decompostion and central decomposition Ergodic decomposition and tradal decompositon

553 559

Chapter 15 (AF)-Algebras

570

15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7.

570 579 585 590 594 602 606

The definition of (AF)-algebras Dimensions and isomorphic theorm The Bratteli diagrams of (AF)-algebras Ideals of (AF)-algebras Dimension groups Scaled dimension groups and stablly isomorphic theorem The tradal state space on an (AF)-algebra

Chapter 16 Crossed Products 16.1. 16.2. 16.3. 16.4. 16.5. 16.6.

613

W*-crossed products Takesaki's duality theorem Group algebras and Group C*-algebras C*-crossed products Takai's duality theorem Some examples of crossed products

613 624 630 640 659 669

Chapter 17 Jones Index Theory

682

17.1. 17.2. 17.3. 17.4.

682 693 698 707

The coupling constant Index for subfactos The fundamental construction The values of index for subfactors

Appendix Weak Topology and Weak

* Topology

714

References

719

Notation Index

729

Subject Index

733

Chapter 1 Fundamentals of Von Neumann Algebras

1.1. Banach spaces of operators on a Hilbert space Let H be a Hilbert space over the complex field (C. We shall denote by F(H), C(H) and B(H) the sets of all linear operators of finite rank, all compact linear operators, and all bounded linear operators on H respectively. The identity operator on H is denoted by 1H, or 1 simply if no confusion arises.

Proposition 1.1.1. If H is separable and infinite dimensional, then C(H) is the unique non-zero proper closed two-sided ideal of B (H). Let I be a non-zero closed two-sided ideal of B(H), and 0 =I- a E I. Then there are €,,., E H such that a€ = ,., =I- o. For any E H, there is bE B(H) such that b,., = e'. Then

Proof.

e,,.,'

ba(c ® ,.,') where

= (hac) ®,.,' = €' ® fI' E I,

e ® ,.,' is the operator of one rank on H

:

€' ® ,.,'(.)

= (" ,.,')

c'.

Thus

F(H) c I and C(H) c I. rr C(H) =I- I, we have t E I\C(H). Then h = (t*t)l/2 E I\C(H). Let {eA} be the spectral family of h. Since h is not compact, it follows that there exists E > 0 such that dim(l - ee)H = 00. Now we can take an isometry v from H onto (1 - ee)H. Then v*hvH = H and o'h» is invertible. Since v*hv E I, we have I = B(H). Q.E.D.

Proposition 1.1.2. If H is infinite dimensional, then C(H) is not the conjugate space of any Banach space.

Proof.

By the Krein-Milman theorem, the closed unit ball of the conjugate space of any Banach space has an extreme point at least. So it suffices to

2

show that the closed unit ball of C(H) has no any extreme point, i.e., for any a E C(H), Hall < 1, we only need to find a non-zero b E C(H) such that

lIa ± bll < 1.

If the rank of a is finite, let HI be the linear span of aH and a· H, and H 2 = Ht, then dimHI < 00, HI and H 2 are invariant for a and a*, and a = a· = 0 on H 2 • Thus it is easily verified that b exists. Now suppose that the rank of a is infinite. Let a = wh be the polar decomposition of a, and write h = .AnPn, where {Pn} is a sequence of

L n

projections of one rank on H, and PnPm = 0, 'Vn i= m,O < .An < 1, 'Vn, and .An ---+ O. Pick Nand g > 0 such that .AN E (0,1), and I.AN ± cl < 1. Now let b = CWPN. Then b i= 0 and Iia ± bll < 1. Q.E.D.

Remark.

A projection P on a Hilbert space, we always mean that it is

. t ·. se lii-a d' l10fn , i.e., P = P = P2 .

We shall denote by S (H) the set of all operators of Hilbert-Schmidt class on H, i.e., a E S(H) if a is compact, and l.A nl 2 < 00, where {.An} is the set

L n

of all eigenvalues of (a*ap /2 (counting the multiplicity). Proposition 1.1.3. Let a E B(H). Then a E S(H) if and only if for some (then for any) normalized orthogonal basis {€I} of H we have IIa611 2< 00.

L l

Proof.

let {€I}, { TJ-) be two normalized orthogonal bases of H. Then

L II(a·a)I/2611 2= L 1((a·a)l j26,TJr)1 2 l l,r L 1(6, (a*a)I/2TJr) 1 = L II (a*a)I/2TJrIl 2 = L IlaTJr1l 2

r

l,r

2

•

r

Hence it suffices to prove this proposition for some fixed basis. Let a E S(H), and {6} be a normalized orthogonal basis of H such that all eigenvectors corresponding to positive eigenvalues of (a*a)I/2 are contained in {€I}. Then II a 6112 = L.A~ < 00.

L l

Conversely, if

L ll a611

n

2

<

00,

where {6}eEA is a normalized orthogonal

lEA

basis of H, then for any e > 0 we can find a finite subset F of A such that lIppapp - all < e, where PF is the projection from H onto the linear span of {€Ill E F}. Thus a E C(H). Now pick a basis {€I} containing all eigenvectors corresponding to positive eigenvalues of (a*ap72 • Then we can see a E S(H). Q.E.D. Let a E S(H) .llal12 =

(L ..\~)1/2 = L IIa611 2 is called the Hilbert-Schmidt n

I

3

norm of a, where {An} is the set of all positive eigenvalues of (a*ap/2 (counting the multiplicity), and {6} is any normalized orthogonal basis of H.

Proposition 1.1.4.

For any a E S(H), se B(H), we have

* two-sided ideal of B(H). l:lla*611 2= l: I{a*e" 6,)12 = l:1{6,ae,,)1 2

In particular, S(H) is a

Proof.

Since

I

follows that lIa1l2 = h = (a*a)1/2. Then

l!a*112'

1,1'

=

l:llacl,1I 2,

1,1'

it

I'

Let a = wh be the polar decomposition of a, where

l: IIba611 2 < IIbl1 2 l: Ilaezl1 2, we have II bal12 < Ilall21l bll. Furl I b*1I IIabl12= Ilb*a*1I2 < lIa*1121I = lIall2l1bll. Q.E.D.

Moreover, since thermore,

Proposition 1.1.5.

Define (a, bh

= l:(a6, b6), where {€I} is a normalized I

orthogonal basis of H. Then (S(H), (, h) is a Hilbert space, and F(H) is dense in it. First, we must show that the inner product (, h in S(H) is independent of the choice of basis {6}. Let {TJr} be another normalized orthogonal basis of H. Since l: I(a6, TJr)(TJr, b6) I < Ila11211 b112' it follows that the series

Proof.

l,r

L (a6, 7},.) 'n-, b6) is convergent absolutely.

Then

l,r

l:(a6, bCl) = l:(ae" 7},.)(TJr, b6) I

I"

l:{b*TJr, 6)(6, a*TJr) = l:{b*TJr, a*TJr)' ~

r

Similarly, l:{aTJr,bTJr) = l:(b*TJr,a*TJr)' Hence the definition of r

Ch

pendent of the choice of {6}. Now let {an} be a Cauchy sequence in (S(H),(,h). Since 11·112 > follows that there exists a E C(H) such that lI an - all ---t O. From am) 611

is inde-

r

2---t O(n,m ---t (0), it is easily verified that a E

Thus (S(H), (, )2) is a Hilbert space.

S(H)

11,11, it L II (an I and Ilan-al12 ---t o.

4

Finally, for any a E S(H) and c > 0, if {6}'EA is a normalized orthogonal basis of H, then there is a finite subset F of A, such that L(lIa~dI2+lla·611 2) < I~F

c

2

Let PF be the projection from H onto the linear span of

•

IlpFapF - all~ = L

II a611 2+ L 11(1 lEF

l~F

But

L 11(1- PF)a611 2 = L L lEF IEF l'~F it follows that IlpFapF - al12 < c.

l(a6, 6,)1 2<

PF)a6tI

L

{€Ill 2

E F}. Then

•

lI a·6,112,

1'~F

Q.E.D.

We shall denote by T(H) the set of all operators of trace class on H, i.e., a E T(H) if a is compact and LAn < 00, where {An} is the set of all positive n

eigenvalues of (a·ap/2 (counting the multiplicity). An is called the trace norm of a.

L

For a E

T(H),llallt

n

Proposition 1.1.6. 1) Let a E B(H). Then a E T(H) if and only if (a·a)l/4 E S(H). Moreover, if {€I} is a normalized orthogonal basis of H, and a E T(H), then

Ilalll = lI(a·a)l/41Ii = L:((a·a)l/26, 6). I

se S(H),

then a, b E T(H). 3) Let a E C(H). Then a E T(H) if and only if 2) If a,

"" I {~n} and {TIn} are arbitrary } sup { ~ I(a~n) TIn) I normalized orthogonal sequences in H <

00.

Moreover, if a E T(H), then Iiall < Ilal12 < lIalib and !Ialll is equal to above sup. 4) T(H) is a * two-sided ideal of B(H), and T(H) C S(H), and T(H) is the linear span of T(H}+.

Proof.

If {An} is the set of all positive eigenvalues of (a·ap/2, then {A~2}

is the set of all positive eigenvalues of (a·a)l/4. That comes to the conclusion 1). 2) Let a, b E S(H), and c = w(C·CP/2 be the polar decomposition of c, where c = abo Since

L((C·C)l/2~h 6) = L(b6,a·w6) = (b,a·wh < l

it follows that c

I

= ab E T(H).

Ila112!1b1l 2,

5

3) Let a E T(H), and a = w(a*a)l/2 = w(a*a)l/4 . (a*a)l/4 be the polar decomposition of a. Then for any normalized orthogonal sequences {Cn}, {TJn} of H,

L

n

(L II(a*a)1/4 cnI12)1/2 . (2: lI(a*a)1/4w*TJnI1 2)1/2

j(aCn, TJn)1 <

n

n

< II (a*aJl/4112 '11(a*ap/4 w*112 < Ilalll'

Moreover, if Cn is the eigenvector corresponding to .An, and TJn = WCn, Vn, then I(acn, TJn) I = L.A n = Iialit. Of course, .A n)2 > .A~. Hence

Ln

Ilalll > IIal12 > [e].

(E n

n

Ln

4) From a = w(a*a)I/4 . (a*ap/4 and 2), it is obvious.

Proposition 1.1.7.

(T(H),

11·111) is a Banach space, and F(H)

Q.E.D. is its dense

subset.

Proof. By Proposition 1.1.6, II . lit is a norm on T(H) exactly. Now let {an} be a Cauchy sequence in (T(H), II ·lld· Since II ·111 > II· II, there exists a E C(H) such that Han - all ~ 0. For any finite normalized orthogonal sequences {6:}, {TJk}, since

it follows that a E T(H) and Ilan- alit ~ 0, and (T(H), II ·lld is a Banach space. Let a E T(H), and a = w(a*aJi/2 be its polar decomposition. Write (a*a)l/2 = .AnPn, where {Pn} is a sequence of projections of one rank on

L n

H, and PnPm = 0, Vn -=I m, and {An} is the set of all positive eigenvalues of (a*a)l/2. Then N

i.e., F(H) is dense in

I

Iia - L .AnWPnlll = L .An ~ 0, n=1 n>N (T(H), 1I·lld.

Q.E.D.

Remark. The completions of F(H) according to the norms 11·11. > ·/1 are T(H) C S(H) C C(H) respectively.

11·112 >

L::(ach Cl)

{cd is

For each a E T(H),tr(a) =

is called the trace of a, where

l

any normalized orthogonal basis of H. Since a is the product of two elements of S (H), the trace of a is well-defined.

6

For any a E T(H), b E B(H), we have

Proposition 1.1.8.

tr(ab) = tr(ba),

Itr(ba)1

< Ilalhllbll.

Proof. From the proof of Proposition 1.1.5, we have tr(ed) = tr(de)(Vc, s e S(H)) indeed. Since a is the product of two elements of S(H), conclude that tr(ab) = tr(ba). Let a = w(a*a)l/2 = w(a*a)l/4 . (a*a)l/4 be the polar decomposition of a. Then

Itr(ab) I =

1((a*a)l/4, (bw(a*a)l/4)*hl < II(a*a)l/4I1illbll = llalllllbll· Q.E.D.

Theorem 1.1.9. 1) C(H)" = T(H), i.e., for any continuous linear functional I on C(H) there exists unique a E T(H) such that

11/11 = IlallI,

I(c) = tr(ac), Ve E C(H);

and

conversely, for any a E T(H) tr(a·) will be a continuous linear functional with norm lIalll on C(H). 2) T(H)* = B(H), i.e., for any continuous linear functional I on (T(H), II . lId there exists unique b E B(H) such that

IIIII = Ilbll,

I(a) = tr(ab), Va E T(H);

and

conversely, for any b E B(H) tr(·b) will be a continuous linear functional with norm II bll on (T(H), II . lid.

Proof.

1) Let a E T(H), {An} be the set of all positive eigenvalues of (a*ap/2, be the eigenvector corresponding to An, II Cn II = 1, Vn. For any positive

and Cn integer N, we define an operator e of finite rank as follows: eui

Ci = Ci,

1

< i < N;

e = 0 on {w Cl, ... , W CN}1. ,

where a = w(a*a)l/2 is the polar decomposition of a. It is obvious that and N

Itr(ae)1 = I L((a*a)1/2 cwCi' ci)1 ~

LAi ~

lIall!

i==l

i

(as N

=

Ilell = 1,

(0). Thus tr( a·) determines a continuous linear functional with norm

lIalll on C(H). Now let I E C(H)*. Since a E B(H) such that

l(c(8lTJ)=(ac,'7),

!/(c (8l '7)1 < 11/1I1Ic1tllTJIl, VC,TJEH,

and

there exists unique

lI all
7

Further from (ae,fJ) = tr(a(e ® fJ)), we have

I(c) = tr(ac),

Vc E F(H).

So it suffices to prove a E T(H). Let a = w(a*a)l/2 be the polar decomposition of a, and {6}'EA be a normalized orthogonal basis of w* H. For any finite 6 ® w6 E F(H). Clearly lIcFl1 = 1. Then subset F of A, define CF =

L

IEF

I l:((a*a)1/26, 6)1 =

Itr(aCF) I = I/(CF)I <

11/11·

lEF

Since F is arbitrary, and w* H = (a*a)l/2 H, it follows that a E T(H). 2) Let bE B(H). By Proposition 1.1.8 and

Ilbli tr(b.) is a continuous linear functional with norm Now let I E T(H)*. Since

=

sup l(bE,fJ)l, lIell=II'1I1=l

Ilbll

on (T(H),

1I·lId.

I/(E e fJ)1 < 1I/1111E e fJlh = 1I/IIlIElIlITJII, VE,TJ E H, there exists unique b E B (H), such that

f(E ® TJ) = (bE,TJ) = tr(b(E ® TJ)),

VE,fJ E

H.

Further from the density of F(H) in T(H), we have f(a) = tr(ab), Va E T(H).

Q.E.D. By natural embedding, B (H) can be regarded as the second conjugate space C(H)** of C(H). If a Banach space X is the conjugate space of a Banach space Y, we shall say that Y is the predual of X, and write Y = X*. Hence, T( H) is the predual of B(H), and T(H) = B(H)*. Remark.

Let JR and (J} denote the real number field and the complex number field, respectively. In this book, a linear space always means that it is over (J}. In special case, we shall write a real linear space. If E is a subset of a linear space, we shall denote by [EJ the linear span of Remark.

E. The symbol "1» will be used extensively in this book. It can be the number 1, or the identity operator on a linear space, or the identity of an algebra. Please don't confuse them.

8

Notes.

The results in this section were obtained by R.Schatten and J. Von Neumann in the study of the tensor product of Banach spaces. A further detailed account of the theory was presented by Schatten.

References. [19],[151],[152],

1.2. Locally convex topologies in B(H) Let H be a Hilbert space. We introduce the following locally convex topologies in B(H): 1) weak (operator) topology. A net a, --+ 0, if

(a'C, 77 )

--+

VC, TI

0,

2)

strong (operator) topology. A net a,

3)

strong

4)

a-weak (operator) topology. A net a,

* (operator) topology.

2:(a, Cn, TIn)

--+

0,

n

5)

--+

0, if

0, if

2)11 enl1 2 + lIT1nI1 2 ) < 00;

a-strong (operator) topology. A net a, --+ 0, if

L: II Cnl1 2 < 00;

V

n

a-sironq

n

* (operator) topology.

A net at

n

7)

--+

n

L: Ii alenl1 2 --+ 0, 6)

0, if

--+

A net a,

V

E H;

--+

0, if

n

a(B(H), T(H)). A net a, --+ 0, if tr(a,a)

--+

0,

Va E T(H),

i.e., a(B(H), T(H)) is the w*-topology in B(H), seeing B(H) as the conjugate space of T(H); 8) s(B(H), T(H)). A net at --+ 0, if net aia, --+ with respect to a(B(H), T(H)); 9) s*(B(H), T(H)). A net a, --+ 0, if nets aja, and a,a; --+ with respect to a(B(H)), T(H);

°

°

9

10)

Mackey topology r(B(H), T(H)). A net a, tr(ala)

---+

0,

uniformly for

---+

0, if

a E E,

where E is any u(T(H), B(H))-compact subset of the Banach space (T(H),

11·111); uniform (operator) topology. It is the topology generated by the operator norm II .II· 11)

In the following, a subset (or a net) of B(H) is called bounded, if the function of operator norm is bounded on it.

Proposition 1.2.1. 1) The *-operation of operators is continuous with respect to above topologies 1),3),4),6),7),9),10),11). 2) For any fixed b E B(H), the maps· ---+ b- and· ---+ -b are continuous in B(H} with respect to all above topologies. 3) If nets {al} and {b r } of B (H) are bounded, and al ---+ a (strongly), b; ---+ b (strongly), then we have albr ---+ ab (strongly).

Proof. If net {all C T(H) and a, ---+ O(u(T(H), B(H))), i.e., tr(a1b) ---+ 0, Vb E B(H), then we also have ai ---+ O(u(T(H), B(H))) since tr(aib) = tr(b*al), Vb E B(H). Also, for any b E B(H), tub ---+ 0 and ba, ---+ 0 with respect to u(T(H),B(H)). Thus, if E is a u(T(H),B(H))-compact subset of T(H), then E*,bE and Eb are also u(T(H),B(H))-compact subsets of T(H), Vb E B(H). Now the conclusions with respect to r(B(H), T(H)) are clear. The rest conclusions are left to the reader. Q.E.D. Proposition 1.2.2. Let f be a linear functional on B(H). following conditions are equivalent: 1) f is u(B(H), T(H))-continuous; 2) There exists unique a E T(H) such that

f(b) = tr(ab), 3) There are {en}, {TJn} C H with

Then the

Vb E B(H);

I:) II Enl1 + IITJn11 2

2

)

<

00

such that f{b)

=

n

L:(bEn,TJn), Vb E B(H). n

Moreover, f > 0 (Le. for any b E B(H) and b > 0, f(b) a > 0 . In this case, we can take En = TJn, Vn.

Proof.

> 0)

if and only if

Since T(H)* = B(H), the equivalence of 1) and 2) is obvious.

10

Let f(·) = tr(a·) for some a E T(H), and the projection from H onto [c]. Then

II cll- 2(ac, e) =

f > O. For any CE H, let p be

tr(ap) = f(p) > O.

This implies a >0. Conversely, if f(·) = tr(a·) and a > 0, it is clear that f > 0 since tr(a·) = tr(a l/ 2 . al/2). 2)==>3). Write a = ala2 such that al E S(H) and a; = a2 E S(H). Let {6}'EA be a normalized orthogonal basis of H. Then there is a countable subset J of A such that alCt = a2Ct = 0, Vl E A\J. Further,

f(b) = tr(a2ba I) = :L(balC" a2c,),

Vb E B(H),

IEJ

and

:L(lIalc,112 + Ila26112) <

00.

Moreover, if f > 0 , we can take al = a2 =

tEJ

al/ 2 • Then alc, = a2c" Vl E J. 3)==> 2). Take a normalized orthogonal sequence al~n = en,

Obviously,

at

{~n}

of H. Define

a2~n = n«, Vn; and al = a2 = 0 on {~n}l..

and a2 E S(H). Then a = ala; E T(H) and

f(b) = 2::(bal~n,a2~n) = tr(a;bad = tr(ab),

Vb E B(H).

n

Q.E.D. Theorem 1.2.3.

The relations between topologies 1)-11) are as follows: top.3)

:)

n top.11) :) top.10) :) top.g) l top.6)

top.2)

::>

n

n

=> top.8)

::>

l top.5)

::>

::>

top.I] top.7) l topA),

where "::>" means that the left side is finer than the right side. Proof. Obviously, we have the following relations: top.3) ::> top.2) ::> top.I}, top.6)::> top.5) ::> topA), topA)::> top.I] , top.5)::> top.2), top.6) ::> top.3), top.11):) top.IO) ::> top.7) (by the Mackey theorem). By Proposition 1.2.2, the equivalence of top.7) and topA) is clear. Now we prove the equivalence of top.8) and top.5). Let a net -+ O(s(B(H), T(H))). Since top.7) topA), we have that aia, -+ 0 (a-weakly) and at -+ 0 (a-strongly). Conversely, let a net at -+ 0 (a-strongly). For any 0 < a E T(H), by Proposition 1.2.2 there exists a sequence {en} C Hwith IIcnll 2 < 00 such that

a,

/"oJ

L n

n

11

Hence tr(a;aza)

= 2: ll azcnll2

-+

0, Va E T(H)+, and a,

-+

O(s(B(H),T(H))).

n

Since * operation is continuous with respect to top.B] and top.6), it follows from top.S] ....., top.5) that top.D] ....., top.6). Further we have top.D] ::J top.S) ::J top.7) from top.6) ::J top.5) ::J top.d]. Now we prove that top.l0) ::J top.S] ::J top.7). By the Mackey theorem and top.5) top.S] ::J top.7), it suffices to show that any top.5)-continuous linear functional f on B(H) must be a(B(H), T(H))-continuous. For such I, there k ) } ; 1) of zero such that is a top.5)- neighborhood U = U(O; {CLl)}, ... "'-J

,{ci

If (b) I < 1, There {c~}}

c

H,

L II c~)112

<

Vb E U. < i < k, and b

00,1

E U means that

n k

LlIbc~)112 < 1,1 < i < k.

Let {en}

=

n

U{c~j)}.

Then

If(b)1 < (2: IlbCnI1 2)1/2, Vb

00

n

i=l

and

2:lIenll 2 <

E

B(H).

n 00

Define

Ji = E9 Hi,

where Hi

= H, Vi.

Then

E= (Cn)

E Ii, and

If(b)1 <

i=l

IIbEII, Vb

E B(H), where b E B(Ii) is defined by bfi = (bTJn), Vfi = (TJn) E H. In particular, f(b) = a if o. Then we can define a linear functional on """" """"" the linear subspace {bElb E B(H)} of H as follows:

1

bE

j(bE) = f(b), Vb E B(H). If(bc)1 = If(b)1 < IlbEII, Vb E B(H), it follows that there is fi = (fin) IV At!,."

""",."

Since such that

f(b) =

l(b[)

=

(be, fj)

= L(bEn, TJn),

..-..,.,

E

H

Vb E B(H).

n

Now by Proposition 1.2.2, f is a(B(H), T(H))-continuous. Finally, since * is r(B(H), T(H))-continuous, it follows that top.l0) ::J top.D]. Q.E.D.

Theorem 1.2.4. In any bounded ball of B(H), we have weak(operator) topology-> a(B(H), T(H)) , strong (operator) topology e- s(B(H), T(H)), and strong * (operator) topology s*(B(H), T(H). "'-J

Proof. Let a net al -+ a (weakly), and lIadl 2 2 C H with 2:(11 Enl1 + IITJnIl ) < 00, we have

< K, Vl. Then for any {En}, {1]n}

n

I ~]aIEn, 'In) I < 'f-1(aIEn,'In} I + ~ L (II Enll' + II'Inll')· n

n=l

n>N

12

It follows that

a,

--+

0 (a-weakly). Similarly, we can prove the rest parts. Q.E.D.

We shall prove that r(B(H), T(H)) ~ s*(B(H), T(H)) in any bounded ball of B(H) (see Section 1.11). However, we have the following.

Proposition 1.2.5. If H is infinite dimensional, then in whole B(H), s: (B (H), T(H)) is not equivalent to r(B(H), T(H)).

Proof. Let {Pn} be an infinite sequence of non-zero projections on H and PnPm = 0, \In i= m. Put K = {Jnpnln = 1,2,,, .}. We say that K n U(Oj at," . ,ak; e) i= 0 for any e > 0,0 < Cli E T(H),1 0, such that K n U(O; at,"', ak; e) = 0, then

2ntr(Pna) > e,

\In,

k

where a

=

L ai· Let P = LPn. Then i=1

n

a contradiction. Since {U(Oj at," . ,ak; e) Ie> 0, k = 1,2,'''' ai E T(H)+} is a s" (B(H), T (H))-neighborhood basis of zero, it follows that 0 belongs to the s*(B(H), T (H))-closure of K. Now it suffices to show that 0 is not contained in the r(B(H), T(H))-closure of K. For each n, take €n E PnH,II€nll = 1. Then, Cn = Jn€n @ €n E T(H), and Ilc n lll = -jn --+ O. Thus L = {O, cnln = 1,2,"'} is a compact subset of (T(H), 1I·lld. Further Lis a(T(H),B(H))-compact. It follows that

L O= {b E B(H)

Iltr(bcn)l < 1, \In}

is a r(B(H), T(H))-neighborhood ofO. Obviously, JnPn rJ. L Osince tr(JnPncn} = 2, \In. This implies that K n LO = 0 and 0 is not contained in the r(B(H), T(H))-closure of K. Q.E.D. By Theorems 1.2.3, 1.2.4 and the standard results of duality theory (see Appendix ), we have the following.

Proposition 1.2.6. Let f be a linear functional on B(H). Then the following statements are equivalent: 1) f is a(B(H), T(H))-continuous; 2)

13

f is s(B(H), T(H))-continuous; 3) f is s*(B(H), T(H))-continuous; 4) f is r(B(H), T(H))-continuous; 5) f is weakly continuous in any bounded ball of B(H); 6) f is strongly continuous in any bounded ball of B(H); 7) f is strongly * continuous in any bounded ball of B (H). On the weak (or strong) continuity of linear functionals, we have the following.

Proposition 1.2.7. Let f be a linear functional on B(H). Then the following conditions are equivalent: 1) f is weakly continuous; 2) f is strongly continuous; 3) There exists unique v E F(H) such that

f(b) = tr(bv), 4) There are

6,' .. , en, and 111,' . "

Vb E B(H); 11n E H such that

n

f(b) = L:(be,,1'J,),

Vb E B(H).

i=1

Moreover, f > 0 if and only if v >0. In this case, we can also take ei = 17,,1 < i < n.

Proof.

It is clears that 3) and 4) are equivalent, and 3) or 4) implies 1), 1) implies 2). Thus it suffices to show that 2) forces 4). Let f be strongly continuous. Then there is a strong neighborhood U = U(O; 171,"·, 17m; 1) = {b E B(H)lllb17ill < 1,1 0, the strong neighborhood V = U(O; 6,' .. , en, g) will be contained in U. In particular, if b E B(H) and bei = 0,1 < i < n, then we have f(b) = O. Of course, f is also s(B(H), T(H))-continuous. By Proposition 1.2.6, we have a E T(H) such that f(b) = tr(ab), Vb E B(H). Let {6} be a normalized orthogonal basis of H containing {€I,"', en}. Take e E B(H) such that eei = 0,1 < i < n, andc6 = a*e"V6 =1= Then 0 = f(c) = 2 tr(ac) = L:(ae6, 6) = lIa* 611 • Hence, a* 6 = 0, V6 =1= 6,"" en'

L:

I

ell· .. ,en.

€f~€6,···,€"

n

Let a*ei = 17,,1 0 is contained in the proof of Proposition 1.2.2 Q.E.D.

14

On the closedness of a convex subset of B(H), by Propositions 1.2.6, 1.2.7, v and the separation theorem, the Krein-Smulian theorem ( see Appendix), we have the following. Proposition 1.2.8. Let K be a convex subset of B(H). Then the following statements are equivalent: 1) K is a(B(H), T(H))-closed; 2) K is s(B(H), T(H))-closed; 3) K is s*(B(H), T(H))-closed; 4) K is r(B(H), T(H))closed; 5) KnAS is weakly closed, VA > 0; 6) KnAS is strongly closed, VA > 0; 7) K n AS is strongly * closed, VA > O. There S is the closed unit ball of (B(H), II ·11)· Moreover, if K is bounded, we can replace K n AS by K in 5), 6) and 7). Proposition 1.2.9. Let K be a convex subset of B(H). Then the weak closure of K is equal to the strong closure of K, or K is weakly closed if and only if K is strongly closed. As the end of this section, we mention the following proposition which we shall use often. Proposition 1.2.10. Let {a,} be a bounded increasing net of self-adjoint = Iladl < M (some constant), Vl, and > elements of B(H), i.e., a" Vl' > l, Then we have a, ~ a = SUPl a, (strongly)

a"

a;

a"

Proof. Since {(a,e, e)} is a bounded increasing net of real numbers for any e E H, it follows that lim(a,e,fJ) exists for any e,fJ E H. Then by l(a,e,1J)1 < MlIellllfJll, we have a E B(H) such that lim(a,e,fJ) = (ae,T/),Ve,T/ E H. Of course, (a€, €) = sup(a, €, e), E H. Thus we have a = sup a, and a, ~ a

ve (weakly). Moreover, for any eE H we also have

I

l

Il(a - a,)e11 2 <

II(a -

a,P/211 2

·11(a -

< 2M((a - a,)e, e) Therefore a,

~

~

aIP/ 2 €1l 2

o.

a (strongly).

Q.E.D.

Notes. In this section, we used many standard results of duality theory in Banach spaces. These can be found in Appendix. The weak topology, the strong topology, the a-weak topology and the astrong topology were introduced by J. Von Neumann. Moreover, Propposition 1.2.5 is due to F.J.Yeadon.

References. [19], [31], [89], [146], [151], [152], [200).

15

1.3. Von Neumann's double commutation theorem Definition 1.3.1. Let H be a Hilbert space. A * subalgebra M of B(H) is called a Von Neumann algebra (or VN algebra simply), if

M = M", where M' = {b E B(H) I ba = ab,Va E M} is the commutant of M, and M" = (M')' is the double commutant of M. If E is a subset of B(H), and M is the smallest VN algebra containing E, then M is called the VN algebra generated by E.

Proposition 1.3.2. Let E be a subset of B(H). Then (E U E*)' is a VN algebra, and the VN algebra generated by E is (E U E*)". In particular, the commutant of any VN algebra is a VN algebra.

Proof.

Obviously, we have (EUE*) C (EUE*)" and (EUE*Y C (EUE*)"'. Now if a E (E U E*)"', then ab = ba, Vb E (E U E*), and a E (E U E*)'. Thus (EUE*)' = (EUE*)'" is a VN algebra. Furthermore, (EUE*)" = ((EUE*)')' is also a VN algebra. Now suppose that N is a VN algebra, and N ~ E. Then N ~ (E U E*). Since N' C (E U E*)', we have N" = N ~ (E U E*)". Therefore, (E U E*)" is the VN algebra generated by E. Q.E.D.

Proposition 1.3.3. 1) Let M be a VN algebra on H. Then M is weakly closed. In particular, M is the conjugate space of quotient Banach space T(H)fMJ.' where MJ. = {a E T(H)

I tr(ab)

= 0, Vb EM}.

2) Let {M,} be a set of VN algebras on H. Then M = algebra, and M' is generated by u,M;.

nM , is also a VN i

Proof.

1) is clear. Now we prove 2). Obviously the following conditions are equivalent: (1) a E M: (2) a E M" VI; (3) a E (Mf)', VI; (4) a E (U,M/)'. Thus M = (UIMt)' is a VN algebra, and M' is equal to (U,M/)", the VN algebra generated by uIM',.

Q.E.D. Proposition 1.3.4. Let M be a VN algebra on H. 1) Let a = vh be the polar decomposition of a, where a E M. Then v, hEM. In particular, the projection vv* from H onto aH belongs to M.

16

2) Let a be a normal element of M, i.e., a*a = aa*, and {e(·)} be the spectral measure of a. Then e (b..) E M for any Borel subset b.. of (C• 3) M is the uniform closure of the linear span of Proj (M), where Proj (M) is the set of all projections of M. And also M is the linear span of U(M), where U(M) is the set of all unitary elements of M(i.e.u E U(M) if u*u = uu* = 1).

Proof. 1) Obviously, we have h = (a*a)l/2 E M. Let b' EM'. If e E (a*H)-1., then ae = ve = O,ab'e = b'ae = 0, and b'e E (a*H)l.. Thus vb'e = = b've. For any

1]

°

E H, we have

b'v(a*a)1/21] = b'a1] = ab'T/ = v(a*a)1/2b'1] = vb'(a*a)1/2T/. Since (a*a)l/2H is dense in ar H; it follows that b'v~ = vb'~,V~ E a*H. Therefore b'v = vb', Vb' E M', i.e., v E Mil = M. 2) It is clear since e{b..)b' = b'e{b..) , Vb' E M'. 3) Since any spectral projection of any self-adjoint element of M belongs to M, it follows that M is the uniform closure of the linear span of Proj (M). Now if h* = hEM and IIhll < 1, then (1_h 2P/2 EM and (h±i{l-h2)l/2) E U{M). Therefore M is the linear span of U{M). Q.E.D.

Proposition 1.3.5. Let M be a VN algebra on H. Then Proj (M) is a complete lattice with respect to the inclusion relation. Moreover, if {PZ} lEA C Proj (M), then sup p, lEA

=

(strongly) -Iim sup p, FIEF

the projection from H onto [UIEApzH], and inf PI lEA

=

(strongly)- lim inf PI F

ZEF

the projection from H onto

nplH,

lEA where F is any finite subset of A, and is directed by the inclusion relation.

Proof.

Let p.q E Proj (M). By Proposition 1.3.4, the projection from H onto [(1 - q)pH] belongs to M. Now noticing

[pH + qH] = qH E9 [(1 - q)pH], so the projection from H onto {pH + qH] also belongs -to M, i.e., sup{p, q} E M. Moreover, since

[pH + qH] = (pH

n qH) E9 [(1 - p)qH] E9 [(1 - q)pH],

17

it follows that inf{p, q} E M. Further, for any finite subset F of A, we have sup PI E M,

inf PI E M.

and

lEF

IEF

By Proposition 1.2.10, we have SUPPI lEA

= sup sup Pl = [stronglyj-Iijn sup p, EM FIEF

F

since M is strongly closed. Considering the family of projections {( 1- inf p,) IEF

lEF

IF

is any finite subset of

A}, we can get the rest conclusion.

Q.E.D.

Proposition 1.3.6. Let M be a VN algebra on H, p E Proj (M). Then M p = pMp and M; = M'p are two VN algebras on pH, and also (Mp ) ' = M;. Proof.

Obviously, (M;)' ~ M p • For a E (M;)' c B(pH), define

_a-- {a,0,

on pH, on (1 - p) H .

Then we have a E B(H). For any b' EM', since pH and (1 - p)H are two invariant subspaces of b', it follows from b'p E M; that ab' = b'a. Thus a E M, and a = pap E M p • Further, M p = (M;)' is a VN algebra on pH. Now it suffices to show that M; = (Mp ) ' , or to prove that a' E M; for any a' E (Mp ) ' since (Mp )' ~ M;. By Proposition 1.3.4, we may assume that a' is a unitary operator on pH. Let q be a projection on H such that qH = [MpH]. It is easily verified that q E M n M'. Define

v' ~ aiei = ~ aia' ei, Vei E pH, at E Mj {

e

,

v' (1 -

$

q) = 0,

VeE H.

Since a' E (Mp )' and a' is unitary on pH, it follows that

II L

2

aia' e.11 =

•

L(aipa' ei, ajpa' ei) i,i

i.i

Hence v' can be extended to a partial isometry on H, which is still denoted by v', and v'·v' = q. For any a E M, since

v'aLase. = Laasa'ei = av'Lasei,Vei E pH,as E M,

i'

i

18

and

v'a(l- q)e = v'(I- q)ae = 0 = av'(I- q)e, Ve E H, we have v' EM'. Obviously, a' = v'p E M~. This completes the proof.

Q.E.D. Definition 1.3.7. Let M be a VN algebra on H. Z = Z(M) = M n M' is called the center of M. If Z = q:1H (or write Z = q: simply), then M is called a factor. Proposition 1.3.8. Let M be a VN algebra on H, Z be its center, and p EProj (M). Then n M; = Z p. In particular, if M is a factor, then and M; are also factors (on pH). Moreover, if q is a central projection of M p , then there is a central projection z of M such that q = zp.

u,

u,

u,

Proof. Obviously, Zp c n M;. Conversely, if a E M p n M~, we have b' EM' such that a = b'p. Let r be the projection from H onto [MpH]. Then r E Z and r > p. Hence a = b'p = (b'r)p. If b' is replaced by b'r, then we may assume that b' = b'r. For any a' EM,' noticing b'a'p = b'p' a'p = aa'p = a'ap = a'b'p and by the definition of r, we have (a'b' - b'a')r = O. But b'r = b' and r E Z, so a'b' = b'a',Va' EM'. Therefore b' E Z and a = b'p E Zp, i.e., MpnM;=Zp. Now let q be a central projection of M p • From preceding paragraph, there is z E Z such that q = zp. Of course, q = !(z + z*)p, so we may assume z* = z, Since q2 = q, this implies (z2 - z)p = 0, and (z2 - z)r = O. Again by the discussion of preceding paragraph, we may assume z = zr. Then Z2 = z since (Z2 - z)r = O. Therefore, z is a central projection of M, and also q = zp.

Q.E.D. Theorem 1.3.9. Let H be a Hilbert space, and M be a w B (H) ,M be the weak closure of M. Then

M

W

= {a E M" I ap« = Paa =

a}

* subalgebra of

= M"po,

where Po is the projection from H onto [M H], and Po E M' n Mit, (1- po)H = {e E H I ae = 0, Va EM}. In particular, if M is also nondegenerate (i.e. W Po = 1), then we have M = M".

Proof. By the definition of Po, it is clear that poa = a, Va E M. Thus W W poa = a,Va E M • Obviously, M c M",po EM' n Mit, and (1- Pa)H is the null subspace of M. Let a EM", poa = apo = a, and U(a; 6,'''' e;) = {b E B(H) b)eill < c,l < i < n} be a strong neighborhood of a. Put

en,

""'-J

H = H ffi ... ffi H

(n times)

I II (a -

19

and let P' = (p"ih::;',i::;n be the projection from H onto [(be.,··. ,ben)lb EM], where P'.; E B(H), Vi,j. For any b E M, define

bfj = (bl1b· .. ,bl1n), ,...,

,...,.,

Vij = (111,· ..

,...,,...,..,,..,,.,,

,1]n) E H. ,...,

Then b E B(H), and p'H = [belb E M], where e = (6,·'" en). Clearly, hp' = p'h, Vb E M. Thus P'.; EM', Vi,j. Since p'be = be, Vb E M, it follows that n n

be. = L. p'.k b6: = b L. p'.kek, k=1

Vb E M,I < i < n.

k=1

n

Hence (e. - L. p'.kek) E (I - Po)H, 1 0 we

e.' .. "

This implies that M n U(a; en; e) 1= 0, Vell···, en E H, and e > O. Hence a belongs to the strong closure of M. By Proposition 1.2.9, we have -:::-:w aEM.

Q.E.D. Now we come to the Von Neumann's double commutation theorem. Let H be a Hilbert space, and M be a * subalgebra of B(H). Then M is a VN algebra on H if and only if M is weakly closed and IH E M.

Theorem 1.3.10.

As the end of this section, we consider the topological problem about a VN algebra. Let M be a VN algebra on H. We have introduced many locally convex topologies in B(H) (see Section 1.2). Naturally, these topologies can be restricted into M. Moreover, from Proposition 1.3.3 M is the conjugate space of the Banach space M. = T{H)/Ml.. Using M., we can also introduce the following topologies: I) a(M, M.). The weak * topology of M for M.; 2) s(M,M.). A net al -+ 0, if net a[a, -+ 0 with respect to a(M,M.); 3) s·(M,M.). A net a, -+ 0, if nets aial and a,a; -+ 0 with respect to u(M,M.); 4) r(M, M.). The Mackey topology of M for M •.

20

Clearly, we have the following relations:

a(M,M.) '" (a(B(H),T(H))IM) '" (a-weak top.IM); s(M,M.) '" (s(B(H),T(H))IM) '" (a-strong top.IM); s·(M,M.) '" (s·(B(H), T(H)) 1M) '" (a-strong r(M,M.)

=:l

* top.IM);

(r(B(H),T(H))jM).

Moreover, the relations a(M,M.) '" (weak top. IM),s(M,M.) '" (strong top. 1M) and s·(M, M.) '" (strong * top. 1M) are possible. See Sections 1.10, 1.13, and 2.11. Finally, the results of Propositions 1.2.6, 1.2.8 and 1.2.9 can be moved to M, but we shall not discuss these in detail. Notes. Theorem 1.3.10 was obtained by J. Von Neumann. This is the first important result on the theory of Von Neumann algebras. It says that the algebraic definition is equivalent to the topological definition for Von Neumann algebras. It was J .Dixmier who recognized the a-weak topology as the weak * topology in the duality between B(H) and T(H), and thus proved that a Von Neumann algebra is the conjugate space of a Banach space. About the topologies in a VN algebra M on H, we have pointed out that r(M,M.) =:l (r(B(H)'T(H))IM). A natural question is that these two topologies are equivalent? It seems still open.

References.[I], [19], {114].

1.4. Tensor products of Von Neumann algebras First consider tensor products of Hilbert spaces. Let HI, H 2 be two complex Hilbert spaces. Let HI 0 H 2 =

{

U

=

e e/

t(l) (2) ~J. ® J• ;=1 n

L:

( )

n = 1,2, ... , E Hi, i = 1,2,1 < j

}

.

If we can introduce a definition of zero element (an equivalent relation), then n

HI 0 H 2 will be a linear space. We say that u =

L: eY) ® e;2) ;=1

element, denote it by u = 0, if for any 111 E HI, 112 E H 2 , n

(U,111 ® 112) =

L(e?), 111)(e;2), 112) = o.

;=1

is the zero

21

Equivalently, u =

°if in Hf, n

~

(1)

L- A;lee;

;=1 m

where e;2) = L Aile~le, 1 of

{ei

1
= 0,

< i < n, and {~b ... ,~m} is a basis of the linear span

1e=1

2

) , ••• ,

e~2)}.

n

For any u

m

= L eJl) @ eJ2)

and v

= L TIll) @1112), we

i=1

define

k=1

(u, v) = L(e;I), 11l1») (e;2) ,1112»). i.1e

Since for any AI, ... ,An E q},

L(e}I), ey») AiAj = iJ

II L Aie}I)11 2 > 0, i

it follows that the matrix

is non-negative. Then we have a n

where J.Li > 0,1

where aile

X

n-unitary matrix (Uij) such that

 0. k,l

i

i

Now by the Schwartz inequality:

I(U,111 @112)1 < (U,U)I/2 ·1111111·1111211,

V111 E HI, 112 E H 2,

22 n

we can see that u

=L

e?} ® e;2) = 0 if (u, u) = O.

;=1

This means that (,) is a

inner product on the linear space HI 0 H 2 • Definition 1.4.1. The completion of HI 0 H 2 with respect to the inner product (,) is denoted by HI ® H 2 , and called the tensor product of HI and' H2• Proposition 1.4.2. Let ai E B(Hi), i al ® a2 E B(HI ® H 2 ) such that

=

1,2. Then there exists unique

(al ® a2)(CI ® e2) = aIel ® a2e2, and Iial ® a211

Proof.

Vei E Hi, i = 1,2,

= IlaIII·ll a 211.

For any u =

L ell) ® e?} E HI 0

H 2, where {e?}} is a normalized

j

orthogonal family in H 2 , we have i

j

Thus at ® 12 can be extended uniquely to a bounded operator on HI ® H 2 • Similarly we have 11 ® a2 E B(HI ® H 2). Now we can define at ® a2 = (al ® 12)(1 1 ®a2) E B(H1 ®H2). Obviously, Iial ®a211 < lIaI!I·lla211. However,

.

smce

Iial ® a211

>

sup{ll at611·ll a26111

Ei

E Hi,

II Ei II = l,i =

1,2}

lI aill ·lI a 211, it follows that Iial ® a211

= IlaIII· Il a 211 ·

Q.E.D.

Using the method of Hilbert direct sum, we can also describe HI ®H2 • Let {edl E A} be a normalized orthogonal basis of H 2, where the cardinal number ~ A of A is dim H 2 • For each lEA, define

Then H(l) is a closed linear subspace of HI ®H2 , and isomorphic to HI, Vi E A. Moreover, HI ® H 2 = EBH(I}.

L

e

lEA

Ve

Suppose that UIC = ® el, E HI, lEA. Then Ul is an isometric map from HI into HI ® H 2 ; and its image is H(l), Vl E A. Clearly, ui is a linear map from HI ® H 2 onto HI such that u; H(l'} = {o}, Vl' 1= 1, and ui maps

23

isometrically H{l} onto HI, \;/1 E A. Moreover, UiUI is the identity map on HI; and UIU; = PI is the projection from HI CiS> H2 onto H(I); and L PI = 1. lEA Let a E B(HI ® H 2), and all' = uiauII(E B(Hd), \;/1,1' E A. Then the operator a will be determined by the operator matrix (all' ),,"EA' In fact, for any E HI ® H 2 , we have

e

ae = L(u;ae)

CiS>

e, = L(L all'6,)

I

I

CiS> eZ,

I'

where 6, = uI,e E HI, \;/1'. Thus, identifying H(l) with HI, \;/1 E A, we can write a = (all')"I'EA' Let a = (aw) and b = (bll') E B(HI CiS> H 2 ) . Then ab = aU"b,,,,, )"I'EA, where for any 1, l' E A the series all"b,,,,1 is convergent

Lemma 1.4.3.

(L

L

I"

I"

with respect to the strong (operator) topology in B(Hd. Proof.

=

Since L PI = LUI u;

topology in topology

lEA B(HI CiS> H 2 ) , it

1 with respect to the strong (operator)

lEA

follows that with respect to strong (operator)

U;abuI' = u;a(

L

ul"U;,,)bul' = L au"bl"",

l"EA

I"EA

Q.E.D.

\;/1, l' E A.

Lemma 1.4.4. Let ai E B(Hi), i = 1,2 and (AW)"I'EA be the matrix representation of a2 with respect to the basis {e,},EA. Then al CiS> a2 = (AlI'al)"I'EA. In particular, al CiS> 12 = (e511'ad"I'EA' Proof. For any E HI, we have

e

u;(al

CiS>

a2)U"e = u;(al

CiS>

ui(ale

u; (al

a2)(e CiS> e,,)

CiS>

a2 el' )

e L Al"l,e,,,) CiS>

I"

L u;(ale

CiS>

Al"l'el") = All'aIe,

I"

Q.E.D. Lemma 1.4.5.

In B(H1

CiS>

H 2 ) , we have

{ulu;dl,l' E A}'

= {al CiS> hlal E B(H1 )} .

24

Proof.

By Lemma 1.4.4, we have

Vs, t,l,l' E A. Thus al ® 12 E {u,u;,jl,l' E A}', Val E B(HI ) . Conversely, let a E {UIU;,j l, l' E A}'. If I f:- 1', then

Thus al = ui aUI E B (HI) is well-defined [i.e, independent of the choice of lEA). Now by Lemma 1.4.4, we have a = (oll/al) = al ® 12 •

Q.E.D. Now we study the tensor product of VN algebras.

Definition 1.4.6. Let M, be a VN algebra on a Hilbert space Hi, i = 1,2. The VN algebra generated by the set {al ®a21l4 E Mi,i = 1,2} (on HI ®H2) is called the tensor product of M I and M 2, and denoted by Mlfi9M2, i.e., M lfi9M2

= {al

®

a2

I ai

E Mi,i

= 1,2}"

For example, by Lemma 1.4.5 we have

Proposition 1.4.7.

Let M I be a VN algebra on HI' Then

Proof. By Lemma 1.4.3, the right side is a VN algebra on HI fi9 H 2 • And also by Lemma 1.4.4 we have M 1fi9B(H2 ) C the right side.

Now let a = (aw)"I'EA E B(H1 ® H 2 ) , and a',l' E MIl VI, I' E A. For any finite subsets E, F of A, define (E,F) _

au,

-

{all" 0,

if 1 E E, l' E F, ot herwi erwise,

25

and

aE,F

(E,F)) ',l'EA. B y Lemma 1.4.4, we h ave = (all' aE,F =

L

ad ® (OsIOW)I,l'EA E M 1 ®B (H 2 ) .

aEE,tEF

Moreover, it is easy to see aE,F

---+

a (weakly). Therefore a E M 1 ®B (H 2 ) .

Q.E.D. We have defined M 1®M2 • Naturally we guess that

For this purpose, we need to do some analysis in detail. Let H be a complex Hilbert space, (,) be its inner product. If we regard H as a real linear space, and define (,)r = Re(,), then (H, (, )r) will be a real Hilbert space. In the rest part of this section, "-i" is understood with respect to (, )r.

Lemma 1.4.8. Let M, N be two * sub algebras containing 1(= In) of B(H), and MeN'. Moreover, suppose that M admits a cyclic vector C (i.e. M C= H). Then the following statements are equivalent:

1) M' = N"; 2) (Mhe + iNhc) is dense in H; 3) (Mhc).L = iNhc, where M h , N h are the subsets of all self-adjoint elements of M, N respectively, and bar "-" is the norm closure.

Proof. Let t' E (M')h, a E M h. Then t'a = at' is still self-adjoint. Thus Im( ac, t' e) = 0, and (ae, it' C)r = o. This implies i( M') hE C (MhE).L. Moreover, by MeN' and N C Nil C M' we have iNhc C (Mhc).L. Suppose that 3) holds. Then

(Mhc

+ iNhC) => Mhc + iNhc = Mhc + (Mh c) l- •

Hence 3) implies 2). Suppose that 2) holds. Since iNhc c (Mhc).L, it follows from 2) that iNhE is dense in (Mhc).L. Hence 2) implies 3). Suppose that 2) or 3) holds. We have pointed out that (M')hC C i(MhE).l. So by 3), (M')hC C Nhc. Then for any t' E (M')h, there is a sequence {bn} C N h such that IWc - bnclI ---+ O. Let s' EN', a, c E M. From MeN', it follows that

(s't'aE, cE) =

lim(s'ab cbnE) nc, cE) = lim(s'ac, n n (s'ac, ct'c> = (t's'ac, cc).

26

Since Eis cyclic for M and a, c E M are arbitrary, we have s't' = t' s', Vs' E N'. Thus t' E N" and M' eN". But M c Nt, so M' = N". This means that 2) or 3) implies 1). Now suppose that 1) holds, i.e., M' = Nil. We must prove that TJ = 0 if

TJ E (MhE + iNkE).L. In the complex Hilbert space H EEl H, define

It is easy to see that

+ iNh E).L. Let P be the projection from H EEl H onto M 2 (~). M~, and we can write P = (~ ;), where p = p.,q = q. and

Fix TJ E (MhE Then P E

rEM'. Since P (~)

= (~), it follows that

pE + rTJ

Further, (TJ, aE)

Thus p(~)

=

By 1].liNhE,

= -(E,a1]), Va

E·

(1)

P1] = rE = O.

(2)

=

E M, i.e.,

0, and

i.e., Re(TJ, ibE) = Im(TJ, bE)

= 0, Vb E N h , we have

(TJ, bE) =

(bE, TJ) = (E, b1]), Vb E N h. Further, we have (TJ, bE) = (E, b1]), Vb E Nil

(3)

From (2) (3), (1) and M' = Nil it follows that

(4) Since P and (1 - P) are projections, this implies that p (1 - p) = (1 - p)2 + rr". Then by (4), we get

o < IlpTJ 11 2 + Ilr*1] W= -( II (1 -

=

p) EI1 2 + II r" E11 2 ) < O.

p2

+ rr"

and

27

This implies that PTJ = (I - p) E = O. Since (1 - p) E M' and vector for M, it follows that (1 - p) = 0, p = 1, TJ = O.

E is

a cyclic Q.E.D.

Let H; be a complex Hilbert space and K; be a real closed linear subspace of H; such that (K; + iK;) is dense in H;,j = 1,2. Then

Lemma 1.4.9.

K 1 ® K 2 +i(Kt ® Kt) = HI ® H 2, where K 1 CiS> K 2, Kt ® Kt are the real closed linear spans of {EI CiS> E2 I El E KI, 6 E K 2}, {TJl CiS> TJ2 I TJI E Kt, TJ2 E Kt} respectively, and Kt is the orthogonal complement of K; in (Hi' (, )r),j = 1,2.

Proof. It is easily verified that (K I CiS> K 2)l..i(Kt CiS> Kt). Thus it suffices to show E= 0 if E E HI CiS> H 2 and el..(Kl CiS> K 2 + iKt CiS> Kt)· For the above E, define a map t from HI into H 2 such that

Clearly, t is bounded and real linear. Since

(t(i6), E2)r

=

(E, iEI CiS> E2)r = (E, 6

®

iE2)r

(t6, i6)r = (-it6, 6)" it follows that

t(i6) = -it6, For any Thus

t"(iE2)

=

-it* E2'

VCi E HI, E2 E H 2 •

(1)

ei E K;,j = 1,2, from El..K CiS> K 2 we have (t6, E2)r = (E, 6®6), = o. 1

uc.

c x],

By el-iKt ® Kt, we have (t(iTJI), 172), Hence

(2)

t*K2CKt.

= (E, iTJI CiS> TJ2)r = 0, VTJj

t(iKt) C K 2,

t*Kt C

E Kt .i

uc;

=

1,2.

(3)

Now by (1), (2), (3), we have

t*tKt

C

iKt, and (t*t)2Kt c Kt.

(4)

Since t*t is a non-negative operator on (HI, (, )r) and t*t can be approximated uniformly by the polynomials of (t*t)2, it follows that t*tKt C Kt. Then from (4), we have t*tKt C Kt n iKt. It is easy to see Kt n iKt = {O} since (K1 + iKt} is dense in HI. Thus

is; = Then (t* K 2, Kt)r =

{O}.

(5)

{o}, i.e., t: K 2 C K 1 • Again by (2), we have

r K2 =

{O}.

(6)

28

From (1) and (6), it follows that (itKt, K 2 )r = (tKt, K 2 )r = {O}. Thus t K, C Kt niKt· Similarly, we have Kt niKt = {O} since (K2 +iK2 ) is dense in H 2 • Hence i K, = (o). Now by (5)' t = O. Therefore, (c,6 ® C2)r = (tE}, C2)r = 0, "16 E HI, 6 E H 2 , and C= O. Q.E.D.

Lemma 1.4.10. Let M, be a VN algebra on Hj, and Ei (E Hi) be a cyclic vector for Mj,j = 1,2. Then we have (MI®M2 ) ' = Mf®M~.

Proof.

C= 6

Let M = M I®M2 , and N = Mf®M~. Clearly, MeN'. Since ® C2 is a cyclic vector for M, by Lemma 1.4.8 it suffices to show that

MhC + iNhC = HI ® H 2 • Let K, = (Mj)hCj,j = 1,2. Clearly, K 1 ® K 2 c MhE. Using Lemma 1.4.8 to the case of {Hj,Mj,Nj = Mi}, then we have ((Mi)hC;).l = i(Mj)hEj,j = 1,2. Thus Kt ® = (MDh6 ® (M~)h6 c NhE·

Ki

Now it suffices to prove that

K 1 e K 2 + i(Kt ® Kt) = HI ® H 2 , or by Lemma 1.4.9, only to prove that (K; + iKj ) is dense in H j , j = 1, 2. Noticing that (K j + iK;) = (Mj)hEj + i(Mi)hEi ~ Mjc;, it follows that (Kj + iKi ) is dense in Hj since Ej is a cyclic vector for Mj,j = 1,2. Q.E.D.

Theorem 1.4.11. Let H j be a complex Hilbert space, and M j be a VN algebra on j = 1,2. Then we have (M1 ®M2 ) ' = M~ ®M~ on HI ® H 2 •

n..

Proof. For fixed Ei E H j • Let pi be the projection from Hi onto MjEi' and K j = pjH; = MjCj, Nj = MjPf,j = 1,2. Then pi E M'j, N j is a VN algebra on K j , and also N j asmits a cyclic vector Ei,j = 1,2. By Lemma 1.4.10, we have (N 1®N2 ) ' = Nf®N~ on K 1 ® K 2 • Let p' = p~ ® p~. Then it is the projection from HI ® H 2 onto K I ® K 2 • Moreover, p' E M'I ®M' 2 and p'(M~®M~)p'

(M1®M2)p' = N I®N2 ,

=

N~®N~.

Let a E (M1®M2 ) ' , b E (Ml®M~)'. Since p' E MI®M~, it follows that

p'bp'

= bp' E

(M1®M~)'p'

=

(p'(MI®M~)p')'

'- N'2 )' = N 1-®N 2 • (N l® Also since p' E Ml ®M~

c (M1 ®M2 )' ,

we have

p'ap' E p'(M1®M2)'p' = ((M1 ® M 2)p')' = (N 1®N2 )' .

29

Thus p'ap' and p'bp' commute. Let e

(abe, e) =

= el ® 6.

Then

(p'ap'. p'bp'e, e) (p'bp'· p'ap'e, e) = (bae, e).

Therefore, we have

(ab6 ® e2' 6 ® e2) = (ba6 ® e2' el ® e2),

V6

E Hb

e2 E H 2.

Further, ab = ba, Va E (M1®M2)', b E (M:®M~)'. This implies that (M1®M2 ) ' ' - 2')" = M ,1®M - ,2 , an d (M 1® -M)' QED C (M1®M 2 = M'-M' l® 2' ••• Proposition 1.4.12. Let M i be a VN algebra on Hi' and Zj be its center, j = 1,2. Then Z = Zl®Z2, where Z is the center of M 1®M2. In particular, the tensor product of two factors (see Definition 1.3.7) is still a factor. Proof.

Clearly, Zl®Z2

c Z. However, Z

C

M 1®M2

C

M 1®B(H2 ) , and

(M1®M2)' = Mf®M~ C Mi®B(H2). Now by Proposition 1.4.7, we have C Zl®B(H2). Thus Z' :J (Zl®B(H2))' = Zf®<1J1 2. Similarly, Z' :J (lJ11®Z~, Therefore, Z' :J Z~®Z~, Z = Z" C (Z~®Z~)' = Zl®Z2, and Z = Zl®Z2' Q.E.D. Z Z

C

Let M j , N j be two VN algebras on Hj,j = 1,2. Then

Proposition 1.4.13.

{(M1®M2 ) (M1®M2 )

U

(N 1®N2 ) ) " = (M1 U NI)"®(M2 u N 2 ) " ,

n (N1®N2) = (M1 n N 1)®(M2 n N 2).

The first equality is clear by the definition of tensor product of VN algebras. Similarly, Proof.

((M~®M~)

u (N~®N~))" = (M~ u N~)"®(M~ u N~)".

Picking commutants of above equality and by Theorem 1.4.11, we get the second equality. Q.E.D.

Notes. The tensor product commutation theorem, Theorem 1,4.11, was first proved for semifinite Von Neumann algebras by Y. Misonou. But the general case remained unsolved for a long time. Using the concepts of generalized Hilbert algebras and modular Hilbert algebras, M. Tomita solved the problem at first. After Tomita's solution, there were several simple proofs depending on the theory of unbounded linear operators. The present proof here is very elementary, and follows the approach of M. Rieffel and A. Van Daele. References. [109], [111], [134], [149], [150], [181J.

30

1.5. Comparison of projections and central cover Definition 1.5.1. Let M be a VN algebra on a Hilbert space H, and p and q be two projections of M. p and q are said to be equivalent (relative to M), and we write p ..... q, if there exists an element v E M such that v" v = p, and vv .. = q. We write p -< q, if there exists a projection ql of M such that q1 < q and p ..... q1' It is easily verified that " ....." is an equivalent relation. And also we shall see that" <" is an order relation (Proposition 1.5.3).

Let M, N be two VN algebras on Hilbert spaces H, K respectively. M and N are said to be spatial * isomorphic, if there is a unitary operator u from H onto K such that uMu· = N.

Proposition 1.5.2. Let M be a VN algebra on a Hilbert space H. 1) Suppose that p, q are two projections of M, and p ..... q. Then the VN algebras M p , M; on pH are spatially * isomorphic to the VN algebras M q , M; on qH respectively. 2) Let {PI}, {ql} be two orthogonal families of projections of M, and PI ql, VI. Then p = Pl ..... q = ql·

L

L

I

I

I"OJ

3) Let a E M, p be the projection from H onto aH, and q be the projection from H onto a· H. Then P"'" q. 4) Let p, q be two projections of M. Then (p - inf {p, 1 - q}) ..... (q - inf {q, 1 - p}},

(sup{p,q} - q) ..... (p - inf{p,q}). Proof. 1) Let p = v·v and q = vv·. Then v is a unitary operator from pH onto qH. Moreover, vpapv· = qaq,

va'pv* = a' q,

Va E M, a' E M'.

Therefore, the VN algebras M p , M; on pH are spatially VN algebras M q , M; on qH respectively. 2) Let PI = vtVl, ql = VIV;, VI. Define v v*v,q = vv*. Thus p q. 3) It is clear by Proposition 1.3~4. t"'.,J

L VI. I

* isomorphic

to the

Then v E M and p

=

31

4) Since

qpH = (q - inf{q, 1 - p})H, pqH = (p - inf{p, 1 - q})H, and (pq)* = qp, it follows by 3) that (q - inf{q, 1 - p}) Furthermore, if write ql = 1 - q, then sup{p,q} - q = f"..I

f"..I

(p - inf{p, 1- q}).

sup{l- ql,p} - (1- qd = ql- inf{qt,l- p} P - inf{p, 1 - ql} = P - inf{p, q}.

Q.E.D. Proposition 1.5.3. Suppose that P, q are two projections of a VN algebra M, and P -< q,q -< p. Then P q. f"..I

Proof. Let P ql < q and q PI < p. From qi < q '" PI, we have a projection P2 E M such that P2 < Pl and P2 '" ql. Thus P > Pl > P2,P '" P2· From PI PI > P2 > Ps, P '" P2,PI '" Ps. Further from P2 < PI '" P3, we have P4 such that P4 < P3, P4 '" P2. Generally, we get f"..I

f"..I

P = Po

> PI > P2 > ... ,

and Po '" P2

'V

•••

Let Pn = u~Un,Pn+2 follows that

'"

P2n '" ... ,

=

unu~,n

Pn+1 = (UnPn+lr(UnPn+l),

PI '" P3 '" . .. '" P2n+ 1

=

'" ...•

0,1,···. By the above discussion, it

Pn+2 = (unPn+d(unPn+d*,

'tin.

Thus we have Pn - Pn+l = (Un(Pn - Pn+I))*(Un(Pn - Pn+d) and (un(Pn - Pn+d)(un(Pn - Pn+l))*

= un(Pn

- Pn+l)U~ = Pn+2 - Pn+3,

i.e.,

Noticing that P=

((Po - pd E9 (PI- P2)) E9 ((P2 - Ps) E9 (P3 - P4)) E9'" E9 inf{Pnln},

PI =

((PI - P2) E9 (P2 - P3)) E9 ((P3 - P4) E9 (P4 - Ps)) E9 ... E9 inf{Pnln}

and by Proposition 1.5.2, it follows that P '" Pl. But PI '" q, thus we have Q.E.D. P '" q.

32

Theorem 1.5.4. Let M be a VN algebra on a Hilbert space H, p and q be two projections of M. Then there is a central projection z of M such that

qz -< pz, and p(l - z) -< q(l - z). In particular, if M is a factor, then we have either q

- p, c(q) > q. If c(p)c(q) = 0, then z = c(p) satisfies our condition. Thus we may assume that c{p)c(q) =f. O. Since M is the linear span of its unitary elements, it follows that there are two unitary elements u, v of M, such that PQ =f. 0, where P, Q are the projections from H onto upH, vqH respectively. Clearly, we have p P and q Q. Let g', hi be the projections from H onto PQH, Q P H respectively. Then a =f. g' < P, a =f. h' < Q, and g' hi. Since p P, q Q, it follows that there are two projections g, h of M such that 0 =f. 9 < p, a =f. h < q, and 9 h. By the Zorn lemma, there are two maximal orthogonal families {g,}, {h ,} of projections of M such that 0 =f. g, < p,O =f. h, < q, g, h" Vl. Let 9 = gl, h = h ,. Then a =f. 9 < p, a =f. h < q, and 9 h. Since the families I"J

I"J

I"J

I"J

I"J

I"J

L

L

I

l

"'-J

I"J

and {ql} are maximal, it follows that c(pdc(qd = 0, where PI = P g,ql = q-h, and c(pd,c(qd are the projections from H onto [Mp1H],[MqlH] respectively. Finally, let z = C(Pl)' Then {PI}

qz = qlc(qdz + hz = hz ,...., gz < pz, p(l - z) =

PIC(Pl)(I - z)

+ g(I -

z)

= g(I -

z) ,...., h(I - z) < q(I - z). Q.E.D.

Proposition 1.5.5.

Let M be a VN algebra, p and q be two projections of

M.

1) There exists a central projection z such that

pz -< qz,

(1 - p)(1 - z)

-< (1 - q)(l - z).

2) There exist three central projections

z},

Z2, Z3,

with

ZiZj

= 0, Vi =f.

i,

3

and

L z; = 1 such that: i=1

(1) pz ,..., qz for any central projection z of M and

z < Zl; (2) pz -< qz and pz f qz for any non-zero central projection z of M and z < Z2; (3) qz -< pz and qz f pz for any non-zero central projection z of M and z < Z3'

33

Proof.

1) By Theorem 1.5.4, there is a central projection z of M such that z inf{p, 1 - q}

(I - z) inf{l - P, q}

-< z inf{I -

P,

(1)

q},

-< (1 - z) inf{p, 1 - q}.

(2)

By Proposition 1.5.2, we also have zinf{p, 1 - q}) ,...,., (zq - zinf{q, 1 - p}),

(zp -

((1- z)(1- p) - (1- z) inf{I- p,q}) ,. .,.,

(3)

(4)

((1 - z){1 - q) - (1 - z) inf{l- q,p}).

Consider (1)+(3), (2)+(4). That comes to

-<

zp

zq,

(1 - z)(1 - p) -< (1 - z)(1 - q).

2) By the Zorn lemma, there is a maximal orthogonal family {Zt} of central projections of M such that pZt ,...,., qzl, Vl. Let Zl = Zl. Clearly, PZl ,...,., qZl.

L I

Suppose that z is a central projection of M such that pz ,...,., qz . Obviously, pz(I - Zl) ,...,., qz(I - Zl). Since z(l - ZdZI = 0, Vl, and the family {Zl} is maximal, it follows that z < Zl. Thus Zl is the maximal central projection of M with the condition pz ,...,., qe, Now in the VN algebra M(l - Zl), using Theorem 1.5.4 for the projections p(I - Zl) and q(I - zd, then we shall find Z2 and Z3. Q.E.D. Theorem 1.5.6 with conditions:

Let

PIP" =

{P'},EA

0,

be a family of projections of a VN algebra M

PI""'" Pl',

Vi

1= r,

and

L Pl =

1.

lEA

Then M is spatially * isomorphic to Mp®B(K), where p ,...,., PI, Vl, and K is a Hilbert space with dimK = aA. Proof. Let H be the action space of M, L = pH, p = vtvz, PI = v,vt, Vl. In terms of {VI}, we can define a unitary operator from H = tBH, onto L® K,

L

lEA

where H, = pi H, Vl, and K is a Hilbert space with dimK = •A. Thus we let H = L®K. For any a E M,a' E M',l,l' E A, we have all' = vtavl' E M p , ,

-

all' -

v, a V" .,

_.

I -

c

I

- v, v"a - vll,a P

E M'p'

By Lemma 1.4.4 and Proposition 1.4.7, it follows that Me Mp®B(K),

M'

c

M;®<EIK.

34

Q.E.D.

Now by Theorem 1.4.11, we have M = Mp®B{K).

Definition 1.5.1. Let M be a VN algebra on a Hilbert space H, and p be a projection of M. Denote by c(p) the projection from H onto [MpH]. Clearly, c(p) E M n M'. And c(p) is called the central cover (or central support) of p (relative to M). Proposition 1.5.8. Let M be a VN algebra on a Hilbert space H. 1) If p is a projection of M, then c(p) is the minimal central projection containing p, and

c(p)

=

sup{q

I q is a projection of M,

and q "" pl.

2) Let p, q be two projections of M, and p -< q. Then c(p) < c(q). In particular, if p "" q, then c(p) = c(q). 3) Let {PI} be a family of projections of M, and p = sup PI' Then c(p) = I

sup C(PI)' I

4) Let p be a projection of M, and z be a central projection of M. Then

zc(p) = c(pz). 5) Let p, q be two projections of M, and p > q. Then the central cover of q in the VN algebra M p is pc{q), where c(q) is the central cover of q in M.

Proof.

2) Let p = v"v and vv* < q. Then c(p)H = [Mv"vH] c [MvH]

i.e., c (p)

C

[MqH] = c(q)H,

< c(q) .

z be a central projection of M, and z > p, Since zaPe = ape, Va E M, E H, it follows that z > c(p). Now by 2), c(p) > sup{qlq,..., pl. Thus, it suffices to show sup{qlq,..., p} is a central projection of M. Obviously, u . sup{ qlq . . . , p} = sup{ qlq p} . u for 1) Let

e

rooJ

any unitary element u of M. By Proposition 1.3.4, sup{ qlq "" p} is central. 3) Clearly, c(p) > sup C(PI) > p. However, sup c(pd is still a central projecI

tion of M. Thus c(p)

= sup c(pz).

I

I

4) It is clear from zc(p)H [zMpHJ [MpzH] = c(pz)H. 5) It is obvious from [MpqpH] = [pMpqpH] = [pMqH] pc(q)H.

p[MqH] =

Q.E.D.

Proposition 1.5.9. Let p, q be two projections of a VN algebra M. Then the following statements are equivalent: 1) c(p}c(q) #- 0; 2) pMq #- {O}; 3) there exist two projections PI, ql of M such that 0 i= PI < p,O #- ql < q, and PI ......, q1·

35

If pMq = {O}, then [MpH]..l[MqH], i.e., c(p) . c(q) = O. Conversely, if c(p)c(q) = 0, then Proof.

paq = pc(p)aqc(p) = paqc(p)c(q) = 0,

Va E M,

i.e. pMq = {O}. Thus statements 1) and 2) are equivalent. 1) ===? 3). It is contained in the proof of Theorem 1.5.4 indeed. 3) ===? 1). Since Z = C(Pl) = C(ql) :/= 0, and z < c(p), z < c(q), it follows that c(p)c(q) :/= O. Q.E.D.

Proposition 1.5.10. Let p be a projection of a VN algebra M. Then: 1) a'p -4 a'c(p)(Va' EM') is a * isomorphism from M'p onto M'c(p); 2) a' -4 a'p(Va' E M ') is a * isomorphism from M' onto M'p if and only if c(p) = 1. It is easily verified that a'p = 0 if and only if a'c(p) = O(a' E M '). Thus 1) and the sufficiency of 2) are obvious. Now let a' -4 a'p be a * isomorphism from M' onto M'p. Since (1 - c(p)) E M' and (1- c(p))p = 0, it follows that 1 - c(p) = 0, c(p) = 1. Q.E.D. Proof.

The material presented in this section and Sections 6.1, 7.1 is today called the Murray-Von Neumann dimension theory. When F.J. Murray and J. Von Neumann laid the foundation for the theory, they first developed it for factors. Despite its importance, it has remained unchanged ever since Murray-Von Neumann's time. Notes.

References.[28], [82], [111].

1.6 Kaplansky's density theorem Theorem 1.6.1. Let N, M be two * subalgebras of B(H), and N c M here H is a Hilbert space. And also suppose that N is weakly dense in M. Then (Nh is r(B(H), T(H))-dense in (Mh, where (Nh, (Mh are the closed unit balls of N, M respectively. By weak (operator) continuity of * operation, N h is weakly dense in Mh' where N h, M h are the sets of all self-adjoint elements of N, M respectively. Futher, from Proposition 1.2.9, Nh is also strongly dense in Mh. We may assume that Nand M are uniformly closed. For any a* = a E (Mh, let a' = a(1 + (1 - a 2)l /2)- 1. Then a' E Mh and a = 2a'(1 + a'2)-1. Pick Proof.

36

a net {b',} C Ni; such that b', -+ a' strongly. Let b, = 2b',(1 b; = bl E (N)l, VI. We prove that b,

-+

a

+ b'~)~l.

Then

(strongly).

In fact,

~(bl - a)

= (1 + b'~)-l[b'I(1 + a,2) - (1 + b';)a'](1 + a,2)-1 (1 + b'~)-l(b~ - a')(1 + a,2)-1

+(1 + b'~)-lb'l(a' - b',)a'(1 + a,2)-1 (1

+ b';)-l(b', - a')(1 + a,2)-1 + b,(a' - b',)a.

Since b', -+ a' (strongly), Ilbtll < 1,11(1 + b';)-lll < 1, Vl, it follows from Proposition 1.2.1 that b, -+ a (strongly). This implies that (Nhh is strongly dense in (Mhh. . In H EI1 H, define M(2) = { N(2)

Similarly, (Nl a net

2)h

tl) b(l)l,) )

b(l) (

~

~

(:~

= { (::

::)

::)

I ai EM,1 < i < 4} ,

I bi

E N, 1 2

is strongly dense in (Mk (2)

in (Nh

h such that

(

)) 1'

< i < 4} .

Thus for any a E (Mh, we have

b(lJ b(l) )

l,) ~tl) ~

-+

(0 0a) (strongly). In •

a particular, b~l) -+ a (strongly) and Ilb~l)ll < I,VI. This implies that (Nh is strongly dense in (Mh. Now by Proposition 1.2.8, (Nh is r{B(H),T(H))dense in (Mh. Q.E.D.

Corollary 1.6.2. Let M be a * subalgebra of B(H). Then the weak closure of M is equal to the r(B(H), T(H))-closure of M. Corollary 1.6.3. Let M be a * subalgebra of B(H), and 1 E M. Then the following statements are equivalent: 1) M is a VN algebra; 2) M is u(B(H}, T(H))-closed; 3) (Mh is weakly closed.

By Corollary 1.6.2 and Theorem 1.3.10, the statements 1) and 2) are equivalent. It is clear that 2) implies 3); and (Mh is u(B(H), T(H))-closed if (Mh is weakly closed. Thus by the Krein-Smulian theorem, it follows that 3) implies 2). Q.E.D. Proof.

37

Proposition 1.6.4 Let M, N be two uniformly closed * subalgebras of B(H), N c M and N is weakly dense in M. Then (Nhh is strongly dense in (Mhh, and (N+h is strongly dense in (M+h, where N+, M+ are the sets of all positive elements of N, M respectively. Proof. In the proof of Theorem 1.6.1, we have pointed out that (Nhh is strongly dense in (Mhh. Now let a E (M+h. Then there is a net {b ,} C (Nhh such that b, - 4 a (strongly). Let f(t) = {t, 0 < t < 1, 0,

-1

< t < O.

Then f(a) = a, f(b l ) E (N+h, Vl. Pick a sequence {Pn(t)} of polynomials on [-1,1] such that

Pn(O) = 0, Then for any

eE H

< Ilf(bt} -

max If(t) - Pn(t) I - 4 O.

-19::;1

= Il(f(bz) - f(a))cll

Pn(b z)II· ll el!

That implies f(b z)

and

, we have

ac!l

Ilf(bdc -

Vn',

-4

+ Ilf(a) - Pn(a)I!·lleJl + II(Pn(bd - Pn(a))CII· Q.E.D.

a (strongly).

Proposition 1.6.5. Let M, be a VN algebra on a Hilbert space Hz,l E A. Then M = LffiMI = {(az)zEA·1 a, E M"Vl E A, and sUPlladl < oo} is a lEA

VN algebra on the Hilbert direct sum H =

L mH

I,

lEA

sum of the VN algebras family {MZ},E'" Moreover,

lEA

and M is called the direct

M' = L mMI,

and the

IE"

projection Zl from H onto Hz is a central projection of M such that M, Mzz, M{ = M'zz, Vl E r:

Proof.

It is obvious form Corollary 1.6.3.

Q.E.D.

Notes. Theorem 1.6.1 is called the Kaplansky's density theorem, and indeed due to him. It is a powerful tool of the theory of Von Neumann algebras. From Corollary 1.6.2, let M be a * subalgebra of B(H), then the closures of M with respect to all topologies introduced in Section 1.2 are only two: the uniform closure of M and the weak closure of M. That is why in the theory of operator algebras we study C*-algebras and W*-algebras mainly.

References. [83].

38

1.7 Ideals in Von Neumann algebras Let M be a VN algebra on a Hilbert space H. From Section 1.3, we have

(a(B(H),T(H))IM) (s(B(H),T(H))IM) (s·(B(H), T(H))IM)

f'J

f'J

f'J

a(M,M.) s(M,M.)

f'J

f'J

s·(M, M.)

(a-weak top. 1M), (a-strong top. 1M),

f'J

(a-strong

* top.

1M).

In the following, we shall denote simply these topologies by a-top, s-top. or a-strong top., s"-top. or a-strong * top. respectively.

Proposition 1.7.1. Let M be a VN algebra on a Hilbert space H, I be a a-closed left (or right) ideal of M. Then there exists unique projection p of M such that 1= Mp (or pM), and also I is weakly closed. Furthermore, if I is a a-closed two-sided ideal of M, then there exists unique central projection z of M such that 1= M z, in particular, T' = I. Proof. Let I be a a-closed left ideal of M. Then E = In I" is a a-closed * subalgebra of M. By Corollary 1.6.2, E is also weakly closed. From Theorem 1.3.9, E has an identity p. Clearly Mp c I. Let a E I and a = wh be the polar decomposition of a. It is easy to see h = (a* a) 1/2 E E. Thus hp = h, a = whp = ap E M p. Therefore we have M p = I. If there is another projection q of M such that I = Mp = Mq, then we have p = pq,p = p" = qp,p = qpq < q. Similarly, q < p, Thus p = q. Now let I be two-sided. From preceding paragraph, there are two projections p and q of M such that 1= Mp

= qM.

Then we have a, b E M such that q = ap, p = qb, Thus q = qp = p. Put z = P = q. Then I = M.z = z M: Hence for any c E M, there exist d, e E M such that cz = zd, zc = ez : Then cz = zcz = zc, Vc E M, i.e., z belongs to the center of M. Q.E.D.

Remark. It M is a factor, then each non-zero two-sided ideal of M must be a-dense in M.

Proposition 1.7.2. Let I be a two-sided ideal of a VN algebra M. Then 1* = I, and I is the linear span of its positive elements, i.e., I = [1+]. Moreover, if a < a E 1, then there is a family {alhEA C 1+ such that

39

a = lim F

L a, = L at, where r tEF

is the weak closure of I; F is any finite subset

lEA

of A., and directed by the inclusion relation. Proof. Let b E I and b = wh be the polar decomposition of b. Then h = ui'b E I, b* = hw* E I. Thus 1* = I. Let h* = h E I. Then there exist two projections p, q E M such that ph > O,qh < 0, and p + q = 1. Hence 1= [I+J. For < a E by the Zorn lemma we have a maximal family {a'}'EA C 1+ such that at < a for any finite subset F of A.. By Proposition 1.2.10,

° L r, lEF

sup La, F

= (strongly)-limLal

tEF

F

lEF

= Lal E

r.

lEA

Denote this element by aI, and let b = a - aI, so it suffices to prove that b = 0. By Proposition 1.7.1,1 = Mz, where z is a central projection of M. From Theorem 1.6.1, we have a net {bt } C I such that

Ilbtll < 1, Vt,

and b, -+ z (strongly).

Then b1 / 2 bt b1/ 2 -+ b (strongly). If b '# 0, then there is an index t such that bl/2btbl/2 #- 0. Thus b1/ 2cb1/ 2 #- 0, where c = btbb;. We may assume that < c < 1. Then '# b 1/ 2cb 1/ 2 < b = a - al = a - L a" It is impossible since the family {a'},EA is maximal. lEA Therefore b = 0, a = a,. Q.E.D.

°

°

L

lEA

Proposition 1.7.3. Let M be a VN algebra on a Hilbert space H, Z be its center, {tii 11 < i,j < I} C M, and {t'i; 11 < i,j < n} eM'. Then the following statements are equivalent: n

1)

L

tikt'ki = 0, VI < i,j < n;

k=1

2) There exists {Zi;

11 < i,j < n} C

n

L

Z such that

n

tikz k; = 0,

k=1

L

Zikt'k; = t'i;,

VI

< i.i < n.

k=1

It is clear that 2) implies 1). Now let 1) hold. Suppose that K is a n-dimensional Hilbert space. Then

Proof,

t = (tiih~i,i~n E M®B(K), t'

= (t'ijhs;i,i~n E M'®B(K).

Let

I'

= {x'

E M'®B(K)

I tx' = O}.

40

Obviously, I' is a a-closed right ideal of M'®B(K). By Proposition 1.7.1, there is a projection z' = (zijh~i,j~n of M'®B(K), where Zij E M',l < i,j < n, such that I' = z'(M'®B(K)). For any a' E M', x' E 1', since

t(a' ® l)x' = (a' ® l)tx' = 0, it follows that (a' ® 1)1' c 1', where 1 = 1K is the identity operator on K. In particular, (a' ® l)z' E 1'. Thus zl(a' ® l)z' = (a' ® l)z'; (a' ® l)z' . . < nj an d Zij E Z ,1 <.. z , (a' ® 1) ; Zija , = a, Zij, \..Jv aI E M' ,1 < ~,] _ ~,] < _ n. Since z' E I', it follows that tz' = 0, i.e. n

.L

tikZkj

= 0,

VI

< i, i < n.

k=l

From condition 1), we have tt' = O. Thus t' E I',z't' = t'; i.e., n

L

Zik

t ' kj

= t'i;, VI < i.i < n.

k=l

Q.E.D. References.[28], [156}.

1.8 Normal positive linear functionals Let M be a VN algebra on a Hilbert space H, ip be a linear functional on M, ip is said to be positive, and denoted by cp > 0, if cp(a) > 0 for any a E M+ (the set of all positive elements of M). For two linear functionals cp,1/; on M, the relation 1/; < cp means that (cp - 1/;) > O. Moreover, a positive linear functional ip on M is said to be faithful, if cp(a) = 0 for some a E M+, then we have a = O. Clearly, if cp > 0, then we have cp(a*) = cp(a), Va E M; and the Schwartz inequality:

Definition 1.8.1.

and Ilcpll = cp(I).

Definition 1.8.2. A positive linear functional cp on a VN algebra M is said to be normal, if for any bounded increasing net {a,} C M+, we have sup cp(al) = cp(sup all. I

I

41

A linear functional ip on M is called a normal state (on M), if it is positive, normal and cp(l) = 1.

Definition 1.8.3. A positive linear functional cp on a VN algebra M is said to be completely additive, if for any orthogonal family {PI} of projections of M, we have cp(I: pz) =
L

l

I

Lemma 1.8.4.

Let cp,1/J be two completely additive positive linear functionals on a VN algebra M, and p be a non-zero projection of M such that cp(p) < 1/J(p), Then there exists a non-zero projection q of M, q < p, such that

cp(a) < 1/J(a), Proof.

Va E M+

and

a < q.

Let

I

£ = {( q,) (qz) is an orthogonal family of non-zero projections of M, }. and 'IjJ(q,) , q, < p, Vl Then £ is a partially ordered set with respect to the inclusion relation. If £ is non-empty, by the Zorn lemma £ has a maximal element {q,hEA at least. Let qo = I: q,. Clearly a i- qo < p. Since

follows that cp(qo) > 'IjJ(qo). However, cp(p) < 1/J(p), thus a i- q = p - qo < P and q # p. Now let r be any projection of M, and r < q. Since {q'}'EA is maximal, we must have cp(r) < 1/J(r). Further, if a E M+ and a < q, then it is easy to see a E qMq. Now we can write a

= ~1 Ade>.,

where e), E qMq, VA.

Suppose that n-1

an

=L

k=O

k

-n (e(k+1)/n -

ekjn).

Then a < a - an < ~q and cp(a n) < 'IjJ(a n) since (e(k+1)jn - elc/n) < q for each k, Vn. From cp(a n) -+ cp(a),'IjJ(a n) -+ 1/J(a), we have cp(a) < 'IjJ(a), Va E M+ and a
cp(a) < 'IjJ(a),

Va E M+ and a

So the projection p is just what we want to find.

< p. Q.E.D.

Proposition 1.8.5. Let ip be a positive linear functional on a VN algebra M. Then ip is a-continuous if and only if

42

Proof. The necessity is obvious. Now let rp be completely additive. First we can find a maximal orthogonal family {ql},EA of projections of M such that rp(.q,) is a-continuous, Vi E A. Let q = q,. We claim that rp(.q) is

2"::

a-continuous also. In fact, for any a E M with F of A, by the Schwartz inequality we have

lEA 11

all < 1 and any finite subset

Irp(aq) -rp(a2"::q,)12 < rp(2"::q,) .rp(a2"::q,a*). IEF ier I~F Since 0

< a 2":: qla* < aa'" < 1, and

ip

is completely additive, it follows that

I~F

rp(aq)

= lim ip (a L FIEF

ql)

=L

rp(aql),

lEA

uniformly for a E M and Iiall < 1. Therefore, rp(.q) is a-continuous. Now it suffices to show that q = 1. If q i- 1, then p = 1 - q i- o. Pick E E H (the action space of M) such that cp(p) < 1jJ(p), where 1jJ(.) = (·E, E). By Lemma 1.8.4, there is a non-zero projection qo of M, qo < p, such that

rp(a) < 1jJ(a) , If a EM and

lIall < 1,

then 0

< qoa*aqo < qo and

Irp(aqo) 12 < rp(l)rp(qoa"'aqo) < rp(I)1jJ(qoa*aqo) = rp(I)llaqoE1I 2 • This implies that rp( ·qo) is strongly continuous on any bounded ball of M. By Proposition 1.2.6, rp( ·qo) is a-continuous on M. It is impossible since the family {q,}IEA is maximal. Therefore we have q = 1. Q.E.D.

Theorem 1.8.6. Let ip be a positive linear functional on a VN algebra M. Then the following statements are equivalent: 1) rp is rr-continuousj, 2) ip is normal; 3) rp is completely additive. Proof. By Proposition 1.8.5, 1) and 3) are equivalent. obvious that 1) implies 2) and 2) implies 3).

Moreover, it is

Q.E.D.

Corollary 1.8.7. Let cp,1jJ be two positive linear functionals on a VN algebra M, and rp > 1jJ. If rp E M*, then 1jJ EM",. Proof.

Let {all be a bounded increasing net in M+, and a

o < 1jJ(a -

=

sup al. Since I

all < cp(a - az)

i.e., 1jJ is normal.

-+

0, it follows that 1jJ(a) = liF1jJ(a,) = s~p1jJ(al)'

Q.E.D.

43

Proposition 1.8.8. algebra M. Define

Let cp be a normal positive linear functional on a VN

Prp = sup{p I p is a projection of M, such that cp(p) = O}. Then Mprp = {a E Mlcp(a*a) = O}, and cp(prpa) = cp(aprp) = 0, \;fa E M.

Proof.

Let L = {a E Mlcp(a*a) = O}. Clearly, L is a s-closed left ideal of M. By Propositions 1.2.8 and 1.7.1, there is a projection Po of M such that L = Mpo. If p is a projection of M such that cp(p) = 0, then pEL and p < Po. Thus Po = Prp' Now by the Schwartz inequality, we have cp(prpa) = cp(aprp) = 0, \;fa E M. Q.E.D.

Definition 1.8.9. Let cp, Prp be as in Proposition 1.8.8. Then (1 - Prp) is called the support of cp, and denote it by s(cp) = 1 - Prp' Clearly, by Proposition 1.8.8, we have

cp(s(cp)a) = cp(as(cp)) = cp(s(cp)as(cp)) = cp(a) ,

\;fa E M.

Proposition 1.8.10. Let cp be a normal positive linear functional on a VN algebra M,a E M+ such that cp(a) = o. Then s(cp)as(cp) = 0. Moreover, cp is faithful if and only if s (cp) = 1. If s{cp)as(cp) #- 0, by its spectral decomposition we can find a nonzero projection p of M and A > 0 such that p < s(cp) and AP < s(cp)as(cp). Then cp(p) = 0, and 0 #- p < Prp = 1- s(cp). It is impossible since p < s{cp). Therefore s(cp)as(cp) = o. The rest conclusion is obvious. Q.E.D.

Proof.

Proposition 1.8.11. Let M be a VN algebra on a Hilbert space H, and E H, cp(.) = ('E, E). Then s(cp)H = M' E·

e

Let p be a projection of M. Then cp(p) = 0 <===> pE = 0 <===> pM'E = O. That comes to the conclusion. Q.E.D.

Proof.

Now let M be a VN algebra on a Hilbert space H, and cp be a positive linear functional on M. Let

Lrp = {a E M

I cp(a*a) ='O}.

Lrp is called the left kernel of ip, By the Schwartz inequality, we can see that Lrp is a left ideal of M. Let a -+ arp = a + Lrp(\;fa E M) be the quotient map from M onto MJ Lrp. Then we can define (arp, brp) = cp(b*a),

\;fa, b E M.

44

Clearly, (,) will be an inner product on M/L'P' Denote its completion by H'P' For any a EM, define

?r'P(a)b'P = (ab}'P'

Vb E M.

It is clear that ?r'P(a) is a linear map on M/ Lip, and

Then ?r'P(a) can be uniquely extended to a bounded linear operator on H'P' Denote this extension still by ?r'P(a). Then we have a * (algebraic) homomorphism i.e.,

?r'P(Aa + J.Lb} = A1f!p(a) + J.L1I"'P(b) { ?r!p(ab) = ?r'P (a) ?r'P(b), ?r'P(a*) = 1I"ip(a)*,

Va,b E M,A,J.L E (C. Moreover, we also have pointed out that 11?r'P(a) II < Iiall, and
Proof.

Thus a E I.

45

Now by Propositions 1.2.8 and 1.7.1, there is a central projection z of M such that I = M(l - z). Therefore, 'lrep is a * isomorphism from M z onto

'lrep(M).

If a E Mz such that 'lrep(a) > 0, we claim that a > O. In fact, since '!rep is injective on Mz, it follows from 'lrep(a)* = 'lrep(a) that a' = a. So we can write a = a+ - a_, where 0 < a± E Mz and a+ . c., = O. Suppose that c., :I O. Then B = 1l"!e(a_) is non-zero and positive on Hep, and there is E Hep such that TJ = B 3/ 2 e:I O. Now we get a contradiction:

e

Thus c., must he zero, and a = a+ > O. Let h* = hE Mz. Since 'lrep(z) = 1 and

it follows from the discussion of preceding paragraph that -11'lrep(h)II'1 < h < lI'lrep(h)II' 1, i.e. Ilhl! < II'lrep(h)B. Thus we have Ilhll = II'lrep(h)I!,Vh* = h E Mz. Furthermore, from II'lrep(a)11 2 = 111I"ep(a*a)1I and !Ia·all = Ilall 2 , we have Ball = lI'lrep(a) II, Va E Mz. So 1I"ep is an isometric * isomorphism from Mz onto

1I"ep(M). In order to prove that 'lrep(M) is a VN algebra on Hep, by Corollary 1.6.3 it suffices to show that the unit ball of1l"ep(M) is weakly closed. Let {A,} be a net of 1I"ep(M) with IIAdl < 1, Vl, and A , ~ A (weakly). From preceding paragraph, there exists a net {all C Mz such that 1I"ep(a,) = At, Iladl = IIAdl, Vl. Since the unit ball of M z is weakly compact, we may assume that at --+ a (weakly) and a E Mz. Then

(Abeplcep) = Vb,c

E

lim('lrep(adbep,cep) = lim
t

M. Therefore A = 'lrep(a) E 1I"ep(M), and the unit ball of 'lrep(M) is

weakly closed. About the a-a continuity of 11" ep' by Proposition 1.2.6 it suffices to show that 1I"ep is weakly continuous on the unit ball of M. This is easy. Finally, let ip be faithful. Then it is easy to see that 11" ep is faithful (injective), and also 1I"ep is isometric. Moreover, let P be the projection from Hep onto 'lrep(M)'lep. Then P E 'lrep(M)" = 1I"ep(M), and there is unique projection p of M such that 1I"ep (p) = P. Since
References.[22], [28], [150].

46

1.9 Polar decomposition and orthogonal decomposition Let M be a VN algebra. For any IfJ E M. and a EM, define

(Ra/{J)(b) = /(J(ba),

(La/{J)(b) = lfJ(ab) ,

Vb E M.

Obviously, RalfJ and LalfJ EM•.

Lemma 1.9.1. Let /{J EM.. , and p be projection of M such that IIRpIfJII = 111fJ11· Then /{J = RplfJ·

Pro01. We may assume that 111fJ11 = 1. If R(l-p)1fJ # 0, then there exists a E M with \Iall < 1 such that (R(l-p)lfJ)(a) = 6 > O. Since M = (M.. )'", we can find bE M with llbll = 1 such that (RplfJ}(b) = 11RplfJII = II/{JII = 1. Notice that Thus

Ilbp + 6a(1 - p) 11 2 = lIbp + 6a(1 - p) II < 1 + 6 2 •

Ilbpb·

+ 62a(1 -

p)a*11 <

1+ 62 •

However,

lfJ(bp+6a(1-p)) = (RpIfJ)(b}+6(R(1_p)/{J)(a) =

1+62 •

This is a contradiction since 111fJ11 = 1. Therefore R(l-p)/{J = 0, IfJ

= RplfJ. Q.E.D.

II all <

1 be such that

Proof. Pick a real number 0 such that ei S l(l-a} > O. Since 1, it follows that 11/11 < I(a) + eie 1(1- a) < 11/11.

Ila+ei 8(1-a)11 <

Lemma 1.9.2. Let I E M" and a E M+ with I(a) = 11/11. Then I > o.

Thus 1(1 - a) = 0, i.e. 1(1) = I(a} = 11/11. Now let bE M+. We need to prove I(b) > o. We may assume that and II III = 1. If I(b) = ,x + ip" where ,x, p, E lR, then

1 That implies ,x

it follows

> 111 - bII > I1(1 - b) I =

(( 1 -

> O. Moreover, since for any r

lIbll < 1

,x) 2 + P, 2 ) 1/2 •

E lR

Ilbl1 2 + r 2 = lIb + irl12 > I/(b + ir)1 2 > p,2 + 2rJ.L + r 2 , that p, = o. Therefore I (b) = x > o.

Q.E.D.

Theorem 1.9.3. Let M be a VN algebra, and IfJ EM... Then we have unique W E M. with w > 0 and unique partial isometry v of M such that /(J

=

R.vw,

v"v = s(w).

47

Moreover, w = Ru· O. Let p = uu*. By a = ap, Proof. Let a*

Then by Lemma 1.9.1, we have ip = Rp

s(w). Let v = us(w). By Definition 1.8.9, we have

Now suppose that there is another 0 < w' E M. and v' E M with ip = Rv'w' and v''''v' = s(w'). It is easy to see that w' = Rv'·'P and I!w'll = 11
thus we have w((v'*v - 1)"'(v'*v - 1)) = O. inequality, w(b(v''''v - 1)) = 0, Vb E M, i.e.,

w(b) =
Then again by the Schwartz

Vb E M, or w = w'.

Let q = v'·v. Then s(w)qs(w) = s(w')qs(w) = q since w' = w. From w((q1)*(q - 1)) = 0 and Proposition 1.8.10, we have

s(w)(1 - q·)(1 - q)s(w) = Thus s(w) = qs(w) = q. Now by v'·v' = s(w')

o.

= s(w) = q, it

follows that

v''''(v'-v) =0.

(1)

o.

(2)

v* (v' - v) = Considering (1) - (2), we obtain v' = v.

Q.E.D.

Definition 1.9.4. The unique expression

w(·) = tr(.h).

48

Proposition 1.9.5. For any ip EM", define cp"(a) = cp(a*), Va E M. Then the polar decomposition of ip" is ip" = Rv.t/J if the polar decomposition of ip is cp = Rvw. Here t/J = Lv·cp. Proof. Clearly, t/J > O,cp* = Rv·t/J,and Iit/JII = Ilwll = lIcpll = !Icp*ll. By Theorem 1.9.3, it suffices to show s(t/J) = vv·. For any projection p of M, the following conditions are equivalent: (1) t/J(p) = OJ (2) w(v*pv) = 0; (3)s(w)v·pvs(w) = 0; (4) v*pv = 0; (5) pv = 0; (6) pvv· = O. That implies s( t/J) = vv·. Q.E.D.

Definition 1.9.6. s(lcpl) and s(lcp*1) are called the left and right support of cp(E M.), denoted by s,(cp) and sr(CP) respectively. Clearly, when cp is hermitian, i.e., cp. = ip , we have SI(CP) = sr(CP)' If ip > 0, then S, ( cp) = Sr (cp) = S(cp ). Proposition 1.9.7. of cpo Then

Let

ip

EM., and

ip

= Rvw be the polar decomposition

= v*v = inf{p I p is a projection of M, sr(CP) = vv" = inf{p I p is a projection of M,

SI(CP)

= cp}, Rpcp = cp}.

and Lpcp and

Let p be a projection of M. The following conditions are equivalent: (1) Lpcp = cp; (2) Lpw = w; (3) P > s(w) = Sl(cp). So we can get the expression of (cp) immediately. The proof of the expression of s, (cp) is similar. Q.E.D. Proof.

s,

Now we study hermitian [unctionals. Let cp be a linear functional on a VN algebra M. Then cp is said to be hermitian, if ip" = cp, where cp*(a) = cp(a"), Va E M. Obviously, cp is hermitian if and only if cp(h) E JR, Vh· = h E

M. Theorem 1.9.8. Let M be a VN algebra, unique CP+, ip : E M*, CP± > 0 such that

ip

= cp"

Moreover, let q.; = s(cp+),q_ = s(cp_). Then

and cp = R(q+_q_) lcpI is the polar decomposition of cpo

E M*. Then there exist

49

Proof. Let ip = R; ICJ?], tp* = Rv-Itp*l be the polar decompositions of CJ?, rp* respectively. Since CJ?* = rp and by the uniqueness of the polar decomposition, we have v* = v. Then we can write

where q+, q: are two projections of M, and q.; . qq.; + q_, it follows that

=

O. Since

s(Jtpl) = v*v =

Itpl = Lq+Rq+ 1CJ?1 + Lq_Rq-ltp[ + Lq+Rq-lCJ?1 + Lq_Rq+ [tpl· In addition, by Proposition 1.9.5, Irp·1 = LvRvlCJ?[ = ltpl. Thus LHRq-ltpl = LHRq_LvRvlCJ?1 = -Lq+Rq-ltpl, i.e., Lq+Rq_ICJ?1 = O. Similarly, Lq_Rq+ltpl = o. So we have and

= RvlCJ?1 = LHRq+ltpj- Lq_Rq-ltpl· Write tp+ LHRH ICJ?I, ip : = Lq_Rq-ltp[. Then tp± EM., tp± > 0 and CJ? = tp+ - CJ?-, Itpl = tp+ + CJ?-. By Theorem 1.9.3, IItp!l = II Itpl II = Itpl(l) = Iltp+ II + Ilrp-ll· Moreover, it is obvious that tp+(l - q+) = O. And ip

also, if p is a projection of M such that tp+(p) = 0, then Itpl(q+pq+) = O,s(ltpl)q+pq+s(ltpl) = O,q+pq+ = O,pq+ = 0, i.e., p < 1 - q+. Thus q+ = s(tp+). Similarly, q: = s(CJ?-)' Now suppose that tph CP2 E M., CJ?h tp2 > 0, such that ip

=

tpl - tp2,

Since Iltp+11 = tp(q+) < tpdq+) < it follows that

Iltpll

Iltplll

=

and

IlcpIII + 11CJ?211· 1ICJ?-1I

= -tp(q-)

< tp2(q-) < !Itp211,

Further, S(CPI) < q+,S(CJ?2) < q-, and s(tp+). s(tp_) = o. So we get ip = R(8(~I)-"(~2»(tpl +tp2)' By the uniqueness of the polar decomposition, it follows that s(tptl- S(CJ?2) = q.; - q-, CPI + tp2 = tp+ + CJ?-. But s(tpt} . S(CP2) = q.; . q.: = 0, it must be that s(cpt} = q+,S(rp2) = q_. Furthermore, tpl = L"{~d(tpl + tp2) = LH(tp+ + tp-) = tp+,tp2 = L8{~2)(tpl + CP2) = Lq_(tp+ + tp-) = CJ?-. Q.E.D.

Corollary 1.9.9. Any a-continuous linear functionals on a VN algebra must be a linear sum of normal positive linear functionals.

50

Definition 1.9.10. The unique expression cP = CP+ - ip : of a hermitian functional 'P (see Theorem 1.9.8) is called the orthogonal (or Jordan) decomposition of cp. The result in 1.9.8 generalizes the ordinary Jordan decomposition of a signed measure. References. [64], [145], [150j, [180].

1.10 Radon-Nikodyn theorems Lemma 1.10.1. Let N be a VN algebra on a Hilbert space K; and N admits a cyclic vector ~ (i.e. N E= K). Define

cp(a) = (aE, E),

Va E N.

If 1/J is a positive linear function on N with 1/J t' E N' with 0 < t' < 1 such that

1/J( a)

=

(at' E,

E),

< cp, then there exists unique

Va E N.

On the dense linear subspace N ~ of K, define

Proof.

[a~, bE]

= 1/J(b*a) ,

Since 1/J < cp, it follows that l[aE, bE]1 exists unique t' E B (K) such that

Va, bEN.

< IlaE11 . Ilb~II, Va, bEN. Thus there

= [aE, bE] = (t'aE, bE), Va, bEN. < 1/J < 'P, we have a < t' < 1. Further, since 1/J(b*a)

From

0

(t'abE,cE)

Va, . b, c E

=

1/J(c*ab)

= 1/J((a+c)*b) = (at'b~,c~)

Nand N E= K, it follows that t' E N'. That comes to the conclu-

sion.

Q.E.D. Lemma 1.10.2. Let 'Po, CPI be two positive linear functionals on a VN algebra N, and CPI = Ra'Po for some a EN. Then CPI < lIallcpo.

Proof.

Let CPn+1 inequality we have

= R a 2 n 'Po, n =

0, I, . ". For any b E N+, by the Schwartz

a
51

Since ~2(b) = rpl(ba) /P2 > 0 and 0 0 and 0 < ~o(b)~+~ '/P3(b)l/4,Vb

o<

/PI (b)

= /Pl(a*b) = rpo(a*ba)

> O,Vb

E N+ it follows that

2,Vb

< /PO(bP/2 '/P2(bP/ E N+. Similarly, we can prove /P2(b) < /po(b)l12rp3(b)l/2,Vb E N+. Thus 0 < /PI(b) < E N+. Generally, we have

< /Po(b)~+"'+2~ '/po(ba2n ) 2~ < /Po(b)4+"'+2~ '1Ialj(llrpoll'llbI1)2~,

Now let n

-+

00, then we get /PI

Vb E N+, Vn.

< lIal1/po.

Q.E.D.

Theorem 1.10.3. Let M be a VN algebra, /p,.,p E M* and /P > .,p > O. Then there exists to E M with 0 < to < 1 such that

.,p(a) = rp(toat o),

Va E M.

Proof. First, let s(~) = 1. If {?rep, Hip} is the * representation of M generated by ~, by Proposition 1.8.13 then N = ?rep(M) is a VN algebra on Hep, and N is * isomorphic to M. -Using Lemma 1.10.1, we have t' E N' with 0 < t' < 1 such that .,p(a) = (?rip(a)t'lip,t'lep), Va EM.

Define two functionals on N' : /p'(a') = (a'lep,lep), Va' E N ', and .,p' = Rt,rp'. Let .,p' = RU'w' be the polar decomposition of .,p'. Then w' = Rv'_t'/P'. By Lemma 1.10.2, we have w' < Ilv'*t'llrp' < /p'. Further, by Lemma 1.10.1 there is to E M with 0 < to < 1 such that

w'(a') = (?rep(t o)a'lip' lip)' Va' EN'. Since w' = Ru'-t'rp', it follows that

(a'?rep(t o) lip' lip) = (a'v'*t'lip, lip)'

Va' EN'.

By Proposition 1.8.13, 1ep is a cyclic vector for N'. Thus ?rep (t 0 ) 1ep=v ''''t'l ep'

(1)

From (1),

(a't'lep,lep) =

.,p'(a')

= (Rvrw')(a')

= (Rv'v'-t,rp')(a')

(a'v'v'*t'lip, lip) = (a'v'?rep(t o)lep, 1ep), Va' EN'. Hence

(2)

52

From (1), (2), for any a E M

=

lP(a)

(1rrp(a)t'1rp,t'1rp)

= (1rrp(a)v'1rrp(t o)1rp,t'1rp)

(1rrp(a)1rrp(to)1rp,1rrp(to)1rp)

=

cp(toat o).

That comes to the conclusion for s(cp) = l. Now consider general cpo Let p = s(cp), M p = pMp. On the VN algebra M p , cp is faithful, and also cp > 11' >0. From the preceding paragraph, there exists to E M with 0 < to s(lP). Then for any a E M, lP(a) = lP(pap) = cp(topapt o) = cp(toat o). Q.E.D.

Theorem 1.10.4. Let M be a VN algebra, cp,.,p E M", and cp > 11' > O. If A is a complex number with ReA > then there exists hEM with 0 < h < 1 such that lP(a) = Acp(ha) + Acp(ah) , Va E M.

!,

Moreover, if cp is faithful, then h is unique.

Proof. Since {h E M M, it follows that

I 0 < h < 1}

is a u(M, M.)-compact convex subset of

£ = {Acp(h.) + Acp(·h)

I hEM, 0 < h < 1}

is a u(M""M)-compact convex subset of M •. If 11' a = a'" E M and a Jit E JR such that

lP(a) > Jit > f(a),

Vf

E

rf. L, then there exists a

t:

Write a = a+ - a_, where a± E M+ and a+ . c., = 0, and pick a projection p of M such that pa+ = a+, pc., = O. Then

lP(a+) >

/-L

> Acp(pa) + Acp(ap) = 2ReA· cp(a+) > cp(a+).

It is impossible since cp o < h < 1 such that

> 11'. Thus 11' E L, i.e., there exists hEM with

lP(a) = A
+ Acp(ah),

Va E M.

Moreover let cp be faithful, and suppose that hand k satisfy our condition simultaneously. Since

(A + A)(h - k)2 =

+ A(h - k)h] -[Ak(h - k) + A{h - k)k), [Ah(h - k)

53

°and h = k.

it follows that tp((h - k)2) =

Q.E.D.

Proposition 1.10.5. Let M be a VN algebra on a Hilbert space H, tp be a a-(or weakly, or strongly) continuous linear functional on M. Then tp can be extended to a a-(or weakly, or strongly) continuous linear functional VJ on B(H), and IIVJII = IIcpll. Moreover, if tp > 0, it can be choosed that VJ > 0.

Proof.

First, we assume t* = t E T( H) such that

° < cp EM..

tp(a)

=

Since M. = T(H)/M.l, we have

Va E M.

tr(ta),

Write t = t+ - i : , where t± E T(H)+ and t+ . t : = O. Then

tr(t+a) > cp(a) > 0, By Theorem 1.10.3, there is to E M with

cp(a) = tr(t+toato)

=

Va E M+.

°< to < 1 such that

tr(tot+toa),

Va E M.

Let VJ (.) =tr( tot+ to') . Then VJ is a a-continuous linear functional on B (H). Clearly, VJ is an extension of cp, and VJ > 0, and Iltpll = cp(l) = VJ(l) = lIVJII· For general tp EM., let cp = Rvw be the polar decomposition of ip • From the preceding paragraph, there is a a-continuous positive linear functional 9 on B(H) such that glM = w, Ilgll = lIwll = Iltpll. Let VJ = Rvg. Then VJ is acontinuous on B(H), and is an extension cp. Clearly, Iltpll < IIVJII < IIvllllgll < Iltpll· Thus IIVJII = Iltpll· When tp is weakly (or strongly) continuous, according to the above procedure it suffices to show there exists t E F( H) such that tp( a) = tr{ ta), Va EM. If cp is weakly continuous, then there exists a weak neighborhood U = U(Oj6,·",en;T/iJ"',T/n;1) = {a E M II(aei,T/i)1 < 1,1 < i < n} of zero such that Icp(a)1 < 1, Va E U. Define a semi-norm p(.) on B(H): n

p(b) =

L l(bEi,T/i)l, Vb E B(H). i=1

Then Itp(a)1 < p(a), Va E M. By the Hahn-Banach theorem, tp can be extended to a linear functional VJ on B(H) such that IVJ(b)1 < p(b), Vb E B{H). Clearly, VJ is weakly continuous on B(H). By Proposition 1.2.7, there is t E F(H) such that VJ(b) =tr(tb), Vb E B(H). In particular, cp(a) =tr{ta), Va E M. If cp is strongly continuous, replacing above U = U(O, Ell"', En; T/l,"', T/n; n

1) and p(.)

= L 1('Ei, T/i) I by i=1

i < n} and p(.) = argument.

U = U(o; 6,"" En; 1)

= {a E M111aEi11 <

1,1

<

n

L II· Ei11 i=1

2

,

we can get the same conclusion by the same

Q.E.D.

54

Remark. By Propositions 1.2.2, 1.2.7 and 1.10.5, each a-continuous functional

where

2:(11 Eil1 2 + II17i 11 2 ) <

00,

and if ep > 0, then we can choose Ei = ru, Vi;

i=1

each weakly or strongly continous functional ep on M has the following form: n

Va E M,

and if

0, then we can choose Ei = Tli, 1 < i

< n.

As an application of Proposition 1.10.5, we have the following. Proposition 1.10.6. Let M be a VN algebra on a Hilbert space H. Then the following statements are equivalent: 1) a(M, M.) ,...., (weak topology IM)i 2) s(M, M.) rv (strong topology 1M); 3) s·(M,M.) rv (strong * topology 1M); n

4) Each a-continuous linear functional on M has the form L('Ei' Tli);

i=1

n

5) Each a-continuous positive linear functional on M has the form L('Ei' Ei)'

i=1

Clearly, 1), 2) and 3) are equivalent; 4) and 5) are equivalent. From 4), each a-continuous functional on M is weakly continuous. Thus (weak topology 1M) -- a(M, M.). By Proposition 1.10.5, it is clear that 1) implies 4). Q.E.D.

Proof.

Remark. The conditions in Proposition 1.10.6 are possible. For example, see Sections 1.13 and 2.11. References.[32], (127], [147], [159].

55

1.11. The equivalence of the topologies bounded ball

8*

and

in a

1"

Let M be a VN algebra on a Hilbert space H, (Mh = {x E M Illxll < I} be its unit ball. In this section, we shall prove that s·(M,M.) """ r(MM.) in any bounded ball of M.

Lemma 1.11.1. Let {a~ = an} be in (Mh and an -----+ 0 (strongly). Then for any 6 > 0 there is a sequence {Pn} of projections of M such that

Pn Proof.

Let an =

-4

l(strongly),

h Ade~n) = /6 de~n), 1

IlanPnll < 6, \In.

and

be the spectral decomposition of an, and define

Pn

qn

-6

Then - 6

(1

= 1- Pn,

\In.

A2

8-2a~ > (/ -1 + 16 ) 6 2 de~n} > qn, \In. Since a~

-4

0 (weakly), it follows that qn

Moreover, IlanPnl1 we want to find.

=

11/:6 Ade~n} I

-4

0 (strongly), Pn

-4

1 (strongly).

< 8, \In. So the sequence {Pn} is just what Q.E.D.

Let 'P be a faithful normal positive linear functional on

Lemma 1.11.2. M, and define

d(a, b) = tp((a- b)*(a- b))1/2,

\la,b E (Mh.

Then ((Mh, d) is a complete metric space, and the topology generated by d is equivalent to s(M, M.) in (Mh. Proof. that

By Propositions 1.2.2 and 1.10.5, there is a sequence {en} C H such

2:\lenI1 2 < 00,

'P(a)

n

Thus d(a, b)

= 2:(aEn,cn),\la EM. n

= (2: ll(a - b) EnI1 2)1/2 is a metric on (Mh. n

We claim that [M' Cn\n] is dense in H. In fact, let P be projection from H onto [M' cnlnJ. Then P E M, and pen = En, \In. Since 'P(1 - p) = 2:({1 n

p) en, En) = 0 and tp is faithful, it follows that P = 1, i.e., [M' En In] is dense in

H. From this claim, our lemma is easy.

Q.E.D.

56

Lemma 1.11.3. Let !P".,!Po be in M., and !p". -+ CPo(u(M., M)). In addition, suppose that {an} is a sequence of (Mh and an -+ O(s·(M,M.)). Then lim!p".{a n) = 0 uniformly for k. n Proof. Obviously, {llcpkll I k} is bounded. So we may assume that 1/2, Vk. By Theorem 1.9.8, for any k we can write

IIcpkll <

CPk = (!p~l) _ cp~2») + i(cp~3) _ !p~4»), where 0
! > Ilcp~l)1I + 11!p~2)11 = IIcp1

) _

CP12 ) II,

= Ilcp13) -

cp14 } 1/,

1

~ > 11!p(3)11 + Ilcp~4}J1 Clearly,

II [cp".JII <

4

1 and 0
=

;:1

f

1". [cp".). Let p = s(cp). Then cpii} (1 - p) 2 k=1 and a EM. Further,

= 0,

and

cp~)(a) = cp~)(pap), Vk,j

cp".(a) = !p".(pap), Vk, and a E M. Clearly, panP -+ O(s·(M,M.)). Considering the problem to pMp, we may assume that p = 1, or cp is faithful. Let d be the metric on (Mh as in Lemma 1.11.2. For any fixed e > 0 and m, define

H m = {a E (Mh Ilcpk(a) - cpo(a)! < s ,

Vk > m}.

Clearly, H m is a closed subset of ((Mh,d). Since cp". follows that 00

(M)l =

-+

CPo(u(M.,M)), it

U u.;

m=l

Now by the Baire category theorem and Lemma 1.11.2, there is a ao E (Mh and a JJ, > 0 and a mo such that

{b E (M)l

I d(ao, b) < JJ,}

C

Hmo·

Since {~(an+a~)}n and {;i(an-a~)}n C (M)l and they converge to 0 with respect to s(M, M.), we may assume that an = a~, Vn. Now pick 6 = ~e. By Lemma 1.11.1, there is a sequence {Pn} of projections of M such that

Pn

-+

l(s(M,M.)),

IlanPnll < 6,

Vn.

57

Let

tP/r, =

ItP/r,(PnanPn) I + ItPle((l- Pn)anPn) I

+ItP Ie (Pn an(1 -

Pn)) 1 + ItP/r, ((1 - Pn) an(1 - Pn)) I

< 36+ltP/r,((1-Pn)an(1-Pn))!,

(

1)

Vk,n.

Let b.; = PnaOPn + (1 - Pn)a n(1 - Pn),Vn. Then bn E (Mh, Vn, and bn ~ ao(s(M,M.. )). Thus there is ni such that bn E Hmo, Vn > ni. By the definition of Hmo, we have ItP/r,(b n) 1 rno, n > ni' (2) Since PnaOPn ~ ao(s(M, M*)), it follows that there is n2 such that PnaOPn E Hmo, Vn > n2' Thus

]tP/r,(PnaOPn) 1
> nb n2, and k > rno,

(3)

Vk > rno, n > n2'

by (1), (2), (3),

< 36 + 11/1 /r, ((1 - Pn) an (1 - Pn)) 1 < 36 + ItP/r,(b n) I + 11/I/r,(Pn aOPn) I < 3c.

Q.E.D.

This means that lim

Lemma 1.11.4. Let A be a a(M*,M)-compact subset of M*. For any e > 0 there exists a 6 > 0 and a finite subset F of A such that if a E (Mh with [ O. Then for ~ and any fixed
[flJO](a~al + aIa~)
}J

Icpl(adl > c.

and {
<

...... Generally, we have {CPnln such that

;2' =

i

= 0, 1,

1 c.

0,1,"'} C A, {anJn = 1,2,"'} C

[flJi](ajaj + ajaj) < ;" { IflJj(aj)I>e:,

~
- 1,

)=1,2,'"

(Mh

58 v

Since A is a(M.. , M)-compact, by the Eberlein-Smulian theorem there is a subsequence {nk} C {O, 1, ...} and .,p E A such that rpn,. --+ .,p (a (Milt, M)) . 00

Clearly, A is bounded. Thus 0 <

tp

L

=

2- k [rp n k] E M*. Notice that

k=l m

Irp(ajaj + ajaj)] <

L

2-

kl[ep

nk](a; aj

+ aja;)1

k=l

+L

k 1

2- + 11[epnk]11

k>m

< 2- j

+ 2- m +2 sup lIrpnll, n

where m is an integer such that nl, ... , n m < j - 1, and nm+l > j - 1. Since m --+ 00 as j --+ 00, it follows that

rp(ajaj

+ uja;)

--+

0,

as

J

--+ 00.

Let P = s(rp). By the Schwartz inequality, it is easy to see that

Since rp is faithful on pMp, by Lemma 1.11.2 we have PUjP

--+

o(s*(M, Milt)).

By Lemma 1.11.3 and p > s([rpnkD (Vk), it follows that lim epn/c (pajp) J

=

lim rpnk (aj) = J

a

uniformly for k. Then we get a contradiction since Irpnk(ank)1 >

E:,

Vk. Q.E.D.

Lemma 1.11.5. Let A be a ai M«, M)-compact subset of M*. Then we have .,p E M*,.,p > 0 with the following property: for any E: > 0, there is 6 > 0 such that if a E (Mh and 'tjJ(a*a + aa"') < 6, then Irp(a) I < e, Vrp E A. For 1, by Lemma 1.11.4 there is a finite subset F; of A and On > 0 n such that if a E (Mh and Irp](a*a+aa*) < 0n,Vrp E Fn, then Irp(a) I < *,Vrp E A. Define 00 'tjJ = 2-(n+m n ) [rp]. Proof.

where m n pick ~o <

L

L

n=l

tpEFn

= •Fn,and the definition of [ep] is as in Lemma 1.11.4. For any E: > 0, E:,

and let fJ = 6no/2no+mno. If a E (Mh and .,p(a*a + aa"') < 6, then

[rp](a*a

+ aa·) < ono'

Vrp E Fn o '

59

By the definitions of F no and 6n o , we have

IIp (a) I < ';0 < s , Vlp E A. Q.E.D.

Theorem 1.11.6. Let M be a VN algebra. Then in any bounded ball of M, we have r(M,M*) .-..; s*(M,M*). Let {a,l be a net of (Mh with a, ~ O(s*(M,M*)), and A be a a(M*, M)-compact subset of M•. It suffices to show that 0, pick 0 < 'IjJ E M* and {j > 0 as in Lemma 1.11.5. Since a, ~ o(s*(M,M*)), it follows that there is an index 10 such that Proof.

t/J(a7al

+ alan < 6,

Vi > 10.

Then by Lemma 1.11.5, IIp(adl < e, Vi > la, lp E A. This means that lp(al) ~ 0 uniformly for

Proposition 1.11.7. Let M be a VN algebra, and dimM = 00. Then in whole M,r(M,M*) f s*(M,M.,). Moreover, in (Mh, uniform topology f r(M, M*). In consequence, if a VN algebra M is reflexive as a Banach space, then diDLh1 < 00. Proof. Let M be infinite dimensional. First we claim that M contains an infinite orthogonal sequence {Pn} of non-zero projections. In fact, if dimZ = 00, where Z is the center of M, then we can easily find such {Pn} C Z. If dimZ < 00, then we may assume that M is a factor. By the theory of factors (see Section 7.1), such {Pn} exists. Now using above {Pn}, similar to the proof of Proposition 1.2.5 we can see that r(M,M*) is not equivalent to s*(M,M.,) in whole M. 00

Again using above {Pn}, let P = LPn. By Proposition 1.2.10 and Theorem n=l

m

1.11.6,

L Pn ~ P with respect to r(M, M*).

On the other hand, we have

n==l m

lip - L Pnll

= 1,

Vm.

n=l

Thus, in (Mh the uniform topology is not equiv.vlent to r(M, M*). If M is reflexive as a Banach space, then r(M, M.,,) .-..; uniform topology. Therefore, M must be finite dimensional. Q.E.D. Remark. If A is a C·-algebra, and A is reflexive as a Banach space, then dimA < 00. Indeed, An is a VN-algebra (see Section 2.11), and is reflexive as a Banach space.

60

Notes. It was S. Sakai who initiated the study of the Mackey topology of a Von Neumann algebra. He showed that the Mackey topology on any bounded ball of a finite Von Neumann algebra M agrees with s·(M,M.). Theorem 1.11.6 was due to C. Akemann who gave an affirmative answer for Sakai conjecture. Further in Section 4.5, we shall study the characterizations of a(M",, M)-compact subsets of M",. References.{2], [146].

1.12. Normal

* homomorphisms

Definition 1.12.1. A * (algebraic) homomorphism from a VN algebra M into a VN algebra N is said to be normal, if for any bounded increasing net {all C M+ we have sup (al) = (su p az) . l

l

Let be a * homomorphism from M to N. First, will preserve the order, i.e., (M+) C N+. This is obvious because any element of M+ has the form a"'a (for some a EM). Secondly, 1I1I < 1. In fact, (l M ) = P is a projection of N. For any h = h'" E M, since -llhll . 1M < h < Ilhll . 1M , it follows that -llhllp < (h) < Ilhllp. Thus 11(h)11 < Ilhll. Further, 11(a) 11 2 = II (a'" a) II < lI a11 2 , Va E M. Hence IIII < 1. From these facts, if {a,l is a bounded increasing net on M +, then { (a,)} is also a bounded increasing net of N+. Now by Proposition 1.2.10, sup c, and sup (al) exist. l

I

Proposition 1.12.2. Let be a * homomorphism from a VN algebra M to a VN algebra N. Then the following statements are equivalent: 1) is a-a continuous; 2) is normal; 3) is completely additive, i.e., (L PI) = L (pz) for any orthogonal I

family {PI} of projections of M. Moreover, (M) is a a-closed

I

* sub algebra of N

if is normal.

Proof. By Proposition 1.2.10, we can easily see that 1) implies 2). Moreover, it is clear that 2) implies 3). Now let be completely additive. Then for any o < cp E N"" cp 0 is a completely additive positive functional on M, and by Theorem 1.8.6, ip o EM",. Further, by Corollary 1.9.9, tp o EM"" Vcp EN",. Therefore is a-a continuous.

61

Now if cI> is normal and N c B(K), let I = {a E M I cI>(a) = O}, then I is a a-closed two-sided ideal of M. Thus there is a central projection z of M such that I = M(1 - z), and cI> is a * isomorphism from Mz into B(K). Similar to the proof of Proposition 1.8.13, the unit ball of cI>(M) is weakly closed. Therefore cI>(M) is a-closed. Q.E.D.

Proposition 1.12.3. Let cI> be a * isomorphism from a VN algebra M onto a VN algebra N. Then cI> is normal and isometric. Let {all be a bounded increasing net of M+, and a = sup a,. Then

Proof.

I

b = sup cI>(al)

< cI>(a).

Moreover, since cI>-1 is a

* isomorphism from

N onto

I

M, it follows that sup cI>-l(cI>(al)) = a

< cI>-l(b).

l

Thus b = cI>(a), i.e., cI> is normal. Another conclusion is clear.

Q.E.D.

Theorem 1.12.4. Let M, N be two VN algebras on Hilbert spaces H, K respectively, cI> be a normal * homomorphism from M onto N. Then

where cI>1 is an ampliation of M, i.e., there exists a Hilbert space L such that cI>l(a) = a ® lL, Va E M; cI>2 is an induction of M ® ®crlL, i.e., there is a projection p' of (M®crl L)' such that cI>2(a ® lL) = (a ® lL)p', Va E M; cI>3 is a spatial * isomorphism from (M®
Proof.

First we assume that N admits a cyclic vector 71. Define

(a)71, 71),

Va E M.

Obviously,

L n

Va EM. Let L =

[2,

E= (En)

n

C H ® L,

(cI>l(a)~,

cI> 1( a) = a ® lL, Va EM. Then

0,

Va EM.

Suppose that pi is the projection from H ® L onto l(M)~. Then

p' E cI>dM)' Let cI>2(a 01L)

= (a ® lL)p', Va E M. 2

0

= (M 0a:'IL)" Clearly,

cI>d(a)~, ~),

Va E M.

62

Define a linear map u from ~(M)TJ to p'(H ® L) as follows: u~(a)TJ

=

(~2

0

~d(a)c

= p'(aCn)

=

(aCn),

Va E M. Since (~(a)TJ, TJ) = ep(a) = ({~2 0 ~d(a)c, c), Va E M, it follows that u is isometric. Moreover, since ~(M)TJ = NTJ and (~20 ~d(M)c = cpdM)€ are dense in K and p'(H ® L) respectively, u can be extended to a unitary operator from K onto p'(H ® L). Clearly, u~(a)u-l = ~2

Then we can define a spatial

0

~da),

* isomorphism

Va E M. ~3

by the operator u, and

~ =

~3 0 ~2 0 ~1.

For general case, write

K =

L

tc, = NTJz, Vt.

tBKz,

l

Let q; be the projection from K onto K l ; then q: EN', Vl. For each l, ~l = q;~ is a normal * homomorphism from M onto N, = N By the above argument ~l = ~~I)o~~l)o~ll),Vl. Define e, = LtB~~I),i = 1,2,3. Then ~ = ~30~2ocpI,

q:.

and

~l, ~2, ~3

I

satisfy our conditions.

Q.E.D.

Proposition 1.12.5. Let ~ be a * isomorphism from a VN algebra M onto a VN algebra N. Then there exists a VN algebra V and two projections p', q' E V' with c(p') = c(q'} = 1 such that M, N are spatially * isomorphic to V q', Vp' respectively, and ~ corresponds to the * isomorphism: vq' -+ vp'(Vv E V). Keep the notations in Theorem 1.12.4, and let V = M®(CIL. Since ~,~1, ~3 are * isomorphisms, it follows that ~2 is also a * isomorphism. By Proposition 1.5.10, the central cover of p' in V' is 1. Further from Theorem 1.12.4, N is spatial * isomorphic to Vp'. Let q' = IH ® q, where q is a rank one projection on L. Then q' E V'. Since Proof.

V'q'{H ® L)

:J

H ® B(L)qL,

it follows that the central cover of q' in V' is 1. Clearly, M is spatially * isomorphic to V q', and ~ corresponds to the * isomorphism: vq' -+ vp'(Vv E V). Q.E.D.

Theorem 1.12.6. Let Mi, N, be the VN algebras on the Hilbert spaces Hi, K, respectively, ~i be a normal * homomorhism from M, onto Ni, i = 1,2. Then there exists unique normal * homomorphism ~ from M 1 ®M2 onto N l ®N2 such that ~(al ®

a2) =

~dad ® ~2(a2)'

Val E M I , a2 E M 2 •

63

Moreover, if «PI and «P2 are * isomorphisms (or spatial «P is also a * isomiphism (or spatial * isomorphism).

Proof.

* isomorphisms),then

By Theorem 1.12.4, we can write

where I, is the identity operator on a Hilbert space Li, p~ E (Mi®
and

=

® U2 is a unitary operator from p'(H1 ® H 2 ® L) onto K 1 ® K 2 • Now define U

UI

~(a) = u(a ® 1L )p'U- 1 ,

Clearly. «P is a normal u- 1 such that

pi(Hl ® Ld ®

P~(H2 ®

L2)

Va E M 1®M2 •

* homomorphism from M 1@M2 onto u(M1@M2®
Since «P is normal and M 1@M2 is generated by {al ® a2 I al E M}, a2 E M 2} , it follows that u(M1@M2®eL"lL)P'U- 1 will be generated by {~l(atl ® ~2(a2)lal E M}, a2 E M 2} . Therefore,

N 1@N2

= u(M1@M2 ®
i.e., «P is the normal * homomorphism from M 1 ®M2 onto N 1 ®N2 which we want to find. From the normality, obviously ~ is unique. Now suppose that
References. [28j, [109], [113], [185].

1.13 Comparison of cyclic projections and spatial * isomorphic theorem Definition 1.13.1. Let M be a VN algebra on a Hilbert space H. For each EE H, we denote by Pe (resp. Pel the projection from H onto M'~ (resp. M~), and call it the cyclic projection of M (resp. M') determined by

c.

64

Theorem 1.13.2. Let M be a VN algebra on a Hilbert space H, E, fJ E H. Then the relations P" -< Pe (in M) and P~ -< Pe (in M') are equivalent.

Proof.

Let P~

-< Pe (in M'), i.e. there is a u' E M' such that

,.,

u u =

Clearly, u'u'·

= P~/'1'P" =

,

P",

u'u'* < Pe'

Pu'". Replacing fJ by u'fJ, we may assume that

TJ E ME·

Let cp( a) = (aTJ, E), Va EM, and cp = R,;w be the polar decomposition of cp. Then w = Rv.cp,cp = Rvv'cp, and (TJ - vV·TJ) E (ME)l.. On the other hand, fJ E ME, (TJ - vV*TJ) E ME. Thus

(1)

TJ = vV·TJ· Since PeE w

=

= E and w = Rv'cp, it follows that w = Lp(w. In addition w > 0, so

R,,(w, i.e.,

(v*TJ, aE)

=

(P(v*TJ, aE),

Va E M.

Thus (v·TJ - P€v*TJ) E (ME)l.· But TJ E ME, (v·TJ - P€v·TJ) E ME, so we have

V*TJ E Pe H = M' e.

(2)

By (2), v*M'TJ = M'v*TJ C M'E; by (1), vv*M'TJ = M'TJ. It follows that v*p" is a partial isometry from M'TJ into M' E. Therefore, P" -< Pe (in M). Similarly, we can prove the converse. Q.E.D.

Definition 1.13.3. Let M be a VN algebra on a Hilbert space H. A vector of H is said to be cyclic for if = H (see Lemma 1.4.9). A vector TJ of H is said to be separating for M, if a E M is such that afJ = 0, then a = O. It is easy to see that a vector TJ of H is separating for M if and only if fJ is cyclic for M'.

e

M, Me

Proposition 1.13.4. Let M be a VN algebra on a Hilbert space H, Ebe a cyclic vector for M, and TJ be a separating vector for M. Then there is a vector ~ of H such that ~ is cyclic and separating for M.

Proof. Since Pe = 1 > p~, it follows from Theorem 1.13.2 that P" -< Pe (in M). But P" = 1, so by Proposition 1.5.3 we have Pe ,...., P", i.e., there exists v E M such that v*v = Pe,vv* = P" = 1. Let ~ = vE. Then M'~ = vM'E = H, and M~ :J Mv*vE = ME = H, i.e., M~ = H. Therefore, ~ is cyclic and separating for M. Q.E.D.

65

Theorem 1.13.5. Let M, be a VN algebra on a Hilbert space Hi, i = 1,2. Suppose that M I and M 2 admit a cyclic and separating vector. Then every * isomorphism ~ from M I onto M 2 is spatial.

Proof.

By Proposition 1.12.3,

~

is normal. Thus

is a VN algebra on HI E9 H 2 • Let p~ be the projection from (HI E9 H 2 ) onto Hi, i = 1,2. Clearly,

Now by Proposition 1.5.2, it suffices to show that p~ ,...., p~ in M'. Let 6(E Hi) be the cyclic-separating vector for M i , i = 1,2. Then pHH I E9 H 2 ) = Hi = MiCi = MCi' i = 1,2. From Theorem 1.13.2, it suffices to show that PI ~ P2 in M, where Pi is the projection from (HI E9H2) onto M' Ci, i = 1,2. Now we prove a more stronger result: PI = P2 = 1. In fact, since

I a~ E M:,i = 1,2} C

{a~ E9 a~

M ',

Mlci

= Hi,i = 1,2,

it follows that Pi > pL (1 - Pi)p~ = 0, i = 1,2. From (1 - ps) E M, i the definition of M, there are a, b E M I such that

a E9

= ~(b)

=

o.

Further, a

b E9 ~ ( b)

= b=

=

=

1,2, and

1 - P2.

°and PI = P2 = 1.

Q.E.D.

Proposition 1.13.6. Let M be a VN algebra on a HIlbert space H, €(E H) be a separating vector for M. Then for any normal positive linear functional Ip on M, there is "I E Mc such that
'""-'

Proof. Let {1rtp,Htp} be the * representation of M generated by Ip,H = H E9 Htp, and M = {a = a E9 1rtp(a) I a EM}. By Proposition 1.8.13, M is a VN algebra on if. Put

a=

""'"

e ,#IV

;v

,.....""

a ffi 1rtp(a),ltp = (O,ltp). Let = (€,O) E H. Obviously, C is a ,......,., separating vector for M, so M' C= H ::J M /l cp . By Theorem 1.13.2, vI'P -<

where

~

~

,-....,

~

Pl

"""'-'~

~

in M'. Since ~ H = Mlcp = 1rcp(M)lcp = Hcp, it follows that P'--- = P~ is the lIP

lIP

66

projection from and

v'v,· < p~.

H onto

HI(>' Then there exists

Noticing that

ME.

!p(a)

E

P'eH = Me = (ME,O), v'II(>

for some '1 E

Vi

= ('1,0) =

M' such

that

v'·v' =

P~

we can write

Tj

Further,

,..,) = (",,,, a11(>,11(>

=

(av'll(>,v'll(»

(",,,, all(>' v I.'''') v 11(>

= (aTj,fj) = (a'1, '1),

Va EM.

Q.E.D. Corollary 1.13.7. Let M be a VN algebra on a Hilbert space H. If M admits a separating vector, then for any ep E M. there are E, '1 E H such that ep(a) = (aE, '1), Va E M. Proof.

It is obvious by the polar decomposition of

ip,

Q.E.D.

Notes. The main results (1.13.2, 1.13.5 and 1.13.6) in this section are the so-called spatial theory of Von Neumann algebras. It is due to F. Murray and J. Von Neumann. B.J. Vowden once gave another proof. The present proof here is taken from R. Herman and M. Takesaki.

References.[69], [111], [177], [192].

1.14. o-F'inlte Von Neumann algebras Definition 1.14.1. A VN algebra M is said to be a-finite (or countably decomposable), if every family of non-zero pairwise orthogonal projections of M must be countable. Proposition 1.14.2. Let M be a VN algebra on a Hilbert space H. Then the following statements are equivalent: 1) M is a-finite; 2) M admits a separating sequence of vectors {En}( C H), i.e, if a E M with aEn = 0, Vn, then a = OJ 3) M' admits a cyclic sequence of vectors {'1n}(C H), i.e., [a''1nla' E M',n] is dense in H; 4) There is a faithful normal positive linear functional on M.

67

Proof. Since a sequence {Cn} is separating for M if and only if {Cn} is cyclic for M', it follows that 2) and 3) are equivalent. Now let {Cn} be separating for M, and {Pl}lEA be an orthogonal family 2 of projections of M. Since LIIPlcnl1 < IlcnI1 2 , \:In , it follows that there is lEA

a countable subset J of A such that PI Cn = 0, \:In and 1 fi J. From {Cn} is separating for M, then PI = 0, \:Il fi J. Thus M is a-finite. Suppose that M is a-finite. Write

H

=L

$H"

H,

= M'TlI = PlH,

\:Il.

l

Clearly, {llpl =f. o} is countable. Thus H =

L

ffiM'Tln, and the sequence {Tin}

n

is cyclic for M'. Therefore 1), 2) and 3) are equivalent. Suppose that there is a faithful normal positive linear functional rp on M. Let {PI}lEA be any orthogonal family of projections of M. Since rp(pd =

L

lEA

rp(L pz) <

00,

it follows that there is a countable subset J of A such that

lEA

rp(Pl) = 0, \:Il fi J. Then Pl = 0, VI fi J, i.e., M is a-finite. Finally, if M is a-finite, then M admits a separating sequence {Cn} of I cnl/ 2 < 00, and let vectors. We may assume that

L n

n

Clearly, rp will be a faithful normal positive linear functional on M.

Q.E.D. Proposition 1.14.3.

Let M be a VN algebra on a Hilbert space H. If M. is separable, then M is a-finite. In particular, if H is separable, then M is a-finite. Proof. Let { 0, n and

L n

°

II 1/J II < 1}, ep(a) = 0,

then rpn(a) = 0, \:In. Further, 1/J(a) = 0, vo < 1/J EM•. By Corollary 1.9.9, a = 0. Therefore

°

Q.E.D.

68

An algebra A is called abelian or commutative, if ab

= ba, \;la, b E

A.

Proposition 1.14.4. Let M be a a-finite abelian VN algebra on a Hilbert space H. Then M admits a separating vector.

Proof.

From the proof of 1.14.2, we can write

H

=L

$Hn,

u; = Pn H =

M'Tln, \;In.

n

Then {Tin} is a separating sequence of vectors for M. We may assume that 2 II Tin 11 < 00. Put TI = n«. Since M is abelian, it is easily verified that n

L

L

n

n

is a separating vector for M.

Q.E.D.

Proposition 1.14.5. Let M be a a-finite VN algebra, and (Mh be its unit ball. Then ((Mh, s(M, M.)) and ((Mh, r(M, M.)) are metrizable and can become two complete metric spaces. Proof.

By the proof of Proposition 1.14.2, tp(.)

= L('€n, €n)

is a faithful

n

normal positive linear functional on M, where {€n} is a separating sequence of vectors for M with II€nIl 2 < 00. From Lemma 1.11.2, ((M1),d,,) is a

L n

complete metric space where ds(a, b) = tp((a - b)·(a - b))1/2 = (L

II(a -

n

b)€nI1 2)1/2,Va,b E (Mh, and the topology generated by d, is equivalent to s(M, M.) in (Mh. Moreover, put d.,.(a, b) = {L

II(a -

n

b)€nI1 2 + L lI(a - br €nI1 2}1/2,

Va, bE (Mh·

n

Clearly, ((Mh, dr ) is a complete metric space, and the topology generated by d; is equivalent to s" (M, M.) '" r(M, M.) in (Mh. Q.E.D.

Definition 1.14.6. LetM be a VN algebra on a HIlbert space H, p be a projection of M. p is said to be a-finite, if M p = pMp is a a-finite VN algebra on pH. Proposition 1.14.7. Let M be a VN algebra on a Hilbert space H. 1) If tp is a normal positive linear functional on M, then s (tp) is a a-finite projection. 2) Let p, q be two projections of M, and p ~ q. If p is a-finite, then q is also a-finite. 3) If {Pn} is a sequence of a-finite projections of M, then p = sup Pn is also n

a a-finite projection.

69

1) It is obvious since rp is faithful on s (rp) M s (rp). 2) Since M p and M q are * isomorphic and M p is a-finite, it follows that M q is a-finite, or q is a-finite. 3) For each n, if {€in)}A:( C PnH) is a cyclic sequence of vectors for M'Pn, n) then {€i I n, k} will be a cyclic sequence of vectors for M' p. Thus p is a-finite.

Proof.

Q.E.D. Proposition 1.14.8. Let P, q be a-finite projections of VN algebra M, N respectively. Then P ® q is a a-finite projection of M ®N. Since P ® q(M®N)p®q = pMp®qNq, it suffices to show that the tensor product of two a-finite VN algebra is a-finite. Let M, be a a-finite VN algebra on a Hilbert space Hi, i = 1,2. If {€~)}(C Hi) is a cyclic sequence of vectors for MI, i = 1,2, then {€~1) ® €i;) I n, m} will be a cyclic sequence of vectors for Mf®M~ = (M1®M2 )' . Therefore M 1®M2 is a-finite.

Proof.

Q.E.D. Theorem 1.14.9. Let M be a VN algebra on a Hilbert space H. Then there is a decomposition M = ffiM,. Here M , is a-finite or is spatially

L I

* isomorphic

to N,®B (Kd, where N , is some a-finite VN algebra and K , is some Hilbert space, v:

Proof.

Let

L

lEA

that

qz -< s(rp)z,

s(rp)(l- z) -< q(l- z).

Since the family {PI}'EA is maximal, it follows that s(
V,

00

1/1 (a) =

L

2- n
Va E M z,

n=O

Then 0

< 1/1

EM". If a E Mz is such that 1/1(a"a) = 0, then rp(v~a·avn) = 0,

\:In

> O.

70

By the definition of Vn, v~a·avn E s(~)M s(~), \in > O. Thus aVn = 0, \in > O. Further, ap,z = aqz = 0, VI E A, and a = az = a(qz + PlZ) = O. Hence w is

L

lEA

faithful on M z, and M z is a-finite. If A is not countable, we can write

A=

U Afj,

Afj

n Afjl = 0,

\if3

-:I f3' E

II,

{jEI

where each

is countably infinite. Fix an index f30 E ll, and let

A{j

'"'I {jo

= q+

L

Ph

IEApo

'"'Ifj =

L

PI,

V{3

i- f3o·

lEAp

Then {"YfjZIf3 E II} is an orthogonal family of projection with '"'Ifjz ,...., '"'Ifj'z, \if3, (3' E II, and '"'IfjZ = z. By Theorem 1.5.6, M Z will be spatially * isomorphic

L

fjEI

to N0B(K), where K is a Hilbert space with dimK = su,» = '"'IfjoMz'"'lfjo' Since Afjo is countable, it follows from preceding paragraph that N is a-finite. Through above procedure, we find a non-zero central projection z of M such that Mz is a-finite or is spatially * isomorphic to N0B(K), where N is a-finite and K is some Hilbert space. Finally, by the Zorn lemma, the proof will be completed. Q.E.D. Let M be a VN algebra. Then there is an orthogonal family {PI} of a-finite projections of M such that L PI = 1.

Proposition 1.14.10.

I

Proof. Let {PI} be a maximal orthogonal family of a-finite projections of M, and P = LPI' If (1 - p) i- 0, then by Theorem 1.14.9 (1 - p)M(1 - p) I

contains a non-zero a-finite projection q. Clearly, q is still a a-finite projection of M and qPI = 0, Vi. This is a contradiction. Therefore Pl = 1. Q.E.D.

L I

References.[I8], [28], [82], [150].

Chapter 2 Fundamentals of C*-Algebras

2.1. Definition and basic properties of C*-algebras Definition 2.1.1. Let A be a Banach algebra over x ---t z" E A with the following properties:

(E.

If A admits a map:

\/x, yEA, A, J.L E (E, then A is called a Banach * algebra, and the map: x ---t x" is. called the »-operaiion of A. If the --operation of A satisfies the following additional condition:

Ilx·xll = Il xlj2, then A is called a C· -olqebra. If A is a C·-algebra, it is easy to see that lIx*1l = Ilxll, \/x E A. Thus the *-operation of A is continuous. In general, a C"'-algebra need not have an identity. If however, the C*-algebra A has an identity (denoted by lA, or 1 simply if no confusion arises), then lA = lA, IllAIl = 1. Let A be a C*-algebra, E be a subset of A. B is called the C*-subalgebra of A generated by E, if B is the smallest C'" -subalgebra of A containing E. Let 0 be a locally compact Hausdorff space, C~ (0) be the set of all continuous functions on 0 vanishing at infinity. With the usual structure, Co (0) is a C*-algebra. The algebra C~(O) has an identity if and only if 0 is compact. The C*-algebra Co(O) is abelian (commutative). Let H be a Hilbert space, A be a uniformly closed * subalgebra of B(H). Then A is a (concrete) C"-algebra. Later we shall show that: any (abstract) C*-algebra is isometrically * isomorphic to a concrete C·-algebra.

72

Proposition 2.1.2. (A+-
Let A be a C*-algebra, and A has no identity. On

IIX + All = sup{lI xy + AYll lyE A, Ilyll < I} Vx E A, A E
It suffices to show that for any x E A, A E
Let 0 < J1. < 1. Then there is yEA, Ilyll < 1, such that J1.211X + AIl 2 < lIxy + Ayl12 = IIY* (x + A)* (x + A)yll

< II(x+A)*(x+A)II· Again let J1.

---4

1-, then we get

IIx + AI1

2

< II(x + A)(X + A)II < II(x + Arll'!lx + All·

Thus lIx + All < II(x + A)*II· II(x + A)*(X + A) II = Ilx + A11 2 •

Remark.

Further Ilx + AI! = II(x + A)*II·

(1) From (1), Q.E.D.

If A has an identity e, on (A+
IIX

+ All = max{llx + Aell, JAI}

Vx E A, A E
For any A E a(h), by le i tAI 2

< lIit hll 2 = II (e it h )* . eithll = 1,

Vt E JR

Then A = A. Thus a(h) C JR. Moreover, by IIhll

= 11 h*hIl 1 / 2 =

IIh 2 111 / 2

= ... = Ilh2 "W/2"

---4

v(h),

73

Q.E.D.

so Ilhll = v(h).

Theorem 2.1.4. Let A be an abelian C·-algebra, fl be its spectral space. Then A is isometrically * isomorphic to Co(fl). Moreover, A has an identity if and only if fl is compact. Proof. Replacing A by (A+eL") if it is necessary, we may assume that A has an identity. Then fl is compact. Let x -+ x(.) be the Gelfand transformation from A into C(fl). By Proposition 2.1.3, x"'(t) = x(t),Vx E A,t E fl, and IIxll2 = IIx"'xll = v(x·x) = max jx*x(t)1 = max Ix(t)12, Vx E A. tEO

tEO

Thus the Gelfand transformation is an isometrical * isomorphism from A to C(fl). Moreover, it is clear that the subalgebra {x(·) I z E A} separates the points of fl [i.e., for any s :I tEfl, there is x E A such that x(s) :I x(t)). So {x(·) I z E A} = C(fl) by the Stone-Weierstrass theorem. Q.E.D.

Lemma 2.1.5. Let A be a C·-algebra with an identity, h = h* E A and o f/. (J (h). Then there is a sequence {Pn (.)} of polynomials with a zero constant term such that lIPn(h) - h- 111 -+ o. Let B be the C·-subalgebra of A generated by {(h->")' (h->,.)-l I >,. f/. (J(h)}. Then B is abelian and 1 E B. By Theorem 2.1.4, B '" C(fl), where fl is the spectral space of B. Clearly, uB(h) = u(h) = {h(t) It E fl}. From Proposition 2.1.3, h(t) = h{ij, Vt E fl. Moreover, since 0 f/. u(h), it follows that min Ih(t)1 = e > O. Define a continuous function / on [-llhll, IIhll] such tEO that /(0) = 0,/(>") = >,.-1, V>,. E [-lIhll,lIhlll\(-e,e),

Proof.

and pick a sequence {Pn{')} of polynomials with a zero constant term such that max{lpn(>") - /(>,.) I I - Ilhll < x < II h l!} -+ O. Then IIPn(h) - h- 111 = max IPn(h(t)) - h(t) -11 tEO

< _lIh~~lIhIlIPn(>") - /(>")1-+

O.

Q.E.D.

Proposition 2.1.6. Let A be a C·-algebra with an identity 1, B be a C"'-subalgebra of A and 1 E B. Then for any b E B, (JB(b) = (JA(b). Proof. By Lemma 2.1.5, (JB(b"'b) = (JA(b"'b),(JB(bb"') = uA(bb"'). Thus if b is invertible in A, then b+ band bb" are invertible in B. So b has a left and a right inverses in B. Therefore, b is invertible in B. Q.E.D.

74

Proposition 2.1.7. Let A be a C*-algebra with an identity 1, u be a unitary element of A (i.e., u*u = uu* = 1). Then a(u) C {A E
Proof.

Q.E.D.

From Theorem 2.1.4 and Proposition 1.3.4, it is easy.

Proposition 2.1.8. Let A be a C*-algebra with an identity 1, a be a normal element of A [i.e., a*a = aa*), B be the C*-subalgebra generated by {1,a}. Then B ,..... C(a(a)), and the function corresponding to a is a(A) A, VA E a(a). In particular, II all = v(a). By Theorem 2.1.4, B .-; C(O). Clearly, the map t -+ a(t) is injective and continuous from 0 onto a (a). Since 0 and a( a) are compact, it follows that 0 .-; a(a). Q.E.D.

Proof.

Proposition 2.1.9. on A such that

Let A be an abelian C*-algebra, 11·111 be another norm

Ilabll l < IlaliI ·llbllb Then

Proof.

Ilxll < Ilxllll Vx E

Va, b E A.

A.

We may assume that A has an identity. Then A .-; C(O). Let 01

= {t E 0 I if {x n} C

A and

lIxnlll

-+

0, then xn(t)

-+

O}.

We claim that 0 1 is dense in O. In fact, suppose that there is a non-empty open subset U of 0 such that 01nU = 0. Then we have a E A with a(t) = 1, Vt E 0 1 and a(s) = 0, Vs E U. Let Al the completion of (A, II· lId· If a is not invertible in All then there is a non-zero multiplicative linear functional p on At such that p(a) = O. Let piA = t. Then t E 0 1 , Thus a(t) = 1 and p(a) = a(t) = O. This is a contradiction. So a is invertible in AI' On the other hand, there is b E A such that suppb(·) C U. Then ab = o. It is impossible since a is invertible in AI. Therefore 0 1 is dense in O. Finally, let X be the spectral space of At. Then for any x E A,

IlxliI > max{lp(x) II p E X} max Ix(t)1 = max Ix(t)\ = lIxl]. tEOl tEO Q.E.D.

Proposition 2.1.10. such that

Let A be a C*-algebra"

II ·111

be another norm on A

75

Proof. By Proposition 2.1.2, we may assume that A has an identity. Let h = h* E A, and B be the C*-subalgebra generated by {1,h},B 1 be the completion of (B, II . II d. Suppose that n l and n are the spectral spaces of B I and B respectively. By the proof of Proposition 2.1.9, {(pIB) I p E n I } is dense in n. Thus

Ilhllt =

max Ip(h)1 = max Ih(t)1 = pEn. fEO

Further, for any a E A, II al1 2 = I[ a* all = Ila* alii =

II hl/ .

II alii, i.e., Iiall = Halil' Q.E.D.

Notes. Theorem 2.1.4 is due to I.M. Gelfand. The aim of the theory of C*algebras is to understand a uniformly closed self-adjoint algebra of operators on a Hilbert space (concrete C*-algebra). A Banach * algebra satisfying the axioms in Definition 2.1.1 is still sometimes called a B*-algebra. The name C·-algebra was coined by I.E. Segal. Presumably the C is meant to indicate that a C*-algebra is a non-commucative analogue of C(n), whereas the * recalls the importance of the *-operation.

References. [51}, [81].

2.2. Positive cones of C* -algebras Definition 2.2.1. Let A be a C*-algebra. An element a of A is said to be positive, denoted by a > 0, if a* = a, and a(a) C lR+ = [0,00). We shall denote by A+ the subset of all positive elements of A. Proposition 2.2.2. Let A be a C*-algebra, h* = h E A. Then there are unique h+, h : E A+ such that

Proof. We may assume that A has an identity 1. Let B be the abelian C*-subalgebra generated by {I, h}. By Theorem 2.1.4, there are h+, h: E B+ such that h = h; - h: and h; . h: = O. From Proposition 2.1.6, h+, h: E A+. Now if there exist other h~, h'- E A+ such that h = h~ - h': and h~ -b': = 0, then the set {h, h~, h'-} is commutative. Let C be the abelian C*-subalgebra generated by {1,h,h~,h'-}. Clearly C => B. Using Theorem 2.1.4 for C, we can see that h~ = h+ l h'; = h..., Q.E.D.

76

Proposition 2.2.3. Let A be a C"'-algebra with an identity 1, and h h: E A with IIhll < 1. Then h > 0 if and only if 111 - hll < 1.

=

Using Theorem 2.1.4 for the abelian C"'-subalgebra generated by {I, h}, it is easy to get the conclusion. Q.E.D. Proof.

Proposition 2.2.4. a, b E A+, we have (a Proof.

Let A be a C"'-algebra. Then A+ is a cone, i.e., if + b) E A+. Moreover, A+ n (-A+) = {O}.

We may assume that A has an identity 1. By Proposition 2.2.3,

11 1 -

a

W" < 1,

Then

+b 111 - ljall + Ilbll ll < a

b

a

Ilall' 111 - W" + Ilbll .11 1 -

Mil

IJal1 + Ilbll

< 1.

Again by Proposition 2.2.3, (a + b) E A+. Moreover, if h E A+ n (-A+), then u(h) = {O}. From Proposition 2.1.3, Q.E.D. h = 0, i, e., A+ n (-A+) = {O}. Remark. We introduce a partial order i. e., a > b if (a - b) > o.

">" in A H

= {h

E A

I h'" = h},

Proposition 2.2.5. Let A be a C",_algebra, and a E A+. Then there exists unique a l / 2 E A+ such that a l / 2 . a = a . a l / 2 and (a l / 2 ) 2 = a. Moreover, this a l / 2 can be approximated arbitrarily by the polynomials of a with zero constant terms. Proof.

Q.E.D.

It is similar to Proposition 2.2.2.

Lemma 2.2.6. Let B be an algebra with an identity, and a, b E B. Then u(ab) U {O} = u(ba) U {O}. Proof.

Let 0 f- A

rt u(ab), and It = (ab -

A)-I. Since

(bua - 1) . (ba - A) = (ba - A) . (bua - 1)

it follows that (ba - A) is invertible.

= A, Q.E.D.

77

Proposition 2.2.7. Let A be a C*-algebra, and a E A. Then a E A+ if and only if there exists b E A such that a = b" b. Proof. The necessity is clear by Proposition 2.2.5. Now let a = b'b. Clearly a'" = a. By Proposition 2.2.2 and 2.2.5, we can write a = 11,2 - v 2 , where 11"v E A+ and 11,V = 0. Then (bv)*(bv) = vav

=

-v 4

< 0.

(1)

Let bv = h + ik, where h* = h, k" (bv)(bv)*

= k, Then by Proposition 2.2.4, = -(bv)*(bv) + 2(h 2 + k 2 ) = v 4 + 2(h 2 + k 2 ) > 0.

From (1)' (2) and Lemma 2.2.6, (bv)*(bv) = -v 4 E A+n(-A+) by Proposition 2.2.5, v = 0, and a = 11,2 E A+.

(2)

= {O}. Further Q.E.D.

Proposition 2.2.8. Let A be a Colo-algebra on a Hilbert space H (Le. A is a uniformly closed * subalgebra of B(H)), and a E A. Then a E A+ if and only if a is a positive operator on H. The necessity is clear from Proposition 2.2.7. Now let a( E A) be a positive operator on H. At least we have a* = a. By Proposition 2.2.2, a = a+ - a_, where a+, e., E A+ and a+ . c., = 0. For any E H, Proof.

e

Therefore, a: = 0, a.,

= 0, and

a

=

Q.E.D.

a+ E A+.

Proposition 2.2.9. Let A be a Colo-algebra. 1) If a, bE A+ and a < b, then II all < Ilbll, and c*ac < c'be, \:Ic E A. 2) A+ is a closed subset of A. 3) If A has an identity, a, b E A+, a < b,and a, b are invertible, then b- 1 a -1 .

°

<

1) we may assume that A has an identity 1. Then < a < b < Ilbll·1. Using Theorem 2.1.4 for the abelian C*-subalgebra generated by {I, a}, we can get that Ilall < IIbll. From Proposition 2.2.7, it is clear that c*ac < c'bc, \lc E A. 2) Let {an} C A+, and an ---+ a. We need to prove a E A+. Clearly, a* = a. Write a = a+-a_, where a+,a_ E A+ and a+'a_ = 0. Let bn = a_ana_ E A+. Then bn ---+ b = a_aa_ = -a: < 0. Further < -b < bn - b < IIbn - bll ---+ 0. Thus b = 0, c., = 0, and a = a+ E A+. 3) Since (a- 1 p /2(b - a)(a- 1 p / 2 > 0, it follows that (a- 1p/2b(a-1)l/2 > 1. By the function representation, al/2b-lal/2 < 1 = al/2a-lal/2. Therefore, b- 1 < a-I. Q.E.D. Proof.

°

78

Proposition 2.2.10. Let A be a C·-algebra, a, b E A+ and a
v;

Proof. We may assume that A has an identity 1. First, suppose that a, bare invertible. Let E = {A E [0,1] I a~ < b~}. Since A+ is closed, it follows that E is closed. Clearly, 0,1 E E. So it suffices to show that for any A, p, E E, we have >';EA E E. Fix A, p, E E. Since

it follows that 11b-~a~b-~11 < 1 and Ilb-~aEAb-~11 < 1. Further, Ila>'/2b~A/211 < 1 and lIaEA/2b-EA/211 < 1. By Lemma 2.2.6, 1

>

Ilb-EA/2a~b-~/211

>

v(b-EA/2a~b-~/2)

>'+,, t: >'+1') >.+,. b- >'+4 "11, v (b - >.+,. 4a2 4 = Ilb- >.+,. 4a2

.i.e.

a < b- >.+,. a >.+,. b- >.+,. 4

2

4

<1,

or

a

>.+,. 2

>.+,. 2

•

Therefore ~;EA E E. For general case, from the preceding paragraph we have (a e)\Ve > O. Let e --+ 0+, then aA < v,

+ e)>' < (b + Q.E.D.

Proposition 2.2.11. Let A be a C·-algebra, S = {a E A Illall < 1} be its (closed) unit ball. Then Int(S) n A+ = {a E A+ Illall < I} is directed with respect to the partial order ">" in A+, i.e. for any x, y E Int(S) nA+ there is z E Int(S) nA+ such that z > x and z > y. Proof.

Consider the problem in (A +
a = x(1 - x)-l,b = y(1 _y)-l, z = (a

+ bH! + a + b)-1 =

1 - ~(!

+ a + b)-I.

Then a,b,z E A+, and z E Int(S) nA+. The inequality z > z is equivalent to the following inequality 1 - -1(1-

2 2

+ a + b) - 1 ->

x

or

By Proposition 2.2.9, the latter is equivalent to the

(I-X)-l < (1+2a+2b). It is clear that (1 + 2a) > (1- x) -1 by the definition of a. Thus (1 + 2a + 2b) > (1 - x)-I. So Z > x. Similarly, z > y. Q.E.D.

79

Notes. Proposition 2.2.4 is due to M. Fukamiya, J .L. Kelley and R.L. Vaught. Proposition 2,2,7 is due to 1. Kaplansky. These results are the key lemmas for the Gelfand-Naimark conjecture (see Section 2.14). Proposition 2.2.10 is known as the Lowner-Heinz theorem, and the proof here is due to G.K. Pedersen.

References. [50], [81], [90], [126], {177J.

2.3. States and the Gelfand-Naimark-Segal construction Definition 2.3.1. Let A be a C--algebra, p be a linear functional on A. p is said to be positive, denoted by p > 0, if p(a) > O,Va E A+.p is called a state if p > a and Ilpll = 1. The set of states of A is denoted by S(A). Clearly, if p > 0, then we have p(a-) = p(a), Va E A; and the Schwartz inequality:

Proposition 2.3.2. Let A be a C·-algebra, and p be a positive linear functional on A. Then p is continuous. Moreover, if A has an identity 1, then

Ilpl[ =

p{I).

Proof. First, we say that p is bounded on S n A+ = {a E A+ Illall < I}. In fact, suppose that there is a sequence {x n } C S n A+ such that

p(xn » n 2 ,

Vn.

By Proposition 2.2.9, A+ is closed. Thus x =

1 I: 2"x n

n

E A+. Then for any

n

positive integer N,

It is impossible. Therefore, there is a constant K such that

p(a) < K,

Va E S n A+.

Further, by Proposition 2.2.2, llpll < 4K, i.e., p is continuous. Now if A has an identity 1, then by the Schwartz inequality we have

80

Q.E.D.

Va E A. Thus Ilpll = p(I).

Proposition 2.3.3. Let A be a C" -algebra, and pEA·. 1) If there is a E A+ with Iiall < 1 such that p(a) = Ilpll, then p > O. 2) Suppose that Ilpll < 1, and there is 0 f a E A+ such that p(a) = lIall. Then p E S(A). Proof, 1) Since p can be extended to a continuous linear functional on (A+O:) with the same norm, we may assume that A has an identity. Now by the proof of Lemma 1.9.2, p > O. 2) By the hypothesis, we have Ilpll = 1, and P(U:II) = !Ipl1. Again from 1), it follows that p E S(A). Q.E.D.

Proposition 2.3.4. Let A be a C·-algebra, p be a positive linear functional on A, and a E A+ with lIall < 1. Then IIpll =

sup{p(b)

I bE A+,llbll < I}

sup{p(b)

I bE

A+, IIbl! < 1 and b > a}.

Proof. Since p(x·) = p(x), "Ix E A, it follows that there is a sequence {hnH C A) with h~ = hn , Ilhnll < 1, "In, such that p(h n ) --+ Ilpll. By Proposition 2.2.2, h n = h;; - h~, where h; E A+ and Ilh; II < Jlhnll < 1, Vn. Replacing {h;} by a subsequence if it is necessary, we may assume that limp(h~) and limp(h~) n n

exist. Since 0 < limp(h~) n

< IlpII, it follows that

limp(h~) = Ilpll, n

limp(h;;) = O. n

Therefore Ilpll = snp{p(b) I bE A+, Ilbll < I}. Now for any e > 0, there is c E A+ with Ilell < 1 such that p(e) > IIpl1 - e. From Proposition 2.2.11, we have b E A+ with IIbll < 1 such that b > c, b > a. It is sure that p(b) > p(e) > IIpll- e. Therefore IIpll = sup{p(b) I b E A+, Ilbll < 1, b > a}. Q.E.D.

Proposition 2.3.5. Let A be a C·-algebra, and p be a positive linear functional on A. Define

,o(a + .\) = p(a) + '\#1-0,

Va E A,.\ EO:

where #1-0 is a fixed positive number with #1-0 > Ilpll. Then jf is a positive linear functional on (A +0:) .

Proof. It suffices to show that p(a) +.\#1-0 > 0 for any positive element (a+).) of (A -i-
81

Q.E.D.

Corollary 2.3.6. Let P be a state on a C*-algebra A. Define p(a p(a) + A, Va E A, A E (C. Then p is a state on (A+a;), and (piA) = p.

+ A)

=

Proposition 2.3.7. Let A be a C*-algebra. Then the state space S(A) of A is a convex subset of A ".

Proof. For any 0, pick a E A+ with lIall < 1 such that 1- c. Now by Proposition 2.3.4, 1

>

I!A
+ (1- A)
> A A(l - e) + (1 - A).

A+, IIbll

< 1,b > a}

I b E A+, IIblll

< 1, b > a}

Since e is arbitrary, it follows that A
Q.E.D.

Definition 2.3.8. Let A be a C"'-algebra, S(A) be its state space. Each extreme point of S(A) is called a pure state on A. The set of pure states is denoted by P{A). Proposition 2.3.9. Let A be a C*-algebra. 1) For p E P(A)' let p{a + A) = p(a) + A, Va E A, A E (C. Then F is a pure state on (A+a;). 2) If 15 is a pure state on (A-t-(C),p = (piA), then either p = or p E P(A).

°

1) Suppose that there are two states PI, 152 on (A+a;) and A E (0,1) such that,o= API+(1-A),o2. Let Pi = (Pi!A),i = 1,2. Since IlpI11 < 1,llp211 < 1

Proof.

82

and

lipII < xII PIli + (1 -

,\) 11 P211

< 1, P E P(A) and P = '\PI + (1- '\)P2

1=

it follows that Pi E S(A), i = 1,2. By we have P = PI = P2· Moreover, .0(1) = .01(1) = .02(1) = 1. Thus .0 = i.e., .0 is a pure state on (A+eL"). 2) Suppose that P f:. o. Let IlplI = J.Lo < 1. By Proposition is a state on (A+eL"), where a(a + A) = J.LoIp(a) + '\, Va E A,'\ E ao(a + ,\) = ,$Va E A,'\ E eL"). Obviously, ao is a (pure) state on Moreover, we have the equality:

on A,

PI = .02, 2.3.5, a eL". Let

(A+eL").

It is a contradiction since .0 is pure. Therefore IIpll = 1, i.e. P E S(A). Now let PI, P2 E S(A) and ,\ E (0,1) be such that P = '\PI + (1 - 'x)P2. PI and P2 can be naturally extended to states .01 and .02 on (A+eL"). Clearly, .0 = ,\.01 +(1-,$.02' Since F is pure, it follows that .0 = PI = .02 and P = PI = P2· Therefore P E P(A). Q.E.D. Proposition 2.3.10. Let A be an abelian C*-algebra. Then P E P(A) if and only if P is a non-zero multiplicative linear functional on A. Proof. By Proposition 2.3.9, we may assume that A has an identity 1. Let P E P(A). Since A = [A+J, it suffices to show that p(ab) = p(a)p(b) for any a, s e A+. If p(a) = 0, then by the Schwartz inequality,

lp(ab) 12 = Ip(a l / 2 • a I/2b)1 2 < p(a)p(bab) =

o.

Thus p(ab) = 0 = p(a)p(b). Now we may assume that

o < a <1, 0 < p(a) <

1.

Let PI(') = p(a)-I p(a'),P2(') = p(1 - a)-I p((1 - a)·). Clearly, Pt,P2 E S(A), and P = p(a)PI + (1 - p(a))P2' Since P is pure, it follows that P = PI = P2· Therefore p(ab) = p(a)p(b). Conversely, let p be a non-zero multiplicative linear functional on A. If A ,..., C(n), then there is to E n such that

p(a)

=

a(t o) = ~ a(t)doto(t),

Va E A.

By the Riesz representation theorem, each state on A corresponds to a probability measure on n. Therefore p E P(A) since p corresponds to a point measure. Q.E.D.

83

Remark. There is another direct proof (without using the Riesz representative theorem) as follows. Suppose that P is a non-zero multiplicative linear functional on A. Clearly P E S(A). Let PI, P2 E S(A) and ,X E (0,1) such that

P = 'xPI

+ (1 - 'x)P2'

For any h = h· E A,

+ (1- 'x)p2(h 2) = p(h2) = p(h)2 [,Xpt{h) + (1- 'x)P2(h)]2.

'xPI(h 2) -

By the Schwartz inequality, Pi(h)2 < pi(h 2), i

= 1,2. Then

+ (1 - 'x)P2(h)]2 + 'xpdh2) + (1 - 'x)p2(h 2) > -[,X2pdh)2 + (1- ,X)2 p2(h)2 + 2'x(1- 'x)PI(h)p2(h)] +'xPI(h)2 + (1 - 'x)p2(h)2

0= -[,Xpt{h)

'x(I- A)[pdh) - P2(h)]2. Thus Pl(h) = P2(h), Vh = h* E A. Further P = PI = P2 and P E P(A).

Proposition 2.3.11. Let A be a C·-algebra with an identity 1, E be a linear subspace [i.e., if a E E, then a· E E) of A and 1 E E. Let

*

1 - {I I I is a state on E, i.e., I is a linear functional on E, } I(a·) = 7[OJ, Va E E, I(b) > 0, Vb E En A+, and 1(1) = 1 . Then we have that: 1) Each element of 1 can be extended to a state on A; 2) Each extreme point of 1 can be extended to a pure state on A. Proof. 1) Let I E 1. For any h = h* f/. E, since and A+ is a cone, we can define I(h) such that

sup{/(b)

I b = b"

E E,b

< I(h) < inf{/(c) I c = c·

-llhll·l < h <

I!hl!·l,l E E,

< h} E E, c > h}.

Then I is a state on E+[h]. In fact, for (a+Ah) E A+, where a E E,'x E ClJ, we need to prove I(a + 'xh) > O. Clearly a* = a, A = 'x. When ,x = 0, it is clear. IT,X > 0, then h > -,X-Ia. By the definition of l(h),/(h) > -,X-l/(a), i.e., I(a + 'xh) > O. If ,x < 0, then h < -,x -lao By the definition of I(h), I(h) < -,X-l/(a), i.e., I(a + 'xh) > O. By the above procedure and the Zorn lemma, I can be extended to a state on A.

84

2) Let

f be an extreme point of 1. Put

c=

{p E S(A)

I (piE)

=

fl·

By 1), £, -=I 0. It is easy to see that £, is a weak * compact convex subset of A"'. From the Krein-Milmann theorem, £, has an extreme point p at least. Now it suffices to show p E P(A). Let PI,P2 E S(A) and A E (0,1) be such that P = API + (1 - A) P2. Clearly, fi =' (Pi IE) E 1, i = 1,2. Since AlI + (1 - A)12 = (piE) = f and f is an extreme point of 1, it follows that f = fl = f". Thus PI, P2 E L, Since P is an extreme point of £', P = PI = P2· Therefore P E P(A). Q.E.D.

Corollary 2.3.12. Let A be a C"'-algebra, and B be a C*-subalgebra of A. Then each state (or pure state) on B can be extended to a state (or pure state) on A. Proof. Each state (or pure state) on B can be extended to a state (or pure state) on (B+cr) (see 2.3.6 and 2.3.9). Further it can be extended to a state (or pure state) on (A+cr) (see 2.3.11). By Proposition 2.3.9, its restriction is Q.E.D. still a state (or pure state) on A.

Remark.

Suppose that A has an identity 1, and 1 E B, cp E S(B). By Proposition 2.3.3, each extension of cp preserving the norm is a state on A.

Proposition 2.3.13. Let A be a C"-algebra, and h = h'" E A. If 0 -=I A E a(h)' then there is a pure state P on A such that p(h) = A. Proof. By Proposition 2.3.9, we may assume that A has an identity 1. Let B ,...., C(n) be the abelian C"-subalgebra generated by {l,h}. Then there is tEn such that h(t) = A. Define f(b) = b(t), Vb E B. By Proposition 2.3.10, f is pure state on B. By Corollary 2.3.12, f can be extended to a pure state p on A. Clearly, p(h) = f(h) = h(t) = A. Q.E.D.

Remark.

If A itself has an identity, then the condition "A -=I 0" is not

necessary.

Corollary 2.3.14. Let A be a C*-algebra, and h = h'" E A. Then there is a pure state p on A such that Ip(h) 1= Ilhll. Consequently, Ilhll = sup{lp(h)11 p E P(A)}. Corollary 2.3.15. Let A be a C"'-algebra, and a E A such that p(a) > 0, Vp E P(A). Then a E A+.

85

Proof. It suffices to show a· = a. If a~r # 0, then there is p E P(A) such that p(a;() # o. Thus p(a) rt. JR., a contradiction. Therefore a· = a. Q.E.D. The GNS construction was once discussed in Section 1.8. The same procedure can be carried for the C·-algebras. Due to its importance, we shall study it again in detail.

Definition 2.3.16. Let A be a C· -algebra. {11'", H} is called a * representation of A, if 1r is a * homomorphism from A into B(H), where H is a Hilbert space, i.e., 1r(Aa + J.Lb) = A1r(a) + J.L1r(b) , 1r (

ab) =

11'" (

a)11'" ( b) ,

1r(a*) = 7f(a)*, \la, b E A, A, J.L E (J).

e

e

e

If there is a vector E H such that 1r( A) = H, then is called a cyclic vector for {1r, H}, and {1r, H} is said to be cyclic. The * representation {7f, H} of A is said to be faithful, if 1r (a) = 0 implies a = O. Two * representations {7fl' Hd and {7f2' H 2 } of A are unitarily equivalent, 7f2, H 2 } , if there is a unitary operator u from HI onto denoted by {1rI,Hl } H 2 such that f'V

{

Proposition 2.3.17. Let {1r, H} be a * representation of a C·-algebra A. Then 111r11 < 1, and 1r preserves order, i.e., 7f(A+) C B(H)+. Moreover, if 1r is faithful, then 11'" is isometric, and 1r- l(7f(A)+) = A+.

Proof.

Consider (A+(J}), and put 1r(I) = In. Thus we may assume that A has an identity 1, and 7f(I) = IH. Then a(7f(a)) C a(a), \la E A. Further,

1I1r(h) II = sup{IAII A E cr(7f(h))}

< Hence

111r(a)11

1r(A+)

C

=

sup{IAII A E a(h)} =

111r(a·a)11 1/ 2 < Ila·aI1 1/ 2

=

Ilhll, \lh = h·

E A.

Iiall, Va E A, i.e., 111r11 < 1. Clearly,

B(H}+.

Now let 1r be faithful. If there is e E A such that 1r(e) = In, then e is just the identity of A. If IH rt. 1r(A) , considering (A+a:) and putting 1r(I) = In, then 11'" is still faithful on (A+a:). In other words, we may assume that A has an identity 1, and 1r(I) = In. Now our conclusion can be obtained from the proof of Proposition 1.8.13. Q.E.D.

86

Let A be a C*-algebra, and

L", = {a E A

I
= O}.

L", is called the left kernel of <po By the Schwartz inequality, L", is closed left ideal of A. Let a ~ a", = a + L", (Va E A) be the quotient map from A onto A/LI{)' On A/LI{)' define

(a""bl{))

=
Then (,) is well-defined, and is an inner product on A/LI{)' Denote by HI{) the completion of (A/Lip, (,)). For any a E A, define a linear map 1rip(a) AI LI{) ~ AIL", as follows: 7fip ( a)bip = (ab) ip' Since b*a*ab

<

Vb EA.

lIaI12b*b, it follow that 11 7f ",(a )bip I12

=
2

lIal1 '!Ibipll2

V b E A. Therefore 7f",(a) can be uniquely extended to a bounded linear operator on Hip, still denoted by 7fip(a). It is easy to show that {1rl{)' HI{)} is a * representation of A.

Proposition 2.3.18. Let A be a C*-algebra, and ip E S(A). 1) If {1rip, Hip} is the * representation of A generated by

7fl{)(a)el{)

= aip'

= (1rl{)(a)eip' el{))'

Va EA.

2) Let r:p be the natural extension of

Proof. Define uaip = a~, Va E A. Then u can be extended to an isometry from HI{) into Hrp. By Proposition 2.3.4, there is a sequence {an} C A+ with I[anll < 1, Vn, such that
87

Proposition 2.3.19. Let A be a C*-algebra, and .t:.. be a subset of S(A) such that sup{~(a) I ~ E .t:..} = Iiall, Va E A+. Then

{ 1r6 =

L

ffi 1rlp,H6 =

IpE6

is a faithful

Proof.

L

ffiH IO }

IpE6

* representation of A.

For any a E A, by Propositions 2.3.17 and 2.3.18 11 a]l2

(a) 11 2 = sup{ 111rI" ( a*a) II

~ E

>

1I 1r6

>

sup{(1rIO(a*a)clO , Clp) I ~ E .t:..} sup{p(a*a)

I ~

E

I

.t:..}

.t:..} = ll a \l 2. Q.E.D.

Therefore, Iiall = 111r6(a)II, Va E A.

Remark. By Corollary 2.3.14, P(A) or any a(A>I<, A)-dense subset of S(A) can be chosen as A. Theorem 2.3.20. Any C*-algebra can be isometrically concrete C*-algebra on some Hilbert space.

* isomorphic

to a

Proposition 2.3.21. Let A be a C*-algebra, {1r, H} be a * representation of A. 1) If 1r admits a cyclic vector C, let p(a) = (1r(a)c, c), Va E A, then {1r p, H p} ~ {1r, H}. 2) There exists .t:.. C S(A) such that {1r, H} is unitarily equivalent to the direct sum of some zero representation and {7f Ip , Hlp} (~ E .t:..). Proof. 1) Let u1r(a) C= ap, Va E A. Then u can be extended to a unitary operator from H onto H p; and it is easy to see that u1r(a)u- 1 = 1r p(a), Va E A. 2) By the Zorn lemma, we can write

H=

L ffiH

I

ffi H o,

lEA

where H, = 1r(A)6, and 11611 = 1, Vi E A, and Ho = {c E H 11r(a)c = 0, Va E A}. For any I E A, let ~l(a) = (1r{a)6, 6), Va E A. Then .t:.. = {~l II E A} satisfies our condition. Q.E.D. The following proposition is a version of the Radon-Nikodym theorem.

Proposition 2.3.22. Let ~,tf; be two positive linear functionals on a C*algebra A, and ~ < ,p [i.e. ~(a) < ,p(a), Va E A+). Then there exists

88

unique t' E 1r,p(A)',O < t' < 1, such that cp(a) = (1r,p(a)t'e,p, e,p), Va E A, where {1r,p, H,p, e,p} is the cyclic * representation of A generated by ,p (as in Proposition 2.3.18). Proof.

On the dense subspace A / t.; of

[a,p,b,pJ

u.; define

= [1r,p(a)e,p,7f,p(b)e,pJ = cp(b·a), Va,b EA.

Ila,pll·llb,pII,Va,b EA.

Thus there is

Now by the proof of Lemma 1.10.1, we can get the conclusion.

Q.E.D.

Since ip < 1/;, it follows that ![a,p,b,pJI < unique t' E B(H,p) such that

Now we study the orthogonal (Jordan) decomposition of a hermiatian functional. Let A be a C"'-algebra, and X = {p E A· I p > 0 and Ilpll < I}. Clearly, X is a compact Hausdorff space with respect to l1(A·, A). Denote by Cr(X) the set of all real continuous functions on X. For a E A H (the set of all self-adjoint elements of A), define a(p) = p(a), Vp E X; then a(·) E Cr{X). By 2.3.14 and 2.3.15, the map: a -+ a(·) is isometric (i.e. Hall = sup la{p) I) pEX

and preserves order [i.e. a(·) > 0 if a E A+) from AH into Cr(X), and also its inverse preserves the order (i.e. a E A+ if a(.) > 0). Suppose that I is a hermiatian continuous linear functional on A, i.e., lEA· and I'" = I, where I· is defined by I·(a) = l(a+),Va E A. Then 11/11 = II/I A HII· Put F{a(·)) = l(a),Va E AH. Clearly, F can be extended to a continuous linear functional on C r (X), still denoted by F, with the same norm II III. By the Riesz representation theorem, we can write

where F+ and F_ are positive on Cr(X). Restricting F+, F_ to {a(·) I a E AN}, we get positive functionals 1+,1- on A H • Let I±(a + ib) = I±(a) + il±(b),Va, b E A H . Then 1+ and 1- are positive on A, and I = 1+ - 1-. Moreover, since 11/11 = II FII = IIF+II + IIF-II, IIF±II > II/±II, and IJIII = 111+ I-II < 11/+11 + III-II, it follows that 11111 = 11/+11 + III-II· The above decomposition is called the orthogonal decomposition (or Jordan decomposition) of the herniation functional I. When A is commutative, it is the ordinary Jordan decomposition of a signed measure exactly. Now we prove that the above decomposition is unique. By Proposition 2.3.18, for each p E X, there is a cyclic * representation {1r p, s ; €p} of A such that

p(a) = (1r p(a)ep, €p),

Va E A.

89

Let

1r

=

L

H

EB1r p,

=

pEX

L

ffiHp-

pEX

Then {1r, H} is a faithful * representation of A. Let M = 1r(A)". Then M is a VN algebra on H. We may assume II/II < 1; so I± E X. Write €± = €f±; then I±(a) = (1r(a)€±, €±), Va E A. Identifying A with 1r(A)' I and I± can be naturally extended as follows:

. I±(b) = (b€±, €±),

I(b) = /+(b) - I-(b),

Vb E M.

Denote by 1IIlIM, 1l/±IIM the norms of /, /± as the functionals on M respectively. By the Kaplansky density theorem, we have 1I/±IIM = 1I/±11. Since IIIII < II/11M < II/+lIM + III-11M = 11/+11 + II/-ll = lIlli, it follows that IIIIIM = 1I/+IIM + III-11M' Now by Theorem 1.9.8, we obtain the following theorem.

Theorem 2.3.23. Let A be a C·-algebra, I be a hermit ion continuous linear functional on A, i.e., lEA· and I(a·) = 7[ii), Va E A. Then there exist unique positive linear functionals 1+ and I-on A such that

1= 1+ - t., Corollary 2.3.24. S(A).

II/II

and

= 11/+11

+ II/-II·

Let A be a C·-algebra. Then A· is the linear span of

Notes. The GNS construction was studied first by LM. Gelfand and M.A. Naimark. Then I.E. Segal gave its perfected form.

References. [52}, [155].

2.4. Approximate identities and quotient C*-algebras Proposition 2.4.1. Let A be a C·-algebra, and L be a left ideal of A. Then there is a net {d l } C L with dl E A+, Ildzil < 1, Vl, and dl < dl" Vl < l', such that IIXdl

-

xli

-t

0, Vx E L.

Proof. Let A be the set of all finite subsets of L. And A is directed by the inclusion relation. For any 1 = {Xl,"', X n } E A, put n

hi

=L i=l

X;Xi,

dl

= nhl(l + nhl)-l.

90

Clearly, hI, dl E L n A+, and Ildlll < 1. Let l' = {Xl'···'X n , · · · , x m } > I = {Xb""X n } , where m > n, and Xi E L,1 ~(~ + hll)-I, it follows that 1(1 + h l )- 1 > 1.-(1.- + hll)-I. Further, n n - m m dl

=

Now suppose l

1 - -1(1 -

=

n n

+ hi )-1 <

1 - -1(1 -

m m

+ hl' )-1 = dl"

{Xb"" Xn } E A. Clearly,

n

Moreover, (1 - dl )hl (1 - dl )

= I)xi(l -

dl )) *(Xi (1 - dz)). Thus

Ilxi -

xiddl <

i=1

(2y!n)-1, 1 0, pick i, E A such that x E l, and > (4c2 ) -1. From preceding paragraph, we have Ilx - xddl < c, VI > leo Therefore, lIxxddl --4 0, Vx E L. Q.E.D.

Definition 2.4.2. Let A be a C*-algebra. A net {dl}(C A) is called an approximate identity for A, if 0 < dl < 1, dl < dl" VI < I', and II ad, - all --4 0, IId l a - all --4 0, Va E A. By Proposition 2.4.1, we have the following.

Theorem 2.4.3.

Every C* -algebra admits an approximate identity.

Proposition 2.4.4. Let A be a C*-algebra, and {d l } be an approximate identity for A. 1) For any 'P E S(A), we have

lim'P(dd = lim'P(d;) = 1. I

I

2) If A has no identity, then the C* -norrn on (A-t-(f) (see Proposition 2.1.2) is also represented by

Vx E A, A E (f. Proof.

I) For any a E A with 1

Itall < 1, by

the Schwartz inequality we have

> 'P (dl ) > 'P (d;) > 'P (di) 'P(a * a) > I'P(dla) 12

--4 1cp (

a) 12 •

91

Iltpll

Since 1 =

= sup{ltp(a) I

I a E A,llalll <

I}, it follows that liFtp(d ,)

=

limtp(d;) = 1. I

2) Fix x E A, AE that

(C.

For any c

> 0, we can pick yEA with

llylll <

1 such

Ilx + All > Ilxy + Ayll > llx + All - c. Since dly ---+ y, it follows that

II x + All > II (x + ).) dd I > II (x + A) d,y II > I x + AII - c fori enough late. Therefore, Ilx+).lI = lim Ilxdl+)'ddl and IIx+AII = !Ix"'+AII = lim Ilx"'d, + Addl = lim Ildjx + Addl, Vx E A,). E (j}. Q.E.D. I I j

Let {1f, H} be a * representation of a Colt-algebra A. The closed linear span of {1f( a) I a E A, E H} is called the essential subspace of {1r, H}. The * representation {1f, H} of A is said to be nondeqenerate, if its essential subspace is H. Clearly, the orthogonal complement of the essential subspace is the null subspace, i.e.

Definition 2.4.5.

e

I 1r(a) 11 = 0, Va E A}. Therefore, the null subspace of a nondegenerate * representation is trivial. {1r(a)e

Ia E

e

A, e E H}l. = {11 E H

In this case, the weak closure of 1r(A) is a VN algebra on H (see Theorem 1.3.9).

Proposition 2.4.6. Let A be a C*-algebra, and {d l } be an approximate identity for A, {1r, H} be a * representation of A. Then 1r(dd ---+ P (strongly), where p is the projection from H onto the essential subspace [1r(A)H] of {1r, H}. In particular, if {1r, H} is nondegenerate, then 1r(dz) ---+ 1 (strongly). Proof.

By Proposition 1.2.10, we have 1f(d,) ---+ P = SUp1f(dd (strongly). Let I

K = [1r(A)H]. Then for any 11 E tc-, 7f(d l ) 11 = 0, Vl, and P11 = 0. On the other hand, for any a E A, e E H, since 117f(d1a)e - 1r(a)ell < Ild,a - all·llell ---+ 0, it follows that p1r(a)e = 1r(a)€. Therefore, pH = K. Q.E.D. If {1r, H} is a nondegenerate * representation of A, by 2.4.4 and 2.4.6, then for any E H with Ilell = 1, (.€, E S(A).

Remark.

e

e)

Proposition 2.4.7. Let I be a closed two-sided ideal of a C"'-algebra A. Then Jolt = {a'" Ia E J} = J.

92

Proof. By Proposition 2.4.1, there is a net {de} C I such that ad, I. Then for any a E I,

II dza* - a*11

lI ad, - all ---t

= II(ad, - a)"'ll =

---t

a, Va E

O.

Since dia * E I, Vi and I is closed, it follows that a* E I, Va E I.

Q.E.D.

Now let A be a C"-algebra, and I be a closed two-sided ideal of A. By Proposition 2.4.7, AI I is a Banach * algebra with respect to the quotient norm. Let {d,} be an approximate identity for I, and a ---t Ci = a + I be the cononical map from A onto AI I. We claim that Iiall

= lim Iladz- all, Va E A. I

In fact, fix a E A. For any bEl, since bd,

limllad l ,

-

all

=

limllad l z

-

b, it follows that

---t

a + bd, -

lirnll(a + b)(l Thus liFlladz - all adz E I, we have

bll

dl)ll

<

Ila + bll·

< inf{lla + bll I bEl} = [a]. On the other hand, since

liFllad, - all

> li~ Ilad l

-

all >

llall·

Therefore, lIall = lim Iladz - all, Va E A. z Now for any a E A, bEl, since bd, ---t b, we have

lim II (1 - dt}a* a(l - dr) I I

liF 11(1

-

d1)(a*a

+ b)(l -

d,)11 < Ila*a + bll·

Hence

Ii Ci l12 < inf{lla*a + bll I bEl} Furthermore, Ila*all =

lI a112 , Va E A.

=

11a*"a11 < lIa* II . Iiall, Va E A.

Therefore we have the following.

Theorem 2.4.8. Let A be a C*-algebra, and I be a closed two-sided ideal of A. Then AI I is a C·-algebra in a natural way. Proposition 2.4.9. Let be a * homomorphism from a C*-algebra A into another C*-algebra B. Then (A) is a C*-subalgebra of B. In particular, if

93

{1r,H} is a

* representation of a

C·-algebra A, then 1r(A) is a C·-algebra on

H. Proof. By Theorem 2.3.20, it suffices to consider the case of {1r, H} and A. Let I = {a E A 11r(a) = O}. Then I is a closed two-sided ideal of A. Define

1T(a) = 1r( a),

Va E AI I and a E a.

Clearly, {1f, H} is a faithful * representation of the quotient C·-algebra AI I. By Proposition 2.3.17, 1r(A) = 1f(AI I) is a C·-algebra on H. Q.E.D.

Proposition 2.4.10. Let A be a C·-algebra, I be a closed two-sided ideal of A and B be a C*-subalgebra of A. Then (B + I) = {(b + c) I bE B,c E I} is a C*-subalgebra of A, and the C·-algebras (B + I) I I and B I (B n I) are canonically isomorphic. Proof. Let a ---+ a = a + I be the canonical map from",A on1;.,o AII. Clearly, it is also a * homomorphism. By Proposition 2.4.9, B = {b I b E B} is a C*-subalgebra of AI I. It suffices to show that (B + I) is closed. Let {x n } C (B + I) and X n ---+ x. Then xn ---+ X. Since B'" = B + I is a C·-subalgebra of AI I, it follows that x E B, i.e., x E (B + I). Now it is easy to see that

---

b + (B n 1) is a

* isomorphicm from

-t

b+ I

B I(B n 1) onto (B

(Vb E B)

+ 1)1 I.

Q.E.D.

Proposition 2.4.11. Let A be a C·-algebra, and I be a closed two-sided ideal of A. If p is a state (or pure state) on A and p(1) = {O}, let ,o(a) = p(a)(Va E AII,a E a), then,o is a state (or pure state) on AII. Conversely, if p is a state (or pure state) on AI I, then there is unique state (or pure state) p on A such that p(I) = {O} and p(a) = p(O:) , Va E A. Let p be a state on A with p(1) = {O}. Then we can define ,o(a) = p(a), va E AII, a E a. Clearly, ,0 is a positive linear functional on AII, and 11,011 < 1. On the other hand, Proof.

1=

sup{lp(a)11 a E A,

11all <

I}

sup{lp(a) I I a E A, Iiall < I} < 11,011. Thus 11,011 = 1 and ,0 is a state on AI I. Conversely, let ,0 be a state on AI I. Define p(a) = ,o(a) , Va E A. Clearly, p is a positive linear functional on A, and p(I) = {O}. From the preceding paragraph, we can see that Ilpll = 11,011, so Ilpll

=

1 and p is a state on A.

94

Now let P be a pure state on A with p(I) = {O}. 'By the preceding paragraph, Pis a state on AI I. Suppose that there are states PI, P2 on AI I and A E (0,1) such that P= API + (1- A)P2. Define pi(a) = Pi(Ci) , Va E A,i = 1,2. Then PI and P2 are two states on A with PI(I) = P2(I) = {O}. Clearly, P = ApI + (1 - A)P2' Since P is pure, it follows that P = PI = P2. Further P= PI = P2. Therefore p is pure on AI I. Finally, let P be a pure state on AI I. Then there is unique state P on A such that p(I) = {O} and p(a) = p(Ci), Va E A. Suppose that there two states PbP2 on A and A E (0,1) such that P = API + (1- A)P2' For any a E In A+, from p(a) = 0 we have pda) = P2(a) = O. Further, PI(I) = P2(I) = {O}. Thus P = Ap + (1 - A)P2' where p; is defined by p;(a) = p;(a), Va E AI I, a E a, i = 1,2. Since p is pure, it follows that P = PI = P2' Further P = PI = P2' Therefore P is pure on A. Q.E.D. Notes. LE. Segal showed the existence of an approximate identity in a C* -algebra. References. [25)' [52], [81], [155], [156].

2.5. Extreme points of the unit ball and the existence of an identity Theorem 2.5.1. Let A be a C*-algebra, and S = {a E A I Hall < I} be its unit ball, xES. Then x is an extreme point of S if and only if

(1 - x*x)A(1 - zz"]

=

{O}.

Moreover, if x is an extreme point of S, then x is a partial isometry, i.e. z" x and xa:" are projections. Proof. Let x be an extreme point of S. First we prove that z" x is a projection. In fact, let B be the abelian C*-subalgebra generated by z" x. Then B '" CO'(O). If there is to E 0 such that x*x(to) E (0,1). By the continuity, we can find an open neighborhood U( C 0) of to and e E (0,1) such that o < z" x (t) < 1 - E:, Vt E U. Pick dEB such that

and TJ E (0,1)

o < d(t) < 1, Vt E 0, d(to) = 1, d(O\U) such that 2TJ + TJ2 < c. Then 2

* ()

( )) xxt 0< ( I±TJdt

=

= {O},

Vt ~ U, 1), < (I+TJ)2(I-c)«I), Vt E U.

{ x* x ( t )( <

95

Since d and x·x commute, it follows that Ilx ± qxdl12 =

II(x(l ± T]d))· . (x(l

± T]d))ll

II (1 ± 7]d)2 . x·xll <

1.

Now from x = ~(x + T]xd) + ~(x - 7]xd), we get xd = O. Further x·x· d = o. It is impossible since (x·x . d)(t o) = x·x(t o) > O. Therefore x·x(t) = 0 or 1, \It E fl, i.e, z" x is a projection. Similarly, zz" is a projection since z" is still an extreme point of S. Let p = x·x, q = zz". If y E (1 - p)A(l - q) with Ilyll < 1, then py = O. Further, 0 = y.py = (xy)· . (xy), and xy = O. By Theorem 2.3.20, IIx ± y"11 2 = Ilx· ± yll2 = II (x* ± y)* . (x* ± y) II Ilxx" + y*yll = IIqxx·q

+ (1 -

q)y·y(1 - q)

II

max{llxx·lI, Ily*yll} < 1.

t

Since x is an extreme point of S and x = ~ (x + y.) + (x - y.), it follows that v' = O. Therefore, (1 - p)A(1 - q) = {O}. Conversely, suppose that (1- x·x)A(l- zz"] = {O}. Then 0= x*(l - xx·)x(1 - x*x) = x·x· (1 - X*X}2.

Thus a(x"x) C {O, I}, i.e., x*x is a projection. Similarly, zz" is also a projection. Let p = x·x, q = zz". Since (xp - x)*(xp - x) = px*xp - px*x - x·xp + z" x = 0, it follows that xp = x,

px·

= z",

(1)

Suppose that there are a, b E S and A E (0,1) such that x = Aa + (1 - A)b, Then p = x*xp = Ax*ap+ (1- A)x*bp. By (1)' P: x*ap = x·ap· p. Thus the set {p, z" ap, z" bp} is commutative, and it can generate an abelian C*-subalgebra with an identity p. By the Gelfand transformation, we can see that (2)

By (1), (2) and q = zz", we have x

From (2), (3), pa*qap 1

>

=

=

qap

= qbp,

(3)

pa'*x = (x·ap)* = p, hence

IIpa*apll

=

Ilpa*qap

+ pa'*(1

lip + pa" (1 -

- q)apll

q)apll·

But pa·(1 - q)ap is a positive element of the C*-subalgebra pAp, and pAp has an identity p, so pa*(l - q)ap = 0, (1 - q)ap = 0, ap = qap, By (3), we obtain x

= ap.

(4)

96

Since x = 'xa + (1 - ..:\)b, it follows that y = ..:\c + (1 - 'x)d, where y = z", c = a*,d = b", Replacing {x,a,b,p,q} by {y,c,d,q,p} in above procedure (1)-(4), we obtain y = cq since y.y = q and yy. = p. Thus

x = qa.

(5)

Since (1- q)a(l- p) E (1- q)A(I- p) = {O}, it follows that a = ap+ qa - qap, Now by (3), (4), (5), x = a. Further, x = a = b. Therefore, x is an extreme point of S. Q.E.D.

Corollary 2.5.2. If a C·-algebra has an identity, then the identity is an extreme point of its unit ball. Theorem 2.5.3. Let A be a C·-algebra, and S = {a E A I lIall < I} be its unit ball. Then A has an identity if and only if S has an extreme point at least. Proof. The necessity is clear from Corollary 2.5.2. Now suppose that S has an extreme point z. Let p = z" x, q = zz", and {dl } be an approximate identity for A. By Theorem 2.5.1, (1 - q)d l (1 - p) = 0, Vi. Thus

dl

-----7

P + q - qp.

Clearly, e = p + q - qp is an identity of A.

Q.E.D.

Proposition 2.5.4. Let A be a C·-algebra, and S(A) be its state space. Then A has an identity if and only if S(A) is compact with respect to the w· -topology a( A *, A). The necessity is clear. Now suppose that A has no identity. We prove U t1rt,lS(A),C1(A"A)) is not compact. It suffice to show that 0 E S{Al , where S A is the a(A*,A)-closure of S(A) in A*. Let U = U(O;al, .. ·,an;g) = {f E A* I If(ai)1 < g,1 < i < n} be any a(A· ,A)-neighborhood of 0, we need to prove that Un S(A) -I- 0. Since A = !A+], we may assume that ai E A+,l < i < n. Let a = at + ... + an' It suffices to show U(O; a; g) n S(A) -I- 0. By Theorem 2.3.20, we may assume that A C B(H) (some Hilbert space H), and A is nondegenerate on H. Since A has no identity, it follows that a is not invertible in B(H). Thus there is €E H with II €II = 1, such that (a€, €) < g. Let p(.) = (.€, €). By Proposition 2.4.6, p E S(A) n U(O; a; g). Therefore, S(A) n U(O; a; g) #- 0. Q.E.D. Proof.

Let E be a linear space, K be a convex subset of E, and F be a subset of E. We shall denote by ExK and CoF the sets of all extreme points of K and convex hull of F respectively.

97

Now let A be a C*-algebra, S(A) and P(A) be its state space and pure state space respectively. For any E c A*, the u(A*, A)-closure of E in A* is denoted by EO. Clearly, {p E A* I p > o,llpll < I} is a u(A*,A)-compact convex subset of A*. And also it is easily verified that Ex{p E A* I p > 0, lip II < I} = {O, P(A}}. By the Kreim-Milmann theorem, we have

I p > 0, Ilpll <

{p EA·

I} = Co{O, P(A)}o.

°

If A has no identity, then from the proof of Proposition 2.5.4, E S(At. Thus Co{O, p(A)}CT C S(At. Clearly, S(Ar c {p E A* 1 p > o,lIplj < I}. Therefore, we get

S(Ar = Co{o, p(A)}CT = {p E A* I p > 0, Furthermore, for any p E S(A) there is a net where ,\~l} > 0, p~l} E P( A), Vi, land

L

Ilpll <

I}.

(l: ,\~l}P~l)}, c

Co{O, P(A)},

i

'\~') < 1, such that

i

L,\~l)P~l}

-t

P

(u(A*,A)).

i

Since

IIY'I! =

1, considering a subnet, we may assume that

(~=,\~l)}-l L'\~')p~l) i

L ,\~l)

-t

1. Then

i - t ip

(u(A*,A)). Thus S(A)

C

Cop(Ar, and we have

i

the following.

Proposition 2.5.5.

Let A be a C*-algebra. Then

Ex{p E A*

I p > 0, Ilpll < I} =

{O,P(A)}

and

{p E A*

I p > 0, Ilpll <

I}

= Co{O, p(A)}CT.

Moreover, if A has no identity, then we have

{p E A*

I p > 0, Ilpll <

Notes.

I} = CoP (At

= S(At.

Theorem 2.5.1 is due to R.V. Kadison. Proposition 2.5.4 was obtained by I.E. Segal.

References. [77J, [156].

98

2.6. Transitivity theorem and irreducible tions

*

representa-

Definition 2.6.1. Let A be a Cli'-algebra, and {1r, H} be a * representation of A. {n , H} is said to be algebraically irreducible, if E is a linear subspace of H such that 1r( a) E E, Va E A, E E [i.e., E is invariant for 1r(A)), then either E = {a} or E = H.{ 1r, H} is said to be topologically irreducible, if E is a closed linear subspace of H such that 1r(a)e E E, Va E A, E E, then either E = {a} or E = H. Clearly, an algebraically irreducible * representation must be topologically irreducible. In this section, we shall show that the converse is also true for a C*-algebra.

e

e

e

Proposition 2.6.2. Let A be a C* -algebra, and {1r, H} be a * representation of A. Then {1f, H} is topologically irreducible if and only if 1r(A) is dense in B (H) with respect to the weak (operator) topology. Let {1r, H} be topologically irreducible. Then 1r(A)' does not contain any projection which is not equal to 0 or 1. Hence, 1f(A)' = (J},1r(A)" = B(H). Of course, 1r is nondegenerate. By Theorem 1.3.9, 1r(A) is weakly dense in B(H). Coversely, if 1r(A) is weakly dense in B(H), then it is clear that 1r(A)' = (E. Therefore {1r, H} is topologically irreducible. Q.E.D. Proof.

Lemma 2.6.3. Let H be a Hilbert space, ei' "1, E H,l < i < n, and (e" ej) = 8ij , 1 < i,j < n. Then there exists b E B(H) with b* = b such that n

bei = iu,

2

1 < i < n, and II bl1 <

L

111]dI 2 •

i=1

Moreover, if there is h* = h E B(H) such that hei can take b E B (H) with b' = b such that

be, =

= 1]i,l n). Now take b E B(H) such that bK.1. = {a}, bK c K, and (bIK) has a matrix representation with respect to Proof.

99

the basis {€I, ... , em} as follows:

o m

where

71, = L

a;ie;, 1

< i < n. Then we have be,

= 71i, 1 < i < n.

;=1

For first case, pick !::J. = (0). Then m

Ilbl! = IlblKl1 = sup{lI bL

m

I L IAil 2 < I}

Aieill

i=1 m

n

;=1

I:IAiI 2 < I} <

sup{IILAiTJilll ;=1

n

(LII71iI1 2)1/ 2.

;=1

;=1

For second case, pick

(!::J.) =

. .1,1 a n. + ( an+l,n

Then we have b" = b since aii aii, VI < i, j < n. Moreover,

= (TJ;, ej) =

(hei' ei)

=

(ei,hei) = (ei,71i)

=

m

< tr(b*b)

=

i=l

n

L IITJil1 i=1

I: Jlbedl 2

2

+

m

n

L L

n

laiil < 2 L 2

i=n+l ;=1

IITJi1l 2 •

;=1

Q.E.D. n

Lemma 2.6.4.

Let HI,···, tt; be Hilbert spaces, H

=

L

fBH; be the

i=1 n

Hilbert direct sum of HI,· .. , H n l and M =

L

ffiB(Hi) (a VN algebra on H).

;=1

Suppose that A is a C· -algebra on H with A eM, and A is weakly dense in M. Assume that tj E B(Hj ), ej is a projection of finite rank on H j, and Pi is the projection from H, onto [eiHj, tje;H j ], 1 < j < n. Then for any e > 0, there is b E A such that

be; = tjej, 1 < j < n, Ilbll < e + 1~~ln II Pi tj Pj II

-,-

100

Moreover, if tj

= ti , 1 < j < n,

then we can choose b E A such that

be; = t jej, 1 < j < n l

b* = b, n

Proof.

n

= LEBtj,p = LE9pj. Then t,P EM. Pick a normalized

Let t

i=1

orthogonal basis

i=1

{Cl"", Cm}

n

of pH =

L

EBpjHj, such that Ci belongs to

i=1

some pjHj , Vi. For el > 0, by Theorem 1.6.1 there is bo E A such that

By the definition of M, bOCi, ptpe, and Ci will belong to the same Hj (some j depending on i), Vi. Then from Lemma 2.6.3 we can find al E M, such that

alCi

= ptpCi -

boEi,

1
m

Il al/1 < 2 L 2

i=1

Similarly, for

e2

> 0 we can find b1

. . . . .. Generally, we have {ao that

2

Ilptpei - bOCil1 < 2mei· E A such that

= ptp, at, ...} c

M and {bo , b1 , ••• }

c A such

l
= 1,2, ... ,llboll < Ilaoli.

Moreover, if tj = tj, 1 < j < 17., then by Lemma 2.6.3 and Proposition 1.6.4 the above ak and bk can be chosen self-adjoint, k = 0,1, .. '. 00

Now take

eJr. = (2m)-!2-

ke.

Then

Ilakll < 2-

ke,

k = 1,2,···. Let b =

L

k=O

Obviously, b E A and

and N

L

bkci = lim(aOci - aN+lci) = ptpCi, k=O N 1 < i < m. In particular, be, = pipe, = tjej, 1 <j < n. bCi = lim N

bk •

101

Finally, if t f = t" 1 < j < n, we can choose a normalized orthogonal basis {6, ... , em} of pH satisfying

where AI,' .. ,Am are real. From b~ = blc , Vk, we have b"' = band bc, = ptpe, = Aic,,1 Ilptpll·

x

IAil < IIptpll, it follows that f(b)Ci = AiCi,I < i < m. Then f(b)"' f(b) E A, Ilf(b) II < IIptpll = 1~~~ lIPi t;p;ll, and f(b)e; = ptpe, = tie" 1 < _1_

Since

j < n.

Q.E.D.

Theorem 2.6.5. Let A be a C"'-algebra, {1I"i' Hi} be a topologically irreducible * representation of A, ti E B( H;), and ei be a projection of finite rank on Hi, 1 < j < n. In addition suppose that {11"" Hd is not unitarily equivalent to {11";, Hi}, VI 0, there is a E A such that

lI all < e + 1~1~n max IlpitiPill, where Pi is the projection from H, onto [eiHi' tieiHj], 1 < j < n. Moreover, if = ti, 1 < j < n, then a can be chosen self-adjoint. 2) If tj is a unitary operator on H j , 1 < j < n, then there is a unitary element u = ei h of (A+
t;

1ri(u)ej = tie;, 1 <j < n. Moreover, if A has an identity, then we can choose n

Proof.

Let H =

L

U

E A.

n

(f)Hj , 1r

;=1 C*-algebra on H, and 1I"(A)

=

L

(f)7fj.

By Proposition 2.4.9, 1I"{A)

IS

a

;=1

n

C

M =

L

(f)B(H;).

j=l

Denote the projection from H onto H, by p~.; clearly, pi E 1I"(A)', 1 < j < n. Let zi be the central cover of pj in 1I"(A)', 1 < j < n. We claim that ZiZ; = 8i i Z i , Vi,j. In fact, if Z;Z/c -::j: 0 for some j -::j: k, by Proposition 1.5.9 there are non-zero projections p'J, p~ of 1r(A)' such that p~ < p~, pi < pi and pi pZ in 1r( A)'. But the * representations 1rj and 1I"/c are irreducible, so pi = pj, p~ = p~. Thus p~. ,...., p~ in 1r(A)'. This is a contradiction since 1I"j is not unitarily equivalent to 1I"/c. Therefore zizi = 8i ; Zi , VI < i, j < n. Further, since "'-J

102 n

2: pi = 1 and

Zj

> pj, Vj, it follows that pj =

Zj,

1 < j < n. Now we have

i=1 n

{L EB1rj(aj) I aj E A, 1 < j

< n} C 1r(A)" C M.

;=1

By Proposition 2.6.2, 1rj(A) is weakly dense in B(Hj), 1 < j < n. Hence 1r(A) is weakly dense in M. By Lemma 2.6.4, we get the conclusion 1). Now let tj be a unitary operator on H j, then dimejH j = dimtieiHi, and there is a unitary operator Uj on pjHi such that ujej = tie;, 1 < J' < n. Pick a normalized basis {cU}} of p.J H·J such that u·J c"k c(i} -- exp(";A (i}) c(i) where Ak(1) c"k k e,.k' is real, VI < k
j) h;ei

= A~)Eij),I < k < dimpjHj,

VI < j < n. By 1), we can find h* n(h)Pi

Since Pj

> ei'

=

=

and hj(I- Pj) = 0,

hE A such that

hiPj,

1

< j < n.

it follows that

Q.E.D.

VI < J' < n.

Theorem 2.6.6. Any topological irreducible algebra is also algebraically irreducible.

*

representation of a C*-

Proof. Let {n, H} be a topologically irreducible * representation of a C·algebra A. Suppose that K is a non-zero proper linear subspace of H such that n(a)K C K, Va E A. Pick 0 i- E E K and TJ E H\K. By Theorem 2.6.5, there is a E A such that n (a) E = TJ. That contradicts n (a) E K. Therefore, such K does not exist. In other words, {n, H} is algebraically irreducible.

e

Q.E.D. Remark. From now on, an irreducible * representation of a C* -algebra means that it is algebraically irreducible or topologically irreducible. Notes.

The transitivity theorem, Theorem 2.6.5, is due to R.V. Kadison.

References. [79].

103

2.7. Pure states and regular maximal left ideals Theorem 2.7.1. Let A be a C·-algebra, and P be a state on A. Then pis pure if and only if the * representation {11'" p, H p } generated by P is irreducible. Moreover, if P is pure, then H p = AI L p, where L p = {a E A I p(a·a) = O} is the left kernel of p. Proof. Let 11" p be irreducible. Suppose that there are states PI and P2 on A and A E (0, 1) such that On AI L p , define

rap, bp] = PI(b"'a), where a

-+

Va, b E A,

ap is the canonical map from A onto AI L p • Since

it follows that there is h = h* E B(Hp ) such that

PI(b·a) = (hap, bp),

It is easy to see h E

Va, b E A.

11" p(A)'.

But 1I"p is irreducible, so that h = J-t, where J-t is some real number, i.e., pdb*a) = J-tp(b·a), Va, b E A. From P,PI E S(A), it follows that J-t = 1, and P = PI = P2. Therefore, p is pure. Conversely, let P be a pure state on A, Cp(E H p) be a cyclic vector for 1I"p(A) as in Proposition 2.3.18. Suppose that there is a projection p' of 1l"p(A)' with p' i- 0, 1. We claim that p'CP i- o. In fact, if pi CP = 0, then

It is impossible since Ep is a cyclic vector for 11" p (A). Thus p' CP (1 - p') p i- o. Put A = lip' Ep 112 • Then A E (0, 1). Define

e

PI(a)

=

i- o. Similarly,

A- I(1I"p(a)p'cp,p'E p),

P2(a) = (1 - A)-l (1I'"p (a)(l - p')Ep, (1 - p')e p), Va E A. By Proposition 2.4.6, PI and P2 are two states on A. Clearly, P = API + (1 - A)P2. Since P is pure, it follows that p = PI = P2. Then

IIA-~p'1I'"p(a)EpIl2 = pda*a) = p(a"'a) =

111I'"p(a)cpI12,

Va E A.

Hence A-!p' is an isometry on H p. But p' is a projection and p' i- 0,1, so this is a contradiction. Therefore 1I"p(A)' = 0:, i.e. 1I'"p is irreducible. Finally, since AI L p is dense in H, and is invariant for 1I"p(A) , it follows from Theorem 2.6.6 that H p = AI L p • Q.E.D.

104

Definition 2.7.2. A left ideat L of a C*-algebra A is said to be regular, if there is Xo E A such that (axo - a) E L, Va E A. In this case, Xo is called a modular unit for L. Of course, if A itself has an identity, then any left ideal of A is regular. Now suppose that A has no identity. Let p be a state on A, p be the natural extension of p on (A+

= O} = L p + L;.

Proof. Let L be a left ideal of A containing L p , and {11" p' H p } be the irreducible * representation of A generated by p, Since LIL p is an invariant subspace for 1r p (A ) and L -:j:. A, it follows that L = L p , i.e. L p is a maximal left ideal of A. Let €p(E H p ) be a cyclic vector for 11"p(A) as in Proposition 2.3.18. If b = b" E N(p), then (b p, Ep) :::: p(b) = O. Thus we can find a self-adjoint element of B(Hp), which maps Ep to 0 and maps bp to bpo By Theorem 2.6.5, there is h = h* E A such that

1rp(h)Ep = 0,

1rp(h)b p = bp.

Let c = b - hb. Since p(c*c) = lI1rp(h)bp - bp lJ2 = 0, it follows that c E L p. Further, b = c + hb, b = b' = bh + c", and c" E L;. Noticing

p((bhf . (bh)) < IIbIl2p(h2)

= IIbll2 '111r p (h)EpI1 2 =

0,

we have bh E L p and b = bh + c" E L p + L;. From N(p) = N(p)*, it follows that N(p) C L, + L;. On the other hand, it is clear that L, + L; c N(p) by the Schwartz inequality. Therefore, N(p) = L p + L;. By Theorem 2.7.1, H, = L p • Thus there is a E A such that a p = €p. Further, 11"p(b)a p = 1r p(b)E p = bp, and

AI

p((ba - b)* . (ba - b))

= II1rp(b)a p - bp l12 = 0,

i.e., (ba - b) E L p, Vb E A. Therefore, L p is regular.

Q.E.D.

Lemma 2.7.4. Let A be a C*-algebra with an identity, L be a closed left ideal of A, and a E A+. If for any e > 0 there is ae E L n A+ such that a < ae + s, then a E L.

105

Proof.

By Proposition 2.2.5, a;/2 E L. Since 0 < a < at

+ e, it follows that

+ cl/2)-la(a~/2 + c 1/2)-1 < (a~/2 + c 1/ 2)-1(ae + c)(a~/2 + c 1/2) - 1 (ae + c)(a e + 2cl/2a~/2 + c)-1 < 1.

o<

(a~/2

Thus we have

Ila 1/ 2(a:/ 2 + cl/2)-la~/2 _ a1/ 211 2 Ilcl/2al/2(a~/2

+ c 1/ 2 )-1112

cll(a:/ 2 + c 1/ 2)- la(a:/ 2 + c1/ 2 )-111< e and But a1 / 2(a e1/ 2

al/2(a~/2

+ el/2)-la~/2

~ a 1/2

as

e ~ 0+.

1/ 2 ELand L is closed therefore a E L + c 1 / 2) - l a e ' , •

Q.E.D.

Lemma 2.7.5. Let L 1 and L 2 be two closed left ideals of a C·-algebra A, and L 1 C L 2 • If for any state p on A with p(Ltl = {O}, we have also p(L 2 ) = {O}. Then L 1 = L 2 • Proof. We may assume that A has an identity. Let a E L 2 n A+. For any e » 0, let Oe = {p E S(A) I p(a) > e}. Then Oe is a u(A*,A)-compact subset of A·. For any p E Oe, since p(L 2 ) :f:. {O}, it follows from the assumption that p(L t ) :f:. {O}. Thus there is a(p) E L 1 such that jp(a(p») I > 1. By the continuity, we can find a u(A*,A)-neighborhood Vp of p such that If(a(p»)! > 1, Vf E v; Since Vp :> n., it follows from the compactness of 0. that

U

pEO. n

there are PI,' .. ,Pn E Oe such that Oe C

UVi, where Vi = V

p ..,

1

 1, Vp E Oe' Put ae = i=1

Iiall I: a;ai'

Clearly, ae E L 1 n A+ and

i=1

p(ae ) >

lIall > p(a) > s,

Vp E Oe'

Thus p(a. + e - a) > 0, Vp E S(A). By Corollary 2.3.15, ae + e > a. Since e > 0 is arbitrary and ae E L 1 n A+ it follows from Lemma 2.7.4 that a ELI' Thus we have L 2 n A+ c L 1 n A+. By Proposition 2.4.1, there is a net {d,} C L 2nA+ such that ad, ~ a, Va E L 2 • But de E L 2 n A+ c L 1 n A+, Vl, so it must be L 2 eLI. Further L 1 = L 2 •

Q.E.D.

106

Q.E.D. Let L be a closed left ideal of a ColI-algebra A. Then L is the intersection of all regular maximal left ideals containing L.

Theorem 2.7.6.

Proof.

Let 0 = {p E A *

I p > 0, IIpll

< 1, and p(L)

= {O}}.

Clearly,

n{LplpEO}~L,

where L p is the left kernel of p, \;/p E O. By Lemma 2.7.5, we have L = n{Lp Ip EO}. Since 0 is a a(A., A)-compact convex subset of A*, it follows from the Krein-Milmann theorem that Co(ExO) is a(A*, A)-dense in 0, where ExO is the set of all extreme points of 0, C 0(' .. ) is the convex hull of (...). If a E A such that p(a*a) = 0, \;/p E ExO, then p(a*a) = 0, \;/p E Co(ExO), and further p(a*a) = 0, \;/p E n. Therefore, we have

n{L p I p E ExO} = n{L p I p E O} = L.

(1)

Noticing that n =I- {O} (otherwise, by Lemma 2.7.5, we have L = A, a contradiction), from Theorem 2.7.3 it suffices to show that p is a pure state on A for each p E ExO and p =I- o. Now let tp E ExO and ip =I- o. Clearly, ip is a state on A. Suppose that there are two states tpl, tp2 on A and A E (0,1) such that ip = Atpi + (1- A)tp2' For any a E L, from (1) it must be a E L"" i.e., tp(a*a) = O. Further, tp,(a*a) = 0, i = 1,2, and by the Schwartz inequality, tpi(a) = 0, i = 1,2. Hence tpi(L) = {O} and tpi E 0, i = 1,2. Since tp E ExO, it follows that tp = tpi = tp2· Therefore, ip is a pure state on A. Q.E.D. Let L be a maximal left ideal of a Coli-algebra A. Then L is regular if and only if L is closed.

Theorem 2.7.7.

Proof.

The sufficiency is clear from Theorem 2.7.6. Now let L be a regular maximal left ideal of A. So there is Xo E A such that (bxo - b) E L, \;/b E A. Let L = L+O:(1 - xo). We claim that L is a maximal left ideal of (A+O:). In fact, suppose that J is a left ideal of (A-i-
J=I.

""

Now L is closed since (A +0:) has an identi ty. Therefore, L = closed.

I nA

is

Q.E.D.

Let L be a regular maximal left ideal of a Coli-algebra A. Then there is unique state p on A such that N(p) = {a E A I p(a) = O} =:) L. Moreover, this p is pure, and its left kernel is L, and N(p) = L + Loll.

Theorem 2.7.8.

107

By Theorem 2.7.7 and the proof of Theorem 2.7.6, there is a pure state P on A such that L p :) Land peL) = {O}. Since L is maximal, it follows that L = Lp- Further, by Theorem 2.7.3, N(p) = L + L*. Now suppose that €p is another state on A with €peL) = {O}. Then €p{N(p)) = {O}. If xo(E A) is a modular unit for Land {d,} is an approximate identity for A, then we have Proof.

p(xo) = limp(d,xo) = limp(dd I

I

and

,}. = lim!p(d,xo} = lim!p(d I I By Proposition 2.4.4, we get p(xo} = tp(xo} = 1. Moreover, A = N(p)+
it is clear that Q.E.D.

Corollary 2.7.9. For any C·-algebra A, there is an one to one correspondence between the set of all pure states on A and the set of all regular maximal left ideals of A. Theorem 2.7.10. Let A be a C·-algebra, p be a state on A, and L be the left kernel of p. If L is regular, then the following statements are equivalent: 1) p is a pure state on Aj 2) L is a maximal left ideal of Aj 3) N(p) = {a E A I pea) = O} = L + L·. Proof. By Theorems 2.7.3 and 2.7.8, it is clear that 1) and 2) are equivalent and 1} implies 3}. Now let N(p) = L+L· and xo(E A) be a modular unit for L. Suppose that there are two states Pll P2 on A and A E (0,1) such that p = API + (1 - A)P2' If x E L, then we have x·x E N(P),Pi(X·X) = 0, and Pi(X) = O,i = 1,2. Further, Pi(N(p)) = {O},i = 1,2. By the proof of Theorem 2.7.8, we can get p(xo) = Pl(XO) = P2(XO) = 1. Moreover, it is clear that A = N(p)+
References. [79], [81], [155], [156].

2.8. Ideals and quotient C*-algebras Definition 2.8.1. Let A be a C·-algebra, and peA) be the set of pure states on A (see Definition 2.3.8). Denote by A the set of all unitarily equivalent

108

classes of irreducible * representations of A, and let Prim(A) be the set of all primitive ideals of A. J is called a primitive ideal of A, if there is an irreducible * representation 7r of A such that kerer = J. Clearly, a primitive ideal is a closed two-sided * ideal. Since two unitarily equivalent representations have the same kernel, so we have a natural map from A onto Prim(A). Moreover, for each p E P(A), the * representation generated by p is irreducible from Theorem 2.7.1. Conversely, if {7r, H} is an irreducible * representation of A, then by Proposition 2.4.6, p(.) = (7r{·)e, e) is a state on A for any e E H with II ell = 1. And also from Proposition 2.3.21, the * representation {7r p,Hp } generated by p is unitarily equivalent to {7r,H}, and is irreducible. Furthermore, p is pure by Theorem 2.7.1. Therefore, we have a map from P{A) onto A by the GNS construction.

Proposition 2.8.2. Let J be a closed two-sided ideal of a C*-algebra A. Then J = n{I E Prim{A) I I ::> J} n{ker7r p

I p E P(A),

and (pIJ)

= O}.

Proof. Since J is also a closed left ideal of A, it follows from the proof of Theorem 2.7.6 that

n{L p I p E P(A), and (pIJ) = O}

J =

::> n{ker 7r p

Ip E

P(A), and (pIJ) = O}.

Now let a E J. For any p E P(A) and (pIJ) follows that

=

0, since J is two-sided, it

Hence, 7rp (a) = 0, and a E ker-r p' Therefore, we obtain

J =

-

n{ker7r p

Ip E

P(A), and (pIJ) = O}

n{I E Prim(A)

II

::> J}.

Q.E.D. Corollary 2.8.3. Let 0 be a compact Hausdorff space, and J be a closed ideal of C(O). Then there is a closed subset 0 0 of 0 such that J

Proof.

= {f E C(O) I f(t) = 0, 'r/t E

Oo}.

It is clear from Propositions 2.8.2 and 2.3.10.

Q.E.D.

109

Let A be a C·-algebra, and J be a closed two-sided ideal

Definition 2.8.4. of A. Define

PJ{A)

= {p E

P{A)

I (pIJ) = o},

pJ(A) = {p E P(A) I (pIJ) ¥= O} = P(A)\PJ(A), A J = A\AJ,

AJ = {'Tr E A I ker 7r :J J}, Primj I.A)

= {I

PrimJ(A)

= Prim

E Prim (A)

II

:J

J},

(A)\PrimJ(A).

Theorem 2.8.5. ..... Let J be a closed two-sided ideal of a C·-algebra A . 1) For any 'Tr E AJ' let 1f(a) = 1f(a),Va E AIJ,a E a, where a ---+ a = a + J is the canonical map from A onto A I J. Then 1f ---+ 1f is a bijection from A onto (AI J)A. 2) tt ---+ ('Tr!J) is a bijection from AJ onto J.

1) It is obvious. 2) Let {'Tr, H} be an irreducible

Proof.

* representation of A, and 7rIJ ¥= O. Since J is a two-sided ideal, it follows that the linear span K of {7r (a) eI a E J, EE H}

is a non-zero invariant subspace for 1f (A). Now since 1f is irreducible, it must be that K = H, and {7r IJ, H} is a nondegenerate * representation of J. If {dl } ( C J) is an approximate identity for J, by Proposition 2.4.6, we have 'Tr(dl ) ---+ 1 (strongly). Then 'Tr(ad/) ---+ 7r(a) (strongly), Va E A, i.e., 'Tr(J) is strongly dense in 7r(A). So {'TrIJ, H} is an irreducible * representation of J by Proposition 2.6.2. Conversely, let {7r, H} be an irreducible * representation of J. By Theorem 2.6.6, H = ['Tr(b)e! s e J,e E H]. For any bt,···,b n E J and 6,···,en E H,

I: ('Tr(bj. bi)ei, Ei) l$i,;$n

is a positive linear functional on (A-t
n

II L'Tr(abdEill < Iiall ·11 I:1f(bi )ei ll, i=1

&=1

Va E A, bt,'" ,b n E J, 6,'" ,en E H. Further, {n , H} can be uniquely extended to an irreducible * representation {1f', H} of A such that

'Tr'(a)7r(b)E = 'Tr(ab)E, Va E A, s « J, e E H. Moreover, if {'Trl,Ht } {'Tr~,

'"'"

{7r2,H2}, then it is easy to see that {'Tri,H1 }

H2 } .

Therefore, 'Tr

---+

('TrIJ) is a bijection from

AJ

onto

J.

'"'"

Q.E.D.

110

Definition 2.8.6. A C·-algebra A is said to be prime, if xAy = {a} implies either x = 0 or y = O. A closed two-sided ideal J of a C*-algebra A is said to be prime, if the quotient C·-algebra AI J is prime. Lemma 2.8.7. Let A be a C·-algebra. 1) Suppose that J 1 and J 2 are closed two-sided ideals of A. Then [J1J2 ] = J1

n J2 •

2) A is prime if and only if for any non-zero closed two-sided ideals J 1 , J 2 of A we have J 1 n J 2 i- {o}. 3) Suppose that J is a closed two-sided ideal of A. Then the following statements are equivalent: (a) J is prime; (b) If x, yEA such that xAy c J, then either x E J or y E J; (c) If J 1 , J 2 are closed two-sided ideals of A and J 1J2 C J, then either J 1 C J or J 2 C J; (d) If J 1,J2 are closed two-sided ideals of A, and J, :> J, and J, f; J, i = 1,2, then J 1 n J 2 i- J.

Proof. 1) Clearly, (J 1J2 ] C J 1 n J 2 • Now let a E J 1 n J 2 • Since J 1 n J 2 is a C·-subalgebra of A, we can write a = b~ - b~ + ib; - ib~, where b; E (J1 n J 2 )+, 1 < j < 4. Obviously, E J 1J2 , 1 < j < 4. Thus a E [J1J2 ] , 'and [J1J2 ] = J 1 n J 2 • 2) Let A be prime, and J ll J 2 be non-zero closed two-sided ideals of A. Pick o i- x E J 1 and 0 i- y E J 2 , and let J z (= [AxA]) , J y (= [AyA]) be the closed two-sided ideals generated by x, y respectively. Clearly, {O} i- J; c JI, {o} iJI/ C J 2 , and xAy C c J 1J2 C J 1 n J 2 • Since A is prime and x, yare non-zero, it follows that xAy i- {O}, and J 1 n J 2 i- {O}. Conversely, if there are x and yEA with x i- 0, y i- 0 such that xAy = {O}, then JzJy C [AxAyA] = {O}, where J x = [AxA] and J y = [AyA] are non-zero closed two-sided ideals of A. By 1), we have JznJy = [JxJ,,] = {O}. Therefore, if A is not prime, then there are non-zero closed two-sided ideals J 1 and J 2 such that J 1 n J 2 = (o}, 3) By Definition 2.8.6 and 2), it is clear that (a), (b) and (d) are equivalent. (d)===>(c). Let J}, J 2 be closed two-sided ideals of A and J 1J2 C J. If J, J,i = 1,2, then Ii = J + J, i- J,i = 1,2. Now by (d), we have

b;

r.r;

ct

J

i- II n 12 ,

J

C

II

n 12 •

However, by 1) and J 1 J 2 C J we have II n /2 = [1112 ] C J, a contradiction. Thus it must be that either J 1 C J or J 2 C J. Now let (c) hold, and x,y E A be such that xAy C J. Then JzJ" C J, where J; = [AxA],Jy = [AyA). By (e), either J'; C J or J y C J. Thus either x E J or y E J, and J is prime. Q.E.D.

111

Proposition 2.8.8.

Let A be a C*-algebra, and J EPrim(A). Then J is

prime.

Let {11", H} be an irreducible * representation of A such that kerer = J. Suppose that x, yEA are such that xAy = {O}. Then 1r(x)1r(A)1r(y) = {o}. By Proposition 2.6.2 we have 1r(x)B(H)1r(y) = (o). Thus either 1r(x) or 1r(y) is zero, i.e., either x or y is in kerrr = J. Therefore, J is prime. Q.E.D.

Proof.

Theorem 2.8.9. Let J be a closed two-sided ideal of a C·-algebra A. 1) I --+ I I J is a bijection from PrimJ(A) onto Prim(AI J). 2) I --+ In J is a bijection from PrimJ(A) onto Prim(J).

Proof. 1) Let I E Prirnj (A), and {11", H} be an irreducible * representation of A such that kersr = I. For any a E A I J, since J c I, we can define 1f(a) = 1r(a), here a E a = a-rJ: Then {n, H} is an irreducible * representation of AI J, and kern = I I J. Thus I I J EPrim(AI J). Moreover, let {1f,H} be an irreducible * representation of AIJ. Define 1r( a) = 1f(a) , Va E A, here a ---t a is the canonical map from A onto AI J. Then {1r, H} is an irreducible * representation of A and ker-r = I :> J, ker1f = I I J. Thus, 1--+ IIJ is a map from Primj I.A] onto Prim(AIJ). Now let I1/J = 12/J, where 11, / 2 EPrimJ{A). For any a E Ii) then we have b E 12 such that (a - b) E J. Since J c 12 , it follows that a E 12 • Thus II c 12 • Similarly, 12 C 11' Thus 11 = 12 . Therefore, I --+ I I J is a bijection from Primj (A) onto Prim (A I J) . 2) Let I E PrimJ(A), and {1r,H} be an irreducible * representation of A such that kerz = I. Since ker-r = I 1J J, it follows by Theorem 2.8.5 that {(1rIJ),H} is an irreducible * representation of J, and ker(1rjJ) = In J EPrim( J). Now if {G, H} is an irreducible * representation of J, then by the proof of Theorem 2.8.5 {G, H} can be uniquely extended to an irreducible * representation of A. So I --+ In J is a map from Prirrr' (A) onto Prim(J). Now let II n J = 12 n J, where 11,12 EPrimJ(A). Then 12 :> II n J :> I 1 J . But 12 1J J, by Proposition 2.8.8 we have 12 :> II' Similarly, II :> 12 , Thus II = 12, Therefore, I --+ In J is a bijection from PrimJ(A) onto Prim(J).

Q.E.D. Theorem 2.8.10. Let J be a closed two-sided ideal of a C·-algebra A. 1) P --+ p is a bijection from PJ(A) onto P(AIJ), where p(a) = p(a), Va E A/J,a E a. 2) p --+ (pIJ) is a bijection from p J (A) onto P( J).

Proof.

1) It is Proposition 2.4.11 exactly.

112

2) Let p E pJ(A), and {tr,H, €} be the irreducible cyclic * representation generated by p. If J ckertr, then p(a) = (tr(a)E, E) = 0, 'Va E J. This is a contradiction since p(J) =f:. {O}. Thus J stkertr. Then by Theorem 2.8.5, {trIJ, H} is an irreducible * representation of J. Further by Proposition 2.3.21, {trIJ, H} is unitarily equivalent to the * representation generated by the state ((trIJ)(·)€, €) = (pIJ)(·). Thus (pIJ) E P{J). Moreover, if {d,} (c J) is an approximate identity for J, by Proposition 2.4.6 we have tr(d ,) ~ 1 (strongly). Then p{a) = (tr{a)E, €) = lim(tr(ad,)€, E), I

'Va E A. This means that the behaviour of p on A is determined by (pIJ). Hence, if PI,P2 E PJ(A) and (PIIJ) = (p2I J), then PI = P2. Finally, let a E P(J), and {trO', H, €} be the irreducible cyclic * representation of J generated by a. By Theorem 2.8.5, {tro , H} can be uniquely extended to an irreducible * representation {tr, H} of A. Then p{.) = (tr(.)€, €)('V. E A) is an extension of a, and the * representation of A generated by p is unitarily equivalent to {tr, H} (see Proposition 2.3.11). Thus p E pJ(A) and (pjJ) = a. Therefore, p ~ (pIJ) is a bijection from pJ (A) onto P{J). Q.E.D. From Theorem 2.8.5, 2.8.9 and 2.8.10, we have the following diagram:

P(A)

P(AIJ)

t

1

(AI J) 1\

1

P(J)

t

A

Prim( AIJ)

~

J

Prim( A)

1

+-

Prim( J)

It is easily checked that this diagram is commutative.

Notes. About the converse of Proposition 2.8.8, J. Dixmier gave the following result. Let A be a separable or Type I (GCR) C·-algebra, and J be a prime closed two-sided ideal of A, then J is also primitive. But the question for the general case is still open.

References. [23], [28}, [33}, [156].

2.9. Hereditary C*-subalgebras Lemma 2.9.1. Let A be a C" -algebra, a, x, yEA and a > O. Suppose that there are number A, J.l > 0 with (A + JL) > 1 such that x* x

Let

Un

= x(~

+ a)-4 y , 'Vn.

Ilull < lI a~±i-lll·

< a>', vv' < a/J..

Then there is

uE

A, such that

Ilun - ull

~ 0 and

113

Let dnm = (~

Proof.

+ a)-4 2

IIU n -

u m l1 =

(~

+ a)-~, 'in, m,

Then

IIxdnmyI1 2 = Ily*dnmx*xdnmyIJ

< Ily*dnmaAdnmYIl = Ila~dnmyl12

Iia ~ dnmYY*dnmat II < Ila ~ dnma J.L dnma ~ = 2 11

2

Ildnma

~ 2

II 2 .

We may assume that A has an identity 1. Let B be the abelian C*-subalgebra generated by {a,1}. Then B ,..., C(!1). For each t E !1, ((~ + a)-4aA~e)(t) / a A+~-l (t). By the Dini theorem, this convergence is uniform for t E !1. Thus IldnmaA~ell-+ 0 and lIun-um" -+ 0, and there is U E A such that lIu n-ulI-+ 0. Similarly, we can prove that Ilunll < II(~ +a)-4aA~ell,'in. Therefore, Ilull <

Iia ~ II.

Q.E.D.

2

Proposition 2.9.2. Then for any A E (0,

!),

Let A be a C*-algebra, x, a E A and a > 0, z" x there is u E A such that

x Proof.

Let Un = x(~

= ua>.,

and

Ilull < Ilai-AII.

+ a)-4 a4->., 'in.

By Lemma 2.9.1, there is u E A such

that

Ilun- ull Moreover, since x* x

< a,

-+ 0,

< a.

and

Ilull < Il a4- A Ii .

it follows ;hat

Therefore, x = ua A •

Q.E.D.

Corollary 2.9.3. Let A be a C*-algebra, x E A, and 0: E (0,1). Then there is u E A such that

Proof.

It is clear by Proposition 2.9.2 and picking a

= x* x, A = 0:/2. Q.E.D.

Remark.

A factorization with 0: = 1 (polar decomposition) is not possible

in a general C·-algebra.

114

Definition 2.9.4. Let A be a C*-algebra. A cone M{C A+) is said to be hereditary, if a E A+ with a < b for some b E M implies a E M. For a hereditary cone M, we define

L(M)

=

{x E A I z" x E M}.

It is easy to see that L{M) is a left ideal of A. A C*-algebra B of A is said to be hereditary, if B+ is hereditary.

Theorem 2.9,5. Let A be a C*-algebra. 1) B ~ B+ is a bijection from the set of all hereditary C*-subalgebras of A onto the set of all closed hereditary cones of A+. Its inverse is M ~ L{M) n L{M)*. 2) M - ? L{M) is a bijection from the set of all closed hereditary cones of A+ onto the set of all closed left ideals of A, and M = L{M)+. Its inverse is L - ? L+. 3) L ~ L n L* is a bijection from the set of all closed left ideals of A onto the set of all hereditary C*-subalgebras of A, and L+ = (L n L*)+. Its inverse is B ~ L(B+). Let B be a hereditary C* -subalgebra of A. By the definition, B+ is a closed hereditary cone of A+. Since B = [B+], so the map B ~ B+ is injective. Let M be a closed hereditary cone of A+. Clearly, L(M) is a closed left ideal of A. Assume that x E A is such that x*x E L(M). By Corollary 2.9.3, we can write x = U(X*X)1/4 for some u E A. Thus x E L(M), and L(M)+ = {x*x I x E L{M)}. By the definition of L(M), we have M = L(M)+. Thus the map M - ? L(M) is injective. Let L be a closed left ideal of A. Clearly, L n L· is a C·-subalgebra of A. Since L+ C L n L* c L, it follows that L+ = (L n L*)+. So in order to prove that L n L· is hereditary, it suffices to show that L+ is hereditary. Suppose that a E A+ and a < b for some b E L+. By Proposition 2.9.2, we have a factorization a~ = vb1 / 3, where v E A. Thus a~ E L+, and a E L+, i.e., L+ is hereditary. Suppose that L is a closed left ideal of A. From the preceding paragraph, M = L+ is a closed hereditary cone of A+. Let x E A be such that x·x E M. By Corollary 2.9.3, we can write x = u(X*x)l/3. Thus x E L. Further, by the definition we have L = L(M). So 2) is proved. Suppose that M is a closed hereditary cone of A+. From preceding paragraphs, L{M) is a closed left ideal of A; L(M) n L(M)* is a hereditary C*subalgebra of A; and M = L(M)+ = (L(M) n L{M)*)+. So 1) is proved. Finally, let B be a hereditary C*-subalgebra of A. Thus B+ is a closed hereditary cone of A+; L(B+) is a closed left ideal of A; L(B+) n L(B+)* is a Proof.

115

hereditary C*-subalgebra of A. Since (L(B+) n L(B+)*)+ = L(B+)+ = B+, it follows that B = L(B+) n L(B+)*. So the map L --+ L n L* is surjective. Moreover, if L is a closed left ideal of A such that B = L n L*, then B+ = (L n L*)+ = L+, and L = L(B+). So the map L --+ L n L* is also injective; and its inverse is B --+ L(B+). Therefore, 3) is proved. Q.E.D.

Lemma 2.9.6. Let cP be a * homomorphism from a C*-algebra A onto a C·-algebra B,a E A+,b E B with b'b < cP(a). Then there is x E A such that b = cP(x) and x*x < a. Proof.

Pick yEA such that

~ (y)

= b. Write

y·y-a=h-k, where h, k E A+ and hk = O. Since l)(y).(y) = b'b < cP(a), it follows that o < cP(h) < (k). But cP(h)~cP(k)(h)1/2 = 0, so we have cP(h) = o. 1 1 Clearly, y*y < a + h,a'2a'2 = a. Let

1 = y(-+a+h)~'2a'2,Vn. 1

Xn

1

n

Then by Lemma 2.9.1, there is x E A such that l[xn we have

Moreover, since y*y X~Xn

< a + h, 1

xII

--+

O. From cP(h) = 0,

it follows that for any n

1

< a'2( - + a + h)-'2(a + h){ - + a + h)-'2a 2 < a. 1

1

n

1

1

n

Therefore, z" x < a, and x is what we want to find.

Q.E.D.

Proposition 2.9.7. Let A, B be two C*-algebras, C be a hereditary C*subalgebra of A, and cP be a * homomorphism from A onto B. Then cP(C) is also a hereditary C·-subalgebra of B.

Proof.

The result is clear by Lemma 2.9.6.

Q.E.D.

Proposition 2.9.8. Let A be a C·-algebra, B be a hereditary C*-subalgebra of A, and cp be a state on B. Then there exists unique state t/J on A such that (?/JIB) = cpo

Proof. Let {de}(C B+) be an approximate identity for B. For any a E A+, since 0 < diad, < lIall2d~ E B+ and B is hereditary, it follows that d,ad, E B, VI. Further, we have d,Ad l C B, Vl.

116

be a state on A such that (t/JIB) = tp. Then Iit/JII = Iltpll = limtp(d ,} = limtp(d;) (see Proposition 2.4.4). Further, by the Schwartz

Now let

=

1

l

t/J

I

inequality we have t/J(a(l - d,)) --+ 0, t/J((1 - d,)a) --+ 0, Va E A. Thus t/J(a) = lim t/J(d,ad ,) , Va E A. From the preceding paragraph d,Ad, C B, so we have I

t/J(a) Therefore, the extension Notes.

t/J

= limtp(dzad z}, z

is unique.

Va E A. Q.E.D.

Theorem 2.9.5 is due to E.G. Effros.

References. [33], [127].

2.10. Comparison, disjunction and quasi-equivalence of * representations Definition 2.10.1. Let A be a C· -algebra, and {w, H} be a * representation of A. If K is a closed linear subspace of H, and K is invariant for w [i.e., w(a}E E K,Va E A,e E K), then {w,K} is also a * representation of A, and {w, K} is called a * sub representation of {w, H}. Suppose that {wI,H t } and {w2,H2 } are two * representation of A. The symbol "Wt -< W2" means that {wl,H t } is unitarily equivalent to a * subrepresentation of {W2' H 2 } . Proposition 2.10.2. Let {?Til HI} and {?T2' H 2 } be two * representations of a C· -algebra A. 1) Let w = Wt ffi ?T2, H = H t ffi H 2 , and let p~ be the projection from H onto Hi (clearly, p~ E ?T(A)'), i=l, 2. Then ?Tl -< ?T2 if and only if p~ -< p~ in w(A}'. 2} If?Tt -< ?T2 and ?T2 -< ?TIl then we have {WI, HI} ,...., {W2' H 2 } , i.e., {wt, H t } and {?T2' H 2 } are unitarily equivalent. Proof. 1) It is clear. Moreover, from 1) and Proposition 1.5.3, we can get 2) immediately. Q.E.D.

Definition 2.10.3. Let {?T}, HI} and {?T2' H 2 } be two * representations of a C·-algebra A. ?Tl and ?T2 are said to be disjoint, denoted by ?Tt..lW2' if any non-zero * subrepresentation of Wt is not unitarily equivalent to any non-zero * subrepresentation of ?T2' Proposition 2.10.4. of a C·-algebra A. Let

?T

Let {?Tl,Ht} and {?T2' H 2 } be two * representations = Wt E9 ?T2, H = HI ffi H 2 ; and let p~ be the projection

117

from H on Hi (clearly, p~ E 7r(A)'), i = 1,2. Then the following statements are equivalent:

1) 7r1J..7r2; 2) c(pi) . c(p~) = 0, where c(pD is the central cover of p~ in 7r(A)', i 3) p~ is a central projection of 7r(A)', i = 1,2.

= 1,2;

Proof. Since pi EEl p~ = 1, it follows that the statements 2) and 3) are equivalent. Clearly, 7r1J..7r2 if and only if there are no projections qi and q~ of 1I"(A)' such that 0 f- q~ < p~, i = 1,2, and qi '" q~ in 7r(A)'. Then by Proposition 1.5.9, the statements 1) and 2) are equivalent. Q.E.D.

Definition 2.10.5. A nondegenerate * representation {7r, H} of a C*algebra A is said to be factorial, if the VN algebra on H generated by 7r(A) (i.e. 1I"(A)") is a factor. Proposition 2.10.6. Let {7rl' Hd and {11"2' H 2 } be two factorial tations of a C*-algebra A. Then one of the relations 11"1-111"2, 7rl -< holds.

* represen7r2, 7r2

-<

11"1

Proof. Let 11" = 11"1 EEl 7r2, H = HI EEl H 2 , M = 7r(A)" and p~ be the projection from H onto Hi,i = 1,2. Then p~,p~ EM'. By the assumption, Mp~ is a factor on Hi, i = 1,2. If c(p:l is the central cover of p~ in M', by Proposition 1.5.10 M p~ and M c(pD are * isomorphic, i = 1,2. Thus M C(pi) is also a factor on H c(pD; and c(pD is a minimal central projection of M (Le., if z is a central projection of M and z < c(pD, then either z = 0 or z = c(p~)),i = 1,2). Therefore, we have either c(pi) . c(p~) = 0 or c(pi) = c(p~). When c(pi) . c(p~) = 0, by Proposition 2.1004 1I"1J..7r2 holds. Now let c{pi) = c(p~) = z, Since M z is a factor, by Proposition 1.3.8 M' z is also a factor. From Theorem 1.504, we have either pi -< p~ or p~ -< pi in M' z (also in M'). Further by Proposition 2.10.2, either 7rl -< 11"2 or 7r2 -< 7rl holds. Q.E.D.

Proposition 2.10.7. Let {7rl, HI} and {7r2' H 2 } be two irreducible * representations of a C"'-algebra A. Then 7r 1J..1I"2 if and only if 11"1 and 11"2 are not unitarily equivalent. Proof. Since any non-zero * subrepresentation of 7ri must be 1,2, the conclusion is obvious from Definition 2.10.3.

Proposition 2.10.8.

If 11" .l7r/) "iI, then 11"1.

L I

EEl7rI.

7ri

itself, i = Q.E.D.

118

Proof.

1T

ffi

I: l

Let H, HI be the action spaces of 1T, 1T1 respectively, Vl. Let a = ffi1T/) K = H ffi ffiHI; and let pi, P~ be the projections from K onto

L 1

H, H, respectively, VI. Then pi, P~ E a( A)', VI. By Proposition 2.10.4 we have c(p') • c(pD = 0, VI. From Proposition 1.5.8,

C(p') 1- supc(pD = c(suppD = c(L:pD. l

Again by Proposition 2.10.4, 1T 1-

1

L: ffi1Tl.

l

Q.E.D.

l

Definition 2.10.9. Let {1Tl,H1} and {1T2,H2} be two nondegenerate * representations of a C·-algebra A, and M, = 1Ti{A)",i = 1,2. 1rl and 1T2 are said to be quasi-equivalent, denoted by 1Tl :::::; 1T2, if there is a * isomorphism cP from M l onto M 2, such that cI>(1TI(a)) = 1T2(a), Va E A. Proposition 2.10.10. Let {1Tl, Ht} and {1T2' H 2} be two nondegenerate * representations of a C*-algebra A. Then the following statements are equivalent: 1) 1Tl : : : ; 1T2; 2) No non-zero * subrepresentation of 1Ti is disjoint from 1Tj, 1 < i =I j < 2; 3) Let 1r = 1Tl ffi 1T2, H = HI ffi H 2, and P~ be the projection from H onto Hi ( E 1T(A)'), i = 1,2. Then c{pi) = c(p~) j 4) There exists an ampliation 1T of 1Tl [i.e., there is a Hilbert space K such that 1T{a) = 1TI(a) ®lK, Va E A) and a projection p' of 1r(A)' with central cover 1 (= IH 1 ®K) such that 1Tp' 1T2; 5) There exist ampliations of 1rl and 1T2 which are equivalent. I"'J

I)===> 4). It is clear Theorem 1.12.4 and Proposition 1.12.5. 4) ===> 1). Define cP3' cP2' cP l as in Theorem 1.12.4; and let cP = cPs 0 cP2 0 cPl. Then it is immediate that 1Tl : : : ; 1T2' 4) ===> 5). From the condition 4), 1T2 is unitarily equivalent to a * subrepresentation of 1Tl ® I K , where K is some Hilbert space. Since 1) and 4) are equivalent, it follows that 1Tl is also unitarily equivalent to a * subrepresentation of 1r2 ® I L , where L is some Hilbert space. Let R be an infinite dimensional Hilbert space, and dimR >dim K, dimL. Then

Proof.

By Proposition 2.10.2, we have 1Tl ® lR ~ 1T2 ® lR. That comes to 5). 5) ===> 2). Suppose that 1Tl ® I R 1T2 ® lR, where R is a Hilbert space. If ai is a non-zero * subrepresentation of 1Ti, then we can regard a; as a non-zero * subrepresentation of 1Ti ® lR. By Definition 2.10.3, a, is not disjoint from 'lrj ® IR. Further, by Proposition 2.10.8 a, is not disjoint from 1r;. 'V

119

2) ==> 3). If c(pD

:I

c(p~), we may assume that c(p~)

"'t c(p'd.

Then

- c(p~) . c(p~) is a non-zero central projection of 1r(A)', and z < c(p~) and z i. c(pi). By Proposition 1.5.8, zp~ :I o. Clearly, c(zp~) i. c(pi). By Proposition 2.10.4, {1r2, ZH2}i.{1rl' HI}. That contradicts the condition 2). Thus c(pi) = c(p~). 3) ==> 1). Let z = c(p~) = c(p~), and M = 1r(A)". By Proposition 1.5.10, Mp~ and M z are * isomorphic, i = 1,2. Thus we have a * isomophism (l) from M; = M'pi onto M 2 = Mp~ such that (bpi) = bp~, Vb E M. In particular, for any a E A, since 1ri(a) = 1r(a)p~, i = 1,2, it follows that (1rt{a)) = 1r(a)p~ = 1r2(a). Therefore, 1rl :;::::: 1r2. Q.E.D. z=

c(p~)

Proposition 2.10.11. Let {1rb HI} and {1r2' H 2 } be two nondegenerate representations of a C'" -algebra A. 1) If 1("1 ,.."., 1r2, then 1rl ~ 1r2. 2) If 1rl and 1r2 are irreducible, and 1rl ~ 1r2, then 1rl ,.."., 1r2.

*

1) It is obvious. Now we prove 2). By Proposition 2.10.10, 1rl is not disjoint from 1r2. Since 1rl and 1r2 are irreducible, it follows from Definition Q.E.D. 2.10.3 that 1rl is unitarily equivalent to 1r2.

Proof.

Proposition 2.10.12. Let {1rl,H1} and {1r2,H2} be two factorial * representations of a C"'-algebra A. Then we have either 1rli.1r2 or 1rl :;::::: 1r2. By Proposition 2.10.6, we may assume that 1rl -< 1r2. Then there is a projection p' E 1r2(A)' such that 1rl 1r2P'. But 1r2{A)' is a factor, so the central cover of p~ in 1r2 (A)' is 1. By Proposition 1.5.10, 1r2 ~ 1r2P'. Therefore 1rl ~ 1("2. Q.E.D.

Proof.

row

References. [28], [104]' [105].

2.11 The enveloping Von Neumann algebra Definition 2.11.1. Let A be a C"'-algebra, and S{A) be its state space. For each ip E S(A), we have a cyclic * representation {1rlp' Hlp' Clp} of A (see Proposition 2.3.18). Then the faithful * representation 1r u

=

L

E91rlp'

lpES{A)

is called the universal

* representation of A.

And 1ru(A)" is called the enveloping VN algebra of the C"'-algebra A, denoted by A = 1ru(A)".

120

Suppose that t/J is a normal state on A. Since A and 1ru(A) are * isomorphic, it follows that there is a state rp on A such that rp(a) = t/J(1r u(a)) , Va E A. By the GNS construction, we have Elp E Hlp C H; such that

Further, t/J(b) = (bcIp' CIp), Vb E A = 1r u (A)" . Therefore, every normal state on the VN algebra M = 1r u (A)" is a vector state. From Proposition 1.10.6, we have a(M, M.) ,. . ., (weak operator top. 1M), s(M, M.) ,. . ., (strong operator top. 1M) and s" (M, M.) ,. . ., (strong * operator top. 1M) . Now we study the relation between the enveloping VN algebra A and the second conjugate space A··. By Proposition 1.3.3, A is the conjugate space of the Banach space A. = T(Hu)j A,L, where

A,L = {t E T(Hu)

I tr(tb) = 0, Vb E A}.

Through the following way, the Banach space A. and the conjugate space A" of A are isometrically isomorphic. For any I E A .. , let

F(a) = 1ru (a)( / ),

Va EA.

Then FE A* and J[FII = 11/11. Conversely, any element of A'* must be of above form. In fact, if I E A .. , by density theorem 1.6.1 11/11 = sup{l1ru (a)(/ ) I I a E A,lIali < I} = IIFII. Now let rp E S(A). Then tp(a) = (1r u(a)ElJ" CIp), Va E A. Let Pip be the one rank projection of H u onto (EIp], and I be the canonical image of Pip in A .. = T(Hu)j A,L. Then

1ru(a)(/) = tr(1r u(a)pIp) = (1r u (a)cIp, cIp) = rp(a),Va EA. Since A * is the linear span of S (A), it follows that for each F E A" there is unique I E A* such that

F(a) = 1ru (a)( / ),

Va E A.

Denote the above isomorphism from A. onto A· by 1r'*' i.e.,

Then (1r.. )* is an isometrical isomorphism from AU onto A, and it is a(A", A*)a(A, A.) continuous. Moreover, since for any a E A c A**

it follows that (1r '*) * IA = 1ru, i.e., (1r.. )'" is an extension of the * isomorphism 1r u from A onto 1r u (A). So we can write (1r.. = 1ru simply. Now we have the

r

following theorem.

121

Theorem 2.11.2. Let A be a C'"-algebra. Then the second conjugate space A *'" of A is isometrically isomorphic to the enveloping VN algebra A of A. So we can introduce a multiplication and a * operation on A""" such that A .... becomes a C'" -algebra, and A becomes a C* -subalgebra of A""". Moreover, if A has an identity, then this identity is also an identity of A "'*. In the above discussion, the multiplication and the * operation on A*'" are defined through A. But we have another way. It depends on A and A'" directly.

Theorem 2.11.3. Let A be a C'"-algebra, and define a multiplication (Arens multz'plication) on A""" as follows:

X"'(F)

=

XY(F) =

X(F"'),

* operation

and a

F"'(a) = F{a"'),

X([Y, F]), [Y, F](a) = Y(LaF), (LaF)(b)

Va, b E A, F E A"', X, YEA""". Then this

* operation

=

F(ab),

and multiplication on

AU are the same as in Theorem 2.11.2.

Proof.

Keep the above notations: A, A"" A.l, 1r", : A*

---+

A"', and (1r*)* = 1ru

:

A .... ~A. For any X E A""", pick a net {Xl} C A such that Xl ~ X(u(A"'''',A''')). Since (X - xl)*(F) = (X - xd(F*),VF E A"', it follows that xi ---+ X"'(u(A .... ,A"')). But 1ru = (1r",)'" is u(A"'*,A*)-u(A,A",) continuous, thus we have

For any t E T(Hu ) and a E A, denote the canonical images of t and t1r u(a)(E T(Hu )) in A* = T(Hu}/A.l by ! and La! respectively. Since for any bE A

(L a1r",(f))(b) = 1r",(f)(ab) = 1ru(ab)(f) tr(t1ru(ab)) = 1ru(b)(L af), it follows that 1r*(L af ) = L a1r",(f). For Y E A"'*, let g be the canonical image of 1ru(Y)t(E T(H u ) ) in A", T(Hu ) / A.l. From

[Y,1r*(!)](a}

=

Y(L a1r",(f)) = Y(1r",(L af)) 1ru(Y)(L af ) = tr(1r u(Y)t1ru(a)) 1ru(a)(g) = 1r", (g)(a),

Va E A,

122

Now for any X, Y E AU by

1ru(XY)(f)

= (XY)(1r*(f)) = X([Y,1r*(f)]) X(1r*(g))

= 1ru(X)(g) = tr(1ru{X)1ru{Y)t)

(1r u(X)1ru(Y))(f),

Vf E

A*,

we have 1ru{XY) = 1ru(X)1ru(Y).

Q.E.D.

Proposition 2.11.4. Let A be a C*-algebra, and B be a C*-subalgebra of f7 A. Then the C*-algebra B** is * isomorphic to the a{A "", A *)-dosure B of B in A **. Proof. For any X E B**, let (X){F) = X(FIB), VF E A*. Then is an f7 isometric linear isomorphism from B** onto B • By Theorem 2.11.3, also Q.E.D. keeps the * operation and multiplication. Notes. The second conjugate space of a C*-algebra is very important since it is a W*-algebra (see Chapter 4). The Theorem 2.11.2 (A** '" A) is due to S. Sherman and Z. Takeda. Moreover, let B be a Banach algebra. We can introduce two kinds of Arens multiplication on B**. The first Arens multiplication is as in Theorem 2.11.3, i.e.,

(XY)(F) = X([Y, F]),

[Y, F](a)

= Y(LaF),

(LaF)(b) = F(ab),

Va, b E B, F E B*, X, Y E B**. The second Arens multiplication is as follows: (X· Y)(F) = Y([X, F]'),

[X, FJ'(a) = X{RaF),

(RaF)(b) = F(ba),

Va, b E B, F E B*, X, Y E B A natural question is when we have XY = X . Y, VX, Y E B"'*. Definition (P. Civin and B. Yood). A Banach algebra B is said to be regular, if XY = X . Y, VX, Y E B*"'. Let H

•

Z(B"'*) = {X E B**

I XY =

X· Y, VY E B U

} .

Z(B"'''') is called the topological center of B**. Clearly, B is regular ¢:::::> Z(B"'*) = B"'*; X E Z(B**) ¢:::::> the map' ----+ X· is continuous in (B**, a(B"'*, B*)); and B C Z{B**) c B*"'. By Sakai theorem (see Section 4.2), any C"'-algebra A is regular, i.e., Z(A*'") = A*. Therefore, for any C*-algebra A, two kinds of Arens multiplication are the same on A ** .

123

References. [16], [64], [161], [168].

2.12. The multiplier algebra Definition 2.12.1. Let A be a C*-algebra, and see the second conjugate space A** of A as the enveloping VN algebra. Let M (A)

= {a

E

A **

I aA U Aa C

A}.

Then M(A) is called the multiplier algebra of A. Clearly, M(A) is a C*-subalgebra of A**; A c M(A) and A = M(A) ¢:::::> A has an identity; A is a closed two-sided ideal of M(A). The C*-algebra Q(A) = M(A)/A is called the out multiplier algebra of A.

Definition 2.12.2. space H. Let

Let A be a nondegenerate C*-algebra on a Hilbert

LH(A)

= {x

E B(H)

RH(A) = {x E B(H)

I xA c I Ax c

A}, A},

and

MH(A)

= LH(A) n RH(A).

LH(A) is called the set of left multipliers of A on H; RH(A) is called the set of right multipliers of A on H; MH(A) is called the multiplier algebra of A on H. Suppose that {d z } is an approximate identity for A. Since A is nondegenerate on H, it follows that d l -+ 1 (= 1H ) (strongly). Thus

LH(A),RH(A),MH(A) Clearly, A C MH(A), and A = M H (A)

¢:::::>

A"

C

= A".

1 E A.

Definition 2.12.3. Let A be a C*-algebra. A linear map p : A called a left (or right) centralizer, if

p(xy) = p(x)y

(or

= xp(y)),

-+

A is

\/x,y E A.

Proposition 2.12.4.

Let p be a left (or right) centralizer of a C*-algebra A. Then p is continuous (bounded).

Proof.

Suppose that there is a sequence {x n } C A such that 1

Ilxnll < -, n

and IIp(x n ) II

> n,

\/n.

124

Let a

Since (x~)· x~ write

< a (p

=

if P is left,

~ X~Xn,

if p is right.

I

left) or x~ X n

or Xn

where

L XnX~,

< a (p

= U n a l/ 3

right), by Proposition 2.9.2 we can

(p right),

Ilunll < lIal/611, Vn. Thus when P is left, we ha~e l 3)11·llu n < IIp(xn)11 < IIp(a / nll < Ila i / 611·llp(a I / 3)11,

when P is right, we have

Q.E.D.

This is a contradiction. Therefore, P is bounded.

Definition 2.12.5. (Ph pz) is called a double centralizer of a C'"-algebra A, if PI and P2 are two maps from A into A such that XPI{Y) = pz(x)y, Vx, yEA. Proposition 2.12.6. Let (Pb pz) be a double centralizer of a C"'-algebra A. Then PI and pz are linear; PI is a left centralizer; P2 is a right centralizer; and IlpIII = IIPzll· Proof. Let {d l } be an approximate identity for A. Since for any z , yEA and A,IL E (J}

dIPI{AX+JLY) =

pz(d,)(AX + JLY) AP2(ddx + JLpz(dl)y dl{API{X)

+ JLPI(Y)) ,

VI,

it follows that pt{AX + JlY) = API(X) + JLPI(Y). Thus PI is linear. Similarly, is also linear. Further, from

oz

we have PI{XY) = PI(X)Y, Vx,y E A. Thus PI is a left centralizer of A. Similarly, pz is a right centralizer of A. Moreover, from dIPI{X) = pz{dl)x,P2{X)dl = xpl(d l),Vl and x E A, we have IIPIII < IIp211 and Ilpzll < IlpI11 respectively. Therefore, IlpI11 = Ilpzll. Q.E.D.

125

Proposition 2.12.7. Let A be a nondegenerate C*-algebra on a Hilbert ,} space H, and {d be an approximate identity for A. Then the map p --+ x = [strongly-] Iimp(d l ) I

is an isometric bijection from left (or right) centralizer set of A onto LH(A)

(or RH(A)). Moreover, the map

is an isometric bijection from the set of double centralizers of A onto MH(A), here II (Pi, P2) II is defined by IlpI11 = IIp211· Proof. Let p be a left centralizer of A. By Proposition 2.12.4, {p( d ,)} is a bounded net of B(H). Since any bounded closed ball of B(H) is weakly compact, it follows that {p( dd} has a weak cluster point x at least. From p(d,y) = p(dl)y, it is easy to see that p(y) = xy, vv E A. If x' is another weak c1 uster point of {p (dl ) }, then we have

(x - x') y = 0,

vu EA.

Thus x = x', and (weakly)-limp(dl ) exists. I

Moreover, from p(y) = xy E

A, Vy E A, it follows that the map p --+ x is injective and x E LH(A). For any x E LH(A), define p(y)

=

xy,

Vy E A.

Clearly, p is a left centralizer of A, and x = (strongly-) lirnxdl = (strongly-) limp(d,) I

I

since A is nondegenerate on H. Moreover, from

lyE A, Ilyll < I} sup{llxyll lyE A, Ilyll < I} < Ilxll and x =(strongly)-p( dl ), we have Ilpll = Ilxll. Therefore, p --+ x Ilpll =

sup[ IIp(y) II

is an isometric

bijection from the set of left centralizers of A onto LH{A). For the right case, the proof is similar. Now let (PI, P2) be a double centralizer of A. From preceding paragraph, Xi =(strongly)-limpi(dz) exists, i = 1,2. We say that Xl = X2' In fact, for any I

y, z E A, we have

126

Thus Xl = X2' Put X = Xl = X2, then X E LH(A) n RH(A) = MH(A), and IIxll = IlpIII = Ijp211· Moreover, if X E MH(A), let Pl(Y) = xy and P2(Y) = yx, Vy E A, then (P1' P2) is a double centralizer of A. Therefore, (PI, P2) --+ x is an isometric bijection from the set of double centralizers of A onto M H (A). Q.E.D. Definition 2.12.8. For any two double centralizers (pr, P2) and (p~, p~), A, J.L E q:, let A(PI, pZ) + Jl(pi, p~) = (API + J.Lpi, AP2 + J.LP~),

(PbPZ)' (pi,p~) = (PI

pi,p~

0

0

pz)

and (Pl,P2)* = (p;,p~), where p;(a) = Pi(a+)*,Va E A,i = 1,2. Then the set of double centralizers of A is a * algebra. Denote it by D(A). Proposition 2.12.9. Let A be a C*-algebra, and {z, H} be a nondegenerate faithful * representation of A. Then M(A),MH(1r(A)) and D(A) are * isomorphic. Moreover, {z, H} can be uniquely extended to a faithful * representation of M (A); denote this extension still by {1r, H}, then

1r(M(A)) = M H(1r(A)). Proof. Let B = 1f(A). Then B is a nondegenerate C*-algebra on H. By Proposition 2.12.7, X --+ (Lx,R x) is a bijection from MH(B) onto D(B), where Lx(Y) = xy,Rx(y) = yx,Vy E B. Since (Lx·,R x·) = (R;.L;) = (Lx,R z)+ and (LxII' Rxll) = (LxL II, R,;R x) = Rx ) ' (L II, Rill, Vx, y E MH(B), it follows that MH(B) is * isomorphic to D(B) as * algebras. Clearly, D(A) and D(B) are * isomorphic. Further by Theorem 2.11.2 M(A),MH(1r(A)) and D(A) are * isomorphic. From the preceding paragraph, we can describe the * isomorphism from M(A) onto M H(1r(A)) as follows:

u.;

x E M(A)

--+

u.; R x) E D(A)

--+

(1f 0

--+

(strongly-) lim 1r

Lx 0 1r-1, 1r 0 Rx 0 l

0

1f-l)

E D( 1r(A))

Lx 0 1r- l(1r(dl ) )

(strongly-) lim1f(xd l ) E M H(1r(A)), l

where {d l } is an approximate identity for A. Clearly, (strongly)-lim 1r(xdl ) =

* representation {1r, H}

1r(xl, Vx E A. Thus the

l

of A can be extended to a faithful * representation of M(A). If this extension is denoted by {1r, H} still, then 1r(x) =(strongly)-lim 7r(xdz) , Vx E M(A), and 1r(M(A)) = M H(1r(A)). I

Moreover, since A is a two-sided ideal of M(A) and {1r, H} is a nondegenerate * representation of A, the extension must be unique. Q.E.D.

127

Example 1.

Let H be a Hilbert space, and A = C(H). By Proposition 2.12.9, M(A) is * isomorphic to B(H).

Example 2. (X). Define

Co

Let X be a locally compact Hausdorff space, and A

H

Vf

= l2(X), 1r(f)E(x) = f(x)E(x)'

e

E A, E H, x E X. Then {1r, H} is a nondegenerate faithful» representation of A. If T E M H {1r{A ))' then it is easily verified that there is a bounded function g on X such that (T~)(x) = g(x)~(x),V~ E H,x E X. Since fg E Cgo(X), Vf E Ccf'(X), it follows that g is a bounded continuous function on X. Denote the set of all bounded continuous functions on X by Cb(X). Then Cb(X) is * isomorphic to C({JX), where (JX is the Stone-Cech compactification of X. Now by Proposition 2.12.9, M(A) is * isomorphic to

C({3X). Proposition 2.12.10.

Let A be a closed two-sided ideal of a C"-algebra B. Then there is a unique * homomorphism o : B --+ M(A) such that u(a) = a, Va E A. Moreover, o is injective if and only if A is essential in B, i.e., for any non-zero closed two-sided ideal J of B, A n J =ft {O}. In particular, A is an essential ideal of M(A); and M(A) is the maximal C*algebra containing A as an essential ideal, i.e., if B is a C'"-algebra containing A as an essential ideal, then B is * isomorphic to a C*-subalgebra of M(A).

Proof. By Proposition 2.11.4, AU is a a-closed two-sided ideal of B "'. Thus there is a unique central projection z of B U such that AU = B .... z. Clearly, b --+ bz is a * homomorphism from B into M(A), and this * homomorphism satisfies our condition. Now if a * homomorphism o from B into M(A) satisfies our condition, then for any bE B, a E A, we have

ba = a(ba) = a(b)a(a) = a(b)a. Pick a net which converges to z a-weakly, then we get a(b) = bz, Vb E B. Therefore, a is unique. Suppose that a is injective. If there is a non-zero closed two-sided ideal J of B such that J n A = {O}, then J A = {O}, and a(b}a = 0, Vb E J, a E A. By M(A) c A.... and Theorem 2.11.2, we have a(b) = 0, Vb E J. It is impossible since a is injective and J is non-zero. Thus A is essential in B. Conversely, suppose that A is essential in B. If kera is non-zero, then there is a non-zero element a E Ankera. But a = a( a) = 0, we get a contradiction. Therefore, kerer = {O}, i.e., 0" is injective. Q.E.D.

128

Definition 2.12.11.

Let A be a C* -algebra.

For any a E A, define a

semi-norm

II ·rla = II . all + lIa. II, on M(A). Then the topology generated by the seminorm family {11·lla I a E A} is called the strict topology in M(A), denoted by s = s(M(A), A). Clearly, s(M(A), A) is a locally convex Hausdorff linear topology in M(A).

Proposition 2.12.12.

Let A be a C*-algebra.

1) A is dense in (M(A), s). 2) (M(A), s) is complete. Consequently, M(A) is the completion of (A, sIA).

Proof. 1) Let {dl } be an approximate identity for A. For any since xdl E A and Ilxdl

-

xlla =

x E

M(A),

lI(xdz- x)all + lIa(xd l - x) II

< IIxl!·lIdla -

all + II (ax)d l -

axil

-+

0

Va E A, it follows that xd, ~ x. Thus A is dense in (M(A),s). 2) Let {Xl} be a s-Cauchy net of M(A). Then for any a E A,{xza} and {axl} are two Cauchy nets of A with respect to the norm. Hence we have two linear maps PI and P2 in A such that

Va E A. Clearly,

apl(b)

= limaxlb = P2(a)b, 1

Va, b E A. Thus (PI, P2) is a double centralizer of A. By Proposition 2.12.9, there is a unique

X

II xl -

E M(A) such that (pI, P2) = (Lx, R x). Further, since

xll a

= II(XI - x)all + lIa(xl

- x) II

llxla - Pl(a)11 + Ilaxl - P2(a)11

-+

0,

Va E A, it follows that Xl -+ x(s(M(A), A)). Therefore, (M(A), s) is complete. Furthermore, M(A) is the completion of (A, sIA). Q.E.D.

Notes.

Propositions 2.12.9 and 2.12.12 are due to R. Busby. About further developments, see references.

References. [4]' [5], [14].

129

2.13. Finite dimensional C*-algebras Lemma 2.13.1. Let M be a finite dimensional factor on a Hilbert space H. Then M is spatial * isomorphic to B{Hn )®
Proof. A non-zero projection P of M is said to be minimal, if a projection q E M with q < P implies either q = 0 or q = p. Since M is finite dimensional, it follows that there is an orthogonal family {PI,' .. , Pn} of minimal projections of M such that E?=l Pi = 1. For each i E {1,··· ,n},Mi = PiMpi is a factor on PiH, and any projection of M, is either zero or pd = 1H ,.) since Pi is minimal. By Proposition 1.3.4, M, = (CPi. For any i,i E {1,"', n}, since M is a factor, it follows from Theorem 1.5.4 that we have either Pi -< Pi or Pi -< Pi. But Pi and Pi are minimal, thus Pi is equivalent to Pi (relative to M). Let P = PI and K = PIH. Since M p = (c1K, it follows by Theorem 1.5.6 that M is spatial * isomorphic to B{Hn )&lC1K. Q.E.D.

Corollary 2.13.2. Let M be a finite dimensional factor. Then M is isomorphic to the algebra of n x n-matrices, where n 2 = dim M .

*

Theorem 2.13.3. If A is a finite dimensional C·-algebra, then A is * isomorphic to a direct sum Lbl ffiMn;, where M n; is the algebra of ni X nimatrices, 1 < i < m. Moreover, the sequence {nl,"', n m } is a complete invariant for the algebraic structure of A, i.e., if B is another finite dimensional C"'-algebra with the associated sequence {n'l"" n ' m , } , then A and B are * isomorphic if and only if{nl,···,n m } = {n'l,···,n'm'}'

Proof. Let {1r, H} be a nondegenerate faithful * representation of A. Then 1r{A) is nondegenerate on Hand dim1r(A) < 00. By Theorem 1.3.10, 1r(A) = M is a VN algebra on H. Since the center Z of M is finite dimensional, it follows that there is an orthogonal family {Zl"" , zm} of minimal central projections of Z such that L~l z, = 1. For each i E {1,···, m}, M Zi is a finite dimensional factor on ZiH. By Corollary 2.13.2, MZi is * isomorphic to M nn 1 < i < m. Therefore, A is * isomorphic to E~l ffiMn j • The proof of the rest conclusion is easy, and is left to the reader. Q.E.D.

Proposition 2.13.4. Let G be a finite group, and R(G) = {L8EG CBS I CB E (C, \Is E G} be its group algebra. Then R(G) is isomorphic to a direct sum E~l ffiMn .. , and UG = ni + ...

n:n.

130

Proof.

Let·

-+

A. be the left regular representation of G on l2(G), i.e.,

\ls,t E G,! E l2(G). Since A; = AS-l,AsA t = Ast,Vs,t E G, it follows that [As I s E G] is a finite dimensional C*-algebra on l2 (G). Clearly, R( G) is isomorphic to [As I s E G]. Therefore, by Theorem 2.13.3 we can get the conclusion. Q.E.D. Remark.

This result is well-known in the text-book of algebras.

2.14. The axioms for C*-algebras The unit ball of a C*-algebra Proposition 2.14.1. Let A be a C*-algebra with an identity 1, S = {a E A I lI all < I} be its closed unit ball, and Au = {v E A I v*v = vv* = I} be the set of unitary elements of A. Then S = C oA u , i.e., S is the closed convex hull of Au. Proof.

Let a E A and

Iiall < 1.

Then

is invertible in A for any A E ([) and IA1 = 1. Since 00

a* (1 - aa*)-1

= a* L (aa*Y~ 00

"::::::0

L (a*a)"a* =

(1 - a*a) -1a*,

"=0 it follows that for any

!(a, A)*f(a, A)

AE ([)

and

IAI =

1,

+ 1 = (1 + Aa*)(1 - aa*)-1(1 + Aa) + 1 (1 - aa*)-1 + Aa*(1 - aa*)-1 + A(1 -

aa") -1 a + [a" (1 - aa*) -1 a + 1]

(1 - aa*)-1 + A(1 - a*a)-1 a* +A(1- aa.. )-1 a + (1 - a*a)-1.

131

Thus we can see that

f(a,A)*f(a,A)

=

f(a*,A)*f(a*,>"),

VIAl

= 1.

Further, Now let

U(A) = (1 - aa*)-1/2(A

+ a)(1 + Aa*)-l(I - a*a)1/2, VIAl <

1.

Clearly, U(A) is analytic in IA) < 1, and U(A) = AU): E Au, VIAl = 1. Moreover, u(O) = a. Thus a=

Since U(A) E Au, VI>.. I = 1, a E CoA u . Further, S = CoA u .

Q.E.D.

Theorem 2.14.2. Let A be a C*-algebra with an identity, and S = {a E A Illall < 1} be its closed unit ball. Then Co{ ei h I h* = h E A} is dense in S. Proof. Let {1ru , H u } be the universal * representation of A (see Definition 2.11.1), and A = 1ru (A)" be the enveloping VN algebra of A. If U is a unitary element of A, then we have the following spectral decomposition:

where {pin) = pe~k) - pe1r(~-l»)

I1 <

k

< n} is an orthogonal family of

projections of A, and Lk=l p~n) = 1. By Proposition 2.14.1, Co{ ei h I h * = h E A} is dense in the closed unit ball of A with respect to the uniform topology. From density Theorem 1.6.1, CO{1r u (ei h ) I h* = hE A} is strongly dense in the closed unit ball of A. Consequently, CO{1r u (ei h ) I h· = hE A} is strongly dense in 1ru (S). Now suppose that Co{e i h I h* = h E A} is not dense in S. Then we can find a E Sand f E A* such that

sup{Ref(i h )

I h· = h

E A}

< Ref(a).

132

However, from preceding paragraph there is a net {al} C Co{e ih I h* = h E A}) such that 7ru (at} ---+ 7r u (a ) (strongly). Then for any state p on A we have

Thus al ---+ a(a(A,A*)). In particular, f(al) ---+ f(a). This is a contradiction. Therefore, Co{eih 1 h* = hE A} is dense in S. Q.E.D. Let A be a C*-algebra with an identity. Then for any

Theorem 2.14.3. a E A, we have

lI all

=

inf{L: i

-

inf{I::

l.xill a = L

= hi

i

l.xill a = I:: .xjuj,

where

Uj

E A, Vi}

E Au,

Vi}.

= hi

E A,

i

g

Proof.

ihj, .xie where hi

It suffices to show the first equality. Let

Ilalll = inf {L l.xi I I a = L .xieihj, i

where hi

Vj}.

i

Clearly, lIalll > Iiali. For any a* = a E A and Iiall < 1, V± [-1,1], it follows that a(v±) i- {.xll.xl such that v± = eih± respectively, and

= a ± i(1 - a 2p /2 E Au. Since a(a) = I}. Thus there are hi: = h± E

C A

1 1 "h a=2 v++2"V_ECo{ e1 I h*=hEA}. Now suppose that x E A and IIxll < 1/2. From preceding paragraph, it follows that (x + x*) and i(x - x"') E Co{e ih I h* = h E A}. But we can write x - z" = e-!i . i(x - z"], so (x - x"') E Co{eih I h* = h E A}. Further x = !(x + z"] + !(x - x"') E Co{e ih I h* = h E A}, i.e., 1 "h 2 SeC o{ e' I h;' = where S = {a E A

! II all <

h E A},

I} is the closed unit ball of A. Hence we obtain

For any 0 i- a E A, by Theorem 2.14.2 there Co{eih I h: = hE A} such that

IS

Ilan-llall- lall ---+ O. Since II . II -- II . lib it follows that Ilan- lIall- Ialll ---+ O. Thus lIalll < lIall. Further, we have lIalll = [e], Va E A.

a sequence {an} C

Clearly,

Ilanlll <

1.

Q.E.D.

133

Proposition 2.14.4. Let A be a C"'-algebra with an identity, S = {a E A I lI a ll < 1} be its closed unit ball, and Int(S)= {a E A I lIall < I} be its open unit ball. Then we have Int(S) C Co{e i h

I h* =

hE A} C S.

Proof. Suppose that a Elnt(S). By Theorem 2.14.3, we can write a L; A;ei hi , where hi = h; E A, A; > 0, V}, and Lj Aj < 1. Then 1-

a= E

~ Ajei hi

,

C o{ei h

+

L

Ai

;

I h* = h E

1-

ei 'O +

L

A;

i 2

ei .1r

A}.

Q.E.D. Theorem 2.14.5. Let A be a C"'-algebra with an identity, B be a normed space, and 4> be a bounded linear map from A into B. Then

Jl4>11

Proof.

= sup{II(e i h ) II

I h'"

= hE

A}.

By Proposition 2.14.4, it follows that

 0, Vj, and LA; = I} i i ih sup{II4>(e ) II I h'" = h E A} < 114>11.

Q.E.D. Strictly positive elements Definition 2.14.6. Let A be a C*-algebra, and S(A) be its state space. An element a E A+ is said to be strictly positive, if cp( a) > 0, Vcp E S(A). If A has an identity, by Proposition 2.3.13 then a(E A+) is strictly positive if and only if a is invertible in A. Lemma 2.14.7. Let A be a C*-algebra, a E A+ be strictly positive, and {1r, H} be a nondegenerate * representation of A. Then 1r(a)H is dense in H.

e

e)

Suppose that there is E H with JI ell = 1 such that (1r{a)1], = 0, V1] E H. Let p(.) = (1r(')E, e). Then p E S(A). But p(a) = 0, this is a contradiction since a is strictly positive. Therefore, 1r( a)H is dense inH.

Proof.

Q.E.D.

134

Theorem 2.14.8. Let A be a C*-algebra. Then there is a strictly positive element in A at least if and only if A admits an approximate identity {dn}~=l such that dndm = dmdn, Vn, m. Proof. Suppose that {dn}~=l is an approximate identity for A, and dnd m = dmd n, Vn, m. Let a = L~=l 2- ndn E A. For any state p on A, since p(d n) --+ 1 (Proposition 2.4.4), it follows that p(a) > O. Thus a is strictly positive. Conversely, suppose that a E A+ is strictly positive. We may assume that 1 Iiall == 1. Put d.; = a~, n = 1,2,···. Clearly, dnd m = dmdn, dm > dn > 0, Ildnll = 1, Vm > n. Now it suffices to show that

Ilxd n Fix x E A+, and let Zn

-

z]

-t

0,

Vx E A+.

= x - x~ dnx 4. Clearly, Zn > Zm > 0, Vm > n. n = {p E A* I p > 0, Ilpl! < I}.

Let

It is a a(A*, A)-compact subset of A*. Let zn(P) = p(zn), Vn, P E O. Then Zn(') E C(O), Vn, and Zl(') > ... > zn(') > .... We claim that

lim zn(P) n

= 0,

Vp E O.

In fact, for any p E 0\ {O}, let {1r p' H p, ~p} be the cyclic generated by p. Then

Zn(P)

=

* representation of A

(1r p(zn) €p, €p)

(1r p(x)€p, ep) - (1r p(x 4dnx 4)ep, ep). By Lemma 2.14.7, 1r p(a)Hp is dense in H p. Moreover, 1r p(dn)1r p(a)'7 = 1rp(al+~) '7 --+ 1r p(a)11 , V11 E H p. Thus 1r p{dn ) - t 1 (strongly), and zn(P) --+ 0, Vp E O. Now by the Dini theorem, we get max{lzn(p) I I P E O} Further,

Ilznll

-t

-t

°

0 by Corollary 2.3.14, i.e. x!dnx~

IIxdn - x1l 2 =

-t

x. Therefore,

11{1 - dn)xll 2 < 411 x ll ·11(1 - dn)1/2 xl/211 2 41Ixll'llxl/2(1- dn)X 1/211 --+ O. Q.E.D.

Theorem 2.14.9. Let A be a separable C*-algebra. Then A has a strictly positive element at least.

135

Proof. Let {x n } be a countable dense subset of A+ n S and a = En 2- n x n , where S = {b E A I Ilbll < I} is the closed unit ball of A. For any state p on A, since p(xn ) > 0 for some n, it follows that p(a) > o. Thus a is strictly positive. Q.E.D.

Proposition 2.14.10.

If A has a strictly positive element, then the set of strictly positive elements is dense in A+. Proof. Let a be strictly positive. For any b E A+, (b + ~a) is also strictly positive, and (b + ~a) --+ b. Therefore, the set of strictly positive elements is dense in A+. Q.E.D.

Banach

* algebras

A is called a Banach * algebra if A is a complex Banach algebra and admits a map: x --+ z" (E A) with the following properties:

Definition 2.14.11.

(Ax + Ity)'" = Ax'"

+ fty"',(xy)'"

= y"'x*,(x*V = x,

Vx,y E A, A,1t E (C. The * operation on A or A itself is said to be hermitian, if for any z" = x E A, its spectrum u(x) C JR. x E A is said to be positive, denoted by x > 0, if z" = x and u(x) c JR+ = [0,00). Moreover, a > b if (a - b) > O.

Lemma 2.14.2. Let A be a Banach * algebra with an identity, and B be a maximal abelian * subalgebra of A. Then B is closed, and uB(b) = O'A(b) , Vb E B. Proof. It is easily verified that B is closed. Now suppose that b E B, A E (C and (b-.\)-l exists in A. Since {(b- A)-I, (b* - A)-l,B} is commutative and B is maximal abelian, it follows that (b - A)-l E B. Therefore, for any b E B we have O'B(b) = uA(b). Q.E.D.

Lemma 2.14.13. Let A be an abelian semi-simple Banach an identity. Then the * operation is continuous automatically.

* algebra with

Proof. Suppose that 0 is the spectral space of A. For any p E 0, define p{a) = p(a*), Va E A. It is easy to see that p E o. Now let {x n } C A and x, yEA be such that

Ilxn

-

xii

--+

0,

and Ilx~

- yll

-t

o.

136

Then for any p E fl,

Ip(X - y*)1 < Ip(xn - x)1 + Ip(x n - y*)1 -

Ip(xn

< Ilxn

-

x)1 +

-

lp(x~

- y)1

xII + IIx~ - yll

~ 0,

i.e., p(x - y*) = O. Since A is semi-simple, it follows that x = y*. Thus the * operation is a closed linear operator on the real Banach space A. Further, the * operation is continuous. Q.E.D.

Theorem 2.14.14. Let A be a Banach * algebra with an identity, a E A, a > 0, and a be invertible in A. Then there is u E A such that: 1) u > 0 and u is invertible in Aj 2) u 2 = a; 3) if B is any maximal abelian of A, and a E B, then u E B too.

* subalgebra

Proof. We may assume that II all < 1. Thus v(1 - a) < 1, and there is E: E (0, I) and a positive integer no such that 11(1 - a)nW/n < 1 - E:, "In > no. Since the complex function 00

(1 + z)1/2 =

L

AnZ

n

n==O

is analytic in

Izl < 1, it

follows that the sequence k

{ak

= LA n(a -

l )n I k

=

0,1,···}

n::::O

is convergent. Suppose that

where u· = u, v·

= v.

Then we have

(u

+ -iV)2 = a.

(1)

Since a· = a, it follows that

(2)

uv = -vu.

Now let B be a maximal abelian * subalgebra of A and a E B. Clearly, ak E B,Vk. From Lemma 2.14.12, we have (u +iv) E B. So it is obvious that

u, v E B,

(3)

and uv = vu.

Summing up (1), (2), (3), we obtain

a

= u2 -

v2,

U,

v E B, and uv =

o.

(4)

137

Let R be the radical of B. Clearly, R* = R. Hence B / R is an abelian semisimple Banach * algebra. By Lemma 2.14.13, the * operation is continuous on B / R. Suppose that b -+ b = b + R is the canonical map from B onto B / R. Then * "-' ,.....,., (ak - u) = (ak - u) ~ (iv) = (£v)*. ~

~

Thus v = 0, and v E R. If O E O'(u), then by Lemma 2.14.12 there is p E n(B) such that p(u) = 0, where n(B) is the spectral space of B. Since v E R, so p(v) = O. Then p(a) = p(u2 - v 2 ) = 0, and 0 E O'B(a). This is a contradiction since a is invertible in A and O'B(a) = O'A{a). Therefore, u is invertible in A, and u- 1 E B. So from (4) we can see that v = u-1uv = 0, and a = u2 • Finally, for any p E O(B), by Lemma 2.14.12 we have that A = p(a) E (0,1). Then k

p(ak) =

L An(A -

I}"

-+

(1 + (A - 1)P/2 = A1 / 2 > 0,

n=1

and p(u) = limj, p(ak) > O. Again by Lemma 2.14.12, O'(u)

C lR+,

i.e. u > O.

Q.E.D. Theorem 2.14.15. Let A be a hermitian Banach * algebra. Then A+ = {a E A I a >O} is a cone, i.e., if a.b E A+, then (a + b) E A+. We may assume that A has an identity. First step. To show the following inequality:

Proof.

lI(x) < 1I(X"X)1/2,

\/x E A.

In fact, fix x E A and e > 0, and let y = (lI(x*x) + g)-1/2 x . Then lI(Y*Y) < l. Since the * operation is hermitian, it follows that (1- y*y) > 0, and (1- y*y) is invertible in A. By Theorem 2.14.14, we have an invertible element w of A such that w > 0 and w 2 = 1 - y*y. Notice the equality:

(1 + y*)(1 - y)

= w(1 + w-1(y* -

y)w-1]w.

Since 0'( iw- 1 (y* - Y )w- 1 ) C lR, it follows that the right side of above quality is invertible. Further, (1 - y) has a left inverse. Suppose that v(y) > 1. Pick AE O'(y) such that IAI = lI(Y). Since lI(Y*Y) < 1, it follows that (1 - ~AI-2y*y) is positive and invertible. Similar to the preceding paragraph, (1 - A-ly) has a left inverse. Let z be the left inverse of (y - A). Since A is a boundary point of O'(y)' we can pick a sequence {An} of regular points of y such that An -+ A. Then Il(y - An)-lll -+ 00, and 1=

Ilz(y - A)(Y - An)-lll·/I(y - An)-lll-l

lIz + (An - A)Z(Y - An)-lll·ll(y - An)-lll-l

< IIzll·ll(y - An)-lll-l + IAn - AI.lI z l1

-+

o.

138

This is impossible. Thus v(y) < 1, and v(x) < (v(x·x) arbitrary, we have v(x) < v(x*x)l/2, "Ix E A. Second step. To prove that v(hk) < v(h)v(k),

Vh*

+ c)l/2.

Since

E:

is

= h, k" = k E A.

In fact, from the first step we have v(hk)2 <

v(kh 2k) = lim II(kh2k)nll* n

lim Ilk(h2k2t-lh2kll* < v(h 2k2). n Generally, we have v(hk) < v(h2nk2n)1/2n < Ilh2nlll/2n ·\Ik2nlll/2n.

Let n - t 00, we obtain v(hk) < v(h)v(k) immediately. Finally, we return to prove the theorem. Let a, b E A+ . Notice that 1+a+b=

-

(1 + a)( 1 + b) - ab

(1 + a)(1 - uv)(1

+ b),

where u = (1 + a)-la,v = (1 + b)-lb. Clearly, v(u) < 1,v(v) < 1. By the second step, v(uv) < 1. Thus (1 + a + b) is invertible, i.e., (-1) tJ. 0'(a + b). For X > 0, similarly we have (-1) rf. 0'(* + f), i.e., -,X rf. O'(a + b). Moreover, O'(a + b) c JR since A is hermitian. Therefore, (a + b) > o. Q.E.D. Theorem 2.14.16. Let A be a Banach and only if a*a > 0, Va E A.

* algebra.

Then A is hermitian if

Proof. Suppose that a*a > 0, Va E A. If there is h'" = h E A such that a(h) ¢- lR, then 0'(h2) ¢- IR+. This is a contradiction since h 2 = h"'h > o. Thus A is hermitian. Conversely, suppose that A is hermitian. We may assume that A has an identity. Suppose that there is x E A such that

(; = inf{'x I ,X

E O'(x*x)} < O.

Replacing x by J.lX (some J.l > 0), we may assume that {; E (-1, y = 2x(1

31 ) .

Put

+ X*X)-l.

Then 1 - v'v = (1 - x*x)2(1 + X*X)-2 > 0, and O'(Y*Y) C (-00,1). Write y = h + ik, where h* = h, k* = k. By Theorem 2.14.15,

139

and a(yy·) C [-1,00). Since a(y*y)\{O} = a(yy*)\{O} (see Lemma 2.2.6), it follows that a(y*y) C [-1,1]. From 8 E a(x*x) and y*y

= 4x*x(1 + x*x)-2, we have 48/(1 + 8)2 E a(y*y).

Thus

1{1~86)21 < 1,

s.e.,

6181 < 1+8 2 • Further 181 < 1/3 since 1+62 < 2.

Then we obtain a contradiction since 6 E (-1, -1/3). Therefore, we have x·x > 0, Vx E A. Q.E.D.

C·-equivalent algebras Definition 2.14.17. Let A be a Banach linear functional p on A is called a state, if

p(l) = 1,

and p(a) > 0,

* algebra

with an identity. A

Va E A+.

If A is also hermitian, then for any h* = h E A, p(h) E JR since Ilhll +h > O. Further p(a*) = p(a), Va E A. Moreover, by Theorem 2.14.16 we have also the Schwartz inequality:

Lemma 2.14.18. and h* = h E A. 0'(h)},A2 = max{1t

Proof.

Let A be a hermitian Banach * algebra with an identity, Then for each A E [AI, A2], where Al = min{1t I It E I JL E a(h)}, there is a state p on A such that p(h) = A.

On the linear subspace [1, h] of A, define

p(a

+ /3h)

= a

+ /3A,

Va,/3 E (E.

Suppose that o:+/3h > 0 for some a,/3 E (E. In particular, a,/3 E JR. Then the real number (0: + /3A) is between (a + /3Ad and (a + /3),.2)' Since (a + /3Ai) E u(a+/3h), it follows that (a+/3Aj) > O,i = 1,2. Thus (a+/3A) > O. This means that p is a state on [1, h]. Now by Theorem 2.14.15 and the fact that the * operation is hermitian, and by a similar proof of Proposition 2.3.11, p can be extended to a state on

A. Q.E.D. Lemma 2.14.19. Let Ll be a compact subset of each A E (C, we have

(E,

and 0 E

~.

Then for

140

Proof.

Since IA

+ III >

max{IA

11l1- IAI, it follows that

+ III I Il

E ~}

> max{11l11 Il

E il} - IAI·

In addition, by 0 E il we have 2 max{IA + III I Il E il} max{IA + III I Il E il} > l(max{11l11 Il E il} + IA1).

Lemma 2.14.20. Let A be a Banach positive constant K such that Kllhll Then the

* operation on

< v(h),

* algebra.

Vh·

> 21AI. Therefore, Q.E.D.

Suppose that there is a

= h E A.

A is continuous.

Proof. Let H = {a E A I a· = a}. It suffices to show that H is closed. For any h E H, there is a sequence {h n } C H such that II hn - h II ---+ O. Then for each E: > 0, we have that v(h} + E: > v(h n ) > Kllhnll if n is sufficiently large. Hence we get v(h) > Kllhll, Vh E H. (1) Now let {h n } C H, hn ---+ k, and k· = -k. Since (hm (hm + hn )2 ---+ (k + hn )2 as m ---+ 00, it follows from (1) that

K11(k

+ hn )211 < -

IIk 2 + h~1I

=

+ hn )2) = v(k + hn )2 v((k + hn}·)2 = v(h n - k)2

!11(k

-

kl1 2

---+

0,

+ hn )2 + (k -

as n

---+ 00.

hn )211

< ~(II(k + h n )211+ Ilk - h n 112) Further, 21lk211 have

Hand

v((k

< Ilhn Thus

+ hn )2 E

< 11k2 + h~11 + IIk 2 -

h~lj

o = II k 211 > v(k 2 )

---+

---+

O.

0, so lIk211 = O. Again by (1) we

= V(k)2

>

K211k1l2,

Q.E.D.

and k = O. Therefore, H is closed.

Definition 2.14.21. A Banach * algebra (A, II·ID is said to be C·-equivalent , if there is a new norm II . IlIon A such that II . lit 11 . II and (A, 1\ . Ih) is a C·-algebra. f"'J

Theorem 2.14.22. Let A be a hermitian Banach a positive constant K such that

Kllh\l < v(h),

Vh·

* algebra.

= h E A,

If there exists

141

then A is C*-equivalent. Proof. If A has no identity, then we consider the Banach * algebra A +(7;'. Clearly, (A +(7;') is still hermitian. Suppose that (h +.x) is a self-adjoin element of (A+
v(h

+ A)

=

max{IJL

+.xl 1JL E a{h)}

> l(v(h) + l.xl) > If(llhll + l.xl)

If(l/h + .xII)· Thus, we may assume that A has an identity. By Lemma 2.14.20, there is a positive constant M such that VaE A.

Let p be a state on A, and L p = {a E A I p(a*a) = O}. By the Schwartz inequality, L p is a left ideal of A. Suppose that a ---+ ap = a + L p is the canonical map from A onto AI L p , and define an inner product on AI L p : Va,b E A. Denote the completion of (AI L p , (,)) by H p • For any a E A, define a linear map 1I'"p{a) on AI L p:

Vb E A. For any e > 0, by Theorem 2.14.14, there is u* = u E A such that

Ila*all + e - a"'a = u2 • Then by Theorem 2.14.16, b*(lla*all + e - a*a)b = (ub)"'(ub) > O,Vb Hence, I/a*allp(b*b) + ep(b*b) > p(b*a*ab). Let e ---+ 0+, then we get 1I11'"Aa)b p 1l 2

E A.

= p(b"'a*ab)

< II a a II . 1/ bp 11 2 < M211 a 11 2 • II bp11 2 , 01<

Vb EA.

So 'lrp(a) can be uniquely extended to a bounded linear operator on H p, still denoted by 'lrp{a). Clearly, Va E A, and {11'" p, H p } is a * representation of A. Let S{A) be the state space of A. Construct the universal of A:

H --

"""" LpES{A}

ffi Q]

H p,.

* representation

142

and let Ilalll = 111I'"(a)ll,Va E A. Suppose that there is a E A such that 11'" (a) = O. Write a = al + ia2' where ai = aI, a; = a2' Then 1I'"(ad = ?T(a2) = O. In particular, p(al) = p(a2) = 0, Vp E S(A). By Lemma 2.14.18, v(aI} = v(a2) = O. But v(aj) > Kllajll,j = 1,2, so al = a2 = O. Therefore, II· 111 is a norm on A. Clearly Ila*alh = Halli, Va E A. Now it suffices to show II . II '" II . lit on A. Obviously, I . Ih < MIl· II· Moreover, let {an} C A and Ilanlh - t O. We may assume that a~ = an since Ila~lh = Ilan ll ll Vn. By Lemma 2.14.18, sup{111I'"p(a n) I

Ilanlh =

> Further, \lanll

-t

I p E S(A)}

sup{lp(a n)! I p E S(A)} = v(a n }

>

Kllanll,

Q.E.D.

0 too. Therefore, 11 ·11 '" II . Ih on A.

Theorem 2.14.23. constant K such that

Let A be a Banach

* algebra.

Vn.

If there is a positive

Klla*all > lIa*II'llall for any normal element a of A [i.e. a*a

Proof.

= aa*), then A is C"'-equivalent.

For any h* = h E A, by the assumption Kllh211

>

Ilh112. Generally,

we have

K2n-Illh2n II > Il h l12n ,

Vn.

Thus Kv(h) > IIhll,Vh* = h E A. Now by Theorem 2.14.22, it suffices to prove that the * operation on A is hermitian. Let h* = h E A. By Lemma 2.14.20, the * operation is continuous on A. Thus f(th ) = e'°th - 1 = E~=l (ith)n n! is a normal element of A, and f(th )'" = f( -th), Vt E JR. Then from preceding paragraph and the assumption, we have

KV(2 - eith - e- ith) =

Kv{f(th)* f(th))

> Ilf(th)* f(th)11 > K-1llf(th)*II'llf(th)11

> K- Iv(f(th))2,

Vt E JR.

Let (3 = nlax{IIm.A1 I .A E u(h)}. Since u{h) = u(h), it follows that there is a E JR such that (a ± i(3) E u{h). Then for t > 0, 2(1

+ e13t ) >

V(2 - eith - e- ith)

>

K- 2 11 _ eit(a- i 13 )12

> K- 2v(f(th))2

K- 2(1 + e213t - 2e 13t cos at).

143

This is impossible if fJ > 0, so it must be that * operation on A is hermitian.

fJ

= 0, i.e.,

C1(h) C JR, and the Q.E.D.

The axioms for C·-algebras Let A be a Banach * algebra with an identity. If there is a positive constant K such that lIeihll < K, \::Ih* = h E A, then A is C·-equivalent. Moreover, if K = 1, then A itself is a C*-algebra.

Theorem 2.14.24.

Proof. First step. To show that A is hermitian. Suppose that h" = h E A, and (0: + ifJ) E C1(h), where 0:, f3 E JR. By C1(h) = a(h), we may assume that fJ < O. Then for any t > 0, K

> 11ei t hl1 > le i t (a +i,8) 1 = e:",

Thus fJ = 0, and C1(h) C JR. Second step. To prove that inf{ II h 21! I h·

= h E A, II h II = I} = e >

O.

= hand Ilhll = 1. Put Ilh211 = 'fl. Clearly 0 < 'fI < 1, and 2n 2n n 1 IIh li < II h211 = 'fin, Il h 2n+ 11 < IIh l! < 'fin, \::In.

In fact, let h*

Let 0 = 'fIl/3. For any n II h2nli

Thus llhnll

> 1, IIh2n+ 1 11

< o3n < o2n,

< on,\::In > 2.

< o3n < o2n+l.

Suppose that t > 0. Then 00

K

>

Ileithil

> ti.e., K

+ et 6 > t.

Pick t = K

>

Ilthll - 1 -

n 1- ~ L-, t

+ 2,

tnllhnll/n!

n=2

2- sn

n=2 n.

L

t6 > - t- e ,

then e6 (K+2) > 2. Therefore,

c: > ((K + 2)-11n2)3 > O. Third step. We claim that

v (h)

> s II h II , .~

Vh· = h E A,

where e is as in second step. In fact, from second step we have II h 211 > ellhl1 2 for any h· = h E A. Generally, IIh2n li > c:2n-l1IhI12n. Therefore, v(h) > ellhll, Vh· = h E A. Now by first, third steps and Theorem 2.14.22, A is C·-equivalent.

144

If K = 1, consider identity map I : (A, II· lit) -+ (A, 11·11), where 11·111 ,..., 11·11 on A, and (A, 1I·lld is a C*-algebra. By Theorem 2.14.5, we have that 11111 < 1, i.e., Iiall < lIall l lVa E A. Suppose that there is ao E A such that Ilaoill > Ilaoli. By Proposition 2.1.8, v(a~ao)

<

Ila~II'llllaoll

<

Ila~111 '1laolll = lIa~aoll = v(a~ao).

This is a contradiction. Therefore, !Iall

=

Q.E.D.

Iiallt, Va E A.

Lemma 2.14.25. Let A be a Banach * algebra with an identity, and Ila*all = Ila"ll . lI all for any normal element a E A. Then A is a C·-algebra. Proof.

Let h:

=h

E A, and un(h)

=

'to ;1 {ih)k,

'
Clearly, un{h) is

normal, and un(h)· = un(-h). Then

Let n -+ 00, we get lIe i h!l·lle- i h II = 1. By Theorem 2.14.23 and the assumption, A is C*-equivalent. In particular, the * operation is hermitian. Thus u(h) C JR. Further, we have Ileihll > 1, Ile-ihil > 1. Therefore, lI~ihll = 1, Vh· = n e A. Now by Theorem 2.14.24, A is a C*-algebra. Q.E.D.

Theorem 2.14.26. Let A be a Banach * algebra. If Ila*all any normal element a E A, then A is a C* -algebra.

=

lIa*II'llall for

Proof. By Lemma 2.14.25, we may assume that A has no identity. By Theorem 2.14.23, A is C*-equivalent. Suppose that II '11' is a norm on A such that 11·11 ,..., II '11' and (A, II '11') is a C·-algebra. By the assumption, we have Ilhll = v(h) = IIhll', Vh* = n e A. In particular, IIddl = Ilddl',Vi, where {d ,} is an approximate identity for (A, II . II')· We say that for any a E A I

lI all =

sup{lI a bll

I bE A, Ilbli < I}.

Indeed, since 11·11' and 11·11 are equivalent, it follows that Ilad, - all -+ O. Then

I b E A, II bll < I} > lI addl I s e A, II bll < I}.

lIali > sup{llabl! Thus lIall = sup{llabll On A define

+0:,

Iia + All = sup{llab + Abll

-+·lI ali.

I s e A, IIbll < I},

145

Va E A, A E (C. Suppose that for some a E A and A E (E we have ab + Ab = 0, Vb E A. Since A has no identity and (A, 11,11') is a C"'-algebra, it follows from Proposition 2.1.2 that a = and A = O. Thus (A+(E, II·ID is a Banach * algebra with an identity 1, and (A, II ·11) is a Banach * subalgebra of (A+(E, II ,11). We need to prove that (A+(E, II . II) is a C"'-algebra. By Theorem 2.14.24, it suffices to show that

°

Vh'" = hE A. Fix h* = h E A. By the definition of the norm on (A+(C) there is a sequence {bn } c A with Ilbnll < 1, \;In, such that lIeihll = lim Ileihbnll. n Let B be the closed * subalgebra of A generated by {h, bn.b~ I n}. Then (B, 11·11') is a separable C"'-subalgebra of (A, 11·11'). If B has an identity p, by Lemma 2.14.25 then (B, II . II) is a C"'-algebra. Thus

('h)i

= lim II (p + L 00

Ileihll =

lim 11 eihb n II n

< lip +

i::=l

n

f

(i~,); II

_z.-, )bn II J.

< 1.

J.

i;;;;l

Now suppose that B has no identity. On B+(E, define

+ Acll (B-t-(E, II· lid is a

lib + AliI = sup{lIbc

IcE B,

llel! <

I},

Vb E B, A E (C. Similarly, Banach * algebra with an identity 1, and Ilbll = IIbllllVb E B. By Theorem 2.14.9, (B, 11·11') has a strictly positive element a. And by the proof of Theorem 2.14.8, {d n = (a/llall')~}n is an approximate identity for (B, 11'11'). Since II b + AliI "Ie E Band IIcll

< 1,

> II( b + A) dn II > II (b + A) dn c II

--t

II (b

+

.A) c II

it follows that lib + .AliI

=

lim lI(b n

+ A)dnll,

Vb E B, A E ~. In particular, Ile-ialil ·lleialll =

lim !Ie- i a dn ll '1leiadnll n lim II (e dn )'" (e dn ) II n ia

-

lim Ild~11 n

ia

= lim Ild~1I' = n

1.

Moreover, since u(a) c JR and lIe±ialll > v(e±ia) = 1, it follows that Ile±iall 1 = 1. By Theorem 2.14.9 and Proposition 2.14.10, the set of strictly positive

146

elements is dense in (B,

IIe'b and

II . II

'" II .

1 -

11

'11')+. Noticing that

e'b> IiI <

:!

~ Ilb~ - b~lI, Vb!> b2 E B

II', we have Ile±iblh = 1,

Vb E (B, II .

11') + .

Now we come back to consider h. Write h = h; - h: where h± E (B, and h ; . h: = O. Since

1

= Ileih+lh = lIeih .eih-Ill <

Ilihlll

< Ileih+11t ·lle- ih-lll

11·11')+,

= 1,

it follows that lIeihll! = 1. Further, lIeihll =

lim lIeihbnll n

Ileihll!

=

< sup{lleihblll bE B, IIbll < I}

1.

Q.E.D.

This completes the proof.

The Submultiplication of a linear C* -norm Definition 2.14.27. Let A be a * algebra over the complex field (J:. A norm II . II on A is called a C* -norm, if it satisfies the following conditions: i) (submultiplication) Ilxyll < Ilxll ·IlYII, Vx, yEA; ii) (C* -condition) Ilx*xll = Ilx112, Vx E A. Clearly, the completion of (A, II . II) is a C*-algebra if II . II is a C*-norm on

A.

Now let II . II be a norm on A. If it only satisfies the above condition i.e., Ilx*xll = I x11 2 , Vx E A, then II . II is called a linear C* -norm on A.

ii),

Theorem 2.14.28. Let A be a (complex) * algebra, and II . II be a linear C*-norm on A, such that (A,II . II) is a Banach space. Then (A, II ·ID is a C*algebra, i.e., II . II satisfies the submultiplication (1lxyll < Ilxll . Ilyll, Vx, yEA) automatically.

Proof

1) We claim that Vx,y E A.

In fact, for x, yEA and numbers A, j.L > 0, noticing that

(Ay*

+ j.LX)(Ay + J-lx*)

2AJ-l(Y*X*

+ xy),

- (AY* - j.LX)(Ay - j.Lx*)

147

and (AY*

+ iP.X)(Ay -

i2AP.( -y* z"

-

ip.x*) - (AY* - iP.X)(Ay + ip.x*)

+ xy),

and by the C*-condition, we have

4Ap.llxyll

+ p.x*11 2 + IlAy - p.x*ll 2 + II AY - z"p.x· 11 2 + II AY + ip.x*1I 2 < 4(Allyll + p.lIx*I))2.

<

IIAY

Let A = IIYII- 1,p. = Ilx*1I- 1, then we get

\:Ix, yEA.

2) We prove that \:Ix, yEA. In fact, if h* have

=

h E A, then IIh211 = IIhll 2 by the C*-condition. Generally, we

Thus for any a E A, by 1) lIa*al12n =

II(a*ap"ll

=

n

lI a*(aa*)2

- l

all

< 4211a1l2 ·11(aa*)2"-111, \:In. 2k Noticing IIh II have that

= lI hl1 2k(\:Ik = 1,2,·, .,

II (aa*) 2"-111

and h*

= h E A)

and the claim 1), we

=

II (aa*) 1+2+ ...+2,,-111

<

4n-11Iaa*II'II(aa*)211" 'l!(aa*)2

_

4 n- 1 1I aa* 11 2 "- 1, \:In.

n- 111

Thus Ila*aI1 2"

<

4n+lllaW~ ·lIaa*112"-I,

\:In.

Further, 11a*all < Ilaa*ll, \:Ia E Aj and lIa*all = lIaa*ll, \:Ia E A. By the C*condition, IIxll2 = Ilx*xll = Ilxx*1I = Ilx*1l2, i.e., Ilx*lI = Ilxll,\:Ix EA. Again by 1), II xy II < 411 x Illl y II, \:Ix, yEA. Now define a new norm II '11' = 411·11 on A. Then by 1), 2)' (A, II ·11') is a Banach * algebra, and Ilx*1l' II xli' ,\:Ix E A. Moreover, by the C*-condition

=

\:Ia E A.

148

From Theorem 2.14.23, (A,

11·11') is C*-equivalent, i.e.,

II . II -- II . II' = I. I on A, where lal = v(a*a)1/2(Va E A) is the C*-norm on A. Furthermore, by the C*-condition of II . II lal 2 = v(a*a) =

I2-n n 2n limll(a*a) I1 2- = Ila*all = ll a11 2 , liml(a*a) n

2n

n

\fa E A. Therefore, (A,

11·11) is a C*-algebra.

Q.E.D.

Notes. In 1943, I.M. Gelfand and M. Naimark proved the following theorem: If A is a Banach * algebra with an identity, and satisfies: 1) lIa*all =

Ila*11 ·llall, \fa E Ai 2) lIa*11 = !I all , \fa E Ai 3) (1 + a*a) is invertible, \fa E A, then A is isometrically * isomorphic to a uniformly closed * operator algebra on some Hilbert space. This is a fundamental theorem for the theory of C*-algebras. Also, they conjectured that the conditions 2) and 3) are not necessary. I. Kaplansky pointed out that the condition 3) can be canceled easily. Then we have the usual definition of C*-algebras (2.1.1) and the Theorem 2.3.20. Proposition 2.14.1 is due to J. Glimm and R.V.Kadison. Using this proposition, they answered affirmatively the Gelfand-N aimark conjecture for unital case. Theorem 2.14.14 is due to J.W.M. Ford, Theorem 2.14.15 is due to V.Ptaki and Theorem 2.14.16 is due to S.Shirali and J.W.M. Ford. These results also have their own interest in the theory of Banach algebras. Theorem 2.14.22 and 2.14.23 are due to R.Arens. Theorem 2.14.24 is due to B. W. Glickfeld. And Theorem 2.14.26 is due to G.A. Elliott. The question on the submultiplication of a linear C·-norm was presented by R.S. Doran. Theorem 2.14.28 is due to H. Araki and G.A. Elliott. References. [6], [7], [30]' [41], [48], [50], [52], [53], [56], [66]' [87], [90], [131], [162], [191], [201].

2.15 Real C*-algebras Definition 2.15.1. Let A be a real Banach * algebra (, i.e., a Banach * algebra over the real field lR ) .A is called a real C* -algebra, if A c = A-t-iA can be normed to become a ( complex ) C*-algebra such that the original norm on A remains unchanged.

149

Proposition 2.15.2. A real Banach * algebra A is a real C"'-algebra if and only if A can be isometrically * isomorphic to a unifomly closed * algebra of operators on a real Hilbert space. Proof. Let 11'" be an isometrical * isomorphism from A into B(H), where H is some real Hilbert space. Consider the complex Hilbert space He = H +iH, and define Iia + ibll = 1111'"(a) + i1r(b)IIHc ' "Ia,b E A

It is easily verified that A c = A-t-iA is a ( complex) C"'-algebra by this norm. Moreover, it is obvious that Iiall = 1I1r(a)lI = 1111'"(a)II Hc'''Ia E A. Therefore, A is a real C·-algebra. Conversely, let A be a real C* -algebra. Then A e = A -t-iA is a ( complex ) C"'-algebra. We may assume that A, C B(He) , where He is a complex Hilbert space. Consider He = H as a real linear space, and define

*

Then H is a real Hilbert space, and A is a uniformly closed operators on H.

Proposition 2.15.3.

Let A be an abelian real C"'-algebra. isometrically * isomorphic to c~(n, -)

= {I

E C~(n) I/(l)

algebra of Q.E.D. Then A is

= l[ij, "It EO},

where 0 is a locally compact Hausdorff space, and bar "-" is a homeomorphism of 0 such that t = t, "It E O. Proof.

Assume that A c

=

A-t-iA "" C~(O), and for each tEO define

l(a + ib) = a(t)

+ ib{ij,

"la, s e A,

where x -+ x(·) is the Gelfand tra~sformation from A onto Cg' (0). Clearly, t -+ l is a homeomorphism of 0; l = t, "It E 0; and {a(')la E A} c {I E C~(O)I/(l) = l[ij, "It EO}. Now let I E CO'(O) with I(l) = l[ij, "It E n. Suppose that 1(,) = a(·) + ib(·), where a,b E A. Then

a(t) - ib{ij = (a + ib)(t) = l[ij

= I(l) Hence, b(t)

=

= a(l)

+ ib(l)

= a(t)

+ ib(ij, "It E

0, "It E O,b = 0, and 1(') = a(.).

Now let A be a real C·-algebra, and define

J(a + ib) = a - ib, "la, b E A.

O.

Q.E.D.

150

Then J is a conjugate linear isometric algebra A c = A +iA, i.e., J(oXx

* isomorphism of the

( complex) C·-

+ J.LY) = >"Jx + j.lJy, Lz" = (Jx)*, Jxy = Jx· Jy, J2 = id, II J xii = Ilxll, 'rIx, y E A c , oX, J.L E q}.

In fact, it suffices to show that J is isometric. We may assum that A C B(H)' where H is a real Hilbert space. Since for any a, b E A, C,,.., E H,

IlaE - b1711 2+ JIbE + a,.., 11 2 = II(a-ib)(E-i,..,)112,

II(a + ib)(E + i17)11 2

=

and liE + i1711 = liE - i1711 = IIEI1 + 11,..,11 , we have IIJxl1 = Ilxll, 'rIx E A c • Conversely, let J be a conjugate linear isometric * isomorphism of a ( complex ) C* -algebra A c , and A = {x E A c IJ x = x}. Then it is easy to see that A is a real C*~algebra, and A c = A-t-iA. Therefore, we have the following 2

2

2

Proposition 2.15.4.

There is a bijection between the collection of all real C*-algebras and the collection {(A c, J) lAc is a complex C*-algebra, J is a conjugate linear isometric * isomorphism of A c } .

Proposition 2.15.5.

Let A be a real C·-algebra, and define

I(a + ib) = I(a)

+ if(b), 'rIa,b E

Then A· can be embedded isometrically into where A c = A+iA.

Proof. We may assume that A

C

A;

A, I EA·. such that

A;

B (H), where H is a real Hilbert space. Since

I( E All<) can be extended to a linear functional on B(H) with the same norm, we may also assume that A = B(H) and {t,} C B(H)* = T(H) such that

Iltdll < 1, 'rIl;

and fl(b)

IIIII =

= tr(t1b)

-t

1. Then there exists a net

I(b), 'rib E B(H).

Clearly, we have II -------+ I in topology a(B(Hc)"',B(Hc)), where H, = H+";H and t l as an element of T( Hc ) maintains its trace norm, so fl as an element of B(Hc}* maintains also its norm, i.e., < 1. Hence, I as an element of B(Hc)· has the norm one. Therefore, A * can be embedded isometrically into A;.

Q.E.D.

Proposition 2.15.6. Let A be a real C·-algebra. P E A* is called a state , if p(a·a) > O,p(a*) = p(a), 'ria E A, and llpll = 1. Denote the state space of A by S(A). Then S(A) = Re S(A c) = {Re Pclpc is a state on A c}, where

151

A c = A-t-iA is a (complex) C"'-algebra. Moreover, for each p E S(A), by the GNS construction there is a cyclic * representation {1I'"p, H p, cp} of A such that

p(a)

=

(1I'"p(a) cP' cp),

\;fa E A,

where H, is a real Hilbert space; and {11'" =

is a faithful

EBpES(A) 11'" p,

H

= EBpES(A)Hp}

* representation of A.

The proof is similar to Section 2.3.

Definition 2.15.7. Let A be a real Banach * algebra. For each x E A, let a(x) be the spectrum of x in the (complex) algebra A c = A+iA, and v(x) = max{I.\II.\ E u(x)} be the spectral radius of x. An element a of A is said to be positive, denoted by a > 0, if a" = a and u(a) C [0,00). A is said to be hermitian, if for each h* = h E A we have u(h) C JR; A is said to be skew-hermitian, if for each k" = -k E A we have u(k) C iJR; A is said to be symmetric, if for each a E A we have a' a > o. Theorem 2.15.8. Let A be a real Banach * algebra. Then A is symmetric if and only if A is hermitian and skew-hermitian. Moreover, if A is symmetric, then A+ = {a E Ala > O} is a cone, i.e., if a, bE A+, then we have (a + b) E A+. The proos is similar to Theorem 2.14.5 and 2.14.6.

Lemma 2.15.9. Let A be a real symmetric Banach identity 1. A linear functional p on A is called a state, if

p(l) = l,p(x·x)

> O,p(x*)

* algebra

with an

= p(x), \;fx EA.

Then for each h'" = h E A there is a state p on A such that p(h) = v(h). Proof. Let A H = {h E Alh* = h} and A K = {k E Alk* = -k}. Then A = AH+A K • Now for h E A H , define a linear functional p on [1, h] = {a +,Bhla,,B E JR} : p(a

Clearly, p(a)

.c =

{

> 0, \;f 0 < (E,PE)

+ ,Bh)

= a

+ ,Bv(h)'

\;fa,,B E JR.

a E [1, h], and p(h) = v(h). Let E is a linear subspace of A H , and l , h E E; } PE is linear on E, andp(a) > 0, \;f0 < a E E; , and PE(a + ,Bh) = a + ,Bv(h), \;fa,,B E JR

152

and (E,PE) < (F,PF), if E C F and PF!E = PE. By the Zorn lemma, £, contains a maximal element (E,PE) . From Theorem 2.15.8 and Proposition 2.3.11 , it is easy to see that E = A B . Further, define p(A K ) = {O}. Then this P will satisfy our conditions. Q.E.D.

Theorem 2.15.10. Let A be a real hermitian Banach 2, Ilx·xll = Il x11 \Ix E A. Then A is a real C·-algebra.

:I<

algebra, and

Proof. By Proposition 2.1.2, we may assume that A has an identity 1. For any k· = -k, since * is isormetric, it follows that

Hence, u(k) C iJR, i.e., A is also skew-hermitian. By Theorem 2.15.8, A is symmetric. For any a E A and e > 0, by Theorem 2.14.14 ( it also holds for real case) there is b E A such that b·b

= IIal1 2 + e - a'»:

Hence, for any state P on A, by the GNS construction there is cyclic sentation {1r p, H p, €p} of A such that

Now by Lemma 2.15.9, A admits a faithful is a real Hilbert space. Define

:I<

* repre-

representation {1r, H}, where H

flxllI = 111f(x)ll, \Ix E A. Then II· lit is a norm on A, and Ilx*xlh = Ilxlli, \Ix E A. For any h· = n e A, clearly we have a(1f{h)) C a(h). Hence, IIhlll = lI(1r(h)) < lI(h) = Ilhll, \lh E A. On the other hand, for any state P on A we have

111r(h)11 > 1(1rp(h)€p, €p)1= Ip(h)l· By Lemma 2.15.9, !Ihllt = J11r(h) II > lI(h) = IIhll, \lh'" = hE A. So Ilhll I = Ilhll, \lh· = h E A. Further, IIxliI = IIx*xll~/2 = lI x*xIl 1/ 2 = IIxll, \Ix E A. Therefore, A is a real C·-algebra.

Q.E.D.

Remark. In above theorem, the hermitian condition is necessary. Indeed, consider (C with norm IIAII = IA I and :I< operation A· = A(\I AE (C), then the real Banach :I< algebra (C is not hermitian. So (C is not a real 'C·-algebra.

153

Definitioln 2.15.11. A real Banach * algebra (A, II . II) is said to be real C"'-equivalent, if there is a new norm 11·111 on A such that 11·11""" II· lib and (A, 11·111) is a real C"'-algebra. Theorem 2.15.12. Let A be a real Banach * algebra. Suppose that A satisfies one of the following conditions: 1) A is symmetric, and there is a positive constatnt K such that KlI(x) > Ilxll for each z" = z or z" = -x of A 2) A is hermitian, and there is a positive constant K such that Kllx"'x!l > Ilx'" II 'llxll for each normal x of A (i.e., x"'x = zz"}; 3) there is a positive constant K such that Kllx"'x + y*yll > Ilx*II'lIxll for any normal z, yEA and xy = yx . Then A is real C'"-equivalent. Proof. 1) By Lemma 2.14.19, we may assume that A has an identity 1. By Lemma 2.14.20, A H = {h E Alh* = h} is closed in A. Now let {k n } C A K = {k E A]k* = -k} be such that kn --+ h E A H . For any k E A K , let Zn = kk n + knk, \In. Then Zn E A H and Zn --+ Z = kh + hk. Since A H is closed, it follows that z E A H n A K = {O}. Hence,

kh+ hk = O,\lk E A K • In particular, knh + hk n = 0, \In, and h 2 = 0. But 0 = 1I(h2 ) = V(h)2 > K-2I1hI12, so h = 0, i.e., A K is also closed. Further, the * operation is continuous on A. For each state p on A, by the GNS construction there is a cyclic * representation {7r p, H p, €p} of A such that

where H, is a real Hilbert space. Further, let

{7r

= EEJ pES(A)1I"p,

H = EEJpES{A)Hp} ,

where S(A) is the state space of A. Since

* is continuous,

it follows that

117r(x) = 1I11"(x*x)f11/2 = 1I(1I"(X*X))1/2

< 1I(X*x)l/2 <

Ilx"'xII 1/ 2 < K'lIxll,

\Ix E A, where K' is some positive constant. Define Ilxllt = 117r(x)lI, \Ix E A. We say that II . 111 is a norm on A. In fact, let IIxlll = and x = h + k, where h* = h, k* = -k. Then for each state p on A we have

°

154

By Lemma 2.15.9, we can see that h = o. Further, from a = 1I"(x·x) = -11"( ~kJ2 we have k = 0, and x = O. Now it suffices to show that I . II is continuous with respect to II . 111. Let IIxnlh ~ O. Since Ilx~lll = Ilxnlh, Vn, it follows that Ilhnll l ~ 0 and IIknll l ~ 0, where = h n , k~ = -kn , Xn = h n + k«, Vn. By IIxlh > lI(x) > K-ll!xll, Vx· = ±x E A, we can see that IIhnll ~ 0, Ilknll ~ 0, and Ilxnll ~ O. Therefore, II· II is continuous with respect to II . IiI, and A is real C* -equivalent. 2) By 1), it suffices to show that A is skew-hermitian. Similar to the 1), * is continuous. For k" = -k E A, consider f(tk) = et k - 1, Vt E JR. Then f(tk) is normal and f(tk)* = f( -tk), Vt E JR. Now by the proof of Theorem 2.14.23, we can see that a(k) C iJR. 3) For h'" = h E A, let x = cos h - 1, y = sin h. Then z, yare normal and xy = yx. Hence

n;

II cos h i.e. II cos h -

111 2

< KII(cos h - 1)2 + sin 2 hll

= 2KII cos h - 111,

111 < 2K, Vh· = hE A. Consequently,

II COS(At)

- 111 < 2K, Vt

E

JR, A E a(h), and h: = h.

Therefore, a(h) C JR, Vh* = h E A, i.e., A is hermitian. Now by 2}, A is real C* -equivalent. Q.E.D. Now consider the unit ball of a real C* -algebra. Proposition 2.15.13. Let A be a real C*-algebra, S be its closed unit ball, i.e., S = {x E Alllxli < I}, S; be the closed unit ball of the ( complex ) C*-algebra A e = A+iA, and Xo E S. Then the following conditions are equivalent: 1) Xo E ExS; 2) Xo E ExSe; 3) (1 - x~xo)A(I - xox~) = {O}. In the case of Xo E ExS, z, is a partial isometry of A, i.e., x~xo and xox~ are projections of A. Moreover, S has an extreme point at least if and only if A has an identity. Proof. Let Xo E ExS. From Theorem 2.5.1 (and notice Proposition 2.15.3), we can prove that Xo is a partial isometry of A, and (1 - x~xo)A(1 - xox~) = {O}. Hence, (1 - x~xo)Ae(1 - xox~) = {O}. Now by Theorem 2.5.1, we have Xo E ExSc • The other conclusions are easy. Q.E.D.

Corollary 2.15.14. 1 in A c •

Let Xo

E ExS, and 0 i= bE A. Then we have Ilxo+ibll >

155

Proof. If lIXQ + ibll

IlxQ

+ ibll <

1.

< 1, then by Proposition 2.15.2 we can see that IlxQ - ibll = This contradicts the fact Xo E ExS e • Therefore, IlxQ + ibll > 1Q.E.D.

Lemma 2.15.15.

Let H be a complex Hilbert space, and A( C B(H)) be a real uniformly closed * algebra. If A n iA = {O}, then A c = A+iA is a( complex) C* -algebra on H, and Iia + ibll > max(llall, Ilbll), Va, b E A.

Moreover, if M is a real weakly closed * operator algebra on H, and M n iM = {O}, then Me = M +iM is also weakly closed. Proof. Since (H, Re(, )) is a real Hilbert space, it follows from Proposition 2.15.2 that A is a real C*-algebra. Then there is a norm II· III on A c such that (Ac,II·lll) is a (complex) C*-algebra, and IIxlh = IIxll,Vx EA. Clearly, the operator norm is also a C*-norm on A e • By Proposition 2.1.10, 11 . 111 = 11·11

on A c , i.e., Ae is a C*-algebra on H. Now suppose that {1r, K} is a faithful * representation of A, where K is a real Hiblert space. Then {11", K c } is a faithful * representation of A e , where K, = K -t-iK. Hence, for every a, b E A,

lIa

+ i bll = 111r(a) + i1r(b) II > sup { 1111" ( a) e + i 11" ( b) e III e E K, II ell < > max {sup 111I"(a)€II, sup 111r(b)€II} 1I€1I:9

lIell::;l

I} = max(llall, Ilbl!)·

Finally, it suffices to show that the closed unit ball of Me is weakly closed. Suppose that (al + ibl) --+ x E B(H) weakly, where aI, bl E M, Vl, and Ilal + ibdl < 1, VI. Since max(lladl, Ilblll) < 1, Vl, and the closed unit ball of M is weakly compact, we may assume that a, -+ a and b, -+ b weakly, where a, s e M. Therefore, x = a + ib E Me. Q.E.D.

Lemma 2.15.16.

Let H be a complex Hilbert space, M be a real weakly closed * operator algebra on H, and 1 = 1H E M. Then Co {ulu E M is unitary} is weakly dense in S = {x E Mlllxli < I}. Proof. Since S is weakly compact, so by the Krein-Milmann theorem it suffices

to show that Xo E Co{ u E

Mlu is unitary }W

for any X o E ExS, where "_w" means the weak closure. Now let z; E ExS. By Proposition 2.15.13, X o is a partial isometry of M, and (1 - x;x o )M (1 - xox;) = {O}. For any projection p of M, denote by c(p)

156

the minimal central projection of M containing p, i.e., c{p) is the projection from H onto [MpH]. By Proposition 1.5.9, we have

c(1 -

x~xo)

. c(l -

xox~) =

o.

Let z = 1 - c{l - x~xo). Then x~xo > z, xox~ > 1 - z: Replacing {xo, M, H} by {xoz,Mz,zH} and {x~(1- z),M(1 - z), (1 - z)H} respectively, we may assume that x~xo = 1, and xox~ = p < 1. Now it suffices to show that for any 6, ... , Em' TIl, ... ,TIm E H there is a unitary element u of M such that

We can write that

00

H = Ho ffi

L ffix~(1- p)H k=O

and let qk be the projection from H onto x~(1 - p)H, k = 0,1, ... , and q be the projection from H onto H o. Pick a positive integer n such that

II L

k>n

and let u

1 2

qkEil1 < -(1

+ l
n+l

I((xo - U)Ei,1Ji)! = I(L(xo - u)qkEi,Tli)1 k>n

< Ilxo - ull·/I L qkEill'!IT1ill < 1, k>n

1 < i < m.

Q.E.D.

Proposition 2.15.17. Let A be a real C*-algebra with an identity, and Xo be a normal element of A with Ilxoll < 1. Then z, belongs to the closure of Co{cosb· eala,b E A,a* = -a,b*

where "z ""'" y" means that xy

=

=

b,a

r'OoJ

b

r'OoJ

xo}

yx.

Proof. We may assume that A is abelian. Then by Theorem 2.14.2, Xo belongs

to the closure of i.e., there are -,;n) > 0,

L

Co{ eklk E A c , k*

= -k}

-,;n} = 1 and k}n)*

= _k;n)

E A c such that

j

/I L i

-,;n) exp(k;n)) - xoll -+ o.

157

Write k~n) = a(n) 1

1

abelian, and A c

+ ib(n) l'

where a(n)", 1

= _a(n) b(n)tIJ = b(n) 1'1 1

= A+iA, it follows that II L A)n) cos b;nl . exp( at l ) -

Xo

I

--+

E A,Vn,J·. Since A is

o.

i

Q.E.D. Corollary 2.15.18. Let H be a complex Hilbert space, M be a real weakly closed * operator algebra on H, 1 = 1H E M, and S = {x E Mlllxlj < I} be the closed unit ball of M. Then the subset

Co{cosb. eB!a,b E M,a* = -a,b'" = b} is weakly dense in S.

Proof. This comes directly from Lemma 2.15.16 and Proposition 2.15.17.

Q.E.D. Theorem 2.15.19. Let A be a real C"'-algebra with an identity, and S = {x E AllJxl1 < I} be its closed unit ball. Then

Co{cosb· eBla,b E A,a* is dense in

= -a,b* = b}

S.

Let {n , H} be the universal * representation of the complex C* -algebra A c = A-t-iA ( see Definition 2.11.1), and M = 7r(A)w. We say that M n iM = {O}. In fact, if x E M n iM, then there exist two nets {aaJ, {b p } of A such that

Proof.

7r(aa) ~ x,

7r(£bJ3) ~ x.

Hence 7r(aa - ibJ3) --+ 0 weakly. In particular, for any c, d E A and any state p on A c we have

(1l"(aa - ibJ3)7r(c) cp, 11" (d"') cp)

p(d(aa - ibJ3) c)

=

--+ O.

Since A; = A * -t-iA'" ( Proposition 2.15.5. ) and A; is the linear span of its state space, we have

f(d(aa - ibJ3)c) i.e., f(daac) we have

--+

--+

0,

Vf

E A*,

0, f(dbJ3c) --+ 0, Vf E A"'. Furthermore, for each state p on A c

(1l"(aa)1l"(C)cp,1l"(d*)ep) = p(daac) (1l"(b p)1l"(c) cp, 1l"(d*) cp)

= p(dbJ3c)

--+ --+

0, 0,

158

i.e., (X1r(c)€p,1r(d"')€p) = 0, \lc,d E A and state p on A c • By the construction of {1r, H}, we have x = o. Now we point out that 7r(8) is r-dense in the closed unit ball 8M of M. Indeed, for any x E 8 M by Theorem 1.6.1 there is a net {al + ibt} of A c such that 7r(al) + i1l"(bz) ~ x weakly, and lIal +ibdl < 1, \ll, where ai, bz E A, Vi. By Lemma 2.15.15, 1, Vl. Since SM is weakly compact, we may assume that

1r(az)

.z,

y EM,

1r(bz) ~

Z

llatll < 1, I]bdl <

E M.

y + iz, i.e., x - y = iz E M n iM = {a}. Therefore, x = y = wlim1r(al) belongs to the weak closure of 7r(S). On the other hand, 11"(8) is a bounded convex set, and its weak closure is equal to its r-closure. So 1r( S) is r-dense in SM. Moreover, we have that CO{1I"(cosb·ea)la,b E A,a'" == -a,b'" == b} is weakly dense in SM. In fact, by Corollary 2.15.18 it suffices to show that its weak closure contains cos u: e", \Ix, y E M with v: = y, z" = -x. From the preceding paragraph, there are two nets {all'}, {b~} of A such that

Hence, x

=

1I"(aa) ~ x,

1I"{b~) ~ y,

and lIaall < Ilxll, \la, Ilb~11 < IIYII, \I{3. Since * operation is continuous in T>: topology, we may assume that a~ = -aa,b; = b~,VCi.,{3. Clearly, for any polynomials P and Q,

Therefore we have

and 1r( cos b{J • ea a ) --+ cos y . e" (weakly). Now if there exists Xo E S \-3:C:;-o~{-co-s-b~.e-a"T1a-,.-.o;b'--E-A-=--,a~"'---a-, b....-:",---'b"'}, then by the Asoli theorem we can find f E A; such that Re f(xo)

> sup{Re f{cosb· ea)la,b E A,a'" = -a,b* = b}.

On the other hand, from the preceding paragraph there is a net {Xl} Co{ cos b . eala, b E A, a* = -a, b* = b} such that 11"( xt}

---t

11"( xo)( weakly) .

In particular, for any state p on A c we have

C

159

Hence, / (Xl) -+ f (xo). This is a contradiction. Therefore, C o{cos b . ea Ia l b E A, a* = -a, b" = b} is dense in S. Q.E.D. Lemma 2.15.20. Let A be a real C*-algebra with an identity 1 , and S be its closed unit ball. Then

~S C 2

Co{cos b· ealal b E A, c"

= -a, b* = b}.

Let xES and write X = h + k , where h, k E A and h* = h, k" = -k. Suppose that B is the real abelian C* -subalgebra of A generated by {I, h}. Then, B '" C(O, -L and h ~ h(t) = h(tJ = h(t), -1 < h(t) < 1, Vt E O. Clearly, arccosh(t) E C(O,-), where arccos A E [0,11"] if A E [-1,1]. Hence, there is bE B with b" = b such that b(t) = arccosh(t),Vt E 0, i.e., h = cosb. Suppose that C is the real abelian C*-subalgebra of A generated by {II k}. Then C::: C(O',-)'k ~ k{t) = -k[ij= -k(t),lk(t)1 < I,Vt EO'. Clearly, -iarc sin(ik(t)) E C(O', -), where arc sin A E [-11"/2,11"/2] if A E [-1,1]. Hence ,there is a E C with a* = -a such that a(t) = -iarcsin{ik(t)), i.e., k ea - e- a ) . Therefore, Proof.

!(

(cos b ea 1 a x = cos b + - (e - e-a) = 2 - - + 224

+

cos 11" • e4

a

)

. Q.E.D.

That comes to our conclusion. Proposition 2.15.21. for any x E A we have

Ilxll = inf Proof.

{

L ,.

Let A be a real C*-algebra with an identity. Then

X

lAil

=

L: Ai cos bj • eai , where ai, bj E A } i

.

and a; = -ai, b; = bj, Vi

By Lemma 2.15.20, we can define Ilxlh = inf{L: 1'\'-11' "}, Vx E A. i

Clearly, II ·111 is a norm on A and II . II < I . 111' Now by Lemma 2.15.20 we have Ilx/211xllllt < 1, so that Ilxll < Ilxllt < 211 x ll,Vx E A. If Ilxll = 1 , then by Theorem 2.15.19 there exists a sequence {x n } C Co{cos beala, b E A, a* = -a, b" = b} such that Ilx n - xII ~ O. Hence, Ilx n - xiiI ~ o. Since Ilxnllt < 1, Vn, it follows that IlxliI < 1. Therefore lIxlll < Ilxll, and IIxlll = Ilxll, Vx E A. Q.E.D.

160

Proposition 2.15.22. Let A be a real C*-algebra with an identity 1 , and 5 be its closed unit ball. Then {h E Alh* = h, lIhll C

<

I} U {k E Alk* =

-k,llkll < I}

Co{cosb'eala,bE A,a· = -a,b· = b}

and Int(S) C Co{cosb. eala,b E A,a· = -a,b· = b}

where Int(S) = {x E Proof.

c

5,

Alllxli < I}.

The former conclusion is contained in the proof of Lemma 2.15.20

indeed. . Now let x E Int(S). By Proposition 2.15.21 we can write x =

L

Ale cos ble . ealr. ,

Ie

where Ale

> 0, aZ = -ale, bZ = bk, Vk, and

L

Ale < 1. Then

Ie

belongs to Co{ cos b . eala, b E A, a·

= -a, b* = b}.

Q.E.D.

Proposition 2.15.23. Let A be a real C*-algebra with an identity, and <1> be a bounded real linear mapping from A to a real normed linear space B. Then lIiPll = sup{II<1>(cosb. ea)llla,b E A,a* = -a,b* = b}. Proof.

From Proposition 2.15.22 we have the right side < 11<1>11

=

sup 11<1>(x)1I < the right side. xEInt(s)

Q.E.D. Now we discuss the axioms of real C*-algebras.

Theorem 2.15.24. Let A be a real Banach there exists a constant K(> 1) such that

Va, b E A and e"

= -a, b'

* algebra with

= b, then A is real C*-equivalent.

an idenity. If

161

Moreover, if the constant K

Proof.

=

1 , then A itself is a real C*-algebra.

For any a, b E A with a* = -a, b' = b , we have

Vt E JR,.A E a(a),j.t E a(b). Hence, a(a) C iJR,u(b) C JR, i.e., A is hermitian and skew-hermitian. Similar to the proof of Theorem 2.14.24, we can show that inf{llx211lx E A,x* = x or (-x), and tJxll = I} >

o.

Hence, there exists a constant C > 0 such that

CII(X) > Ilxll, Vx E A with z" = ±x. Then by Theorem 2.15.12, A is real C*-equivalent. Now let K = 1. From the preceding paragraph, there is a new norm II . III on A such that 1\ . lit '" II . 1I and (A, II . III) is a real C*-algebra. Consider the identity map I : (A, II . Ill) -+ (A, II· II). By Proposition 2.15.23, we have 11111 < 1, i.e., Ilxll < Ilxlh, Vx E A. If there is Xo E A such that II xoll < IIxollt, then lI(x~xo) . Ilxoxolll = Ilx~lh ·lI xolll

> This is a contradiction. C*-algebra itself.

IIxoll . Il xoll >

Therefore, Ilx!!

Ilx~xoll

=

> lI(xoxo),

llxllb Vx E A, and A is a real Q.E.D.

Theorem 2.15.25. Let A be a real hermitian Banach * algebra with an identity 1. If for any normal element x of A we have Ilx*xll = Ilx*lI '\lxll, then A is a real C*-algebra. From our assumption, we have IIxll = lI(x), Vx· = ±x E A. In particular, /[ cos bll = II (cos b) < 1, Vb* = b E A. By the proof of Theorem 2.15.12.1) , the * operation is continuous on A. Hence, for every a* = -a E A we have a!l·lIe-all. 1 = v(l) = 11 111 = II [e"]" . eall = !le

Proof.

By the proof of Theorem 2.15.12.2), A is skew-hermitian. Thus, Ile±all > le±).1 = 1, V,\ E a(a), and Ileal! = 1. Then from Theorem 2.15.24 A is a real C·-algebra. Q.E.D.

Theorem 2.15.26. x E A we have

Let A be a real hermitian Banach

* algebra.

If for any

162

then A is a real C'" -algebra.

Proof. By Theorem 2.15.12, A is real C'" -equivalent, i.e., there is a new norm I . IlIon A such that II . 111 -- II . II, and (A, II . II d is a real C·-algebra. Clarly, IIxlll = Ilxli = v(x), Vx'" = ±x E A. Let {e,} be an approximate identity for A, and define Ilx + All

= sup{llxy +

Ayilly E A,

Ilyll

< I}

Vx E A, A E JR ( here we assume that A has no identity, otherwise, the conclusion is obvious by Theorem 2.15.25). Then (A+ JR) is a real hermitian Banach * algebra with an identity 1, and the original norm 11·11 on A remains unchanged ( since Iledl = Ile,llt < 1, Vl. ) Similar to the proof of Proposition 2.4.4, we have Ilx + All

= lim Ilel(A + I

x)11 = lim II(A + x)edl, I

Vx E A, A E JR. Hence II(x + A)"'II·llx

+ All = lim IIAel + elx"'ll·lI xel + I

Aedl

= lim Ilel(x + A)"'(X + A)edl I < liF II(x + A)"'(x + A)edl =

II(x + A)'" (x + A)II

< lI(x+A)"'II·llx+AII, and II(x+A)"'. (x+ A)II = II(x+ A)"'II·llx+AII, Vx E A, A E JR. Now by Theorem 2.15.25, (A+JR)) and A are real C"'-algebras. Q.E.D.

Notes. Theorem 2.15.10 is due to L. Ingelstam and T.W.Palmer. Lemma 2.15.16 is due to B.Russo and H.A. Dye. Theorems 2.15.19 and 2.15.26 are due to B.R.Li, and Theorem 2.15.26 gives an affirmative answer for the GelfandNaimark conjecture in real case. However, it is still an opern question: does Theorem 2.15.25 hold in the absence of the identity? References. [72], [100], [101], [124], [141].

Chapter 3 Tensor Products of C*-Algebras

3.1. Tensor products of Banach spaces and cross-norms Let X b

·· .

,Xn be (complex) Banach spaces, and IO.n

'OIi=l

I Xi(i)

Xi -- {'" IO.n (i) LJ 'OIi=lx i

E Xi, \J' vZ,J'} .

i

U

=L

®~=l X}i) is called zero, if for any Ii E Xt, 1 < i < n,

j

(®~=l/i)(U) = L j

n

II /i(x;i l ) = o. i""l

Then ®?=1 Xi is a linear space, and is called the algebraic tensor product of X b · . - ,Xn - This is a generalization of the algebraic tensor product of Hilbert spaces (see Section 1.4). If 0:(') is a norm on ®~=lXi, then the completion of (®?=lXi, 0:(-)), denoted by 0: -®~=lXi, is called the tensor product of XI, ... ,Xn with respect to 0:(')'

Definition 3.1.1. A norm a(·) on the algebraic tensor product of Banach spaces X b · · - ,Xn is called a cross-norm, if a( ®~=l Xi) = IlxII!-' ·llxnll, VXi E X i,l < i < n. Let Xl"", X; be Banach spaces, and ®f=lXi be their algebraic tensor product. 1) -X(u) = sup{] ®?=l li(u)11 /i E xt, II/ill < 1,1 < i < n}(Vu E ®f=lXi ) is a cross-norm on ®?::IXi ,

Proposition 3.1.2.

n

2) i(U)

= inf{L II IlxY)11 I u = L i i=l

cross-norm on ®~IXi'

j

®f=lX}il}(VU E

®~lXi) is the largest

164

Proof.

It is easy, and we leave it to the reader.

Q.E.D.

Let XI,' .. ,Xn be Banach spaces; and consider the algebraic tensor product ®?=IXi* of conjugate spaces X;"" ,X~. Since Xi is a(Xt*, Xt)-dense in Xr,l < i < n, it follows that u*(E ®?=lXn is zero if and only ifu*(®?=lxi) = 0, \/Xi E Xi, 1 < i < n. Now if a(·) is a norm on ®?=IXi such that a*(u*) = sup{lu*(u)11 u E ®?=lXi,a(u) < I} < oo,Vu· E ®i=lX:' then a*(·) becomes a norm on ®~lXi*; and a*(.) is called the dual norm of a(·).

Proposition 3.1.3. Let XI,' .. ,Xn be Banach spaces. 1) The dual norm of "'Y(') on ®?=lXi is just the norm ).(.) on ®~=lXt (about the definitions of "'Y(') , ).(-), see Proposition 3.1.2) . 2) Let a(·) be a cross-norm on ®~=lXi' Then a*(.) is a cross-norm on ®i=l X; if and only if >..(.) < a(·) < "'Y(')' In this case, we also have "'Y(') > ).*(.) > a·(·) > >..(.) on ®?=lX;, Proof. 1) For any u* E ®?=lX;, since the unit ball of Xi is a(X;*, Xt)-dense in the unit ball of Xr, 1 sup{ju"' (®?=l Xi) I I Xi E Xi, IIXil1 < 1,1 ..(u*).

However, for any u E ®~lXi with "'Y(u) < 1 and e: > 0, we can write u = n

E ®?=lXY> such that E II IlxY)II- "'Y(u) < s. i

j

lu*(u)1

<

Then

i=l

E IU"'(®?=l i

X

}i)) I < E >..(u*) i

n

II II x}i)II

i=1

< >,,(u*)("'Y(u) + e:) < >..(u*)(l + s]. Thus "'Y*(u*) < (l+e:)>..(u*). Since e: is arbitrary, it follows that "'Y*(u*) < ).(u*). Therefore, "'Y*(u*) = ).(u*),Vu* E ®?=lX,~. 2) For any u E ®i=1 Xi, by the definition of ).(.) it is easy to see that

Hence if a* (.) is a cross-norm on ®?:1 X;, then we have >..(.) < a(.) . Obviously, a(·) < "'Y(') since "'Y(') is the largest cross-norm on ®?=lXi.

165

Conversely, suppose that A(') ®?=l X i*, by 1) we have ..\(u*) =

< 0:(') < ,),(.)

on ®?=lXi , For any u* E

sup{ju*(u)11 u E ®?=IXi,')'(U) < I}

< sup{lu*(u)11 o:(u) < I}

= o:*(u*)

< su P{ Iu *(u) I I A(u) < I} =

x*( u *) n

< sup{I: I ®?=l f?l(u)11 A(U) < I} < I: II Ilfpll!, ;

i i=1

l n f(i • • n X*i ' h were u * -- '" LJ®;=1 i an d f(i ; l E X*;, \..I' v't,), i.e., on ®i=l ;

Since ,),(.) and ..\(.) are cross-norms, it follows that a*(.) is also a cross-norm. Q.E.D. Notes. The tensor products of Banach spaces was first studied by R. Schatten and J. Von Neumann. The general theory of tensor products was further developed by A. Grothendieck, which led him to the discovery of nuclear spaces.

References.[63], [151).

3.2. Tensor products of C*-algebras and the spatial C*norm Let A I, ·

· · ,

An be C*-algebras. Naturally, define

Definition 3.2.1. Let AI,' .. , An be C*-algebras. A norm 0:(') on ®7=1 Ai is called a C* -norm, if

Vu, v E ®?=IA i . The completion of ®?=IA i with respect to 0:(')' denoted by a®?=l A., is called the tensor product of At, ... , An with respect to 0:(')' Clearly, O:-®?=IA i is also a C*-algebra.

166

Proposition 3.2.2. Let a:{-) be a Cot-norm on 0?=IAi' Then a(.) < ')'('), where ')'(-) is as in Proposition 3.1.2. First, notice the following fact: Let A be a Cot-algebra, and a E A+. Then Hall < 1 if and only if a2 < a. Now let b, E Ai, a < b, < 1, 1 < i < n. 8ince Proof.

in a-0~=I Ai, it follows from the above fact that a( ®:'=I bi ) < 1. Further, for any at E Ai, 1 < i < n, we have a(0?=I a&) 2 = a(0?=I a;ai)

Q.E.D.

Proposition 3.2.3. Let a(·) be a C*-norm on 0r=IAi, and {d~:)} be an approximate identity for Ai, 1 < i < n. Then {d l = 0:'=1 d~~) I l = (lb' .. , In) } is an approximate identity for a-0?=IA,. n

From Proposition 3.2.2, a(dd

Proof.

< ')'(dd

=

II Ildf:)11 < 1, Vl =

(lI, "',

,=1

In). 80 it suffices to show that

Va, E Ai, 1 < i < n. Notice the following equalities: n

®?=Ibi a, - ®i=Iai =

L

i=I

and

0j~~aj 0 (biai - ail 00j=i+I bjaj

n

0i=lai bi - 0:'=lai =

L

i=1

0~~~aj 0 (aibi - ad ® 0j'=i+I ai b,..

Now from a(·) < 1(-) and the property of {dJ:)}(1~i~nb we can get the conclusion immediately. Q.E.D.

Definition 3.2.4. Let AI,"', An be C*-algebras, and 0?=IAi be their algebraic tensor product. u E ®i=I Ai is said to be positive, denoted by u > 0, rn

if u =

l::: ujUj, where Ul,'"

;=1 ®:'=l A i

I U > a}.

,Urn E ®:'=IAi. Moreover, let (0:'=IA,)+ = {u E

167

A linear functional cP on EB?=l Ai is said to be positive, denoted by cP

> 0, if

cp(u) > O,Vu E (®~lAi)+' Clearly, (®?=lAi)+ is a cone; ®?=lAi is the linear span of (®?=IAi)+, and for any C·-norm a{) on ®i:lAi, (®?=l Ai)+

c (a-

®?=1 Ai)+.

Moreover, if cP is a positive linear functional on ®?=lAi, then we have cp(u*) = (cp (u)) and the Schwartz inequality:

Icp(v·u)1 2 < cp(u*u)cp(v*v),

Proposition 3.2.5. Let CPi be a positive linear functional on a C·-algebra A i , l < i < n. Then ®i=lCPi is positive on ®?=lAi, where ®i:1CPi is defined by n

(®?=lCPi)(®j=laj) =

II CPi(ai), Va, E Ai, 1 < i < n. i=1

Proof.

Q.E.D.

It is similar to the Section 1.4.

Theorem 3.2.6. Let Ai be a C·-algebra, and S, = B(A i ) be the state space of Ai, 1 0

Then ao(') is a C*-norm on ®i=lAi, and oX(.) ao(') is a cross-norm on ®i=lAi.

< ao(') < ,),(.). In particular,

Proof. Let CPi E Bi , and {11" lpi' H lpi' elpi} be the cyclic * representation of Ai generated by cpi,l < i < n. Naturally, we can define a * representation {®i=l11"cp;, ®i=lHcpJ of ®i=l A i ' Since AilLtpi is dense in H'I'i' where L'I'; is the left kernel of 'Pi, 1 0 , ®i=ICPi v v

Vu E ®i=l Ai. Thus we have

ao(u) = sup{11 ®i=11l"'I';(u)1I1 CPi E Bi , 1 < i < n}, Vu E ®i=IA,. If u E ®i=lA i is such that ao(u) = 0, then 0= (®?=I11"'I';(U) ®i=l e'l'i' ®k=lecpk) = ®i=lCPi(U), = 0 since A: is the linear span of Si, 1 < i < n. Therefore, ao(') is a C·-norm on ®i=lAi' By Proposition 3.2.2, ao(') < ,),(.) on ®?=lAi.

VCPi E Bi , 1

< i < n. Hence

U

168

Notice the following facts. Let A be a C"'-algebra, and f E A'" with II/ll = 1. By Theorem 2.11.2 and 1.9.3, there is the polar decomposition I = Rulp, where Ip is a state on A, and u is a partial isometry of A*"'. Since the unit ball of A is cr(A "'* , A'" )-dense in the unit ball of A*"', it follows that there is a net {a,} C A with Iladl < 1,Vl, such that a, ---t u(cr(A"'''A''')). Consequently, Ra,((J ~ Ru(lp) = l(cr(A"', A"'''')). Now for any u E ®i=IAi, by the above facts and the Schwartz inequality 2 A(u)2 = supj] ®7=1 li(U)1 I Ii E A;, II Ii II = 1,1 < i < n}

I Ilaill Ipi E s., < 1,

n R () 1 sup { I®i=1 a/Pi U

2

n

<

(

sup { ®i=1 Ipi UU

"')

n

(n

as

E Ai, and }

1
'"

. ®i=1Ipi ®k=1 akak

)

suP{(®7=I11"lpi(UU"') ®i=1 Clpj,®I:=IClpJ;)

I IpiII aillE B
I Ipi E

Si,l

< i < n}

< O:o(U)2.

Q.E.D. Let Is E Ai, 1 < i < n. Then the linear functional ®i=l/i on ®i=lAi can be uniquely extended to a linear functional on O:O-®i=IAi with

Corollary 3.2. T. n

the norm

II

111s11· i=1 Proof. For any u E ®i=lA i, by the definition of A(') and A(-) ®i=lAi we have

I ®i=1

fi(U)1

n

n

i=1

i=1

on

< II IIfiII A(u) < II Ilfillao(u).

Q.E.D.

That comes to the conclusion.

Proposition 3.2.8.

< ao(')

The above norm ao(') on ®i=IAi is also represented

by

aa{u) = sup{11 ®i=l11"i(u)lll11"i is a sup{11 ®i=l 11"Pi(u)11

* representation of A i,l tiP}i) I Ai > 0, p}i) E L\i' Vi},

A:

and 11"Pi is the Proof.

{l:

* representation of Ai

;

generated by Pi, VPi E L\i' 1 < i < n.

Let

a(u)

= sup{lI

®;=1 11"Pi(U) II I Pi E L\i' 1

< i < n},

169

Vu E- ®f=l Ai. Since any * representation of a Clio-algebra is a direct sum of a zero * representation and a family of cyclic * representations, and any cyclic * representation is unitarily equivalent to a * representation generated by a state, it follows from Theorem 3.2.6 that

ao(u) = sup{lf ®f=111"tpi(U)111 CPi E 8 i , 1 a(u), Vu E ~?=lAi' a(·) is also a Clio-norm on ~?=lAi' In fact, if a(u) = for some u E ®f=lAi, then ®?:lPi(U) = O,VPi E Doi,1 < i < n. By the definitions of Doh···,Do n, we have u = 0. For any Pi E Do;, 1 < i < n, since

°

it follows that ®?=lPi can be extended to a positive linear functional on a®?=lAj. Further, for any u,v E ®f=lAi,v*(a(u*u)-u*u)v is a positive element of a-®f=lAi. Thus we have

VPi E Do;, 1 0,

VCPi E 8 i,1 ao(u), Vu E ®f=lAi . Therefore, ao(') = a(·) on ®i=lA i. Q.E.D. Let A be a Colo-algebra on a Hilbert space H, and Do = {we(.) = (ee, e) leE H, Ilell = I}. Then the a(A'" ,A)-closure of CoDo contains the state space 8(A) of A.

Lemma 3.2.9.

Proof. Let tp E 8(A). By Corollary 2.3.12, cP can be extended to a state on B(H), still denoted by cp. Since the unit ball of T(H) is w*-dense in the unit ball of B(H)*, it follows that there is a net {t,l C T(H) with IItllll < 1, Vl, such that tr(t,b) --+ cp(b), Vb E B(H).

tt - t,-,

tt

By cp* = cp, we may assume that ti = t" Vl. Write t, = where E T( H)+ and tt = 0, VI. Clearly, 111 < !ltdll < 1, Vl. Since the unit ball of B(H)* is w*-compact, so we may assume that the nets {ttl and {tl} are w*convergent in B(H)*. Let Ittl = tt +t Vl. Then there is t/J E B(H)* such that

tt .

Iitt

"

170

tr(ltzlb) ---t 1/J(b),Vb E B(H). Obviouslyv e is positive on B(H)+. Moreover, since cp(l) = 1 = liptr(tz) < liptr(ltzl) = ?jJ(1) and IIltzllll = Iltzlll < 1, VI,

1/J is a state on B(H) and limtr(t l ) = O. Further, for any t a E B(H)+,O < tr(t, a) < Iialltr(tl) ---t o. Thus, tr(tt b) ---t cp(b), Vb E B(H), and we may assume that tt > 0, VI. Now write it follows that

n

where

I c~) II

:=;

1, A~) > 0, Vn, 1, and let N

N

L(L A~))-lA~)(·e~),e~)),

WN,t(·) =

VN,I.

n=1 k=1

Clearly, WN,l E CoD., and WN,t(a)

Theorem 3.2.10. Let {1I"i' Ai,l < i < n. Then

ao(u) where Proof.

=

---t

cp(a), Va EA.

Q.E.D.

Hd be a faithful * representation of a C·-algebra

II rg;?=111"i(U)!I,

Vu E ®7=IAi,

ao(') is defined by Theorem 3.2.5. By Proposition 3.2.8,

sup{11 rg;?:l 1I"Pi(U) 11

2

I

Pi E D.i' 1

< i < n},

Vu E rg;?=IAi, where ~i = {(1I"i(·)ei, ei) 1 6 E Hi,lleill = I}, and 11" Pi is the * representation of Ai generated by Pi, VPi E D.i,1 < i < n. Further, from Proposition 3.2.8 and Lemma 3.2.9 it follows that

Vu E ®?=IAi. That comes to the conclusion.

Q.E.D.

From this theorem, the geometric sense of the C·-norm ao(') on rg;?:l Ai is given. Therefore, ao(') is called the spatial C·-norm on rg;?=IAi. Later, we

171

shall see that Cto(') is the minimal C* -norm on ®?=1 Ai indeed. So sometimes we also denote ao - ®?=1 Ai by min -®?=1 Ai'

Proof. From Corollary 3.2.7, ®?=lA; c (ao-®?=l Ai) *. We may assume that Ai is a C*-algebra on a Hilbert space Hi, 1 < i < n. By Theorem 3.2.10, ao®?=lAi is the uniform closure of ®i=IAi in B(®?:IHi). Further, from Lemma 3.2.9, the w*-closure of Co{ ®?=1 (·ei' ei) I ei E H i,l < i < n} contains the state space of aO-®?=IAi. Therefore, ®?:IA; is w*-dense in (ao-®?:IA i)·. Q.E.D. Notes. The tensor products of C·-algebras was studied first by T. Turumaru. aO-®i=1 Ai is also called the injective tensor product of AI,"', An. Proposition 3.2.2 seems very simple, but we got it quite later. B.J. Vowden gave first proof. The present proof here is taken from C. Lance. Theorem 3.2.10 is due to A. Wolfsohn.

References. [95], [171], [185], [193], [198].

3.3. The maximal C*-norm Let AI,"', An be C·-algebras. From Theorem 3.2.5, there is a C·-norm ao(') on ®?=1Ai' By Proposition 3.2.2, we have a(.) < ,),(.) for any C·-norm a(·) on ®?=lA•. Thus we can define the maximal C'<norm al(') on ®?=1Ai:

\/u E ®i=lAi. And sometimes we also denote Ctl-®?=1Ai by max-®i=lA i. Clearly, a1(') < 1(') on ®i=IAi' As in Definition 2.4.5, a * representation {-Jr, H} of ®?=lA i is said to be nondegenerate, if [1I"(u)e leE H, u E ®i=IAil is dense in H.

Let {11", H} be a nondegenerate * representation of ®?=IAi. Then there is unique nondegenerate * representation {1I"i, H} of Ai, 1 < i < n, such that 11"i ( a~ 11"j ( aj) = 11"; (a j ) 11"i ( a;) { 1I"{®i=1 ai) = 11"1 (ad" '1I"n{an)

Lemma 3.3.1.

'Vai E A i , l < i =I j < n. Consequently, 111I"(u)1I < 1(u), \/u E ®?=lA i- Moreover, if 11" is faithful, then 1I"i is also faithful, 1 < i < n.

172

Let AP) = Ai+a:1i,1 < i < n. For any i E {I",., n} and any € = L:k 7r(Uk)€k, where Uk E ®i=l Ai ' €A: E H, Vk, define

Proof.

ep.(b,) = L:\1r(®;,ti1; ® bi · Uk)€k,1r(Ul)6) Je,l

Vb, E AP). Clearly, €Pi is a positive linear functional on A!l), Vi. Then

"L: 7r(®i,ti l ; e bi . Uk) 6: I! =

€Pi(b;bi P /2

k

Since 1r is nondegenerate, it follows that there is unique 1ri(bs) E B(H) such that

1ri(bi) C=

L:

11"

(®j;ti1j ® b, . Uk)€A:,

k

Vb i E AP). It is easy to see that {1ri, H} is a * representation of A!l), 1 < i < n, and 1rl(bd .. '1r n(bn)€ =

L: 1r(®?=l bi . Uk)€k k

Vbi E AP), 1 < i < nand C= Lk 1r( Uk) Ck' Therefore 1r(®?""lai)

= 11"1 (ad

.. '1I"n(a n),

1ri(ai)1rj(aj) = 1rj(aj)1ri(a,), Vai E Ai, 1 < i

1= i < n.

CE H

is such that 1I"i(ai)€ = 0, Vat E A, for some i. Then 1r(®i=la,.)€ = II1I";(aj)1r,(ai)€ = O,Va,. E Ai,1 < i < n. But 1r is nondegenSuppose that

j,ti

erate, so € = 0. Thus 11", is nondegenerate, 1 < i < n. If {1r'i}i=l satisfies the same conditions as well as {1r,}i=I' then we have

1rH ai)C =

L: 1I"(®j,ti 1

j

® a,' uk)6 = 1I",(a.)€,

k

Va, E Ai and C = Lk 11"( Uk) €k. Since 11" is nondegenerate, it follows that . 7r,i -- 1r" 1 < _ ~. < _ n, .l.e., {}n' 1r, ,=1 IS unIque. Finally, let 11" be faithful. By 1r(®?=lai) = 1rl(al) .. '1r n(an)(Va, E Ai,l < i < n), clearly 1ri is also faithful, 1 < i < n. Q.E.D. In Lemma 3.3.1, if we don't assume that 11" is nondegenerate, then we have still a decomposition {1ri }f=l of 1r, but {1I"i }i=1 is not unique possibly.

Proposition 3.3.2.

For any U E ®i=lA i, we have

01(U) = sup{II1r(u)lll1r is a

*

representation of ®i=1 Ad·

173

Let 1ro be a faithful lI 1ro(u}II · Hence

Proof.

* representation

al(u} < sup{II1r(u)lll1r is a

*

By Lemma 3.3.1, 111r(·)11 < ,),(.) for any sup{ll1r(·)III1r is a

*

of al-®?=IAi. Then adu) =

representation of ®~=1 Ai}.

* representation 1r of ®~=lAi, so

representation of ®~=l Ai}

is a C·-norm on ®f=l Ai. Further the conclusion follows from the definition of aI(·), Q.E.D.

Proposition 3.3.3.

For any u E ®f=lAi, we have

al(u} = sup{a(u) 1a(.) is a C·-seminorm on ®?=1 Ai}. Here a(.) is called a C'<seminorm on ®?=lA.. , if a(·) is a seminorm on ®i=lAi, and a(uv) < a(u)a(v), a(u*u) = a(u)2, Vu, v E ®?:lA i. Clearly, the right side > the left side. Now let a(·) be a C· -seminorm on ®~=lAi' Then I = {v E ®?=l Ai I a(v) = o} Proof.

is a two-sided

* ideal of ®i=lAi.

Suppose that

v-tv=v+I is the canonical map from ®?= 1 Ai onto ®?= 1Ad I, and let

a(v) = a(v),

Vv E ®~=lAi/ I,

v E v.

Then a(.) is a C·-norm on ®?=lAi/ I. Pick a faithful a-( ®f=1 Ad I), and define

1r(v) = 1f(v), Then 1r is a

* representation of ®?=1 Ai. at{u) > 111r(u}11 =

* representation

1f of

Vv E ®?::IAi. Now from Proposition 3.3.2,

111f(u)II = a(u} =

Vu E ®?=1 Ai' That comes to the conclusion.

a(u),

Q.E.D.

Proposition 3.3.4. There is a bijection from the set {a(·) I a(·) is a C· - norm on ®?=1 Ai} onto the set {I I I is a closed two-sided ideal of al®~=lAi' and In ®i=lAi = oj as follows: Let a(.) be a C·-norm on ®i=lAi. Then there exists unique closed twosided ideal I a of al-®?=IAi such that the map u - t U = u + Ia(Vu E ®i=lAi)

174

can be extended to a * isomorphism from a-®i::;::lAi onto al-®i=lAd I a . In addition, this L; satisfies the condition: lOt n ®i=lAi = {O}. Conversely, for each closed two-sided ideal I of al-®i=lA i with I n ®i=lA i = {O}, there is unique C*-norm a(.) on ®i=lA i such that I = lOt. Consequently, for any C*-norm a(·) on ®?=l Ai,

Proof. Let a(·) be a C"'-norm on ®i=lA i. Clearly, a(.) < ad,), Thus the identity map id on ®?=l Ai can be uniquely extended to a * homomorphism from al-®i=lA i to a-®?=lA i. But id(ar®i=lAi) is a C"'-subalgebra of a®i=lAi containing ®?=l A j, so id (al-®?=l Ai) = a-®i=l Ai' Denote the kernel of this * homomorphism by lOt. Clearly, lOt is a closed two-sided ideal of o:r®i=lA i. Moreover, since id(u) = u,Vu E ®i=lA i, it follows that lOt n ®?=lAi={O}. Then we get a natural * isomorphism from a-®?=lA i onto al®i=lAiIIOt such that u ---+ U = u + IOt,Vu E ®i=lAj. Now let I be a closed two-sided ideal of al-®i=lAi, and q> be a * isomorphism from o:-®i=lAj onto al-®i=lAd I such that q>(u) = it = u + I, Vu E ®?=l Ai' Define \[I(a) = ()-l(a), Va E al- ®i=l Ai,

where a = a+ I is the canonical image of a(E O:l-®i=lAi) in O:l-®?=IAt/I. Then \[I is a * homomorphism from Ct:l-®?=lAi onto o:-®i=lA i , and its kernel is I, and \[I(u) = u, Vu E ®i=lA i • Thus \[I is the extension of the identity map on ®i=IAi, and I =kerW = lOt. Conversely, let I be a closed two-sided ideal of al-®i=l Ai, and In ®?=1 Ai = {O}. Then ®i=lA i can be embedded into al-®i=tAd I, and the norm on 0:1®i=lAj/I determines a C"'-norm 0:(') on ®i=lA j. Then o:-®i=tAi and O:t®i=lAil I are * isomorphic, and this * isomorphism maps u to u = u + l, Vu E ®i=lA i . Then from preceding paragraph, we obtain that I = lOt. Q.E.D. Remark. furthermore.

In Section 3.9, we shall discuss the maximal C"-norm al(')

Notes. Lemma 3.3.1 is due to M. Takesaki, al-®i=lAj is also called the projective tensor product of At,' . " An; it was considered first by A. Guichardet. The maximal C·-norm al(') is important and is more natural than ao(')' But very little is known about o:d')' The theory of nuclear C'" -algebras will be important in this aspect (see Section 3.9 ).

References. [65], [95], [171]-

175

3.4.. States on algebraic tensor product Let AI,' .. ,An be C· -algebras, and ®?=:l Ai be their algebraic tensor product.

Proposition 3.4.1. IT rp is a positive linear functional on ®?=IA i, then there is a positive constant K such that

Proof. Fix ai E (A i)+l2 < i < n. Then rp(. ® ®?=2ai) is a positive linear functional on AI, so it is continuous (Proposition 2.3.2). Similarly, rp( ®?=la;) is continuous for each variable. Now the result follows from the principle of Q.E.D. uniform boundedness.

Proposition 3.4.2. Let AP) = Ai+a:lill < i < n, and rp be a positive linear functional on ®~=IAi' Then rp can be extended to a positive linear functional ep on ®?=IAP) such that for any subset II of {I,···, n} and a; E Ai,ifJ.ll, rp( ®iEl1i ®itll ail = 1,',sEU lim rp( ®iEldf~) ® ®itlllai) ' I

where {df:)} is an approximate identify for Ai, 1

< i < n.

Proof. For any ai E A i,2 < i < n, by Proposition 2.4.4 the following two limits exist and are equal:

Now rp can be extended to a linear functional

rp

on AP) ® ®?=2Ai such that

176

= L:j (Ajdf:) + a;I}) e ®~2a~i). In this way, we can obtain a positive Q.E.D. linear functional f; on ®7=IA~I) satisfying the conditions.

where Yh

Proposition 3.4.3. Let ep be a positive linear functional on ®7=IAi, and {df:)} be an approximate identity for Ai, 1 < i < n. Then

Proof.

Let ep be an extension of ep as in Proposition 3.4.2.. Then -- ep ""(lOIn I·im ep (n ®'=1 d(')) I· 'OIi=1 1) , . '

11," ,,1..

Moreover, for any

at

E Ai, Hail!

< 1,1 < i < n, it follows that

lep (®?=1 ad 12 = If; (t8l?=1 ail 12 < ep (®7=1 1,) ep (®7=1 a;ad

< sup {ep (®7=lbd21 bi E (A i )+, Ilbill < 1,1 < i < n} by the Schwartz inequality. That comes to the conclusion.

Q.E.D.

Definition 3.4.4. A positive linear functional ep on ®7=lAi is called a state, if sup {ep (®7=lai) lai E (A,)+, Ilaill < 1,1 < i < n} = 1. Denote by S(®f=lA,) the set of states on ®?=::lA i .

Proposition 3.4.5. Let ep be a state on ®i=IAi. Then there exists unique state f; on ®7=lAP) which is an extension of ep, where A~l) = A i +a:1" 1 < i < n.

177

Proof. The existence of $ follows from Proposition 3.4.2. Now let 1/; be a state on ®?=I AP), and also be an extension of ip . Then

where {dl} is an approximate identity for AI. By Proposition 3.4.2,

1/;(1 1 e ®i=2ai) > ep(1 1 ® ®i:=2 ai),

Vai E (Ai)+,

Suppose that there is a ai E (Ai) + with II ai 11 < 1, 2 O. Then for bi E (A i )+) Ilbill < 1, b, > ai, 2 < i < n,

1/;(1 1 ® ®i=2 bi) - $(h ® ®~=2bi) [1/;(1 1 ® (b2 - a2) ® ®?=3 bi) - $(1 1 ® (b2 - a2) ® ®?=3 bd] +[1/;(1 1 ® a2 ® (b 3 - a3) ® ®i=4 bi) - $(1 1 ® a2 ® (b3 - a3) ® ®~=4bi)] + ... + [1/;(1 1 ® ®?=2ai) - D > 0, i.e., Further,

But by Proposition 2.11 and Definition 3.4.4,

We get a contradiction. Thus 1/; and

Proposition 3.4.6.

Let cp be a state on ®?=IAi. Then cp can be uniquely extended to a state on 01-®?=1 Ai' Consequently,

By Proposition 3.4~5, cp can be uniquely extended to a state $ on n A(I) h A(l)' 1 ®i=l i ) were i = A i +a: i , 1 < t. < n. Let Proof.

178

x

It is a left ideal of ®~=lA~l) by the Schwartz inequality. Suppose that x --t = X + L is the canonical map from ®~=1A~I) onto ®r=1 A~l) / L. Define an inner product on the quotient space:

(x, Y)

Vx, Y E ®r=l A

= <,C5(y*x),

P)·

Denote the completion of (®~=lAP) / L, (,)) by H. For any x E ®?=1 A!l), let ,..,

1r ( X) y

,-..., = xy,

w

n

vy E ®i=l

A(l) i

.

n

Since ®7=lX;Xi <

II II xil1 2 ®?=l i, in ®~=lA~I), it follows that

i=1

n

111r( ®~=l xdjJI1 = <,C5(y* . ®7=1 x; Xi . y) < 2

II II xil1

2

•

11Y'11 2 ,

i=1

Vy E ®?=lA!l). Thus 1r(®7=IXi) can be uniquely extended to a bounded linear operator on H, VXi E AP), I < -£ < n. Consequently, we obtain a * representation {1r,H} of ®f=lA!l) such that <,C5(x) = (1r(x)I, I),

where

I

= ®?=lli

+L

Vx E ®?:=lAP) ,

= ®~li. Furthermore, by Proposition 3.3.2

IfP(u) [ = ItP(u)j < 1I1r(u)1I < al(u), Vu E ®7=lA~1). Therefore, fP can be uniquely extended to a state on al-®?=l AP). From Proposition 3.2.3, (1/1 I ®?= 1 Ai) is a state on ®?=1 Ai for any state 1/J on al-®?=lA i. Then by the preceding paragraph, we have ad u )2 = sup{fP(u*u)1 fP E S(®?=lAd},

Vu E ®?=lAi. Q.E.D.

From Proposition 3.4.6 and 3.2.3, "p S(al-®?=lAd onto S((®?=lA i ) .

---t

("pI ®7=1 Ai) is a bijection from

Let fP be a state on ®7=1 Ai' Then by the GNS construction, we can get a cyclic * representation {1r l H, e} of ®7=1 Ai such that

Proposition 3.4.7.

Proof. By Proposition 3.4.6, tp can be uniquely extended to a state on at®f=l Ai' which is still denoted by tp, Let

L

= {u E ®i=lAil

fP(u*u) = O}

179

and

Lrp = {a E 0:1- ®7=1 Ail cp(a*a) = O}. Clearly, L c Lrp. Suppose that U -+ fi = u + L and a -+ arp = a + Lrp are the canonical maps from ®?=IAi onto ®7=lA i / L and from al-®?=lA i onto al-®i=lA~I}/ Lrp respectively. Then we have the following diagram: uEAi/L

---+

H

!U where {7r, H} is the * representat ion of ®i= 1 Ai generated by cp (the GNS construction, H = (®?=lAi/L,{,))-, see the proof of Proposition 3.4.6), and {7rrp,Hrp, €rp} is the cyclic * representation of al-®?=lA i generated by tp: It is easily verified that U can be uniquely extended to a unitary operator from H onto Hrp, and 1r(U) = U- l7rrp(u)U, Vu E ®7=lA i. Now let

e= U-1€rp.

Then {7r, H, €} satisfies our conditions.

Q.E.D.

Proposition 3.4.8. Let a(.) be a C*-norm on ®?=lA i , and f = {cp E 8(®7=lAi) I cp is continuous with respect to a(·)}. Then for any U E ®?=lAi,

a(u) = sup{cp(U*U)1/2Icp E r) = sup{ll1rrp(u)lIlcp E f}, where {7rrp, Hrp, €rp} is the cyclic cp, Vcp E r.

*

representation of ®f=l Ai generated by

Proof· By Proposition 3.2.3, r = {(,pI ®?=l A~l»)I,p E 8(a- ®~l Ai)}' So we can get the first equality. Replacing ad,) by 0:(') in the proof of Proposition 3.4.7, we get II 1l'rp (u) II < a(u), V

Q.E.D. References. [37], [95], [96].

3.5. The inequality ,\(.) < ao(') < a(·) < ,),(.) Lemma 3.5.1. Let At, ... ,An be C*-algebras, and An be with no identity. If x E ®i~lAi ® A~l), where A~l) = A n-t-a: l n, such that xv = 0, Vv E ®i=lAi, then x =

o.

180

We may assume that Ai C B (Hi) and Ai is nondegenerate on Hi, 1 < i < n. Since An has no identity, we may also assume that In is the identity operator on H n' Then Proof.

Thus x is a zero operator on ®f=IHi. By Theorem 3.2.10, Ilxll Therefore, x = o.

°

= = ao(x). Q.E.D.

Proposition 3.5.2. Let AI, ... ,An be C·-algebras, and An be with no identity. Suppose that a(.) is a C·-norm on ®i=IAi. Define a(x) = sup{a(xu)lu E ®f=IAi,a(u) < I},

"Ix E ®?:lA i ® A~I),

where A~I) = An+
Proof. By Proposition 3.2.2, a( ®f=1 ad is continuous for each variable ai in Ai, 1 0, if (ill' .. ,in) is sufficiently later, then we have

®f=1 dt) . v) + e < a(x . ®f=1 dt)) + e < sup[ a(xu) I u E ®f=IAi, a(u) < I} + c.

a(xv) <

a(x·

Therefore,

, I1m

11 .....1..

a

(lOIn d(i)) x . 1cY.=1 Ii '

Consequently, a(·) is an extension of a(·). Similarly, from a(x*) =sup{a(ux)lu E ®i=IAi,a(u) a(ux' ®i=ld~?) + e < a(x· ®?=ldt)) + e we can see that a(x*)

= a(x),

"Ix E ®i;11A• ® A~I).

< I} and a(ux) <

181

t£

By Lemma 3.5.1, a(.) is a norm on ®i;;lAi ® Ail). Moreover, since for any E ®i=lA i with a(11.) < 1

a{xyu) = lim a(x· ®i=ldt)· yu) < a(x)li(y), h,"'I'n

it follows that a(xy) < a(x)a(y),\lx,y E ®i~}Ai ® Ail). Noticing that

a(x)2

=

a(x·)a(x) > a(x*x)

= sup{a(x·xu)lu E ®i=lAi,a(u)

> sup{a(u*x·xu)I11. E ®i=lA"a(11.) <

< I}

I} = a(x)2,

we get a(x*x) = a(x)2, \Ix E ®i;;lAi ® Ail). So ii(·) is a C·-norm on ®i:::l Ai ® Ail). Q.E.D.

Proposition 3.5.3. Let Ai "" C(O,), where Oi is a compact Hausdorff space, 1
Then ip(®i=lai) = 0, \lip E O. This is a contradiction since ®i=lai element of a-®i=IAi. Therefore, n = 0 1 X ••• x On, i.e., a-

and so a(·)

®i=l Ai

r-.J

C(OI

= ao(')' Moreover, for any 11.

sup {] ®r=l

-

fi(u)11 t,

X •..

-=f=.

0 as an

x On),

E ®i=lA i,

E

Ai, II/ill

< 1,1 < i < n}

'\(u).

By Theorem 3.2.6, ao(') = ,\(.) on ®i=IAi'

Q.E.D.

Proposition 3.5.4. Let Ai be a C*-algebra with an identity Ii, 1 < i < n, and a(·) be a C·-norm on ®i=IAi. Suppose that ,8(.) is the restriction of a(·) on 11 ® ®i::2Ai. If ip is a state on a-®i=IAi such that X(,) = ip(' ® ®i=21i)

182

is a pure state on Ai) then there is unique state tf; on ip = X ® tf; on ®7=lA i.

{1-®~2Ai

such that

Proof. Define tf;(v) = ep(l l®v), Vv E ®7=2Ai. Clearly tf; is a state on ®7=2Ai, and Itf;(v) I < {1(v), Vv E ®7=lAi. Thus tf; can be uiniquely extended to a state on {1-®~=2Ai' Now it suffices to prove that ep(®~=la,) = x(adtf;(®~=2a,),

VO

i= a, E

(A.)+,

1 < i < n.

Fix 0 i= a, E (Ai) + ,1 < i < n. If tf;(®~=2a,) = 0, then by the Schwartz inequality we have

0< ep(®7=la,) = ep(al e ®7=2 a;/2 . 11 e ®7=2 a;/2)

< ep( ai ® ®~=2ad 1/2 . tf;(®7=2ai)

=

o.

= X(at}tf;(®7=2 ai) = o. n If tf;(®7=2 ai) = Il a211 .. '!lanll, let v = II II a, II ®~=2 i, - ®~=2ai' then v E i=2 (®?=2Ai)+, and ep(al®v) = 0 since 0 < ep(al®v) < Il alllep(1l®v) = Ilallltf;(v) =

Thus ep( ®~=l a,)

O. Thus n

ep(®i=lai)

= ep(al ® ®i=2 1i) . II f1 a ifl = x(al)tf;(®i=2 a.). i=2

= tf;(®~=2ai) < Il a21! "n '!lanll· Let p, = Il a211.. ·11 = ..\-lep(·®®i=2 ai), and P2(') = p,-I[ll lIaillep('®®~=21i) -ep('®

Now suppose that 0 < ..\

anI! -..\,PI(·)

i=2

®~=2ai)]' Then PI and P2 are two states on AI, and n

X(·) =

(ll i=2

n

lI a,ln- l ..\pl (·) + (ll lI ai l[) - Ip,p2(') i=2

on AI. Since X(·) is a pure state on AI, it follows that X = PI = P2. Therefore, we obtain

Q.E.D. Corollary 3.5.5. Let Ai be a C·-algebra with an identity l i,1 < i < n, and ip be a state on ®i=IA i. If Xi(') = ep(. ® ®;#,1;) is a pure state on A,,1 < i < k (some k < n), then ep = Xl ® ... ® Xk ® tf;, where tf; is a state on ®k+IAi. Moreover, if ep is continuous with respect to a C·-norm a(·) on ®i:IA" then tf; is continuous with respect to the C·-norm {1(.) on ®k+IAi' where {1(v) = a(®:=ll i ® v), Vv E ®~+IAi'

183

Proposition 3.5.6. Let Ai be a C*-algebra with an identity l i,1 < i < n,and AI'" ,A k be abelian (some k < n). Suppose that a(·) is a C·-norm on ®?=l Ai and ip is a pure state on a-(8)?=l Ai. Then cP = Xl (8) .•. (8) Xk (8) tP,

where Xi(') = cp(. (8) (8)j#i1;) is a pure state on A i , l < i < k, and tP(v) = cp( (8);=11, (8) v)(\iv E (8)~+IAd can be uniquely extended to a pure state on ,B-(8)~+lAi' and ,B(v) = a «(8);::::: 1 I, (8) v), \iv E (8)~+IAi' Let {11", H, €} be the irreducible cyclic * representation of A = a®?=IAi generated by cp. Since 1I"(A)' = (CIH and Ai is abelian (I < ". < k), it follows that there is a pure state (non-zero multiplicative linear functional) Xi on Ai such that 11" (. (8) (8) j #i 1 j) = Xi(.)1H , \i. EA.,

Proof.

By Corollary 3.5.5 and Xi(') = (11"(' (8) (8)j~ilj)€,€) = cp(. (8) (8);#i 1; ), 1 < i < k, we obtain cp = Xl (8) •.• (8) Xk (8) tP, where tP is a state on ®~+IAi' and is continuous with respect to ,B(.). Now it suffices to prove that tP is a pure state on ,B-(8)k+1 Ai' Let tPb tP2 be two states on ,B-(8)~+1 Ai and A E (0, 1) such that 1

< i < k,

Then for any U E (8)i:::::IA. and j IXI

(8) ••• (8) Xk (8)

tPi(U) 12 <

<

= 1,2, X (8) •.• (8) Xk (8)

tPj(U*U)

(t + 1 ~ )Jcp(u·u) < (t + 1 ~ )Ja(u}2.

Hence XI(8)" . (8) Xk (8) tPj can be uniquely extended to a state cp; on a-(8)i=lA i ,J' = 1,2. Clearly, cp = ACPI + (1 - A)CP2' Since cP is a pure state on a-(8)?=lAi, it tPl = tP2' i.e., tP is a pure state follows that cP = CPI = CP2' Furthermore, 1/J on ,B-®k+IAi' Q.E.D.

=

Lemma 3.5.7. Let A be a C*-algebra with an identity, and S(A) be the state space of A. Suppose that E is a a(A·, A)-compact convex subset of S(A) such that for any h· = hE A there is cP E E with cp(h) = max{A I A E a(h)}. Then E = S(A). Proof. Suppose that p E S(A)\E. By separation theorem, there is h* hE A such that

p(h) > sup{cp(h)

I cP E E}.

184

Clearly, p(h) < max{Al A E a(h)}. However, by the assumption sup{cp(h) I cp E E} = max{A I A E a(h)}. This is a contradiction. Therefore, E = S(A).

Q.E.D. Lemma 3.5.8. Let Al,···.A n be C"'-algebras, and a(·) be a C"'-norm on ®~=lAi' Then a(.) > ao('} on ®7=lA i if and only if for each CPi E S, (the state space of Ai), 1 < i < n, ®~=lCPi is continuous with respect to a(·).

Proof.

The necessity is clear from Corollary 3.2.7. Conversely, if ®?=l CPs is continuous with respect to o{), VCPi E Si, 1 ao(')'

Q.E.D. Proposition 3.5.9. Let Ai C(Oi), where OJ is a compact Hausdorff space, 1 < i < n - l,and An be a C"'-algebra with an identity In. Then there is only one C·-norm ao(') on ®~=lAi, and ao(') = A(')' and r-J

A- ®?=l Ai

r-J

C (0 1

X ••• X On-lJ

An).

Proof.

Let a(.) be a C·-norm on ®?=1 Ai' Fix Xi E Oi (Le., Xi is a pure state on As), I < i < n - I, and put

E =

{I Xn is a state on An such that } Xn ®?=lXi is continuous with respect to a(·) .

Clearly, E is a a(A~, An)-compact convex subset of Sn, where Sn is the state space of An. For any h"' = h E An, let B be the abelian C· -subalgebra of An generated by {In, h}. Pick a state tPB on B such that tPB (h) = max{ A I A E a(h)}. By Proposition 3.5.3, there is only one C"'-norm on ®~;ll Ai®B. Further by Corollary 3.2.7, ®7;lxi ® tPB is continuous on ®7;11 Ai ® B with respect to a(.). Thus ®~;llXi ® tPB can be extended to a state cP on a-®~=lAi' Clearly, cp(. ® ®j:;ei1j) = Xi('), 1 < i < n - 1. By Corollary 3.5.5, ip = ®?=lXi, where Xn is a state on An, and is an extension of tPB' In particular" Xn E E and Xn(h) = tPB(h) = max{A I A E a(h)}. From Lemma 3.5.7, E = S(A n) = Sn' From the preceding paragraph, ®?=lXi is continuous with respect to a(·), VX, E Oi,1 ao(') on ®?=lA i. However, if cP is a pure state on a-®?=l Ai, then by Proposition 3.5.6 we have cP = ®?=lXi, where Xi is a pure state on Ai, I < i < n. Further for any

185

a(uF = -

a(u*u) sup{cp(u*u)

I ip

is a pure state on a- @?=I Ai}

sUP{@?=IXi(U*U)

I Xi

is a pure state on A i,1

< i < n}

Therefore, there is only one C*-norm ao(') on @i=lAi' For each u E @~=I Ai, we can uniquely write that u = u (t b ... , t n - d, where u(tt,' .. ,tn-I) is a continuous map from 0 1 x ... x On-I into An. Clearly,

is a C*-norm on @~lAi' Therefore, ao(u) = llull,Vu E @~=IAi, i.e.,

ao- @?=I Ai ,...., C(OI

X .•. X

On-I, An).

Finally, since for any u E @i= I Ai

<

sup

t ))11 fn E A~,llfnll < 1 and} ( (t ... ,n { If nUl, t, E O. 1 < i < n - 1

sup

{I @i=1

l

fi(U) II

.,

_

_

t. E l Al'~1~lIn< 1, } =

-\(u),

Q.E.D. Theorem 3.5.10. Let Ai be a C*-algebra, 1 C*-norm on @i=IAi. Then on ®i=lA i,

Consequently,

<

t

< n,

and a(.) be a

of) is a cross-norm on @i=IAi .

Proof. It suffices to prove that ao(') < a(·). By Proposition 3.5.2, a(.) can be extended to a C"'-norm on ®i=lA'i, where A~ = Ai if Ai has an identity; and A~ = A i + ao(') on @i=IAi if k( < n - 1) C'"-algebras of {AI,' .. , An} are abelian. And we want to prove that the assersion holds for (k - 1). Let AI, ... , A k - 1 be abelian. Fix a pure state Xi on Ai, 1 < i < n - 1, and put E = {Xn I Xn E Sn, and @~=l Xi is continuous with respect to a(.)}, where

186

Sn = S(A n) is the state space of An. Then E is a a(A~, An)-compact convex subset of Sn' For any h'" = hE An, let B be the abelian C"'-subalgebra of An generated by {In, h}. Pick a state tPB on B such that tPB(h) = max{A I ,\ E a(h)}. Since k C"'-algebras of {All"', An-I, B} are abelian, it follows from the induction that a{) > ao(') on ®7;l Ai®B. Further by Lemma 3.5.8, ®7;lxi® 1PB is continuous on ®?;ll Ai ® B with respect to a(.). Thus ®r;;lXi ® tPB can be extended to a state cP on a-®?=IA i. From Corollary 3.5.5, ip = ®i=IXi, where Xn is a state on An, and is an extension of tPB' In particular, Xn E E and Xn(h) = tPB(h) = max{,\ I ,\ E a(h)}. By Lemma 3.5.7, E = S(A n) = Sn' Therefore ®i=IXi is continuous with respect to a(.), where Xi is any pure state on Ai, 1 ao(') on ®7=IA i. Q.E.D. Lemma 3.5.11. Let «I> be a * homomorphism from a C"'-algebra A onto a C"-algebra B. Then «I>'" is an isometric map from B'" to A"'.

Proof. Let I = {a E A I <1>(a) = O}. Then I is a closed two-sided ideal of A, and A/I is * isomorphic to B. Thus for any b E B,llbll =inf {lIalll a E A, (a) = b}. In consequence,

{b E Billbil < I} C <1> ({a E

Al II all <

I}).

Now for any 9 E B" , 1I«I>"'(g) II =

sup{lg(<1>(a)) I I a E A, Iiall < I}

> sup{lg(b)ll bE B, Ilbll < I} However, 1I <1>'" II = II <1> II < 1. Therefore, <1>'" is isometric.

=

IIgll. Q.E.D.

Proposition 3.5.12. Let a(.) be a C"'-norm on ®i=tAi. Then a"'(') is a cross-norm on ®i""IA;, and is independent of the choice of a(.). By Theorem 3.5.10 and Proposition 3.1.3, a"'(') is a cross-norm on ®?=IA;. From a o(') > a"'(') > ai('), it suffices to show that a;(·) = a o(') on ®i=IA;. Clearly, there is a * homomorphism <1> from al-®i=l Ai onto ao-®7=t Ai such that «I> (u) = u, Vu E ®i=l Ai. For any w E ®?=l A;, it is easy to see that a~(w) is the norm of w as an element of (ao-®i=lAi)"'. Now by Lemma 3.5.11,

Proof.

ai(w)

=

sup{lw(u)llu E ®~lAi, al(u) < I}

-

sup{lw(<1>(u))II u E ®i=lAi,at{u) < I} sup{I<1>*(w)(u)llu E ®i=:lAi, at{u)

II "'(w) II = a~(w),

< I}

Vw E ®i=lA;.

Q.E.D.

187

Notes.

Propositions 3.5.6, 3.5.9 and Theorem 3.5.10 are due to M. Takesaki.

References. [96], [150], [171]' [193].

3.6. Completely positive maps Let n be a positive integer, H n be a n-dimensional Hilbert space, and M n B(Hn ) be the algebra of n X n matrices.

=

Lemma 3.6.1. Let A be a C"-algebra, and n be a positive integer. Then there is only one C·-norm eto(') on M n ® A, and ao -(Mn ® A) = M n ® A. Moreover, if A is a C*-algebra on a Hilbert space H, then M n ® A is * isomorphic to the C·-algebra Mn(A) on H EB ••• EB H(n times), where

Mn(A) = {(aijh~i,j~nlaij

E

A, \:fi,i},

and

Mn(A)* where ((lij), (aij))

:=

Mn(A *) = {(lij h~i,j~nlhj E A*, \:fi,i},

= L:ij lij(aij)'

Proof. Let {eijll < i,j < n} be a matrix unit of M n , i.e.,

Then each u E M n ® A can be uniquely expressed by u = L:ij aij ® eij , and we get a * isomorphism from M n ® A onto Mn(A) : (u) = (aij)' Clearly, Mn(A) is a C·-algebra on H EB'" EB H(n times ). Define Ilull = 11~(u)!1 = II(aii)ll, \:fu E Mn®A. Then Mn®A is a C·-algebra. By Proposition 2.1.10, there is only one C·-norm eto(') = 11·11 on Mn®A, and eta - (Mn®A) =

Mn®A. The rest conclusion is obvious.

Q.E.D.

From now on, we shall identify M n ® A with Mn(A).

Proposition 3.6.2. Let n be a positive integer, A be a C·-algebra, and a = (aij) E Mn(A). Then the following statements are equivalent: 1) a is a positive element of Mn(A); 2) a is a sum of matrices of the form (a;aj) with al, ... an E A; 3) x;aiixj > 0 in A, \:fxI,"', X n E A.

L

i,j

188

aij = L b;ibli;j, Vi,j. Ii;

Put

CIi;

= (bkibli;j) , 1 < k < n.

Then a

= Cl + ... + Cn'

2) => 3). It is obvious. 3) => 1). For any cyclic * representation {1f,K,c} of A, define a sentation {1f, K E9 •.. E9 K(n times) } of Mn{A) :

1f((bi j ) ) = (1f(b ij)),V(bij) For any

6,'" Cn E K, Pick

x!:2

E

* repre-

Mn(A).

E A such that

1f(x~)c ~ Ci, 1 O. m ij

Thus 7f(a) is a positive operator on K E9 .•• EB K(n times ). Now let {1ft} be a family of cyclic * representations of A such that 1f = Ll EB 1f l is faithful for A. Then 7f = Ll E9it-, is also faithful for Mn{A). From the preceding paragraph, ;r(a) > O. Therefore, a is a positive element of Mn(A).

Q.E.D. Definition 3.6.3. Let be a linear map from a C*-algebra A to a C*algebra B, and n be a positive integer. Naturally, define a linear map n from Mn(A) to Mn(B) :

n((aij)) = ((aij)), ~ is said to be

positve, if

~

V(aij)

n-positive, if ~n(Mn(A)+) C

E

Mn(A).

Mn(B)+.~ is said to be

completely

is n-positive for any positive integer n.

Proposition 3.6.4. 1) If is a * homomorphism from A to B, then is completely positive. 2) The composition of completely positive maps is completely positive. 3) Let {11", H} be a * representation of A, and v be a bounded linear map from a Hilbert space K to H. Then (.) = v*1I"(')v is a completely positive map from A to B( K).

Proof. 1) It is clear since

n is also a

* homomorphism from Mn(A)

for any positive integer n. 2) is obvious.

to Mn(B)

189

3) For any n, at,' .. an E A and b1 ,

bn E B(K), Lb;CJ)(a;aj)bj = (L1T(ai)vb i (l::1T{ai)vb i) > o. i,j i t ' ",

r·

Now by Proposition 3.6.2, we can see that CJ) is completely positive.

Q.E.D.

Lemma 3.6.5. Let A, B be C*-algebra, and CJ) be a positive (i.e. I-positive) linear map from A to B. Then CJ) is continuous. Proof. It suffices to show that is a closed operator. Suppose that an ~ 0 in A, and ~(an) ~ b in B. For any positive linear functional f on B, f 0 CJ) is a positive linear functional on A. Thus I 0 is continuous ( Proposition 2.3.2) , and I 0 ~(an) ~ o. Therefore I(b) = 0, VI E B*, and b = 0, i.e., ~ is closed. Q.E.D. Proposition 3.6.6. Let A, B be C*-algebras, and be a positive linear map from A to B. If either A or B is abelian, then ~ is completely positive. Proof. Let B :::: Co(O), where 0 is a locally compact Hausdorff space. For any n, ah'" ,an E A, b1 , ' .. bn E Band tEO, notice that

(Li,i b:~(a:aj)bi)(t) = Li,i ~(a;ai)(t)bj(t) = CJ)({Li bi(t)ad* . (Li bi{t)ai))(t)

> O.

Therefore, ~ is completely positive. C;'(O), and B C B(H). For any n, ah"', an E A, Now suppose that A and 6,"" €n E H, we need to prove that f"<J

L(CJ)(a;aj)€j, €i) > o, i,i By Lemma 3.6.5, for any t.i E {I"", n} there is a finite Radon measure J.lii such that

((a)€;, €i) =

£a(t)dJ.Lij(t) ,

Va E A.

Let J.L = Li,jIJ.Lijl· Then there is!i; E L1(O,J.L) such that J.Lij = Fix )q, ... ,An E (C. Since CJ) is positive, it follows that

/ii ·J.L,Vi,i·

In la(t)12 d(Li,i Ai AjJ.Lij (t)) ( (a*a)(Li Ai€i), (Li Ai€i))

> 0,

Va E A. Thus Li,i AiAjJ.Lij is a positive measure on O. Further, Li,j AiAj/ij(t) 0, e.e.u, So we can find a Borel subset 0 0 of n such that

J.L(00) = 0,

and

L i,j

Ai>"j fij (t) > 0,

>

190

Vt tJ. 0 0 and any complex rational numbers AI,· .. ,An. But any complex number can be approximated arbitrarily by complex rational numbers, hence we have

L

i,j

AiAjftj(t) > 0, Vt (j. 00, AI,"·, An E (C.

Therefore, L:i,j ( ( L:i,j

a;aj) €j, €i)

Jo (a;aj)( t) Iii (t) dJ.L( t)

J0 (L:i J ad t)aj ( t) I, j ( t)) dJ1. (t) > o. Q.E.D. Theorem 3.6.7. Let A be a C* -algebra, K be a Hilbert space, and ~ be a completely positive linear map from A to B(K). Then there exists a * representation {1r, H} of A , a normal * homomorphism \II from the VN algebra B = ~(A)' to B(H) .and a bounded linear operator v from K to H, such that

cI>(a)

= v*1r(a)v, Va E A, (b)v = ub, Vb E B,

and <1>(B) C 7f(A)', H = [7f(A)vK], Ilvll = 11<1>11 Moreover, if A has an identity 1, and (1)

1

(2.

= lK, then

v can be isometric.

Proo]. Let A ® K be the algebraic tensor product of the Banach spaces A and K. Define

(L a, ® ~i, L b ® 7]j) = j

i

Vai, bj

E

A, Ei' 7]j

E

i,j

j

K. Since

L(<1>(bj ai)€i' 7]j),

~ is completely positive, if follows that (,) is a

non-negative inner product on A ® K. Let N = {x E A ® K I (x, x) = O}, and let x ---t X = X + N be the canonical map from A ® K onto (A ® K)IN. Then we get an inner product on (A ® K)IN,

(x /[1) = (x, y), Vx,:y E

A ® KIN, x Ex, Y E y.

Denote the completion of (A ® KIN, (,)) by H, and let

7f(a)L: i ~® Ei = L:i ~e €i,

\II(b)Ei;;(® €,

=

Ei a7® bE"

Va E A, b E B = (A)'( C B(K)), and «; E A, €i E K, Vi. Since is completely

191

positive, it follows that

a

o

•

<

o

o

o

a

Thus 7r(a) can be uniquely extended to a bounded linear operator on H, still denoted by 7r(a) . Clearly, {7r, H} is a * representation of A. From B = cI>(A)', we have n

n

L (b·b~(a;aj)ej, ei)'

\II(b)L ai ® ei i=l

i,;=l

But (cI>(a;aj)h$i,j$n is a positive element of Mn(B'), so we can write that

where b~i E B', 1

< i,y" < n.

2

n

\II(b)

Then

L

a,

e ei

i=l

(C'

b :

b.J (b:;l

< Ilbll' ((b:;l (

CJ CJ} ,(b:;l

;~ ) ,(b: ;~) 1= Ilbll' ·11L:i=l7." 0 cill' . j ) (

Hence \II(b) can be also extended to a bouned linear operator on H, still denoted by \II(b). Clearly, \II is a * homomorphism from B to B(H), and

\II(B) C 1r(A)'.

192

If {btl is a bounded increasing net of B+, then {\lI(b,)} is a bounded increasing net of B(H)+. For any ai E A, Ei E K, --.-.

,-....;

(\lI(bdLai ® Ei,La; ® E;) = L(~(a;ai)bIEi' E;)· i,;

i

i

Picking the limit for I, we can see that

i.e., \lI is normal. Now suppose that {d ,} is an approximate identity for A . By Lemma 3.6.5, {~(dt)} is a bounded increasing net of B(K)+. Thus SUP, ~(d,) = (strongly) - lim, ~(d,), Let K ----+ H be as follows

v, :

--.-.

vIE = d, ® E, 'IE E K. It is clear that IIvlEI12 = (<1>(dnE, E) < !I<1>!I·IIEI1 , and Ilvdl < 11~111/2 ,VI. If I' > l, then (dl' - dz)2 < d - d, and " 2 II(VI' - v')E1I < ((<1>(d ,,) - <1>(d,))E, E) ~ 0, 'IE E K. 2

1/2

Thus there is v : K ----+ H with Ilvll < 11<1>11 such that Vt --+ v ( strongly) . Since (vt a® E, TJ) = (<1> (d, a) E, 71) --+ (<1> (a) E, 71 )

----

e

VTJ E--.-. K, VI, it follows that v*a ® E= <1>(a)E, Va E A, E K. Thus v*7r(a)vIE = v*ad, ® E= ~(ad,) E, Vl, 'Ie E K and a EA. Further,

<1>(a) = v*7r(a)v, Va E A.

2

Consequently, 11<1>11 < llvl1 . Hence Ilvll = 11<1>11

(7r(a)vE, b017) = lim(ad;® l

1/ 2 • For any a, s e A, E, TJ

E

K,

E, b®TJ)

= (~(b*a) E, TJ) = (a® E, b~ E). --.-.

Hence 7r(a)vE = a ® E, Va E A, E E K . Consequently, [7r(A)vK] = H, and {7r,H} is nondegenerate for A. By Proposition 2.4.6 7r(d ,) ---4 IH (strongly). Notice that 7r (

d,) \lI ( b) v E= \lI ( b) 7r ( d,) v E= d, ® bE =

7r (

d,)vbE,

Vb E B, cE K, Vl. Therefore, \lI(b}v = vb, Vb E B. Finally, if A has an identity 1, and ~ (1) = lK, we can pick d, = 1, VI. Then vc = 1® C, 'IE E K. Thus v is isometric. Q.E.D.

193

Proposition 3.6.8. Let A, B be C·-algebras, and linear map from A to B. Then ~(a)*~(a)

<

II~II ~(a*a),Va

~

be a completely positive

EA.

Proof. We may assume that B C B(K) for some Hilbert space K. Then by Theorem 3.6.7, ~(a)·~(a)

= v*~(a*)vv*~(a)v < IIvll 2 v·~(a*a)v

= II~II ~(a*a),

Va E A.

Q.E.D. Lemma 3.6.9. Let A be a C·-algebra, B be a C*-subalgebra of A, and {~, H} be a * representation of B . Then there is a Hilbert space H 1 and a * representation {~l, Hd of A such that HI ::) Hand ~r(b)E

=

~(b)E, Vb E B,

E E H.

Proof. We may assume that {~, H} is generated by a state lp on B. Then lp can be extended to a state1/; on A> Let {~l' H d be the * representation of A generated by 1/;. Then {~l , HI} is what we want to find. Q.E.D.

Proposition 3.6.10. Let A be a C*-algebra, B be a C* -subalgebra of A , and ~ be a completely positive linear map from B to B(K) ( K some Hilbert space). Then ~ can be extended to a completely positive linear map from A to B(K). Proof. By Theorem 3.6.7, there is a * representation {~, H} of B and a v : K --., H such that ~(b) = v*~(b)v, Vb E B. Let {~b HI} be a * representation of A satisfying Lemma 3.6.9, and P be the projection from HI onto H. Define

w(a)

=

v· P~I(a)Pv, Va E A.

By Proposition 3.6.4, W is a completely positive linear map from A to B(K). Clearly, W is also an extension of ~. Q.E.D.

Proposition 3.6.11. Let ~i be a completely positive linear map from Ai to B"l i can be extended to a completely positive linear map from Qo - 1817=1 Ai to Qo - 1817=1 Bi, Proof. Let B, C B(K,), 1 < i < n. By Theorem 3.6.7, there is a tion {~i, Hi} of Ai and Vi : K, ~ Hi such that ~,(ai) = v:~,(ai)vi, Vas E

Ai, 1 < i < n.

* representa-

194

By Proposition 3.2.8, ®7=171"i can be extended to a ®7=1 Ai' Suppose that

* representation

of ao -

(a) = (®f=1Vi)*' ®f=171"i(a) . ®f=1 Vi, Va E ao- ®f:1 Ai' Then is a completley positive linear map from ao - ®;=1 Ai to B( ®7=1 K i ) , and is an extension of ®7=1i. Moreover, since (®i=1Ai) c ®7=1Bi, and ao - ®i=1 B, is the uniform closure of ®7=1 B, in B( ®7=1 Kd, it follows that (ao -®i=1Ai) C aO-®7=1Bi' Q.E.D.

(3») E M n (A) +,1 < L emma 3.6.12. L et as = (aij

i, k,l and

8

i= 8'. Then

8

< m, aij(s) akl(s')

(s') (s)

\..I'

= akl aii ' vs,

Proof. It suffices to prove this lemma for m = 2. Let x = (xii), Y = (Yii) E Mn(A)+, and xiiYkl = YklXij,Vi,i, k, I, we need to prove (XiiYij) E Mn(A)+. Suppose that Band C are the C*-subalgebras of A generated by {Xii I ~',i} and {Yi,i I i, i} respectively. Since x:j = Xji, Ykl = Ylk Vi, i. k, I, it follows that be = cb, Vb E B, c E C. Clearly, x E Mn(B)+, Y E Mn(C)+. Then we can write Xij

= Lbkibkj,Yij = LC:iCkj,Vi,j, k

k

where bki E B, Cki E C, Vi, i, k. Therefore,

(XijYij)

= L((bkicu)* . (bkicli)) k ,I

Q.E.D.

is a positive element of Mn(A) by Proposition 3.6.2.

Proposition 3.6.13. Let i be a completely positive linear map from Ai to B , and i(aj)j(ai) = j(aj)i(ai)' Vai E Ai,aj E Aj,l < i i= j < n. Dentine (®i=lai) = ni=l(ai),Vai E Ai,l < i < n. Then can be extended to a completely positive linear map from al - ®i=1 Ai to B.

Proof. Let B, be the C*-subalgebra of B generated by (Ad, 1 < i < n. Then bibj = bibi, Vbi E Be, bi E B j, 1 < i i= i < n. First we say that is positive from ®7=1 Ai to B. In fact, let u = L~I n aj(i) , were h • Th ®i=1 aj(i) E A i, \..I' vt,J. en m

om( * ) -~ "" om ( a (l)* a(I}) ... 'J.'n om ( a (n)* ak (n») 'J.' U U L- 'J.'1 j j k

i,k=l

.

195

n)), Define bile = da~l)* a11 ) ) ••• n(a~n).a1 Vj, k. By Lemma 3.6.12, (bj le) E M m (B)+. Further by Proposition 3.6.2, Erle:1 dlbjled l E B+, VI, where {d l} is an approximate identity for B. Thereofore, (u*u) = Ei,1e bile E B+. Now for any positive linear functional p on B, po is positive on ®i=1Ai' By Proposition 3.4.6, po
can be uniquely extended to a bounded linear functional on al -®i=l Ai for any I E B*. Then we can define a linear map '(/)

=f

0

<1>, VI E B*.

We claim that <1>' is closed. In fact, let lie (E B*) ---+ 0, and '(IIe) fir. 0

F(®?=lai) = li,F/le(n?=li(ai)) = 0, it follows that F = 0, i.e., <1>' is closed. So <1>' must be continuous. For any U E ®i=l Ai, notice that

= sup{lf 0 (U) I I IE B*, IIIII < I} = sup{I'(/)(u) I 1IE B*, 11/11 < I}

1I<1>(u)II

< 11<1>'11 al(u), Hence,
. Finally, we prove that «I> is completely positive from al-®i:IAi to B . By the continuity of and Proposition 3.6.2, it suffices to show that for any m, Ut,' .. ,U m E ®i=IA i and b1 , ' .. ,b m E B, m

L

i ,i =1

b;(u;uj)bi E B+.

Suppose that Ui = E1=1 ®~=l a~Z), 1 < i is completely positive , it follows that

< m, where a~~) E As, Vi, k , s. Since i

(8). (8))) ( 8 ( aile ail l$i,i$m,I$Ie,I$p E

M mp( B ) +,

1 < s < n. By Lemma 3.6.12. (nn ,,=1 Let bile = bi, VI

8

(a~8). a'.8))) 1 ,Ie

1

1
< i < m, 1 < k < p. m

L i,j=1

m

b;(u;uj)bj =

E M

pm

(B)

+.

Then n

p

L L i ,i = 1 1e,I=l

b;/c

II 8(a~~)*a~:))bil' ,,=1

196

By Proposition 3.6.2, it is a positive element of B.

Q.E.D.

Notes. Theorem 3.6.7 is due to W. Stinespring. The recognition of the importance of completely positive maps in the tensor products was due to E.Effros and C.Lance. Proposition 3.6.10 and its generalization are due to W.Arveson.

References. [8], [37], [95], [163].

3.7. The inductive limit of C*-algebras Let JI be a directed index set, and A a be a C·-algebra for each a E JI. And also there is a * isomorphism tf.>f3a from A a to Af3 if a < {3( a, (3 E JI) such that tf.> 7f3tf.>f3a = tf.>7 a, Va,{3"

E JI and a <{3 < ,.

Let

J = xaEUAa = {(aa)aEI lao E A a, Va Ell}. By the addition, multiplication and naturally a * algebra. Let

t

= {(all E J

I

* operation on each component, J

there is an index a such that af3 = tf.>f3a(aa), V{3

becomes

> a.}

Clearly, l is a * subalgebra of J. For a = (aa) E L, we can define that lI all = lima Ilaall· Obviously, 11·11 is a C*-seminorm on L. Further, let

-a

Illall = O} E J I there is an

=

{a E l

=

{(al)

index a such that ap

= 0, V{3 > a.}

Clearly, -a is a * two-sided ideal of L. Denote the canonical map from l onto l/{} by a ~ a = a + {}(Va E l). Then Iiall = lIall (Va E l/{), a E a) is a C·-norm on l/{}. Further, we get a C*-algebra A = (l/{), 11'11)-.

Definition 3.7.1.

Denote the above C·-algebra A by

It is called the inductive limit of {A a

Ia

E JI} defined by the family of

isomorphisms {~f3la I (a,p) E II X JI,and a <{3}. Now for any a E II, let

*

197

* subalgebra of L, and £Q/f} = {a E £/f} I there is (a,) E a such that

Clearly, £Q is a

AQ =

a, = cI>lQ(a Q) if 1 > a; a, = 0 if 1 l a.}

For any a E A Q

Q ,

define if 1 > a, if I l a.

---

Then cI>Q(a Q) = (al)'EI(Va Q E A Q ) determines a * isomorphism from AQ onto A Consequently, A becomes a C*-subalgebra of A. We claim that cI>Q = cI>,BcI>,BQ' Va < {J. Q •

Q

In fact , for any a Q E AQ and {J

>a

we have

and

Moreover, since £ = UQEI£Q, it follows that £/f} = UQEllAQ• Thus, we have the following.

Theorem 3.7.2.

Let A = 1ig1{A Q, cI>,BQ I (a,{J) E JI X JI,and a <{J}. Then a

there is a family {A Q I a E JI} of C*-subalgebras of A, and a * isomorphism cI>Q from AQ onto AQ for each a E II, such that: 1) AQ C A,8, Va < {J; 2)cI>Q = cI>,BcI>,BQ,Va < {Jj3) UQEI AQ is dense in A. Conversely, we also have the following.

Theorem 3.7.3.

Let A = li'!p{AQ,cI>,B,Q

I a,{3

E JI, and a <{J},B be a

a

C* algebra, {B rx I a E II} be a family of C* -subalgebras of B, and \Ifrx be a * isomorphism from AQ onto Bet) Va E II. Assume that: l)B C B,B, Va < {J; 2) \Ifrx = \If,BcI>,BQ' Va < {J; 3) UrxEJ" B Q is dense in B. Then there is a * isomorphism \If from A onto B such that \If(AQ) = B rx, \IfcI>rx = \IfQ' Va E JI, where {A Q , cI>Q I a E II} is defined by Theorem 3.7.2. Q

198

Proof. For any a E JI, Wa
In fact, wt1~~l(a) = Wt1
Clearly, \}J is isometric. Thus W can be uniquely extended to a * isomorphism from A onto B, still denoted by W. Then W is what we want to find.

Q.E.D. Corollary 3.7.4. Let A be a C*-algebra, and {Ace I a E JI} be a family of C*-subalgebras of A with A a C A t1, Va < (3, and UceA cr = A. Denote the embedding map from Ace into A,a by «I> t1ce, Va < (3. Then A is * isomorphic to 1ig1{Ace,
Proof. Using Theorem 3.7.3 to the case: B = A, B ce = A cr, and Wce = Ice ( the identity map on A a ) , Va E II, we can get the conclusion immediately.

Q.E.D. Theorem 3.7.5.

Let A

=

lim{A cr, «I>t1ce

I

a,(3 E Il, a < (3}, and B =

a

lig1{B ce , W,sce

I a,(3 E JI, a < (3}. Suppose that there exists a * isomorphism /\a

a

from A a onto Bart/a E II) such that /\,s
= wt1ce

/\ce (Va < (3). Then A is

*

Proof. By Theorem 3.7.2, A = UcrA a, B = UceB cr, and there is a * isomorphism
Then wa/\ce is a

* isomorphism from

;v

A a onto B ce, Va E Il, and

Now by Theorem 3.7.3, A is * isomorphic to B.

Q.E.D.

Let C, D be two algebras with the identities lel ID respectively. A linear map (Ie) = lD.

199

= 1ig1{A n ,
Let A

Proposition 3.7.6.

1,2,"" m

> n},

and

n

B

= 1igl.{Bn , \limn I m, n = 1,2"", m > n}. n

Suppose that An and B n are * isomorphic to M pn , where M p n is the algebra of Pn X Pn matrices, Vn, and n. Then A and B are * isomorphic. Proof. By Theorem 3.7.2, we can write A and Bn are * isomorphic to M p n , Vn, and l A E Al C ... C

An

IB E

C .. "

Now it suffices to construct a n such that

= UnA n and III C

* isomorphism /\n

/\n+ 1

I An

... C

B

= UnB n,

Bn C

where An

....

from An onto B n for each

= /\n, Vn.

Suppose that /\1,"', /\n satisfy the conditions. We need to construct /\n+l such that /\n+l I An = An· Let {ei; 11 < i, j < Pn} be the matrix unit of An , i.e.,

We identify ..4n+1 with M p n +1 = B(H), where H is a Hilbert space of Pn+1dimension. Then {e;i I 1 < i < Pn} is a family of pairwise orthogonal projections on H, and L; e;; = I H , and ei; ~ ei; (relative to the VN algebra B(H)) , Vi,j. Put Hi = e;;H, Vi, then dim H; = m is independent of i. Clearly, m = p;:1pn+b i.e., v« I Pn+1' Now we can pick {VIi,lj I 1 < i, j < m} C An+1 such that V~i,lj =

VI < i,j, k, 1 < m, and

Vli,1;, Vli,liVlk,ll

I:~l vIi,li =

=

Ojk Vli,lh

en. Further, let

VI < i, k < Pn, 1 < j,l < m. It is easy to check that {Vij,kl 11 < i, k < Pn, 1 < i, 1 < m} is a matrix unit of An +1 • Put Iii = I\n(e;j) , Vi,j. Clearly, {Iii 11 < i,i < Pn} is a matrix unit of B n. Similarly, we can find {Uli,I; 11 < i, i < m} C Bn + l such that

VI <

i,i, k, I < m,

and

Uii,kl

L~I uli.li =

In.

Furhter , let

= /;lUli,l, Ilk , 1 < i, k < Pn, 1 < i, I < m.

Then it is a matrix unit of

Bn + l .

200

Finally, define

Then I\n+l is a

Now let A

* isomorphism from

=

1i!!I{Ace,~pce

I a,{3

A n+ 1 onto E ll,a

Bn+ll

and I\n+l

I An =

An·

Q.E.D.

< {3}. And suppose that Pa is a

Q

state on Ace, Va. E II, such that

Further , define a state Pce on Ace as follows:

<Pa(aa) Va E ll, where {Ace, ~a

Ia

= Pa(~~l(ace)), Vaa E Aa ,

E II} is defined in Theorem 3.7.2. Then we have

pp(~~l(ace)) = pp(ep~l~ce~~l(aa))

= pp(~,Bl
Vace E Ace, a < {3. Thus, we obtain a linear functional P on UceA a defined by {<Pa I a E II} : p I Ace = tPa' Va E ll. Clearly, p can be uniquely extended to a state on A, still denoted by p.

Definition 3.7.7. The above state

ip

on A = 1ig1{Ace ,

I a.,{3 E

II, a <,B}

a

is called the inductive limit of {Pce'
Notes.

The inductive limit of C·-algebras was studied first by Z.Takeda.

References.

[150], [167].

3.8. Infinite tensor products of C*-algebras Let 1\ be an index set, and Ai be a C*-algebra with an identity Iz, Vl E 1\. Let

201

We say that U = UF ® ®z{lF1 z = 0, if ®ZE"CPZ(U) = ®ZEFCPZ(UF) = 0, \/cpz E S, ( the state space of Az), l E 1\. Clearly, ®IE"A z becomes a * algebra in a natural way, and is called the algebraic tensor product of {A, Ii E I\}. A norm 11·11 on ®IE"A, is called a cross-norm, if for any finite subset F of A and a, E Az, \/i E F, we have

For example,

A(u) =

sup{I®IEFfz ® ®Z{lFCPI(U) I I F is any finite subset of 1\;

II E Ai, IIfdl < 1, \/i E F; CPZ E Sf, \/t ¢ F}

and I(U)

= inf{2:TIzEF Ila;l} I I U =

L

j

®zEFa;l) ® ®'{lF 1,}

j

are two cross-norms on ®lE"Az, and 1(') is the largest cross-norm. A norm a{.) on ®'E"A , is called a C'--norm , if a(uv)

< a(u)a(v),a(u·u) = a(u)2,

Vu, v E ®IE"A I • Then the completion of (®IE"Az, a{')), denoted by a - ®lE"A z , is called the tensor product of {A, It E I\} with respect to a{·).

Similar to the discussion in Sections 3.1 -3.5, we can see that: there is a spatial Ct-norm ao(') on ®IE"A ,; and any C" -norm a(.) must satisfy the inequality A(.) < ao(') < a(·) < 1(') on ®'E"A ,; in particular, any C·-norm a(·) is a cross-norm ; and etc. Now let II = {F I Fis any finite subset of I\}. II is a directed index set with respect to the inclusion relation. Suppose that a(·) is a C·-norm on ®IE"A ,. For any F E II, let aF(uF)

= a(uF

® ®l{lF11), \/UF E ®IEFA ,.

Then aF(') is a Ct-norm on ®IEFAz, \/F E II. Put B F = aF - ®IEFAI, \/F E ll, and for any F', FEll with F ' ::J F, define ~F'F( UF)

= UF ® ®'EF'\F1

"

\/UF E ®'EFA,.

Then fPF'F can be uniquely extended to a unital * isomorphism from B F into BF' , and fPFIIF = ~FItF'~F'F, \/F" ::J F ' ::J F. By Theorem 3.7.3, it is easy to prove the following.

Proposition 3.8.1.

a - ®IE"AI is

1ig1{BF , ~F'F F

* isomorphic to

IF', FEll, and

F c F'}.

202

As an example, we consider the following. Let A be a C*-algebra with an identity. A is said to be (U H F) ( unIformly hyperfinite ) , if there exists an increasing sequence ( i.e. An CAm, Vn < m) of C*-subalgebras An containing the identity such that the uniform closure of UnA n is A and An is * isomorphic to B(HnH dimHn < oo),Vn. If dimHn = p«, Vn, A is also called a (UHF) C*-algebra of type {Pn I n = 1,2", .}. Let mn be the embedding map from An into Am, Vn < m. Then A is * isomorphic to 1i.!:"9{A n,mn I m,n = 1,2,···,n < m}

Definition 3.8.2.

n

by Corollary 3.7.4. Moreover, from Proposition 3.7.6 the (UHF) C*-algebras of same type are * isomorphic.

Proposition 3.8.3. A (UHF) C*-algebra of type {Pn} is existential if and only if Pn I Pn+b Vn. And it is * isomorphic to the infinite tensor product aD ®~lMmn' where M mn is the algebra of m n x m.; matrices, and ml = PI, m n = P;;~IPn, Vn > 2. Proof. The necessity is contained in the proof of Proposition 3.7.6 indeed. Conversely, let Pn I Pn+h Vn. Since aD - @i=lMm.. = M pn, Vn, it follows that ®~lMmn is a (UHF)C*-algebra of type {Pn}. Moreover, all (UHF) C*-algebras of same type are * isomorphic. Therefore, the (UHF)C*-algebra of type {Pn} is * isomorphic to aD -®~IMmn'

Q.E.D. Notice that ®i=IMm i is finite dimensional, Vn. Thus from Proposition 2.1.10 there is only one C*-norm ao{') on ®~=lMmn indeed. In chapter 15, we shall discuss the (UHF) algebras furthermore.

Definition 3.8.4. Let /\ be an index set , and HI be a Hilbert space , Vt E /\. Fix EI E HI with II elll = 1, Vi E /\. The tensor product of {HI It E I\} with respect to the vector of (0fEI\H I , (,)), where 0;EI\HI

=

e = (EI)IEI\, denoted by @fEI\Hz, is the completion

U{0 1EFHI

®

@lgF6

IF c 1\, # F

<

oo}

and (@IEFTJl @ @lgF6, VTJL, ~l E

Hl,l E F, and Fe

1\, #

@IEF~1 @ @lgFEI) = II IEF ( 7] I , ~l)'

F <

00.

203

Lemma 3.8.5 Let M, N be VN algebras on Hilbert spaces H, K respectively, and f E M .. , 9 E N .. . Then I Q9 9 can be uniquely extended to a a-continuous linear functional on Mfi9N with norm 11/11 . IIgll, where (I Q9 g)(x fi9 y) = f(x)g(yL Vx E M, yEN. Proof. Pick s E T(H), and t E T(K) such that I(x) = tr(sx),g(y) = tr(ty), Vx E M,y E N.

It is obvious that sfi9t E T(HQ9K), lis Q9 till =

IIslll·lItlll' and tr((sQ9t)(XQ9Y)) =

f(x)g(y), Vx E M, YEN. Clearly, cp(.) = tr((s fi9 t)·) E (Mfi9N) .. , and cp is the unique extension of I e g on M Q9 N with Ilcpll > IIIII . Ilgli. However, IIcplI < lis e till = IlslIl ·lI tlll , and Il s111' Iltll l can be approximated arbitrarily to 11/11 ,llgll respectively. Therefore, Ilcpll = 11/11· Ilgll· Q.E.D.

Proposition 3.8.6. Let H = Q91EI\Hz, M , be a VN algebra on H,(Vl E /\), and M be the VN algebra on H generated by {M, e ®lt~ll,t Il E /\}. Then we have the following: 1) M' is generated by {M! Q9 Q9lt~1111 Ii E A}; 2) M = B(H) if and only if M, = B(H,}, Vi E /\; 3) M is a factor if and only if M , is a factor, VI E /\. Proof. 1) Let F be a finite subset of /\. Clearly, H = (Q9 IEFH,) Q9 (Q9f~FHI)'

M = M Ffi9MI\\F,

where M F is the tensor product of {M, It E F}, i.e., M F = fi9 IEFMI, and M"\F is generated by {Ml Q9 Q9jt~F,II~111' It (j. F}. By Theorem 1.4.12, M'

= M'-M' Ffi9 I\\F'

Let gF(') = (. Q91~F 6, Q91~F6) E (M~\F)'" By Lemma 3.8.5, there is a linear map ~F from M' to M} such that ~F(x)(IF) = (IF Q9 gF)(X), Vx EM', IF E (M~)*.

For any '1 = '1F ® ®1~F6, ~ = ~F Q9 ®1~F 6 E 0;E"H" where '1F, ~F E 0 IEFH" define t· = (', TJ)~, V· E H, tF' = (', TJF)~F' V· E ®,EFH,

and t"\F' = (', fi91~F6) Q91~F

6, v·

E

®f~FH"

Clearly, t, tF, t"\F are one rank operators on H, ®'EFH" ®f~FHI respectively, and t = tF ® t"\F, and gF(') = tr(t"\F')' V· E M~\F' Let IF{') = tr(t F·) E (M~)., and

PF

=

fi91~Fl"

204

Then for any z E M' , we have

tr(t· cI>F (x)

@

PF) = tr(tF . ~F (x)) . tr(t/\\F . PF)

= ~F (x)(IF)

= (IF ®

ar )(X)

= tr(tx).

From the preceding paragraph, we can see that

lim tr(t . ~F(X) F

@

PF) = tr(tx),

"Ix EM', and anyone rank operator t on H with form t- = (', TJ)~ for some t7,~ E 0;E/\H,. Since {~F(X) @PF I F} is bounded, it follows that ~F(X) @

PF

---4

x(u(B(H), T(H)), "Ix EM'.

By Theorem 1.4.12, cI>F(X) c M} = @,EFM!, "Ix E M'. Thus

M' c {M:

@ @":;t,ll'

It E A}".

However, it is obvious that M,' @ @l':;tlll' eM', "It E A. Therefore, M' is generated by {M{ @ @1':;tll' 1 It E A}. 2) M = B(H) if and only if M' = (CIB' By 1), This is equivalent to M; = (cl" "It E A, i.e., M 1 = B(H,), Vt E A. 3) M is a factor on H if and only if {M,M'}" = B(H). By 1) , this is equaivalent to the fact

{(M1 U M;) @

@ll:;t,l"

It E

A}" =

B(H).

By 2) , the fact is equivalent to (M1 UM{)" = B(H,), Vt E A, i.e., M , is a factor on H" "It E A. Q.E.D.

11" \p

A state cP on a C·-algebra is said to be factorial ,if the * representation is factorial ( Definition 2.10.5) , where 11" \p is generated by cpo

Proposition 3.8.'1.

Let A be an index set, A, be a C·-algebra with an identity 1, , and fIJI be a state on AI, Vi E A. If a(-) is a C·-norm on @'E/\A" and A = a-®IE/\AI, then @IEt,CPI can be uniquely extended to a state fIJ on A. Moreover, cP is pure or factorial if and only if CPI is pure or factorical , VI E A. Proof. Let {7I"z, HI, €I} be the cyclic A, and

* representation of A, generated by CPl, VI E

H = @tE/\Hz,

11"

=

®IE/\ 11""

where € = (€,),E/\' By Proposition 3.2.8 and a(·) > ao(')' {11", H, * representation of A.

€}

is a cyclic

205

By Corollary 3.2.7 and a(·) > ao('), l8llE"~l can be uniquely extended to a state ~ on A. Let {1I"cp,Hcp,€cp} be the cyclic * representation of A generated by ~. Since

Vu E l8llE" A" {11", H} is unitarily equivalent to {11" CP' Hcp}. That comes to the conclusion from Proposition 3.8.6 immediately. Q.E.D. Remark.

By Proposition 3.8.1 , A = a - I8l,E"A j is

B = 1ig1{BF , (}) F IF I F',F E 1I and

* isomorphic

to

Fe F'}.

F

Clearly, l8llEF~l can be uniquely extended to a state ~F on BF = aF-l8llEF~I, V FEll. By Definition 3.7.7, the state ~ = l8llE"~l on A is corresponding to the inductive limit state lim{~F' (})F'F

I F, F'

E lI, and

F

c F'}

F

on B. Notice that the algebra M n of n x n matrices is simple, i.e., M n contains no non-zero closed two-sided ideals (see Proposition 1.1.1 ) . So each non-zero * representation of M n is faithful. In consequence, each state on M n is factorial. Therefore, we have the following.

Proposition 3.8.8. @~=l Mm.,.. is factorial.

Each product state on a (UHF) C·-algebra ao -

Notes. (UHF) algebras first appeared in the thesis of J.Glimm. The corresponding notion for VN algebra goes back to F.J. Murray and J.Von Neumann, see Chapter 7. J.Dixmier considered inductive limits of matrix algebras without the demand that the embeddings preserve units. Later, O. Bratteli gave a study of C·-algebras which are inductive limits of arbitrary finite-dimensional C·-algebras with arbitrary embeddings, see Chapter 15.

References. [26), [54], [1161, [150], (167].

3.9 Nuclear C*-algebras Definition 3.9.1. A C·-algebra A is said to be nuclear, if for every C·algebra B, there is only one C·-norm on A I8l B ( equivalently, the maximal C·-norm 01(') on A I8l B coincides with the spatial C·-norm ao{'))'

206

Proposition 3.9.2. Let A be a finite dimensional C·-algebra. Then A is nuclear, and for any C·-algebra B, A @ B is complete with respect to the unique C·-norm on A @ B. Proof. If A = M n , the algebra of n x n matrices, then the conclusion is the Lemma 3.6.1 exactly, and M n @ B = Mn(B) for any C·-algebra. Now if A = ED~=lMnk' then for any C·-algebra B, A ® B = ED~=lMnA:(B) is a C·-algebra with respect to a natural C·-norm. Further by Proposition 2.1.10 there is only one C·-norm on A @ B. Q.E.D.

Lemma 3.9.3. Let A be a C·-algebra with no identity, A(l) = A+a:l, and 0:(-) be a C·-norm on A(l) @ B, where B is a C·-algebra. Then a-(A(l)

@

B)

= a-(A @ B)+1 @ B,

and

a(x) = a(x) = sup{a(xu)lu E A ® B, a(u) < I}, ® B. In consequence, any C·-norm on A ® B can be uniquely

"Ix E A (1) extended to a C·-norm on A(l)

@

B.

Proof. Clearly, a - (A @ B) is a closed two-sided ideal of a - (A (1) @ B), and 1 ® B is a C·-subalgebra of a-(A(l) ® B) . Hence, by Proposition 2.4.10 we have a_(A(l} @ B) = a-(A @ B) + 1 ® B. Now let {un} be a sequence of A@B, and bE B such that a(u n -1@ b) ~ O. Suppose that f is a pure state on A(l) with flA = O. By Corollary 3.2.6 and Theorem 3.5.10, f e 9 E (a-(A(l) @ B»+, Vg E B·. Then we have

g(b)

=f

@

g(1 ® b - un)

---+

0,

Vg E B"

and b = 0 . Therefore, a-(A(l)

e B)

= a-(A ® B)+1 ® B.

For any x E a-(A @ B) and b E B, pick a sequence {un} of (A that a(un - x) ~ O. Then

a((un - urn) . (1

@

b)) < a(u n - u rn )a(1 @ b)

~

@

B) such

0,

i.e., {un' (1 ® b)ln} is a Cauchy sequence of a-(A ® B). Clearly, the limit of {un' (1 @ b) In} is independent of the choice of {un} with a( Un - x) ~ O. Thus we can define x· (1 @ b) = lim Un . (1 ® b), and C = a -(A @ B)+l ® B n becomes a * algebra naturally. Now C is a C·-algebra with respect to a(.) and a{·). By Proposition 2.1.10, it must be a{·) = a(·) on A(l) @ B. Q.E.D.

207

Lemma 3.9.4. Let A be a C"-algebra with no identity, A(l) = A+
Proof. (i) Let 13(,) be any C"-norm on AQ9B. Then by P(·) < a(.) on A(l)Q9B, we have 13(,) < o{)I(A Q9 B). Hence, a(·))(A Q9 B) is the maximal C* -norm on A®B. (ii) Let a(·) be the maximal C*-norm on A(1) Q9 B. Then by (i) a(')I(A Q9 B) = ,8 (.). Further, from the uniqueness of extension we must have 0:(') = on A(l) Q9 B. (iii) It is obvious by Theorem 3.2.9. (iv) Let a(·) be the minimal C*-norm on A(l) Q9 B. Then by (iii) a(')I(A Q9 B) = ,8(.). Again by the uniqueness of extension, we obtain that a(·) = P(·) on A(l) e B. Q.E.D.

P(.)

Proposition 3.9.5. Let A be a C*-algebra with no identity. Then A is nuclear if and only if A(l) = A+
Proposition 3.9.6.

Let A be an abelian C·-algebra. Then A is nuclear.

Proof. By Proposition 3.9.5, we may assume that A has an identity. If B is a C·-algebra with an identity, then by Proposition 3.5.9 there is only one C*norm on A®B. Now let B be a C"-algebra with no identity, and B(I) = B+
208

Then there is only one C'"-norm a(.) on A@B(I). By Lemma 3.9.4 (i) and (iii) a:(·)I(A @B) is the maximal and the minimal C"'-norm on A @ B. Therefore, there is only one C"-norm on A @ B, and A is nuclear. Q.E.D. Now consider the maximal C"'-norm. Let AI" .. ,An be C·-algebras, and AP) = Ai if Ai has an identity Ii; AP) = Ai+tTli if Ai has no identity, 1 < i < n. If a(.) is a C·-norm on @i'=lAi, then by Proposition 3.5.2,

li(x) = sup] a(xu) lu E @i=IAi, o:(u) < I} --

I'1m a (JOIn x· lOIi=1 d(i») Ii

h,···,l..

'

u

vX

E

n

@i=l

A(l) i

,

is a C·-norm on ®i=IAP) and is an extension of a('), where {df:)11i} is an approximate identity for Ai, 1 < i < n. We shall say that a(·) is the canonical extension of a(.). If 0:(') is a C·-norm on ®i'=IAi, and II is a nonempty subset of {I,'" ,n}, for any v E ®iEIAi define

at (v) = a( v @ @iflll Ii) = lim a(v @ @iflldf~»), li,.ill

•

then a,(,) is a C·-norm on ®iE1Ai. We shall say that a,(,) is the C·-norm induced by a(·) on @iE1Ai.

Lemma 3.9.7. Let {lI t ll < t < k} be a partition of {I,'" ,n}, i.e., 0 -# Il t C {I,·", n}, Il t n Ht, = 0, VI < t -# t' < k ; and U:=IIlt = {I,···, n},.Bt(·) be a C"'-norm on @iEJ'tAi, 1 < t < k, and a(·) be a C·-norm on @:=lB t , where B t =.Bt - @iEltAi, 1 < t < k. Then a(·)1 @i=l Ai is a C·-norm on @i'=lAi. Proo]. It suffices to show that: if u E @i'=1 Ai, then a( u) = 0 if and only if u = O. First, let a(u) = O. By Corollary 3.2.6 and Theorem 3.5.10 we have @:=lFt E (a-@:=lBt)\ \1Ft E B;, 1 < t < k. Hence, @:=lFt(u) = 0, VFt E B;, 1 < t < k. Since @iEltli E B;, VIi E A:, i E Il t , 1 < t < k , it follows that

@i'=l/i(u) = 0,

Vii E A;, 1 < i < n,

and u = o. Now let u = O. We need to prove that u is also the zero-element of ®~=l B t . It is equivalent to show that: if Il is a non-empty subset of {I" .. ,n}, and .0 is a C·-norm on @iEI Ai, then u is also the zero -element of @iil Ai ® B, where B = .0 - @iEllAi. We use the induction for m, where m = n - # Il. If m = 0, i.e., Il = {I,' .. ,n}, then the conclusion is obvious. Now let the conclusion hold for

209

(m - 1), where m > 1. For m and II C {I"", n - I}, we can write U = ® Xi' where {xiIi} is linearly independent in An' and ui E @?:lAi,Vj.

LUi i

Since U = 0, it follows that ui = 0 in @7:11Ai, Vi By induction assumption, u; is also the zero-element of @{Aili f/. II,I < i < n - I} @ B, and hence,

vi.

(®{lili f/. ll, 1 < i < n - I}

Ii

Further, for any

@

F)(u;)

A;, i f/. ll, 1 < i < n,

E

(@{/i Ii

lit II, 1 < i < n}

L(@{/;li

lit II, 1 < i <

@

= 0,

Vfi E

A;, F E B\ vi.

and F E B*,

F)( u)

n - I}

@

F)(ui) . In(xi) = O.

;

Therefore,

U

is also the zero -element of @illJ' Ai @ B.

Q.E.D.

Proposition 3.9.8.

Let o{) be the maximal C*-norm on @?=lAi, II be a non-empty subset of {I"·,, n}, and alT(') be the C*-norm on @iEIA i induced by a{·). Then a,(,) is the maximal C·-norm on @iEIA i. Proof. Let fJ(·) be a C·-norm on (max-@iEHA;) @(max -@illJ'A;). By Lemma 3.9.7, fJ(·)1 @~=l Ai is a C*-norm on ®7=lAi. Let fJH(') be the C·-norm on @iEIAi induced by (fJ(·)1 @?=l Ai)' Then fJI (v) = lim fJ( v ® @illldt))

< lima(v ® @illJ'd~:)) = al(v), Vv E ®iEIA i. However,let a~(.) be the maximal C·-norm on @iEIA i. Since fJ(·) is a C·-norm on (max-@iElTAs) ® (max -®illIAi), it follows that a~(v) = limfJ(v @ dl ) ,

Vv E @iEJ'A i,

where {dl } is an approximate identity for max - ®illl Ai' By Proposition 3.2.2, we can pick {d l } = {@iflldf:)}. Therefore,

al(v)

< a~(v) = fJI(V) < al(v),

Vv E @iEIA i,

i.e., a,(,) is the maximal C*-norm on @iElTAi.

Q.E.D.

Let {II t]l < t < k} be a partition of {I"", n}; fJ(·) be the maximal C·-norm on @;=lBt, where B, = max- ®iEJ'tAi, 1 < t < k; and a(·) be the maximal C*- norm on ®~=lAi' Then fJ(·)1 ®7=1 Ai = a(.), and fJ(·) is the unique extension of a(·) on®;=l B t • Formally, we can write that

Theorem 3.9.9.

max- @7=1 Ai

= max- @;:=1

(max- @iElt Ai)'

210

Proof. By Proposition 3.2.2, the extension of a(.) on@~=l B, is unique. Assume that a(.) can be extended to a C"-norm a(-) on ®~=lBt. Clearly, a(.) < ,8(.). Then ,8(·)1 @f=:l Ai > a(·). But a(.) is maximal on @f=lAi, so we have ,8(,)1 ®f=l Ai = a(·) . Further, by the uniqueness of extension we get ,8(.) = a(·) on @:=l Bt. Now it suffices to show that there is an extension of a(·) on @:=lBi,

Let It be a state on Bi, 1 < t < k , and U E @f=lAi. Write U =

L ®~=1 U)t} , j

where u}t} E @iE.ltAi, V)'. Similar to the definition of the tensor product of Hilbert spaces ( see Section 1.4 ), we have

@t=l f t (U *) U k

/r:

II Jt ui(t)* u (t)) > O.

= ~ L..., i,i t=l

f (

j

Hence, @:=l/t is positive on @f=lAi. By Propositios 3.4.3 and 3.4.6, ®~=llt E (a -@f=lAi)*. Further, @~=lB; C (a - @7=lAi)*' For any U = @:=lx;t) E @:=lBt, where x}t) E B, Vi and t, then there

L i

exist x}~ E @iE.ltAi such that

vi

and t,

where at(') is the maximal C"-norm on @iE.l,Ai, i.e., B, = at [email protected]., 1 < t < k. By Proposition 3.9.8, at(') is exactly the C*-norm induced by a(·) on @iE"A., Vt. Noticing that /r:

a(@f=lad =

a(II (@iE'tai /r:

<

@

@ig.l,li))

t=l

II at(@iE.ltai),

Va. E A i,1 < i < n,

t=l we have a(u m

-

um ' }

--t

0, where

Um

= L @~=lX~~

E

®?=lA., Vm. So that

j

we can define

a(u) = lima(u m } m which is obviously independent of the choice of equivalent {x)~lm} (Vj, t). Now it suffices to prove that a( u) = 0 if and only if u = o. Let a(u) = 0 i.e., a(u m ) - - t O. For each It E B;, 1 < t < k, since ®~=l/t E

211

k

1~@tft(L:@tx}~) = L:III~ft(x;~)

0=

i t=1

j

k

L: II ft(x~t») = t=1

j

@~=lft( U),

i.e., U = 0 in ®~=lBt. Conversely, let u be the zero-element of @:=IBt. If u has the form @~=1 x(t), where x(t) E B t, I < t < k, then there must exist to such that x(to) = O. Now let x~ E ®tEIIAi such that adx(t) - x~) ---+ 0,1 < t < k. Then ato(x~o») -----4 o. Therefore, we have

We make the induction assumption that if u is the zero-element of ®:=1B, which has the form @:=lX}t) @ @~=S+l X(f), where X)t) E s; Vj, 1 < t < s, and

L i

x(t} E Bt,S

+ I
then a(u) = O. Now let

u

= ' " IOIs+I X (t ) L- IC>'t=1 1 i

IC>'t=8+2 x(t) = 0 .

101 IOIk IC>'

Thus we can find linearly independent elements {er } in B s+ 1 such that Vj

and then

We may assume that Ur

x(t)

=L

i:

Ajr

0, S

+ 2 < t < k, so that

®:=1 x}t) @ e,

@

@~=8+2X(t)

= 0, Vr.

i

By the induction assumption, we have a(ur ) = 0, Vr. Therefore, a( u) < a(u r ) = O. Q.E.D.

L r

Remark. By Theorem 3.2.9, it is obvious that • n Ai mIn - ®i=l

for any partition {lltil

. = mm

k

(

•

®t=l mm - ®iEH,

< t < k} of {I," . ,n}.

A) i

212

Now we study the tensor product of nuclear C*-algebras.

Proposition 3.9.10. Let AI,' .. ,An be nuclear C*-algebras, and B be any C*-algebra. Then there is only one C*-norm on ®f=:::IAi ® B. In particular, min - ®f=IA i = max - ®f=IA i is a nuclear C*-algebra. Proof. When n = 1, the conclusion is apparent. Now assuming that there is only one C*-norm on ®f=2Ai 0 B, then by Theorem 3.9.9 we have

max- (®f=IA i ® B)

= max- (AI ® max -

(®f=2Ai ® B))

= min- (AI ® min-(®f=2Ai ® B)) = min-(®?=lAi ® B),

and hence there is only one C*-norm on ®i=IA i ® B. Moreover, max- ((max- 0f=1 Ai) ® B)

= max-(0i=lA i 0 B)

= min -( ®?=l Ai ® B) = min -( (min - ®?=l Ai) ® B),

and therefore max - 0?=1 Ai

=

min - 0i=1 Ai is also a nuclear C*-algebra.

Q.E.D. Theorem 3.9.11. Let A b · · · , An be C*-algebras. Then min-®?=IA i is nuclear if and only if each Ai is nuclear. Proof. By Propositio 3.9.10, it suffices to prove the necessity. Let min - ®?=l Ai be nuclear. Since min - ®i=l Ai = min -(AI ® min - 07=2 Ai), we may assume that n = 2. Suppose that ao(') l al (.) are the minimal C*-norm, the maxmimal C ..norm on A 10B respectively, where B is any C*-algebra. Fix any u E A I 0 B . We need to prove that ao(u) = al(u).

If f3(.)is the minimal C*-norm on Al ® A 2 0 B, then we have

where {d ,} is an approximate identity for A 2 • Since min -(AI ® A 2 ) is nuclear and min-(A 1 ® A 2 0 B) = min-((min-(A 1 ® A 2 ) ) 0 B) max- ((min- (AI ® A 2 ) ) 0 B),

213

it follows from Proposition 3.4.6 that

ao(u ) = lim,8(u

e dl )

= limsup{g(u*u ® didt)1/2Ig E

S(min- (AI ® A 2) ® B)},

c. (

where S( ®~l C i ) is the state space of ®~l see Definition 3.4.4 ) for any C*-algebras C 1 , ' • " C m . For any f E S(A 1 ® B) and h E S(A 2 ) , it is easy to see that f ® h can be uniquely extended to a state on min - (AI ® A 2 ) ® B, still denoted by f ® h. Hence, by Proposition 3.4.6 we have

ao(") >

!iF"UP {(f &I h)(u'u &I an'/'I f

= sup{f(U*U)I/2If E S(A I ® B)}

E

S1~'S(:i and}

= al(u).

Therefore, ao(u) = al(u), Vu E Al ® B, and Al is nuclear. Similarly, A 2 is nuclear. Q.E.D.

Theorem 3.9.12. Let 1\ be any index set; Al be a C*-algebra with an identity, VI E Ai and B be any C*-algebra with an identity. If Al is nuclear, Vi E A, then there is only one C*-norm on ®IEAAl ® B. In particular, min - ®IEA AI = max - ®IEA Al is a nuclear C" -algebra. Moreover, min - ®IEA AI is nuclear if and only if Al is nuclear, VI E A. Proof. It is immediate from Theorem 3.9.11 and the definition of ®lEAA I • Q.E.D.

Theorem 3.9.13.

The inductive limit of nuclear C*-algebras is nuclear.

Proof. Let A = li!Jl{Aa,cPl:/a!a,,8 E 1I,a a

< ,8}, and A a be nuclear, Va E 1I.

By Theorem 3.7.2, there is a family {Aala E 1I} of C"-subalgebras of A such that : 1) Aa is * isomorphic to A a( so Aa is also nuclear) , Va E 1l; 2) Aa C AI:/' Va < (3i 3) UaE.lA a is dense in A. If B is any C*-algebra, and a(·)',8(·) are two C*-norms on A ® B , then it must be that a(·) = ,8(.) on (UaE.lA a) ® B. Further, a(.) = (3(.) on A ® B. Therefore, there is only one C*-norm on A ® B, and A is nuclear. Q.E.D. Remark. By Theorem 3.9.13, we can see that any (UHF) algebra and any (AF) algebra ( see Chapter 15) are nuclear. In particular, C(H) is nuclear. Moreover, we can prove that any type I ( GCR) C*-algebra ( see Chapter 13) is nuclear. In Section 7.4, there are the examples of non-nuclear Colo-algebras. Notes. The tensor product of VN algebras is defined naturally. M. Takesaki discovered first that there are two possible C* -norms on the algebraic tensor

214

product of C*-algebras. Thus, the theory of C*-tensor products is complicated. M. Takesaki introduced a class of C*-algebras with the property (T). Later, C. Lance gave such class of C*-algebras another name: nuclear C*algebras, since this class of C*-algebras has some connections with the theory of nuclear spaces. The theory of nuclear C·-algebras is very rich and deep. C. Lance once gave a nice survey. Theorem 3.9.9 is due to B.-R. Li .

References. [37]' [95]' [97]' [102), [171].

Chapter 4 W*-Algebras

4.1 Projections of norm one Definition 4.1.1. Let A be a C·-algebra with an identity 1, and B be a C*-subgebra of A, and 1 E B. P is called a projection of norm one from A onto B, if P is linear, P A = B, Pb = b, Vb E B, and IIPa\1 < llall, \;fa E A. Lemma 4.1.2. Let cI> be a positive linear map from a C·-algebra A to a C·-algebra B. If A has an identity 1, then l cI>11 = 11cI>(I)I\.

Proof.

By Theorem 2.14.5, it suffices to show that

Thus we may assume that A is commutative. By Proposition 3.6.6, cI> is completely positive. Let B C B(K). By Theorem 3.6.7, there is a nondegenerate * representation {1f, H} of A and an operator v : K ---+ H such that cI>(a)

Then

1lcI>(I) 11

=

Ilv·vll

= v~1f(a)v, \;fa E =

Ilv1l 2 = llcI>lI·

A,

and

\Ivll = IIcI>W/ 2 • Q.E.D.

Proposition 4.1.3. Let A be a C·-algebra with an identity 1, P be a positive linear map on A, PI = 1 and

Then PA = B is a C·-subalgebra of A, 1 E Band P is a projection of norm one from A onto B.

216

By Lemma 4.1.2, IIPII = 1. Clearly, tv« = Pa (Va E A) by the assumptions. Now it suffices to show that B is a C*-subalgebra of A. Since P is positive, it follows that B* = B. Moreover, we have

Proof.

Pa- Pb = Pa . PI . Pb

= P{Pa . 1 . Pb),

Va, b E A.

Thus, B is a * subalgebra of A. Finally, if an, a E A, and Pan -+ a, then p 2 an = Pan ~ Pa = a. Therefore, B is closed. Q.E.D.

Lemma 4.1.4.

Let P be a projection of norm one from A onto B. Then p can be extended to a projection of norm one from A ** onto B**, and this extension is o-o continuous.

Proof.

By Proposition 2.11.4, B** is the u(A**,A*)-closure of B in AU. Hence B** is C*-subalgebra of A ** . Clearly, P** is a linear map of norm one from A** onto B+*, and is a-a continuous. Moreover, p .... is a projection since pUlA = P. Q.E.D.

Theorem 4.1.5.

Let A be a C*-algebra with an identity 1, and B be a C*-subalgebra of A, 1 E B, and P be a projection of norm one from A onto B. Then: 1) P is completely positive, in particular, P A+ c B+ and Po" = (Pa)*, Va E A; 2) P{Pa· b) = Pa- Pb

= P{a . Pb), Va, bE A;

3) (Px)* . (Px) < P(x·x), 'Ix E A.

Proof.

First we claim that P is positive. In fact, let B C B(K) and 1 = lK. For any € E K, put

wda) = (P{a)€, e),

Va EA.

Since PI = 1, IIPII = 1, it follows that wdl) = IIell 2 = Ilwell. By Proposition 2.3.3, we(·) is a positive linear functional on A. Thus P is positive. Further, we have Po" = (Pa)·, Va E A. By Lemma 4.1.4, we may assume that B is the closed linear span of projections of B. So for the conclusion 2), it suffices to show that

P(pa) = P: Pa, where p is a projection of B, and a E A. If y E A+ and Ilyll < 1, then p > pyp. Further, p pP(pyp)p = P(pyp). Generally, we have

P{pxp) = pP(pxp)p,

'Ix E A.

= Pp >

P(pyp). Thus

(1)

217

Replacing p by (1 - p), we get

P((I - p)x(1 - p))

(1')

= (1- p)P((1 - p)x(I- p))(I- p), \Ix E A. Let a E A and Iiall

< 1.

Then

IIpa(I - p) ± nPIl

II(pa(I- p) ± np)' (pa(l- p) ± np)*11 1 / 2 -

IIpa(I - p)a*p + n 2p111/2 < (1

Put a' = P(pa(I - p)) and b = ~(pa'p 0"1= .\ = .\ E a(b). Since

II a' ± nPIl >

+ n 2)l/2.

+ pa'*p).

If b

-#

0, then there is

npll > lib ± npll > .\ ± n, it follows that (1 + n 2)l /2 > Ilpa(I - p) ± npll > IIP(pa(1 - p) ± np) II = Ila' ± nPIi > A ± n. But this is impossible when Inl is sufficiently large. Therefore, !(pa'p + pahp) = O. Replacing n by in, we can prove that i (pa'p - pa'*p) = 0 Ilpa'p ±

similarly. So we obtain that

pa'p = O.

(2)

Since a'* = P((I-p)a*p), it follows from a similar discussion that (l-p)a'*(1p) = O. Further we get that

(1 - p)a'(1 - p) = O. Suppose that (1 - p)a'p -#

Ha'

+ n(I -

o.

p)a'pll = = =

(3)

By (2), (3), we have

+ (n + 1)(1 - p)a'pll max{llpa'(1 - p)ll, (n + 1)11(1- p)a'pll} (n + 1)11(1- p)a'pll IIpa'(1 - p)

if n is sufficiently large. However, since (1 - p)a'p E B, it follows that

Ila'

+ n(l- p)a'pll

+ n(l- p)a'p)11 < Ilpa(1 - p) + n"(l - p)a'pl[ = nil (1 =

IIP(pa(1 - p)

p)a'pll

if n is sufficiently large. That is a contradiction. Therefore, we have (1 - p)a'p = O.

(4)

By (2), (3), (4), a' = pa'(1 - p), i.e.,

P(pa(l - p)) = pP(pa(1 - p))(1 - pl.

(5)

218

Replacing p by (1 - p), similarly we have

P((1 - p)ap) = (1 - p)P((1 - p)ap)p.

(6)

By Pa = P(pap) + P(pa(l - p)) + P((l - p)ap) + P((l - p)a(l - p)) and (1), (1'), (5), (6), we can see that p .

P a . (1 - p) = P (pa (1 - p))

and p.

P a . p = P (pap).

Therefore P: Pa = P(pa). That comes to the conclusion 2). For any n, bl , " ' , bn E B, al,"', an E A, by the conclusion 2) and P(A+) C B+, i,i

i,j

P((2:: aibd* . (2:: aibd) > O. i

Therefore, P is completely positive. Finally, for any x E A,

P(x*x) - (Px)* P(x) P(x*x) - P(Px* . x) - P(x* . PX)

+ P(Px* . 1 . Px)

P((x - Px)* . (x - Px)) > 0, r.e. (Px)* . (Px)

< P(x*x).

Q.E.D.

Proposition 4.1.6. Let M, N be VN algebras on Hilbert spaces H, K respectively. Then there is a (J"-(J" continuous projection

Proof.

Fix a normal state tp on M. By Lemma 3.8.5, we can define

= x(tp 0 I),

Then

Notes.

Vx E M&JN,

I

E N*.

Q.E.D.

In general, a linear map from a C* -algebra A onto its C* -subalgebra B satisfying the conditions 1), 2), 3) of Theorem 4.1.5, is called a conditional expectation, and was studied first by H. Umegaki. A conditional expectation is clearly a projection of norm one from A onto B. Conversely, J. Tomiyama proved that a projection of norm one from A onto B is automatically a conditional expectation.

219

References. [183], [184], [187].

4.2. W* -algebras and their

* representations

Definition 4.2.1. A C'"-algebra M is called a W'" -algebra, if there is a Banach space M", such that (M",)'" = M. For a W"'-algebra M, M", is called the predual of M (see Section 1.1.) From Proposition 1.3.3, every VN algebra is a W'" -algebra. By Theorem 2.11.2, if A is a C"'-algebra, then A .... is a W"'-algebra. Let M be a W"'-algebra. Then M has an identity.

Lemma 4.2.2.

Proof.

Since the closed unit ball S of M is a(M, M",)-compact and convex, it follows from the Krein-Milmann theorem that S admits an extreme point at least. Now from Theorem 2.5.3, M has an identity. Q.E.D. Now let M be a W"'-algebra, 1 be its identity. By Theorem 2.11.2, M"'''' is also a W*-algebra; 1 is also an identity of M"'*; and M is a C"'-subalgebra of M'", Let M", be the predual of M. See M", as a closed linear subspace of M"', and let P : M** ~ M as follows

P(X) = XIM""

\IX EM"''''.

Clearly, P is a projection of norm one from M""" onto M, and is a (M"'''' ,M"') - a(M, M",) continuous. Let

I={XEM"""1 PX=o}. Clearly, I is just the orthogonal complement M",.1 of M* as a closed linear subspace of M"'. Thus, I is a(M*"',M"')-cIosed. By Theorem 4.1.5, P(aXb)

= a' PX· b;

\IX E M .... ,a,b E M.

By Theorem 2.11.2, M*'" M (the enveloping VN algebra of the C*-algebra M). Thus, the multiplication on M*'" is a(M**, M"')-continuous for each variable. Moreover, M is a a(M"'''', M"')-dense subset of M""". Therefore, I is a a(M**,M*)-closed * two-sided ideal of M H • By Proposition 1.7.1, there is a unique central projection z of M .... such that ro.J

Since P

= p 2 and

I is an ideal, it follows that

(PX - X) E I,

and

(P X - X)Y E I

220

\IX, Y E M H

•

By Theorem 4.1.5, we can see that

P(XY)=PX·PY,

\lX,YEM**.

Thus, P is a * isomorphism from M*"' z onto M. Let Q(: M - ? M··z) be the inverse of (PJMhz). Since P(xz) it follows that

Q(x) = xz,

= x,\lx EM,

\Ix E M.

For any X EMu, we can write X = PX + (X - PX), where PX E M, (X - PX) E I = Mt = M (1 - z). If z E M n I, then x = Px = O. Therefore, we get U

M·· = M+M;.

For any F E M*, RzF and R(1-z)F (see Section 1.9) E M· since the multiplication on M is ai M:", M*)-continuous for each variable. Hence, M· = RzM*+R(l-z)M*. Now we claim that U

M* = RzM*. In fact, since M* is a closed linear subspace of M*, it follows that M. = (M;-).l = (MU(l- z)).l . Thus, M. ::> RzM*. Conversely, if f E M*, then by the definition of z, f(X{l - z)) = 0, f(X) = f(Xz) = (Rzf)(X) ,\IX EM*". Thus f = Rzf E RzM*, and M. = RzM·. We say that the * isomorphism Q(: M - ? M'I'" z) is also a(M, M.) a(M·" M·) continuous. In fact, let {Xl} be a net of M, and Xl - ? O(a(M, M.)). Then for each F E M· ,

since

f = RzF E M•.

From the above discussion, we obtain the following.

Proposition 4.2.3. Let M be a W·-algebra, and M. be its predual. Embedding M, M* canonically into M**, M· respectively, then we can find a central projection z of M·* and a projection P of norm one from M** onto M such that: 1) P is also a * homomorphism from M onto M, and is a(M**, M*) a(M, M.) continuous; 2) P is a * isomorphism from MHZ onto M. If Q(: M - ? M**z) is the inverse of (PIMHz),then Q(x) = xz,\lx E M, and Q is also a(M,M.) a(M*·, M*) continuous; 3) Mt = M**(l- z),M. = RzM·, and U

M *·

=

M+' M.':. ,

M*

M +' R (l-z) M* . =.

221

Definition 4.2.4. Let M be a W'"-algebra, and M", be its predual. {a , H} is called a W'"-representation of M, if tr is a * homomorphism from M to B(H), and tr is a(M, M",)-a(B(H), T(H)) continuous. If {1r, H} is a W*-representation of M, then tr(M) is a weakly closed * subalgebra of B(H) by the proof of Proposition 1.8.13. Moreover, if tr is nondegenerate, then 1r(lM ) = I H , and tr(M) is a VN algebra on H. If 1r is faithful, by Proposition 1.2.6 and a(M, M",)-compactness of the closed unit ball of M, then the * isomorphism tr- 1 from 1r(M) onto M is also a(B(H), T(H))u(M,M",) continuous. Theorem 4.2.5. Let M be a W"'-algebra. Then M admits a faithful nondegenerate W"'-representation. In consequence, M is * isomorphic to a VN algebra on some Hilbert space, and this * isomorphism is a-a continuous. Proof. Let {tr, H} be the universal * representation of M as a C" -algebra. By the discussion of Section 2.11, {1r, H} can be extended to a faithful nondegenerate W'"-representation of the W* -algebra M""", which is denoted by {1r, H} still. By Proposition 4.2.3, there is a a-a continuous * isomorphism Q from M onto M U z, where z is a central projection of M**. Then {1roQ, 1r(z)H} is a faithful non degenerate W*-representation of M. Q.E.D.

By Theorem 4.2.5, we can regard a W"'-algebra as a VN algebra. In particular, we have Proposition 4.2.6. Let M be a W'" -algebra, and M", be its predual. Then the * operation on M is a(M, M",)-continuous; the multiplication is a(M, M",)-continuous for each variable; M. is the linear span of normal positive linear functional on M, in consequence, M", is unique; for any normal positive linear functional ip on M, by the GNS construction there is a cyclic W"'-representation {trrp, Hrp, lrp} of M.

Proposition 4.2.6. the normal universal

Denote the normal state space on M by Sn(M). Then * representation

H=

L

rpESn(M)

aJHrp}

is a faithful nondegenerate W· -representation of M. Now we discuss some properties of W"'-representations.

Theorem 4.2.7. Let A be a C·-algebra, and {tr,H} be a * representation of A. Then there exists a unique W*-representation {1f, H} of A"'· such that 1f is an extension of tr, and 1f(A **) is the weak closure of tr(A). Consequently,

222

there is a bijection between the set of W* -representations of A **.

* representations

of A and the set of

Proof. Notice that 1r : A ~ B(H) and 1r* : B(H)* ~ A*. Regard T(H) as a closed linear subspace of B(H)* (since T(H)* = B(H)), and let 1r* = 1r*IT(H). We claim thatjf = (1r,,)"(A** ~ T(H)* = B(H)) satisfies the conditions. In fact, since 1r*: T(H) ~ A* and n = (1r*)*, it follows that 1f is u(A*"',A")u(B(H), T(H)) continuous. Notice that

7f(a)(t) = a(1r*(t)) = a(1r*(t))

=

1r(a)(t),

Thus n is an extension of zr. Further, 1f is a W*representation of A ** since A is a (A ** , A *)-dense in A ** and 1f is a - o continuous. Clearly, 1f is the unique a - a continuous extension of 1r, and 1f( A **) is the weak closure of 1r(A). Q.E.D.

'it E T(H),a E A.

Proposition 4.2.8. Let {1rb HI} and {1r2' Hz} be two nondegenerate W*representations of a W"-algebra M, and ker-r, = {a E MI1ri(a) = O},i = 1,2. If ker1rl cker1r2, then {1rz, Hz} is unitarily equivalent to an induction of some amplication of {1rI, HI}, i.e. there is a Hilbert space K and a projection pi of (1rl (M )®
Proposition 4.2.9. Let {1r1l HI} and {1rz, Hz} be two W*-representations of a W*-algebra M, and ker 1ri = {a E M I 1ri(a) = O}, i = 1,2. If 1ri(M) admits a cyclic-separating vector in Hi, i = 1,2, and ker 1r1 = ker 1r2, then {1rI, HI} :::: { 1r 2 ' Hz}. Proof. Let M, = 1ri(M), then M, is a VN algebra on Hi, i = 1,2. Since kerx, =ker1r2, thus there is a * isomorphism iP from M 1 onto M2, such that iP 0 1r1 = 1rz. Now by Theorem 1.13.5, the conclusion can be obtained. Q.E.D.

Notes. We have a definition of abstract C·-algebras (see Chapter 2). A natural question is how to define abstract VN algebras. This question received considerable attention during 1950's. Theorem 4.2.5 is due to S. Sakai, and it gives an answer for the above question. However, the proof of Theorem 4.2.5 presented here is due to J. Tomiyama based on his result, Theorem 4.1.5. The uniqueness of the predual of a W*-algebra, due to J. Dixmier, answered

223

completely the question concerning to what extent the algebraic structure of a W"-algebra determines its topological structure. References. [19], [143], [183], [184].

4.3. Tensor products of W* -algebras Let M, N be two W"'-algebras. We want to define the tensor product Mti;N of M and N such that Mti;N is still a W*-algebra. If we regard M, N as VN algebras, using the tensor product of VN algebras and by Theorem 1.12.6, we can define M®N. But in this section, we shall define M®N from M and N themselves. Let M, N be two W· -algebras, and M., N", be their preduals respectively. As C"'-algebras, there is a spatial C"'-norm ao(·) on the algebraic tensor product M ® N. Then we get a C·-algebra ao - (M ® N). Let a~(·) be the dual norm of a(·) on M· ® N*. Then

(ao-(M ® N))* ::J

a~-(M'" ®

N*) ::>

a~-(M*

e N.),

where a~-(M'" e N"') is the completion of (M'" e N"', a~(·)); and a~-(M* ® N",) is the completion of (M. ® N., a~(·)), and is equal to the closure of M", ® N", in a~-(M'" ® N·). Let 1= (a;-(M. ® N*))-l(c (ao-(M

e N))U),

i.e., I is the orthogonal complement of ao-(M. ® N",) which is regarded as a closed linear subspace of (ao-(M ® N))"'. Suppose that Y E I, X E (ao(M ® N))"'*. Pick a net {Xl} C M ® N such that Xl -+ X with respect to the w"-topology in (ao-(M ® N))". For any f E M. ® N"" since LZ1f and R:t 1 f E M" ® N., it follows that

\:Il. Taking the limits, we get XY and Y X E I. So I is a a-closed two-sided ideal of the W·-algebra (ao-(M e N))". Therefore, (ao-(M e N))** /1 is a W"'-algebra, and its predual is ao-(M. ® N.). Definition 4.3.1. The W· -algebra (ao-(M ® N)) "'' ' / I is called the tensor product of W"'-algebras M and N, which is denoted by Mti;N. From preceding paragraph, M0N = (ao-(M. ® N.)*, and (M®N)", = a o(M", ® N",).

224

Lemma 4.3.2.

M. ® N. is w"-dense in (ao-(M ® N))·.

By Proposition 3.2.10, M" ® N" is w"-dense in (ao-(M ® N))". Notice that the unit balls of M., N. are w·-dense in the unit balls of M·, N· respectively, and aD (.) is a cross-norm on M" ® N·. Then it is easy to see that M.®N. is dense in M*®N" with respect to the w·-topology in (ao-(M®N))"'. Therefore, M. e N" is w*-dense in (ao-(M ® N))*. Q.E.D. Proof.

Proposition 4.3.3. ao-(M ® N) n 1 = {O}, in consequence, ao-(M ® N) can be embedded in M®N. Moreover, ao-(M ® N) is w·-dense in M®N. Proof.

Let x E ao-(M ® N)

n 1.

/®g(x)=O,

Then

V/EM",

gEN•.

By Lemma 4.3.2, we have x = o. Now if X E M®N = (ao-(M e N))** / 1, and X E X, then there is a net {Xl} C ao-(M ® N) such that XI --7 X with respect to the w*-topology in ao(M@N))**. Further, for any F E (M®N)" = a~-(M*@N*) c (ao-(M@N))"',

I(XI - X)(F) 1= I(Xl - X)(F) I --7 O. Therefore, ao-(M @ N) is w·-dense in M®N.

Q.E.D.

Theorem 4.3.4. Let {1ri' Hi} be a nondegenerate W"-representation of a W*-algebra Mi, i = 1,2. Then there exists a unique W*-representation {1r, H} of M 1®M2, where H = HI @ Hz such that

1r(al

@

az)

= 1r1(ad ® 1r2(a2),

Vai E M i,

i = 1,2,

and 1r(M®N) = 1r(M)®1r(N) ( the tensor product of VN algebras 1r(M) and 1r(N)). Moreover, if 1ri is faithful, i = 1,2, then 1r is also faithful. By Proposition 3.2.7, there is a unique ao-(M1 @ M z) such that Proof.

1ro(al @a2) = 1rl(ad @7fz(aZ),Val

E

*

representation {1ro, H} of

M1,az

E

M z.

By Theorem 4.2.7, {1ro, H} can be uniquely extended to a W·-representation {1fo, H } of (ao-(Mt @ M z))** . For any Ci,TJi E Hi, let Ii(') = (1ri(·)f.i,TJi) E (Mi)*,i = 1,2. Then It @ /z E (Md" e (Mz). C (ao-(M1 ® M z))". By the definition of I (see 4.3.1), we have

It e 12(1)

= {O}.

Since HI 0 Hz is dense in HI @ Hz, it follows that ?f o(1) = {O}. Thus {?fo, H} induces a W*-representation {1r, H} of M 1®M2 = (ao-(M1@M2 )) .... /1. Clearly,

225

{1r, H} satisfies the conditions. Moreover, the uniqueness of such {1r, H} is also obvious. Now suppose that 1ri is faithful, i = 1,2. For any Ii E (Mi ) . , since M, is * isomorphic to 1ri(Mi), there are two sequences {e~i)} and {TJ~)}(C Hi) with En(lle~)1I2 + IITJ~i)112) < 00 such that

Ii (.)

=

L (1ri ( .) e~i) , T] ~i)) ,

V· E

u;

i

=

1, 2.

n

Thus for any x E M t ®M2 , we have

(It e 12)(x) =

L (1r(x) e}t) ® ei2) ,

TJ?) ® TJi2 ) ) .

i,k

If 1r(x) = 0, then (It e 12) (x) = 0, VIi E (Mi ) . , i = 1,2. But (Mt ). ® (M2 ) . is dense in a~-((Md. ® (M2 } . ) = (Mt®M2 ) . , so X = 0, and 1r is also faithful.

Q.E.D. Corollary 4.3.5. Let M, be a VN algebra on a Hilbert space Hi, i = 1,2. Then the W·-tensor product of W·-algebras M t and M 2 is * isomorphic to the VN tensor product of VN algebras M, and M 2 • Proposition 4.3.6. Let CPi be a normal positive linear functional on a W·algebra M«, i = 1,2. Then there is a unique normal positive linear functional tp on Mt ®M2 such that

cp(at ® a2) = cpt(adcp2(a2),

Vai E M i ,

i = 1,2,

and s(cp) = s(cpt) ® S(CP2)'

Proof. Let {1ri, Hi, ei} be the cyclic W· -representation of M, generated by tpi, i = 1,2. By Theorem 4.3.4, 1rt ®7r2 can be extended to a W· -representation {1r, H} of M t@M2 , where H = H, ® H 2 • Now let cp(x) = (1r(x) 6 ® e2' 6 ® 6),

Vx E M, ®M2.

Then cP is what we want to find. Moreover, by Proposition 1.8.11 and Theorem 1.4.12, we can see that s(cp) = s(cpt} ® S(cp2)' Q.E.D.

Proposition 4.3.7. Let «Pi be a completely positive linear map from a W·algebra M, to a W·-algebra N i , and also «Pi be a - a continuous, i = 1,2. Then there exists a (J- o continuous completely positive linear map «P from M t®M2 to N t®N2 such that

«P(at ® a2)

=

«Pt(ad

e «P2(a2),

Vat E MI,

a2 E

M 2.

226

Proof. By Proposition 3.6.11, there is a completely positive linear map cI>o from ao-(MI ® M 2 ) to ao-(NI ® N 2 ) such that

epO(al ® a2) = cI>1(ad ® cI>2(a2), For any that

Ii

E (Ni ) . , i

=

a2 E M 2·

Val E MIt

1,2, since epj is a - a continuous, i = 1,2, it follows

e 12) = cI>~(fd e cI>;(f2) E (MI ) . e (M2)•. cI>~(a;-((Nd. e (N2 ) . )) C a;-((Md. e (M2 ) . ) . Let e = (ep;la;-((N1 ) . ® (N2 ) . )) · . cI>;(/1

Further,

Then cI> is a a-a continuous linear map from M l0M2 to N l 0 N 2 (see Definition 4.3.1), and

cI> (al ® a2) = 4>t{ ad ® 4>2( a2),

Vai

E M«,

i

=

1,2.

Finally, we prove that ep is completely positive. Assume that N; ())N2 B (H). Then we need to prove

c

L(cI>(x;Xj)€j,€i) > 0 i,i

for any n, Xl,' .. ,Xn E M 1 ®M2 and €l,"', €n E H. This is immediate from cI>IMI e M 2 = cI>o and Theorem 1.6.1. Q.E.D.

References. [109], [1501.

4.4. Completely additive functionals and singular func-

tionals Let M be a W·-algebra, and M. be its predual. Proposition 4.2.3, there is a central projection z of M** such that

Definition 4.4.1.

M· = M.

+ R(I-z)M· ,

M.

By

= RzM*.

Any element of M. (a a(M, M.)-continuous functional on M) is called a normal functional on M, and any element of R(l-z)M* is called a singular [unctionalon M. For any F E M*, we have the unique decomposition

F

= F; + Fs'

F; = RzF EM.,

F,

= R(l-z)F,

Fn , FlJ are the normal, singular functionals on M respectively. It is easy to see that

IIFlI = IIFnl1 + IIFsll·

227

Theorem 4.4.2. Let F be a positive linear functional on a W"'-algebra M. Then F is singular if and only if for any non-zero projection p of M, there is a non-zero projection q of M with q < p such that F( q) = o. By Proposition 2.3.2, F E M"'. Write F = Fn + F! as Definition 4.4.1. Sufficiency. If Fn i=- 0, then s(Fn ) = p is a non-zero projection of M. By the assumption, there is a non-zero projection q of M with q O. Clearly, Fs{q) > O. Then we get a contradiction. Therefore, Fn = 0, and F = Fa. is singular. Necessity. Let F n = 0, F = F s ' and p be a non-zero projection of M. We may assume that F(p) > O. Pick a normal positive linear functional f on M such that f(p) > F(p). Suppose that

Proof.

£,

= {qlq is a projection of M,q <

p ; and

f(q) < F(q)}.

With the inclusion relation of projections, L is a non-empty partially ordered set. Let {q,} be a totally ordered subset of L, and q = sup, qi. Since f is normal, it follows that

F(q) > sup F(ql) > sup f(qd = f(q). i

i

Thus q E L, By the Zorn lemma, E has a maximal element Po. But p rf. L, so qo = P - po i=- o. For any non-zero projection q of M and q < qo, we have

F(q) < f(q) since Po is maximal. Further, F(qoxqo) < f(qoxqo), Vx E M+. By Proposition 1.6.4, F(qoXqo) < f(qoXqo)) , VX E M~·. In particular, F(qo{l - z)) < f(qo(l - z)). Since f E M. = RzM*, it follows that f(qo(l - z)) = 0 and F(qo(l - z)) = O. Moreover, F is singular, i.e., F = R(1-z)F. Therefore, F(qo) = F(qo{1- z)) = 0, and qo satisfies the condition. Q.E.D.

Corollary 4.4.3. Let F be a singular positive linear functional on a W"'algebra M, and p be a projection of M. Then there is an orthogonal family {Pi} of projections of M such that L:I p, = p, and F(pt} = 0, Vi. Definition 4.4.4. Let M be a W* -algebra, and f E M"'. f is said to be completely additive, if for any orthogonal family {PI} of projections of M, we have f(p) = L:I f(PI) ,where p = L:,PI. The following theorem is a generalization of Proposition 1.8.5.

Theorem 4.4.5. Let M be a W*-algebra, and f EM"'. Then f is normal if and only if f is completely additive.

228

The necessity is obvious. Now let I be completely additive and I = In + I.· We need to prove that Itt = O. By Theorem 2.3.23. Write I = 1(1) - 1(2) + i/(3) - i/(4), where IU) > 0,1 < i < 4. Then I. = I!l) - 1!2) + il,(3) - il!4) , where I,(j) is singular and positive, \:Ii. Define gtt = 2:1=1 I!i). Then gtt is also singular and positive on M. Let P be a projection of M. By Corollary 4.4.3, there is an orthogonal family {PI} of projections of M such that P = 2:1 PI and Ys (Pi) = 0, \:Ii. Then Itt(PI) = 0, \:II. Since I is completely additive and In E M., it follows that Proof.

I,{p) =

I{p) - In{P) = 2:[/(PI) - In(PI)] 1

L !.,(pd = o. I

Q.E.D.

Therefore, III = 0 since P is arbitrary.

Now let A be a set, and v(.) be a bounded additive complex valued function defined on all subsets of A, i.e., sUPJcA Iv(J) I < 00, and v(A 1 U A2 )

= v(Ad +

v(A 2 ),

VAl, A2 c A and Al n A2 = 0. Denote the set of all such v by BV (A). Clearly, BV(A) is a linear space. 1) Let v E BV(A). Define

v(v){ J) = sup{L Iv( J i ) IIJi c J, i, n i, = 0, \:Ii #- i}, I

VJ c A. Then v(v) E BV(A). In fact, let JI,"', I n C J and i, n J,. = 0, Vi 11 U 12 = 13 U I ... , where II = {i

I Re

n

L

!v(Ji)1 <

i:::1

L

Write {I,'" ,n}

12 = {i I Re v(Ji ) < O},

v ( J i ) > O},

13 = {i I 1m v(Ji ) > O}, Then

#- i-

14

= {i I Im v(Jd < O}.

Rev(Ji )

L

-

Rev(Jd

iEl2

iEll

2: Imv(Ji ) - L Imv(Ji ) iEI3 iEl. Rev( U Jd - Rev( U Ji ) +

iEI.

+Imv(

<

U Ji ) -

iE12

Imv(

iE13

4 sup Iv(J') I < PeA

UJ

iEl. 00.

i)

=

229

Clearly, v(v) is additive. Thus v(v) E BV(A).

2) For any v E BV(A), define Ilvll = v(v)(A). Then (BV(A),

II ·11)

is a

Banach space. The proof is easy. We leave it to the reader.

3) Denote the set of all bounded complex functions on A by lOO(A), and define 1I111 = SUPlEA 11(1)1, VI E lOO(A). Then lOO{A)$ = BV(A). First, let I E lOO(A) and I be simple, i.e., there is a partition A = Ut=l Ai (where Al n Ai = 0, Vi i=- j), and complex numbers All·'·, An such that

I (I) = Ai,

V1 E Ai,

1

< i < n.

For any v E BV(A), define

1I(t) =

l t(l)dll(l)

=

~ A;II(A;).

Clearly, Iv(/)1 < Ilvll . 11/11· For any IE lOO(A), pick a sequence {In}(C 100(A)) of simple functions such that Il/n - III -4 o. Since Iv(ln - 1m) I < Ilvll . Il/n - Imll -4 0, then we can define v(f) ::::: limv(ln), Vv E BV(A). n

Clearly, this definition is independent of the choice of {In}, and Ilvll ,11/11, VI E 100 (A), t/ E BV(A). Also, it is easy to see that IIvl!

=

sup{lv(/)11

Iv(/) 1 <

f E 100 (A), II/II < I}.

Thus, BV (A) can be isometrically embedded in lOO(At. On the other hand, let F E 1OO(A)*, and define v{J) = F(XJ), where XJ is the characteristic function of J, VJ C A. Then 1/ E BV(A) .and v(/) = F(/), VI E lOO(A). So lOO(A)$ = BV(A).

4) Let v E BV (A). Clearly, we have

L Iv( {l}) I < Ilvll· lEA

5) Let {vn} C BV{A},and SUPn Ilvnll

<

00,

lim v n { J) = 0, n

Then limn LIEA Ivn { {I}) 1 = o. In fact, suppose that there is a e V

VJ

c A.

> 0 such that

L I n ( {I} ) I > s , lEA

and

Vn

(1)

230

(replacing {n} by a subsequence in necessary case). For nl finite subset F 1 of A such that

Since

lin

(l)

1, there is a

~ 0, Vl E A, there exists n2 such that

L

IlI

e

n2

({l } )1< -. 20

lEF l

Thus there is a finite subset F2 C A\F1 such that

L IlI

n2

({ l })1>

L IlI

lE F 2

e

({l })I - - ·

n2

10

lEA

Generally, we can find a sequence {F/c} of finite subsets of A and a subsequence ink} such that

and F k n F, = 0, Vk =I [, Fix m such that m > leO SUPnI11Inll. Let E1 = F1,j.ll = lIn l • If

1
111-l111 > v(l-ld (U;::o:l U~l Fm j +p ) m

L

V(l-ll)(U~l Fm j + p ) >

c m . 10

p=l

>

Ilvnll >

1I1-l111· This is a contradiction. Thus, there is an integer PI with 1 < PI < m such that cO· V (I-ll )(U~l F m j +Pl ) < 1 sUPn

Let E 2 = F m+ P1 , 1-l2 = vnm+Pl' and F; integer P2 with 1 < P2 < m such that

=

Fmj+Pl,Vj.

V(1-l2) (u~ 1 F' mj +p~J

V(1-l2)(U~1Fm (mi +P2)+PJ < . ' -, Generally, we have

ips I s =

1,2, ... } such that 1

<

p,

< m,

lC · O

Similarly, there is an

231

L:

c

L: IJLs({I})I- 10'

IJLs({l})1 >

lEE.

lEA C

(2)

V(JLs)(Uj>sEj) < -

10

(noticing (1)), Vs, where JLs = vnb.,E s Ph' .. .b, = mb s - 1 + Ps, . , '. Now define a I E 100 (A) as follows:

o

00

Fb.., and b1

if I tt U~l e; if I E E j , vi.

I ()I - { ' argJLi({l}), By • E; <

=

(Vj) and (2), for any s we have

L:

IJLs (I) -

IJLs-( {I} } II

lEE.

<

lL:l ;<8

!dJLsl+ ll

Ej

IdJLs-L:IJls({I})11 E"

LEE ..

+Ilu.1>8 E.ldJLsl 1 C < L: L: IJLs( {I}) I + V(JLs)(Uj>sE j ) < S· ;<sIEEj

Thus from (1), (2),

IJLs(/)1 >

L:

IJls({l})I-

e

5

LEE ..

>

3 7 L IJLIl({l})I- -c > -c, lEA 10 10

Vs. However, from SUPn Ilvnll < 00 and vn(J) -+ 0, VJ c A, it must be that vn(/) ~ O. Then JLs(/) ~ 0, a contradiction. Therefore,

Proposition 4.4.6. Let M be a W*-algebra, {!,Ik I k = 1,2,"'} C M* and f k ~ I (a(M* ,M)). Then the normal part {!k} and the singular part {I:} of {Ik} converge to the normal part In and the singular part I' of I respectively in a(M*, M)-topology. We may assume that O(a(M*,M)). Since SUPk < Proof.

Il/rll

I = 00,

O. So it suffices to show that it is enough to prove that

I::(p) for any projection p of M.

~ 0

Ik

-+

232

For any hEM"', by Theorem 2.3.23 we can uniquely write h = hI - h 2 ih s - ih 4 , where hi > 0,1 < j < 4. Define [h] = hI + h 2 + h s + h 4 •

+

f

lA: [I:]. Then g is a singular positive functional on M. Fix 2 a projection P of M. By Corollary 4.4.3, there is an orthogonal family {PI}IEA of projections of M such that

Now let g =

A:=I

Vi E A. Thus lie (PI) = 0, Vk, t. Define VA: E BV (A) as follows

Vk( J)

= IA:

CL, PI),

c A.

VJ

IEJ

Since

SUPA:

II IA: II < 00,

it follows that sup{lvdJ)j

Clearly, lim VA: ( J) = 0, VJ

lie(PI)

A:

= 0, Vk, l, we

c A.

I k, J

c A} <

00.

Then by the discussion 5) about BV(A) and

have lim A:

L 1/~(pd I =

lim A:

lEA

L IfA:(PI) I lEA

limLlvA:({i})1 =0. k

lEA

Further, from I~ E M", (Vk) ,

lim I~(p) n

= lim L k

lEA

f~(PI) = 0.

Q.E.D. Let M be a W"'-algebra,and M", be its predual. Then M", is weakly sequentially complete.

Theorem 4.4.7.

Proof. Let {/A:} be a weakly Cauchy sequence in M",. Since SUPk Il/kll < 00, it follows that there is a I E M'" such that IA: ~ f{a(M"', M)). By Proposition 4.4.6 I~

-t

In(a(M*, M)),

and

I:

-t

1 (a (M *, M)). 8

But II: = IA;, lie = 0, Vk, therefore IS = O,and I = In E M*.

Q.E.D.

The decomposition of M* into the normal part and the singular part was given by M. Takesaki. Theorem 4.4.2 is due to M. Takesaki. Theorem

Notes.

233

4.4.7 is due to S.Sakai and C.A. Akemann. Moreover, J.F. Aarnes proved that (M,r(M,M*)) is a complete locally convex space for a W*-algebra M.

References. [1], [2], [3], [170].

4.5. The characterizations of weakly compact subsets in predual Let M ba a W*-algebra. Lemma 1.11.5 gives a charaterization of a weakly compact subset of its predual M •. Further, we have the following.

Theorem 4.5.1. Let M be a W*-algebra, M. be its predual, and A eM•. Then the following statements are equivalent: 1) The a(M*, M)-closure of A is a(M., M)-compact; 2) There exists a normal positive linear functional t/J on M with the property that for any e > 0 there exists E = E(e)(> 0) such that l
2). It is Lemma 1.11.5 exactly. 2) ==> 3). Pick t/J as in the Condition 2) and e suppose that m is sufficiently large such that

t/J(b*b where b =

aim, and Ilbtl < 1.

=

+ bb*) < 6 = 6(1),

Then by 2), we have

sup{I
1. For any a E M,

234

Now let {an} be a sequence of M with an --+ O(s*(M,M.)). Clearly, SUPn Ilanll < K (some constant). For any e > 0, since (a~an + ana~) --+ O(u(M,M.)), there is no such that

K- 2 t/J (a~an

+ ana~) < 8 =

8(c),

Vn

> no.

By 2), Icp(an)1 < e K, Vcp E A and n > no. Therefore, limcp(an) = 0 uniformly n for cp E A. 3) ==> 4). It is obvious. 4) ==> 5). Let {Pn} be an orthogonal sequence of projections of M. Then {qn = Pk}n is a decreasing sequence of projections of M with infn qn = O.

L

k>n

By 4), limn cp(qn) = 0 uniformly for cp E A. Then

CP(Pn) = cp(qn) - cp(qn+d

~ 0

E A. 5) ==> 1). Regard A as a subset of M*, and let A be the u(M*, M)-closure of A in M*. Since A is bounded, it follows that A is u(M*, M)-compact. Now it suffices to show that A c M*. By Theorem 4.4.5, we need to prove that f is completely additive for any f E A. Also this is equivalent to prove that

uniformly for

ip

L

cp(pd ~ cp(p)

cp E A,

uniformly for

IEF

where {PI Ii E A} is any given orthogonal family of projections of M, {F I F c A,· F < oo} is directed by the inclusion relation, and p = Pl. In fact, let

L

lEA

f E A and {CPa} be a net of A with CPa

-t

f(u(M*,M)). For any e > 0, then

there is a finite subset Fo such that I

L

CPa (PI) - CPa(P) 1 < s ,

VF ~ Fo

and a.

IEF

Thus

IL

f(PI) - f(p) I < e, VF ~ Fo. Further, f is completely additive.

IEF

Now suppose that there is a e we can find CPF E A with

IL

>

0 such that for any finite subset F of A

CPF(pdl > e.

letF

Then we can pick a disjoint sequence {Fn} of finite subsets of A and {CPn} C A such that I CPn(PI)! > s , Vn.

L

IEF..

Since {qn = from 5) that

L

PI} is an orthogonal sequence of projections of M, it follows

IEF"

lim cp( qn) = 0 n

uniformly for

cP E A.

235

This is a contradiction. Therefore,

L tp(Pl) ~ tp(p)

uniformly for tp E A.

tEF

6) ==> 5).

Let {Pn} be an orthogonal sequence of projections of M, and N be a maximal abelian W"'-subalgebra containing {Pn}. By 6), the u(N., N)closure of AN = {(tpIN) I tp E A} is o(N., N)-compact. Since 1) implies 5), it follows that tp{Pn) = (tplN){Pn) ~ o uniformly for tp E A. Similarly, we can prove that sup{ltp(h) 11 tp E A} < 00 for any h· = hEM.

Thus, A is bounded. 1) ==> 6). Let A be the u(M.,M)-closure of A. Then A is a o(M.,M)compact subset of M«. Suppose that N is a maximal abelian W·-subalgebra of M. Clearly, AN = {(tpIN) I tp E A} is a a(N""N)-compact subset of N",. But {(tpIN) I tp E A} c A, so 6) holds. 2) ==> 7). Let {Pt} be an increasing net of projections of M,and P = sup, Pl' Pick 'tjJ as in the condition 2). For any e > 0, since

'tjJ((p - Pl}*(p - Pt)

+ (p -

p,)(p - pd*)

= 2'l/;(p - pd

~ 0,

there is lo such that 'tjJ(p- PI) < 8 = 8(e:), VI > lo. By 2), we have Itp(p- PI)I < E,Vtp E A,l > Ie. That is limtp(p,) = tp(p) uniformly for tp E A. Moreover, by I the same proof of 2) ==> 3), A is bounded. 7) ==> 4). Let {Pn} be a decreasing sequence of projections of M, and inf Pn = 0. Then {(I - Pn)} is an increasing sequence of projections of M n and sUPn(l- Pn) = 1. By 7), limtp(lPn) = tp(l) uniformly for tp E A, i.e., n tp(Pn -4 0) uniformly for
7). It suffices to notice that: if {PI} is an increasing net of projections of M, then {qj = Pl + (1 - p)} is an increasing net of projections of M, and SUPtq, = 1, where P = SUPtPI. 2) ==> 8). Let {PI} be an increasing net of projections of M, and sup I PI = 1. Pick w as in the condition 2). For any e: > 0, there is l{e:) such that ¢(1 - Pl) < l8(c), VI > l(e). Then for any a E M with Iiall < 1 and l > l(e), we have ¢((1- Pl}a'" (1 - PI)a(l - p,)

+ (1 -

PI)a{1- p,)a"'(I- Pt)) < 2'tjJ(1- PI) < 8(e)

By 2), 1 l(e),lIall < 1. Therefore, IIL(l-PI)R(l-PI)tpll ~ 0 uniformly for tp E A. Moreover, A is bounded by the same proof of 2) ==> 3). Q.E.D.

Proposition 4.5.2. Let M be a W*-algebra, M", be its predual, and A C (M",)+. Suppose that the a(M., M)-closure of A is u(M", , M)-compact. Then

236

the a(M.,M)-closure of E = {Racp 1a EM, compact.

IIall < l,cp E

A} is also a(M.,M)-

Proof. Clearly, E is bounded. Let {Pn} be a decreasing sequence of projections of M and inf'., Pn = O. By Theorem 4.5.1, CP(Pn) -+ 0 uniformly for rp E A. From the Schwartz inequality,

IRaCP(Pn) I < cp(a·a)1/2CP(Pn)1/2 < IlcpJll/2CP(Pn)1/2 Va E M and lIall < 1. Thus P(Pn) -+ 0 uniformly for pEE. Now again by Theorem 4.5.1, the a(M.,M)-closure of E is a(M.,M)-compact. Q.E.D.

Notes. Theorem 4.5.1 is a combination of results due to several mathematicians: A. Grothendieck, S.Sakai, M.Takesaki, H. Umegaki and finally, C.A. Akemann.

References. [2], [62], [146], [169], [188}.

Chapter 5 Abelian Operator Algebras

5.1. Measure theory on locally compact Hausdorff spaces Let 0 be a localy compact Hausdorff space, and B be the collection of all Borel subsets of 0 [i.e. the a-Bool ring generated by compact subsets of 0). Define B,oc = {E c n lEn K E B, VK compact CO}.

B,oc is a u-Bool algebra. Each subset in B,oc is called a locally Borel subset. Clearly, E E B,oc if and only if E n FEB, VF E B. A complex function f on 0 is said to be measurable, if it is B-measurable. f is said to be locally measurable, if it is B,oc-measurable. Clearly, A measurable function is locally measurable. And a locally measurable function f is measurable if and only if {t E 0 I f(t) ::I O} E B. Let v be a regular Borel measure on O. F( c 0) is called t/-zero, if FEB and v(F) = OJ E(C 0) is called locally u-zero, if E E Bloc and v(EnK) = 0, VK compact C O. A Proposition about P(t) on 0 holds almost everywhere with respect to v (a.e.v), if {t E 0 I P(t)does not hold} is a subset of some v-zero set; P{t) on 0 holds locally almost everywhere with respect to v (l.a.e.v), if {t E 0 I P( t) does not hold} is a subset of some locally v-zero set. Let u be a regular Borel measure on o. Then Va = U{V

c n ] V is open and locally v-zero}

is the maximal locally v-zero open subset. Let

suppv = (0\ Vo) . It is called the support of u, and clearly it has the following property. Let U( C 0) be a Borel open subset. Then v(U) = 0 if and only if Un suppv = 0.

238

Lemma 5.1.1. Let v be a non-zero regular Borel measure on n. Then there is a non-empty compact subset K( c n) such that v(K n U) > 0 for any open subset U of n with U n K ¥ 0. Proof. Since supp v is a non-empty closed subset of fl, we can find an open subset V such that V compact and K = V n suppv ¥ 0. Then K is what we want to find. In fact, suppose that there is an open subset U with Un K =I- 0 such that v(K n U) = o. Then v(U n V n suppv) = 0 and v(U n V) = v(E), where E = (U n V)\suppv. But E is open-and En suppv = 0, so we have

v(U n V)

= v(E)

=

o. = 0.

From the definition of supp v, U n V n supp v On the other hand, pick t E un K. Since U is an open neighborhood of t and t E K = V n suppz/, it follows that U n V n suppv ¥ 0. We get a contradiction. Therefore, K is what we want to find.

Q.E.D.

Proposition 5.1.2. Let v be a non-zero regular Borel measure on n. Then there is a disjoint family {Kl}IEA of non-empty compact subsets of n such that N = n\ UIEA K, is a locally v-zero subset, and the family {KI}'EA has the locally countable property, i.e., for any compact subset K of n the index set {I E A I tc, n K ¥ 0} is countable. Proof. By Lemma 5.1.1 and the Zorn lemma, there is a maximal disjoint family {K1hEA of non-empty compact subsets of n such that v(KI n U) > 0 for any open subset U of n with Un K, ¥ 0, VI. Suppose that V is an open subset of nand V is compact. Then

L »i«, n V) < v(V)

<

00.

lEA

Thus {I E A I v(K, n V) > o) is countable. However if some I E A is such that v(K, n V) = 0, then K , n V = 0 by the property of s; Thus {I E A I K, n V =I0} is countable. From this discussion, it is easily verified that the family {K'}'EA has the locally countable property. In consequence, UIEAK, E Bloc and N = n\ UIEA K 1 E Bloc' Now we prove that N is locally v-zero. Suppose that there is a compact subset HeN such that v(H) > o. Applying Lemma 5.1.1 to Hand (vIH), we can find a non-empty compact subset K C H such that v(UR n K) > 0 for each open subset UR of H with UR n K =I- 0. Thus for any open subset U of n with Un K =I- 0 we have also v(U n K) = v((U n H) n K) > o. Clearly, K n K , = 0,VI E A. This is a contradiction since the family {K'}'EA is maximal. Therefore, N is locally v-zero. Q.E.D.

239

Let I be a locally measurable function on n, and v be a regular Borel measure on 0. f is said to be locally essentially bounded with respect to u, if there is a constant C such that

I/(t)1 < C,

l.a.e.t/.

The minimum of such C is called locally essentially supremum of f, denoted by 11/1100. Let L 00 (0

,v)

Clearly, (LOO(O, v),

Theorem 5.1.3. Prool.

=

{II andf isis locally measurable on 0, } locally essentially bounded .

11·1100) is an

abelian C*-algebra. Indeed, it is a W*-algebra.

L 1 (0 , £I)'" = LOO(O, v).

Suppose that f E LOO(O, v). Define F(g) =

In I(t)g(t)dv(t),

Vg E L 1(0, v).

Clearly, FE L 1(O, v)t and IIFll = 11/1100. Now let F E L 1(0, v)"'. For any compact subset K of 0, since v(K) < there is unique IK E LOO(K, £11K) such that

I/K(t)1 < IIFII,

Vt E K,

and

00,

F(g) = IK fK(t)g(t)dv(t),

Vg E Ll(K,vIK) ( see [178] Theorem 7.4-A). Then we can write K,lIIK). By Proposition 5.1.2, n = N U U1EAK,. Then for any I E A,

IK = Fj

L1(

Let

f(t) = LXK/(t)fl(t). lEA

Then I/(t)1 < IIFII,Vt E 0, and f E £00(0,£1). For any 9 E L 1(0,v), since suppg = {t E n I g(t) # o} E B, it follows that J = {I E A I K, n supp 9 is countable.

Let g, = XK,g. Then 9 =

# 0}

L: g,. IEJ

Now by the continuity of

F and the bounded convergence theorem, we have F(g) = ~ f(t)g(t)dv(t). Therefore, L 1(0, v)t = Loo (0, v). Q.E.D.

240

°

Let v be a regular Borel measure on 0. A function I on is said to be non-negative locally v-integrable, if I is non-negative locally measurable, and for any compact subset K of 0, XK I E L 1 (0, v). In this case, define

J.L(E) =

L

fdv = / !XEdv,

VE E 8.

Then J.L, denoted by J.L = ! . u, is also a regular Borel measure on 0, and is absolutely continuous with respect to i/, denoted by J.L ~ t/, i.e., if E E 8 with v(E) = 0, then J.L(E) = o. Moreover, if g is a measurable function on 0, then g E Ll(O,J.t) if and only if Ig E Ll(O,V). And we have

In gdJ.L = In I gdv. Theorem 5.1.4. Let J.L, v be two regular Borel measures on 0. Then the following statements are equivalent: 1) There is a non-negative locally v-integrable function f such that J.L = f -z>; 2) If N is a locally v-zero subset, then it is also locally J.t-zero; 3) If K is a compact subset and v(K) = 0, then J.L(K) = 0, i.e. J.L -< u, Proof. The equivalence of 2) and 3) is obvious. And also it is clear that 1) implies 2). Now let 2) hold. By Proposition 5.1.2, n = N U U,EAK,. Since N is locally v-zero, it follows from 2) that N is also locally u-zeso. For each I E A, from v(K,) < 00, J.L(K,) < 00 and the Radon-Nikodym theorem there is o < I, E Ll(K/, vIK,) such that

J.L(E) = Let f =

L XK1f,·

L

I,dv,

VE E Band E C K,.

Then f is non-negative locally measurable since the family

lEA

{K'}'EA is locally countable. For any E E 8, since liN = O,J.L(E n N) = 0 and J = {I E A I K, n E f:. 0} is countable, we have

L J.L( K, n E) IEJ - L: r flE Idv =

J.L( E) =

IEJ

i.e., J.t =

f . u,

JKr

r fdv,

JE

Q.E.D.

J.L and v are said to be equivalent, denoted by J.L ,...,. t/, if J.L ~ v and v ~ J.L. In this case, clearly a.e.J.t = a.e.u.La.e.ii = l.a.e.v, and there is a non-negative locally v-integrable function f and a non-negative locally J.t-integrable function g such that J.L = I . v, and v = g . J.L.

241

And also, !(t)g(t) = 1, La.e.ti or La.e.u, Let J.L and v be two regular Borel measures on O. J.L and v are said to be singular each other, denoted by J.LJ..v, if there is A E B,oc such that A is locally u-zen» and (O\A) is locally v-zero. Let J.L, v be two regular Borel measure on O. Then we can uniquely write that J.L = !. v+ J.Ll' where! is non-negative locally v-integrable, and J.L1J..v.

Theorem 5.1.5.

Proof.

By Theorem 5.1.4, there is a non-negative locally (J.L + v)-integrable function g such that J.L = g. (J.L + v) and 0 < g(t) < 1, Vt E O. Let

A = {t E 0

I g(t) = I}, I 0 < g(t) < I},

J.tl = J.LIB,

J.Lo = J.LIA.

B = {t E 0

Clearly, A is locally J.tl-zero. If K is a compact subset and K c B, then

J.L(K)

= IK gd(J.L + v) = J.L(K) + v(K)

and v(K) = O. Thus B is locally v-zero,and J.L1J..v. Now suppose that K is a v-zero compact subset. Then

J.Lo(K) = J.L(K n A)

r gdJ.L + r s-:

lKnA

i.e., {

lKnA

gdv

={

lKnA

(1 - g}dJ.L=O. By the definition of A, J.Lo(K)

gdJ.L

= J.L(K n A)

= O. Thus

J.Lo --< t/, and there is a non-negative locally v-integrable function! such that J.Lo=!·v. SOJ.L=!·V+J.LI' Finally, we prove the uniqueness. Let J.L = /i·V+J.Li, where Ii is non-negative locally v-integrable,and J.Li.lV, £ = 1,2. Then there is Ai E B,oc such that Ai is locally v-zero and (O\Ad is locally J.Li-zero, i = 1,2. Clearly, (AI U A 2) is locally v-zero, and

is locally J.Ll-and J.L2-zero. Let K be a compact subset and K C (AI U A 2). Then v(K) = 0, and J.LdK) = J.t2(K). Thus J.LII(A 1 U A 2 ) = J.L21(A 1 U A 2),and J.Ll = J.L2· Further, It = 12 l.a.e.u, Q.E.D.

242

Notes. Proposition 5.1.2 is taken from N. Bourbaki. Theorem 5.1.3 is indeed a characterization of a localizable measure space (I.E. Segal). Moreover, on the measurability (of subsets, functions and etc.) we follows the treatment of P.R. Halmos, i.e., the measurability is independent of the measures. So in this book, the expresion of some results are slightly different with some standard books.

References. [12], [67], [157], [178).

5.2. Stonean spaces Definition 5.2.1. A Hausdorff space is said to be extremely disconnected if the closure of every open subset is also open. A compact extremely disconnected space is called a Stonean space. Proposition 5.2.2. Let 0 be a Stonean space. Then the linear span of projections of C(O) is dense in C(O). Proof.

Let / E C(O), / > 0 and e > O. Consider the following partition:

o = Ao < Al < ... < An = II/II + 1 such that (Ai+l - Ai)

< e,O < , ; < n E1={tEO

is an open subset of O. Then G 1 induction, define

=

1. Clearly,

I /(t)
E 1 is an open and closed subset of O. By i-I

s, = {t E

0

I / (t) < Ai, t rt.

UG

j }

j:;1

and Gi = E,;, 2

< i < n. n

We say that 0 =

Then E, is open,and G i is open and closed, 1

UG

i.

In fact, suppose that there is t E 0\

';=1

< i < n.

n

UG

i .

In

i:;1

n-l

particular, t

rt. Gn ,

and t

rt.

U G,;.

On the other hand, since /(t)

< II/II < An'

i=l

it follows from the definition of En that tEEn C G n. This is a contradiction. n Thus

n=

Uc..

i:;1

243

We have also G i n G,.

= 0, Vi '# i. Indeed, we may assume i >

i-I

j. Then

i-I

U GJc is an open subset containing G,.. Clearly, (0\ U G Jc) is a closed subset

Jc=l

Ie=l

i-I

containing E«. Thus G i C (0\

UG

k),

and G i n G,.

= 0.

1e=1

Now let Xi be the characteristic function of G t • Then Xi is a projection of C(O),l < i < n. Notice Ai-l < I(t) < Ai, Vt E Gi , 1 < i < n. Therefore, n

Ilf -

L -Xix.11 < c.

Q.E.D.

1'=1

Let 0 be a compact Hausdorff space, and Cr(O) be the set of all real continuous functions on O. Then the following statements are equivalent: 1) 0 is a Stonean space. 2) For any bounded increasing net of non-negative function in Cr(O), there exists its least upper bound in Cr(O). 3) For any bounded subset of Cr(O), there exists its least upper bound in

Theorem 5.2.3.

C,(O). 4) Let 9 be a bounded real valued lower semicontinuous function on O. Then there is I E Cr(O) such that E = {t E 0 I I(t) '# g(t)} is a first category Borel subset of O. Moreover, the function I in 4) can be chosen as I(t) = limg(t'), Vt E O. t'-t

Proof.

4)

=}-

3). Let A be a bounded subset of Cr(O). Then

g(t)

= sup{f' (t) I I'

E A}

is a bounded lower semicontinuous function on O. By 4), there is I E Cr(O) and a first category subset E of 0 such that I(t) = g(t), Vt ~ E. Clearly, (g - f) is also lower semicontinuous. Then G = {t E 0 I g(t) > I(t)} is open and GeE. Since 0 is a Baire space and E is first category, it follows that G = 0, i.e., I(t) > g(t), Vt E O. Suppose that h E Cr(O) and h > I', VI' E A. Then h(t) > g(t), Vt E 0, and h(t) > I(t), Vt ~ E. In addition, (O\E) is dense in O. Thus h(t) > I(t), Vt E 0, and I is the least upper bound of A. 3) => 2). It is obvious. 2) => 1). Let U be an open subset of O. Let

A =

{I' E Cr(O) 10 < I' < 1,supp !' c U}.

With respect to the partial order in Cr(O), A is a bounded increasing net of non-negative functions in Cr(O). From 2), A has its least upper bound I in C,(O). Suppose that

g(t)

= sup{f' (t) I I' E A},

Vt E

o.

244

Clearly, g(t) < f(t),Vt E O. For any tEO, there is f' E A such that f'(t) = 1. Thus glU = 1. On the other hand, since I' < 1, Vf' E A, it follows that f < 1. So flU = 1. Suppose that there is to fJ. U such that f(t o) > O. Pick hE Cr(n) such that h > 0, h(t o) = 0 and h(t) = 1, Vt E U. Then

f' < inf{h, f} -# I,

Vf' E A.

This contradicts the fact that f is the least upper bound of A. Therefore, f(t) = 0, Vt tI. U. Further, U is open. 1) ==* 4). Let g be a bounded real valued lower semicontinuous function on n. We may assume that 0 < g(t) < 1, Vt E O. For any real number A,F(A) = {t E 0 I g(t) < ..\} is a closed subset of O. Put G(A) =Int(F(..\)). Noticing that

= O\(O\F(>.)),

G(>.)

G(A) is an open and closed subset of 0 from 1). Thus the characteristic function XA of G(>') belongs to Cr(O). Let k

2"

2"-1

In = " L..Jt d 2 Xi 2"

k=l

Xk-l) 2"

=

"L..J Xi· 2n

1-

k=l

Fox fixed nand tEO, we assume that k = min{i

I 0
Then X~(t) = 1, Vm > k, and fn(t) = 2~' Moreover, clearly X 2 t:'+ i (t )

= 1,

and

Vm>2k;

Thus

k

X 2t ; ' f l ( t ) = O , V m < 2 ( k - 1 ) .

k

1

-2 n - -2n+1 < f n+ l(t) < - 2n ' Further, we get 1

Illn+1 - Inll < 2 n+ 1 ' Therefore, there is

f

E

c, (0) E=

I In - III

such that 00

2

11

~ O.

k

k

2

2

Let

U U(F(n)-G(n))' n=1 k=1

Then E is a first category Borel subset of

I (t)

= g( t ) ,

For fixed n, put N = 2n,Fk = F(~),Gk Ek = n\Fk • Then

F I C F2 C ...

Vn.

c FN

= 0,

n.

We claim that

Vt

fJ.

E.

= G(~)

=Int(Fk)), and

G 1 C G 2 C ...

c GN = n

245

and E I :J E 2 :J ... :J EN

Thus G s n Ei

= Gi\Fi = 0, Vi < j. An (B

U

= 0.

By the formula

C) = (A

n B)

U

(A n C)

we have

(GlUEd

n (G 2 U E 2 ) =

[G1 U E 2 U(EI n G 2 ) ] n (G 3 U E 3 )

and

n(Gk

G} U E 2 U (E 1

=

N

G I U E 3 U(G 2 n Ed U(G g n E 2 ),

N-I

U

U U (Gk+ I \ Fk) .

E k) = GI

k=l

1ft E G}, then

n G2 ) ,

k=l

° 1

1 and

1

fn(t) = N - Nxo(t). If t E G k + l \Fk(1

1), then

-;, < g(t) < k X w(t)

= 1,

X~(t) = 0,

and fn(t)

=

tJ 1;

Vm > k

+ I;

Vm < k;

k~l. In other words, we obtain

1

Ig(t) - In(t) I
n(G 2'"

Vt E

2

k

U E k ) , Vn.

k=l

Since In ~ I in Cr(fl), it follows that g(t) = l(t),Vt tJ. E. Now it suffices to show that

f(t) = limg(t'),

Vt E fl.

t'-t

From preceding discussion of liin - In+lll < 1/2 n+ I (Vn ), In(t) ~ I(t), Vt E O. If t' E E, then there is some n and some k(1 < k < 2 n ) such that t' E F(2kn)\G(2~'). Since X

2Pk

2"+1'

(t') = X k (t') = 2fr

°

for any

p,

it follows that fn+p(t') > 1 2

k 1+p 2"+"-} n 1 > 2 n > g(t').

L

j>2 Pk

246

Thus I(t) > g(t), \It E O. Further

limg(t') < lim/(t ') = I(t), t'--+t

\It

t'-.t

E O.

On the other hand, for any tEO and E > 0 there is a neighborhood U of t such that I(t") > I(t) - E, \It" E U. Since E is first category, U\E -# 0. Pick t' E U\E. Then g(t'} = I(t'} > I(t) - E. Further

limg(t') > I(t) -

t'-.t

and limg(t') > I(t) since t'--+t

E

E,

is arbitrary. Therefore

I(t) = limg(t') t' --+t

1

\It E O.

Q.E.D. Definition 5.2.4. Let 0 be a Stonean space, and JL be a regular Borel measure on 0 (i.e. a positive linear functional on C(O)). JL is said to be normal, if J1(/) = SUPI J1(/d for any bounded increasing net {II} of non-negative functions of Cr(O), where I is the least upper bound of {It} in Cr(O). Proposition 5.2.5. Let 0 be a Stonean space and J1 be a normal regular Borel measure on O. Then JL(F) = J1(E) = a for any rare closed subset F and first category Borel subset E.

Proof. Let F be a rare closed subset of O. Then (O\F) is open and dense in 0, and O\F = U{suppl I I E C(O), a < I < 1, suppl C (O\F)}, where suppl = {t E 0 follows that

I I(t) f::. a}, \II E C(O).

O\F = U{ G C O\F

IG

Since 0 is a Stonean space, it

is open and closed}.

By the inclusion relation with respect to G, {XG I G C O\F, and G is open and closed} is a bounded increasing net in C;(G). Clearly, the least upper bound of {XG} in C;(G) is 1. Thus we have

JL(O) = sup{JL(G)

IG

is as above}

since JL is normal. Further, J1( F) = a. Now suppose that E is a first category Borel subset of O. We can write E = F n , where each F n is rare. Then F n is closed and rare, \In. Therefore

U n

J..L(E)

=

a from

the preceding paragraph.

Q.E.D.

247

Proposition 5.2.6. Let 0 be a Stonean space, and J.,t be a normal regular Borel measure on n. Then supp J.,t is an open and closed subset of n.

Proof.

Let F := sUPP/L. Then F is a closed subset, and F\ Int(F) is a rare closed subset. By Proposition 5.2.5, J.,t(F) = J.,t(lnt(F)). Let E be the closure of Int{F). Then E is open and closed, and Int(F) c E c F. Thus Jt(E) = Jt{F). By the definition of SUPPlI, we have E = F = suppz/, Q.E.D."

Proposition 5.2.7. Let n be a Stonean space, and h be a bounded measurable function on O. Then there is f E C(O) such that

= h(t)" a.e.ti

f(t)

for any normal regular Borel measure j1 on O.

Proof.

We may assume that h is real valued. Then g(t) = limh(t ') is a t'-t bounded real valued lower semicontinuous function on n. By Theorem 5.2.3, there is a f E Cr(O) and a first category Borel subset E of 0 such that

f(t) = g(t),

Vt fj. E.

For any normal regular Borel measure J.,t on 0, by the Lusin theorem there is a disjoint sequence {Kn } of compact subsets of 0 such that h is continuous on K n , Vn, and n

Then

h(t)

= g(t), n

Since (Kn \ Int (Kn )) is rare and closed, it follows from Proposition 5.2.5 that Jt(K\Int(Kn )) = 0, Vn. Thus j1(E U(O\ Int(Kn )) = 0 and

U n

f(t)

= h(t),

Vt E

(U Int(K

n))

n (O\E),

n

i.e., f(t) = h(t), e.e.u,

Q.E.D.

Definition 5.2.8. 0 is called a hsjperetonean space, if it is a stonean space.and for any a < f E 0(0) and f ;:f. 0 there is a normal regular Borel measure J.,t on n such that J.,t(f) > o. Proposition 5.2.9. Let 0 be a hyperstonean space. Then there is a family {J.,tl} of normal regular Borel measure on 0 such that SUPP/Ll nSUppJ.,tI' = 0, Vi -=I l', and USUPPJtI is dense in n. l

248

Let {Ill} be a maximal family of normal regular Borel measures on such that SUPPlll n supprzi. = 0, Vl 1'= l',

Proof.

o

Put

r

=

USUPPlll'

By Proposition 5.2.6,

r

is on open subset of O. Then

r

l

is open and closed. If E = O\f ;f. 0, then 0 < XE E C(O) and XE 1'= O. From Definition 5.2.8, there is a normal regular Borel measure Il' on 0 such that 1l'(E) > O. Let V Borel subset

6.

Clearly, Il is a normal regular Borel measure on 0, and

oi- SUPPIl C

E =

O\r.

This is a contradiction ,~ince the family {J.ll} is maximal. Therefore,

r=

O.

Q.E.D. Notes. The concept of Stonean spaces was introduced by M. Stone. The presentation here follows a treatise due to J. Dixmier. References. (20), [164], [177].

5.3. Abelian W*-algebras Theorem 5.3.1. Let Z be a a-finite abelian W*-algebra, and 0 be its spectral space. Then 0 is a hyperstonean space, and there is a normal regular Borel measure v on 0 such that suppz/

= 0,

and

Z '"'-' C(O)

= LOO(O, v).

Proof.

Suppose that Z C B(H), here H is some Hilbert space. By Proposition 1.14.5, Z admits a separating vector Eo(E H). Let f ~ mf be the * isomorphism from C(O) onto Z. From Theorem 5.2.3 and Proposition 1.2.10, o is a Stonean space. Clearly, there is a regular Borel measure v on 0 such that (m,Eo, Eo) = f(t)dv(t), VI E C(O).

fo

By Proposition 1.2.10, v is normal. Suppose that there is a non-empty open Borel subset U of 0 such that 1I(U) = O. Pick I E C(O), I > 0, f 1'= O,and suppj' C U. Then (m,Eo, Eo) = o.

249

eo

Since is separating for Z, it follows that I = 0, a contradiction. Thus suppz/ = O. In consequence, 0 is a hyperstonean space, and C(O) can be embedded into Loo(O, v). Let {I,} be a net of C(O), Il/dl < 1, Vi and I, ~ I(E Loo(11,v)) with respect to w* -topology in Loo(O,v). Put mi = m,,(E Z), Vl. Then Ilmdl < 1. Replacing {ml} by its subset if necessary, we may assume that ml ~ m g weakly, where g E C(O). Then for any hE C(O),

1/(1, -

g)hdvl = I((ml - mg)mheOl eo) I ~ O.

Since C(O) is dense in L 1(0, v), it follows that It ~ g with respect to w*topology in Loo(O,v). Hence f(t) = g(t), a.e.v. From above discussion, C(O) is w*-closed in Loo(O,v). Clearly, C(O) is w*-dense in Loo(O,v). Therefore, C(11) = Loo(O, v). Q.E.D.

Proposition 5.3.2.

Let 0 be a compact Hausdorff space, and v be a regular Borel measure on O. Then Loo (0, v) is a a-finite abelian W*-algebra.

Proof.

By Theorem 5.1.3, Loo(O,v) is an abelian W*-algebra. Let

w(/) =

In I(t)dv(t),

Since 1 E L 1(0,v), it follows that w(·) is a faithful a-continuous positive functional on Loo(O, v). From Proposition 1.14.2, Loo (11, v) is a-finite. Q.E.D. Let 0 be a hyperstonean space. Then C(O) is an abelian W*-algebra. Moreover, if there is a normal regular Borel measure v on 0 with suppv = 0, then C(O) = Loo(O, v) is a-finite.

Theorem 5.3.3.

Proof. First suppose that there is a normal regular Borel measure v on 11 such that suppz- = O. Then C(O) can be embedded into Loo(O, v). Moreover, for any h E Loo(O,v), by Proposition 5.2.7 there is I E C(O) such that I(t) h(t), a.e.v. Thus C(O) = Loo(O,v). Further, C(O) is a a-finite abelian W*-algebra from Proposition 5.3.2. Generally, by Proposition 5.2.9 there is a family {VI} of normal regular Borel measures on 0 such that SUppVln supp Vp = 0, Vl =i l' .and I' = Usupp

=

v, is dense in O.

,

By Proposition 5.2.6, sUPP VI is open and closed, Vi. Then I' is a locally compact Hausdorff space. Let v = EBvl. Then v is a regular Borel

L I

measure on f and supp v = f. Consequently, I ~ Ilf is an injective map from C(11) to Loo(r,v). Moreover, for any hE Loo(f,v), let h(t) = O,Vt E O\f. Then by Proposition 5.2.7 there is I E C(O) such that f(t) = h(t), a.e.z/,

250

Thus f(t) = h(t), l.a.e.v on is a W*-algebra.

r.

Further C(O) is

* isomorphic to LOO(f,v), and Q.E.D.

Theorem 5.3.4. Let Z be an abelian W* -algebra, and 0 be its spectral space. Then 0 is a hyperstonean space, and there is a locally compact Hausdorff space r and a regular Borel measure v on I' with suppv = r such that Z is * isomorphic to Loo [I' , v). Proof. Let Z c B(H), and I ~ mf be the * isomorphicm from C(O) onto Z. Then for any EE H, there is a regular Borel measure ve such that

(mfE, E) =

~ f(t)dve(t), VI

E C(O).

From Theorem 5.2.3 and Proposition 1.2.10, 0 is a Stonean space, and ve is normal, VE E H. If I is a non-zero positive element of C(O), then there is EE H such that (m fE, E) > 0, i.e., Ltdf) > 0. Therefore, 0 is hyperstonean. The rest conclusion is contained in the proof of Theorem 5.3.3 indeed. Q.E.D. Definition 5.3.5. Let M be a W* -algebra. E( C M) is called a generated subset for M, if M is the smallest W*-subalgebra containing E. Moreover, if M admits a countable generated subset, then M is called countably generated. A generated subset for a C*-algebra is understood similarly. Lemma 5.3.6. Let n be a compact Hausdorff space. If the C*-algebra C(O) is generated by a sequence {Pn} of projections, then C(O) can be generated by an invertible positive element. Proof.

Let 00

h

=

L

1

3n (2pn

1

+ 2)'

n=l

Then h is an invertible positive element of C (0). For any t 1 , t 2 E 0 and t, # t2, there is a minimal positive integer k such that

Pk(t 1 ) ~ Pk(t 2 ) since {Pn} is a generated subset for C(O). Thus

Now by the Stone-Weierstrss theorem and Lemma 2.1.5, C(O) is generated by {h}. Q.E.D.

251

Theorem 5.3.7. Let Z be a countably generated abelian W· -algebra. Then Z can be generated by an invertibel positive element. In particular, every abelian VN algebra on a separable Hilbert space is generated by a single operator. Let {an} be a generated subset for Z. Replacing an by ~(an+a:rJ, we may assume that a~ = an, Vn. From the spectral decomposition of {an}, Z can be generated by a sequence {Pn} of projections. Let A be the C"" -subalgebra of C·-algebra Z generated by {Pn}. By Lemma 5.3.6, A is generated by an invertible positive element a. Clearly, A is also a generated subset for Z. Thus, Z is generated by a. Moreover, each VN algebra on a separable Hilbert space is countably generated. That comes to the rest conclusion. Q.E.D.

Proof.

Theorem 5.3.8. Suppose that Z is an abelian VN algebra on a separable Hilbert space H, and Z contains no minimal projection (a projection P of Z is said to be minimal, if p -# 0 and any projection q of Z with q < P implies either q = 0 or q = p ). Then Z is * isomorphic to LOO([O, 1]), where measure on [0, 1] is Lebesgue measure.

Proof.

Let 0 be the spectral space of Z. By Theorem 5.3.1, 0 is a hyperstonean space, and there is a normal regular Borel measure v on 0 with suppv = 0 such that Z ::: C(O) = LOO(O, v). From Theorem 5.3.7, Z is generated by a positive element a. We may assume that 0 < a < 1. Put 1= [0,1], and let z ---+ z(.) be the Gelfand transformation from Z to C(O). Then a(.) is a continuous map from 0 to I. Define a Borel measure j.,t on I and a * homomorphism ~ from LOO(I,j.,t) to LOO(O,v) as follows:

j.,t(E) = v(a- 1 (E )), ~(/)(t)

= I(a(t)),

V

Borel subset

Eel,

Vt E 0, IE LOO(I,j.,t).

Clearly, 4>(p) = p(a) for any polynomial p(.) on I. Thus 4>(LOO(I,J.l)) is w""dense in LOO(O, v) since Z is generated by {a}. We claim that 4>(Loo(I,j.,t)) is dense in Ll(O,V). In fact, suppose that there is some g E LOO(O, v) such that

In g(t) 4> (f)( t) dv (t) =

0,

V f E L 00 (I, j.,t) .

Since 4>(LOO(I,J.l)) is w"'-dense in LOO(O,v), there is a net {Il} c LOO(I,j.,t) such that

252

Clearly, 9 E L 1(O, v) too. Thus

°= ~ g(t)(fl)(t)dv(t)

--+

~ Ig(t)1 2dv(t)

and 9 = 0. Therefore, (LOO(I'J-l)) is dense in L 1(0,v). Now we say that is a-a continuous. It suffices to show that {fz) --+ (w*-topology) for any net {fl} C L OO(I, J-l) and Ilfdl < 1, Vi, and I. --+ (w*topology). For any 9 E L 1{0, v) and e > 0, from preceding paragraph we can pick f E L OO{I,J-L) such that

°

~ Ig(t) Then

(f)(t) Idv(t)

°

< e.

I ~ 9 (t) (II) (t) du (t) I

h

< I II (A)f(A}dJ-l(A) I + llg(t) - (/)(t)ldv(t) < 2e

°

if I is sufficiently later. Thus (ft) ----t (w*-topology), and is a-acontinuous. Thus, we get (L OO (I, J-l)) = L oo (0, v). Suppose that I E LOO(I,J-l) such that (f) = 0. Then ( f g)

and

= 0,

V9 E C (I) ,

hf(A)g(A)dJ-l(A} = ~ (fg)(t)d1/(t} = OO(I'/1)

0, Vg E C(I), and f = 0. There-

fore, is a * isomorphism from L onto LOO(O,v). The measure J-l on I is not atomic, i.e., J-l({A}) = 0, VA E I. In fact, suppose that there is A E I such that /1({A}) > 0. Put E = a- 1 ({ A}}. Then 1/(E) > O. So XE is a non-zero projection of LOO(O, 1/). Since (X{A}) = XE and X{A} is a minimal projection of L OO(I,J-l), it follows that XE is a minimal projection of L OO {O, 1/)(...... Z}. This contradicts the assumption. Let f(A) = J-l{[0, Aj), VA E I. Then I is a continuous increasing function with f(O) = and f{l) = 1 (we may assume that v(O) = 1). Further, let

°

g(A} = min{A'

E

I

I f(A'} =

A},

VA E I.

Then 9 is a left continuous strictly increasing function on I, and has countable jump points at most. Suppose that {AI < .\.2 < ... < An < ...} is the set of jump points of g. then there is a sequence {A~} with Al < Ai < A2 < .\.~ < ... < An < A'n < ... such that for each n

f{A} = f(A n}, Then go f{A} = A, VA E

VA E [An,A~J;

1\ UlAn' A~l. n

f(A} > f{A n},

On the other hand

VA > A~.

253

Vn. Thus g 0 1(>') = >., e.e.u, Let m be the Lebesgue measure on I. For any 0

< >'1 < >'2 <

1, we have

Thus m = )1, 0 1-1. Further, )1, = m 0 g-l. Now define a * homomorphism W from L oo(l) = Loo(l,m) to L oo(I,)1,) as follows:

Vh E L oo(I).

w(h) = hoi, If k E Loo(I,)1,), then k

0

g E Loo(I), and

w(k 0 gH>') = k(>'), a.e.u, So W(Loo(I)) = Loo(I,)1,). Moreover, suppose that h E L oo(l) such that '11 (h) = O. Since J w(hh)(>.)d)1,(>') = J Ih(>.)j 2dm(>.), it follows that h = O. Thus '11 is a * isomorphism from L OO (1) onto L OO (1,)1,). Further, 0 W is a * isomorphism from Loo([O, 1]) onto L oo (0, v). Q.E.D.

Corollary 5.3.9. Let H be a separable Hilbert space and .Ita be the collection of all abelian VN algebras on H. For any Z E .Ita, define [Zl = {Y lYE .Ita, and Y is * isomorphic to Z}. Then {[Z] I Z E .Ita} is countable. Definition 5.3.10. An abelian VN algebra Z on a Hilbert space H is said to he maximal abelian, if there is no abelian VN algebra on H which contains Z properly. Clearly, Z is maximal abelian if and only if Z = Z'.

°

Definition 5.3.11. Let be a locally compact Hausdorff space, and v be a regular Borel measure on 0. For any f E Loo(O, v), define

Clearly, m/ is a bounded linear operator on L 2(0, v). {m/ called the multiplication algebra on L 2(0, v).

II

°

E Loo(O, v)} is

Lemma 5.3.12. If is a compact Hausdorff space, and v is a regular Borel measure on 0, then the multiplication algebra Z is a maximal abelian VN algebra on L 2(0, v).

Proof.

Let a' E Z'. Then for any

f

E

Loo(O,v)(c L 2(0,v))

,- 1 =m/a -. 'I = a '1 =am/ Put a'l = 9(E L 2(0,v)). Then a' !

I · a'I.

= gl,VI E Loo(O,v).

254

We say that Ig(t) < Ilalll, a.e.v. In fact, suppose that there is e > compact subset K of such that

°

v(K) > 0, and Ig(t) I >

Ila'll + s,

°and a

Vt E K.

Then

v(K)(lla'll + e)2 < /19(t)XK(t)1 2dv(t)

= lIa'xKl12 < Ila'11 2 v (K ). This is a contradiction. Thus Ig(t) I < Ila'll, a.e.v, and 9 E Loo(O, v). Now from 9 E Loo(O, v ) and a'i = gl,VI E Loo(O, v ), we have a' = mg since LOO (0, v) is dense in L 2 (0, v). Therefore, Z' = Z. Q.E.D. Theorem 5.3.13. Let Obe a locally compact Hausdorff space, and v be a regular Borel measure on 0. Then the multiplication algebra Z is a maximal abelian VN algebra on L 2(0, v).

Proof. If mj = 0 for some I E Loo(O,v), then m/XK = IXK = 0 (a.e.v) for each compact subset K of 0. Further, I = 0, l.a.e.v. Thus I -+ mj is a * isomorphism from Loo(O, v) onto Z. Also, this * isomorphism is a-a continuous. Consequently, Z is a VN algebra on L2 (0, v). By Proposition 5.1.2, = NUUKI , where N is locally v-zero,and {Kd is

°

1

a disjoint family of compact subsets of Then

L 2(0, v)

°

with the locally countable property.

= L EBL 2(K1, vd I

where

L

2(K I,

=

vlKI, Vl. For any a' E Z', since a'h l = a'xK1h/ = XK,a'h l E VI), Vhf E L 2(Kt,vz), it follows that L 2(KI,vz) is invariant for a',Vl. By

VI

Lemma 5.3.12, for each l there is 91 E Loo (Kt, vz) such that

a' IL 2 (Kl, VI)

= mg/ .

Let 9

=

LXK/gi. I

Then 9 E Loo(O,v), and a' =

mg.

Therefore, Z'

= Z.

Q.E.D.

Proposition 5.3.14. Let Z be an abelian VN algebra on a Hilbert space H, Eo (E H) be a cyclic vector for Z ,and 0 be the spectral space of Z. Then there is a regular Borel measure v on 0, and a unitary operator u from H onto L 2(0, v) such that suppv

= 0,

z,..... C(O)

= Loo(O, v),

255

where

f

~

m, is the Gelfand transformation from C(O) onto Z.

Proof. Since Z C Z', it follows that Eo is also separating for Z. Let v be the regular Borel measure on 0 such that

(m,Eo, Eo) =

fo f(t)dv(t),

Vf E C(O).

Then by the proof of Theorem 5.3.1 we have suppz-

= 0,

C(O)

= Loo(O, v).

Now define um,Eo = f,Vf E C(O). Then u can be extended to a unitary operator from H onto L2(0, v) since Eo is cyclic for Z. Further umtu-1 = Vf E C{O) = Loo(O, v). Q.E.D.

mt,

Proposition 5.3.15. Let Z be an abelian VN algebra on a Hilbert space. Then Z is maximal abelian and a-finite if and only if Z admits a cyclic vector. Proof. The sufficiency is obvious from Proposition 5.3.14, Theorem 5.3.13 and Proposition 1.14.2. Now suppose that Z is maximal abelian and a-finite. By Proposition 1.14.5, Z admits a separating vector Eo. Further, Eo is also cyclic for Z since Z' = Z.

Q.E.D.

Corollary 5.3.16. Let Z be an abelian VN algebra on a separable Hilbert space. Then Z is maximal abelian if and only if Z admits a cyclic vector. Theorem 5.3.17. Let Z be a maximal abelian VN algebra on a Hilbert space H. Then there is a locally compact Hausdorff space 0 and a regular Borel measure v on 0 with suppv = 0 such that Z is unitarily equivalent to the multiplication algebra on L 2(0, v). Proof.

We can write HI = ZEl,

VI.

Let PI be the projection from H onto Hi, Vl. Then PI E Z' = Z, Vl. Suppose that 0' is the spectral space of Z,and f ~ mt is the * isomorphism from C(O') onto Z. Then for each 1 there is an open and closed subset 0 1 of 0' such that PI = mxI' where Xl is the characteristic function of 0 1• Since PIPI' = 0, it follows that 0 1 n OJ' = 0, Vl ¥- l', For each l, Zl = ZPI admits a cyclic vector 6, and OJ is its spectral space. By Proposition 5.3.14, there is a regular Borel measure VI on 01 with suppz-, = 0 1,

256

and a unitary operator

C() 0,

Ul

from Hi onto L 2 (01, vd such that

= L oo( 0h Vl,)

-1 Ulm (l) f U,

= -(l) mf ,

WI "\ Vl ) , vEL 00(1'HI,

where I -+ m~) is the * isomorphism from C (Od onto Z" and m~) is the multiplication operator of I on L 2 ( Oh VI). Put n = U 0 1• Then n is an open dense subset of 0'. So n is a locally l

compact Hausdorff space. Let V

=L I

on 0, and suppv

= 0.

Further, let

EBVl. Then v is a regular Borel measure

=L

U

EBul. Then

U

is a unitary operator

I

from H = LEBHl onto L (O, v) = LEBL 2 (OL, vz) . Denote the multiplication 2

I

l

algebra on L v) by Z = {m g ! g E Loo(O, 1/)}. For any f E C(O'), it is easy to see that umfu- 1 = g , where g = flO E Loo(O, v). Thus c Z. Since Z is maximal commutative, it follows that «z«:! = Z. Q.E.D. 2(0,

m

«z«:

Let M be a W··algebra, and p be a projection of M. p is said to be abelian (commutative), if pMp is abelian (commutative).

Definition 5.3.18.

Proposition 5.3.19. Let M be a W· -algebra, p and q be two projections of M, and p be abelian. 1) If p """ q, then q is also abelian. 2) pMp = Zp, where Z is the center of M. 3) If q < p, then q = c(q)p, where c(q) is the central cover of q in M. we may assume that M is a VN algebra. 1) It is immediate from Proposition 1.5.2. 2) Since M; is commutative, it follows that M; eM;. Now by Proposition 1.3.8, u, = M p n M; = Zp. 3) By Proposition 1.5.8, the central cover of q in M p is c(q)p. But M p is commutative, so q = c(q)p. Q.E.D. Proof.

Notes. Theorem 5.3.8 is due to P. Halmos and J. Von Neumann. Theorem 5.3.7 is due to J. Von Neumann. There is a general conjecture: if M is a VN algebra on a separable Hilbert space, then M is generated by a single operator? This conjecture is still open now, but we have rich results on it, see T. Saito's Lectures.

References. [20], [68], [157], [142].

257

5.4.

* Representations

of abelian C*-algebras

In this section, let A be an abelian C* -algebra with an identity. Then A """ C{O), where 0 is the spectral space of A, a compact Hausdorff space. Let {a, H, e} be a cyclic * representation of A. Then there is unique (in the sense of equivalence) regular Borel measure J.L on 0 such that

Theorem 5.4.1.

{7r,H} """ {~,L2(0,JL)}, where (~(a)f)(t) = a(t)f(t),Vt E O,f E L 2(0,1-l), a E A, and a -+ a{·) is the Gelfand transformation from A onto C (0) . Let I-l be the regular Borel measure on 0 such that

Proof.

Va E A.

e

Further, define u7r(a) = a('), Va E A. Then U can be extended to a unitary operator from H onto L 2(0,1-l), still denoted by u. Clearly, u7r(a)u-1 = ~{a), Va E

A.

Now suppose that v is a regular Borel measure on 0, and v is a unitary operator from L 2(0,1-l) onto L 2(O,v) such that v~(a)v-l =

Put vI

=

a(E L 2(0, v)). Then va

v(a),

Va E A.

= vtL(a) 1 = v(a)a,

! la(t)12dl-l(t) = Jla(t)a(t)1 2

dv (t ),

Thus JL

= 10:1 2 • u, and

I-l

-<

t/,

and

Va E A.

Similarly, v -< u, Therefore, I-l ,...... u,

Q.E.D.

Each * representation of A is a direct sum of a zero representation and a family of cyclic * representations. In this section, we study a * representation {1r, H} of A such that 7r is a countable direct sum of cyclic * representations. By Proposition 1.14.2, 7r is as above if and only if 1r(A)' is a-finite. Let {7r, H} be a there is unique regular Borel measure

Definition 5.4.2.

* representation of u« on 0 such that

A. For any

eE H,

VaE A.

e

We introduce a partial order ">" on H as follows. > 7] means that JLe >- JL,.,. Moreover, E H is said to be maximal, if > 7], V7] E H.

e

e

258

If 77 E He

Lemma 5.4.3.

= 7r(A) E, then

77

< E.

Proof. By Theorem 5.4.1, there is a unitary operator u from He onto 2 L (0 , J.l e) such that

u(7r(a)IHdu-1 = ~e(a), Let

Va E A.

1 = U77 (E L 2 (0, J.l~)). Then (7r(a)77,77) = / a(t)l/(t) 12dJ.l d t) ,

Thus, J.l'7

= 1/1 2 • l1e

and 11f7 -< l1e, T} <

If H =

Lemma 5.4.4.

Va E A.

e.

Q.E.D.

L ffiJh, where n, =

7r(A) Ek and II Ekll < 1, Vk, then

k

is maximal. Proof.

L2- J.l ek· Fix 17 E H,and write T} = L77k, where Then J.l'7 = L J.lrl/<' and for each k, J.l'7k -< J.lf.1t by Lemma 5.4.3.

Clearly, J.le

k

=

k

T}k E Hv, Vk.

k

k

If E is a Borel subset of 0 such that J.le(E) = 0, then J.lek(E) = 0, Vk. Thus JL'7It{E) = 0, Vk, and 11f7(E) = O. Therefore, J.l'7 < J.le, i.e., 77 < E· Q.E.D.

If 7r(A)' is a-finite, then the set of maximal vectors is dense

Lemma 5.4.5. in H.

Proof. By Lemma 5.4.4, there is a maximal vector E( E H) at least. Define He = 7r(A)E, and fix T} E H. We can write T} = 771 + 772, where 771 E He, 772 E Let HI = 7r(A)T}1 C He, and write

Ht.

E= 6 + E2' Suppose that J.li

JL'71+c6 = J.l'71

+

where

= J.le.. , i

6

= 1,2. Then J.le €2J.l2' By Lemma 5.4.3,111

E Hh

6

E H

t.

= J.ll + J.l2'

For any e > 0, clearly -< 11'71' Thus

+ C 2 J.l2 ,. . ., J.lI + J.l2 = J.le and (771 + €E2) is also maximal since E is maximal. Notice that El So C2 = E- 6 E He and 771 + CC2 E Hi, Then 11'71+c6

>-

P,l

E HI

c Hi,

259

and TJ

+ eC2 =

TJl

+ E:C2 + TJ2

is maximal. Clearly,

Q.E.D.

Therefore, the set of maximal vectors is dense in H.

If 1r(A)' is o-finite, then we have a decomposition

Lemma 5.4.6.

00

H

=L

n, =

ffiHk,

'JT"(A) 6,

k=l

and

6 > C2 > ... >

Proof.

Ck

> ., '.

Let {~n} be a cyclic sequence of vectors for tr(A), and {TJk

Ik =

1,2,"'}

By Lemma 5.4.5, pick a maximal vector

6 (E H)

such that

Denote the projection from H onto HI = 1r(A)€I by Pl' Similarly, there is a maximal vector C2 in H t = (1 - pd H such that

Again let P2 be the projection from H onto H 2 = 1r(A)6 . . . '. Generally, suppose that we have 6,' .. , Ck-l' and Pi is the projection from H onto Hi = k-l

1r(a) Ci, 1 < i < k -

1. Then we can pick a maximal vector

6

in

(L: ffiHi ) 1. = i=l

k-l

(1 -

L

Pi)H such that

i=l

k-l

116 -

(1 -

L pd TJ k II < 1/ k.

i=l

Further, let Pk be the projection from H onto H k sequence {Ck} satisfies:

Cl> 6>· .. > 6 > "', Now it suffices to show that H

=L k

Hil-Hj,

ffiH k •

7r(A) Ck' Clearly, the

Vil-i,

260

For fixed k, by the definition of {11m} there {I, 2,"'} such that 11k,. = ~b Vn. Then

IS

a subsequence {k n} of

kn-l

II(Ek,. + L

Pi~k) - ~kll

i=1

1

k,.-1

116.. - (1 -

L

pj)11k"'l <

k

~ O.

n

j=1 00

Thus ~k E H =

L

L ffiHi , Vk.

Since [1r(a)~k

Ia E

A, k] is dense in H, it follows that

i=1

Q.E.D.

ffiH k •

k

Lemma 5.4.7. Let J1" v be two regular Borel measures on 0, and v bea bounded linear operator from L 2(O, J.L) to L 2(O, v) such that v~(a) =

Then vf

= af, Vf E

Proof.

Since va

v(a)v,

L 2(0,J.L), where a

= vI

Va E A.

E L 2(0,v).

= v~(a)I = v(a)a = aa,

! la(t)a(t)1

2dv(t)

< IIvl12

! la(t)1

it follows that

2dj.L(t),

Va E C(O).

Then liV112J.L > lal 2 ·v, and a] E L 2(0,v) for any f E L 2 (O, J.L ). Further, from va = aa(Va E C(O)) and the density of C(O) in L2 (0, J.L) we get v f = af,Vf E L 2 (O, J.L ). Q.E.D.

Lemma 5.4.8. on 0, and

Let {J.Lk}, {Vk} be two sequences of regular Borel measures

Suppose that there is an isometry u from H to K such that

where H(a)(fl"" ,fk"") = (afb'" ,afb"') for any a E A and (it"", fkl ...) E H (i.e., fk E L 2(0 , J.L k), Vk), and K(a) is defined similarly. Moreover, if we assume that

J.Ll >- J.L2 >- ... >- J.Lk >- ... , V2 >- V3 >- ... >- Vj >- ... , then vi >- J.Lj, Vj > 2.

261

Let Pk be the projection from H onto Hk = L 2(O, I-lk),and qi be the projection from K onto K, = £2(0, Vj), and Ujk = qiUPkl vi, k. It is easy to see that

Proof.

Ujk tik (a) Put aik = ujk1(E L2(0,Vj)

= l.'j(a)ui k' vi, k, a E

A.

= K j). Then by Lemma 5.4.7,

u(O,···,!k,O···)

Since

U

is isometric, it follows that

!

Ifk(t)12dl-ldt) =

Vlk E s ; Now let E be a Borel subset of 0, vi > 2. Then for any a E A,

~! laik(t)fd t )12dvj(t)

(1)

,

° such that v2(E) aaUXE

0

= 0. Clearly, l/j(E) =

aa1kXE 0

U : (k-th)

aXE

(2)

---+

0

aajkXE

Further, by (2)

! lYn(t)a12{t)a(t)XE(t)dvl(t) (CUlIXE) ° , ( .

(

aXE

0

°

aXE

°

aXE

aU12XE) o

) = 0,

0

) = (u

,U

Va EA.

Thus an(t)a12(t) = 0, a.e.z-j , tEE. Put

E1

= {t EEl all(t) f O},

E 2 = E\E1 •

Then a12(t) = 0, a.e.z-j , t EEl' By (2),

U(XE z ' 0, ...) = (lYnXE2, 0, ...) =

°

°

aXE 0

)

262

°

since all(t) = on E z. Thus 111(Ez) Again by (2), we have

o= so a12(t)

=

0, and I1z(Ez)

u(0, X E2 , 0, ...) = (a 12 X E2 ,

=

° since J.Lz -<

Ill'

°...),

=

0, a.e.z/j , t E E 2 , and aI2{t) = 0, a.e.z-j , tEE. Now from U(O,XE,O,' ..) = (a12XE,O,"') = 0,J.L2{E) = 0. Therefore, we get

Vz >- J.Lz >- J.L3 >- . . . . By (1) for any Ik E Hi; k

= 1,2,' . "

(3)

we obtain

Jlalklk{t) 12dvdt ) < JI/k{t) 12dJ.L k(t ). Suppose that E is a Borel subset of n such that vz{E) J.LdE) = 0, Vk > 2. From (4), we have

IE lalk(t) 2dv l (t) = 0,

Vk

1

(4)

= 0.

Then by (3),

> 2.

Thus laul 2 • VI -< V2, Vk > 2, By Theorem 5.1.4, there is a non-negative measurable function flk on n such that

Define v

IQIkl 2 • VI = flk . Vz, Vk > 2. : H e HI -+ K e K 1 as follows: v(O,"', IbO"') = (0, (fh + I 2kIZ) l j ZI», a3klk"")

(5)

Q

Vlk E Hk,k > 2. By (5) and (1), v is isometric. Clearly, we have V~HeHl (a) = ipKeK I (a)v,

Va E A,

and 112 >- J.L3 >- "', V3 >- V4 >- .... From the discussion of preceding paragraphs, we obtain V3 >- J.L3 and a relation between V2 and V3 which is similar to (5). Cotinuing this process, we get vi >- J.Lj(VJ > 2) generally. Q.E.D.

Lemma 5.4.9.

Let u be an isometry from H

= LEBL2(fl , J.L k)

to K

=

Ie

U~H(a)

=

ipK(a)u,

Va E A.

If J.LI >- J.L2 >- .. " and Vj >- Vj+l >- . , " where j is an integer with j > 2, then Vk >- J.Lk, Vk > J'. Proof. When J' = 2, it is exactly the Lemma 5.4.8. Now we assume that the Lemma holds for (j - 1), (j > 2).

263

L

$L 2(0, Vk) is invariant for the * representation epK. By Lemma 5.4.6

Ie~i-l

and Theorem 5.4.1, there is a sequence {/!c I k measures on with Ii -1 >- Ii >- ... such that the and {L,II>L} are unitarily equivalent, where

°

= L

K'

$L 2 (0 , Vic),

L

L

=

>

j - 1} of regular Borel * representations {K', II> K' } 2

$L (O' l k)'

k~i-l

k~i-l

From Lemma 5.4.8, Vk >- lie, Vk

> j. Now for Hand

j-2

L

$L 2 (0 , Vk) $ L

Ie=l

we have Ik >- /-Lk(Vk

> } - 1) by induction. Therefore, Vk >- /-Lk, Vk > j. Q.E.D.

Lemma 5.4.10.

Let u be a unitary operator from H =

K

=L

L

$L 2 (0, /-Lk) onto

Ie

$L

2(0,

Vk) ,and

Ie

Proof.

Let

e= u(1, 0,' ..).

! a(t)

Then for any a E A,

dill (t) =

(ep H ( a)( 1, 0, .. '), (1, 0, ... )

(epK(a)E,E) =

E and

where ve is the measure determined by Clearly, (epK(a)171e,1/k)

=/

!

a(t)dve(t),

ep K· Thus /-Ll = ve'

a(t)dvk(t),Va E A, where 17k

= (0, .. ·,1,0''')

(

E K). Put 17 = L(II171e1l2~) -1 17k. Then 17 is maximal in K by Lemma 5.4.4. k

So v'1 >- ve

= /-LI.

Since v,.,

k

= L(II17kI122)-lVk

and VI >- vk(Vk > 2), it follows

A:

that v'1 "" VI' Thus VI >- /-Lt. By Lemma 5.4.9, Vk >- Ilk, Vk > 2. So we get VA: >- /-Lie, Vk > 1. Similarly, Ilk >- Vic (Vk > 1) since u is unitary. Therefore, /-Lk "" Vic, Vk > 1.

Q.E.D.

°

Theorem 5.4.11. Let A be an abelian C"'-algebra with an identity, be its spectral space, and {11", H} be a * representation of A such that 11"(A)' is

264

u-finite. Then there is a sequence {Ilk} of regular Borel measures on 0 with III >-- 112 >-- •.• such that

{1r,H}

'"V

{, L EBL 2 (0 ,llk)} , k

where <J(a)(/l"'" Ik"") = (all" ", alk,"') and (alkHt) = a(t)lk(t), Vk, Va E A and (11,'" ,lb' ..) E H. And the sequence {IlA;} is unique in the sense of equivalence. Moreover, for each k > 1 the measure Ilk is equivalent to • {

mm

T/ is a maximal vector in

Il,.,

(I: EB1f(a)ej)l.

}

j=::l

V6,"', 6:-1 E II such that 1r(A) ei-.L1r(A) ej, Vi "# j. Where " min" is taken according to the absolute continuity of measures. Proof. The result follows immediately from Lemma 5.4.10, 5.4.6., 5.4.9 and Theorem 5.4.1. Q.E.D. Remark. The determination of {J.ld is very similar to the Courant principle. If a is a completely continuous non-negative operator on a Hilbert space H, and {AI > A2 > ...} is the sequence of eigenvalues of a, then for any k, \ . = mIn

I\k

max

€l,"',€k-lEH O:;ir]E[€l,···.€k_d.L

JaT/,T/)

\

(T/, TJ)

.

Proposition 5.4.12.

With the assumptions and notations of Theorem 5.4.11, the * representation {1r, H} is faithful if and only if SUPPlll = O.

Proof. Suppose that SUPPIlI = n. If a E A is such that cI> (a) = 0, then a] = 0, VI E L 2(0, Ild· Picking I = a, we can see that a(t) = 0, a.e.lll' Put U = {t EO I a(t) "# O}. Then U is open and Ill(U) = O. But SUPPlll = 0, so U = 0, i.e., a = o. Thus 1r is faithful. Conversely, if there is a non-empty open subset U such that III (U) = o. Then Ilk{U) = 0, Vk > 1. Pick a E A such that suppct-] C U. Then cI>(a) = o. Thus 1r is not faithful. Q.E.D.

Definition 5.4.13. A function n(-) on 0 is called a multiplicity function, if n(.) is measurable, and n(t) E {1,2,···,oo},Vt E O. For any given regular Borel measure Il and multiplicity function n(·) on 0, define a * representation {Il,n,HIl,n} of A(::: C(O)) as follows: HIl,n =

Ilk

L

= XEk

EBHb

n, = L 2(0, Ilk),

k

E k = {t E 0

I n(t) > k},Vk > 1,

265

and

he,·") = (all"'" aile," '),

~~,n(a)(h,""

V(fb" . , flc' ...) E H and a E A.

Lemma 5.4.14. Let Ji, be a regular Borel measure on 0, a O}. Then v XE . u, "-J

Proof. Let F be a Borel subset of n such that v(F) = O. Since v(F) = p(t)dJi,(t) and p > 0, it follows that p(t) = 0, a.e.p. t E F. Thus there is a Borel subset F I C F such that Ji,(Ft} = 0 and p(t) = 0, Vt E F\FI . Then

k

(XE . Il)(F) = Ji,(E

n F) = Il(E n (F\Ft}).

But p(t) > 0, Vt E E, and p(t) = 0, Vt E F\F1 , so E n (F\F1 ) = 0, and (XE • Il}(F) = Ji,(E n (F\Fd) = 0. Thus XE . Ji, -< u, Conversely, let F be a Borel subset such that (XE • Ji,)(F) = Ji,(E n F) = O. Clearly, p(t) = 0, Vt E F\E. Then from p E LI(O, Ji,),

v(F) =

iFr p(t)dJi,(t)

Hence, v -< XE . Ji,. Therefore v

"-J

=

r p(t)dJi,(t) = O. iFnE

XE . Ji,.

Q.E.D.

Let A be an abelian C*-algebra with an identity, 0 be its spectral space, and {1r, H} be a nondegenerate * representation of A such that 1T'(A)' is a-finite. Then there is unique (in the sense of equivalence) regular Borel measure Ji, on 0, and unique (in the sense of a.e.rz] multiplicity function n(·) on 0 such that

Theorem 5.4.15.

Moreover,

Proof.

1T'

is faithful if and only if sUPPIl = O.

Pick the sequence {Ji,Ie} as in Theorem 5.4.11. By Theorem 5.1.4, there is < Pie E L1(O, Ji,) such that Ji,1e = Pie . u; Vk > 1, where Ji, = Ji,l and PI = 1. Let E lc = {t EO! Ple(t) > O}, Vk.

°

Since Ji,1e >-

Now let

Ji,1e+1,

we may assume that

266

Clearly, n(.) is a multiplicity function 0, and E k = {t E 0 I n(t) > k}, \/k. By Lemma 5.4.14, Ilk XE/c . J.L, \/k. Thus, {1r, 1I} {IL,n, HIL,n}' From the uniqueness of {J.lk}, it is easily verified that Il and n(·) are unique. Finally, by Proposition 5.4.12, 1r is faithful if and only if n = SUPPJ.ll = sUpPJ.l. Q.E.D. r-.i

r-.i

Notes. Using the theory of type (I) VN algebras, we can also obtain the main results in this section. The presentation here follows a treatment due to A.A. Kirillov.

References. [10], [28], [91].

Chapter 6 The Classification of Von Neumann Algebras

6.1. The classification of Von Neumann algebras Definite 6.1.1. Let M be a VN algebra. A projection p of M is said to be finite, if any projection q of M with q < P and q ,..., p implies q = p . p is said to be infinite, if it is not finite, i.e., there exists a projection q of M such that q < p, q ,..., p and q "# p. p is said to be purely infinite, if p contains no non-zero finite projection, i.e., if q is a projection of M with q ~ p and q "# 0 , then q is infinite. Moreover, M is said to be finite, infinite, purely infinite, if its identity is a finite, infinite, purely infinite projection respectively. Proposition 6.1.2. projection Z1'

In a VN algebra M, there is a maximal finite central

Proof. Let ZI = sup{z I z is a finite central projection of M}. It suffices to show that ZI is finite. Suppose that p is a projection of M with p < ZI and p ,..,., ZI' If z is any finite central projection of M, then z = zz! ,..,., zp < z. Thus zp = z, i.e, p > z . Further, p = z! and ZI is finite. Q.E.D.

Proposition 6.1.3. Let p, q be two projections of a VN algebra M, q and p be finite. Then q is also finite.

Proof. Let v be a partial isometry of M such that v~v = q and vv* = ql < q. Define u = v + (p - q). Then u~u = p, uu* = (p - q) + ql < p, Since p is finite, it follows that (p - q) + ql = p, i.e., ql = q . Therefore, q is also finite. Q.E.D.

268

Proposition 6.1.4. In a VN algebra M, there is a maximal purely infinite central projection Z3' Proof. Let Z3 = sup{zlz is a purely infinite central projection of M}. It suffices to show that Z3 is purely infinite. Suppose that p is a finite projection of M with P < Z3' If z is a purely infinite central projection of M, then pz is finite by Proposition 6.1.3. Since pz < z and z is purely infinite, it follows that pz = O. Further, P = pZs = O. Therefore, Z3 is purely infinite. Q.E.D.

Definition 6.1.5. A VN algebra M is said to be semifinite, if Z3 = 0 , where Zs is defined by Proposition 6.1.4, i.e. any central projection of M is not purely infinite. M is said to be properly infinite, if Zl = 0, where Zl is defined by Proposition 6.1.2, i.e, any non-zero central projection of M is infinite. Moreover, a projection P of M is said to be semi-finite, or properly infinite, if the VN algebra M p is semifinite, or properly infinite. Theorem 6.1.6. position:

Let M be a VN algebra. Then there is a unique decom-

M = M 1 EJ3 M 2

EJ3

M3 ,

where M 1 = M Zl is finite, M« = M Zs is purely infinite, M 2 finite and properly infinite, and Zl + Z2 + Z3 = 1.

= M Z2

is semi-

Proof. From Propositions 6.1.2 and 6.1.4, such decomposition exists. Now suppose that M = M PI EJ3 M P2 EJ3 M Ps is another such decomposition. Clearly, PI < Zt, P3 < Z3, and the central projection (Zl - pdpi is finite, i = 2,3. Then we have (Zl - PI)Pi = O(i = 2,3) since MP2 and Mps are properly infinite. So Zl = Pl' Moreover, if the central projection (Z3 - PS)Pi is not zero, then it is purely infinite, i = 1 or 2 . But M PI and M P2 contain no purely infinite central projection, so (zs -PS)Pi must be zero, i = 1,2, and Zs = P3 . Therefore, Zi = Pi, i = 1,2,3. Q.E.D.

Definition 6.1.7. A VN algebra M is said to be discrete, if for any non-zero central projection z , there is a non-zero abelian projection q ( see Definition 5.3.18) such that q < z. M is said to be continuous, if M contains no non-zero abelian projection. Moreover, a discrete VN algebra is also said to be type (I) j a purely infinite VN algebra is also said to be type (III) ; a semi-finite and continuous VN algebra is said to be type (II) j A finite type (II) VN algebra is also said to be type (lId, and a properly infinite type (II) VN algebra is also said to be type (1100 ) • Clearly, each abelian projection is finite. Thus a type (I) VN algebra is semi-finite.

269

Theorem 6.1.8. position:

Let M be a VN algebra. Then there is a unique decom-

M = M 1 EEl M 2 EEl M 3 , where M, = M Zi, i = 1,2,3 are type (I), (II), (III) VN algebras respectively, and Zl + Z2 + Zs = 1.

Proof. By Proposition 6.1.4, there is a maximal purely infinite central projection Zs in M. Then M s = M Zs is type (III). Let Zl = sup{zlz is a central projection of M such that M z is type (I)}. We claim that M Zl is also type (I). In fact , suppose that p is a non-zero central projection of M with p < Zl . Then there is a central projection Z of M such that pz i- a and M Z is type (I) . So pz is a non-zero central projection of type (I) VN algebra M z . By Definition 6.1. 7, there is a non-zero abelian projection q of M Z such that q < pz . Clearly, q is also a non-zero abelian projection of M Zl • Therefore, M Zl is type (I) . Since each abelian projection is finite, it follows that ZlZS = O. Now let Z2 = 1 - Zl - Z3. Clearly, M 2 = MZ 2 is semi-finite. If p is a non-zero abelian projection of M 2 , then c(p) < Z2. We say that M c(p) is type (I) . In fact, suppose that Z is a non-zero central projection of M c(p). By Proposition 1.5.8, zp i- O. Since (pMp)z = zp(Mc(p))zp , z contains a non-zero abelian projection zp . Thus, M c(p) is type (I). By the definition, we have p < Zl . This contradicts the fact that c(p) < Z2 • Therefore, M 2 contains no non-zero abelian projection, i.e., M 2 is type (II). Moreover, since Zl and Zs are maximal, this decomposition is unique.

Q.E.D. Theorem 6.1.9. position:

Let M be a VN algebra. Then there is a unique decom-

M = Mn EB M 12 EB M 21 EB M 22 EB M 3 , where M u is finite type (I) , M 12 is properly infinite type (I) ( and semi-finite also), M 21 is type ( II 1 }, M 22 is type (II co ) ( and semi-finite also) ,Ms is type (III) ( purely infinite). Consequently, there are only five classes of factors.

References.

[21], [28], [82], [111].

6.2. An ergodic type theorem for Von Neumann algebras

270

Let H be a Hilbert space, h'* = h E B(H), and p be a projection on H with hp = ph. Let

Mp(h)

=

sup{(he, e)le E pH, Ilell = I},

mp(h) = inf{(he, e)le E pH, Ilell = I}, wp(h)

=

Mp(h) - mp(h).

Clearly, Mp(h), mp(h) are the maximal, minimal spectral points of (hlpH) respectively. If p = 1 ,we denote MI(h),mdh),wdh) by M(h),m(h),w(h) respectively. If 1 is a family of projections on H with ph = h.p; Vp E 1, then we define

wl(h)

=

sup{wp(h)lp E 1}.

Lemma 6.2.1. Let M be a VN algebra on H, Z = MnM', and h'" = hEM. Then there is a projection z E Z and a self-adjoint unitary operator u E M such that

Proof. Let n(h) = l(M(h) + m(h)), be and let h = J )"de~ be the spectral decomposition of h. Clearly, e = en(h) and I = 1 - e are two projections of M, and Me(h) < n(h) < mf(h). By Theorem 1.5.4, there is a central projection z of M such that ez

-< I z,

f z' -< ez',

where z' = 1 - z. Thus there are partial isometries v, w of M such that

w"'w = [z",

Let u = v

+ v* + w + w* + (1 H=

(ezH

ez EI:3

EI:3(1 z'H

uno" = el

11 - Iz' - ell.

hH) EEl

< ez',

EI:3

elH)

Since

(Iz - h)H EI:3

(ez' - edH,

it follows that u is a self-adjoint unitary element of M. Now we prove that the above u and z satisfy the condition. Since

hz > m(h)ez + n(h)lz

= m(h)ez + n(h)h + n(h)(lz - fd and

(uhu-l)z > m(h)iI + n(h)ez + n(h)(fz - fI),

271

it follows that ~(h + uhu-1)z

> ~(m(h) + n(h))(h + ez) + n(h)(fz - ft}

>

~(m(h)

+ n(h))z.

Noticing that

we have

M(h) - ~w(h) = ~M(h)

+ ~m(h)

= ~(m(h)

+n{h)),

M(h)z > !(h + uhu-l)z

> (M(h) -

~w(h))z,

i.e.,

Similarly, from

hz' < n(h)ez' + M(h)fz' = n(h)el

and

+ M(h)fzl + n(h)(ez' -

{UhU-1)ZI < n{h)fzl + M(h)el

et)

+ n(h)(ez' - ed,

we have

+ uhu-1)z' < ~(n(h) + M(h))(fzl + el) + n(h)(ez' - el)

m(h)z' < !(h

< !(n(h) + M(h))z' = (m(h) +

~w(h))z'.

Thus

Q.E.D. Lemma 6.2.2. Let M be a VN algebra, Z = M n M', h· = hEM, and 1 be a finite orthogonal family of projections of Z with z = 1 . Then

L

zEl

there is a finite orthogonal family l' of projections of Z with a self-adjoint unitary element u of M such that

L z'El'

z' = 1 and

272

Proof. Let 7 = {Zit' •• ,zn} . For each i E {I, ... ,n}, by Lemma 6.2.1 there is a central projection Cil of M, = M z; and a self-adjoint unitary element Ui of M, such that

n

where hi

=

h.z,

C,2

= z,

~

Cil'

LUi'

=

Define u

Then

U

is a self-adjoint

i=1

unitary element of M and

wcij(l(h

+ uhu- I )) <

~wzi(h)

< ~w7(h), VI

< i < n,j = 1,2. Now let 'I' = {ciil1 < i < n,j = 1, 2}. Then

W71(~(h + uhu- I ))

<

~W7{h). Q.E.D.

Definition 6.2.3. Let M be a VN algebra, and G = U(M) be the set of all unitary elements of M. Denote by Q the set of all functions I on G satisfying: 1 > 0, #{u E GI/{u) i= O} < 00 and I(u) = 1.

L For 1 E Q and a E M, let I· a = L I(u)uauuEG

I

•

uEG

For

t.s

E Q, define (/*g)(·)

(I * g) . a = I . (g . a), Va E M.

=

L

I(u)g(u- I . ) . Clearly, I *g E Q, and

uEG

Lemma 6.2.4. Let h* = hEM and e > 0 . Then there is some some Z E Z = M n M' such that

III. h -

I E

Q and

a] < c.

Proof. By Lemma 6.2.1, there is a central projection p of M and such that 3

II

E Q

W71 (11 . h) < -w(h), 4

where 7t = {p,1 - p}. Now we assume that for some positive integer j there is a finite orthogonal family 7; of projections of Z with p = 1 and Ii E Q

L

pE7j

such that

273

Using Lemma 6.2.2 for /; . h and ~ , then there is a finite orthogonal family Ji+l of projections of Z with P = 1 and g E Q such that

L

pE7;+1

W 7i+ 1 (g

. (ff . h)) <

i

w 7; (Ii

. h)

< (i)i+IW(h). Therefore, for any positive integer k there is a finite orthogonal family fie of projections of Z with P = 1 and fie E Q such that

L

pE7..

W 7.. (

3 k fie . h) < (4) w(h).

Pick k such that (~)kw(h) < E. For any c E fk' let Ac = lI(fle . h)lcHII(H is the action space of M). Then II (fie . h)c - Accll < wc(fle . h). Now let f = fie and Z = Acc. Then we have

L

cE7J,

Ilf . h - zil = max{II(1 . h)c - Acclll cE

Ji:}

< wh(lk' h) < (~)kw(h) < c.

Q.E.D. Lemma 6.2.5. Let {al"", an} C M and e > O. Then there is alE Q and {Zh ..• ,zn} C Z = M n M' such that II I . ale - Zle II < e, 1 < k < n. Proof. We may assume that a: = ale, Vk. When n = 1 , this is just Lemma 6.2.4. Now we assume that the conclusion holds for n. For at, ... ,an+l E M and c > 0 , first pick Zll ... ,Zn E Z and I E Q such that III· ale - zlell < e , 1
Again by Lemma 6.2.4, there is g E Q and Zn+l E Z such that zn+lll < c. Since Zle E Z, it follows that

Ilg . (I . ak)

- zkll =

<

Ilg . (I· an+d -

Ilg . (I· ak - zle)11 II f . ak - Zk 11 < e,

VI < k < n. Therefore,

Q.E.D.

274

Lemma 6.2.6. Let {ak} {In} C Q such that

C

M. Then there is {Zk}

C

Z = M n M' and

Proof. By Lemma 6.2.5, for al we can pick gl E Q and Zll E Z such that

1

lliI· al - znll < 2". For gl . al and gl . a2, there is g2 E Q and Z12, Z22 E Z such that

.. '. Generally, we have gl, ... ,gn E Q and Zln, ... ,Znn E Z such that

II (gn * ... * gI) Now let In

= gn * ... * gi'

. ak -

zknll <

1 2n '

1 < k < n.

Then for 1 < k < n

Illn+l . ak -

Zkn!l = \Ign+I . (fn . ak - Zkn) II

< Ilfn·ak-zkn!l < 2~' Thus Il/n+l' ak - In' akll < 2,Ll' 1 < k < n, and {In' ak}n is a cauchy sequence for each fixed k, Further, {Zkn}n is also a cauchy sequence for each k, Suppose that Zkn

(n)

--'--'-+

Zk(E Z), then II/n . ak - Zkl!

Theorem 6.2.7.

~ 0, Vk.

Let M be a VN algebra, Z

K(a)

= {f . a I f

= M n M',

E Q}

Q.E.D. and a EM. Let

n Z,

where the closure is taken with respect to uniform topology. Then K( a) Proof. By Lemma 6.2.6, there is Z E Z and {In} C Q such that zil ~ 0. Therefore, Z E K(a} and K(a) f- 0.

Proposition 6.2.8.

f:. 0.

Il/n . a Q.E.D.

With the notations of Theorem 6.2.7, we have

1) K(al + a2) C K(ar) + K(a2), Val, a2 EM, 2) K(za) c zK(a), Vz E Z, a E M.

°

1) Let Z E K (al + a2)' Then for any E: > there is I E Q such that III' (al + a2) - zll < c. By Lemma 6.2.6, there is g E Q and al E K(/· ail C K(aj) such that Proof.

275

Since IIg·(f·(al+a2))-zl[ < lIf·(al+a2)-zll < e, it follows that Ilz-(ZI+Z2)1I < 3c. Therefore, K(al + a2) C K(al) + K(a2)' 2) Let C E K(za). Then for any E: > 0 there is f E Q such that

Ilf' (za) -

c] < c.

By Theorem 6.2.7, there is g E Q and Ilg· (I' a) - clll < c. Then

Cl

E K(I . a) C K(a) such that

Ilzcl - clI < Ilz((g * f) . a) - cll + lI z ((g * I) . a) - ZCII! < Ilg . (I . za - c) II + I zll . IIg . (f . a) - cllI < c(1 + Ilzll). Therefore, K(za) C zK(a).

Q.E.D.

Notes. Theorem 6.2.7 is due to J. Dixmier. Through this approach he proved the existence of the central valued trace (see Section 6.3).

References.

[18], [28].

6.3. Finite Von Neumann algebras Proposition 6.3.1. Let M be a VN algebra. 1) M is finite if and only if v E M and volov = 1 imply vvolo = 1. 2) Suppose that M is finite, p and pi are projections of M and M' respectively. Then M p and M p ' are finite. ffiM,. Then M is finite if and only if M, is finite 3) Suppose that M =

L:

for each 1.

I

Proof. 1) It is obvious since vvolo is also a projection. 2) By Proposition 6.1.3, p is a finte projection. Thus M p is finite. Now let C(pl) be the central cover of p' in M'. Then M p ' and M C(p/) are * isomorphic. Clearly, M C (p') is finite, so is M p " 3) The necessity is clear by 2). Conversely, let Mi be finite, and M , = M Zh Vl. If P is a projection of M with p "" 1, then PZI Vl. Since Zl is finite, it follows that PZI = zi, VI. Thus p = 1 and M is finite. Q.E.D. I"V

Proposition 6.3.2.

z"

Let M be a finite VN algebra, and {Pl' P2, qt, q2} be projections of M satisfying

276

Proof.

By Theorem 1.5.4, there is a central projection z of M such that

(P2 -

pd z -< (q2 -

(q2 - ql)(1 - z)

qI)z,

-< (P2 - Pl)(l - z).

If (P2 - pdz "'-' q < (q2 - qdz and q f:. (q2 - ql)Z, then P2 Z = (PI Z + (P2 - pl)Z) "'-' (qlZ

+ q) <

q2 z, and (ql Z + q)

f:.

q2 Z

But P2Z "'-' q2Z, so q2Z "'-' (Q1Z+Q) < Q2Z and (qlZ+Q) t= q2Z. This is impossible since Q2Z is finite. Thus (P2 - Pl)Z "'-' (Q2 - Qdz. Similarly, (P2 - Pl)(l - z) "'-' (q2 - ql)(1 - z). Therefore, (P2 - pd "'-' (q2 - ql)' Q.E.D.

Definition 6.3.3. called a trace , if

A positvie linear functional

Va EM.

If

=
=
Therefore,
Lemma 6.3.4. Let ip be a positive linear functional on M, and K be a positive constant such that
for any projections P, q of M with P rv q. Then
Proof.

Let a E M and

Va E M.

lIall < 1, and

be the spectral decomposition of a*a , where p!n) = es. n

fi-l, n

1 < i < n. If

a = uh is the polar decomposition of a, then p~n) < u*u, Vi, n. Since n

aa *

= ua * au * = li1mL: n

;=1

. Z (n) u, * -up-

n'

277

=

n

.

i=l

nn

.

i=l

n

liR! L ~cp(p~n))

< K lim L ':'cp(up~n)u*) = Kcp(aa*). n

Q.E.D.

Corollary 6.3.5. Let cp be a positive linear functional on M. Then ip is a trace if and only if cp(p) = cp(q) for any projections p, q of M with P q. rv

Lemma 6.3.6. Let M be a finite VN algebra, p be a non-zero projection of M, and n be a positive integer. Then there exists a non-zero projection Po of M and a faithful normal state CPo on M o = M po such that Po < p,

cpo(a*a) < (1 +

.!.n )cpo(aa*), Va E Mo.

Proof. Pick a normal state 'if; on M p , and let cp(x) = 'if; (pxp) , Vx E M. Then cp is a normal state on M and s(cp) < p. Replacing p by s(cp), we may assume that s(cp) = p ; i.e. there is a faithful normal state ip on M p • If for any projections ql, q2 of M p with ql rv q2 we have cp(qd = cp(q2), then by Corollary 6.3.5, CPo = cp and Po = P satisfy the condition. Otherwise, by the Zorn lemma there are two maximal orthogonal families {ell, {f,} of projections of M p such that el Put el

II <

=

L e" I, I

p and

h f-

=

L

f'J

I"

cp(el) > cp(fd,Vl.

fl. Then el

f'J

11 and cp( ed > cp(fd. Consequently,

I

It and M is finite, it follows by Proposition (p- II). Thus, we get eI 0, and cp (e) Ji e and f with e

f'J

I

with e

f'J

f,

< Jicp(f) for any projections } f, and e Jio, then there is Ji E [Jio, cp(p - el)) such that cp( e) < Jicp(f) for any projections e and I with e f, and e 1. But cp(p - It) < 1, a contradiction. Thus, 0 < cp(p - ed < Jio· Now pick e > 0 such that 0 < (JiO - c)-IJio < 1 + ~. By the definition of J.Lo, there are projections e2 and [z with e2 [z, and e2 (tLo -c)cp(/2). Clearly e2 and f2 are not zero. We claim that there are non-zero projections e3 and 13 with e3 ,..., 13 , and e3 < e2, 13 < 12, such that cp(e) > (tLo - c)ep(/) for any projections e and I with e ,..., I, and e < e3, I < 13. In fact, if such e3 and 13 don't exist, then e2 and f2 are not such e3 and 13. Thus there are projections e and I with e ,..., I, and e < e2, I < 12, such that cp(e) < (tLo - c)cp(/). Further, (e2 - e) and (/2 - I) are not such e3 and 13 , we have also· By the Zorn lemma, we can write e2 = E!1e, and f2 = L>~B/, such that

L

I

I

e, ,...,

Ih

and

cp(e,) < (tLo - c)cp(/l)'

Vl.

Since cp is normal, it follows that cp(e2) < (tLo - c) cp(/2)' This contradicts that cp(e2) > (tLo - c)cp(/2)' Thus e3 and 13 exist. Let v E M p be such that v*v = e, vv* = 13 , and define

1/J(x) = ep(v*xv), Vx E 13M 13. Clearly, 1/J(/3) = cp(e3), and cp(e3) > 0 since cp is faithful on M p • If r and q are projections of 13M/3 with r ,..., q, by (v*q)*(v*q) = q we have r ,..., q ,..., v*qv in M p and v*qv < e3. By the property of (e3,/3) and the definition of tLo, we get (tLo - c)cp(r) < cp(v*qv) < tLocp(r). In particular, (tLo - c)cp(r) < cp(v*rv) < tLoep(r). Then

1/J(q)

=

ep(v*qv) < tLoep(r)

<

Il~~ e cp (V * rv)

< (1 + ~) 1/J (r).

Finally, let Po = 13( < p) and

CPo(x)

= 1/J(/3)-11/J(x),

Vx E M o = M p o '

Clearly, CPo is a normal state on Mo. If x E M« is such that CPo(x*x) = 0, then vx = 0 since cp is faithful on M p • Further, x = xl3 = xvv* = 0, i.e., CPo is faithful on Mo. From the preceding paragraph, we have

CPo (q) < (1

1

+ -) CPo (r ) n

for any projections rand q of Mi, with r ,..., q. By Lemma 6.3.4., we obtain

cpo(a*a) < (1 + ~)cpo(aa*), Va E Mo.

Q.E.D.

Lemma 6.3.7. Let M be a finite VN algebra. Then for any positive integer n there is a normal state 1/Jn on M such that

279

Proof. By Lemma 6.3.7, for fixed n there is a non-zero projection Po of M and a faithful nonmal state CPo on M p o such that

CPo ( a· a) < (1 + .!:. )CPo( aa·), Va E M p o ' n

Let {PI, ... ,Pm} be a maximal orthogonal family of projections of M such that Pi Po,1 < i < m ( notice that m is finite since M is finite). By Theorem 1.5.4, there is a central projection Z of M such that ,-.,J

(1 -

L pd z -< POZ,

Po(1 - z)

-<

(1 -

L

Pi)(1 - z).

i

i

Since {Pi} is maximal , it follows that PaZ i- O. Let V;Vi = POZ, ViV: = PiZ, 1 < i < m, and =

Vm+lV:n+1

(I - LPi)Z,

v:n+lvm+1 < POZ,

i

and define

m+l /Pn(x) = L /Pa(V;XVi), i=l

Vx E M.

Then for any x E M, m+l

CPn{x·x)

=L

m+l

CPo(v;x*xvd

i=1

=L

CPa(v;x·V;vixvd

i,;=1

< (1 + ~) L CPO(V3~XViVi·X·Vj) i,i

= (1

+ ~) ~ cpo(vixx·v;)

= (1

+ ~)CPn(XX·).

j

Moreover, CPn(l) > mcpo(poz), and cpo(Paz) > 0 since PaZ i- 0 and CPo is faithful on M p o ' Therefore, tPn(') = CPn(1)-lcpn(') is what we want to find. Q.E.D.

Theorem 6.3.8. Let M be a finite VN algebra. Then for any a E M, # K(a) = 1, where K(a) is defined as in Theorem 6.2.7. Proof. By Proposition 6.2.8 and Theorem 6.2.7, we may assume that a > 0 and Ball < 1/2. Suppose that there are CI,C2 E K(a) and CI i- C2. Clearly, Cl,C2 > 0 and

Ilel - c211 < 1. Let Cl -

C2

=

ill J1dzp. be the spectral decomposition of (CI -

C2)

, where zp is a central projection of M, \:j /1. Since CI i- C2, there is oX > 0 such that either Z_>. i- 0 or (1 - z>.) i- O. Let Z = Z_>. for the case of Z_>. i- 0, or z = 1 - z>. otherwise. Then we have

280

By the symmetry we may assume that CIZ > C2Z + Az. Since CtZ t= C2Z and CIZ, C2Z E K(az), and replacing M by Mz , we may also assume that z = 1. Pick {tPn} as in Theorem 6.3.7. Then for any unitary element u of M,

tPn(u*au) = tP((aiur(aiu)) < (1 + .!. )tPn(a), n

tPn{a) = tPn{{a!u)(aiur) < (1 + !.)tPn{u*au). n

Thus for any

I, g E

Q ( see Definition 6.2.3),

tPn(/' a) < (1 + !.)tPn(a) < (1 + !.)2 tPn(g · a). n

Let {fA:}, {gA:}

Further, from

c

Cl

Q be such that

>

C2

n

IA: . a

---+ Cb

gA:' a ---+

C2'

Then

+ A it follows that

When n is sufficiently large, we get a contradiction since A > O. Therefore, K(a) contains only one element, \;fa E M.

Q.E.D.

Remark. In the end of section 6.4, we shall prove that: if # K(a) = 1, \;fa

E

M, then M is finite. Now we start to characterize finite VN algebras by normal tradal states.

Lemma 6.3.9. Let M be a finite VN algebra. Then there is a normal tracial state on M at least.

Proof. By Theorem 6.3.8, we can define a map T(.) from M to Z = MnM' such that

K(a) = {T(a)},

\;fa

E

M.

From Proposition 6.2.8 and the definition of K(.), T is linear, and

T(z) = z,

\;fz

E

Z,

T(M+)

C

Z+.

Since K(u*xu) = K(x) for any x E M and any unitary element u of M , it follows that T(u·xu) = T(x). Further, T{xy) = T(yx), \;fx, y E M. Let TjJ be the same as the tPl in Lemma 6.3.7, and define

cp(a)

= tP(T(a)),

\;fa E M.

281

From the preceding paragraph, ip is a tradal state on M. Now it suffices to show that ip is normal. Let {b,} be a bounded increasing net of M+, and b = suPlb,. Put al = b - b" VL. Then al ---+ O(u(M, M.)). We need to prove that cP (ad ---+ O. For any e > 0 , there is Lo such that

o < t/;(al) < e,

vi > La

since t/; is normal. Pick j, E Q such that 6.3.7, for I > 10 we have

1111' a, -

T(al)11 < e, VI. By Lemma

o < cP (a,) = t/; (T (at)) < t/; (I, . a,) + e

L: I, (u)t/;(u· at u) + e < 2 L: ft(u)t/;(al) + e < 3c.

=

u

u

Q.E.D.

Therefore, cp(a,) ---+ O.

Theorem 6.3.10 A VN algebra M is finite if and only if there is a faithful family of normal tradal states on M , i.e., for any non-zero a E M+ there is a normal tradal state cp on M such that r.p(a) > O. Proof. Suppose that M is finite. Then there is a normal tradal state ip on M by Lemma 6.3.9. Let z = s(r.p). Then z is a non-zero central projection of M, and cp is faithful on M z . Again we continue this process for finite VN algebra M(l - z),·· " and so on . By the Zorn lemma, there is a family {CP/} of normal tradal states on M such that

S(CPI)' s(r.pl') = O,Vl i-/',

and

L:S(CP1) = 1. I

It is easily verified that the family {r.p/} is faithful. Conversely, let '7 be a faithful family of normal tracial states on M. If w E M is such that w"w = 1, then r.p(I- p) = cp(w"w) - cp(ww"') = 0, Vcp E 1, where p = ww·. Since '7 is faithful, it follows that p = 1. Therefore, M is finite.

Q.E.D. Another characterization of finite VN algebras is as follows. Lemma 6.3.11. be such that

Let p be a projection of a VN algebra M, and let v E M v*v = p,

vv*

<

p and vv*

i-

p,

Put en

=

qn - qn+b

n

= 0,1,2"

...

282

Then {en} is an orthogonal sequence of non-zero projections of M with en em, Vn, m, and en ----+ (strongly).

°

~

Since ql < p, it follows that pv = v and v*Rvn = p, Vn > 1. Thus qn is a projection for each n. From qnqn+l = qn+l we have Proof.

p

=

qo

> ql > q2 > ....

Further, enem = 0, Vn "I m, and en ----+ Let Un = vq«, n = 0,1,2,' ... Then

i.e., en ""-' en+ll Vn > 0. Moreover, by eo

° (strongly). =

p - vv* "lOwe get en

"I 0, Vn > 0. Q.E.D.

Theorem 6.3.12. A VN algebra M is finite if and only if the is strongly continuous in any bounded ball of M.

* operation

Let M be finite. By Theorem 6.3.10, there is a faithful family 1 of normal tradal states on M. Suppose that {Xl} is a net of M with Ilxdl < 1, VI, and X, ----+ (strongly). Then for any a E M + and ip E 7, Proof.

°

ILacp(Xjx;) I = Icp(x;axj) I < Ilallcp(x;Xj)

----+

0.

°

If [Lacpla E M, cp E 7] is dense in M* , then XIX; ----+ (weakly) . Further, xi ----+ (strongly) . So the * operation is strongly continuous in any bounded ball of M. Now we need to prove that [Lacpla E M, cp E 7] is dense in M* . Let b E M be such that Lacp(b) = 0, Va E M, cp E 7. Then cp(bb*) = 0, Vcp E 7. Since 7 is faithful, it follows that b = 0. So the above assertion holds. Conversely, suppose that the * operation is strongly continuous in any bounded ball of M. If M is not finite, then there is v E M such taht v*v = l,vv* "11. By Lemma 6.3.11, there is a sequence {en} of non-zero projections of M with en' em = 0, en ,. . ., em, Vn "I m, and en ----+ (strongly). Let Wn E M be such that w~wn = en, wnw~ = el, Vn. Clearly, W n ----+ (strongly), IJwn11 < 1,Vn. By the assumption, w~ ----+ (strongly). Thus, el = wnw~ ----+ (weakly) , a contradiction. Therefore, M is finite. Q.E.D.

°

°

°

°

°

In the proof of Lemma 6.3.9, we introduce a map T(·) from a finite VN algebra to its center. Now we discuss the properties of that map in detail.

Definition 6.3.13. Let M be a finite VN algebra. The map T from M to Z = M n M' is defined by {T(a)} = K(a)(Va E M) and is called the central valued trace on M, where K(·) is defined as in Theorem 6.2.7.

283

Proposition 6.3.14. Let M be a finite VN algebra, and T : M ---+ Z = M n M' be the central valued trace. Then: 1) T is a projection of norm one from M onto Z, and is a-a continuous. Consequently, T(a} > O,Va E M+;T(za) = zT(a),Va E M,z E Z; T(a)*T(a) < T(a*a)' Va E M;

2) T(ab) = T(ba), Va, b E M; 3) T(a*a) = 0 if and only if a = 0; 4) {~(T('))I~ is a normal state on M} is a faithful family of normal tracial states on M; 5) p -< q if and only if T(p) < T(q}, where p and q are two projections of M. Proof. 1) From the proof of Lemma 6.3.9, it suffices to show that T is a -a continuous. Since T is positive, this is equivalent to prove that ~(T(.)) is normal for any normal state ~ on M, i.e., to prove

T(a)

= suPIT(al).

Clearly, T( a) > supzT(al)' If they are not equal, then there is a non-zero central projection z and a positive number ..\ such that

zT(a) > zsuPIT(az) + AZ. Let 7 be a faithful family of normal tradal states on M. For any e > 0 and I, pick I, E ~ such that lllz' (a - al)z - T((a - at}z) II < c. Then for any 7J; E 1,

I7J; (T ((a -

al) z}) I

Since 7J; is normal, it follows that

7J;(T( a - ad z) i.e., 7J;(T(a)z) = lim 7J;(T(at}z) l

-+

0,

V7J; E 7,

= t/J(zsuPZT(al)), Vt/J E 7. However, 7J;(T(a)z) >

,p(zsup,T(az)) + A7J;(Z) , V7J; E 1. Thus, t/J(z) = 0, V7J; contradiction. Therefore, T(a} = sup,T(al). 2) It is contained in the proof of Lemma 6.3.9.

E 7, and

z

= O.

This is a

284

3) Let I = {a E MIT(a*a) = a}. Clearly, I is a s(M,M*)-closed two-sided * ideal of M. By Proposition 1.7.1, there is a central projection z of M such that I = Mz. In particular, z E I, i.e.,

z = T(z) = T(z*z) = O. Therefore, I = 0, This is just our conclusion. 4) If a E M+ is such that
pz -< qz, q(1 - z) -< p(1 - z). Thus, T(q)(l-z) < T(p)(l-z) < T(q)(l-z), i.e., T(q)(l-z) = T(p)(l-z). Let q(1 - z) PI < p(1 - z). Then T(pd = T(q)(1 - z) = T(p)(1 - z), and T(p(1 - z) - pd = o. By 3), Pl = (1 - z)p, i.e., q(1 - z) ,. . ., p(1 - z). Therefore, ro.J

p

-<

Q.E.D.

q.

Now consider some properties of a- finite and finite VN algebras.

Proposition 6.3.15. Let M be a VN algebra. Then the following statements are equivalent: 1) M is a-finite and finite; 2) M is finite, and Z = M n M' is a-finite; 3) There is a faithful normal tradal state on M. Proof.

3) => 1). It is immediate from Theorem 6.3.10 and Proposition

1.14.2. 1) =} 2). It is obvious. 2) =} 3) . By Proposition 1.14.2, there is a faithful normal state t/J on Z. Let
Proposition 6.3.16. Let M be a finite VN algebra. Then M is a direct sum of {Mz } , where M, is a-finite and finite, Vl. Consequently, each finite factor is a-finite. From the proof of Theorem 6.3.10, there a family {
L

t=

z

M =

L I

fJjM"

U = M S(
Vi.

285

Since CPl is a faithful normal tracial state on M" it follows from Proposition 6.3.15 that M, is a-finite and finite, Vi. Q.E.D.

Notes. Using a fixed point theorem, F. J. Yeadon gave another proof of the existence of a trace. Theorem 6.3.12 is due to S.Sakai. References.

[18], [78], [144], [ISO}, [1991.

6.4. Properly infinite Von Neumann algebras Proposition 6.4.1. only if

M"

Let M =

L EflMI . Then M

is properly infinite if and

I

is properly infinite, VI.

The necessity is obvious. Now let M 1 = M z, be properly infinite, Z be some finite central projection of M. Then z z, is a finite central projection of M Vl. Thus Zz, = 0 since M 1 is properly infinite, Vl, and Z = 0 " infinite. . So M is properly Q.E.D.

Proof. Vl, and

Proposition 6.4.2. A VN algebra M is properly infinite if and only if there is no normal tradal state on M.

Proof.

Let cP be a normal tradal state on M. Then s( cp) is a non-zero central projection, and Ms(cp) is finite by Proposition 6.3.15. Thus M is not properly infinite. Conversely, if M is not properly infinite, then there is a non-zero central projection Z of M such that M Z is finite. There is a normal tradal state 1/J on Mz at least. Let cp(.) = tjJ(.z). Then cp is a normal tradal state on M. Q.E.D. Proposition 6.4.3. If M is a properly infinite VN algebra, then the operation is not strongly continuous in unit ball of M.

Proof.

It is immediate from Theorem 6.3.12.

*

Q.E.D.

Theorem 6.4.4. Let M be a VN algebra. Then the following statements are equivalent: 1) M is properly infinite; 2) There is an orthogonal infinite sequence {Pn} of projections of M such that LPn = 1, Pn '" 1, Vn; n

286

3) There is a projection p of M such that

(1- p) "'" 1.

P"'"

Proof. 2)

00

*

3) , Pick P = L P2n+l. Then we have n=O

p "'" (1 - p) "'" 1.

3)

*

1) . Let z be any non-zero central projection of M. Then pz "'" (1 - p)z ~ z, Thus z is not finite, and M is properly infinite. 1) * 2). Since the identity 1 is infinite, it follows that there is v E M such that v·v = 1 and vv* < 1, and vv· =1= 1. Let

By Lemma 6.3.11, en =f. 0, enem = 0, en ~ em, Vn =1= m. By the Zorn lemma, there is a maximal orthogonal family {ezle E /\} of non-zero projections of M such that {ezle E A} => {en}, and e, '" e",VI,I' E /\. Define p = 1- Lel' By eEA

Theorem 1.5.4, there is a central projection z of M such that

pz -< eoz,

°

eo(1 - z) -< p(1 - a}.

Clearly, z =1= by the maximum of the family {ezle E /\}. Since # A is infinite I then we can write 00

A

U Ai,

=

i=1

where /\i n Aj = 0, Vi =f. i, and # /\i rl = pz

+

L

=

# A,

eiz ;

Tj

lEAl

Vi. Let

=

L

elz, Vj > 2.

IEl\i 00

Clearly,

TiTj

= 0, Vi =1= i, Tl

~

T2 "'" •.• "'"

z, and z

=L

Tj.

Since M(1 - z) is

;=1

still properly infinite, we can make the same process, .... Then by the Zorn lemma, there is an orhtogonal family {z,} of non-zero central projections of M with Zl = 1 such that for any 1 there exists an orthogonal infinite sequence

L I

{Tlnln = 1,2, , ..} of projections of M satisfying rll "'" rl2 ", ... "'" zi,

and

L r ln

=

Zl.

n

Now let Pn

= L TIn, n =

1,2", '. Then

I

PnPm = 0,

Vn

f:. m, LPn = 1,

and

Pn "'" 1, Vn.

n

Q.E.D.

287

Using Theorem 6.4.4, we can get a property of finite projections. Proposition 6.4.5. Let p, q be two finite projections of a VN algebra M. Then sup{p, q} is also finite. Proof.

We may assume that sup [p, q} = 1. By Proposition 1.5.2,

(1- p)

("oJ

(q - inf{p, q}) < q.

Thus (1 - p) is also finite. Suppose that z is a non-zero central projection of M such that M z is properly infinite. Noticing that pz and qz are finite and sup{pz, qz} ::::: z, we may also assume that M is properly infinite (1 - r) ,...- 1. By Theorem 6.4.4, we can write 1 = r + (1 - r), where r From Proposition 1.5.5, there is a central projection z such that ("oJ

rz -< pz,

(1 - r)(1 - z) -< (1 - p)(1 - z).

Since pz and (1 - p)(1 - z) are finite, it follows that z("'" rz) and (1 - z)(""(1 - r)(1 - z)) are finite. This contradicts that M is properly infinite. Therefore, sup{p, q} is finite. Q.E.D. In the end of this section, we prove the conclusion of the Remark after Theorem 6.3.8. Proposition 6.4.6. M is finite.

Let M be a VN algebra. If

#

K(a)

= 1, Va E M,

then

Proof. Suppose that z is a non-zero central projection of M such that M z is properly infinite. By Theorem 6.4.4, there is a projection p of M such that p < z and p ,...- (z - p) ,...- z. Let u, v E M z be such that

u * u = v *v = z,

uu * = p,

vv * = z - p.

Define t1>: M -----+ Z = MnM' such that K(a) = {t1>(a)},Va E M. By the definition of K(·) , it is clear that t1>(ab) = t1>(ba), Va, bE M. Then

z = t1>(z) = (z) =
[18], [82}.

+
This contradicts that z =I- O. Q.E.D.

288

6.5. Semi-finite Von Neumann algebras Definition 6.5.1. Let M be a VN algebra. A trace on M+ is a function cp on M+, taking non-negative, possibly infinite, real values, possessing the following properties: 1) cp(a + b) = cp(a) + 0 ( with the convention that O· +00 = O)j 3)
Proposition 6.5.2.

Let

JI = {x E MI
= )/2 = [xylx,y

E

)/J

( Later, M is called the definition ideal of

* two-sided

= {a E M+I
where M+ = M n M+; and

a E M,b EM,

or

a,b E)/.

Moreover, the tracial condition
Va E M+ and unitary element u E M. Proof. Clearly,)/ is a * two-sided ideal of M, and so is M. If a E M+ with
289

where Xj, Yj EN, vi- By the polarization, 4xjYj =

(Xj

+ Yjt(x; + Y;)

- (Xj - Yj)*(Xj - Yi)

-i(xj + iYj)'*(xj + iYj) + i(xj - iYi)*(Xj - iYj).

So we have that

a=

~(a + a*)

<

~ ~(Xj + Yj)*(Xj + Yj), )

and tp(a) < 00. Thus M+ = {a E M+ltp(a) < co}. Further, from the polarization we can see that M = [M+ I. Now let x EM, and x = uh be the polar decomposition of x. Clearly, h = u'*x E M+,h1 E N, and x = uh4 ·hL Hence , M = {xylx, YEN}. From M = [M+ J, it is easily verified that cp can be uniquely extended to a linear functional on M, still denoted by cp . If a E M+ and u(E M) is unitary, then

cp(uau'*) = tp((ua 1/ 2)* . (ua~)) = cp(a). In particular, cp(a) = cp(uau*) , Va E M+ and unitary element u E M. Further, tp{a) = cp(uau*) ,\:Ia E M and unitary element u E M. Since M is a two-sided ideal, it follows that cp(ua) = cp(au), Va E M and unitary element u E M. Hence, we have cp(ab) = cp(ba),Va E M,b E M. Moreover, ifa,b E N, by the polarization of ab and tp(xx'*) = cp(x*x)(Vx EM), we can obtain that tp(ab) = cp{ba). Finally, let cp : M+ -----+ [0, +ooJ satisfy the conditions 1) , 2) of Definition 6.5.1 and

tp(a) = cp(u*au), \:Ia E M+ and u E M is unitary. Similarly, we can also define Nand M, which possess the above properties. If x E M is such that x'*x E M+, then zz" = w(x· x)w· E M+ also, where x = wh is the polar decomposition of x, Thus, cp(x·x) < 00 if and only if cp(xx*) < 00. If cp(x*x) and cp(xx·) are finite, then we have

cp(xx*) = cp(w(x·x)w*) = cp(w·w(x*x)) = cp(x·x). Therefore, tp is a trace on M+.

Q.E.D.

Proposition 6.5.3. Let cp be a normal trace on M+, and M be the definition ideal of ip: Then cp(a.) EM. for any a E M. Proof. We may assume that a E M+. By Proposition 6.5.2, cp(a.) = cp(a~ . a4). Since cp is normal, it follows that tp(a~ . a~) is normal. Therefore,

tp(a.) EM•.

Q.E.D.

290

Propostiion 6.5.4. Let tp be a trace on M+, and M be the definition ideal of ip, Then tp is semifinite if and only if M is a(M, M.} -dense in M. Moreover, if tp is a semi-finite trace on M+, and p is a projection of M, then p = sup{qlq is a projection of M,q > p,and tp(q) < oo}. Let tp be semi-finite, and M be the a(M, M.)-closure of M. Since M is a a -closed two-sided ideal of M, there is a central projection z of M such that M = Mz. If 1 - z"# 0, then there is 0"# a E M+ such that a < (1 - z), and tp(a} < 00. So a E M+ c M z, a contradiction. Therefore, z = 1, i.e., M is a(M, M . . )-dense in M. Conversely, suppose that M is a(M, M . . )-dense in M. By Proposition 1.7.2, for any 0 "# a E M+ there is an increasing net {at} C M+ such that a = sup,al. If I is sufficiently late, then 0 "# al < a and tp(a,) < 00. Therefore, tp is semifinite. Now let tp be semi-finite, and p be a projection of M. Put Proof.

e = sup{qlq is a projection of M,q

< p,tp(q) < oo}.

If e < p and e "# p, then there is a non-zero projection ql with ql < p - e such that tp(qd < 00. This contradicts the definition of e. Therefore, p = e.

Q.E.D. Proposition 6.5.5. Let tp be a normal trace on M+. Then z = sup{plp is a projection of M, and tp(p) = O}

is a central projection of M, and

Mz and

ip

Proof.

= {x E Mltp(x"'x} = O},

is faithful on M+ (1 - z).

It is clear that u'" zu = z for any unitary element u of M. Thus, z

is a central projection of M. By Proposition 1.5.2, (sup{p,q} -p} '" (q-inf{p,q}). Thus tp(sup{p.q}} tp(inf{p, q}) = tp(p) + tp(q). Further, we have tp(sup{p, q}) = 0 if tp(p) tp(q) = o. Hence

+ =

{pip is a projection of M, and tp(p) = O} is an increasing net of projections with respect to the inclusion relation. Since tp is normal, it follows that tp(z) = O. Further tplM+z = O.

291

Now if 0 =f. a E M+(l-z), then there is a non-zero projection p of M(l-z) and a positive number A such that a > Xp, Further,
A
o. Therefore,

Definition 6.5.6. Let M be a VN algebra, and

Proposition 6.5.7. A VN algbebra M is semi-finite if and only if there is a faithful family of semi-finite normal traces on M +. Let M be semi-finite. Then there is a non-zero finite projection p of M at least. Suppose that {PI}IEI\ is a maximal orthogonal family of projections of M such that PI P, VI E 1\. By Theorem 1.5.4, there is a central projection z of M such that Po z ~ pz, P (1 - z) ~ Po (1 - z), Proof.

.-...J

where Po = 1 -

L Pl. Since the family {Pi} is maximal, it follows that pz =f. O. lEA

Let VI, Vo E M with VoV o* = paZ,

For a normal tracial state

t/J(a) =

L

Va E (Mz)+.

IEI\U{O}

Since

L

vlvt = z and

IEAU{O}

t/J(a*a)

L

l,rEI\U{O}

L

=

t/J(aa*)

l,rEAU{O}

t/J

is a normal trace on (Mz)+. Suppose that .M is the definition ideal of t/J. It is clear that PIZ E M+, VI E /\ U {O}. Moreover, from LPIZ + PaZ = z, M is a-dense in Mz, and from Proposition 6.5.4, t/J is semi-

Va E (Mz)+. Thus,

lEA

finite. Now, if 0 =f. a E M z, then there is an index I E 1\ U {O} such that aVI =f. O. Noticing that M pz is finite, by Theorem 6.3.10 there is a normal

292

tracial state cP on M pz such that cp(vta*avz} I: 0. Then 1jJ(a*a} I: 0, where 1jJ is definied by cp as above. Therefore, there is a faithful family of semi-finite normal traces on (Mz)+. Further, since M(1 - z) is still semi-finite, we can continue this process. Agaim by the Zorn lemma, there is a faithful family of semi-finite normal traces on M+. Conversely, let 1 be a faithful family of semi-finite normal traces on M+. If z is a non-zero purely infinite central projection of M, then there is 1jJ E 1 such that 1jJ(z) > 0. By Proposition 6.5.4, we can find a projection p of M with p < z and < 1jJ(p} < 00. Thus there exists a normal tradal state on M p • From Propositon 6.4.2, Mp is not properly infinite. This is a contradiction since z is purely infinite and i- p < z . Therefore, M must be semi-finite.

°

°

Q.E.D.

Theorem 6.5.8. A VN algebra M is semi-finite if and only if there is a faithful semi-finite normal trace on M+. Proof. The sufficiency is clear from Proposition 6.5.7. Now suppose that M is semi-finite. By the Zorn lemma, there is a family {CPI} of semi-finite S(CPI) = 1. normal traces on M+ such that S(CPI) • S(CPII) = 0, VI i- 1', and

L

Then cP

=L

I

cpz is faithful semi-finite normal trace on M+.

Q.E.D.

l

Remark. If cP is a faithful semi-finite normal trace on M+, then we can get a faithful W*-representation of M by the GNS construction. In fact, let N, M be as in Proposition 6.5.2. Define an inner product on N:

(z, y) =

ip (y *x)

= cP ( xy *), Vz, yEN .

Denote the completion of (N, (,))- by Hip . For any a EM, define 7r

= (ax)

Vz EN,

where x --+ x
= cp(y*azx) =
--+ 0

by Proposition 6.5.3. Therefore, 7r

Proposition 6.5.9.

Let M be a VN algebra.

293

1) If M is semi-finite, P and p' are the projections of M. and M' respectively, then Mp and M p' are also semi-finite. 2) If M = I: EBMi, then M is semi-finite if and only if M, is semi-finite l

for each l. Proof. 1) Let cp be a faithful semi-finite normal trace on M+. Clearly ,cp is still a faithful semi-finite normal trace on (Mp )+. Thus, M p is semi-finite from Theorem 5.3.8. Moreover, M p ' is a * isomorphic to Mc(p'), where c(p') is the central cover of p' in M'. Since Me (p') is semi-finite, it follows that Mpt is also semi-finite. 2) It is obvious. Q.E.D.

Theorem 6.5.10. Let M be a VN algebra. Then the following statements are equivalent: 1) M is semi-finite; 2) There is an orthogonal family {PI} of finite projections of M such that LPI = 1; I

3) There is an increasing net {ql} of finite projections of M such that

(strong)-lim ql = 1; 4) There is a finite projection P of M such that c(p) = 1, where c(p) is the central cover of P in M.

Proof. 1) =} 2) . By the Zorn lemma, there is a maximal orthogonal family {PI} of finite projections of M. If P = 1 PI ¥- 0, then there is a non-zero

L I

finite projection q of M p since M p is still semi-finite. Clearly, qPI = 0, Vl. This is a contradiction to the maximum of {PI}, Therefore, I: Pl = 1. I

2) =} 3) . It is immediate from Proposition 6.4.5. 3) =} 1). Let z be any non-zero central projection of M. Then there is an index l such that zq; ¥- O. Since zq, is finite,' it follows that z is not purely infinite. Therefore, M is semi-finite. 1) =} 4). By the Zorn lemma, there is a maximal family {PI} of finite projections of M such that c(pz) . c(PI') = 0, Vi ¥- l', Put P = I: Pl. We claim I

that P is also finite. In fact, if r is a projection of M with r < P and r ,...., P, then we have that

rC(PI) ,...., pc(pt} = PI,

rc(pt} < pc(pd = Ph Vl.

Since PI is finite , it follows that rc(pz) ;;:::: Pl, Vl. Further,

P = I:Pl I

= r I:c(pt} = rp LC(PI) I

l

= rp = r,

294

Thus P is finite. Now it suffices to show c(p) = 1. If 1 - c{p) i- 0, then there is a non-zero finite projection q < (1- c(p)) since M{l- c{p)) is semi-finite. Clearly c{ q) . C (PI) = 0, Vl. This is a contradiction since the family {PI} is maximal. 4) *1}. Let z be a non-zero central projection of M. By Proposition 1.5.8, zp i- 0, where P is a finite projection with c(p) = 1. Thus z contains a non-zero finite projection zp, and z is not purely infinite. Therefore, M is semi-finite.

Q.E.D. Lemma 6.5.11. Let N be a VN algebra on Hilbert space K, and e be a cyclic and separating vector for N with 1I~11 = 1. If
e

J'ajbc~

Thus J'N J'

c N'.

= bca" ~ = bjaic~.

Further, noticing that

Va, b, c E N, we get (faJ')"'

=

ja'*J', Va E N. It is clear that JaJ'

a = 0. Now it suffices to prove J'N J' = N'. Let a' E N' with

tjJ(a) =

°

(aa'~,

E),

°< a' <

=

0 implies

1. Define

Va E N.

°

Then < tjJ < 'P. By Theorem 1.10.3, there is to E N with < to < 1 such that 1jJ(a) =
e

Q.E.D.

e

Lemma 6.5.12. Let M be a VN algebra on a Hilbert space H, E H, and P, P' be the projections from H onto M' ~, M~ repectively. Then p is finite if and only if p' is finite. Proof. Clearly, p E M, p' E M' ( see Definition 1.13.1) . Suppose that P is a finine projection of M. Consider the VN algebra L = pp'Mp'p on pp'H. Then ~{E pp'H} is cyclic and separating for L. By Proposition 6.3.1, L is finite. Clearly, L is also a-finite. Hence, there is faithful normal tracial state

295

tp on L from Proposition 6.3.15. Let

{7rIp' Hlp, elp} be the faithful cyclic W*-

representation of L generated by ip : Put N = 7rlp(L). Then N is * isomorphic to L, and is finite. Further, by Lemma 6.5.11, N' is also finite. Since N, L admit cyclic and separating vectors €Ip, € respectively, it follows from Theorem 1.13.5 that N is spatially * isomorphic to L. Therefore, L' is also finite. Now if x' E M' is such that pp' x' p'p = 0, then

0= yp'x'p'p€ = yp'x'p' € = p'x'p'y€,

Vy E M.

Me

But = p'H, so p'x'p' = O. Thus, p'x'p' ----+ pp'x'p'p(Vx' EM') is a * isomorphism from p' M' p' onto pp' M' p'p. Further, p' M' p' is finite since L' = pp'M' p'p, i.e., p' is a finite projection of M'. Similarly, p is finite if p' is finite. Q.E.D.

Propositon 6.5.13. Let M be a semi-finite VN algebra on a Hilbert space H. Then M' is also semi-finite. Proof. Suppose that there is a non-zero central projection z of M n M' such that M' z is purely infinite. Clearly, M z is still semi-finite. Therefore, we may assume that M is semi-finite , and also M' is purely infinite. By Theorem 6.5.10, there is a finite projection p of M with c(p) = 1. Then M' is * isomorphic to M;. So we may further assume that M is finite, and also M' purely infinite. Pick a non-zero vector € E H, and let p, p' be cyclic projections of M, M' determined by respectively. Clearly, p is finite. Then by Lemma 6.5.12, p' is also non-zero finite. This contradicts that M' is purely infinite. Therefore, M' is semi-finite. Q.E.D.

e

Proposition 6.5.14. A VN algebra M is semi-finite if and only if M is isomorphic to some VN algebra N so that N' is finite.

*

Proof. The sufficiency is obvious from Proposition 6.5.13. Now let M be semi-finite. Then M' is also semi-finite. By Theorem 6.5.10, there is a finite projection p' of M' with c(p') = 1. Pick N = Mpl. Then M is * isomorphic to N, and N' = M;, is finite. Q.E.D. Proposition 6.5.15. Let M be a semi-finite VN algebra, and p be a projection of M. Then p is finite if and only if there is a faithful family 1 of semi-finite normal traces on M+ such that
Sufficiency. Let q be a projection of M with q < p and q ,..., p. Then

296

i.e., tp(p - q) = 0, Vtp E 1. But 1 is faithful, so p = q. Thus p is finite. Necessity. Suppose that P is finite. By the proof of Proposition 6.5.7, there is a non-zero central projection z of M and a faithful family 1z of semi-finite normal traces on (M z)+ such that tp(pz) < 00, Vtp E 1z. By the Zorn lemma, we can find an orthogonal family {Zj} of central projections of M and a faithful family 1i of semi-finite normal traces on (M Zj)+ for each 1 such that

I: z, =

1,

tpl(PZI)

<

00, Vtpj

E

Ji. and I.

l

Since for every I, each tpl (E Ji.) can be naturally extended to a semi-finite normal trace on M+, so 1 = U, Ji. satisfies the conditions. Q.E.D. Proposition 6.5.16. Let P be a finite projection of a VN algebra M. Then the * operation is strongly continuous in any bounded ball of M p.

Proof. Replacing M by M c(p), we may assume that M is semi-finite. By Propositon 6.5.15, there is a faithful family 1 of semi-finite normal traces on M+ such that tp{p) < 00, Vtp E 1. Now let a net X, --+ 0 (strongly), and lIxdl < 1,XIP = xi, VI. For any tp E 1 and 0 < a E .Mit', where Mit' is the definition ideal of tp, by Propositions 6.5.2 and 6.5.3, we have

IILatp(XIX;) I = Itp(axjXnl = Itp(xiaxI}1

< lIalltp(x;x/) = Ilall(Lptp)(X;XI)

--+

o.

Hence, it suffices to show that the set [Latplip E 1,a E (.MIt')+] is dense in M*. Let X E M be such that

tp(ax) = 0,

Vtp E 1, a EMit'.

Then tp(x*ax) = 0 since x*a E .Mit' for any a E .Mit', and tp E 1. Since tp(E 1) is semi-finite and normal. it follows from Propositions 6.5.4 and 1.7.2 that tp(x*x) = 0, Vip E 1. But 1 is faithful, so X = O. That comes to the conclusion.

Q.E.D. As preliminaries of next section, we study the semi-finite and properly infinite VN algebras on a separable Hilbert space. Lemma 6.5.17. Let H be a separable Hilbert space, and M be a semifinite and properly infinite VN algebra on H. Then there is an orthogonal infinite sequence {Pn} of finite projections of M with Pn '" Pm' Vn, m, such that »« = l.

I: n

297

Proof. Let q be any non-zero finite projection of M, and {ql}lEA be a maximal orthogonal family of projections of M with ql ....... q, VI E /\. Put

By Theorem 1.5.4, there is a central projection pZ

Clearly, qz and

i-

-<

qz,

-<

q(l - z)

Z

.such that

p(l - z).

0 since the family {ql} is maximal. Further, Zl

=

L qlzl +

PZl,

Zl

= c(q)z

i-

0,

-< qz«.

PZl

lEA

If # A is finite, then by Proposition 6.4.5 Zl is a non-zero finite central projection. It is impossible since M is properly infinite. Thus # /\ is infinite. Indeed # /\ is countably infinite since H is separable. Then Zl ....... qlzb i.e., there is

L

v E

lEA

M such that vv· =

Zb

v*v =

L

qlZl'

lEA

For each 1 E /\, let PI = vq,Zl u". Then PIPlt = 0, Vt

1= I',

and

Clearly, {PI },EA is an orthogonal infinite sequence of finite projections of M, Pl ,..", PI', VI, 1', and PI = Zl i- O. Further, by the Zorn lemma we can get the

2: lEA

Q.E.D.

conclusion.

Proposition 6.5.18. Let M, N be two VN algebras on a separable Hilbert space H, and M', N' be semi-finite and properly infinite. If c) is a * isomorphism from M onto N, then c) is also spatial. Proof. By Proposition 1.12.5, we may assume that there is a VN algebra L on H, and projections p', q' of L' with c(p') = c(q') = 1 such that M = L p" N = Lqt , and c)(ap') = aq', Va E L. By Lemma 6.5.17, there are two orthogonal infinite sequences {p~}, {qH of

finite projections of L' such that

~ .... q*

Write {I, 2, ...} = Un/\n such that An countably infinite.

q'.l '

n Am

w; J' VII,.

=

0, Vn i- m, and each

An

IS

298

Fix n. Then there is a central projection z of M such that zq'n-< z L"'"' pi," -

(1 - z)

iEl\n

L

p~

-< (1 - z)q~ .

iEl\n

L p~ ~ L p~. Then

Fix sEAn, and write A~ = An \ {s}. Clearly,

iEI\~

iEl\n

(1 - z)

L

p~ ~

(1 - z)

L

p~

iEI\~

iEl\n

L

< (1 - z)

p~

-< (1 - z)q~ .

iEl\n

But q~ is finite, so (1 - z)

L

L

p~ = (1 - z)

iEI\~

p~, i.e.

iEl\n

(1 - z)p~ =

o.

Moreover , c(p~) = c(p'} = 1. Thus z = 1 and

q~ -<

L

p~.

iEl\n

Since n is arbitrary, it follows that q' -< p'. Similarly, p' p' q', and is spatial. ,o,J

References.

-<

q'. Therefore, Q.E.D.

[28], [144], (150].

6.6. Purely infinite Von Neumann algebras Proposition 6.6.1. Let M be a VN algebra. 1) If M = L EBMt, then M is purely infinite if and only if M, is purely I

infinite, Vi. 2} If M is purely infinite, then so is M'. 3} Let M be purely infinite, and p, p' be projections of M, M' respectively. Then M p and Mpl are also purely infinite.

Proof. 1) The necessity is obvious. Now let M, = M Zl be purely infinite, VI. If p is a finite projection of M, then pZz = 0, Vi, and p = O. Therefore, M is also purely infinite. 2) It is immediate from Proposition 6.5.13. 3) By 2) , it suffices to show that Mpl is purely infinite. But Mpl is * isomorphic to M c{p'}, and M c(p'} is purely infinite obviously, so Mpl is purely infinite. Q.E.D.

299

Proposition 6.6.2. A VN algebra M is purely infinite if and only if there is no non-zero semi-finite normal trace on M+. Proof. IT there is a non-zero semi-finite normal trace ep on M +, then its support s (ep) is a non-zero central projection of M. By Theorem 6.5.8, M s (ep) is semi-finite, i.e, M is not purely infinite. Conversely, if there is a non-zero central projection z of M such that M z is semi-finite, then we have a non-zero semi-finite normal trace on (M z)+. Therefore, there is a non-zero semi-finite normal trace on M+. Q.E.D.

Proposition 6.6.3. A VN algebra M is purely infinte if and only if the * operation is not strongly continuous in any bounded ball of M p for any non-zero projection p of M. Proof. Let M be purely infinite, and p be a non-zero projection of M. Since p is infinite, there is v E M such that u" v = p, vv· < p, and vv· ip, By Lemma 6.3.11, we can find an orthogonal sequence {en} of non-zero projections of M such that en '" em, 'TIn, m, en < p, 'TIn, and en -----+ 0 ( strongly ) . Suppose that w~ Wn = en, wnw~ = et, 'TIn. Clearly, Wn ---+ 0 ( strongly) , IIwnll < 1, WnP = Wn, 'TIn. However, {w~} does not converge to 0 strongly, so the * operation is not strongly continuous in any bounded ball of M p. Conversely, if M is not purely infinite, then M contains a non-zero finite projection p. By Proposition 6.5.16, the * operation is strongly continuous in any bounded ball of Mp. Q.E.D.

Pr-opoalt.ion 6.6.4, Let M be a a-finite and purely infinite VN algebra, and p,q be two projections of M with c(p) = c(q). Then p '" q. Proof. From the Zorn lemma, we can take a maximal orthogonal family {ZI}'El\of central projections of M such that qZ, -< pz" 'TIl E /\. Let Z

=L

zi,

p' = p{1 - z),

q' = q(1 - z).

lEA

By Theorem 1.5.4 and the maximum of the family {ZI}'EI\,we can see that p' -< q' ( relative to M{l - z)). Suppose that p' -=1= O. Pick a maximal orthogonal family {q~}8EJ' such that q~ '" p', q~ < q', \Is E 1/. For Theorem 1.5.4, there is a central projection z' of M with z' < (1 - z) such that

(q' -

L

q~)z'

-< p'z',

8EJ'

p'(1 - z') -< (q' -

L 8EJ'

q~)(1 - z').

300

Clearly , p' z' f:- 0 since the family {q~} sEll is maximal . Since Mp is purely infinite, from Theorem 6.4.4 there is an orthogonal infinite sequnce {en} of projections of M p such that 00

L

en =

v,

en.-v p, \In.

n=l

Then Since M is a-finite, it follows that the index set II is countable, and 00

L

enz' ,..,.,

n,.,,2

L

q:z'.

sEll

In addition,

elZ' ,..,., p'z' >- (q' -

L q~)z', sEI

So we have pz' >- q'z' = qz', Moreover, 0 f:- z' < 1- z . This is a contradiction since the family {zlhEA is maximal. Therefore p' must be zero, i.e., p < z. Now from z > c(p) = c(q) > q, we have p

= pz =

L lEA

PZI

>-

L qz, =

q.

lEA

Similarly, we can prove that p -< q. Therefore, p ,..,., q.

Q.E.D.

Proposition 6.6.5. Let M be a a-finite and purely infinite VN algebra, and a be a non-zero element of M. Then K (a) f:- {O}, where K (a) is defined in Theorem 6.2.7.

Proof. By Proposition 6.2.8. and K(a*) = K(a}*, we may assume that a* = a. Also we may assume that Iiall < 1 and a+ f:- 0 ( otherwise, replace a by -a ) . Then we can find a non-zero projection p of M and a positive integer n such that 1 q>-p-(1-p). n

If p contains a non-zero central projection z, then a > n~l Z - 1. Further, for any b E K(a), we have b > n~l Z - 1. Thus, K(a) f:- {O}. Now suppose that p contains no non-zero central projection. Replacing M by Mc(p), we may assume that c(p) = 1. Since p > 1 - c(1 - p], it follows that c(1 - p} = 1. By Proposition 6.6.4, we have p.-v (l-p),..,., 1.

301

From Theorem 6.4.4, there are pairwise orthogonal projections such that

{ft,' •. ,fn+l}

n+l

P=

L

fi,

and

fi"""

p, Vi.

i=1

Pick

Vi

where and

E M such that

fO

=

1- p. Let Vn+l

= Va ••• v~. Then u =

Let b = (n + 2)-1

VI

+ VI +... + Vn+l

is unitary,

n+l

L

uiau-;.

i=o

Since

n+1

L

u i eiu - i = 1,

0

n

+ 1,

;=0 it follows that n+l

(n

+ 2)b > L

1

.

u' (-(f1 . 0 n 1=

+ ... + en+l)

.

-

1

fo)U-' = -, n

and c > n(n~2)' Vc E K(b). Clearly, K(b) c K(a). Therefore, K(a)

f:- {O}. Q.E.D.

Proposition 6.6.6. Let M be a purely infinite VN algebra on a separable Hilbert space H. Then M admits a cyclic and separating vector.

Proof.

Let e be a non-zero vector of H, and p be the cyclic projection of M determined bye, and {PI} be a maximal orthogonal family of projections of M such that PI ,..... p, Vi. From Theorem 1.5.4, there is a central projection z of M such that

(1 -

:L pd z -< pz, l

p(1 - z)

-< (1 - LPI)(l - z). I

Clearly, z

f:- 0, zc(p) = c(pz) >

:L PlZ. On the other hand. I

c(p)z = c(pz) > (1 -

L PI)Z. I

Thus c(p)z > z, and c(pz) = z, By Proposition 6.4.4, we have pz ,..... z. Further , there is 1] E H such that z is the cyclic projection of M determined by 1].

302

By the Zorn lemma and the separability of H, there is an orthogonal sequence {zn} of central projections of M such that Zn = 1 and s.H = M'Tln,

L n

where TIn E H, Vn. We may assume that

lI1]nll < 2- n,Vn.

Then TI =

L TIn is a n

cyclic vector for M'. Similarly, there is a cyclic vector for M since M ' is also purely infinite. Now from Proposition 1.13.4, M admits a cyclic and separating vector. Q.E.D. The following proposition is a generalization of Proposition 6.5.18.

Proposition 6.6.7. Let M, N be two VN algebras on a separable Hilbert space H , and M', N' be properly infinite. Then each * isomorphism 4> from M onto N is spatial.

Proof. Let Z be the maximal central projection of M such that M z is purely infinite. Then N4>(z) is also purely infinite, and M'(I- z),N'(1 - 4>(z)) are semifinite and properly infinite. From Proposition 6.6.6, both M z and N 4> (z) admit a cyclic and separating vector. Thus 4> : M z - 4 N(z) is spatial by Theorem 1.13.5. Moreover, 4> : M(1 - z) - 4 N(1 - (z)) is also spatial from Proposition 6.5.18. Therefore, the * isomorphism from M onto N is spatial. Q.E.D. References.

[18], [28], [144].

6.7. Discrete ( type (1 )) Von Neumann algebras Theorem 6.7.1. Let M be a VN algebra on a Hilbert space H. Then the following statements are equivalent: 1) M is discrete ( type (1)); 2) M' is discrete ( tyep (1)); 3) M is * isomorphic to some VN algebra N such that N ' is abelian; 4) there is an abelian projection P of M such that c(p) = 1 j 5) any non-zero projection of M contains a non-zero abelian projection of

M.

*

Proof. 1) 4). Pick a maximal family {Pt} of non-zero abelian projections of M such that c(Pt) . c(Pt') = 0, Vi # l', and put P = LPt. If c(p) # 1, then t

(1 - c(p)) contains a non-zero abelian projection since M is discrete. This contradicts the maximum of the family {Pi}, Thus c(p) = 1. Moreover, by Proposition 1.5.9, PtMPt' = {O}, Vi #i'. Therefore, P is also abelian.

303

*

2) 3). Suppose that M' is discrete. From the preceding paragraph, M' admits an abelian projection p' with c(p'} = 1. Now let N = Mpl. Then M is * isomorphic to N, and N' = M;, is abelian. 3) 5). Suppose that ep is a * isomorphism from M onto N: where N is a VN algebra on a Hilbert space K, and N' is abelian. Let p be any non-zero projection of N, be a non-zero vector of pK, and q be the cyclic projection of N determined bye. Since the VN algebra N~ on qK = N'e is abelian, and admits a cyclic vector it follows from Proposition 5.3.15 that N; = (N;Y = N q , and q is abelian. Clearly, 0 f:. q < p, Therefore, any non-zero projection of M contains a non-zero abelian projecton of M. 5) 1) . It is obvious by Definition 6.1.7. 4) 2). Suppose that p is an abelian projection of M with c(p) = 1. Let L = M;. Then M' is * isomorphic to L, and L' = M p is abelian. Now by 3) 5) 1), M' is discrete. Q.E.D.

*

e

e,

*

* * *

Remark. For any VN algebra M, the VN algebra N generated by MuM' is discrete. In fact, N' = M n M' is abelian.

Proposition 6.7.2. Let M be a VN algebra. 1) If M = $Mz , then M is discrete if and only if M, is discrete, VI.

L z

2) Let M be discrete, and p, p' be projections of M, M' respectively. Then M p, Mpl are also discrete. Proof. 1) It is abvious from Definition 6.1.7. 2) Since M p ' is * isomorphic to M z, where z = c(p')' it follows that M p ' is Q.E.D. discrete. Moreover, M p is also discrete by (Mp)' = M; .

Proposition 6.7.3. Let M be a discrete ( type (I) ) factor. Then M is isomorphic to B (K), where K is some Hilbert space.

*

Proof. From Theorem 6.7.1, M can be * isomorphic to a VN algebra N on some Hilbert space K such that N' is abelian. Clearly, N is a factor, and N' is an abelian facer. Therefore, N' = (C, and N = B(K). Q.E.D.

Remark. From this proposition, we can get the results on finite dimensional C· -algebras ( see Section 2.13), too.

Lemma 6.7.4. abelian, p < c(q)

Let p, q be two projections of a VN algebra M, and p be . Then p -< q.

304

From Theorem 1.5.4, there is a central projection z such that

Proof.

qz -< pz,

p(1 - z} -< q(1 - z).

Let qz ,..., PI < pz. Then the central cover of PI in M p z is C(Pl)PZ by Proposition 1.5.8. Since M p z is abelian, it follows that PI = c(pdPz = c(qz)pz = c(q)pz = pz.

Thus qz ,..., pz, and P -< q.

Q.E.D.

Lemma 6.7.5. Let M be a VN algebra on a Hilbert space H, and {pdl E A}, {qrlr E lI} be two orthogonal families of abelian projections of M such that PI ,..., PI', Vl, l' E A, qr ,..., qr', Vr, r' E lI, and L PI = L qr = 1. Then # A = # lI. lEA

rEI

Proof. Clearly, c(pd = c(qr) = 1, Vl, r. So by Lemma 6.7.4, we have PI ,..., qr, Vl E A, r E lI. If # /\ < 00, then M is finite by Proposition 6.4.5, and # II must also be finite. Thus # /\ and # II are finite or infinite simultaneously. Consider the case that # /\ and # II are finite. We may assume that # /\ < # lI. Then

I=LPI"'" Lqr
r

rEI'

lEI

r

where c II and # = # A . Since M is finite, it must be ][' = lI, and #A = # lI. Now let both A and II be infinite index sets. Fix P E {PI}' The abelian VN algebra M p is finite obviously. So by Propositon 6.3.16 and 1.3.8, there is a non-zero central projection z of M such that M pz is a -finite. Considering MZ,{PIZ},{qrz}, we may assume that z = 1. For any I E A,Mpl is a a-finite VN algebra on PIH. By Proposition 1.14.2, there is a countable subset .M , of PlH such that [M'.Md is dense in PlH. Let III = {r E lllqr.M 1 #- {O}}. Since {qrlr E lI} is pairwise orthogonal, III must be a countable subset of lI, VI E A. Moreover, if there exists r E lI\ U' E A lI" then qr.M 1 = {O}, VI E A. Further, qrM'.M, = {O}, i.e. qrPI = 0, VI E A. This is a contradiction since PI = 1 and qr #- O. Therefore, II = U'El\lI" and # II # A = # lI.

<#

L,

A . Similarly, # A

<

# lI. So

Q.E.D.

Definition 6.7.6. A VN algebra M is called type ( In) or ti-homoqeneous , where n is a finite or infinite cardinal number, if there is an orthogonal family {Pill E I\} of abelian projections of M such that Pl ,..., PI', VI, I' E A and pi = 1, and # 1\ = n.

L

lEA

By Lemma 6.7.5, the definition of type (In) is independent of the choice of the family {pdI E A}.

305

Proposition 6.7.7. Let M be a VN algebra. 1) If M is type (In) , then M is type (I). 2) M is type (In) if and only if M is spatially * isomorphic to N®B(Hn) , where N is an abelian VN algebra, and H n is a n-dimensional Hilbert space. Consequently, A factor of type (In) must be * isomorphic to B(Hn)' Proof. 1) Let {PI} be as in Definition 6.7.6. Then c(pz) = 1,Vl. By Theorem 6.7.1, M is discrete. 2) The necessity is immediate from Definition 6.7.6 and Theorem 1.5.6. Conversely, suppose that {edt E A} is a normalized orthogonal basis of H n , where # A = n. Let Pl be the projection from H n onto led. Then {pdl E A} is an orthogonal family of abelian projections of B(Hn ) with PI PI', V/,l' and PI = 1. Therefore, N®B(Hn) is type (In) since N is abelian. Q.E.D. f".i

L I

Lemma 6.7.8. Let M be a VN algebra, and {z,} be an orthogonal family of n-homogeneous central projections of M where n is a cardinal number. Then Z = ZI is also n-homogeneous.

L I

z,

Proof. Let {p~) 10: E Il} be an orthogonal family of abelian projections of M with pCI) p(l] Vo: 0:' and "pO) = "Vi where # Il = n . Put Per = 'L.....,; " p(l) er er'" L.....,; er er' er I Then {Perlo: E H} is an orthogonal family of abelian projections of M such that Per Per', Vo:, 0:' , and LPer = z. Therefore, Mz is type (In)' er Q.E.D.

z,

.-..J

.-..J

Lemma 6.7.9. Let M be a VN algebra, and z; be a ni-homogeneous central projection of M, i = 1,2. If nl =1= n2, then ZlZ2 = o. Proof. Let {p~i) It E Ai} be an orthogonal family of abelian projections of wt l' ,an d '" h # Ai -- ni, Z. -- 1,. 2 Th en M Zi sue h t h at P,(i) PI'(i) ,v, L.....,; P,(i) -- Zi, were f".i

lEI\,

AI} and {p~2)ZII1 E A2} are also two orthogonal family of pairwise equivalent abelian projections of M ZlZ2 with pP) Z2 = p~2) Zl = ZIZ2. {pP)Z2Il E

If ZlZ2 =1= 0 , then # 1\1 = Therefore, ZlZ2 = O.

#1\2

L

L

IEl\l

IEI\2

by Lemma 6.7.5. This contradicts

nl

i-

n2.

Q.E.D.

Theorem 6.7.10. Let M be a type (I) VN algebra. Then there is unique decomposition M = EBMn , where E is some set of different candinal num-

L

nf1E

bers, and M n is type tIn}, Vn E E.

306

Proof. Pick a non-zero abelian projection P of M, and let {PI} be a maximal orthogonal family of projections of M such that PI ,...., P, Vl. By Theorem 1.5.4, there is a central projection Z such that

(1 - q)z -< pz, where q z

=L

=L

p(I - z) -< (1 - q)(1 - z),

Pl· By the maximum of {Pl}, z is not zero. If (1 - q)z

= 0, then

I

PIZ is homogeneous. Now let (1 - q)z =I- O. Then

Zl

= zc(l -

q) is a

I

non-zero central projection. Suppose that

(1 - q)z ...... ql < pz. Since pz is abelian , it follows from Proposition 1.5.8 that ql c((1 - q)z)pz = PZl' Clearly, (1 - q)Zl = (1 - q)z. Thus

(1 - q)Zl ...... ql = PZl ....., and

Zl

=

L

PlZl

+ (1 -

PlZl,

Vl,

q)Zl is homogeneous.

1

From above discussion, we get a non-zero homogeneous central projection of M. Further, by the Zorn lemma there is an orthogonal family {zr} of homogeneous central projections of M such that z; = 1. Finally, by Lemma

L r

6.7.8 and 6.7.9, the conclusion can be obtained.

References.

Q.E.D.

[82), [86], (88].

6.8. Continuous Von Neumann algebras and type (II) Von Neumann algebras Proposition 6.8.1.

A VN algebra M is continuous if and only if there is no non-zero central projection Z such that M Z is discrete. In consequence, a purely infinite VN algebra must be continuous.

Proof. If M is not continuous, then M contains a non-zero abelian projection p. Let Z = c(p). By Theorem 6.7.1, Mz is discrete. Conversely, suppose that z is a non-zero central projection such that M z is discrete. Then M z contains a non-zero abelian projection p. Clearly, P is also an abelian projection of M. Thus, M is not continuous. Q.E.D.

Proposition 6.8.2. Let M be a VN algebra. 1) If M = EBMt, then M is continuous ( or type (II)) if and only if M,

L I

is continuous ( or type (II) ) , Vl.

307

2) Suppose that M is continuous ( or type (II) ), then so is M'. 3) Let M be continuous ( or type (II) ) and P, P' be projections of M, M' respectively. Then M p and are continuous ( or type (II) ).

u,

Proof. 1) It is obvious by Definition 6.1.7. 2) Let M be continuous. If there is a non-zero central projection z of M' such that M'z is discrete. Then by Theorem 6.7.1, (M'z)' = Mz is also discrete, a contradiction. Therefore, M' is also continuous. Moreover, by Proposition 6.5.13, M' is type (II) if M is type (II). 3) From (Mp )' = M; and the conclusion 2), it suffices to show that Mpl is continuous. Since Mpl is * isomorphic to M c(p'), the conclusion is obvious.

Q.E.D. Theorem 6.8.3. A VN algebra M is continuous if and only if every projection P of M can be writen as P = PI + P2, where PI, P2 are projections of M with PIP2 = 0, and PI ""' P2.

Proof. Sufficiency. If P is an abelian projection of M, then from the assumption we can write P = PI + P2, where PI, P2 are projections of M with PIP2 = 0, PI ""' P2. Clearly , C(Pl) = c(p). Since PI < P and P is abelian, it follows from Proposition 1.5.8 that PI = C(Pl)P = p. Therefore, P = 0, and M is continuous. Now let M be continuous, and P be any non-zero projection of M. Then Mp is not abelian, and there is a non-zero projection q of M p such that q (/. Mp nM;. By Theorem 1.5.4, we can find a central projection z of M such that

qz -< (p - q)z,

(p - q)(l - z) -< q(l - z).

If qz = (p - q)(l - z) = 0, then q = p(1 - z) E M p n M;, a contradiction. Thus we have either qz i- or (p - q)(1 - z) -# 0. If qz -# 0, let qz = rl '2 < (p - q)z, then 'Ir2 = 0, and (rl + r2) < p. If (p - q)(1 - z) -# 0, let (p - q)(l- z) = rl r2 < q(1 - z), then rlr2 = 0, and rl + r2 < p. Therefore, p contains two non-zero projections rl, r2 such that rl r2 = and r1 r2' Continue this process for (p - (rl + r2)) ..... Then by the Zorn lemma, we can get a decomposition P = PI + P2 with PIP2 = and PI P2.

°

/"V

°

/"V

°

f"oI

I'"V

Q.E.D.

A VN algebra M is type (II) if and only if there is a decreasing sequence {Pn} of projections of M such that PI is a finite projection with C(Pl) = 1, and (Pn - Pn+1) Pn+1l Vn.

Theorem 6.8.4.

I'"V

Proof.

Let M be type (II) . From Theorem 6.5.10, there is a finite projection

308

PI of M with c(Pt) = 1. By Theorem 6.8.3., we can write

Pn = Pn+l

+ qn+l,

where

Pn+Iqn+1 = 0,

and

Pn+l

f"J

qn+b

Then {Pn} satisfies the conditions. Conversely, suppose that {Pn} is a decreasing sequence of projections of M, where PI is a finite projection with c(pd = 1, and (Pn - Pn+d ,...., Pn+b Vn. From Theorem 6.5.10, M is semi-finite. If M P1 is continuous, then (MpJ' = M~l is also continuous. Further, M' is continuous ( since M' is * isomorphic to M~l ) and M is type (II) . So it suffices to show that M P 1 is continuous, and we may assume that PI = 1, i.e., M is finite. By Propositions 6.3.16 and 6.8.2, we may also assume that M is a-finite. Then there exists a faithful normal tradal state ip on M. If P is an abelian projection of M, then by Theorem 1.5.4 there is a sequence {Zn} of central projections of M such that

PnZn ,...., qn

< PZn,

p(l - zn)

-< Pn (1 - zn), Vn.

Since P is abelian, it follows that qn = c(qn)pzn = c{Pn)pzn. Noticing Pn ,...., (Pn-l - Pn), we have C(Pn) > Pn-l. Thus, c{Pn) = 1, and qn = PZn i.e., PnZn PZn, and P -< Pn, \In. On the other hand, from f"J

it follows that CP(Pn) = 2CP(Pn+l)' Noticing that cp{pd = cp(l) = 1, we get that CP(Pn) = 2- n +1 , Vn. In addition, cp(p) < CP(Pn) = 2- n+ 1 , Vn, thus cp{p) = o. Further P = 0 since cp is faithful. Therefore, M contains no non-zero abelian projection, and M is continuous.

Q.E.D. References.

[18], [82].

6.9. The types of tensor products of Von Neumann algebras Let M, be a VN algebra on a Hilbert space Hi, i = 1,2. Their tensor product M I @M2 is a VN algebra on If1 @ H 2 • In this section, we consider the relations between the types of M}, M 2 and the type of M I @M2 •

309

Proposition 6.9.1. The VN algebra M 1 &;M2 is finite if and only if both VN algebras M 1 and M 2 are finite. Proof. Let M 1 &;M2 be finite. Since M 1 is * isomorphism to M 1 &; 12 , it must be that M 1 is finite. Similarly, M 2 is finite too. Now suppose that both M 1 and M 2 are finite. By Proposition 6.3.16, we may also assume that both M 1 and M 2 are a-finite. Then there are faithful normal tradal states PI, P2 on Ml, M 2 respectively. We can write Pi (.) =

L (.e~i) , e~i)), V· E M i , n

n

n,m

(·e~l) &; e~), e~l) &; e~»). Clearly, !PI &;!P2 is a normal tradal state on M 1&;M2 • Since Pi is faithful on M i , {e~i)} is a cyclic sequence of vectors for MI, i = 1,2. Thus {e~l) &; ei:)}n,m is cyclic for (MI &;M2)', and CPI &; P2 is also faithful on

M 1 &;M2 • Therefore, M 1 &;M2 is finite.

Q.E.D.

Proposition 6.9.2. M 1&;M2 is properly infinite if and only if either M 1 or M 2 is properly infinite. Proof. The necessity is immediate from Proposition 6.9.1. Now assume that M 1 is properly infinite. If M 1 ®M2 is not properly infinite, then there is a normal tradal state P on M 1 &;M2 • Clearly, (p IM1 &; 12 ) is also a normal tradal state on MI' It is impossible since M 1 is properly infinite. Therefore, it must be that M 1 ®M2 is properly infinite. Q.E.D.

Proposition 6.9.3.

If both M 1 and M 2 are semi-finite, then M 1 ®M2 is

also semi-finite.

Proof. By Proposition 6.5.14, we may assume that both Mf and M~ are finite. Then (M1 &;M2 )' = M: ®M~ is finite, and M 1 &;M2 is semi-finite. Q.E.D.

Lemma 6.9.4. Let N be a VN algebra,

--t

ba' (Va E N) is strongly continuous in any bounded ball of N.

Proof. Suppose that {all is a net of N with Ilalll < 1 and a, ---+ 0 (strongly) . We need to prove that alb*bai ---+ 0 ( weakly) . Since aib'ba; E Ns(cp), Vl, we may assume that s(cp) = 1, i.e., P is faithful. Then there is a faithful W *-representation {71" cp, Hlp} of N generated by P (

310

see Section 6.5 ) . It suffices to show

(7rrp(alb*banxrp,xrp) =
---+ 0

Vx E )/ by Ilalll < 1, VI ( see Propositon 6.5.2 ) . In fact, from Proposition 6.5.2., 6.5.3 and b E )/, it follows that

tp( z" aib"balx) =

ip (bal xx* al b*)

< IlxI12tp(baialb*)

= IlxI12tp(b*ba;al)

---+

0,

Vx E )/.

Q.E.D.

Proposition 6.9.5. M 2 is purely infinite.

M 1®M2 is purely infinite if and only if either M 1 or

If both M 1 and M 2 are not purely infinite, then by Proposition 6.9.3, M I ®M 2 is not purely infinite. Now let M I be purely infinite. If M 1®M2 is not purely infinite, then there is a non-zero semifinite normal trace tp on (MI®M2 )+. Pick 0 i- s e (M1®M2)+s(tp) and tp(b 2 ) < 00. Write Proof.

H 1®H2 =

L: fBHz,

b = (bill )I,I/EA,

lEA

where HI and H 2 are the action spaces of M I and M 2 respectively, and H, = HI, Vl, # 1\ = dirnH2 , bill E B(H1 ), Vl, l'. Since b > 0 and b i- 0, there is an index 10 such that bl = blolo > 0 and bl i- O. Now consider the following chain

M 1 ~ M 1®M2 ~ M I®M2 .i; M I , where a(ad = al ® 12, Val E M I,{3(a) = ba" and ,(a) = alol o' Va E M 1®M2 • By Lemma 6.9.4, the map

(,0 {3 0 a) : al

---+ b1a;,

(Val E

Md

is strongly continuous in any bounded ball of M 1 . Moreover, we can find a non-zero projection PI of M 1 and a positive number Asuch that bl > API' Then the map al -----+ PI ai(Val E A1d is also strongly continuous in any bounded ball of M 1 . Consequently, the map al ---+ ai is strongly continuous in any bounded ball of MIPI. However, since PI i- 0 and M I is purely infinite, we get a contradiction from Proposition 6.6.3. Therefore, M I ®M2 must be purely infinite. Q.E.D.

Corollary 6.9.6. If M I®M2 is semi-finite, then M 1 and M 2 are semi-finite.

311

Proposition 6.9."1. Let M i be type (In;), i = 1,2. Then M 1 ®M2 is type (In1n2). Consequently, if both M 1 and M 2 are discrete, then M 1®M2 is also discrete. Proof. By Proposition 6.7.7, we may assume that M, N. is abelian, and dim K, = ni, i = 1,2. Then

= Ni®B(Ki ) , where

Q.E.D.

Proposition 6.9.8. and only if n < 00.

Let M be a type (In) VN algebra. Then M is finite if

Proof. By Proposition 6.7.7, we may assume M = N®B(K), where N is abelian, and dim K = n. Clearly, N is finite. Thus by Proposition 6.9.1, M is finite if and only if B(K) is finite, i.e., n < 00. Q.E.D.

Proposition 6.9.9. M 1 ®M2 is type (II).

Let M 2 be semi-finite, and M 1 be type (II). Then

Proof. By Theorem 6.8.4, there is a decreasing sequence {Pn} of finite projections of M 1 with c(pd = 1, and Pn+l -- (Pn - Pn+d, Vn. From Theorem 6.5.10, there is a finite projection q of M 2 with c(q) = 1. Now let en = Pn ® q, Vn. Then {en} is a decreasing sequence of finite projections of M 1 ®M2 by Proposition 6.9.1. From Definition 1.5.7, it is easy to see that the central cover of fl in M 1 ®M 2 is 1 . Clearly, en+l ,.., (en - en+l), Vn. Therefore, M 1®M2 is type (II) by Theorem 6.8.4. Q.E.D.

Proposition 6.9.10. is continuous.

M 1 ®M2 is continuous if and only if either M 1 or M 2

Proof. Since any purely infinite VN algebra is continuous, we may assume that both M 1 and M 2 are semifinite. Thus the sufficiency is immediate from Proposition 6.9.9. Now let M 1®M2 be continuous, and M h M2 be semi-finite. If M 1 and M 2 are not type (II), then by Proposition 6.9.7 M 1 ®M2 is not continuous, a contradiction. Therefore, either M 1 or M 2 is continuous. Q.E.D.

Corollary 6.9.11. 1) If M 1®M2 is discrete, then M 1 and M 2 are discrete; 2) If M 1 ®M2 is type (II), then both M 1 and M 2 are semi-finite, and either M 1 or M 2 is continuous.

312

Summing up above, we have the following.

Theorem 6.9.12. 1) M 1®M2 is finite, or semifinite, or discrete if and only if both M 1 and M 2 are fnite, or semi-finite, or discrete. 2) M 1 @M2 is properly infinite, or purely infinite, or continuous if and only if either M 1 or M 2 is properly infinite, or purely infinite, or continuous. 3) M 1 @M2 is type (II) if and only if both M 1 and M 2 are semi-finite, and either M 1 or M 2 is continuous. Notes. The tensor product of semi-finite VN algebras was proved to be semifinite by Y. Misonou. The case involving algebras of type III was settled by S.Sakai. Thus we have now the full result of Theorem 6.9.12.

References. [28], [109], [144].

Chapter 7 The Theory of Factors

7.1. Dimension functions From the classification in Chapter 6, there are only five classes of factors: 1) Type (In) factors, i.e., discrete finite factors. It must be * isomorphic to B(Hn }, where dim H n = 00); 2)Type(I oo ) factors, i.e. discrete infinite factors. It must be * isomorphic to B(H), where dimH = 00; 3) Type (lId factors. i.e., continuous finite factors; 4)Type (1I 00 ) factors, i.e., continuous infinite factors; 5)Type (III) factors, i.e., purely infinite factors.

n«

Definition 7.1.1. Let M be a factor. A trace t.p on M+ is called satisfying the condition (R), i.e., if M contains a non-zero finite projection, then there is a non-zero finite projection Po such that t.p(Po) < 00. Proposition 7.1.2. Let M be a factor, and t.p be a faithful normal trace on M+ satisfying (R). 1) Let P be a projection of M. Then P is finite or infinite if and only if t.p(p) < 00 or t.p(p) = +00. 2) Let P, q be finite projections of M. Then P -< q if and only if t.p(p) < t.p(q). 3) If M contains a non-zero finite projection, then t.p is semi-finite. 4) t.p is uniquely determined up to multiplication by a positive constant. Proof. 1) If P is infinite, then it must be properly infinite. By Theorem 6.4.4, we can write P = PI +P2, where PIP2 = 0 and PI ,...., P2 ,...., p. Then t.p(p) = 2t.p(p). Since ip is faithful, it follows that t.p(p) = +00. If P is finite, and P f:. 0, then by Definition 7.1.1, there is a non-zero finite projection Po of M such that t.p(Po) < 00. We have either P -< Po or Po -< P

314

since M is a factor. Clearly,
PI '" Po,

Pl < P,

'Ill E A,

and

(p -

L

Pi) -< Po·

LEA

Since P is finite, # A must be finite. So it is easy to see that Ap. Since M is semi-finite, there is a non-zero finite projection q of M with q < p. Then, a > Aq, and 0 <
Va E M.

Then

cp(tab) =
Since {ql} is increasing, it follows that A/ is independent of the index 1. Put A = Al''Ill. Then

Va E M+,

'Ill.

Moreover, by Proposition 6.5.2,
Q.E.D.

315

Remark.

For type (I) factor B(H), the unique (up to multiplication by a positive constant) faithful semi-finite normal trace on B(H)+ is as follows

L:('6, €,),

tr(·) =

V· E B(H)+,

I

where {e,l is a normalized orthogonal basis of H. For a finite factor, there is unique faithful normal tradal state on it. For any semi-finite factor, there is unique (up to multiplication by a positive constant) faithful semi-finite normal trace on its positive part. For purely infinite factors, the case is trivial. Proposition 7.1.3. Let M be a factor, P =Proj (M) be the set of all projections of M,tp be as in Proposition 7.1.2, and 0 = {ep(p)lp E Pl. Then multiplying ep by a proper positive constant, we can get the following: 1) D = {O, 1",' ,n}, when M is type (In) (n finite or infinite). In particular, if M = B(Hn ), where dimHn = n, then

ep(p)

= dim pHn ,

Vp E P;

2) 0 = [0,1], when M is type (IIdj 3) D = [0, +00], when M is type (IIoo)j 4) 0 = {O, +oo}, when M is type (III). Proof. 1) and 4) are obvious. 2) Let tp be the unique faithful normal tradal state on a type(II 1 ) factor M. By Theorem 6.8.3, {2-nkll

< k < 2n, n

= O,I,"'} C

O.

For any A E [0,1], pick Pn E P such that ep(Pn) = An / A. From Proposition 7.1.2, Pn -< Pn+b Vn. Let qt = PI, and ql '" q < P2' By Proposition 6.3.2, (1 - ql) ,...... (1 - q). Thus, (P2 - q) -< (1 - qd. Let (P2 - q) ,. . . r < (1 - qd· Then P2 '" ql + r. Put q2 = ql + r , Then q2 > ql, qi ,. . . Pi, i = 1,2. Generally, we can get {qn} with qn < qn+ll qn '" Pn, 'lin. Let q = sUPn qn' Then ep{q) = sUPn ep{Pn) = A. Therefore, 0 = [0, I}. 3) Suppose that {PI} is an increasing net of finite projections of M with SUP,P, = 1. Clearly, ep(pt} / ep(l) = +00. By 2), [O,ep{p,)} C 0, 'Ill. Therefore, [) = [0, +00]. Q.E.D.

Lemma 7.1.4. Let M be a finite factor, and P =Proj{M) be the set of all projections of M. Suppose that D : P ---+ [0, +00) satisfies: 1) if PllP2 E P and PIP2 = 0, then D(PI + P2) = D(pt} + D(P2)j 2) for any unitary element u E M and pEP, D(upu'") = D(p)j

316

3) D(I) > O.

Then D =
O. We may assume that). = 1. Since each projection of M is an orthogonal sum of several minimal projections of M, it follows that D(P) = {O, 1"", n}. By Proposition 7.1.3, D = tpiP. Now let M be a type (IId factor. We may assume D(l) = 1. Suppose that
D(Pm) = tp(Pm) -+ tp(p). Similarly, D(1 - p) > tp(I - p). Therefore, D(p) = tp(p) , and D = tpiP. Q.E.D.

Proof.

/"'oJ

J",J

Definition 7.1.5. Let M be factor, and P be the set of all projections of M. A function D : P --+ [0, +00] is called a dimension function, if: 1) D(p) = o ¢=} P = 0; 2) for any unitary element u E M and PEP, D(upu"') = D(p); 3) if P, q E P and pq = 0, then D(p + q) = D(p) + D(q); 4) if M contains a non-zero finite projection, then there is a non-zero projection Po of M such that D(po) < 00. Theorem 7.1.6. Let M be a factor, P be the set of all projections of M, and Dr) be a dimension function on P. Then D = tplP, where tp is as in Proposition 7.1.2.

Proof. First, we claim that: if p(E P) is infinite, then D(p) = +00. In fact, P is properly infinite. By Theorem 6.4.4, we can write p = En Pn, where PnPm = On,mPn,

Pn

J",J

P,

Vn, m.

Clearly, we have a unitary element Unm of M such that UnmPn u~m = Pm' Vn, m. Thus D(Pn) = D(Pm),\In, m. Of course, D(Pn) > 0, \In. Therefore, D(p)

+00.

From the preceding paragraph, we get D = tpJP if M is purely infinite. Now let M be semi-finite, and Po be as in Definition 7.1.5. Clearly, Po is finite (otherwise, D(po) = +00, a contradiction). Pick tp as in Proposition

317

7.1.2 such that
DI(P n M q ) =
Q.E.D.

and D(p) =
Corollary 7.1.7. The dimension function is uniquely determined up to multiplication by a positive constant. References. [28], [111].

7.2. Hyperfinite type (Ill) factors Let M be a type (III) factor. Then there is unique faithful normal tracial state cp on M. Define

IIXI12 = cp(x*x) ~, Then

II . 112 is a norm on M,

\/x E M.

and

From Lemma 1.11.2, the topology generated by 11·112 is equivalent to the strong (operator) topology in the unit ball (Mh of M.

Lemma 7.2.1.

Let p be a projection of M, and a* = a E {Mh. Then there 1

is a spectral projection q of a such that a > 0, then

Proof.

IJq - pl12 < gila - pili.

Moreover, if

Let c E (O, 1/2), a = J~l >'de>., and

IA 2 - AI > e a)q2112 < IIa2- al12

When>. (j. [-c,c] U (1 - e, 1], we have

!cllq2112 < II (a2 -

e 2 > e/2. Then

< lI(a - p)all2 + IIp(a - p)1I2 + lip - all2 < 311p - a112'

318

i.e.,

I!q2!12 <

~llp

- all2'

!Iaqlll < e, IJaq - qll < e, so

On the other hand,

6

lI a - ql12 < Ilaq - ql12 + IIaqll12 + IIaq21!2 < 2c + -lip - a1l2' e 1 1 1

If

Iia - pili <

2' put e

= Iia - pili, then 1

lip - ql[2 < IJa - P1l2 + Iia - qll2 < 911 a ": pll?· If

lla -

pll~/2 > 1/2, then we have immediately

Ilq - pl12 < IIql12 + Iia - plb + lI al12 < 2 + (11a112 + IlpI12)1/21Ia - pll~/2 < Iia - pll~/2( 4 + (11a1l2 + IlpI12P/2) < 911a _ pll~/2. Now let a > 0, and keep above notations. Then J

1

1

IIa 2q - qll < s, IIa 2 ql li < c2 • From

IIq21!2 <

~lla

- pl12

(see preceding paragraph), we have that

1

1

Ija 2 - qlb < IIa 2 q-

qll2

1

1

+ IIa 2qll12 + lIa 2q2112

+ c1/ 2 + IIq2112 < C + c l / 2 + ~lla - pp112. If Iia - pll~/2 < 1/2, put e = Iia - pll~/2, then Ila 1/2 - ql12 < 711a - pll~/2 + Iia _ pll~/4 < 611a _ pll~/4. Since I[q - pl12 < 911 a - pll~/2 (see preceding paragraph), it follows Ila 1/ 2 - pll < lI al /2 - ql12 + lip - qll2 <

e

< 611 a-

1

that

1

pill + 911a - pll?

1

If

lIa - pili > 1/2, then we have immediately Q.E.D.

319

Lemma 7.2.2. Let p, q be two projections of M. Then there is a partial isometry w of M such that 1

w·w < p,

unu"

<

q,

and

Ilw - pl12 < 1411p - qJlj.

Let qp = wb be the polar decomposition of qp. Then 0 < b < 1, w*w < p, ww· < q. Since Proof.

it follows from Lemma 7.2.1 that

Noticing that wp = w, we have

I[w - pl12 < IIw - qpl12 + Ilqp - pl12 Ijw(p <

b) 112

+ II (q - p)p112

lip - bl1 2 + Ilq -

p1l2

1

<

1311q - pllj + Ilq - p112' 1

If

Ilq - pll2 < 1, then Ilw - pl12 < 1411q - pill·

If

Ilq - pl12 >

1, then we have

immediately 1

Ilw - pl12 < 2 < 1411q - pll;· Q.E.D. Lemma 7.2.3. Let u be a unitary element of M, w be a partial isometry of M, and uw·w = w. Then Ilu - wlli < 211 w - 1112. Proof.

Since (u - w)( u - w)'"

lIu -

wll~ =

=1-

ww*, it follows that

cp(l - ww*)

< Icp(l - w)j + Icp(w(l - w*))1 <

111 - wl12 + 111 - w"'112

=

211 w-

1112. Q.E.D.

320

Lemma 7.2.4. Let p, q be projections of M, and p "'" q. Then there is a unitary element u of M such that

lIu -

q:;:= upu*,

Proof.

IJ12

1

3611p -

<

qll~.

By Lemma 7.2.2, there is a partial isometry W·W

< p,

WW·

<

q,

and

W

of M such that

Ilw - pl12 < 141lq -

1

pll~.

Since M is finite and p "'" q, by Proposition 6.3.2 there is v E M such that v*v = p - w*w, vv· = q - ww"'. Again from Lemma 7.2.2, there is a partial isometry WI of M such that

and

1

(1 - p) 112 < 1411p - qll:· By Proposition 6.3.2, (1 - p) '" (1 - q). Thus we can pick IIWI -

Now let u = W + V q = upu": Notice that

+ WI + VI'

VI

E M such that

Then u is a unitary element of M, and

1

< 2S11p - qll: and u(w

+ wt}*(w + wd

Ilw + WI -

= w

+ WI'

By Lemma 7.2.3, it is clear that 1

1

ul12 < v2ll w+ WI - llli < slip - qlll·

Thus, 1

< slip - qlll + 2sllp If

lip - qll2 <

1

1, then

lIu - 1112 < 3611p - qlll·

If

1

qll~

lip - qlJ2

> 1, then we have

immediately 1

Ilu - 1112 < 2 < 3611p - qlll· Q.E.D. In the following Lemmas 7.2.5-7.2.S, M possess the following property:

321

(*) For any elements

of M and e > 0, there exists a finite dimensional * subalgebra B of M and elements bb"" bm E B such that a},"', am

II ai -

b, 112

< s,

 0, there is a type (12 ,, ) subfactor N of M (n sufficiently large) and b},· .. , bm E N such that

II ai -

bi 112 <

e,

1

m.

First for c/2, there is a finite dimensional • , Cm E A such that

Proof. Cl,"

Ilai - cil12 < c/2,

1

* subalgebra A

of M, and

< i < m.

We may assume 1 E A, where 1 is the identity of M. By Section 2.13, there is an orthogonal finite set {Zi} of central projections of A with 2:i Zi = 1 such that Ai = AZi is a finite dimensional factor, Vi. Suppose that {p~.i)}; is an orthogonal set of minimal projections of Ai with 2:; pY) = Zi, Vi. Clearly, i) p)il ,... p1 (relative to Ai), vi, k, Thus, we have {W;i)} C Ai such that W~i)

= p~i),

W;i)* W;i) = p~i),

wY)wy)*

= p~.i),

Vj.

Then {w)i)wii)*h,k is a matrix unit of Ai, Vi. Further, {e}2 = wli)wii)*ji,j, k} is a basis of A. Now it suffices to show that: for enough small 8 > 0(8 depends on e and Cll ••. , cm), there exists a type (1 2,, ) subfactor N of M(n sufficiently i large) and {vJ ) } C N such that

IlwJ(i ) -

v(i) I2 J

< s: _ u,

w·· v'l,).

Pick sufficiently large n, such that 2- n < 8 2 and 2- n <
Lqt») k

Now let N be a type (I 2tl ) subfactor of M such that {wy) qii) Ii, j, k} eN, and define v~i) = W)i) 2:k qii), Vi,j. Then (i) ( wi -

(i») * ({i)

v;

WJ

-

(i») -_ PI(i) -

vi

'"

(i)

L.- qk k

322

and

II w ji) - vY)II~

=

cp(p~i) - I:qii)) < 2- n < 0 2 ,

Vi,j.

k

Q.E.D.

Lemma 7.2.6. For any al,"', am E M, any projection p E M and e > 0, if cp(p) = 2- n , then there exists a type (I 2 r) subfactor N of M (where r > n), bb ... ,b m EN, and a projection q E N such that

By Lemma 7.2.5, we can find a type ((I 2r) subfactor N of M and bb ... ,b m + 1 E N such that Proof.

where 0(> 0) will be determined later, and b:n+l = bm+ 1 • Let b = 2bm+ 1 {1

Clearly, bEN, Ilbll < 1. Since p = 2p(1

2l( b -

+ b:n+l)-I.

+ p)-t,

it follows that

p) = (1 + b2m+ 1 )~1 (bm+ 1 - p)(1 + p) -1

b - bm+t}p. + 4"(p

Thus, lib - pl12 < ~o. By Lemma 7.2.1, there is a spectral projection qt of b such that Then,

Icp(qd - 2- nl = Icp(p -

qdl < lip -

1

q1112 < 1502".

Now pick a projection q of N such that cp(q) = 2- n and either q > qt or q < qt. Then Ilq - qtll~ = (cp(qt) - 2- nl < 150 4, and

If take 0 > 0 is such that 8 < conclusion.

€

and

ISo! + v'15o i <

s, then we can get the Q.E.D.

Lemma 7.2.7. Let at,'" am E M and p be a projection of M with cp(p) = n 2- , and pa, = aiP = ai, 1 0 there is a type (12 r ) subfactor N of M with r > n, and b},···, bm E N such that

323

Moreover, if pEL, where L is a type (I2n) subfactor of M, then we can choose the above N :> L. Proof. From Lemma 7.2.6, there is a type (I2 r) subfactor A of M, where r > n, and cr.: .. ,C m , q E A, where q is a projection, such that

where 6(> 0) will be determined later. Then p '" q. By Lemma 7.2.4, we have a unit ary element u of M such that

Let N = u* Au, b, = PU*CiUP, 1 < i < m, Then N is also a type (I2 r ) subfactor of M,b},"',bm,p E N,pb i = biP = bi,l 0) such that

6

+ 728 1/8 1~~~Jlaill < c.

Now let pEL, where L is a type (I2n) subfactor of M. Suppose that {PI = P,P2,'" ,P2n } is an orthogonal set of minimal projections of L. By Theorem 1.5.6, M is spatially * isomorphic to Mp(g)B(K), where dimK = 2n • This spatial * isomorphism also maps L to Lp(g)B(K) = (c1 pH(g)B(K) , where H is the action space of M. From the preceding paragraph, there is a type (I2 r )( r > n) subfactor A of M with pEA, and bI,"', bm E A such that IlUi - hi 112 < s, pb; = biP = b,l < i < rn, Clearly, p, bi-: ", b-« E A p • Since fP{p) = 2- n , A p should be * isomorphic to a matrix algebra of order zr-n. Let N = ~-l(Ap(g)B(K)), where is the above spatial * isomorphism from M onto Mp(g)B(K). Clearly, LeN, and P, b1 , " ' , bm E N, and N is type (I2r).

Q.E.D. Let L be a type (12 n) subfactor of M, a l l " ' , am E M, and e > O. Then there is a type (12 r) subfactor N of M, and bv.: .. ,bm E N such that r > n, LeN, II ai - b, 112 < c, 1 < i < m

Lemma 7.2.8.

324

Suppose that {Pi I 1 < i < 2 n } is an orthogonal set of minimal projections of L, and {Wj} c L such that Proof.

WI

= PI,

WiWj

= Ph

= Ph Vj.

WjW;

Let P = PI, aijk = W;akWj. Then paiik = aijkP = aijk, VI < i,j < 2n , 1 < k < m. From Lemma 7.2.7, there is a type (I2 r ) subfactor N of M with r > n, and bii k E N such that

LeN, pbii k = biikP = bi j b

Vi,j, k, where

s > 0 and 22nS < C. bk

L

=

Haijk - bijk 112

< b,

Put 1

WibijkWf'

< k < m.

1$i ,j:52 n

Clearly, bh

...

,bm E N. Notice that

L PiakPi = L akwi w ; .,' i,i L wiaiikw;, 1 < k < m.

ak =

Wi Wi·

i,i

Q.E.D. Proposition 7.2.9. Let M be a countably generated type (lId factor. If for any at, ... , am E M and c > 0, there is a finite dimensional * subalgebra B of M and bh · . " bm E B such that Ila; - b, 112 < c,l < i < m, then we have an increasing sequence {Mn } of subfactors of M such that: M n is type 12 n ,Vn, and UnMn is a(M, M.)-dense in M. Let {an} be a generated subset of M.

Proof.

construct

M r 1 C ... C

u.; C

...

c

By Lemma 7.2.8, we can

M,

where for each k, M r" is a type (I2rk) subfactor of M, and also there exists b(k) ••. b(k) E M such that I'

'k

ric

(k)

II bi

-

ai 112

1

< k'

Clearly, UkMrlc is a(M,M.)-dense in M. {Mrlc}, we can get the conclusion.

1

< i < k.

Further, making a refinement of

Q.E.D.

Definition 7.2.10. A VN algebra M is said to be hyperfinite, if there is a sequence {Pn} of positive integers and 1 E M P1 C ... C M p n C ... c M, where M p n is a type (Ip n ) subfactor of M, Vn, such that UnMp n is a(M, M.)-dense in M.

325

From Proposition 3.8.3, it must be PnlPn+b Vn.

Definition 7.2.11. A VN algebra M is said to be approximately finitedimensi- onal, if there is an increasing sequence {An} of finite dimensional * sub algebras of M such that UnA n is a(M, M.)-dense in M. Theorem 7.2.12. Let M be a type (lId factor. Then the following statements are equivalent: 1) M is hyperfinite; 2) M is approximately finite-dimensional; 3) M is countably generated,and for any aIt ... ,am E M and e > 0, there exists a finite dimensional * subalgebra B of M and bt , ' " , bm E B such that lIai - bi l12 < e,l 0, there exists a subfactor N of M and bt , " ' , bm E N such that llai - bi ll 2 < e,1 < i< m. Proof. It is clear that 1) implies 2), 2) implies 3), and 4) implies 3). From Lemma 7.2.5, 3) implies 4) obviously. Moreover, 3) implies 1) immediately from Proposition 7.2.9. Q.E.D.

Lemma 7.2.13. Let A be a (UHF) C·-algebra. Then there exists unique tradal state cP on A, i.e., ip is a state on A and cp(ab) = cp(ba),Va,b EA. Proof. By Proposition 3.8.3. A = ao - ®:=t M m " . For each n, there is unique tradal state CPn on M m". Therefore, ®nCPn is the unique tradal state on A.

Q.E.D. Theorem 7.2.14.

All hyperfinite type (Ill) factors are

* isomorphic.

Proof. Let M, be a hyperfinite type (lId factor, CPi be the unique faithful normal tradal state on Mi,and {1l"i,Hi , ei } be the faithful cyclic W*representation of M generated by CPi, i = 1,2. Then 1l"i(Mi) is also a hyperfinite type (IIIl factor on Hi, i = 1,2. Let A be a (UHF) C*-algebra of type {2 n } . From Proposition 7.2.9 and Theorem 7.2.12, there is a * isomorphism fl>i from A into 1l"i(Mi) such that CPi(A) is a(M,M.)-dense in 1l"i(Mi ) , i = 1,2. Thus, (fI>i(')Ci, Ci) is a tradal state on A, i = 1,2. By Lemma 7.2.13, (fI>t(a)6,6)

= (cI>2(a)6, 6),

Va E A.

Let ufl>l(a) 6 = fl>2(a) 6, Va E A. Then u can be uniquely extended to a unitary operator from H t onto H 2, still denoted by u. Clearly, ufl>t(a)u· =

326 ~2(a), Va E

A. Therefore, U7rt{MdU* = 7r2(M2),

and M 1 is

* isomorphic

to M 2 •

Q.E.D.

Proposition 7.2.15. Let M be a finite VN algebra on a Hilbert space H. If M is also hyperfinite, then M is a factor. Proof. Let z be a central projection of M, and z C,71 E H with Ilell = 117111 = 1 such that

ze=c,

i-

0, 1. Then there exist

Z71=O.

Since M is hyperfinite, there is a (U H F)C*-algebra A C M with 1 E A, and A is a(M, M.)-dense in M. From Proposition 6.3.14, we have the central valued trace T: M ~ Z = Mn M'. Then (T(·)c,c) and (T(')TJ,TJ) are two tradal states on A. By Lemma 7.2.13, (T(a)c, c) = (T(a)TJ,TJ), Va E A. Further, this equality holds on whole M. In particular,

1 = (zc,c) = (T(z)e, e) = (T(z)71,TJ) (ZTJ,71) = 0, a contradiction. Therefore, M is a factor.

Q.E.D.

Proposition 7.2.16. Let M he a hyperfinite type (lId factor,and {Pn} be any sequence of positive integers with PnlPn+b \in, and Pn - t 00. Then there exists an increasing sequence {Mp,j of subfactors of M, where M pn is type (IpJ, Vn, such that UnMp " is a(M, M.)-dense in M. Proof.

From Proposition 7.1.3, we can pick 1 E Nt C ...

c n;

C ... C

where N n is a type (I p ,.) subfactor of M, of UnNn. Clearly, N c M, and N is also pn - t 00, N is also a hyperfinite type (lId have a * isomorphism from N onto M. {Mp ,. } is what we want to find.

M,

Vn. Let N be the weak closure finite. By Proposition 7.2.15 and factor. From Theorem 7.2.14, we Now let M pn = (Nn ) , Vn. Then Q.E.D.

Notes. Contrary to the W· -case, there are uncountably many non-isomorphic (UHF) C·-algebras (see Chapter 15). References. [491, [110], [113], [196].

327

7 .3~ Construction of factors of type (II) and type (III) Definition 7.3.1. (M, G, a) is called a dynamical system, if M is a VN algebra, G is a discrete group, and a is a (group) homorphism from G into Aut(M), where Aut(M) is the group of all * automorphisms of M. In Chapter 16, we shall study general W*- and C*-dynamical systems. For the aim of this section, Definition 7.3.1 is enough. Now let (M, G, a) be a dynamical system, and H be the action space of M. Consider Hilbert space H 0[2 (G), and define

= ag-da)~(g),

(1r(a)~)(g)

Va E M,

g,

h E G, and

~(.) E

H

@

(.\(h)~)(g)

= ~(h-lg),

[2(G).

Proposition 7.3.2. {1r,H @ [2(G)} is a faithful W*-representation of M, {A, H ® l2(G)} is a unitary representation of G, and

.\(g)1r(a).\(g)* = 7r(ag{a)), Proof. Clearly, (weakly). Since

1r

is faithful.

1(1r(a,)e, e)! =

Va E M,

9 E G.

Let a net {c.} C M,lIadl < 1 and a, ~ 0

I L(a9-l(a,)€(g), ~(g))1 gEG

<

I:: I(ag-l(a,)€(g),e(g))! + L Ile(g)11 2 , g~F

gEF

Ve E H ® l2(G), where F is any finite subset of G, it follows that 1r(al) ~ O(weakly). Thus 1r is also a W*-representation of M. Moreover, we can check

the equality:

.\(g)1r{a).\(hf = 1r(ag(a)),

Va E M,

9E G

directly.

Q.E.D.

Definition 7.3.3. The VN algebra on H®[2(G) generated by {1r(a),.\(g)la E M, 9 E G} is called the crossed product of dynamical system (M, G, a), denoted by M X a G, i.e.,

M x , G = {1r(a), .\(g) la E M, 9 E G}". Now let

if =

H ® [2(G), and write

Ii =

L $H

g,

9EG

n, =

H,

Vg E G.

328

Ji onto

Let Pg be the projection from matrix representation

Hg, Vg E G. Then any x E B(il) has a

x = (Xg,h) g,hEG, where Xg,h = P9xPh E B(H)), Vg, h E G. For any a E M and kEG, it is easy to see that

In the following, we assume that

ag(a)

= ugau;,

Va EM,

g E G,

where 9 --+ ug(Vg E G) is a unitary representation of G on H, and ugMu; = M,Vg E G.

Lemma 7.3.4. such that

For any x E M PgXP~ =

Xa

G, there is unique function b : G

U;bgh-1U g,

M

vo, h E G.

H let ep(x) = be, where e is the unit of G, then linear map from M X a G to M. Proof.

--+

is a a - a continuous positive

For a E M, kEG, since pg1r(a)A(k)p'h = Dh,k-1gU;aug, we have pg1r(a)A(k)p~ =

where

b = g

Generally, for

U;bgh-1U g,

{a,0,

if g if g

Vg,h E G,

= k,

-# k.

L 1r(a.).\(ki ) , where ai E M, k. E G, and k, -# k

j ,

Vi

-# i,

let

i

b = {ai, 0,

9

if g if g

= ki for

-# k i , Vi.

some i,

Then

Since

{L 1r( ail .\(ki) I ai E M, k, E G}

is a-dense in M

Xa

G by Proposition

i

7.3.2, thus for any x E M xQ! G, there is b : G --+ M such that u;bgh-l Ug, vo, h E G. Notice that UgPg 1r(ai)A(ki)p~U; = L:Dki,9h-1ai

L: •

•

PgXP~

=

329

in M

Xa

i

G, we can see that

Vx E M X a G and gl h 11 = g2 h;l. Therefore, for each x E M X a G the function b : G -+ M is unique. Since q.(x) = Pexp;(Vx E M X a G), it follows that q. is a-a continuous. Moreover, if x = (u;bgh-lu 9 ) EM x a G, then q.(xx*) = bgb;. Therefore, •

L

gEG

is positive.

Q.E.D.

Lemma 7.3.5. Let rp be a faithful semi-finite normal trace on M+. If rp is G-invariant ,Le. rp(ag(a)) = rp(a), Va E M+,g E G, then t/J = rp 0 ~ is a faithful semi-finite normal trace on (M X a G)+, and ip = t/J 0 n . Moreover, t/J is finite if and only if ip is finite. Proof.

Let x = (u;bgh-lu 9 ) E M

q.(xx*) =

Xa

L bgb;, gEG

G. Then

q.(x*x) =

L: u;b;bgu g. 9EG

Thus, t/J = rp 0 q. is a trace on (M X a G) +. Clearly. t/J is normal since q. is a-a continuous. If t/J(xx*) = 0, since ip is faithful and q. is positive, then q.(xx*) = 0, i.e., bg = 0, Vg E G, and x = 0. Hence, t/J is faithful. The equality rp = t/J 01r is obvious. Since rp is semi-finite, it follows from Proposition 6.5.4 that there is an increasing net {all of M+ such that sup, a, = 1 and cp(a,) < 00, VI. Then {1r(a,)} is also an increasing net of (M X a G)+,sup/1r(a,) = 1, and t/J(1r(a,)) = ~(a,) < 00, VI. For any ¥ x E (M X a G)+, there is an index 10 such that x!7I"(a)x! ¥ 0, where a = a, o • Then by Proposition 6.5.2,

°

Hence, t/J is semi-finite also. Finally, from t/J = ip 0 q. and

ip

=

t/J 0 1r, t/J is finite

¢=:>

rp is finite.

Q.E.D.

Lemma 7.3.6. Suppose that M is abelian, and 1r(M) is maximal commutative in M X a G. Then M X a G is semi-finite if and only if there exists a G-invariant faithful semi-finite normal trace on M+. Proof. The sufficiency is immediate from Lemma 7.3.5 and Theorem 6.5.8. Now let M x , G be semi-finite. Then there is a faithful semi-finite normal trace t/J on (M X a G)+. Let rp = t/J 0 n . It is easy that cp is a faithful normal

330

trace on M+. By Proposition 7.3.2 and

t/J is a trace, we have

= t/J(1r(ag(a))) = t/J(,X(g)1r(a).\(g)*) t/J(1r(a)) =
Va E M+, g E G, i.e,
(x)) E K:r; w, where K; = CO{1r(U)*X1T(U) I U is a unitary element of M}, and K z W is the weak closure of K:r;. In fact, suppose that U is a unitary element of M, and x = (U;bgh-lUg) is as in Lemma 7.3.4. Then

Further,

PgYP;

=

u;(x)ug,

v E K:r; ".

Vg E G,

Since K z W is a compact convex subset of (B(il) , weak top.) and M is abelian, it follows from the Kakutani-Markov fixed point theorem (see [31] ) that there is Xo E K:: such that Xo = 7r(U)*X01T(U) for any unitary element U of M. But 1r(M) is maximal commutative in M X a G, so Xo E 1T(M), i.e. Xo = 1r(a) for some a E M. Since pgxoP; = u;(x)u g = l"X g- l ( (x)), Vg E G, then a = (x), i.e. 7r(~(x)) = Xo E K z w. Now we prove that ep is semi-finite. Let 0 -# a E M+. Then 1r(a) is also a non-zero positive element of M x , G. Thus there is 0 -=I x E (M X a G)+ with x < a such that t/J(x) < 00. Since M is abelian, it follows that

o < Y < 1T(a), vv E K z ". In particular, 0 < 1r((x)) < 1T(a). Hence 0 < (x) < a. ~ is faithful on (M Xa G)+ (see the proof of Lemma 7.3.4). So (x) -# o. Now it suffices to show that (x)) < 00. Put 1r((x)} = xo. Then there is a net {z.] C K; such that Xl --t Xo (weakly). By Proposition 6.5.4 and since t/J is semi-finite, there exists an increasing net {Yt} C (M x , G)+ with SUPtYt = 1, and t/J(Yt) < oo,Vt. By Proposition 6.5.2, 1

1

t/J(YtXI) = t/J(xlYtXl) < t/J(XI) =t/J(x). From Proposition 6.5.3, t/J(YtXo) =

!iF t/J(YtXI)

< t/J( X). Further, since 1/; is

normal, it follows from Proposition 6.5.2 that

cp((X))

=

1/;(xo) = limt/J(x~/2YtX~/2) t

lim t/J(YtXo) t

<

00.

Q.E.D.

331

Lemma 7.3.7. If M is maximal commutative on H, and M n MUg = {o}, Vg i- e, then 1r(M) is maximal commutative in M X a G.

Proof. Let x = (U;bgh-lUg) E (M X a G) n 1r(M)'. From x1r(a) = 1r(a)x, we get bguga = abgu g, Va E M, 9 E G. Since M is maximal commutative on H, it follows that bgu g E M n MUg, Vg E G. By the assumption, bg = 0, Vg i- e. Therefore, x = 1r((x)) E 1r(M). Q.E.D. Lemma 7.3.8.

Suppose that 1r(M) is maximal commutative, and

{a E M

I ag(a) = a,Vg E G} = (clH·

Then M x , G is a factor.

Proof. Let x be a central element of M X a G. In particular, x1r(b) = 1r(b)x, Vb E M. Thus, z = 1r(a) for some a E M. By Lemma 7.3.4, it is easy to see

u(k) = (Ok,gh-1Uk) E (M x a G)', Consequently, u(k)1r(a) = 1r{a)u(k). Hence, Ukau~

= a,

Vk E G.

Vk E G.

By the assumption, a = AIH for some A E

(E.

Therefore, M x , G is a factor. Q.E.D.

Definition 7.3.9. (G, 0, Il) is called a group measure space, if 0 is a locally compact Hausdorff space satisfying the second countability axiom, Il is a regular Borel measure on 0, G is countable discrete group of homeomorphisms on 0, and let Il be quasi-invariant under G, i.e., /1-g -< u; Vg E G, where dJ.Lg(·) = dIL(g-l.), and the action corresponding to 9 on t is gt, vs E G, tEO. Clearly, J.Lg ~ IL, Vg E G. Then there exists a measurable function T g ( ' ) on such that

°

Vg E G. Since G is countable, we may assume that

Definition 7.3.10. Let (G, 0, IL) be a group measure space. 1) (G, 0, IL) is said to be free, if for each 9 E G with 9 =I e, we have /1-( {t E 0

I gt = t}) = 0;

332

2) (G,O,J.L) is said to be ergodic, if a Borel subset E of

= 0,

J.L((E U gE)\(E n gEl)

° satisfies:

\/g E G,

then we have either J.L(E) = 0 or J.L(O\E) = OJ 3) (G,O,J.L) is said to be measurable, if there exists a a-finite measure v on all Borel subsets of such that v J.L and v is G-invariant, i.e., dv(g·) = dv(.), \/g E G; 4) (G,O,J.L) is said to be non-measurable, if it is not measurable.

°

r-.J

Now let (G,O,J.L) be a group measure space. Let

H = L 2(O,J.L),

M = {m,

If

E LOO(O,J.L)},

where m,. = f·, \/f E Loo(O,J.L), \/. E L2(O,J.L). From Theorem 5.3.13, the multiplication algebra M is a maximal commutative VN algebra on H = L 2(O,J.L). For each 9 E G, define

(ugf)(·) = Tg(.)1/2 f(g-l.), Clearly, 9

---+

\/f E L 2(O,J.L)

= H.

u g is a unitary representation of G on H, and

u;m/ug = mig,

\/f E LOO(O,J.L),

9 E G,

where fg(') = f(g·). Let ag(mf) = ugm/u;, \/f E LOO(O,J.L),g E G. Then ,..., (M, G, a) is a dynamical system, and there is a VN algebra M X a G on H =

H e l2(G).

Lemma 7.3.11. If (G, 0, J.L) is free and ergodic, then 7r(M) is maximal commutative in M X a G, {a E M 1 Qg(a) = a, \/g E G} = cI:1 H , and M X a G is a factor.

Proof.

First, we claim that M

n MUg

= {O},

\/g

EG

and

9

i- e.

In fact, let 9 E G with 9 i- e, and Fg = {t E 0lgt = t}. Clearly, Fg is by (O\Fg ) , we may assume that a J.L-zero closed subset of 0. Replacing Fg = 0. Now let mh = mhug E MnMu g, where fl' f2 E LOO(O,J.L}, and define E = {t E 0Jh(t) #- O}. For each tEO, since gt #- t and 9 is a homeomorphism of 0, there is an open neighborhood Vt of t such that Vt n gVt = 0. Clearly, {vt I tEO} is an open cover of 0. But satisfies the second countability axiom, so admits a countable open cover {Vn } such that Vn n gVn = 0, \/n. If J.L( E) > 0, then we have J.L(V n E) > 0 for some n, where V = Vn • Further, pick a Borel subset F of n with F c V n E and 0 < J.L(F) < 00. From mhXF = mhUgXF, we get

°

°

°

ft{t)XF(t) = f2(t)r g (t )1/2XF(g-lt), a.e.u,

333

From F n g F = 0 and FeE, the above equality does not hold at each t E F. This is a contradiction since /-L(F) > O. Therefore, /-L(E) = 0, i.e., 11 = 0, and MnMu g = {O}. Now from Lemma 7.3.7, 1T"(M) is maximal commutation in M X a G. Let I E LOO(O, /-L) be such that u;m/u g = m/, Vg E G. Then I(gt) I(t), a.e.u, Vg E G. We may assume that I is real. If I is not a constant function, then there are real number TI, T2 with rl < T2 such that /-L( E) > 0 and /-L(O\E) > 0, where E = {t E I T1 < I(t) < T2}' On the other hand, since I(t) = l(gt),a.e'/-L,Vg E G, and G is countable, it follows that

°

/-L((E u gE)\(E n gE))

= 0,

Vg E G.

Then we get either /-L(E) = 0 or J.l(O\E) = 0 since (G, 0, /-L) is ergodic, a contradiction. Therefore, I is a constant function, i.e.,

{a E M

I ag(a)

= a, Vg E G}

= (J:I H • Q.E.D.

F'inarrly, by Lemma 7.3.8, M x a G is a factor.

Lemma 7.3.12. Let (G, O,/-L) be a free and ergodic group measure space, and v be a G-invariant a-finite measure on all Borel subsets of 0 with v ,..., /-L and v( {t}) = 0, Vt E G. 1) If v(O) < 00, then M X a G is a type (Il.) factor. 2) If v(O) = +00, then M X a G is a type (IIoo ) factor.

Proof.

Define

Then ep is faithful on M+ since v ,..., u, Let {mit} be a bounded increasing net of M+, and mf = sUPI mit. By Theorem 5.3.13, II - ? I with respect to w*-topology in LOO(O,/-L) or LOO(O,v). Since v is a-finite, we can write o = UnEn, where {En} is an increasing sequence of Borel subsets of 0, and v(En ) < 00, Vn. Thus XE,. E L 1(0, v) and

f IIXE,. du Further, sup I

-?

fiXE .. du,

f 11dv f s~p =

Vn.

[idu,

i.e., ip is normal. The semi-finiteness of ep is obvious from the a-finiteness of u, Moreover, since v is G-invariant, it follows that

ep(u;m/u g )

=

JI(gt)dv(t)

= f I(t)dv(t)

= ep(m/),

334

\/m, E M+,g E G, i.e., cP is also G-invariant. Now by Lemma 7.3.6 and 7.3.11, M X a G is a semi-finite factor. If v(o) < 00, then cp is finite, and M X a G is also a finite factor from Lemma 7.3.5. If v(o) = +00, then cp is not finite, and M X a G is an infinite factor from Lemma 7.3.5 and Proposition 7.1.2. Now it suffices to show that M x a G is ciontinuous. Let p be any non-zero projection of M with cp(p) < 00. Then by Lemma 7.3.5, t/J = cpoib is a faithful semi-finite normal trace on (M x a G) + and cp = t/J 0 7T". Thus t/J (7T" (p)) < 00, and 7T"(p) is a non-zero finite projection of MXaG by Proposition 7.1.2. If MxaG is not continuous, we may assume that MxaG = B(K), where K is some Hilbert space. Then dim7T"(p)K < 00, and M contains a nonzero minimal projection « p). This contradicts the assumption: v( {t}) = 0, \/t E 0. Therefore, M X a G is continuous. Q.E.D. Lemma 7.3.13. Let (G,O,Jl) be free and ergodic. measurable, then M X a G is a type (III) factor.

If (G,O,Jl) is non-

Proof.

If M X a G is semi-finite, then by Lemmas 7.3.11 and 7.3.6, there is a G-invariant faithful semi-finite normal trace cp on M+. For any Borel subset E of 0, define

Then v is a measure on all Borel subsets of 0. Since tp is faithful, it follows that II '" u, From the G-invariance of cp, v is also G-invariant. By the Zorn lemma and the semi-finiteness of cp, there is an orthogonal family {PI},EA of projections of M such that EIEA PI = 1 and cp(PI) < 00, \/1. Since H = L 2(0, Jl) is separable, A is countable. Suppose that PI = m XE / , where E l is a Borel subset of 0, \/1. Then v{EI ) = cp(PI) < 00, \/l, and

v(O\ UIEA E,)

= cp(l -

L

pz)

= O.

lEA

Thus, II is a-finite. From Definition 7.3.10, (G, 0, J.t) is measurable, a contradiction. Therefore, M x a G is not semi-finite, and is a type (III) factor. Q.E.D.

Lemma 7.3.14.

Let (G, 0, J.t) be a group measure space, and Go

= {g E GITg(t)

Then Go is a subgroup of G. (G, 0, J.t) is non-measurable.

Proof. Go

¥-

= 1, a.e.J.t}

If (Go,O,Jl) is ergodic, and Go =I- G, then

Clearly, Go is a subgroup of G. Now let (Go, 0, J.t) is ergodic, and G. Suppose that v is a a-finite measure on all Borel subsets of with

°

335

v "" J..L, and v is G-invariant. For any 9 E Go, since J..Lg = J..L "" v = vg , it follows that

~(t)dv(t) = dJ..L(t)

= dJ..L(g-lt)

~(g-lt)dll(9-lt)

_

=

~(g-lt)dv(t).

Thus, ~(t) = ~(g-lt), Vg E Go. Now (Go,O,J..L) is ergodic, so ~(t) = constant (a.e.J.l) by a similar discussion of Lemma 7.3.11. Further, J..L is also G-invariant, i.e., G = Go, a contradiction. Therefore, (G,O,J..L) is nonmeasurable. Q.E.D. From above discussions, we have the following.

Theorem 7.3.15. Let (G, O,J..L) be a free and ergodic group measure space, and H = L2(0,J..L), M = {mf I f E LOO(O,J.l)}

(ugf)(t) = T g(t )I/2f(g-lt), Qg{m,) = ugmfu;,

Vf E H,g E G,

Vf E LOO(O,J.l),g E G,

where rg ( · ) = (dJ..Lg/dJ..L)(') and dJ..Lg(·) = dJ..L(g-l.) , Vg E G. 1) If there is a a-finite G-invariant measure on all Borel subsets of 0 with v "" J..L and v({t}) = 0, Vt E 0, then M X a G is a type (lId factor when o < v(O) < 00, and M X a G is a type (II oo ) factor when v(O) = +00. 2) Let Go = {t E G I rg(t) = l,a.e.J..L} (a subgroup of G). If (Go,O,J..L) is ergodic and Go f:. G, then M X a G is a type (III) factor.

Example 1. Let = {z E (C I Izi

°= 1},J..L

be one dimensional circle group (compact group), i.e., o be the Haar measure on 0 with J..L(O) = 1,G be a countable infinite subgroup on 0, and the action a of G to 0 be the multiplication of numbers. Clearly, (G,O,J..L) is free, J..L is G-invariant, and J..L({z}) = O,Vz EO. Suppose that E is a Borel subset of such that

°

J..L((E

U

gE)\(E n gE))

= 0,

Vg E G.

Write n

where {zn

I n E LZ}

is a normalized orthogonal basis of L2(0, J..L). Then

L AnZ

n

= XE(Z) = XE(gZ)

Angnz n,

a.e.u,

n

n

Vg E G. Thus, An (G, O,J..L) is ergodic.

=L

= 0, Vn f:.

0, i.e., either J..L(E)

= ° or

J..L(O\E)

= O,and

336

Now by Theorem 7.3.15, M x a G is a type (lId factor. Let 0 = IR (a locally compact abelian group), J..L be the Haar Example 2. measure on 0, G be a countable infinite dense subgroup of 0 {for example, G = {r E IR I r is rational}), and the action a of G on 0 be the addition of numbers. Clearly, (G, O,J..L) is free, J..L is G-invariant, and Jl(O) = OO,J..L({7]}) = 0, V7] E

O. Suppose that E is a Borel subset of 0 such that Jl({E U (E

+ 1]))\(E n (E + 1])))

= 0,

V1] E G,

i.e., u~mXEutI = m XE' V1] E G, where 7] ---+ uti is the regular representation of o on L2(O, J..L). Since G is dense in 0, it follows that mXEu tI = utlm XE' V7] E O. Thus, we have either J..L(E) = or J..L(O\E) = 0, i.e., (G, 0, J..L) is ergodic. Now by Theorem 7.3.15, M x a G is a type (II oo ) factor.

°

Example 9. and

Let (0, Jl) be as in Example 2, G = {(p, a) I p > 0, p, a rational},

a(p, a) 7] = P1] + a,

V(p, a) E G,1] E O.

Clearly, (G,O,J-t) is free, and J..L is quasi-invariant under G. Let Go = {(I,a) I a rational}. By Example 2, (Go, O,J..L) is ergodic. Clearly, Go :f- G. Now by Theorem 7.3.15, M X a G is a type (III) factor.

Theorem 7.3.16. On a separable Hilbert space, there exist five classes of factors: type (In), (1 00 ) (lId (11 00 ) (III) factors. Type (11 00 ) factors can be indeed constructed through type (lId factors.

Proposition 7.3.17. A factor M is type (II oo ) if and only if M = N®B(H oo ), where N is a type (lId factor, and H 00 is a infinite dimensional Hilbert space.

Proof.

The sufficiency is obvious from Theorem 6.9.12. Now suppose that M is a type (II oo ) factor. Pick a non-zero finite projection P of M, and let {PI}'EA be a maximal orthogonal family of projections of M such that PI P, Vl. Then q= 1PI -< P by Proposition 6.4.5, •A = 00. Thus, lEA f'V

L

1=

L Pl + q "" L PI' lEA

lEA

Further, there exists an orthogonal family {ql }'EA of projections of M such that

Lq, = 1, q, "" P,VI.

lEA

337

Now by Theorem 1.5.6, M and dimHoo = •A = 00.

= Mp®B(Hoo ) ,

where M p is a type (lId factor, Q.E.D.

We have another method to construct type (Ih) factors. Let G be a discrete group, and g ---+ Ag , Pg be the left, right regular representations of G on 12( G) respectively, i.e.,

(Agf)(·) = f(g-l.), Vf E 12(G), g E G. Let R(G)

Lemma 7.3.18.

(pg(f))(·) = f('g),

= {A g I g E G}".

R{G) is a a-finite and finite VN algebra on 12(G).

Proof. For each g E G, let cg(k) Let e be the unit of G, and define cp{ a)

= Sg,k.

Clearly

Cg

is a unit vector of 12(G).

= (ace, ce), Va E R(G).

Then cp is a normal state on R(G). If a E R( G) satisfies ae, = 0, then

I g E G]

is dense in [2(G), so a = 0, i.e., cp is faithful. Moreover, since cp(AgAh} = cp(AhAg}, v«, h E G, it follows that cp{ab) = cp(ba) , Va, b E R(G). Thus, tp is also a trace. Now by Proposition 6.3.15, R( G) is a-finite and finite. Q.E.D. But

[cg

Definition 7.3.19. An infinite countable discrete group G is said to be of infinite conJugacy class, if for any e f::. g E G, the conjugacy class {hgh- 1 I h E G} of 9 is infinite. We often abbreviate such a group as an ICC-group. For example, the group of all finite permutations of IN = {I, 2,' ..} is an ICC-group, and the free group of two or more generators is also an ICCgroup,and etc. Proposition 7.3.20. on P{G). Proof.

Let a E R(G)

If G is an ICC-group, then R( G) is a type (lId factor

n R(G)'. Then for any g E G,

i.e., (ace)(') = (ace)(g-l. g),\/g E G. Since (ace) E 12(G) and G is ICC, it follows that (ace)(h) = 0, Vh i- e, i.e., ae, = ACe" for some A E (C. By the proof of Lemma 7.3.18, a = A. Thus, R(G) is a factor. Moreover, R(G) is infinite Q.E.D. dimensional. Now by Lemma 7.3.18, RiG) is type (lId.

338

Proposition 7.3.21. Let G be an ICC-group,and {G n } be an increasing sequence of finite subgroups of G with G = UnG n. Then R(G) is a hyperfinite type (lId factor on l2 (G). Proof. Clearly, for each n, [.x g I g E G n] is a finite dimensional * subalgebra of R(G), and Un[.x g I g E Gn] is a(M,M.)-dense in M. Now by Theorem 7.2.12 and Proposition 7.3.20, we get the conclusion. Q.E.D. Remark. Let G be the group of all finite permutations of IN = {I, 2, ... , }, and G n be the finite subgroup of all permutations of {I"", n}, Vn. Then G = UnG n. Notes. the construction of factors in the section is standard. It is called the group measure space construction (of Murry-Von Neumann).

References. 1113], 1119J, [132].

7.4 The existences of non-hyperfinite type (lId factors and non-nuclear C*-algebras Consider a discrete group G. Let eg(h) = bg,h' vs, h E G. Then {eglg E G} is an orthogonal normalized basis of [2 (G) . Suppose that g -----+ .x g, Pg are the left, right regular representations of G on [2(G) respectively, and R(G) = {.xglg E G}". By Lemma 7.3.18, R(G) is a a-finite and finite VN algebra on l2(G); cp(.) = ('ell' Cll) is a faithful normal tracial state on R(G), where e is the unit of G; and e; is a cyclic-separating vector for R(G).

Proposition 7.4.1. Define [xe, = x·ce, Vx E R(G). Then i can be uniquely extended to a conjugate linear isometry on [2 (G), still denoted by i and l

Moreover, JR(G)J = R(G)'

= {pglg

E G}".

Proof. Since cp(.) is tracial and e, is cyclic for R(G), i can be uniquely extended to a conjugate linear isometry on [2(G). Clearly, i 2 = I, and i.xgi = Pg,Vg E G. By UXce,JYce) = (Yce,Xee),Vx,y E R(G), we have U€,iTJ) =

339

(t},€),V'€,11 E l2(G). For any x' E R(G)',x E R(G),

Hence, jx'c~ = x'*c~, V'x' E R(G)'. Further, by

Vx',y',z' E R(G), we have jR(G)'j c R(G). On the other hand, jR(G)j = {pglg E G}" C R( G}'. Therefore, we obtain that jR(G)j = R(G)' = {pglg E G}". Q.E.D. For any €, 11 E [2 ( G), let

(€ * t})(g)

L

=

€(h)11(h- 1 g),

C(g)

= €(g-l),

Vg E G.

hEG

Clearly,

I(€ * 11)(9) 1<

II€*II = II€II·

1I€11·1!1111,

Proposition 7.4.2. Let

B - {b E l2( G) -

I

there is a positive constant K = K(b) } such that (b * e) E [2(G)andllb * ell < KI1cII, Ve E l2(G).

Then:

(i)

B = {b E l2(G)1 b * e E l2(G), Ve E l2(G)}

I

= {b E l2(G) there is >'(b) E B(12(G))such that }; A(b)eg = b * Cg = p;b, V'g E G

* algebra,

and b -----t >.(b) is a faithful * representation of B on l2(G), where >.(b)e = b * e,Vb E B,c E [2(G); (iii) R(G) = A(B) = {>.(b)lb E B}. Consequently, if we define Ilbll = II A(b) II, V'b E B, then B is a a-finite .and finite W* -algebra, and cp(b) = (A(b)e~, c~)(Vb E B) is a faithful normal tradal state on B; (iv) jA(b)j = p(b), where p(b)e = c * b\Vb E B,c E l2(G). And R(G)' =

(ii) B is a

p(B) = {p(b)lb E B}.

(i) Let b E l2(G)' and b * c E linear operator A(b) on [2(G) : Proof.

A(b) c

[2 (G),

= b * c,

Ve E l2( G). Then we can define a

VeE

[2 ( G).

340

We claim that A(b) is continuous. It suffices to show that A(b) is a closed operator. Suppose that {En} is sequence of l2(G) such that

En

~ O,and

A(b)En = b * en

in [2(G), where TJ E l2(G). We need to prove TJ (TJ,

= 0.

~ TJ

For any g E G,

cgl = lim(b * En' cgl n = lim n

L hEG

b(h) En(h-1g) = lim(En,cI n

= 0,

where c(·) = b(g.-l) E [2(G). Hence, TJ = 0, and A(b) is continuous. Further, we have B = {b E [2(G))b * c E [2(G), Vc E [2(G)}. Now let b E z2(G), and suppose that there is A(b) E B([2(G)) such that )'(b)cg = b * Cg, Vg E G. Clearly, such A(b) is unique, and A(b)c = b * c, Vc E [cglg E GJ. For any c E l2(G) and any finite subset F of G, let

CF =

L (c, Cglcg.

gEG

* CF)' In particular, (A(b)c)(g) = (A(b)c, cgl = lim(b * CF, Cgl = (b * c)(g) since I(b * CF)(g) - (b * c)(g) I < Ilbll·llcF - cll --+ O,Vg E G. Hence, b * c = )'(b)c E l2(G), Vc E [2(G), and s « B. (iii) For any s e B and x E R(G), we have xA(b)cs = xp;b = p;xb, Vs E G. By (i) , we get xb E Band A(xb) = xA(b), Vx E R(G), se B. In particular, x = XA(ce) = A(Xce) E A(B), Vx E R(G). Then CF ~

C

in [2(G), and A(b)c = limA(b)cF = lim(b

On the other hand, for any b E B,

A(b)pgCh

= A(b)Chg-1

= p~g-lb = pgA(b)Ch'

vs, hE G.

Hence, A(b) E {pglg E G}' = R(G), Vb E B. (ii) For any a, bE B,

A(a)A(b)Cg = A(a)p;b = p;A(a)b, Vg E G. Then by (i) we have A(a)b = a * b E B, and A(a * b) = A(a)A(b). Moreover,

(A(b)*c g , ChI

= \cg,b * ChI = (b* * Cg,Ch/'

341

\lg,h E G,b E B. Hence, A(b)*cg = b* * c g , and by (i) we get b" E B and A(b)* = A(b*),\lb E B. Therefore, B is a * algebra, and b --+ A(b) is a * representation of B on l2 (G). Moreover, if A(B) = 0 for some b E B, then 0= .:\(b)ce = b e e, = b. (iv) It is obvious. Q.E.D.

Definition 7.4.3. Let M be a finite factor, and ep(.) be the ( unique) faithful normal tracial state on M. We say that M has the property (f) , if for any X l l " ' , X m E M, and e > 0, there is a unitary element U of M such that cp(u) = 0, and Ilu*XiU - xil12 < C, 1 < .,; < m, where Ilxlli = cp(x*x), \Ix E M. Proposition 7.4.4. property (f).

If M is a hyperfinite type (IId factor, then M has the

By Proposition 7.2.10, there is an increasing sequence {Mn/n > o) of subfactors of M such that unMn is weakly dense in M, where M n is type (I2n), \In. Now for any Xl,"', X m E M, we can find nand Yll"', Ym E M n such that IIxi - Yil12 < c/2,1 < i < m. M n + 1 is * isomorphic to the tensor product of M n and a type (12 ) subfactor. Hence, there is a unitary element U of M n + 1 such that ep(u) = 0 and UYi = YiU, 1 < i < m. Further, 1(Xi 2+ Ilxi - Yill2 < s, - Yd uI1 Ilu- 1xi u - Xi 112 < Ilu-

Proof.

1 < i < m. Therefore, M has the property (f).

Q.E.D.

Proposition 7.4.5. Let G be an ICC group. If there is a non-empty subset F of G and elements 01, g2, g3 of G with following properties: (i) F U glFg11 U {e} = G, (ii) the subsets F, g2Fgii and g3Fgii are disjoint, then R( G) has no the property (f). Therefore, by Proposition 7.3.20 and 7.4.4 R(G) is a non-hyperfinite type (IId factor. Suppose that R( G) has the property (f). Then for c > 0, by Proposition 7.4.2 there is b E B such that: A(b) = U is a unitary element of R(G); cp(u) = (uce' ce) = 0; and Ilxi - u*xiul12 < e, where Xi = A(cgJ = Agi' l < i < 3. Noticing that

Proof.

we have

lIXi -

u*xi ulI2

= llu =

lib -

x; u xi l12 = IIA(b - c;i

c;i * b * cgill < e,

* b * CgJII2

1 (L + L 1 + L )lb(g)1 2 gEF 9E92F g"i gEg3Fg;1 2 = L Ib(g)1 + L 1(C:;2 * b * c: g2 )(g) 2 + L 1(C:;3 * b * cga)(g) 2 • gEF gEF gEF

1

1

1

<

<

L Ib1(g}1 2 = L l(bI(g) - b(g)) + b(g)1 2 gEF gEF L Ibdg) - b(g)1 2+ L Ib(g)1 2 + 2Re L (bdg) - b(g))b(g) gEF gEF gEF 2 c + L Ib{g)1 + 2{L Ib 1(g) - b(g)I L Ib{g)1 2)1/2 gEF gEF gEF 2 c + 2c + L Ib(g)12, Z

2

•

gEF

and

L

lbi{g} 12 >

yEF

Further,

.i.e.

L

Ib(g)12 - c 2

-

2c,

i

= 2,3.

yEF

3 L Ib(g)j2 - 4c - 2c 2 < 1 <2 L Ib(g)1 2 + 2c gEF

+ c2 ,

gEF

2

1 - 2; - 0

<

L

Ib(g)1 2 < 1 + 4: + 20

2 •

gEF

This is a contradiction if c:( > 0) is small enough. Therefore, R( G) has no the property (f). Q.E.D.

Theorem 7.4.6. There exists a non-hyperfinite type (III) factor on a separable Hilbert space.

Proof. Let G = F2 be the free group generated by two elements gl and g2, F be the subset of all reduced words of G ending with g(, n = ±1, ±2,' ... If b is a reduced word of G ending with g~ and m i- 0, then c = g11bg1 E F. Hence, F U gtFgll U {e} = G. Further, let s« = gil. Then F,gzFgi l and gsFgg l are disjoint obviously. Now by Proposition 7.4.5, the conclusion is obvious. Q.E.D.

343

Now we construct examples of non-nuclear C*-algebras. Let R be a type (III) factor on a Hilbert space H such that R' is also type (Ill)' Suppose that A, Bare C·-subalgebras of R, R' respectively, and C is the C*-algebra on H generated by A U B. Define a map q> : A 18I B -----+ C as follows: q>(L a, 18I bi) = L a,bi, i

Vai E A, b, E B.

i

L: ai 18I b, =

The map q> is well-defined. In fact, if

0, let {bj} be a basis of

i

[bili], and b;

= L Aijbj, Vi, then we have L Aiiai = 0, Vj. Hence, i

j

L tub, = L: ai L Aiibj = L bi(L A'i a, ) = 0. iii ; i Further, we say that q> is injective. In fact, if L aibi = 0, let aik = ak, Vk; bki = i

bl"

vi. then by Proposition 1.7.3 there are numbers {Aii} such that L aikAki = L Akjak k

{

L: A,kbkj = k

= 0, VJ',

k

L Aikbk

= bij

= b.; Vi.

k

Hence, = L at 18I (L Aikbk) k

i

= L(L: Aikai) 18I bk = 0. k

i

Define a(L ai 18I bi) = i

II L

a;bill, \fa; E A, bi E B.

i

Then a(·) is a C*-norm on A 18I B , and a - (A 18I B) is * isomorphic to C. Rl8IR' is still a type (Ill) factor on H 18I H. So there is a faithful normal tradal state on Rl8IR'. a()(A@B) is a C*-subalgebra of R@R'. Hence, there is a faithful tradal state on ao- (A 18I B). If we can prove that there is no faithful tradal state on a-(A 18I B), then o{) =I ao(') on A 18I B. Therefore, A and B are non-nuclear C*-algebras. If there is a non-zero projection p of C and an infinite sequence {Uj} of unitary elements of C such that {Ujpuj Ij} is pairwise orthogonal, then we claim that there is no faithful tradal state on C ( and a-(A 18I B)). In fact, let T be a faithful tradal state on C. Then by n

r(1)

> Lr(ujpuj) i=l

= nr(p), Vn

344

we get r(p) = o. This is a contradiction since p =f. 0 and r is faithful. Therefore, there is no faithful tracial state on C { and 0: - (A l&J B)). Consider G = F2 , the free group generated by two elements gl and g2. Then R = R(G) and R' = R(G)' are tyep (III) factors on H = l2(G). Let A, B be the C*-subalgebras of R, R' generated by {A,lg E G}, {p,lg E G} respectively. Since R is a factor, the C*-algebra C generated by (A U B) is irreducible on H. Let p be the projection from H = 12(G) onto [eel. 1) Let a = >"91P'l + >";lP;l +A,:aP,:a + >";:aP;:a' Clearly, a* = a E C, and Noticing that '

lIall < 4.

(aee)(g) = ee{gllggd + ee(glggll) + ee(g2 1gg2) + ee(g2gg2 1 ) = 4ce{g),

we have ap = 4p,pa = (ap)* follows that lIall = 4. 2) Let

= (4p)* = 4p,

eE H = l2(G), e(e)

= 0, and

L le(g) -

and ap

=

pa = 4p. Since p

lIeli = 1. Then for j = 1 or

=I 0, it

2 we have

e(gjgg;l) 12 > e 2 ,

gEG

where E: = 1/25. In fact, for any subset E of G, denote the norm on 12(E) by II . liE, and let p,{E) = Ilell~ = le(g)1 2 = le(g) - e(gjggjl)!2,ej = >";jP;je,j = 1,2.

L

gEE

,t;

L

gEG

Then for any subset E of G we have

t; > II e- ei II ~ , By (1) and

II ell

(1)

j = 1, 2.

= 1, for any subset E of G we get

Ille;ll~ -llell~1 = 1p,(yjEgjl) - p,(E) I

= III eillE - II erlEI· (II eillE + II elIE) <

i

(2) 2tj,

= 1,2. Replacing E by gil Egj, we have

Vi

= 1,2, and any subset E of G. Now let E be the subset of all reduced words of G = F2 beginning with gl. Then the subsets E, g2Egil, gil Eg 2 are disjoint. By p,( G) = II el1 2 = 1,

minVt(E),tt(g2Eg;1),tt(g;IEg2)} < 1/3.

(4)

345

Since c(e) = 0 and EUg 11Eg l = G\{e}, it follows that p,(E) +p,(glIEg) > p,(E U gil Egd = p,(G) = 1 . Hence, max{p,( E), p,(gll Egt}}

> 1/2.

(5)

l

Now by (4) and (3) , J1(E) < + 2t2; and by (5) and (3) , J1(E) > ~ - 2t l . Hence, 2t l < + 2t 2 , i.e., (tl + t 2 ) > 112, Therefore, for j = 1 or 2 we have

l-

l

=

tj

L

le(g} - e(gjgg;l} 12 >

gEG

~ = C. 25

3) a(1- p) < (4 - g-2}(1- p), where g- = 1/25, and a is the same as in 1). In fact, by ap = pa it suffices to show that

(ae, e) < (4 - c 2 ),

Vc E (1 - p)H and lIell = 1. Fix (e, g-e) = (pc, g-e) = O. Since

L

Ilell

= 1.

Clearly, c(e) =

le(g} - e(gjggi 1)!2 =

gEG

IIel\2 + 11,x;jP;jeI1 2 -

Ilc - A;jP;jeI1 2 2Re(,x;jP;je, e)

2[1 - Re(,x;jP;je, e)),

-

p)H and

e) = 2Re[(,x;lP;1 e, e) + (,x;2P;2C, C)]

(ac, and

eE (1 -

j = 1,2,

it follows from the conclusion 2) that 2

(ae, e)

=4 -

L L: le(g) -

C(gjggjl) 12 < 4

- g-2.

j=1 gEG

Now we prove that p E C. Let b = (4 + a}/8. Then by ap = pa = 4p, a* = a < 4 and the conclusion 3) we have bE C, bp = pb = p,O < b(1 - p) < 8(1 - p), where 8

= (8 -

g-2) /8, and c = 1/25. Hence, for any positive integer n,

b"p Further, lib" limb"EC.

-

= p,

pil = lib" -

and 0 < b"(1 - p) < 8"(1 - pl.

b"pll = Ilb"(1 - p) I < 8" ~ O. Therefore, p =

n

Moreover, {,xgp,x;lg E G = F2 } is an infinite orthogonal sequence of projections of C obviously. Therefore, there is no faithful tracial state on C, and A, B are non-nuclear C"'-algebras.

346

Theorem 7.4.7.

There exist separable non-nuclear C*-algebras.

Notes. The property (f) was introdced by F.J.Murray and J.Von Neumann. Theorem 7.4.6 is also due to them. The examples of non-nuclear C*-algebras presented here are due to M.Takesaki. References. [28], [80], {113], [171], [194).

Chapter 8 Tomita-Takesaki Theory

8.1 The KMS condition Definition 8.1.1. Let (H, ( , )) be a complex Hilbert space. Define ( , )r = Re( , ). Then H, = (H, ( , )r) is a real Hilbert space (see H as a real linear space). Suppose that K is a closed real linear subspace of H, K is said to be nondegenerate, if K n iK = {O}, and (K +iK) is dense in H. Lemma 8.1.2. Let K be a nondegenerate closed real linear subspace of H, p, q be the projections from H; onto K, iK respectively (self-adjoint on H r), a = p + q,and p - q = [b be the polar decomposition of (p - q) on Hr. Then 1) pi = iq,ip = qij 2) a is a positive linear operator on H,O < a < 2, and {O,2} are not eigenvalues of Aj 3) b is a positive linear operator on H, b = a! (2 - a)!, and 0 is not an eignevalue of b. Moreover b commutes with p, q, a and i, 4) i is a self-adjoint unitary operator on H, and i is a conjugate linear operator on H, i.e., ii = -iJ·. Moreover,

UC,TJ)=UTJ,c), and J'p

= (1 - q)i, i« = {I - p)J', [a =

VC,TJEH, (2 - ali·

Proof. 1) Let TJ E K, and iTJ = ~ + ~-l be the orthogonal decomposition with respect to H; = K ff) K 1., i.e. p(iTJ) = ~. Then -TJ = i~ + i~1. is the orthogonal decompositive with respect to H, = iK ff) (iK) 1. , and -qrJ = i~

= ip( iTJ).

348

Now if €,7] E K, then

ip(e + il1) = ip€ + ip(i7]) q(i

=

ie - q7]

e- 7]) = qi (€ + i 7]).

Since (K+iK) is dense in Hr, it follows that ip = qi. Further, pi = iq. 2) From 1), a is linear on H. Clearly, a is self-adjoint on HI'" and

Thus, a is also self-adjoint on H. Since

it follows that 0 < a < 2. If a€ = 0, then p€ = qe = 0 from above equality, i.e., e-l(K+iK) in HI'" But (K+iK) is dense in HI'" so € = 0, and 0 is not an eigenvalue of a. Let K.L be the orthogonal complement of K in HI'" Then K.L is also a nondegenerate closed real linear subspace of H. Considering K.L, we can see that 0 is not an eigenvalue of (2 - a), i.e., 2 is not an eigenvalue of a. 3) Clearly, (p_q)2 is linear on H from 1). Similar to the proof of 2), (p_q)2 is positive on H. Thus, b is a positive linear operator on H. Since (p - q)2 and p or q commute, it follows that b commutes with p, q and a. The equality b = a~ (2 - a) ~ is obvious. So 0 is not an eigenvalue of b by 2). Moreover, since (p - q) is self-adjoint on Hr, bi = jb. 4) Since (p-q) is self-adjoint on HI" and 0 is not an eigenvalue of b, it follows that j is self-adjoint and unitary on HI'" Noticing that bi = ib, (p - q)i = -i(p - q), we get Ji = -iJ·. For any €,7] E H,

U€,7])

U€, 7]),. + i (i (i e), 7]) (€,J'l1),. + i(i€,J·7]),. = U7], e). q)p = (1 - q)(p - q) = b(1 - q)j

=

Finally, from bJ"p = (p eigenvalue of b, we get ip ja = (2 - a)j.

I"

=

and 0 is not an (1 - p)j. Similarly, Jq = (1 - p)j. Further, Q.E.D.

Lemma 8.1.3. Keep the assumptions and notations of Lemma 8.1.2, and 1 let ~ = (2 - a)a- = a- 1 (2 - a). Then ~ is a (unbounded) positive invertible linear operator on H ,and for any everywhere finite measurable function f on

[0, +oo),J1(~)J

= f(~-l).

Proof. By Lemma 8.1.2, [a = (2 - a)J'. Thus J'~J ji = -ij, we obtain jf(~)J' = f(~ -1).

= ~-l.

Further, by Q.E.D.

349

Lemma 8.1.4. define

Keep the assumptions and notations of Lemma 8.1.3, and

s(e+i77)=e-i77, S+(ie1 + 77d

Ve,77EK,

= i6 -

O(s)=K+iK;

V6,771 E K1.,

77b

O(s+)

= iK1.+K1.,

where K 1. is the orthogonal complement of K in H r (it is also a nondegenerate closed real linear subspace of H). Then: 1) s and s+ are two conjugate linear closed operators on H with a dense domain; 2) s+ is the adjoint of s on H r , s is the adjoint of s+ on H; and jsj = s+; 3) s = j/::,.I/2,S+ = i/::,.-1/2 are the polar decompositions of s,s+ on H r respectively. Consequently, 0(/::,.1/2) = K+iK. 1) It is obvious. 2) Clearly, s" C the adjoint of s on Hr. If~,~' satisfy

Proof.

(e - i77, ~)r = (e + i77, ~/)r, Let 77

6 ==

= O. Then

(~

- ~') E K 1., Let

;i (~ + ~') E K 1.,771 =

l(~

- ~') E K

~ = i6

+ 77,

e = O. 1.,

Ve,77 E K.

Then i(~

+

~/) E K 1..

Thus,

and

~' = i 6

-

771'

Now we can see that s" is the adjoint of s on Hr. Moreover, since s is closed, it follows that s is also the adjoint of s" on H,.. From Lemma 8.1.2, [K

=

jpH

== (1 - q)jH = (iK).L = iK1..

Similarly, j(iK) == K1.. Thus J'sj = s". 3) If 6,771 E K.L, then P771 = O,qi6 = ip6 = 0 and as+(i6 + 77d (p - q)( i6 + 771)' Thus, as+ C P - q = jb = bJ' , Since s+ = i»i, it follows that ajs C b, i.e., [s C /::,.1/2. But J's and /::,.-1/2 are self-adjoint on H, so s = J'/::,.1/2. By Lemma 8.1.3, s+ = J'/::,.~1/2. Now from s+s = /::",ss+ = /::"-1, thus s = i/::,.I/2,s+ = j/::,.-1/2 are also the polar decompositions. Q.E.D. Lemma 8.1.5, {/::,.it I t E JR} is an one-parameter strongly continuous group of unitary operators on H, and satisfies the following: , A

it

A

it '

Ju. =u. J,

/::,.it K = K,

Vt E JR.

By Lemma 8.1.3, J'/::,.it J' = /::,.it, Vt E JR. Moreover, from ab = ba, we have /::,.itb = b/::,.it. Further, /::,.it and jb = p - q commute. Clearly, /::,.it and a = P'+ q commute. Thus, /::,.it p = paCt, i.e., aCt K = K, Vt E JR. Q.E.D. Proof.

350

Definition 8.1.6. The above operators i, ~ are called the unitary involution, the modular operator (relative to the nondegenerate closed real linear subspace K of H) respectively. They will play an important role in the theory of this chapter. Now we discuss the KMS condition. Let K be a nondegenerate closed real linear subspace of a (complex) Hilbert space H, and keep above all notations. Definition 8.1.7. An one-parameter strongly continuous group of unitary operators {Ut It E lR} on H is called satisfying the KMS condition (relative to K), if for any e,7] E K, there is a complex function f(z) which is continuous and bounded on 0 < Imz < 1 and is analytic in 0 < Imz < 1 such that

f(t) = (7], Ute),

f(t

+ i) = (Ute, 1]) = T(iJ,

TIt E lR.

Clearly, this f is unique, and is called the KMS function corresponding to

e,7]· Proposition 8.1.8. An one-parameter strongly continuous group of unitary operators {Ut I t E lR} on H satisfies the KMS condition (relative to K) if and only if for any t,» E K, there is a complex function f(z) which is continuous and bounded on 0 < Imz < 1/2 and is analytic in 0 < Imz < 1/2 such that

Proof.

(179]) .

The sufficiency is obvious by the Schwartz reflection principle (see

e,

Now let f be the KMS function corresponding to 7], and g (z) = f (z - i). Clearly, g is also a KMS function corresponding to e,7]. Thus, f = g. In particular,

Q.E.D. Definition 8.1.9. Let {Ut I t E lR} be an one-parameter strongly continuous group of unitary operators on H. E(E H) is said to be analytic (with respect to {Ut}), if there is a vector valued analytic function e(z) : (C ~ H such that e(t) = Ute, Vt E JR. '

351

Lemma 8.1.10. Let h be a non-negative invertible self-adjoint operator on H. For any 0 > 0, define

A(o) = {€(z) I €(z) is continuo~ and bo~n~ed from - 0 < Imz < 0 } to H, and IS analytic In - 0 < Imz < O. If

€ E H,

e,

€(t) = have €(z)hiz €. Proof.

e

then E D(h6 ) if and only if there exists €(z) E A(o) such that hit \:It E Dl. Moreover, in this case, for any z with -0 < Imz < 0 we

Suppose that

€E

D(h6),and z E
D(hiz) = D(h-

1m Z )

=> D(h6 )

and € E D(hiz). If {e A } is the spectral family of h, then

IlhiZ(e n

-

iz€1l 2 e.d€ ,. - h

(f + E»: 'n'dlle,ell' z-

< lle~el12 +

£00 e261nAdlleA€1l2 ~ 0

uniformly for z with -0 < Imz < O. But for each n, z ~ hiz(en - e.d€ is an ,. analytic function from (C to H, thus €(z) = hiz € is continuous in -0 < Imz < 0, and is analytic in -0 < Imz < O. Moreover,

Ilhizell2 = <

(ll + loo)e-2Imz'lnAdlleA€1I2 II €1I 2 + I h6,11 2 ,

\:Iz with -0 < Imz < O. Therefore, ,(z) = h iz , E A(o). Now let €(z) E A(o) be such that ,(t) = hit C, \It E Dl. For any 1] E D(h6 ), 1] (z) = hiz1] E A(~) from the preceding paragraph. Since f(z) = (,(z), 1]) and g(z) = (" h- iz1] ) are continuous and bounded on -0 < Imz < 0 and analytic in -0 < Imz < O,and f(t) = g(t), \It E Dl, it follows that f = g. Consequently,

(,(-io),1]) = ("h 6 1]), Therefore, , E D(h6 ) .

\11] E D(h6 ) . Q.E.D.

Proposition 8.1.11. ,(E H) is analytic with respect to {~it} if and only if , ED, where ~ is the modular operator (relative to K), and D = n{D(~Z) I z E
352

Using Lemma 8.1.10 to ~ and ~ -1, the conclusion is clear. Q.E.D.

Proof.

Proposition 8.1.12. Let {tLt I t E lR} be an one-parameter strongly continuous group of unitary operators on H ,and E H. For each r > 0, define

e

rr 1 v;

00

er = Then

e-00

rs2 tL.eds.

is analytic with respect to {tLt}, and II er -

€r

Proof.

Clearly er(z)

function:

(C -+

L:

e-

rs2

-+

a as

r -+

+00.

= ~ [ : e-r(z-.r~tL.€ds is a vector valued analytic

H .and

Thus, er is analytic with respect to {tLt}, Vr > O. For any e > 0, pick 6 > a such that II (tLt - 1) ell

#

ell

< s, Viti < 6. Then since

ds = 1, it follwes that

if r sufficiently large.

Q.E.D.

Theorem 8.1.13. Let K be a nodegenerate closed real linear subspace of H, and ~ be the modular operator relative to K. Then {~it I t E JR} is the unique one-parameter strongly continuous group of unitary operators on H, which satisfies the KMS condition relative to K and is invariant for K.

Proof. From Lemma 8.1.5, {~it} is invariant for K, i.e., ~it K = K, Vt E JR. Now let €, TJ E K. Then by Lemma 8.1.4 and Lemma 8.1.10 K C D(~1/2) and

is continuous and bounded on a < Imz < ~ and is analytic lin For any t E JR by Lemmas 8.1.2, 8.1.4, 8.1.5, '

f(t +~) = (TJ,~it~4€) = (~ite,i'1), 2

t)

a<

Imz < ~.

Vt E lR.

But ~ite E K,iTJ E iK.L, so f(t + is real, Vt E JR. Now by Proposition 8.1.8, {~it} satisfies the KMS condition relative to K.

353

If {Ut} satisfies the KMS condition relative to K and is invariant for K, we need to prove Ut = d it, Vt E JR. Since (K+iK) is dense in H, it suffices to show that UtTl = d it T1, Vt E JR, TI E K.

From Proposition 8.1.12, we may assume that TI is analytic with respect to {Ut}, and T1(Z) is bounded on every horizontal strip. Notice that

tt:

i-V~

e- r 8 2 d i 8 eds

-00

r 82 = V; . rrfoo e- d i 8 j eds, j i = -ij, and -00

j( K -t-iK) is dense in H. So we need only to prove (ditjc, UtTl) =

(ie, T1),

e

where TI is as above, E K, and c,jc are analytic with respect to {d it}. Let g(z) = (d iz j e, T1 (z)). Then g(z) is analytic on (C, and is bounded on every horizontal strip. If t E JR, then TI (t) = Ut TI E K, and d itj = j d it E jK = iK..l. Thus, g(t) is real, Vt E JR. Fix s E JR. For T1,di 8e(E K), we have a KMS function 1 (relative to {Ut}) such that I(t) = (di 8 C, UtTl) = I(t + i), Vt E JR.

e

By Proposition 8.1.8, I(t

+ i)

= I(t

+ i), Vt E

e

JR. Notice that

is analytic on (C,and h(t) = I(t), Vt E JR. Using the Schwartz reflection principle to (I - h), we can see that I(z) = h(z),O < Imz < 1. In particular, h(s + = I(s + is real. Thus

i)

i)

. 9 (s + -) = 2 I

.8

(d·

e,

. . I TI (s - -)) = h (s + -)

is real, Vs E JR.

I

2. 2

i,

Now g(z) is real on Imz = 0 and Imz = is continuous and bounded on o < Imz < 1/2, and is analytic in 0 < Imz < 1/2. By the Schwartz reflection principle, g(z) can be extended to a bounded analytic function on (C. So g(z) is a constant function. Consequently,

Vt E JR.

Q.E.D.

Notes. The KMS condition was initially proposed by R. Kubo, P.C. Martin and J. Schwinger. Theorem 8.1.13 is due to M. Takesaki. References. [93], [107], [127], [135], [174].

354

8.2. Tomita-Takesaki theory In this section, let M be a VN algebra on a Hilbert space H, and H, II II = 1) be a cyclic-separating vector for M.

eo

eo(E

Proposition 8.2.1. Let K = {xeo I x E M,x· = x}. Then K is a nondegenerate closed real linear subspace of H, and

{x'eo I x'

E

M',x'· = x'}

C

(iK).l = iK.l,

where" 1." is in the sense of H, (see Section 8.1).

Proof. If x· = x E M, x'· = x' E M', then (x'eo, xeo) is real. Thus x' eo E (iK).l ,and M'eo C (iK).l + K.l = (K n iK).l. But M'eo is dense in H, so K n iK = {O}. Moreover, since Meo c K +iK, it Q.E.D. follows that (K+iK) is dense in H. In the following, for the above K we keep the notations of Section 8.1: p, q, a.], b,~, s, s", and etc.

Proposition 8.2.2.

qeo = 0; peo = aeo = jeo

=

beo = eo; ~iteo =

eo,Vt E JR;Meo c D(~1/2), and the operator s is the closure of the operator: xeo --+ x·eo(Vx EM). Moreover, for each x'. = x' E M, there is x· = x E M such that (p - q)x'eo = xeo. Since eo E K n (iK).l, it follows that qeo = O,peo = eo, aeo = eo. From (p - q)2 eo = eo, we have also beo = eo. Further, j eo = j'beo = (p - q) eo = eo. By eo = seo = j~ 1/2 eo, we get ~eo = eo, ~it eo = eo, Vt E 'JR. By the definition of the operator s, it is clear that M C D(~1/2) and s is the closure of the operator: xeo --+ z" eo(Vx EM). Now let x' E M', 0 < x' < 1, and

Proof.

eo

Then cp,1/J EM., and 0 o < x < 1 such that

< 1/J < cp. From Theorem 1.10.4, we have x E M with

(yeo, x'eo) = i((x y + yx) eo, eo),

vu E M.

In particular, (Yeo,x'eo) = (Yeo,xeo)r,Vy· = y E M. Thus, (x' - x) eo E and xeo = px'eo. But x' eo E (iK).l, so xeo = (p - q)x'eo.

«-,

355

Therefore, for any x'· = x' E M' there is z" = x E M such that xco =

(p - q)x' Co.

Q.E,D.

Lemma 8.2.3. x E M such that

For each x' E M' and each A E

(E

with ReA > 0, there is

bix'jb = A(2 - a)xa + Aax(2 - a). Proof.

We may assume that 0 < x' < 1. Let

cp(.) Then that

(·co, co),

=

1/1(.) = (·co, x'co).

'P,1/1 EM., and 0 < 1/1 < cp. By Theorem 1.10.4, there is (Yco, x'co) = ((AXY + AYX)CO, co),

x E M+ such

Vy E M.

Replacing y by z·y, we have

(YCO, x' zco) = A(YCo, zxco) +A(YXCo, zco), For any y'. = y', z,· z E M such that

= z'

Vy, z E M.

E M', by Proposition 8,2.2 there are y.

(1)

= y, z" =

= YCo,

jbz' C = ZCo· Now from (1), the property of j, and f).1/2b = (2 - a), we get jby' Co

(bjx'jbz' Co, y' co)

+ A(YXCO' jbz' co) A(iby' Co, if).1/2 x z CO) + A(if).1/2 x y Co , jbz' co) AUby' co, zxco)

+ A((2 - a)z'co, xyCo) A(xjbz'co, (2 - a)y'co) + A((2 - a)z'co,xjby'co), A(XZCO' (2 - a)y'co)

\ly'. = v', z,· = z, EM'. Since a - J'b

= 2q and qe'co = 0, Ve'''' = e' EM', it

follows that

(bi x'J" bz' co, y' co) A(xaz'co, (2 - a)y'co)

+ A((2 -

a)z'co, xay'co)

((A(2 - a)xa + Aax(2 - a))z'co, y'co), Vy'·

z' EM'. Further, the above equality is valid for any y', z' EM'. But co is also cyclic for M', so we obtain =

y', z,·

=

bJ'x'J'b

=

A(2 - a)xa + Aax(2 - a). Q.E.D,

356

Lemma 8.2.4. Let A = ei 8 , with 101 < 1r .and on (E and bounded in {z E (C I IRez I < t}. Then

11

«:"

00

h{) (A/(it

1(0) = -

2

c

-00

1rt

1 be

an analytic function

1 1 + -) + A/(it - -))dt. 2 2

Proof. Consider g(z) = s::.et;:) I{z). In {z E (E I [Ree] < 1/2}, g(z) just has a pole at z = 0, and the residue is /(0). Moreover, when Izl --+ 00 and [Ree] < 1/2, g(z) converges to a rapidly. Thus

/(0) = ~1°O (g(it +!) 21rl

2

-00

- g(it - !))idt. 2

Now through a computation, we can get the conclusion. Let x', A, x be as in Lemma 8.2.3, and A = ei 8 with

lemma 8.2.5. Then

1

x Proof.

= 2"

Q.E.D.

101 < 1r.

-8t

00

1

e d i t . I ' d -it dt ch(1rt) JX J .

_00

Suppose that e,71(E K) are analytic with respect to {dit}. Define

I(z)

= (bxbd- ZE,d%71).

Clearly, /(z) is analytic on (E and is bounded on every vertical strip. Since d~b = 2 - a, bd -4 = a, it follows that

/(it

(bxbd-itd-4E,~-itd471)

+!) =

(d it (2 - a) xa~ -it /(it -~) =

e,"),

(bxbd-itdie,d-itd-~71) (d itax(2 - a)d-itE, 71).

By Lemma 8.2.3,

A/(it + ~)

+ A/(it - ~)

(ditbjx'jb~ -it e,

71).

Further, from Lemma 8.2.4,

(bxbE, 71) =

/(0)

11

2"

(1 2"

-8t

00

_00

e (ditbjx'jbd-ite,71)dt ch(1rt)

00

-8t

_00

e d ch( 1rt)

1

itjx'jd- itdtbe,b71)

357

since b and ~it commute. Now from Proposition 8.1.12 and since (K+iK) is dense in Hand b is invertible, we can get the conclusion. Q.E.D. DYjrj~-it EM,

Lemma 8.2.6.

Proof,

Let y' E M',

g(t)

Vr EM', t

E JR.

e, 1/ E H, and define

= (( ~it jx'j ~ -it y' - y' ~itjx'j~ -it) e, 1/}.

By Lemma 8.2.5, we have

1

00

-00

Put f(z)

«"

~hg(t)dt = 0, c 1rt

=!~ c~~., )g(t)dt. 1rt

V8 E JR

and

181

<

1r.

Then [(z) is analytic in IRezl

< 11",

and [(6)

=

-00

0, V8 E 1R and 181

<

1r.

Thus,

f

= 0. Consequently,

I ~ c~~;'\ g(t)dt = 0, -00

'
1rt

By the uniqueness of Fourier transformation, g = 0, i.e.,

(li.it jx'j~-it y'e, 1/) = (y' ~it jx' i ~ -it e, 1/) Vy' E M', e, 1/ E H. Therefore, ~itjx'j~-it E M, Vx' E M', t E JR. J'Mj =

Theorem 8.2.7.

M',li.itM~-it =

Q.E.D.

M',Vt E JR.

Proof. From Lemma 8.2.6 with t = 0, we get jM'j c M. For any x .. = x, y. = Y EM, since xeo E K,jYeo E (iK).l, it follows that (xjyeo, eo) = (jYeo, xeo) is real. Further, from the property of i, we get

(yjxeo, eo) = (xjyeo, eo) = (eo, xjyeo). Generally, we have

(bjaeo, eo) = (eo, ajbeo), Va, b E M. In particular, for z"

=

x, y.

= Y EM,

and y' E

M',

(y(jy'j)jxeo, eo) = (eo, xjy(jy'j) eo) since jy'j E M. Noticing that jeo = eo, we get

(xjyeo, y'eo)

=

(iyjxeo, y' eo).

Since eo is .cyclic for M', it follows that

xjyjeo = jyjxeo,

Vx· = x,y· = Y EM.

358

Clearly, the above equality holds also for any x, y E M. Thus

jyjxzeo

= xzjyjeo = xjyjzeo,

Vx, y, z E M.

Since eo is cyclic for M, it follows that XJYJ

= JYJX,

Vx, Y E M,

i.e., jMj eM'. Therefore, jMj = M'. Now from jM'j = M and Lemma 8.2.6, we can see that flit M fl- it = M, Vt E JR. Q.E.D.

Definition 8.2.8. The * automorphism group {O't(') = flit. fl- it of M is called the modular automorphism group of M.

It

E JR}

Definition 8.2.9. Let 'Po(·) = (·eo, eo). Then CPo is a faithful normal state on M. An one-parameter strongly continuous * automorphism group {at(-) 1 t E .JR} of M [i.e. for each x E M,t -+ at(x) is strongly continuous) is called satisfying the KMS condition relative to CPo, if for any z, y EM, there is a complex function f(z) which is continuous and bounded on 0 < Imz < 1 and is analytic in 0 < Imz < 1 such that

Clearly, this f is unique, and is called the KMS function corresponding to x,y. When z" = x and y. = y, we can see that 7[iJ = f(t + i),V E m. By Proposition 8.1.8, in this case we have also f(t + ~) = f(t + ~), Vt E JR.

Theorem 8.2.10. CPo is invariant for the modular automorphism group {at I t E m} of M, i.e., 'Po(ue(x)) = CPo(x), Vx E M, t E JR. And also {at I t E m} is the unique one-parameter strongly continuous * automorphism group (of M) satisfying the KMS condition relative to CPo.

Proof. From Proposition 8.2.2, 'Po is invariant for {Ut I t E JR}. For any z" = x, y" = y E M, by Theorem 8.1.3 there is a KMS function f such that f(t) = (yeo, flitxeo) , f(t + i) = (/litxeo, yeo), Vt E JR. Since /l-it eo

= eo, it

follows that

Vt E .JR. Further, we can see that {at} satisfies the KMS condition relative to CPo· Now let {at I t E m} be an one-parameter strongly continuous * automorphism group of M, and {at} satisfies the KMS condition relative to CPo.

359

First, we claim that CPo is invariant for {at}. In fact, for x E M+ and Y = 1, there is a KMS function I such that

I(t) = I(t + i) = cpo{at(x)) > 0,

Vt E JR.

By the Schwartz reflection principle, I can be extended to a bounded analytic function on (C. Thus I is a constant function . In particular,

cpo{at(x))

= I(t) = 1(0) = CPo(X) ,

Vt E JR,

i.e., CPo is invariant for {at}. Now define

UtXeO = at (x) eo,

Vt E M.

Then Ut can be extended to a unitary operator on H, still denoted by Uh Vt E JR. Clearly, {Ut I t E JR} is an one-parameter strongly continuous group of unitary operators on H, and utK = K, Vt E JR. We say that {ue} satisfies the KMS condition relative to K. In fact, for any e,11 E K, we can pick two sequences {x n}, {Yn} of M with X~ = Xn, v; = Yn, Vn, such that xneo ---+ e, Yneo --+ n, Since {at} satisfies the KMS condition relative to tpo, then for each n there is a KMS function In such that

In(t) = tpo( at(xn)Yn) = (Yneo, at{x n) eo) (Yneo, UtXn eo), In(t + i) = tpO(Ynat{xn)) = (at(x n) eo, Yneo) (UtXn eo, Yn eo) , Vt E JR. By the maximum modulus theorem (see [137]), we have sup

I/n(z) - Im(z)1 = sup I/n(t) - Im(t)1 tER

O::::;lml:::::;l

<

IIYnEoll'llxne - xmEolI + Ilxmeoll ·11 . IIYneo - Ymeoll

and

Thus, there is a KMS function

In(z)

---+

I(z),

I such that

uniformly for z with 0

< Imz < 1.

Clearly, I(t) = (TJ,utE),/(t + i) = (Ute, 11),Vt E JR. Therefore, {Ut} satisfies the KMS condition relative to tpo. From Theorem 8.1.13, we get Ut = die, Vt E JR. Then for any x E M and t E JR, we have

360

Since

eo is separating for M, it follows that at(x) = at(x) ,'Ix E M, t E IR.

Q.E.D.

Notes. Theorem 8.2.7 is due to M. Tomita. The first version of Tomita's theory is given by M. Takesaki. The Proof presented here is due to M.A. Rieffel and A. Van Daele. Theorem 8.2.10 is due to M. Takesaki.

References. [127J, [135], [174], [181], [182].

8.3. The modular automorphism group of au-finite W*-

algebra Let M be a a-finite W* -algebra, Then there is a faithful normal state cp on M. Suppose that {'/rrp, Hrp, erp} is the faithful cyclic W*-representation of M generated by cp. Clearly, erp is a cyclic-separating vector for the VN algebra '/r rp(M) on Hrp. Using the theory of Section 8.2 to {'/rlp(M), Hrp, erp}, we have the modular operator drp (a non-negative invertible self-adjoint operator on Hrp) such that d~1rrp(M)d;it = 7l"rp(M), 'It E JR. Since M and 1rrp(M) are

* isomorphic, we can define

A it ( ) A -it) atrp (x ) -_ 1rrp-1 ( ulp7l"rp X ulp ,

'Ix E M, t E JR.

Clearly, {ai I t E lR} is an one-parameter s(M,M*)-continuous phism group of M.

* automor-

{ai I t E JR} is called the modular automorphism group of M corresponding to cp. By Theorem 8.2.10, cp is invariant for {at}, i.e., cp(ai(x)) = cp(x) , 'Ix E M, t E IRj and {at} satisfies the KMS condition relative to cp, i.e., for any x, y EM, there is a KMS function f (continuous and bounded on 0 < Imz < 1 and analytic in 0 < Imz < 1) such that

Definition 8.3.1.

f(t)

= cp(at(x)y) , f(t + i) = cp(yat(x)),

'It E JR.

Moreover, {at I t E JR} is the unique one-parameter s(M,M.)-continuous automorphism group of M satisfying the KMS condition relative to cp.

*

Let cp be a faithful normal state on a W*-algebra M,

Proposition 8.3.2. and

M'"

= {x E M

lai(x)

= x, 'It E IR}.

361

Then x E Mf(J if and only if cp(xy - yx) = 0, vu E M. Let x E Mf(J. For any y EM, there is a KMS function

Proof.

I

such that

I(t) = cp(ai(x)y) = cp(xy), I(t + i) = cp(yaf(x)) = ep(yx), Thus,

I

Vt E JR.

is a constant function. In particular,

cp(xy) = 1(0) = I(i) = cp(yx). Conversely, let x E M with ep(xy - yx) = 0, vu E M. We may assume that z" = x. For any y. = y EM, there is a KMS function I such that

I(t) = ep(af(x)y),

f(t

+ i)

= cp(yaf(x)),

Since z" = x and v' = y, it follows that I(t) = I(t hand, since ip is invariant for {ail, we have

f(t) =

cp(xa~t(Y))

=

Vt E JR.

+ i}, Vt E JR.

On the other

cp(a~t(Y)x)

ep(yaf(x)) = I(t + i),

Vt E JR.

Thus, by the Schwartz reflection principle I can be extended to a bounded analytic function on (C, and I is a constant function. In particular,

cp((ot(x) - x)y) = I(t) - 1(0) = 0, Since

ip

Vy· = Y E M, t E JR.

is faithful, it follows that af(x) = x, Vt E JR, i.e., x E Mf(J.

Q.E.D.

Proposition 8.3.3. Let M be a a-finite W·-algebra, cp, t/J be two faithful normal states on M, and {ai}, {at} be the modular automorphism groups of M corresponding to cp, t/J respectively. Then there exists an one-parameter s(M, M. )-continuous family {Ut I t E JR} of unitary elements of M such that

at(a}

=

utaf(a}u;,

Ut+6 = Utaf(u,) ,

Va E M, t, s E JR. Proof.

Consider the following W·-algebra

and define a functional 8 on M 2 as follows

o((: ~))

= cp(a) + t/J(d) , Va,d,c,d

E M.

362

Clearly, (J is a faithful normal state on M 2. Let morphism group of M 2 corresponding to (J. Put en

= (~ ~).

{C1:}

be the modular auto-

Then it is easy to see that (J(el1x - xen)

= 0, 'Ix E M 2.

By Proposition 8.3.2, we have a:(ell) = ell, 'It E JR. Notice that

Va E M, t E JR. Thus for each t E JR we can define at (.) : M ---+ M such that at (a) a - at6 ( ( aa a) a ) ' \..Iv a E M . ( o It is easily verified that {at I t E JR} is an one-parameter s(M, M.)continuous * automorphism group of M. Since {at} satisfies the KMS condition relative to (J, it follows that {at} satisfies the KMS condition relative to cp. By Theorem 8.2.10, at = af, Vt E JR.

0) _

By a similar discussion for

e22 = (~ ~ ) , we can also get

a: (( ~ ~)) = (~ C1/< a) ), Since that

(~

0 _- (00 0)

a: (( ~ From (

~))

=

0 (0 1) 1

1 0 (0 0)

Va EM, t E JR.

0) a ,for each t we have

(~ ~) u: (( ~ ~)) (~ ~)

=

Ut E

M such

(~t ~).

~ ~) = a% ( (~ ~) (~ ~ ) ) = ( ~t

~) at ((~ ~

n (~ u~;), =

we get UtU; = 1, Vt E JR. Similarly, UtU; = 1, Vt E JR. Hence, {Ut 1 t E JR} is an one-parameter s(M, M.)-continuous family of unitary elements of M. Since

(~ ~) = (~ ~) (~ ~) (~ ~ ), it follows ( ~ uf~

that

a)) = at ((~ ~) ) ( Uta

0) (af(a) 0) (0 U;) a o a a '

0

i.e., ut(a) = utC1f(a)u;, Va E M,t E JR.

363

Moreover, from

Q.E.D. Remark. Let a be a * automorphism of M, a is said to be inner, if there is a unitary element u of M such that a(x) = u·xu, Vx E M. By Proposition 8.3.3, for each t E JR the innerness of the * automorphism or of M is independent of the choice of the faithful normal state ep on M.

Lemma 8.3.4. Let ep be a faithful normal state on a W· -algebra M, h E M'P n M+, and the spectral family of h be s(M, M·)-continuous at o. Then ,p(.) = ep(h·) is also a faithful normal positive functional on M, and

of (x) = hitor(x)h- it, where {of to ,p.

It E

Vx E M, t E JR,

JR} is the modular automorphism group of M corresponding

Since h E M'P n M+, it follows from Proposition 8.3.2 that ,p(.) = ep(h·) = ep(·h) = ep(h 1/ 2 • h 1/ 2 ) . Thus w is also a faithful normal positive functional on M (noticing that 0 is not an eigenvalue of h). Fix x, y E M. For each positive integer n, let

Proof.

X

n=

Then from Proposition 8.1.12,

xn(z) =

~1 -

1T

Xn

1T

e- n3 o;(x)ds. 2

-00

is analytic with respect to {or}, i.e,

~1 -

°O

°O

2

e- n(3-Z) u;(x)ds

-00

is analytic: (C ---+ M and xn(t) = or(x n ) , Vt E JR. Moreover, clearly xn(z) is bounded on every horizontal strip. Thus, the function

is continuous and bounded on 0 < Imz < 1 and is analytic in 0 < Imz < 1. Now we compute the boundary values of In. For t E JR, by hit E M'P we have

364

and

fn(t

+ i)

=

lp(hitxn(t + i)h-ithy) lp(xn(t + i)h-ithyhit) {1rrp (h -it hyhit) erp, 1rrp( xn(t

+ i))· erp),

where {1rrp,Hrp, erp} is the faithful cyclic W·-representation of M generated by lp. Put TJ = 1rrp(x·) erp( E Hrp). Then by Proposition 8.1.12,

is analytic with respect to {.6.~}. From Proposition 8.1.11, we have Vz E

Thus, for any z E (C we get

1rrp(Xn(z)·)erp =

~ ~

I: I:

(C.

e-n(6-z}~1rrp(a~(x·))erpds e- n(, - z)2 .6.~TJds

f7n(Z) = .6.*;TJn = .6.~1rrp(X~) erp· Further,

fn(t

+ i)

(1r rp(h-ithyhit) erp, .6.rp.6.~7r rp(X~) .6.; it erp)

=

(.6.~21r rp(h-it hyhlt ) erp, .6.~21r rp (aF (X~)) erp)

{1rrp (ar (x n)) erp, 1rrp (h-ity. hhit) erp)

lp(h-ithyhitar(x n)) = lp(hyhitar(xn)h- it ) "'(yhitot (x n)h- it). From the maximum modulus theorem and

lIxn - xII

---+

0, we have

and sup Ifn(z) I < n O~lmz~l

Hence, there is a KMS function

fn(z)

---+

f(z),

11"'11 . Ilyll . sup llxnll < 00. n

f such that

uniformly for z with 0

< Imz < 1,

365

and

I(t)

=

+ i) = t/l(yhitaf(x)h- it) , * automorphism group {hitaf(·)h- it I t E

t/l(hitaf(x)h-ity) ,

Vt E JR. Therefore, the

I(t

JR} of M

satisfies the KMS condition relative to t/l. Now by Theorem 8.2.10, we obtain that at(x) = hituf(x)h- it , Vx E M,t E JR. Q.E.D.

Lemma 8.3.5. Let cp be a faithful normal state on a W*-algebra M. If ai(x) = x,Vx E M,t E JR, i.e., M/P = M, then cp is also a trace. Proof.

For any x EM, there is a KMS function

I(t) I(t

=

I

such that

cp(af(x*)x) = cp(x*x) > 0,

+ i) = cp(xuf(x*))

= cp(xx*)

> 0.

Thus, I must be a constant. Therefore, cp(x*x) = cp(xx*).

Q.E.D.

Theorem 8.3.6. Let cp be a faithful normal state on a W*-algebra M, and {uf I t E lR} be the modular automorphism group of M corresponding to cp. Then M is semi-finite if and only if {ai I t E JR} is an inner * automorphism group of M, i.e., there exists an one-parameter s(M, M*)-continuous group {u, I t E JR} of unitary elements of M such that

ai(x) = UtXU;,

Vx E M, t E JR.

Proof. Let {Ut I t E JR} be an one-parameter s(M, M*)-continuous group of unitary elements of M such that ai(x) = UtXU;, Vx E M, t E JR. We may assume that M is a VN algebra on a Hilbert space H. By the Stone theorem, there is a non-negative invertible self-adjoint operator h such that Ut = h- it, Vt E JR. Since Ut E M/P, Vt E JR, it follows that each spectral projection of h belongs to M/P. Let Pn be the spectral projection of h corresponding to the interval [n-t, n]. Then Pn, hpn E M/P. Define tPn(') = cp(hPn') on M p n • By Lemma 8.3.4 the modular automorphism group of M p n = PnMpn will be

(hPn)ituf(x)(hPn)-it = hitUtxu;h-it = x, Vx E PnMpn, t E JR. From Lemma 8.3.5, t/ln is a faithful normal tradal state on PnMPn. Thus, P« is a finite projection of M, Vn. Clearly, sup Pn = 1. n

Therefore , M is semi-finite. Conversely, suppose that M is semi-finite. Then there is a faithful semifinite normal trace r on M+. 1) Let P be a projection of M with r(p) < 00. Then r can be uniquely extended to a faithful normal trace on pMp, We say that there exists unique

366

h E pMP with 0

< h < 1 such that T((1 - h)x) = cp(hx) = cp(xh),

In fact, using Theorem 1.10.4 to T, cp o < h < 1 such that

T(X) =

\Ix E pMp.

+ T and pMp, there is hE pMp with

~(cp + r)(xh + hx)

~cp(xh + hx) + T(hx) , Le., r((1 - h)x) = lcp(xh

+ hx), \Ix

~cp(xh2 + hxh) =

E pMp. Then

T((1 - h)xh) T(h(1 - h)x) = r((1 - h)hx)

~cp(hxh + h 2x), i.e., cp(xh 2 ) = cp(h 2x), \Ix E pMp. Generally, cp(xh 2n ) = cp(h 2nx), \In and x E pMp. Since h can be approximated arbitrarily by the polynomials of h 2 , it follows that cp(xh) = cp(hx) , Vx E pMp. Therefore, we have r((1 - h)x) = cp(xh) = cp(hx), \Ix E pMp. 2) There exists unique hEM with 0 < h < 1 and the spectral family of h is s(M, M*)-continuous at the points 0 and 1 such that

T(1 - h)

<

00,

r((1 - h)x) = cp(hx) = cp(xh) ,

\Ix E M.

In fact, since r is semi-finite, we can pick an increasing net {PI} of projections of M such that SUPPI = 1,T(pl) < 00,\11. By 1), for each 1 there is I

hi E p,MPI with 0

< h, < 1 such

T((1 - h,)XI)

that

= cp(x,h ,) = cp(h,XI),

\lxl E p,Mpl.

By Proposition 6.5.2, we have

r((1 - h,)PIXPI) Vx E M, and

t.

= r(PI(1 - h,)xp,) = T((1 - h,)XPI),

Thus,

T( (1 - h,)XPI)

= cp(h,xp,) = cp(Plxh,),

(1)

Vl and x E M. Since the closed unit ball of M is u(M, M*)-compact, replacing {h,} by a its subset in necessary case, we may assume that h, ---+ h E M(u(M, M*)). Clearly, 0 < h < 1. If PI < PI', then by (1) we have

T((1 - h,,)XPI) =

r((1- h,,)XPIPI') cp(h"xPIPI') = CP(hI'XPI).

367

Since 1"(Pl) < 00, it follows from Proposition 6.5.3 that cp( ·Pl) EM.. Now picking the limit for l', we get Vl and x E M.

1"((1 - h)XPl) = cp(hXPI)'

(2)

In particular, cp(hpl) = 1"((1 - h)PI) = 1"((1 - h)l/2 p,(1 - h)l/2) , Vl. Further, we have 1"((1 - h)) = cp(h) < 00 since 1" is normal. Again by Proposition 6.5.3 and picking the limit for 1 in (2), we obtain

1"((1 - h)x) = cp(hx),

Vx E M.

(3)

Further from (1), we have 1"((1 - h,,)XPI) = cp(p"xp,h,,) when Pl' > Pl. By the Schwartz inequality, it is easy to see

Icp(Pl,Xp,h ,,) - cp(xp,h) I

< Icp(Xpl(h - h))1 + Icp((I- Pl,)xp,h ,,)I "

< Icp(Xpl(h - h))1 + Ilxllcp(l- Pl,)l/2.

" Then picking the limit for l', we get 1"((1 - h)xp,) = cp(xPlh) , Vl. Now by Proposition 6.5.3 and picking the limit for l, we have

1"((1 - h)x) = cp(xh),

Vx E M.

(4)

Moreover, since Cp,1" are faithful, it follows from (3) and (4) that the spectral family of h is s(M, M.)-continuous at 0 and 1. 3) Let at(x) = h- it(1 - h)if xhit(1 - h)-it, \Ix E M, t E JR. Then it suffices to show that CTr = a" Vt E JR. For any x, y E M, let

<

1 and is analytic in

cp(at((1 - h)xh)y) , cp(hat(x) (1 - h)y) 1"((1 - h) at(x) (1 - h)y) r ( (1 - h)y (1 - h) at(x)) cp(y(1 - h) at( x)h).

(5)

Clearly, I(z) is continuous and bounded on 0 < Imz o < Imz < 1. For any t E JR, by (3) and (4) we have

I(t) I(t + i)

Pick y = l,x = ha(l - h), where a· = a. Then

is real, \It E JR, where Yo = h(1 - h). Thus,

cp(at(yoayo)) = cp(yoayo),

I

Va·

is constant. In particular,

= a E M, t E JR.

368

Let Yo = t' >"de)., and Pn = e1_l. - e i , Vn. Since {ex} is s(M,M.)-continuous Jo n n at 0 and 1, it follows that v« /' 1. Put

Then yoanyo = Pnapn, and ~(at(PnaPn)) = ~(PnaPn), Vn. Picking the limit for n, we get (6) ~(at(a)) = ~(a), Va E M, t E JR. For general z, y E M, let

and

In(Z)

= ~(h-iZ(l - h)iz+lhxn(l -

h)h i Z+1 (l _ h)-iz y ) .

By (5), we have

In(t) = In(t

~(at(PnXPn)y),

+ i) =

~(YO:t(PnxPn)),

Vt E JR.

From the maximum modulus theorem, the Schwartz inequality and (6), we get sup I/n(z) - Im(z) I 0::; ImZ::; 1

and sup

O$lm.1l$l

I/n(z)1 < Ilxll·llyll·

n

Thus, there is a KMS function

In(z)

~

I(z),

I such that

uniformly for z with 0

< Imz < 1.

Clearly, I(t) = ~(at(x)y), I(t + i) = ~(yat(x)), Vt E JR. Therefore, {at} satisfies the KMS condition relative to ~, and at = Vt E JR. Q.E.D.

or,

Notes. The Unitary cocuele theorem (Proposition 8.3.3) is due to A. Connes. Theorem 8.3.6 is due to M. Takesaki. As noted by Takesaki, this theorem shows that every type (III) factor on a separable Hilbert space has outer automorphisms.

References. [17], [127], [174].

Chapter 9 The Connes Classification of Type (III) Factors

9.1. Preliminaries Consider some properties of Ll(lR). Clearly, with convolution

11/111 = IE 1/(s)lds and the

Ll(lR) is an abelian Banach algebra. Let n Zn (t) = { ~ , 0, \In.

l
- n - . -

otherwise,

Then Zn > O,lIznlll = 1, \In, and {zn} is an approxianate identity for

Ll(lR), i.e.,

For each / E L1(lR), define its Fourier transform

Then

j

1:

E Cgo(lR). HIE Ll(lR), then we have inversion formula:

/(t) = - 1

21T

l e- .

J6 /(s)ds t"'"

(a.e.).

E

For each / E L 2 (lR), define its Fourier transform ;/:

(;/)(t) =

.~

y21T

l.i.m.N_oo

t"

J-N

eillt/(s)ds.

370

Then 7 is a unitary operator on L 2(lR), and

1 (7- 1 / )(t ) = . ~1.i.m'N_oo y21r

IN e-id I(s)ds -N

(the Plancherel theorem). In particular, we have the Parseval formular: (/,g) = (71,7g),

V/,g E L 2 (lR).

Lemma 9.1.1. Let K, U be two subsets of lR, K be compact, U be open, and K C U. Then there exists k E L 1 (lR) such that

o < k < 1; k = 1 on tc,

and suppk C U.

Proof. Pick an open neighborhood V of 0 such that K + V - V cU. Let 2 g,h E L (lR) with 7g = XK-v,7h = Xv respectively, and k(·) = g(.)h(·)JIVI. Clearly, k E L 1 (lR} and ~

k(t)

=

1

wr(XV

* XK-V)(t) =

wr Jr 1

v

XK-V(t - s)ds.

Therefore, we have

o < k < 1., k

1

on K; and suppk C K

+V

- V c U.

Q.E.D. Lemma 9.1.2. Let to E lR, I E L 1(lR} with [(to) = 0, W be a neighborhood of to,and e > O. Then there exists k E L 1 (lR) such that IIkl11 < 2, supp k c W, k 1 on some neighborhood of to, and III * kill < C.

=

Proof.

11/11t),

Without loss of generality, we may assume to = O. Let fJ = cJ4(1 and pick a compact subset E of JR such that

r

JR\E

I/(s) Ids

+

< 6.

11-

Further, pick T] > 0 such that [-3T],3T]] C W, and eidl < fJ,Vs E E and ItI < 3T]. Let K = V = [-T], T]], and k = gh/2T], where g, h E L 2 (lR) with 7 g = XK-V = X[-2".2"b 7h = Xv = X[-",,,I respectively. Clearly, K + V - V = [-3T],3T]] C W. Then by Lemma 9.1.1, we have

o < k < 1; Moreover,

k

1

on [-T], T]]; and supp

kC

W.

371

Since

IIR I( s )ds = 1(0) = 0, it follows that (I

Then

III I(s)[k(t - s) - k(t)]ds, II I * kill < III II (s) I. II killds

* k)(t)

=

ks

where E'

-

(rlE + lEI r )1/(s)I·llk" -

=

= lR\E, k,,(·) = k(· -

Vt E lR.

killds,

s), Vs. Clearly,

r

lEI II (s] I. II k s - killds < 48, and

r If(s)I·llks - klilds < 11/111 . sup Ilks -

lE

"EE

kill.

Then by the definition of 8, it suffices to show that sup "EE

Since k

= gh/2TJ,

Ilk" - kill < 48.

it follows that

2TJ(k" - k)

= g(h"

- h)

+ (gs

- g)h".

Notice that for any sEE,

and similarly,

Ilh s-

hll~

< 21]8 2 • Therefore,

Ilks - kill < ~{IIJ"gI12

< 48,

·llh" - hl1 2 + Ilgs - gll2 ·lIjh,,112}

Vs E E.

Q.E.D. In the following, we put

Kl(lR) = {I E Ll(lR) Lemma 9.1.3.

I suppf is compact}

Kl(lR) is dense in L 1(lR).

Proof. Denote all continuous functions with a compat support on lR by K(lR). Clearly, K(lR) isdense in L2(lR). Then {I E L 2 (lR) E K(lR)} is 2 dense in L (lR).

11

372

For any 0

E L 1(lR) and e

> 0, pick 9 E L 2(lR) such that

Ilg - f1/2112 < e

amd

9 E K(lR).

Then we have

IIg 2- fill < Ilg· f1/2 - fill + Ilg· f1/2 - g2111 < Ilfl/2112 ·lIg - f1/2112 + IIgl12 ·llg < c(21If 1/ 2112 + c).

f1/2112

Moreover, since 9 * 9 E K(lR) and the inverse Fourier transform of 9 * 9 is g2 E L1(lR), it follows that (g2)" = g * 9 E K(lR). Q.E.D. For a closed ideal 1 of L1(lR), let

t-

{t E lR I j(t) = 0, Vf E I}

=

n{ AI (])

If

E I},

where AI(]) is the zero point set of f. Clearly Conversely, for any closed subset E of lR, let

I(E) = {f E L1(lR)

I (fiE)

t-

is a closed subset of lR.

- O}.

Then I(E) is a closed ideal of L1(lR). Lemma 9.1.4. 1) If E is a closed subset of lR, then I(E)l- = E. 2) If E is a compact subset of lR, then I(E) is a regular closed ideal of Ll(lR) , i.e., I(E) is a closed ideal of Ll(lR), and I(E) admits a modular unit 1 [i.e., (f - f * l) E I(E), Vf E L1(lR)).

Proof. 1) Clearly, E c I(E)l-. Conversely, if t rf. E, then by Lemma 9.1.1 there exists f E L 1 (lR) such that j(t) = 1 and (fIE) - O. Thus f E I(E} and t rf. I(E)l-. Furhter, I(E)l- C E,and I(E)l- = E. 2) Since E is compact, by Lemma 9.1.1 there exists 1 E L1(lR) such that (nE) - 1. Then (f - f * l)"IE 0, i.e., (f - f * l) E I(E), Vf E L 1(lR}. Q.E.D.

=

Lemma 9.1.5. Let Ut, U2 be two open subsets of lR, K be a compact subset of lR with K c U2 , f E L l(lR),1 be a closed ideal of L1(lR), and h,f2 E I such that fi=f on Ui, i=1,2. A

A

Then there exists gEl such that "...

9= f

on

U1 U K.

373

elK -

1 and supp e C U2 • Let g = 12 * e + II * (1 :: e) .... II - II * e + 12:! e. ~hen 2. E I, . .a nd [ = = on K(C U2 )jg = h = I on UI\U2 jg = !2e+ II - lie = le- I + Ie = I on U1 n U2 • Notice that UI = (UI \U2 ) U (U1 n U2 ) . Therefore, we have g = on

Pick e E LI(lR) such that

Proof.

b l f

U1UK.

Q.E.D. Lemma 9.1.6. Let I be a closed ideal of LI(lR), E = T\ and If 1 vanishes on a neighbourhood of E, then I E I.

I E

L1(lR}.

=

Proof. 1} Suppose that to rf. E. We say that there is h E I such that It 1 on some neighborhood of to. In fact, pick a compact neighborhood K of to with E n K = 0. Let J = I(K). Then by Lemma 9.1.4 J is a closed ideal of L1(lR), and J admits a modular unit 1 with [ - 1 on K. Clearly, (I + J) is still a regular ideal of LI(lR} ,and 1 is its modular unit. If (f + J) 1= L 1(lR), then there is a maximal regular ideal L of LI(lR} such that (f + J) c L. Since L = I({s}) = {g E LI(lR} I y(s) = o} for some s E lR, it follows from Lemma 9.1.4 that s E EnL. But En K = 0, a contradiction. Thus, 1+ J = L1(lR). Consequently, we can write 1 = l] + l;, where

h

E I, lJ E J. Now let h = 1] E f. Then

hlK

= (nK) -

([JIK)

= nK =1.

2} If to rf. E, then there is gEl such that g =

f

on some neighborhood of

to. In fact, pick h as in 1), and let g = I * h. Then g satisfies the condition. 3} Suppose that K = supp1 is compact. Since 1 0 on some neighborhood of E, it follows that En K = 0. By 2) for each t E K, there is a gt E I and an open neighborhood U, of t such that

=

{-

gt = f

on

u;

Further, for each t E K, pick an open neighborhood Vi of t such that t E Vi C V t C Ut and V t is compact. Now by the compactness of K, there are t 1 , . ,tn E K, gi = gti E I, Vi = lIti , U, = Uti' 1 < i < n, such that 0

•

U?=I Vi::J K,

and

gi

= 1

on

o., 1 < i < n.

1

Further, let 0 = gn+l( E I}. Clearly, Yn+l = on lR\K = Un + l . By Lemma 9.1.5, there is g~ E I such that

g~ =

1,

on

U1 U V 2:J VI U V2 •

374

Aganin by Lemma 9.1.5, there is g~ E I such that .,."

g2

....

= I,

on

-

U1 U V2 U V 3 J VI U V2 U V3 •

. . '. So we can get g' E I such that

.g. , = I.. .,

on

VI U ... U Vn J K.

Now by Lemma 9.1.5, there is g E I such that

g = 1,

on

Un +1 uK

= JR.

From the uniqueness of Fourier transform, we obtain 1 = g E I. 4) General case. Since L1(lR) admits an approximate identity, for any e > we can pick Z E L1(lR) such that III - I * zlll < c/2. From Lemma 9.1.2, there is u E Kl(lR) such that Ilu - zlll < c/211/Ih. Then .

°

!II - 1* ulll < III - I * zlll + 11/111 ·lIu - zlll < c. supp (I * u)" (Csuppu) is compact, and (I * u)"(= lu) A

Clearly, a neighborhood of E. By 3), closed, it follows that I E I.

vanishes on 1 * u E I. Now since e(> 0) is arbitrary and I is Q.E.D.

References. [136].

9.2. The Arveson spectrum For our purpose, we just consider a W*-system

(M, lR, 0"), where M is a W· -algebra; for each t E lR,O"t is a * automorphism of M; and t ~ p(O"t(x)) = (at(x),p) is continuous on lR, '1x E M,p EM•. Denote the collection of all bounded Radon measures on lR by M(lR), i.e., M(lR) = Co(R)*. By the convolution ((JL * £1)(/) = II I(s +t)dJL(s)dv(t), '1JL, £IE M(lR), IE Co(lR)) ,M(lR) is an abelian Banach algebra with an identity 60 ( / ) = 1(0),'11 E Co(lR). Moreover, Ll(lR) is a closed ideal of M(lR).

Proposition 9.2.1. that

For each JL E M(IR), there exists O"(JL) E Ba(M) such

375

VX· E M,p EM., and lIu(JL)1I < IIJLII, where Bo(M) is the set of all u(M,M.)u(M, M.) continuous linear operators on M. In particular (u(f)(x),p)

III p(Ut(x))f(t)dt,

=

Ilu(f)11 < IIfllb

Vf E L 1(Dl), x E M,p EM•. Clearly, Ilu(JL) II < IIJLII, VJL E M(Dl). Now it suffices to show that for any normal positive functional p on M, the positive functional (u(JL)('),p) on M is normal, where JL E M(Dl)+. Let {Xl} be a bounded increasing net of M+, and x = sup X,. Then for any

Proof.

I

t E Dl,(Ut(Xl),p) /' (Ut(x),p). By the Dini theorem, (Ut(XI),p) /' (u,(x),p) uniformly for t E K, where K is any compact subset of Dl. Now by the regularity of JL and the boundedness of [z.}, we can see that (U(JL)(x),p) =

I p(ut{x))dJL liF I p(u,(xz))dJL

=

s~p

I p(ut(xl))dJL

sup (u(JL) (xt), p). I

Q.E.D. Definition 9.2.2. Let (M, n, u) be a W· -system. 1) Define the Arveson spectrum of U by sprr

= {f

E L 1 (Dl )

I u(f) = O}l.

{t E 1R I if f E L1(R) with u(f)

n{)/(1)

If

If

= O}

t. and clearly {f E L 1 (Dl) I u(f) =

O} is

I u(f)(x) == O}l.

{t E Dl I if f E L 1 (Dl) with u(f)(x)

n{)/(l)

then j(t)

E L 1 (Dl) and u(f) = O}

where )/(1) is the zero point set of a closed ideal of L 1 (Dl). 2) If X E M, let

sPo(x) = {f E L 1(Dl)

= 0,

E

= 0,

then j(t)

= O}

L 1(Dl) and u(f)(x) = O}.

3) If E is a closed subset of Dl, define the associated "spectral subspace" by

M(u, E) = {x E M

I sPo(x) C E}.

376

Clearly, bX- the definition spa is a closed subset of JR and 0 E spa (since a(/)(l) = 1(0), Vf E L 1(JR)); sPo(x) is closed and sPo(x) C spa, Vx E M; sPo(O) = 0 (from Lemma 8.4.1) j and 0 E M(a, E) for any closed E C JR.

Proposition 9.2.3. Let (M, JR,a) be a W*-system. Then: (a) spa = UZEMSPo(X); (b) sPo(x*) = -sPo(x), spo(at(x)) = sPo(x), Vx E M,t E JR; (c) sPo(x) = 0 <==> x = 0; (d) spo(a(/)(x)) C sPo(x) n sUPP1, Vx E M, I E L 1 (lR); (e) at(M(a, E)) = M(a, E), Vt E JR and closed E C JR; (f) for any closed E c JR, x E

M(a, E) <==> a(/)(x) = 0,

f=

VI E L 1 (JR)

and 0 on some neighborhood of E. Consequently, M( a, E) is a u(M, M* )-closed linear subspace of M; (g) if x E M and J-L E M(JR) satisfy fi, - 0 on some neighborhood of sPo(x) , then a(J-L)(x) = 0, where it(t) = JEt ei'tdJ-L(s). Moreover, if I E L 1(lR) satisfies either 1 0 or 1 on some neighborhood of sp; (x), then either a(/) (x) = 0 or u(/)(x) = x.

1=

Proof.

(b) Since a(f)(x)* = a(f)(x*), it follows that .....

sPo(x*) = n{)/(f) I a(f)(x*) = O}

- n {)/(j) I a(/)(x) = O} = -sPo(x), Vx EM. Moreover, by a(/){at(x)) = a(ft){ z}, j; (s) = ei d

j( s)

and )/ (it) =

)/(1), where It(s) = I(s - t), we have spu(at(x)) = n{)/(l) I a(/){at(x))

=

O}

n{)/(,ft) I a(lt)(x) = O} = sPo(x), Vt E lR, x EM. (e) It is immediate from (b). (f) Let x E M(a, E), and I E L 1 (JR ) with 1 - 0 on some neighborhood of E. Let I = {g E L 1 (JR ) I a(g)(x) = O}. Then sPo(x) = [1. C E. Now by Lemma 9.1.6 we have I E I, Le., a(/l.{x) = O. Conversely, let x E M and a(/)(x) = 0 for any I E L1(JR) with I - 0 on some neighborhood of E. If .....there is s E (sPo(x)\E), then by Lemma 9.1.1 ..... we can find k E L 1 (JR ) with k(s) = 1 and supp keF, where F is a closed neighborhood of sand F n E = 0. Then a(k)(x) = 0 since k - 0 on the open

377

neighborhood (lR\F) of E. But s E sPo(x), so k(s) must be 0, a contradiction. Therefore, sPo(x) C E and x E M(u, E). Consequently, M( a, E) is a linear subspace of M. Aganin by Proposition 9.2.1, M(u, E) is u(M, M.)-closed. (c) Clearly, sPo(O) = 0. Now let x E M and sPo(x) = 0. By (f), we have u(f)(x) = 0, Vf E L 1 (JR ). Thus,

III f(t)p(ut(x))dt =

0,

Vf E L1(JR),

P EM•.

p(Ut (x)) is a bounded continuous function on lR, it follows that p(Ut(x)) = 0, Vt E lR,p EM• . Therefore, x = O. (a) Clearly, E = UZEMSPO"(X) C sper, Now if s rf. E, then by Lemma 9.1.1 we can find k E Ll(JR) such that k(s) = 1, and k - 0 on some neighborhood of E. By (f), we have u(k)(x) = 0, Vx E M, i.e., u(k) = o. Since k(s) = 1, it Since t

---+

follows from Definition 9.2.2 that s rf. spo. Thus, sprr C E, and E = sprr. (d) If u(g)(x) = 0, then u(g)(u(f)(x)) = u(g * f)(x) = ai] * g)(x) u(f)(u(g)(x)) = O. Thus,

{g E L 1(JR) I u(g)(x) = O} C {h E L 1 (JR ) I u(h)(u(f)(x))

= O},

and sPO"(x) ::J sPo(u(f)(x)). Moreover, if s rf. supp[, then bI Lemma 9.1.! "!!e can pick k E Ll(JR) such that k(s) = 1 and supp kn suppf = 0. Thus, kf = 0 and k * f = f * k = o. Further u(k)(u(f)(x)) = 0, and sPO"(u(f)(x)) C .AI (k). But k(s) = 1, so s rf. sp 0" ( U (f)( x)). Therefore, sp; (u (f)( x)) C supp ..... (g) Since f * J1, E L 1 (IR) and f * J1, = fJl - 0 on some neighborhood of spo(x), it follows from (f) that u(f)(u(J1,)(x)) = u(f * J1,)(x) = 0, Vf E L 1 (JR ). Thus by the proof of (c), we have a (J1,)( x) = O. Now let f E Ll(lR) and 1 on some neighborhood of sPO"(x). Pick J1, = 60 , Clearly, U(IL){X) = z , and Jl(s) = 1, Vs E lR. Thus (I - J1,)A - 0 on some neighborhood of sPo(x). By the preceding paragraph, we get

-

f.

1-

0= ai] - J1,)(x) = u(/)(x) - x.

Q.E.D. Proposition 9.2.4. Let (M, lR, u) be a W·-system, E 1 and E 2 be two closed subsets of lR, E = E 1 + E 2, Xi E M(u, E i ), i = 1,2, and x = XIXZ' Then x E M(u,E). Consequently.

SPo(XlX2) C sPo(xd

+ SPO"(X2) ,

Vx}, X2 E M.

378

Proof. 1) First, we assume that SPo(Xi) is compact, i = 1,2. Replacing E, by sPo(xd, we may assume that E, is compact, i = 1,2. Then E = E I + E 2 is also compact. By Proposition 9.2.3 (f), it suffices to show that

u(/)(x) = 0 for any 1 E LI(JR) with f - 0 on some neighborhood of E. Fix 1 E LI(JR) with f 0 on (E + V + V)' where V is a compact neighI borhood of O. Pick Ii E L (lR) such that 1 on some neighborhood of E«, and sUPPh C E, + V, i = 1,2. From Proposition 9.2.3 (g), we have

=

h-

Then for any p E M., by the Fubini theorem we get

(u(/)(x),p)

=

fll p(u6(x))/(s)ds fll I(s) (u (XI) . U(/2)(X2)), 6(u(/t}

p)ds

III I(s) II (td/2( t2)(U6+t l (XI) . U8 +t2 (X2), p)dsdtldt2 III l(s)/dsl - S)/2(S2 + Sl - S)(U61(xdu61+'2(X2),P)dsdslds2' Let

I l(s)/l(sl - S)/2(S2 + Sl - s)ds

k(SI,S2) =

(I * It . 121-,~J (Sl) Fix S2, and take Fourier taansform for Sl' Then k(t, S2) = j(t)(fl * g)(t) , where g(.) = 12,~6(')

= 12(' + S2)'

supp(11 * g)

c

Noticing that

supph.

+ suppy

....

....

supp/l + SUPP/2 C E + V + V .... .... and lOon E + V + V, we have k(t, S2) = 0, Vt. Thus for any S2, we get k(St,S2) = 0, a.e. for SI. Further, by the Fubini theorem we can see that (u(/)(x), p) = 0, Vp EM., i.e., u(/)(x) = o. 2) Let {zn} be the approximate identity for L I (JR) as the beginning of this section. Then it is easy to see that u(zn)Y ---+ y(u(M, M.)), Vy E M. Thus Y E {u(/)(Y) I IE LI(JR), IIIIII < 1}(1, vv E M. Now by Lemma 9.1.3 and Proposition 9.2.1, we have

Y E {u(/)(y)

I I EKl(JR)

and

IIIIII < If',

379

Vy E M. Further, from Proposition 1.2.8 and 1.2.1 we have

x E {u(f){xI} . U(g){X2}

I i, g E KI(lR), Ilfllt and

!Ig1l1 < 1}(7.

Now by 1) and Proposition 9.2.3 (d),

sPa(U(f)(XI) . U(g)(X2))

C

E,

Vf, g E K1(lR)' IIflll and IIglh < 1. Finally, since M(u, E) is u(M, M.)-closed, it follows that x E M(u, E}. Q.E.D. Lemma 9.2.5. Let t E lR, K be a compact subset of lR, and e > O. Then there exists a compact neighborhood V of t such that

Vs E K, and x E M(u, V).

Proof.

Pick a compact neighborhood WI of t and 1 on WI. For each s E K, let

1=

f

E Kl(JR) such that

f6(r) = f(r - s) - ei6t I(r). Then [6(t) = 0, Vs E K. By Lemma 9.1.2, there is k' E Ll(lR) and some neighborhood W" of t such that kll 1 on W" and IIf6 * k"lIl < e. Since K is compact and s ---* fll is continuous from lR to Ll(lR), there is a compact neighborhood W 2 of t such that for each s E K, we can find k E Ll(lR) with k - 1 on W 2 and III" * kill < c. Now let W = WI n W2 , and V be a compact neighborhood of t with V C the interior of W. For each s E K and x E M(u, V), pick k E L1(lR) such ..... that k 1 on Wand Ilf" * kill < c. Clearly, I - 1 on W, and 1* k 1 on W. By Proposition 9.2.3 (g), we have al] * k)(x) = x. Then we obtain

=

.....-

=

Ilu,,(x) -

eilltxll

lIu,,(u(f

* k)(x))

Ilu(fll

* k)(x) II <

- eidu(1 * k)(x) II IIf"

* kill . II xii < cllxll· Q.E.D.

Theroem 9.2.6. Let (M, lR, u) be a W· -system. Then the following statements are equivalent: I} t Espuj 2) For any closed neighborhood V of t, M(u, V) -=I {o}; 3) There exists a net {Xl} C M with !lxdl = 1, Vl, such that

Ilull(XI) - ei lit x, II

---* 0,

380

uniformaly for s E K, where K is any compact subset of lR; 4) If(t) I < lIO'(/)II,VI E L1(lR).

Proof. 1) ~ 2). Let t E spzr, If there is a closed neighborhood V of t such that M(O', V) = {O}, then pick I E L1(lR) with f(t) = 1 and supp f C V. By Proposition 9.2.3 (d), sPo(O'(f)(x)) c V, Vx E M, i.e., O'(/)(x) E M(O', V), Vx E M. Since M(O', V) = (o), it follows that O'(/} = o. By the definition of SpO', f(t) must be 0, a contradiction. Therefore, M(O', V) '# {a} for each closed neighborhood V of t. 2) => 3}. For each closed neighborhood V oft, we can pick Xv E M(O', V) with Ilxv I = 1 by the condition 2}. From Lemma 9.2.5, the net {xv I V} satisfies the condition 3}. 3)~ 4}. Let {XI} be as in 3}, and I E L 1(lR). Then

110'(/) II >

II O' (/ )(x,}II =

> II

II fR O'

3 (

Xl) l(s)dsIJ

fIR eid I(s)dsxlll - fR IIO' (xl) 3

eidxzll

'1/(s) Ids.

Now by the condition 3) and I E L 1 (lR), we can see that li(t} I < 4) ~ 1). It is immediate from the definition of sprr,

110'(/) II. Q.E.D.

Theorem 9.2.7. Let (M, lR, 0') be a W*-system, A be the abelian Banach subalgebra of B(M) generated by {O'(/) I I E L 1 (lR) } , and O(A) be the spectral space of A. Then spzr ,..,.. O(A). Proof. Clearly, 0' is a cotinuous homomorphism from L 1 (lR) to A, and the image of 0' is dense in A. For each P E O(A), (p, 0'(.)) is a non-zero multiplicative linear functional on L 1 (lR). Thus, there is unique t E IR such that

(p,O'(/)) = f(t), Clearly, f(t) = 0 if 0'(/) = from O(A) to spo:

O'*(p)(/)

=

o.

VI E L1(IR).

So t EspO'. Put t = O'*(p). Then 0'* is a map

(p,O'(f))

=

A.

f(tl,

1

v! E L (lR).

We say that 0'* is injective. In fact, if O'*(pt} = 0'* (P2) for some PbP2 E O(A), then

(P1 - P2,O'(/)) = 0, But O'(L 1(IR)) is dense in A, so P1 = P2. Now if t E sprr, define (p,O'(/)) = f(t),

VI E L1(lR).

Vf E L 1(lR),

381

then P is a non-zero multiplicative linear functional on u(L 1 (lR)) . From Theorem 9.2.6 and t E spo, we have

!(p,u(fnl = If(t)1

< Ilu(f)ll,

Vf E L

1(lR).

Thus, P can be uniquely extended to a non-zero multiplicative linear functional on A, i.e., the map o" is also surjective. If U*(PI) = tl ---* t = u*(p) in sprr, then (pl,u(f)) = [(tl) ---* [(t) = (p,u(f)), vi E L 1 (Hl ) . Since u(L 1 (lR)) is dense in A and Ilpll = Ilpdl =.... 1, Vl, it follows that PI ---* P in fl(A). Conversely, if PI ---* P in fl(A), then I(t l ) ---* I(tl, Vf E L 1 (lR), where tl = U*(PI)' Vl, and t = u*(p). Further, by Lemma 9.1.1 we can see that t l ---* t in spu. Therefore, a" is a homeomorphism from fl(A) onto spu. Q.E.D.

Theorem 9.2.8. Let (M, Hl, u) be a W* -system. Then t continuous if and only if spu is compact.

---* Ut

is uniformly

1=

Proof. Let sprr be compact, and pick I E K 1 (lR) such that 1 on some open neighbourhood of sprr. By Proposition 9.2.3 (g), we have u(/)(x) z, Vx E M. Further IIUt(x) - xii

* I)x - u(/)(x) II < 11ft - fill '11xll ---* O(as t ---* 0),

=

Ilu(6t

uniformly for x E M with [Ixll < 1, i.e., IIUt - idll ---* 0 as t ---* o. Conversely, let t ---* Ut be uniformly continuous, and {zn} be an approximate identity for Ll(lR). Then

Ilu(zn)x - z] <

!

IIUt(x) -

xii· zn(t)dt

< sup IIUt(x) - xl! ---* O. Itl$~

i.e., Ilu(zn) - idll ---* 0, where id is the identity operator on M. Therefore, id E A, and fl(A) is compact. Finally, by Theorem 9.2.7 sp« is also compact. Q.E.D.

Theorem 9.2.9. Let (M, Hl, u) be a W*-system, and t ---* Ut be uniformly continuous (i.e. spu is compact). Then there exists h* = hEM such that Ut(x) = UtXU;, where Ut = eiht, Vt E lR and x E M. Proof.

1) For any

eA

x E n, define

_ {I p

-

is a projection of M, and pulf)(x) = 0, } sup P Vx E M, I E Kl(lR) and suppl c (oX,oo) .

382

Clearly, (1 - e.x) is the minimal projection q" E M such that qa(/)(x) o(/)(x) , Yx EM, I E KI(lR) and supp j C (A,oo). In other words, ( 1 _ e )M = .x

[a(/)(x),

y

I x, v E M, [

E

KI(lR) and]a .

suppl C (A, 00)

where the right side is a a-closed right ideal of M. 2) e.x < elJ , YA < u; e.x = 1, if A > max{p, I p, E spo}; e.x = 0, if A < min{p, I p, E spo}. Moreover, A ~ e.x is strongly right continuous. In fact, clearly e.x < elJ , YA < u: If A > max{p, I p, E spa}, then for any IE K1(lR) with suppj C (A.oo),j is zero on a neighborhood of spa. From Proposition 9.2.3 (a) and (g), 0(/) = O. Thus by 1), ex = 1. Similarly, e.x = 0 if A < min{p, I p, E spa}. Now let An > A, Yn and An ~ A, then q > e.x, where q = inf e~n. H n

I

K1(lR) with suppj C (A,oo), then there is rno such that suppj C (Am' 00), Ym > mo. Thus by 1), e.xmo(/)(x) = 0, 'Ix E M, m > mo. Further, qo(/){x) = 0, Yx E M and I E KI(lR) with suppj c (A,oo). Now from E

the definition of e.x, we have e.x

III

>

q, and e.x = q

= inf e.xn• n

3) Let h = Ade.x. Then h* = hEM. Further, let Ut = eiht = Yt E lR. Then clearly we have:

III eiude.x,

III l(t)Ut dt III j(s)de!H YI LI(lR); e~ III h(t)Ut dt = III h(t)Ut dt, Y/I E K1(lR) =

E

A

and SUPP/l C (-00, A], (1 - e.x)

III gl(t)Ut dt III gl(t)Ut dt, =

AE

u,

Yg i E K1(lR) and

suPpgl C (A,oo),

A E lR.

4) Let "'0,'" E lR,1 E K1(lR) with suppj C (.,.,00) and x E M with sPa(x) C (-00''''01. Then by Proposition 9.2.3, spO'(x*) C [-1"0,00). Further from Proposition 9.2.4,

C

°

[-"'0,00)

+ [.,. + c,oo)

C (.,. - TO'oo),

where e > is such that suppj C I.,. + e, 00), Yy E M. Clearly, spO'(z) is compact, where z = x·a(/)(y). Now pick g E KI(lR) with suppg C ("'-"'0,00) and g _ 1 on a neighborhood of spO'(z). Then a(g)(z) = z, By 1), we have e,._,.ou(g)(z) = 0, i.e

383

'tIy E M, IE K 1(lR) with suppj C (T,oo). Further, from 1) we get e,._,.ox·(le,.) = 0, i.e., (1 - e,.) xe,._,.o = 0, 'tiT, TO E lR and x EM with spc(x) C (-00, TO]. 5) Pick I,g E K 1 (lR) with suPPY C (-00,0), suppy C (0,00) respectively. Let T,TO E lR, and 11(·) = I(·)e-i-{"-"o),gt{.) = g(·)e- i.". Then 11"gl E K 1(lR) and supph C (-oo,T - TO)' SUPpgl C (T,oo). By 3), we have f,._,.o I It{s)u.ds = I 11(S)u.ds and (1- e,.) I gl(t)Utdt = I gl(t)Utdt. Now let M C B(H) ,and x E M with spc(x) C (-00, To]. Then by 4) we get (x

I ft(s)u.ds€, I gdt)UtdtfJ) =

0,

'tI€,fJEH. Notice that (x

f 11 (s)u.ds€, I gl(t)Ut dtfJ)

II(u;xUtu.-t€,fJ)/l(s)~dsds

II (fJt(x)u fJ) II( s - t)gl (-t)dsdt J h(s)e- ill("-l"o)ds = h(TO - T), ll€,

where fJt(x) = UtXU;, and

......

Since h(TO - T) = O,'tIT E lR, it follows that h(s) = O(a.e.). continuous, so h(s) = 0, 'tis E lR. In particular,

°

= h(0)· =

But h(·) is

f (fJt (x) €, fJ) k (t ) dt = (fJ (k) (x) €, fJ )

'tI€,fJ E H, where k(t) = I(-t)g(-t)e-it'l"o. Thus, fJ(k}(x) = 0, and spp(x) ...... N(k). Notice that k(s)

=

IE g(r)j(To -

1 g(r)j(TO -

s

C

+ r)dr

00

s

+ r)dr,

where e > 0 is such that suppg C (c,oo). Since suppj C (-00,0), if follows that k(s) = 0 if s < TO + c. Further, since spp(x) is closed and e, I, g are arbitrary, we can see that spp(x) C (-00, To],'tIx E M with sPc(x) C (-00, TO],

.

i.e.,

M(u, (00,1"0]) C M(fJ, (-00, TO]),

vr«

E lR.

384

Now by Proposition 9.2.3 (b), we have also

M(oo, [TO, 00)) c M({J, [TO, 00)), E JR,and I,g E Kl(JR) with supp

VTo E JR.

1 Ci1" (-00,0), suppy C

(0,00) respectively. Let 11(') = g(.)e- .. Then 11,gl E K 1 (JR ) and suppl1 C (-oo,T), SUPPYI C (T,OO). For any x E M, since u(/t}(x) E M(oo, (-00, TJ) C M({J, (-00, T)), it follows that gl on a neighborhood of sPp(oo(/l)(X)). Thus, (J(gl)(oo(/l)(X)) = 0. Notice that 6) Let

T

1(·)e-i1"·,gl(·) =

=°

(J(gtl (00(/1)( x))

II {Jt(oolJ(X))gl (t)1t (s)dsdt II ~t(ulJ+t(x))gl(t)/I(S)dsdt II ~t(oo,,(x))f(s - t)g(t)e- i a1" dsdt I h(s)e- i"1"ds = h(-T), where

~t =

(Jt 0 oo-t, Vt E JR, and h(s) =

J~t(oolJ(x))/(s - t)g(t)dt.

Now since h( -T) = 0, VT E JR, and h(·) is continuous, it follows that h(s) 0, Vs E JR. In particular, 0= h(O)

=

J~t(x)k(t)dt

where k(t) = I(-t)g(t). Thus, sp~(x) C

=

=

~(k)(x),

JJ(k). Notice that

! y(r)j(r - s)dr 1 g(r)l(r - s)dr, 00

k(s) =

=

where e > 0 with suppy = (c,oo). Since suppf C (-00,0), it follows that k(s) = 0 if s < c. Furhter, since sp~(x) is closed, and s , t, 9 are arbitrary we can see that sp~(x) C (-00,0], Vx E M. Similarly, from (J(/l)(oo(gd(x)) = 0, we have sp~(x) C [0,00), Vx E M. Thus, sp~ (x) C {o}, Vx E M. Finally, by Lemma 9.2.5 we get ~t = id, i.e., oot = (Jt, or oot(x) = UtXU;, Vt E JR, and x E M.

Q.E.D. Notes:

Spectral subspaces were introduced by R. Godement. It may be viewed as an attempt to extend the Stone theorem. A systematic study of spectral subspaces and their applications to dynamical systems was presented

385

by W.B. Arveson. Theorem 9.2.6 is due to A. Connes. And Theorem 9.2.8 and 9.2.9 are due to D. Oleson.

References. [9), [17], [57), [122), [123].

9.3. The Connes spectrum Let (M, JR, 0") be a W*-system, and denote by MeT the fixed point algebra:

MeT

= {x E M I O"t(x) = z, Yt E JR}.

Clearly, MeT is a W*-subalgebra of M. For a projection e E MeT,O" induces an action o' on Me such that u;(exe) = eO"t(x)e, Yt E JR, x E M. Then we obtain a W·-system (Me = eMe, JR, o' = O"IMe), and denote its Arveson spectrum by spo", The Oonnes spectrum of W*-system (M, JR, 0") is defined

Definition 9.3.1. by

r(O") = n{spO"e

I 0 i= e E Proj

(MeT)},

where Proj(MeT) is the collection of all projections of the fixed point algebra Mer. Clearly, r(O") is a closed subset of JR and 0 E r(O"). Lemma 9.3.2.

For any e EProj (Mer) with e

i= 0,

and a closed subset E of

JR, we have Me(O"e, E ) = M(O",E) n Me. where Me(u e,E) = {x E Me

I SPeT"(x)

C

E}.

Proof. If x E Me, then O":(x) = O"t(x) , Yt E JR and ue(f)(x) = 0" (f)(x) , Yf E L 1 (JR ). Thus by Definition 9.2.2, we have sPer(x) = sPer"(x),

Yx E Me.

That comes to the conclusion.

Proposition 9.3.3.

Proof.

Q.E.D.

Let (M, JR, u) be a W*-system. Then r(O")+spO" = sper.

First, since 0 E r(O"), it follows that spzr C sprr + r(O").

386

Now let Al E I'[o], A2 E spo. We need to prove A = Al + A2 E spo. From Theorem 9.2.6, it suffices to show that M(o, V) =1= {o} for every compact neighborhood V of A. Fix a compact neighborhood V of A, and pick compact neighborhood ~ of Ai,i = 1,2, such that VI + V2 C V. Since A2 E V2 , it follows from Theorem 9.2.6 that M(u, V2 ) =1= (o). Let X2 E M(u, V2) with X2 =1= o. Then Ot(xi) =1= 0, Yt E JR. Let ot(xi) = Vtht be the polar decomposition of ot(xi),et = VtV;, Yt E JR, and e = sup [e, It E JR}. Clearly, e, =1= 0, Yt E JR, and e =1= o. We say that e EProj (MO). In fact, if M c B(H), then eH = [ot(x;)H I t E lR]. Thus olJ(e)H = «n, i.e., olJ(e) = e, Ys E JR. Now Al E r(o) = n{spop I 0 =1= p E Proj (MO)}. In particular, Al E spo", From Theorem 9.2.6 and Lemma 9.3.2, we have

M(o, VI) n Me = Me(oe, VI)

=1=

{O}.

Then there is Xl E M(u, VI) n Me with Xl =1= o. By the definition of e and eXI = XI =1= 0, there exists t E JR such that etXl #- o. So we can find €, fJ E H such that 0 #- (etXI€,fJ) = (Xl€' etfJ) , where M C B(H). But etfJ E ot(x;)H, then there is ~ E H such that (XI€,Ot(X;)~) #- o. Thus Ot(X2)· Xl #- O. Put x = Ot(X2) . XI(#- 0). By Propositions 9.2.3 (b) and 9.2.4, we get sPo(X C SPo(X2)

+ SPo(Xl)

C

V2 + VI

C

V.

Therefore, x E M(o, V), and M(o, V) #- {O} . Proposition 9.3.4. subgroup of JR.

Q.E.D.

Let (M, JR, 0) be a W*-system. Then r(u) is a closed

For any e EProj(MO) with e #- 0, from Proposition 9.3.3 we have r(oe) + spe" = spo". Clearly, r(o) c I'{e"}, and r(u) c spo". Then r(o) + r(u) c r(u e) + spo" = spe", Proof.

YO #- e EProj (MO). Further from definition 9.3.1, we obtain that r(o) I'(e] c I'(e). Moreover, from Proposition 9.2.3 (a) and (b) we have

+

U sPo<:(x) = spo" = U sPu.:(x*) = -spoe, zEAl.:

zEAl.:

YO #- e EProjMu. Thus, r(u) = n{spu e

I0

#- e E Proj (MO)} = -r(o). Finally, since r(o) is closed and 0 E r(o), r(o) is a closed subgroup of JR.

Q.E.D. Lemma 9.3.5. such that

If H is a proper closed subgroup of JR, then there is A > 0

H = 7ZA.

387

Proof. We may assume that H such that

H

n (O,A)

f:. {O}. Then we claim that there is A > 0 = 0,

and

A E H.

In fact, if such A does not exist, then there is a sequence {An} C H with An > 0, \In, and An ~ o. For any Jl E JR and n, we can find N(n) such that N(n)A n < Jl < (N(n) + l)A n. Thus dist(Jl, H) < An ~

o.

Since H is closed, it follows that Jl E H ,and H = JR, a contradiction. Hence such A exists. Clearly, H =.> mA. If Jl E (H\mA) with Jl > 0, then there is n such that

nA < Jl < (n+ l)A. Now 0 < (Jl - nA) < A and (Jl - nA) E H. This contradicts H Therefore, H = mA.

n (0, A) = 0. Q.E.D.

Remark. From Proposition 9.3.4 and Lemma 9.3.5, the Connes spectrun r(u) of a W*-system (M, JR, u) is one of the following forms:

JR,

{O},

and

mJl

(some Jl > 0).

So the subgroup er(a) of the multiplicative group (0,00) is one of the following forms: (0,00),{1}, and {AnlnEm} (someAE(O,l)).

Lemma 9.3.6. Let {Vj liE A} be an open cover of JR, and x E M with x f:. o. Then there exists a / E L1(JR) and some j E A such that .... supp f c Vi, and u(f)(x) f:. o.

i

=

Proof. Let 10 = {f E K 1 (JR ) I supp c Vj, for some j E A}, and I 10 • Clearly, 10 is an ideal of L1(lR). For any t E JR, there is j E A such that t E Vi. By Lemma 9.1.1, we can find / E K1(JR) such that j(t) = 1 and suppj C Vi. Then / E 10 c I, and t ft 11.. Thus 11. = 0, and 1= L 1 (JR ) by Lemma 9.1.6. Moreover, by the proof of Lemma 9.2.5 u(zn)(x) ~ x f:. 0, so there is ~ with u(zn)(x) f:. o. Now since 10 is dense in L1(JR), thus we can find f E 10 such that u(/) (x) f:. o. Q.E.D.

Lemma 9.3.7. Let el,e2 E Proj (Ma),and el f:. O,e2 f:. (relative to M), then

o.

H el - e2

388

Proof. Let A E r(u e 1 ) . We must prove A E r(ut: 2 ) , i.e., A E spu h , '10 =I 12 E Proj (MO) and 12 < e2' By Theorem 9.2.6, for 0 =I 12 E Proj (MO) with h < e2 and a compact neighborhood V of A, we need to show that

M(u, V) n M h = Mh(u h , V) =I {Ole Let U, W be compact neighborhoods of A, 0 respectively such that U + W C V. Pick an open cover {Vj I j E A} such that V; - Vi c W, vi. Since el ,...., e2, there is U E M such that u·u == el, uu* == e2. By huu· = 12e2 ==. h =I 0, we have that 12u =I o. From Lemma 9.3.6, there exists g E Ll(JR) such that suppg c Vj for some i, and u(g)(/2u) =I o. Since 12, el E MO and 12u == 12 . 12U . eb it follows that x == 12xet, where x == u(g)(/2u). Clearly, spo(x) C supp g, and (sPo(x) - sPo(x)) C (suppg - suppg) c (Vi - Vj) C W. Let M c B(H),and It be the projective from H onto [Ut(x*)Hlt E JR]. Then 11 E Proj (MO) and It =I o. Moreover, by elUt(x*) == Ut(elx·) == Ut(x·), 'It E JR, we have 11 < el. Now A E U and A E r(u tl 1 ) C spu h, then from Theorem 9.2.6 and Lemma 9.3.2 we have

M(u, U) n Mil == Mil (u h , U) =I {O}. Pick 0 =I y E M(u, U) n M h . By the proof of Prosition 9.3.3, we can also see that

(x)y =I 0, for some t l E JR. Again by the definition of II, there is t 2 E JR such that UtI (X)YUt2 (x·) =I O. Put z == UtI (X)YUt2(X*). Since 12x = x,x*f2 == x*,and ut(/2) = 12, 'It, it follows that z == 12zh E M h . Now by Propositions 9.2.4 and 9.2.3 we have UtI

(x)yh ==

UtI

sPo(z) C C

Therefore, M(u, V) n M h

sPo(x)

+ spo(y)

- sPo(x)

U+WcV.

=I {Ole

Q.E.D.

Definition 9.3.8. Two actions a, T of JR on M are said to be outer equivalent, if there is an one-parameter strongly continuous group {Ut I t E JR} of unitary elements of M such that

Ut+& == Utu(U,,) ,

Tt(X) == UtUt(x)u;,

VS,t E JR,x EM. Lemma 9.3.9. Suppose that 0' and exists an action 1 of JR on Ai such that

T

are outer equivalent. Then there

1t(x ® en) == Ut(x) ® en, { 1t(x ® e22) == Tt(X) ® e22,

389

"'It E JR.,x E M, where M = M {eii [I

Proof.

<

e M 2 ((C )

=

{(~

: ) la,b,c,d EM}, and

t.i < 2} is the matrix unit of M 2 ((C ). Let {Ut

It

E JR.} be as in Definition 9.3.8, and define

Xu X12)) _ (a t(xll) ( ( "Yt X21 X22 - Utat (X21)

at( Xl2)u; ) Tt (X22) ,

"'It E JR., xii E M, 1 < i, i < 2. It is easily verified that "Y does the job.

Q.E.D. Proposition 9.3.10. Let a, T be two actions of JR. on M, and let a and be outer equivalent. Then r(u) = I'{r},

Proof.

Let (M,JR.,,) be as in Lemma 9.3.9. Clearly, If write it = 1 @ e21, then

l@ell

and

l@e22

T

EM"'.

...... '" = 1 @ en,

U U

So by lemma 9.3.7, we have obvious that and Therefore, r(u)

r("Yl~el1) = r("Yl~e22).

On the other hand, it is

(M, JR., a) '"'" (Ml~e2Z' JR., ,l~e22).

= I'{r].

Q.E.D.

Notes. The Connes spectrum was defined by A. Connes. Propositions 9.3.4 and 9.3.10 were also established by A. Connes. References. [17].

9.4. The Connes classification of type (III) factors finite case)

«(7"-

Let M be a a-finite W·-algebra. Then there exists a faithful normal state lp on M. Let {1I'"cp,Hcp,~cp} be the faithful cyclic We-representation of M generated by!p. Clearly, Ecp is a cyclic-separating vector for the VN algebra 11'" cp(M) on H cpo From Tomita-Takesaki theory (see Section 8.2), there is the modular operator Acp (a non-negative invertible self-adjoint operator on Hcp),and a W·system (M, JR., acp), where {af I t E lR} is the modular automorphism group of M corresponding to
390

Proposition 9.4.1.

s E sp aCP

{:=:>

e" E sp.6.rp. Consequently,

eSPu" = sp.6.cp\{O} = sp.6.cp n

Proof.

Let

f

(0,00).

E L 1(lR), and x E M. Then

j(ln .6.rp)1rcp (x) ell'

=

! .6.~f(t)dt1l'"cp(x)ecp 11'"

cp(acp(f)(x)) ell"

Hence,

aCP(f) = 0

{:=:>

j(ln .6.cp)1I'"cp(x)acp

{:=:>

[(In .6.cp) = 0

{:=:>

[(InA) == 0,

==

0,

Vx E M

VO < A E sp.6.CP'

Since {In AI 0 < A Esp .6.rp} is closed, it follows that spall' = n{.AI(j) I acp(f) oj = {In A I 0 < A E sp .6.cp}, i.e., e 8 P U " == sp

==

s;\ {O}. Q.E.D.

Definition 9.4.2.

Let M be a-finite W· -algebra, and define r(M) = r(arp),

where 'P is a faithful normal state on M. From the Connes unitary cocycle theorem (Proposition 8.3.3) and Proposition 9.3.10, r(M) is well-defined, i.e. r (M) is independent of the choice of 'P. If 'P is a normal state on M, let p ==sUPP'P, then 'P is a faithful normal state on M p • So there is the modular operator .6.cp for M p • Define

S(M) = n{sp .6.cp Proposition 9.4.3. e$ E S(M), i.e. er(M)

I 'P is a normal state on M}.

Let M be a a-finite W*-algebra. Then s E r(M)

{:=:>

= S(M) n (0,00) n{eSPu"

I 'P is a normal state on M},

where {ai I t E JEl} is the modular automorphism group of M p corresponding to 'P, and p =sUPP'P.

391

Proof. Let ep be a normal state on M, and p =supptp. Pick a normal state ,p on M with supp,p = 1 - p. Then p = (tp + ,p) is a faithful normal state on M. Let {O"t It E B}, {u( It E lR} be the modular automorphism groups of M, M p corresponding to p, ep respectively. By the KMS condition and the uniqueness of the modular automorphism group, we can see that

l

UtlMp = ur,

\It E B.

Moreover, p E MD. In fact, from

tp(xp) = tp(pxp)

= ep(px) ,

,p(xp) = ,p(xp(l - p))

=

0 = ,p((l - p)px) = ,p(px),

we have p(xp - px) = 0, \Ix E M. Now by Proposition 8.3.2, we get p E MD. Noticing that r(u) = n{spu

q

10 =f. q E Proj(M

and

r(M) = r(u)

c

) }

10 =f. q E Proj(MD),q < p},

r(u'P) = r(uIMp ) = n{spO"q we have

D

r(O"'P)

c spO"'P

= {In.\

I 0 < A E sp..6.'P}

by Proposition 904.1. Since tp is arbitrary, it follows that er(M)

C S(M) n (0,00).

Conversely, let s E Band e8 E S(M). Pick a faithful normal state tp on M, and let {O"t = O"f 1 t E B} be the modular automorphism group of M corresponding to tp, and 0 =f. p E Proj (M D). Clearly, {(O"tIMp) I t E B} is the modular automorphism group {O"t I t E lR} of M p corresponding to ,p = (tpIMp)/tp(p). Then from e" E S(M) and Proposition 8.4.26, we have e8 E sp..6.,p and s E spO",p = sPO"P. Since p E Proj (MD) is arbitrary, we get

s E n{spuP

I 0 =f. P E Proj(M D)} = r(u) = r(M).

Therefore er(M) =

s(M) n (0,00). Q.E.D.

Remark. Let tp be a faithful normal state on M, and {O"t I t modular automorphism group of M corresponding to ip: Then r(M) = r(O") = n{spu e

I 0 =f. e E Proj (M

D

) } .

E

lR} be the

392

Since (
=

n{sP~e 10=1- e E Proj(MO')} n (0,00).

Moreover, from the remark under Lemma 9.3.5 f(M) is one of following forms: lR, {O}, and mp, (some p, > 0), and

er(M)

is one of following forms:

(0,00),

{I},

and {An

I n E m}

(some

A E (0,1)).

Theorem 9.4.4. Let M be a o-finite factor. Then the following statements are equivalent: 1) M is semi-finite;

2) S(M) = {I}; 3) lit S(M).

°

Proof. Let M be semi-fiinite. Clearly, 1 E S(M). Now pick a non-zero finite projection p of M. From Proposition 6.3.15, there is a normal state

°

°

Definition 9.4.5 Let M be a o-finite type (III) factor. M is said to be type (1110)' if r(M) = {O}, or S(M) n (0,00) = {I}. Let A E (0, 1).M is said to be type (III>.), if r(M) = mIn A, or S(M) (0,00) = {An I n Em}. M is said to be type (IIId, if r(M) = lR, or S(M) ::,) (0,00). Now from Theorem 9.4.4, we have immediately the following.

Proposition 9.4.6. Let M be a a-finite factor. Then: M is type (IIIo)~ S(M) = {O, I}; M is type (lIh)~ S(M) = {O, An 1 n Em}, VA E (0,1); M is type (11I1)~ S(M) = [0,00).

n

393

Notes. The theory of factors in Chapter 7 was presented by F.J. Murray and J.Von Neumann in 1930's. After near 40 year, A. Connes gave a new essential development for the classification of factors.

Reference. [17].

9.5. Examples of type (IlIA) factors Proposition 9.5.1.

Let (M, lR, a) be a W* -system. Then we have

where Z(M U) is the center of the fixed point algebra M U.

Proof. By Definition 9.3.1, it is obvious that the left sided is contained in the right side. Now it suffices to show that for any 0 i= e E Proj (MU) there is o i= e E Proj(Z(MU)) such that spo" =spat . For 0 i= e E Proj(MU), let

e = sup{ueu* I u is a unitary element of M

U

} .

Clearly, 0 i= e E Proj(Z(MU)). Now by Theorem 9.2.6 and Lemma 9.3.2 we need to prove that

M(a, E)

n Me i= {O} -<==> M(a, E)

n Me

i= {O}

for any closed subset E of lR. Since e < e, the "==>" is obvious. Suppose that x = exe is a non-zero element of M(a, E). By the definition of e, there are unitary elements u, v of MO such that

(ueu*)x( vev*)

i= o.

Let y = eu*xve. Then 0 i= y E Me. Since for any z E MO, u(f)(z) = l(O)z, Vf E L 1(R), it follows from Definition 9.2.2 that SPoz = {a}, Vz E MU\{O}. Now from Proposition 9.2.4 we have that spoy = spox c E. Thus, M(u, E) n Me i= {O}. Q.E.D.

Corollary 9.5.2. Let M be a a-finite factor, p be a faithful normal state on M, and {at I t E lR} be the modular automorphism group of M corresponding to p. Then we have r(M) = n{spa e I 0 i= e E Proj(Z(M U))}.

394

Proposition 9.5.3. Let M, tp, {Ut} be as in Corollary 5.2, and .6. e be the modular operator for Me corresponding to (tpIMe ), '10 i= e E Proj(Z(MCT)). Then we have

S(M) = n{sp .6. e I 0

i= e E Proj(Z(MOO))}.

By Corollary 5.2, Proposition 9.4.1, Proposition 9.4.3 and its Remark, we can see that Proof.

er(M) =

S(M) n (0,00)

i= e E Proj(Z(MCT))} n (0,00). each 0 i= e E Proj (Z(MOO)), Me

n{sp.6. e 10

If M is type (III), then for is also type (III). By Theorem 9.4.4, it follows that 0 E S(Me ) and 0 E sp.6.e. Hence

S(M) = n{sp.6. e I 0

i= e E Proj

(Z(M CT))}.

If M is semi-finite, then by Theorem 9.4.4 we have 0 (/. S(M). We must exhibit a non-zero e E Proj (Z(MCT)) such that 0 (/. sp.6.e, or, equivalently, such that .6. e is bounded. From the proof of 8.3.6, we can write O't(x) = h-itxhit, 'It E JR., x E M, where h is a non-negative invertible (maybe unbounded) operator on H (here, assume that M C B(H)) ,and each spectral projection of h belongs to MO". Further, since xh it = hit, 'It E JR., x E Moo, each spectral projection of h belongs to Proj(Z(MOO)) indeed. Pick n(> 1) such that e of h.

= If de>. i=

0, where h

= roo Ade).

is the spectral decomposition

" 10 Consider the functional ",(.) = ip(he·) on Me. ut(x) = hitUt(x)h-it = x,

'It E JR.,

By Lemma 8.3.4, we have

x E Me.

Hence, '" is a trace on Me from Lemma 8.3.5. Consequently,

tp(X·x) < n"'(x·x) n"'(xx*) < n 2ip(xx*),

'Ix E Me.

If {1re, He) Ee} is the cyclic * representation of Me generated by (tpIMe ) , then by Proposition 8.2.2 we have

11.6.~/21re(x)EeI12 =

lIie.6.~/21re(x)Ee\12 = II1r e(x*)EeIl 2

tp(xx*) < n 2tp{x·x) = n2111re(x)€eI12, 'Ix E Me. Therefore, .6. e is bounded.

Q.E.D.

395

Now let (G, O,J.L) be a group measure space. By Definition 7.3.9 we have

o<

rt{') <

00,

r,d(') = rt(s-l')r a ( ' ) ' Let H = L2(O,J.L),M H,and

VS,t E G,a.e.J.L.

= {mf I IE LOO(O,J.L)} be the multiplicative algebra on

at(mf)

•

= UtmfU; = mft-p

where It-I(') = l{t-1'),VI E LOO(O,Jl),t E G. Then, t --+ Ut is a unitary representation of G on H, and (M, G, a) is a dynamical system. Let H = H ® 12 (G) and define ,....."

,....."

Va E M, s, t E G, e(·)..-E H. Then we have the crossed product M X a G = {1r(M) , >'(G)}". Put M = M X a G simply. Now suppose that (G, O,J.L) is free and ergodic (see Definition 7.3.10). Then ~ Lemma 7.3.11, 1r(M) is maximal,Sommutative in M,and M is a factor on H. By Lemma 7.3.4, for each x E M there is unique function b.(: G --+ M) such that Psxp; = u:bd-lU S , Vs,t E G . Let q>(x) = be. Then q> is a unital a-a continuous positive linear map from M to M. Assume that Ip is a faithful normal state on M. Then rp = Ip 0 q> is a faithful normal state on M. Let {u. I . E lR} be the modular automorphism group of M corresponding to rp. ~

,....."

~

,....."

Lemma 9.5.4.

Proof.

Z(MO) C 1r(M)

Let a E M,

€

C

MO.

eE H, s, t E G. By Lemma 7.3.4 we have

€(.)

where E Hand = D.,te. Hence, the function b.(: G --+ M) corresponding to 1r(a) is b. = D.,ea, where e is the unit of G. Let y E M, and c. be the function (: G --+ M) corresponding to v, i.e.,

396

Then

(1r(a)y).,t =

L 1r(a).,kYk,t

kEG

for s,t E G. So the function (: G ---+ M) corresponding to 1r(a)y is ac., and $(1r(a)y) = I(J 0 ep(1r(a)y) = l(J(ac e). Similarly, $(Y1r(a)) = I(J(C eex e0)) = l(J(cea). Since M is abelian, it follows that $(1r(a)y) = i(v1r(a)) , Vy E M,a E M. Now by Proposition 8.3.2 we obtain that 1r(M) eM". Further, we have

since 1r(M) is maximal commutative in

M.

Q.E.D.

Now we assume further that: 0 is compact; p, is a probability measure on 0; and for each t E G there are two positive constants Ct and '7t such that o < Ct < rt(') < '7t < 00, V· E G. Define l(J(mf) = f jdp" 'If E LOO(O,JL). Clearly, I(J is a faithful normal state on M. Let .6. be the modular operator corresponding to the faithful normal state $ = I(J 0 ep on M.

il = L

tEG

.6.

is spatially isomorphic to the operator h = ffhEGht on H, where ht is a bounded invertible positive operator on H = L 2(0, p,)

Lemma 9.5.5.

corresponding to multiplication by the function rt l ( . ) = rt-1('), and ff)tEGh t is 2 IIhtetl1 < a self-adjoint operator on il with the domain D = {(e,l E ill

L

tEG

oo}.

Proof. It is easy to check that h = ffitEGh t is a self-adjoint operator on jj with the "domain D.

Let x, y E

M,

and b., c. be the function (: G

---+

M) corresponding to z, y

397

respectively. Then

L

(xY) ..,t =

X ..

,kYk,t

/cEG

L

U:b./c-l u e ' UiC/ct- 1 U/c

/cEG

U:(L b./c-l • U ../c-1Ckt-1U:k-du. /cEG U: C~= b./c-l • a ..k- 1 (Ckt- 1 ))U. /cEG

U:(L bat-l/c-l • a",-lk-1(CIe))U., /cEG

(Y~)8,t =

P.Y~P; =

(PtYP;)*

(U t*Ct6- 1 Ut )* =

* Ut U t*C,..-l

\/S, t E G.

L

Hence, the functions (: G -+ M) corresponding to xy, Y· are b. 1e • a./c (C/c-l ), d. = a. (c~- d respectively. Further, the function (: G -+ M)

tEG

corresponding to y. x is

L

d./c·

a./c(b/c-l) =

IeEG ".....,

L

a,/c(ck-1.-1b/c-l).

IeEG

;y

Let {1f,K, E} be the cyclic have that

* representation

,....,

of M generated by $. Then we

(i(x) €, 1f(y) €) = $(y*x)

/cEG

10

/cEG

10

,....,

\/x E M,

where k -+ b/c = m J.. is the function (: G -+ M) corresponding to,....,x. Then ,....,,...., U can be uniquely extended to an isometry from K to H. Let M o be the * subalgebra of M generated by {1r(M), >.( g}. Clearly, the number of nonzero components of U1f(x)E is finite, \/x E Mo. Moreover,~OO(n,cl. is dense in H = L 2(n,p,). Therefore, U is a unitary operator from K onto H.

398

Let Ho = {(6l~EG E H I 6: = 0 except fiinite k's}, and he = hlHo. It is easy to see that h is the operator closure of ho• Hence, h is the unique self-adjoint extension of he. Furhter, we can see ........ that,., h is the operator closure and the unique self-adjoint extension of (hIU1f(Mo)€). ........ ........ For any x E Mo and y EM, let b. = mi. (Ik = 0 except finte k' s) and c. = mg. be the functions (: G ~ M) corresponding to x and y respectively. b.klX.(CZ), it follows Since the function (: G ~ M) corresponding to xy· is

L

kEG

that

(Xl/21F(x)e, Xl/21F(y)E) =

(jXl/21F(y) e,jXl/21f(X) E) tP(xy·) =

L

f fkYk dJ1-

k

(h(lkr~/2), (gl:r~/2))

= (U- 1hU1f(x) e, 1f(y) e).

By Proposition 8.2.2, Xl/2 is the operator closure of (Xl/211f(M)e). Hence,

(Xl/21f(x)l, Lil/2i7) = (U- 1hU1f(x)l,fJ), \/x E

Mo,ij E

D(Xl/2). Therefore, we can see that Xl/21f(x)l E D(Lil/2) and X1f(x)l = U- 1hU1f(x)l,

\/x E

Mo.

Now UXU-l is a self-adjoint extension of (hIU1f(Mo) E). By the preceding paragraph, we have uXU- 1 = h. Q.E.D. Now let E be a Borel subset of 0, and J1-(E) > O. Then p = m X E is a non-zero projection of M. By Lemma 9.5.4, if = 1r(p) is a non-zero projection of MU. Clearly, cp; = ~IMp is a faithful normal state on M;, and the modular ........ ,., ........ automorphism group of M p corresponding to CPp is uP = alMp. Suppose that ,., ,....., /)"r; is the modular operator corresponding to cpr;. Let x E M and b. = mi' be the function (: G ~ M) corresponding to x. Then it is easy to see that ~tI'tJ"..."

x E

~

Mp <==> supplk C En kE,

\/k E G.

Thus, similarly we have the following. ,..,

Lemma 9.5.6. £lp is spatially isomorphic to the operator b» = ffhEGht,E 2 on EB t EGL (Ot , vd, where Ot = En tE, Vt = J1-10t, and -ht,E is the operator on L 2 (Ot, Vt) corresponding to multiplication by (r,l(.) lOt), \/t E G. Lemma 9.5. 'T. A E S(M) if and only if for each Borel subset E of 0 with J1-( E) > a and c > 0, there exists a non-Zero projection q of M and t E G such that sup{q, lXt(q)} < P and

Sp(ht-lqlqH)

C

(A - s, A + c),

399

where p = m X E ' and ht-l is as in Lemma 9.5.5.

Proof. By Proposition 9.5.3 and Lemma 9.5.4, we have S(M)

= n{spLi p I p =

1r(p), 0

# p E Proj(M)}.

Hence, from Lemma 9.5.6 we obtain that A E S(M) -<==> A E spLi p,

-<==> A E sph E , -<==> A E

Vp = 1r(p), p =

m X E ' and

J1.(E) > 0

Vp,(E) > 0

U Spht,E,

Vp,(E) > 0

tEG

Vp,(E) > 0 and e > 0, there exists t E G

¢:::::>

such that (A - s, A + e) n sp be-v,s

# 0.

......,

If A E S(M), then for each Borel subset E of 0 with p,(E) > 0 and e > 0, there exists t E G such that

(A - e, A+ e) n sp ht-l,E # 0. Let F = {s E En t-IE I rt(s) E (A - s, A + e)}. Then q projection of M such that

= m XF

is a non-zero

where p = m X E ' Conversely suppose that A has the following property: for any Borel subset E of 0 with J1.(E} > 0 and e > 0, there exists 0 # q EProj{M) and t E G such that

We may assume that q = mxp' and FeE. Since sup{q,at(q)} < p, it follows that FeE n t- I E. Moreover, Sp(ht-lqlqH) = (A - s, A + e) n sp ht-l,E # 0. Therefore, A E S(M). Q.E.D. Let (G, 0, IJ) be a group measure space. Define the ratio set r(G)(C [0,00)) as follows. A(> 0) E r(G) if and only if for any Borel subset E of 0 with p,(E) > 0 and e > 0, there exists a Borel subset F of 0 and t E G such that p,(F) > 0, F u tF c E, and

Definition 9.5.8.

dp, 0 t dp,

- - (s) - A < s, Noticing that rt(s)

=

Vs E F.

dE£~~-l (s), from Lemma 9.5.7 we have the following.

400

Proposition 9.5.9. Let (G,O,JL) be a free and ergodic group measure space. Further, suppose that 0 is compact, JL is a probability measure on 0, and for each t E G there are two positive constants Ct and!lJ such that o < Ct < d:;t(s) < Tit < 00, Vs E G. Then there is a factor M such that

S(M) = r(G). Let On(n = 1,2,"') be the additive groups of integers, reduced mod 2, i.e., On is a compact (discrete) group composed of two elements {O, I} as follows: 0+0 = 0,0+1 = 1+0 = 1, and 1+1 = O. Let n = x~=10n be the direct product of {On I n = 1,2,"'}' Thus 0 is a compact Hausdorff space satisfying the second countability axiom, and 0 is a compact group. Let G be the set of those a = (an) E n for which an =I 0 occurs for a finite number of n only. Then G is a countable group. For bEG, define a homeomorphism of n : a --+ b(a) = a + b(Va E 0). Let /.Ln be a probability measure on On with JLn ({O})

= Pn,

JLn ({ I}) = qn,

where Pn E (0,1), and Pn + qn = 1, Vn. Let JL = X~=1JLn be the infinite product measure of {JLn} on O. Then, JL is a probability measure on O. If a is a permutation of {O, I}, i.e., u(O) = 1, u(l) = 0, then it is easy to see that dJLn 0 o (s) = 2.-1, S = 0,1, dJL qn Vn. Let Cn be the element of G such that the n-th component of Cn is 1 and other components of Cn are 0, Vn. Then we have

(pn)

JL

0

Cn

=

Xk"lnJLk

X

(JLn

0

u)

and dJL 0 Cn (a) __ (Pn) 2a.. -1 , dJL qn

(

)

Va = ak k E 0,

Vn. For any bEG and b =I 0, there is unique finite sequence {i 1 < ... < i k } of positive integers such that b = Cil + ... + Cit' Hence, we have d b JL 0 ( a) dJL

=

IT 00

n=l

(

Pn

)

(2a.. -1)6..

,

qn

Va = (an) E n,b = (bn) E G; and (G,O,JL) is a group measure space. Let bEG and b =I o. Clearly, {a E 0 I b(a) = a} = 0. Thus, (G, 0, JL) is free. For any Borel subset E of 0, let F = b(E). Clearly, F is also a Borel

U

bEG

subset of 0, and b(F) = F, Vb E G. Then for any a E F we have

aE

F

401

whenever a is obtained by changing any finitely many components of a. Thus, for any positive integer n, F has the following form:

where Fn is a Borel subset of Xk=n+IOk. Now if C = C n X x~n+IOk is a cylinder subset of 0, where Cn is a (Borel) subset of Xk=IOk, then it is obvious that J-L(F n C) = J-L( O« x Fn) = J-L( C)J-L(F). If K is a compact subset of 0, then we can see that K

= n(1I"n(K) X X~n+IOk), n

where 1I"n is the projection from 0 onto Xk=IOk. Thus,

J-L(K n F) = limJ-L(1I"n(K) x Xk=n+IOk)J-L(F) n J-L(K)J-L(F) Further, by the regularity of J-L we have

p,( C n F) for any Borel subset C of

=

J-L( C)J-L(F)

o. Now if E satisfies the following: p,(bE~E) =

0,

Vb E G,

then we have

J-L(E)

=

p,(F) = J-L(F n F)

=

p,(F)2

=

J-L(E)2,

and either p,{E) = 1 or J-L(E) = o. Therefore, (G, 0, J-L) is ergodic. Now let A E (0,1), and Pn = A(1 + A)-I, Vn. Then from Definition 9.5.8 we have r{G) = {O, An I n E LZ}. Thus by Propositions 9.4.6 and 9.5.9

Proposition 9.5.10.

M=

M x a G is a type (III>.) factor.

Type (III>.) factors (0

< A < 1) do

exist.

Remark. By Proposition 9.5.9 and above construction, we can also obtain the examples of type (1110) and (III') factors (see [165]). Now keep the above notations: On = {O,I},Vn;O = X~=10n;G;P,n({O}) = Pn E (O,I),J-Ln({I}) = qn,Pn + qn = I,Vn;p, = x~=lJ-Ln; and the element en of

G,Vn. Let

402

and

E

{

E(

(11" ( X) E) (b) = ab-1 ( x) (b) = ab(x) b), ab(mf) = ubmfut = mil" 16(,) = f(· + b),

(A(c) E) (b) =

E(b + c), ""

,....,.

\:f,!, c E G, x E M, f E LOO(O, p,), and € E H. Clearly", the crossed product M = M X a G = {1I"(M) , A(G)}" admits a cyclic vector eo:

C(b) = {I,0,

if b = 0, otherwise,

vo

where "1" is the constant function 1 on O(E L 2(O,p,) = H). For any ai E {O, I}, 1 < i < n, let p(al" .. ,an) be the operator on H = 2(O, L p,) corresponding to multiplication by XE(al,···,a .. ) , where E(a},"', an) = (al"'" an) X Xr=n+l0k(C 0), and put

Vn. Then it is easy to check that

V~

E {O, I}, 1 < i < n, 1 < k < n. Thus

generates a type (12,, ) subfactor of Now define

Ai, Vn.

w((~ ~)®~1) =Ak,

Vk,

..1:-1< 00

w(ea 1 ® ...

@

ea .. ® ® 1) = p{ab'" ,an), n+l

where eo =

1 ' Va, E {O, I}, 1 < l.< n, Vn. Then Wcan be (°1 °0) ,el = (0° 0) 00

uniquely extended to a

* isomorphism from min-® M 2 (
It is easily verified that:

(p(at, ... , an) Eo, Eo) = p,(E(aI, ... ,an)),

(Akp(al" .. ,an)€o, Eo) = 0,

(AkEo, Eo) = 0,

M

Xa

G.

403

Vai E {O,!},! < i < n,1 < k < n,Vn. Thus, we have

&.""0,1

1<.<..-1

+6- - L (jJ( aI, ... , an-I, 1) Eo, lo) &.""0,1

l::5i::5,,-l

+,8{A nEo, lo) + ('"1 -,8)

L

(AnP(al"" ,an-bO)Eo, Eo)

"i""O,l 1::5;::5,.-1

'It (~

~)

E M 2 (a:'j , 'ltn. So

(i1i(·jlo, lo) is the product state

~ 'Pn on

00

min-@ M 2 (
00

Theorem 9.5.11.

Let A = min-@ M 2 (
{2 n

In = 1,2···}, where M 2 (
For any A E (0, ~),

00

define a product state

t/J). =

@ CPn on A as follows: n=l

CPn ( (

v (~ ~)

*

~ ~))

= An + (1 - A)6,

E M 2 (
is the representation of A generated by t/J).. Then, for any A, J.L E (O,~) and A i- J.L the factors R). and R~ are not isomorphic.

*

From above discussion, we have {7r)., H).} ~ {w, H}, where Pn A,qn = 1 - A, Vn, VA E (0, ~). Thus, R). is indeed a type (111).,) factor, where A' = l~).,VA E (O,~). That comes to the conclusion. Q.E.D.

Proof.

Notes. The ratio set r( G) was introduced by W. Krieger. Theorem 9.5.11 is due to R.T. Powers. The problem of finding non-isomorphic factors on a

404

separable Hilbert space is one of central problems in the theory of factors. P.T. Powers proved the existence fo uncountably many type (III) factors. The existence of uncountably many type (Ill) and type (1100 ) factors were obtained by D. McDuff and S. Sakai.

References. [92], [108], [130], [150], [165].

Chapter 10 Borel Structure

10.1. Polish spaces Definition 10.1.1. A topological space is said to be Polish, if it is homeomorphic to a separable complete metric space. Let E be a Polish space. A metric don E is said to be proper, if (E, d) is a separable complete metric space, and the topology generated by d is equivalent to the original topology in E. Let E be a set, and d be a metric on E. We shall denote the topology generated by d in E by d-topology (or d-top. simply). Let (E, d) be a metric space, and FeE. We shall denote the diameter of F with respect to d by Dd(F), i.e.,

Dd(F) = sup{ d(x, y) lx, y E F}. Example. Let H be a separable Hilbert space, and S be the closed unit ball of B(H). Then S is a Polish space with respect to the weak (operator) toplogy, the strong ( operator) topology, or the strong * topology. In fact, let {~n} be a countable dense subset of the closed unit ball of H. For any a, b E S, define dCa, b) =

1 L 2n+m ·I((a n,m

b) ~n, ~m) \,

L 21nll(a - b)~nll, 1 p*(a, b) = L 2n(11(a - b)~nll + II(a p(a,b) =

n

b)* ~nll)·

n

Clearly, d-top, p--top., p* -top. are equivalent to weak, strong, strong * top. in S respectively. Further, since B(H) is countably generated, then by the density Theorem 1.6.1 we can 'get the conclusions.

406

Proposition 10.1.2. The countable direct product and the countable direct sum of Polish spaces are still Polish. Proof. Let {En} be a sequence of Polish spaces, and dn be a proper mectric on En,\ln. On E = unEn ( disjoint union) , define d(x

)

.u

=

{min{l,dn(x,y)}, 0,

if x,y E En for some n, otherwise.

Clearly, E is a Polish space with d-topology, each En is an open and closed subset of E, and in each En the d-top. is equivalent to dn-top. On xnEn, define

V(x n) , (Yn) E xnEn. Then we can see that xnEn is a Polish space with respect to the product topology. Q.E.D.

Proposition 10.1.3. Let E be a Polish space, and FeE. Then F is Polish with respect to the relative topology if and only if F is a Gr-subset of E ( a countable intersection of open subsets). Proof. Let d be a proper metric on E. We may assume that Dd(E) < 1. Suppose that U is an open subset of E. Let 8(x, y)

=

d(x, y)

+ Id(x, U,)-1 -

d(y, U')-II,

\Ix, y E U, where U' = E\U. Clearly, 8 is a metric on U, and the 8-top. is equivalent to the relative top. in U. If {x n } ( C U) is Cauchy with respect to 8, then it is also cauchy with respect to d. Thus , there is x E E such that d(xn,x) ~ O. Since {d(xn,U')-l} is Cauchy and Dd(E) < 1, it follows that lim d(x n, U') = A > o. Thus, d(x, U') = A > 0, i.e., x E U and 8(xn, x) ~ o. n So U as a subspace of E is also a Polish space. Now let F = nnUn, where {Un} is a sequence of open subsets of E. Suppose that 8n is a metric on Ui; as in the preceding paragraph, \In, and define dF (x, Y)

=

,,1 L- n

2n

.

8n ( x, y) c ( ) , \Ix, y E F. 1 + Un z, y

Since 8n-top. and d-top. are equivalent in F, \In, it follows that dF-top. is equivalent to the relative top. in F. If {x n } (C F) is a Cauchy sequence with respect to dF , then for each k there is Yle E Ule such that 81e (x n,YIe) ~ o.

407

Clearly, we have also d(x n, Yk) -+ 0, \lk. Hence, there is x E F such that Yk = x, \lk, and dp(xn,x) --+- O. Therefore, F as a subspace of E is Polish. Conversely, let F( C E) be Polish as a subspace of E, and dp be a proper metric on F. For each n , put

I

F. = {x E F there is an open neighborhood Uof x, } n

such that DdF(U n F) < lin

.

Clearly, F c nnFn. Conversely, if x E UnFn, then for each n there is an open neighborhood U'; of x such that DdF (Un n F) < lin. We may assume that U1 => U2 => ... , and Dd(Un) --+- O. Pick X n E Un n F, \In. Then {x n} is Cauchy in (F, dp). Thus, there is Y E F such that dp(x n, y) --+- O. Clearly, d(xn,y) --+- o,nnUn = {x}, and d(xn,x) --+- O. Hence, x = Y E F, i.e.,

F

nnFn. H x E Fn , then there is an open neighborhood U of x such that DdF(U n F) < lin. By the definition of Fn, it is obvious that U n F c Fn. Thus , =

F n is an open subset of F, i.e., there is an open subset G n of E such that F; = F n G n , \In. Put

Um = {x E Eld(x,:F) < 11m}, \1m. Clearly, Um is open, \1m, and F = nmUm. Therefore,

Q.E.D.

is a Gs-subset of E.

Proposition 10.1.4. Any Polish space must be homeomorphic to a G ssubset of [0,1]00 ( the countale infinite product of [0,1] ). Proof. Let E be a Polish space , d be a proper metric on E, and {an} be a countable dense subset of E. Then d(an, x) ) ( x --+- 1 + d(a n , x)

n

(\Ix E E)

is a homeomorphism from E into [0,1]00, and also by Proposition 10.1.3, its

image must be a Gs-subset of [0,1]00.

Q.E.D.

Proposition 10.1.5. Let 0 be a locally compact Hausdorff space. Then 0 is a Polish space if and only if 0 satisfies the second countability axiom. Proof. The necessity is clear. Now let 0 satisfy the second countability axiom, and 0 00 = OU {oo} be the compactification of O. Clearly, 0 00 is a Polish space. Now n is an open subset of 0 00 , so n is also a Polish space. Q.E.D.

408

Definition 10.1.6.

{n

The Polish space ]Noo is the set

= (nk)lnk

non-negative integer, k

= 1,2,"'}

with the topology generated by the metric

~ . Ink - mkl - ~ 2 k 1 + Ink - mkl'

d(n m) _ " ,

Vn

= (nk), m = (mk)

E ]Noo. Clearly , {n = (nk) I the number of non-zero components of n is finite } is a countable dense subset of ]Noo. Moreover, for any n = (nk) E IN°O,

IN:::..... ,n/c = {m = (m6 ) E IN°Ojmi = nit 1 < i < k}, k = 1,2" .. is a neighborhood basis of n.

Proposition 10.1.7. Let E be a Polish space. Then there exists a continuous map from ]Noo onto E. Proof. Let d be a proper metric on E, and D d (E) < l. For nl = 0,1,"" let F(nt} = E. For each nl, pick a countable closed cover {F(nt,n2)ln2 = 0,1,· ..} of F(nl) such that D d (F (nl , n2)) < 1/2,Vn2' Further, for each (nt,n2)' pick a countable closed cover {F(nt,n2,ns)lns = 0,1,' ..} of F(nt, n2) such that D d(F (nll n2, ns)) < 1/22, Vns,'" , Generally, we have a family {F(nt,' .. , np ) Ini = 0,1, ,1 < i < p, P = 1,2,' ..} of closed subsets of E such that F(nd = E, F(n}, , np ) = U~=OF(nb"" np , k), and Dd(F(nl , ... , np )) -< 2-(p-l) " \lnl ... 2 ... , np, p = 1" . Since (E, d) is complete, it follows that #{n~=IF(nb''''nk)} = 1,

Vn = (nk) E ]Noo.

Let {f(n)} = n~IF(nb .. ·,nk),\ln = (nk) E N?", Clearly, f is a map from IN°O onto E. Suppose that n(k) -----+ n in ]Noo. For any e > 0, pick p such that 2-(p-l) < e. Then we have n~k) = ni,1 < i < p, if k sufficiently large. Hence, f(n) and f(n(k)) E F(nt, ... ,np ) and d(f(n(k»),f(n)) < 2-(P-1) < e if k sufficiently large. Therefore, f is also continuous. Q.E.D.

Lemma 10.1.8. Let E be a Polish space with no isolated point, and d be a proper metric on E. Then for any e > there is an infinite sequence {En} of non-empty G 6-subsets with no isolated point of E such that Dd(En ) < s, Vn; En n Em = 0, \In #- m; and UnE n = E.

°

Proof. We may assume that e < Dd(E). Pick a countable open cover {Vn } of E such that Vn #- 0, Dd(Vn ) < s, Vn. Let E 1 = VI. Clearly, E 1 is a G6 subset with no isolated point. By induction, define En = V n\Fn, where Fn =

409

Uk::E k , 'In >

1. Since E; = Uk~:V k is closed, it follows that En is a G ssubset. Moreover, from (Vn\Fn) C En C Vn\Fn , En has no isolated point. If #{nIEn i= 0} = 00, then {En} satisfies our conditions. Otherwise, notice that 0 =I- E 1 i= E ( since D d(E 1 ) < e < Dd(E)), then the same process can be carried to E 1 ( for some e' with 0 < e' < Dd(Ed). In this way, we can complete the proof. Q.E.D.

Lemma 10.1.9. Let E be a Polish space with no isolated point. Then for any non-negative integers nt,"', nk , there is a non-empty Gs-subset E~~~....nl with no isolated point such that 1) if(nt,"',nk) =I- (mb"',mk), then

Ei:~""nk n E!:1....,mk

=

0;

2) E(k) 00 E(k+l) nl,"',nk -- Up=o n....·,nk,p' Vnl,"', nk,. 3) if d~~),,,,,nk is a proper metric on E~~~"'Jnk' then the diameter of E~~~.~~nk,nk+l with respect to (d + d~ll) + ... + d~~)""Jnk) is less than (k + 1)-1, Vnl,'" ,nk+b where d is a proper metric on E.

Proof. Using Lemma 10.1.8 to (E, d) and e = 1, we get {E~~)lnl = 0,1," .}. Again using Lemma 10.1.8. to (E~~), d + d~l2) and e = 1/2, we get {E~~~n2In2 = 0,1", '}, Vnl. Continuing this process, we can get the conclusion. Q.E.D. Proposition 10.1.10. Let E be a non-empty Polish space. Then there exists an injective continuous map from IN°O onto E, if and only if , E has no isolated point.

Proof. Since IN°O has no isolated point, the necessity is obvious. Now suppose that E has no isolated point. Pick {Ei:~ ...,nl} as in Lemma 10.1.9. Since IN:;:'II ... , nL is homeomorphic to IN°O , it follows from Proposition 10.1.7 that there is a continuous map fA~~ nk from IN::', ... ,n! onto E~~~...,nk' V'nI,"', nk. Fix k, Since {IN:::', ... ,n! In}, ,nk} is a closed and open cover of JNOO, we can define a continuous map f(k) from IN°O onto E, such that f(k)lE~~~...,nk = f~~~,,,,nk . For any n = (nk) E IN°O and integers p, q with p < q, noticing that I<

f(q) (n) E E~ql)'" n'I C E:I'l), ... nP , ,

I

I

so by Lemma 10.1.9 we have d(f(p)(n),f(q)(n)) < p-l. Thus, {f(k)(n)}k is a Cauchy sequence of (E, d), and there is f(n) E E such that d(f(k)(n), f(n)) ---+ 0, uniformaly for n E IN°O. Further, f is continuous. By Lemma 10.1.9, {f(p}( n) }p~k is a Cauchy sequence of (E~:~""n!, d~~.> ,nJ, Vk. Hence, f(n) E nr=lE~:~ ....n!,Vn = (nk) E IN°O. But Dd(Ei~~....nJ <

..

410

k-l,\lk, so {f(n)} = nk=IE~~~ ...,nk,\ln = (nk) E JNOO. Now if n = (nk) -# m = (mk), then there is r such that (nl"'" nr ) -# (ml"", rn,]. SinceE~~ ... ,nr n E!:L... ,mr = 0, it follows that f(n) -# f(m), i.e., f is injective. Finally, for any x E E, by Lemma 10.1.9 there is n = (nk) E )Noo such that x E nk=IEi:~ ...,nk' i.e., f( n) = x. Therefore, !(JNoo) = E. Q.E.D. Proposition 10.1.11. Let E be a Polish space. Then we can write E = F U G, where F n G = 0, G is a countable open subset of E, and either F = 0 or there is an injective continuous map f from JNOO to E with f(JN OO) = F. Proof. Let {Vn} be a countable basis for the topology of E, G = U{VnIVn is countable }, and F = E\G. Suppose that F f::. 0. By Proposition 10.1.10, it sufficies to show that F has no isolated point. Let x E F, and V be any neighobrhood of x. Then there is n such that x E Vn C V. Since x f{. G, it follows that Vn is not countable. Now we can pick u E Vn\ G C V n F and y -# z , Therefore, F has no isolated point. Q.E.D.

References. [10], [13], [190].

10.2. Borel subsets and Sousline subsets Definition 10.2.1. Let E be a Polish space. A subset of E is said to be Borel ,if it belongs to the o--Bool algebra generated by all open subsets of E. A subset A of E is said to be Sousline (or analytic ) , if there is a continuous map f from JNOO to E with f(JNOO) = A. Lemma 10.2.2. Let E be a Polish space, and 1 be a family of subsets of E such that: 1) 1 contains any open subset and closed subset of E; 2) if {En} C f, then nnEn E r, 3) if {En} C r and En n Em = 0, \In -# m, then UnEn E f. Then f contains any Borel subset of E. Proof. Let B = {V C EIV and (E\ V) E fl. Clearly, B contains any open subset and closed subset of E. If VI, V2 E B, then V1\V2 = VI n (E\V2 ) E r, andE\(V1\V2 ) = (E\Vdu(V1 n V2 ) E f. Thus, (V1\V2 ) E B. H{Vn} C B and Vn n Vm = 0, \In f::. m, then UnVn E f, and E\ u, Vn = nn(E\Vn) E f. Thus Un Vn E B. So B is a a- Bool algebra containing any open subset of E. Therefore, B, then r ( since B C f), contains any Borel subset of E. Q.E.D.

411

Propositon 10.2.3. Let E be a Polish space. 1) If P is a Polish space, and I is a continuous map from P to E, then f(P) is a Sousline subset of E. 2) If B is a Borel subset of E, then there is a Polish space P, and an injective continuous map from P to E, such that I(P) = B. 3) Any Borel subset of E is Sousline. Proof, 1) It is obvious from Proposition 10.1.7 and Definition 10.2.1. 2) Let 7 = {F c EI there is a Polish space P, and an injective continuous map from P to E, such that I (P) = F}. Now it suffices to check that 7 satisfies the conditions of Lemma 10.2.2. Any open subset or closed subset of E is a Polish space itself. Thus, 7 contains any open subset and closed subset of E. Now let {En} C 7. Then for each n, there is a Polish space Pn and an injective continuous map In from Pn to E such that In(Pn) = En. Define I : xnPn ~ xnE as follows:

f(Pb'" ,Pn,"') = (ft(pd,"" In(Pn) , .. '), \lpn E Pn, n. Clearly, I is continuous and injective. Put 6. = {(x"", x, .. ,) Ix E E}. Then 6. is a closed subset of xnE. Further, Q = f-1(6.) is also a closed subset of xnPn. Let 1r be the projection from xnE onto its first component. Then 1r 0 I maps injectively Q to nnEn. Hence, E 7. Finally, let {En} C 7, and En n Em = 0, \In -=f. m. Suppose that Pn is a Polish space and In is an injective continuous map from Pn to E such that In(Pn) = En' \In. Define I : P = UnPn ( disjoint union ) ~ E such that IIPn = In' \In. Clearly, I(P) = unEn. Thus, unEn E 7. Therefore, 7 satisfies the conditions of Lemma 10.2.2. Q.E.D. 3) It is clear from the conclusions 2) and 1).

«»:

Proposition 10.2.4. Let E be a Polish space, and B be a Borel subset of E. Then either B is countable, or there is an injective continuous map from JNOO to E such that I(JNOO) C Band (B\/(JN°O)) is countable. Proof. It is an immediate result of Propositions 10.2.3 and 10.1.11.

Q.E.D.

Proposition 10.2.5. 1) The continuous image of a Sousline subset is Sousline, i.e., if E and F are Polish spaces, f is continuous from E to F, and A is a Sousline subset of E, then f(A} is a Sousline subset of F. 2) The countable intersection and the countable union of Sousline subsets are Sousline.

412

Proof. 1) It is obvious from Definition 10.2.1. 2) Let {An} be a sequence of Sousline subsets of a Polish space E. Then for each n, there is a continuous map In from JNOO to E such that In(JN OO) = An. Define I : P = UnPn ( disjoint union) ---+ E such that IIPn = In' where Pn = JNOO, \In. Then I(P) = UnAn, and UnAn is Sousline. Let n = xnPn, where Pn = JN OO, \In, and Clearly, M is a closed subset of n. Then define g(x) nnAn = g(M) is Sousline.

= Il(xd, \Ix E

M. Hence, Q.E.D.

Definition 10.2.6. Let E be a Polish space, and A, B be two subsets of E. A and B are said to be Borel-eeparated, if there is a Borel subset F of E such that A c F and B C (E\F). Lemma 10.2.1. Let {An}, {B m } be two sequences of subsets of a Polish space E, and let An and B m be Borel-separated, \In, m . Then A = UnAn and B = UmBm are also Borel-separated.

Proof. Let Fnm be a Borel subset of E such that An (E\Fnm) , \In, m. Then An

C

nmFnm,

n, C

E\Fnk C E\ n m Fnm,

C

Fnm and B m

C

\In, k,

Let F = Un n m Fnm. Then A C F, and B c nn(E\ n m Fnm} Therefore, A and Bare Borel-separtated,

(E\F). Q.E.D.

Proposition 10.2.8. Let E be a Polish space, A and B be two Sousline subsets of E, and A n B = 0. Then A and B are Borel-separated.

Proof. Let I, g be two continuous maps from JNOO to E such that I(IN OO) = A and g(JNoo) = B. If A and B are not Borel-separated, then by JNOO = U~oJNr and Lemma 10.2.7, there are nl and ml such that I(JN::,) and g(JN~J are not Borel-separated. Continuing this process, generally we have n = (nk) and m = (mk) E JNOO such that I(JN::.....,nJ and g(JN::1, ... ,m,) are not Borelseparated, \lk. Since An B = 0, it follows that I(n) 1= g(m). Pick two open subsets U and V of E such that

I(n)

E U,g(m) E V,and

U nV = 0.

Then I(JN::',... ,n,) C u, g(JN:lI ... ,mJ c V, if k sufficiently large. This contradicts that I(JN::......n,J and g(JN:1,.... m,J are not Borel-separated, \lk. Therefore, A and B are Borel-separated. Q.E.D.

413

Proposition 10.2.9. Let {An} be a disjoint sequence of Sousline subsets of a Polish space E. Then {An} is Borel-separated, i.e., there is a disjoint sequence {B n} of Borel subsets of E such that An C B n, Vn. Proof. By Proposition 10.2.5 and 10.2.8, for any n there is a Borel subset Fn of E such that An C Fn, Uk>nAk C (E\Fn). Now let B 1 = Ft, and B n = r;\ ui:::-l B i , Vn > 1. Then An C s.; Vn. Q.E.D.

Theorem 10.2.10. (Sousline criterion) Let E be a Polish space, and BeE. Then B is Borel if and only if Band (E\ B) are Sous line. Proof. The necessity is obvious. Now if Band (E\B) are Sousline, then by Proposition 10.2.8, Band (E\B) are Borel-separated. Therefore, B must be Q.E.D. Borel.

Theorem 10.2.11. Let E be a Polish space, B be the collection of all Borel subsets of E, and A be a Sousline subset of E. Then for any a-finite measure v on B, there are B and FEB such that A C B, (B\A) C F, and v(F) = O. Proof. Replacing v by an equivalent finite measure on B, we may assume that v is finite. For any 8 C E, we say that there exists a minimal Borel cover T of 8 relative to v , i.e., 8 C T E B, and for any 8 C FEB we have v(T\F) = O. In fact, since v is finite, we can find {En} C B such that E 1 => E 2 => .•• => 8 and limv(En) = inf{v(F) 18 C FEB}. Let n

T = nnEn, and Then 8

v(T n F)

C

=

A = inf{v(F) 18 C FEB}.

T E B, and v(T) = A. Now if 8 c FEB, we have v(T) = A, and v(T\F) = O.

Now we prove the theorem. By Definition 10.2.1, there is a continuous map f : IN°O -------t E such that f(JN°O) = A . For any non-negative integers nb' .. ,nk, let Enl,.."nJc be a minimal Borel cover of f(JN:;:,.",nJ relative to u, and Enl,.."n/c C f(JN::' .... ,n,J. Define

and Then

414

By the Definition of Enl, ....nk' we have

Hence, v(F) = O. Now it suffices to show (B\A) c F. Let x E (B\A). Then there is nl such that x E E nl \A. Suppose that x f/. F. Then x f/. (Enl \ u:=o Enl.n:a) , and x E E nl n (U~=oEnl,n2)' Thus, there is n2 such that x E E nl,n2' ... , generally, we can find n = (nk) E IN°O such that x E E nl nk , Vk. We claim that

{/(n}} = n~l/(JN::, ,nJ. Indeed, let y -:/= I(n). Pick a closed neighborhood V of I(n) such that y f/. V. Since I is continuous, it follows that I(JN::, ... ,n,.} c V if k sufficiently large. Then from I(JN::, .... c V = V and y f/. V, we have y f/. n~l/(JN::, ...,nJ. Thus, {/(n}} = n':=l/(JN::..... n Now x E Enl,...,nk C I(JN~,,,,,nJe)' Vk, so x = I(n) E A. This contradicts x E (B\A). Therefore, x E F, and (B\A) C F.

nJ

J.

Q.E.D. Corollary 10.2.12. Let E, B, A, v be as in Theorem 10.2.11. Then there are C and G E B such that C C A, (A\C) c G, and v(G) = O. Proof. Pick B, F as in Theorem 10.2.11. Let C = B\F, and G and G satisfy our conditions.

=

F. Then C

Q.E.D.

References. [10], [94], [190].

10.3. Borel maps and standard Borel spaces Definition 10.3.1. (E, B) is called a Borel space, if E is a set, and B is a a-Bool algebra of some subsets of E. A subset B of E is called B-Borel ( or Borel simply if no confusion arises ), if B E B. B is also called the Borel structure of the Borel space (E, B) . For example, let E be a Polish space, and B be the collection of all Borel subsets of E ( see Definition 10.2.1). Then (E, B) is a Borel space, In the following, we understand a Polish space as a Borel space always in this meaning. A map I from a Borel space (E, BE) to another Borel space (F, BF ) is said to be Borel ,if 1- 1 (B F ) E BE,VB F E BF • If I is a bijective Borel map from (E, BE) onto (F, BF ) , and 1-1 is also Borel, then we say that Borel spaces (E,BE ) and (F,BF ) are Borel isomorphic, and I is called a Borel isomorphism from (E, BE) onto (F, BF ) . Let (E, B) be a Borel space, and PcB. P is called a generated set for B, if B is the minimal a-Bool algebra containing P.

415

Proposition 10.3.2. 1) Let (E, BE) and (F, BF ) be two Borel spaces, and P be a generated set for BF • Then a map 1 : E ---+ F is Borel if and only if 1- 1 (B F ) E BE, YBF E P. 2) Let 1 be a continuous map from a Polish space E to another Polish space F. Then 1 is Borel. 3) The composition of two Borel maps is still Borel.

Proof. 1) Let B1 = {/- 1 (B F ) IBF E BF } . Clearly, 8k is a e--Bool algebra. If B~ is a a -Bool algebra generated by {/- 1 (BF ) IBF E P}, then B~ C B1. Thus, P C {B F E BF 1/- 1 (BF ) E B~} BJ;. C BF • Since Bj.. is a a-Bool algebra and P is a generated set for BF , it follows that Bj;. = BF • Further, B~ = B1. If 1 is Borel, then for any B F E P, it is obvious that 1- 1 (B F ) E BE. Conversely, let 1- 1 (B F ) E BE, YBF E P. Then B~ C BE' From the preceding paragraph, for any BF E BF we get 1- 1 (BF) E Bk = B~ c BE. Therefore, 1 is

=

Borel. 2) Since 1- 1 (U) is an open subset of E for any open subset U of F, it follows from 1) that 1 is Borel. Q.E.D. 3) It is obvious.

Definition 10.3.3. A Borel space (E, B) is said to be standard, if we can introduce a topolog r in E such that (E, 'f) is a Polish space and the collection of all Borel subsets of the Polish space (E, 'f) is equal to B. Clearly, a Polish space regarded as a Borel space is standard. Conversely, for a standard Borel space, maybe, we can introduce several topologies such that they become different Polish spaces with original Borel structure ( see Proposition 10.3.14 ). Proposition 10.3.4. 1) Let (E, B) be a standare Borel space, and 1 be a Borel map on E. Then {x E Elx = I(x)} E B. 2) Let (E, BE) and (F, BF ) be standard Borel spaces, and 1 be a Borel map from E to F. Then the graph {(x,/(x))lx E E} of 1 is a Borel subset of E X F, where the Borel structure of E X F is generated by {BE X BFIB E E BE, B F E BF}( and E X F with this Borel structure is also a standard Borel space).

Proof. 1) We may assume that E is a Polish space. Then ~ = {(x, x)lx E E} is a closed subset of E X E. Define a map (I x id) from E to E X E : x ---+

(/(x), x)(Yx E E). Clearly, (I X id) is Borel. Thus, {x E Elx = I(x)} = (I X id)-l(~) is a Borel subset of E. 2) Define a map tp on E X F : (x, y) ---+ (x, I(x)). It is easily verified that cp is Borel. Now by 1), {(x, I(x)) Ix E E} = {(x, y) I(x, y) = cp(x, y)} is a Borel

416

subset of E x F.

Q.E.D.

Proposition 10.3.5. Let E, F be Polish spaces, f be a Borel map from E to F, and A be a Sousline subset of E. Then f(A) is a Sousline subset of F. Proof. Let 9 be a continuous map from IN°O to E such that g(JN OO ) = A. Then fog is a Borel map from IN°O to F, and by Proposition 10.3.4 {(n, fog(n)) In E JNOO} is a Borel subset of IN°O x F. Define a map 9 X id from IN°O x F to Ex F : (n,y) ~ (g(n),y),Vn E JNoo,y E F. Clearly, 9 x id is continuous. Then by Proposition 10.2.3, (g x id)({(n, f 0 g(n))ln E JN OO } ) = {(x, f(x))lx E A} is a Sousline subset of E x F. Let 11" be the projection from E x F onto F, then by Proposition 10.2.5 1I"({(x,f(x))lx E A}) = f(A) is a Sousline subset of F. Q.E.D.

Definition 10.3.6. A family '7 of some subsets of a set E is said to be separated (for E) , if for any x, y E E with x -=1= y, there is F E '7 such that either x E F and y fj. F or y E F and x fj. F. Lemma 10.3.7. Let (E, B) be a Borel space, and PcB. Then the following statements are equivalent: 1) B is separated ( for E) , and P is a generated set for B; 2) P is separated ( for E) , and P is a generated set for B. Proof. It suffices to show that 1) implies 2). Suppose that 1) holds. If P is not separated, then there are x, y E E and x -=1= y, such that for any F E P we have either x, y E F or z, y fj. F. Let £ = {B E B I either x, y E B or x, y fj. B}. Clearly, P c £ c B. If {B n } C £, obviously we have c;»; E t: For any B I , B 2 E £, one of the following relations holds: 1) x, y fj. B«, 2) x, Y E B 1 \B2 , 3) x, Y E B 1 n B 2 • Thus (B 1 \B2 ) E t: Further, £ is also a a-Bool algebra, and £ = B, and B is not separated, a contradiction. Therefore, P is separated. Q.E.D.

Definition 10.3.8. A Borel spaec (E, B) is said to be l-standard , if B is separated ( for E) , and B contains a countable generated set. From Lemma 10.3.7, (E, B) is l -standard if and only if B contains a countable generated set P such that P is separated ( for E). Clearly, A standard Borel space is l-standard. Define M = x n{O,I} = {a = (at,"·,an, .. ·)la n = 0 or I,Vn}. M is the countable infinite product of the discrete compact space {O, I}, and M is a compact Polish space.

417

Theorem 10.3.9. A Borel space (E, B) is l -standard if and only if it is Borel isomorphic to a subspace of M.

Proof. Let Fn = {a E M I the n-th component of a is 1 } . Then Fn is an open and closed subset of M, and {Fnln} is a generated set for the Borel structure of M. If (E, B) is l-standard, then there is a sequence {B n } C B such that {B n } is separated ( for E) and is a generated set for B. Define f : E ---+ M as follows: if x E n; I the n-th component of f(x) = { if x (/. B :

0:

n

n = 1,2,···, 'Ix E E. Since {B n } is separated, it follows that f is injective. Notice that f(B n) = Fn n f(E), 'In. Thus by Proposition 10.3.2, f is a Borel isomorphism from E onto f(E), where the Borel structure of f(E) is {F n f(E)IF is any Borel subset of M}, i.e., is generated by {Fn n f(E)ln}. Conversely, suppose that (E, B) is Borel isomorphic to a subspace of M. Since any subspace of M is ~-standard, it follows that (E, B) is l-standard.

Q.E.D. Lemma 10.3.10. Let E be a Polish space, f be an injective continuous map from IN°O to E. Then f(JN OO ) is a Borel subset of E.

Proof. For any k, {f(JN::....,n/)In}, ... ,nk} is a disjoint sequence of Sousline subsets of E. From Proposition 10.2.9, there is a disjoint sequence {Fnl,·..•n&: In}, ... , nk} of Borel subsets of E such that f(N::, ....nJ c Fnl .....nk , Vnl'···' nk. By induction, define AnI = F nl, Anl ...·,n&: = F n l ...·•n.\: n f(JN:; ....,nJ n Anl,....n.\:_l' Vnl, .", nk and k > 1. Then the family {Anl .....nklnI,···,nk,k > I} of Borel subsets of E has the following properties:

1) Anl, ,nl: n Aml, mk = 0, V(nb· .. ,nk) t= (ml,' .. ,mk); 2) Anl, ,nHI C A nl nk ,'In}, .. " nk+l; 3) f(JN::., ....n'J c Anl,...,nk C f(JN:; ....,nJ' 'In},·· . ,nk. In fact, we can prove f(JN::, ...,nJ c Anl,...,n! by induction, and the rest facts are obvious. Since f is injective, it follows that f(n) = f(nr=lJN::, ...,n.\:) = n~lf(IN::', ...,nk)' 'In = (nk) E IN°O. By the proof of Theorem 10.2.11, we have also {f(n)} = n%"=l f(JN;::, ...,nJ. Further, from above property 3) we get {f(n)} = n~lAnl ....,nl:' 'In = (nk) E IN°O. Now we prove that f(lN°O) = n~l Unl, ...,nk Anl,...,nk. In fact, since {f(n)} = OO n~lAnl •... ,nk' 'In = (nk) E IN°O, it follows that f(JN ) c n~l Unl, ..·,n! Anl •... ,nk. Coversely, let x E n~l Unl,.",nk Anl,....nk. For k = 1 , there is ml such that

418 x

E AmI' From above properties 1) and 2) , we have x

E n~l U n1,···,n" (Anl ....,n"

n Ami)

= AmI n (nf=2 U n 2, .. ·,nlc A ml,n2 ....,n"). Repeating this process, there is m = (mk) E JNOO such that x E n~lAml,.~.,mlc = {/(m)}. Hence, x = I(m) E I(JN OO ) , and I(JN°O) = nf=l Unl,·.. ,nlc Ant. ... ,n". Therefore, I(JNOO) is Borel. Q.E.D.

Lemma 10.3.11. Let E, F be two Polish spaces, and I be an injective Borel map from E to F. Then I(E) is a Borel subset of F, and I is a Borel isomorphism from E onto I (E) . Proof. It suffices to show that for any Borel subset B of E, I(B) is a Borel subset of F. Fix a Borel subset B of E. Let G = {(x, I(x)) Ix E B}, d be a proper metric on F, and {an} be a countable dense subset of F. Put

Vn, k. We claim that G

= nn Uk (vt

U:). Indeed, since UkU: Conversely, let (x, y) E n, u, X

= F, Vn, it (vt X U~),

follows that G c n, u, (Vk X U~). i.e., for each n, there is k = k(n) such that (x, y) E Vkn X Uk' Then I(x) E Uk n I(B). Since I is injective, we have x E Band I(x) E Uk' By y E UJ:, we get d(/(x), y) < lin. But n is arbitrary, so I(x) = y i.e., (x, y) E G. Thus , G = nn Uk (Vt xU;). Further, since I is injecive, it follows that Vt = f-l(UJ:) n B,Vn,k. Hence, G is a Borel subset of (E X F). If G is countable, then I(B) is Borel obviously. Now suppose that G is not countable. By Proposition 10.2.4, there is an injective continuous map g from IN°O to E X F such that g(JN OO ) C G, and (G\ (g(W OO )) is countable. Denote the projection from E X F onto F by 1r. Since g(JN OO ) C G and I is injective, it follows that 1r 0 g is injective and continuous from Woo to F. From Lemma 10.3.10, 1r 0 g(JN OO ) is a Borel subset of F. Therefore, I(B) = 1rG = 1r 0 g(lN oo ) U 1r(G\g(JNoo ) ) is a Borel subset of F. Q.E.D. n

Theorem 10.3.12. Let E be a standard Borel space, F be a ~ -standard Borel space, and I be an injective Borel map from E to F. Then I(E) is a Borel subset of F, and f is a Borel isomorphism from E onto I(E). Proof. By Theorem 10.3.9, we may assume that F C M. Then by Lemma 10.3.11, we can get the conclusion. Q.E.D.

419

Theorem 10.3.13. Let (E, B) be a standard Borel space. If a sequence {B n } of B is separated ( for E) , then {B n } is a generated set for B.

Proof. Let Bo be the a -Bool algebra generated by {B n } . Clearly, Bo c B, and {E, Bo} is ~-standard. Now the identity map id is an injective Borel map from (E, B) onto (E, Bo). Thus by Theorem 10.3.12, we have Bo = B. Q.E.D. Proposition 10.3.14. Let H be a separable Hilbert space. Then the weak ( operator) topology, strong ( operator) topology, strong * ( operator ) topology, o(B{H),T(H)),s(B(H),T(H)),s*{B(H),T(H)) and r{B{H),T (H)) in B{H) will generate the same strandard Borel structure, where the Borel structure genreated by a topology means that the o-Bool algebra is generated by all open subsets with respect to that topology. In particular, the Polish spaces of 8 = {a E B(H)lllall < I} with respect to weak ( operator) topology, strong ( operator ) topology, and strong * ( operator ) topology ( see the example in Section 9.1) are the same as the standard Borel spaces.

Proof. Denote one of above topologies by a,8n = {a E 8 1 , Vn +1

B(H)lllall <

n}, VI =

= 8 n +1 \8n , Vn >

1. By the example in Section 9.1, (8",0) is Polish is an open subset of (8n , a), it follows that (Vn , a) is also

, Vn. Since Vn Polish. Denote the topological union of {(Vn,o)ln > I} by (B(H),a'). Clearly , (B(H), a') is Polish, and a subset U of B(H) is a' -open if and only if Un Vn is an open subset of (Vn , 0), Vn. Let Ba " Ba be the Borel structures of B(H) generated by a', 0 respectively. Since 0' ::J a, it follows that Ba C BeY" On the other hand, if U is a a'-open subset of B(H), then it is obvious that U E Ba • Thus, Ba , = Ba , and (B(H), Ba ) = (B(H), Ba , ) is standard. Moreover, obviously we have B1" ::J BeY, where r = r(B(H),T(H)). Then by Theorem 10.3.13, we get BeY = Bn Va ( one of above topologies in B(H)). Q.E.D. Proposition 10.3.15. Let E be a standard Borel space, and BeE. Then B as a Borel subspace of E is standard if and only if B is a Borel subset of E.

Proof. Let B be standard, and id be the embedding of B into E. Then by Theorem 10.3.12, B is a Borel subset of E. Conversely, let B be a Borel subset of E. We may assume that E is a Polish space. Then by Proposition 10.2.3, there is a Polish space P and an injective continuous map / from P to E such that /(P) = B. Now by Theorem 10.3.12, / is a Borel isomorphism from P onto B. Therefore, B is standard. Q.E.D. Theorem 10.3.16. The cardinal number of a standard Borel space is either countable or continuum, and the standard Borel spaces with the same cardinal

420

number are Borel isomorphic. Proof. By Proposition 10.1.11, it suffices to show that E and JR are Borel isomorphic, where E is a standard Borel space and its cardinam number is continuum. By Proposition 10.1.10, there is an injective continuous map from N OO onto JR. Then by Proposition 10.1.11 and Theorem 10.3.12, we have a Borel isomorphism f from JR to E such that (E\f(JR)) is countable. Pick a closed subset T of JR such that #T = #(E\f(JR)). Clearly, there is a Borel isomorphism ep from Tonto (E\f(JR)). From Propositon 10.1.10, we have also a Borel isomoprphism .,p from (JR\T) onto JR. Now let

t ) _ {f o.,p(t), if t E (JR\T), ( g ep(t), iftET. Then g is a Borel isomorphism from JR onto E. References.

Q.E.D.

[10J, [106], [190].

10.4. Borel cross sections Lemma 10.4.1. Define a total order in !Noo as follows: n < m if either n = m or there exists j such that nk = mk,1 < k < j and nj < mj, where n = (nk),m = (mk) E !N oo. Then there exists a minimal element in any non-empty closed subset of !N OO • Proof. Let F be a non-empty closed subset of ]NOO. And put al = min{ nlln = (nk) E F},FI = {n = (nk) E Flnl = all; a2 = min{n2ln = (nk) E F I } , F2 = {n = (nk) E F lln2 = a2};···. Then we get F :> F I :> F2 :> .... Clearly, Da(Fj ) < 2- j ( see Definition 10.1.6 ) . Thus, njFj = {n}, and this n is the minimal element of F. Q.E.D.

Theorem 10.4.2. Let E be a Polish space, and r-- be an equivalent relation on E such that : 1) for any x E E, {y E ElY""" x} is closed; 2) if F is a closed subset of E , then F = {y E EI there is x E F with y ,..,., x} is a Borel subset of E, or replacing 2) by the following 2') if V is an open subset of E , then V = {y E EI there exists x E V with y x} is a Borel subset of E. Then there exists a Borel subset B of E such that #(B n x) = 1, Vx E E, where x is the equivalent class of x, i.e., x = {y E ElY""" x}, Vx E E. /"oJ

421

Proof. Let d be a proper metric on E. If 1) and 2) hold, then we pick a family {B(nb"', nk)} of non-empty closed subsets of E such that: (1) E = U~=oB(nd, (2) B(nt,"" nk) = U~OB(nb'''' nk'p), Vk, (3) Dd(B(nb''', nk)) < 2- k, Vk. If 1) and 2') hold, then we pick a family {B(nb"" nk)} of nonempty open subsets of E such that: (1) , (2) , (3) are as above, and (4) B(nl,"" nk+d C B(nI,"', nk). Since E is Polish, the family {B(nl,"" nk)} can be found. In each case ( either 1) , 2) or 1) , 2') hold), define f : IN°O ~ E such that {f(n)} = nr=lB(nl,"', nk), Vn = (nk) E N?', Clearly, f(JN OO ) = E, and f is continuous. Let B(nl,'''' nk) = {y E EI there is x E B(nr, ... , nk) with y "'" z}. By the assumption, B(nl"'" nk) is Borel, Vnt,"', nk. Now by induction, define a family {A(nl"", nk)} of Borel subsets of E as follows:

A(nl) = B(nd n [E\ A(nl"" ,nk+l)

Um1
B(mtll,

= B(nl,' .. ,nk+d n A(nl,' .. ,nk) n[E\

Umk+1
B(nb···,nk,mk+l)],

Vnt,.· ., nk+l, and k > 1.

For each equivalent class X of E, by 1) t:' (X) is a non-empty closed subset of IN°O. From Lemma 1004.1, f-l(X) has a minimal element (Pk) = P = p(X). Then we claim that: a) A(pl,'" ,Pk) n X = B(Pl,'" ,Pk) n X i= 0, Vk; b) if (nl"", nk) i= (PI,'" ,Pk), then A(nl, "', nk) n X = 0, Vk. In fact, since f(p) E X and {f(p)} = n~lB(Pb "', Pk), it follows that B(Pb ... , Pk) n X i= 0, Vk. By the definition, A(pl," . ,Pk) n X c B(pr, "', Pk) n X, Vk. Now let x E B(Pl) n X. Suppose that there is ml < PI such that x E B(ml)' Pick y E B(md with y "'" x, Clearly, there is m = (mk) E IN°O such that y = f(m). Then m E f-I(X). But P is the minimal element of f-I(X), so PI < ml . This contradicts ml < Pi- Thus, x E B(pd\ Um1 p, By (nl,"·,nk) i= (Pl,"',Pk), there exists j « k) such that ni = Pi, 1 < ,: < j and Pi < ni' Then

A(Pb'" c

[E\

l

Up
Pi-b nil n B(Pl'''' ,Pi) B(Pl" .. ,Pi-l, p)] n B(Pb' .. ,Pi)

On the other hand, since B(Pb' .. ,Pi) n X

= 0.

i= 0, it follows that X c B(Pl' ... ,

422

Pi)' Then from i < k, we have x E A( nl, ... , n;) n X =

A(PI,'" ,Pi-I, nil

nX

c A(PI,'" ,Pi-I, nil n B(Pb'" ,Pi) = 0, a contradiction. Thus A( nl, ... ,nk) n X = 0, V(nl, ... ,nk) # (PI,' .. ,Pk), and any k. Let B = nr;=l Un1,... ,nj; A(nl,"',nk)' Clearly, B is a Borel subset of E. For each equivalent class X of E, by above a) and b) we have

B nX

= n~l U n ll ...,nj;

(A(nI,"', nk) n X)

=

n~l (A(PI,'

=

(n~l(B(Pb'"

. " Pk) n X) = nr;=l (B(Pb' .. ,Pk) n X) ,Pk)) n X = {/(p)},

where P = (Pk) is the minimal element of x) = 1, 'Ix E E.

1- 1 (X ). Therefore, we

have #(B n Q.E.D.

Theorem 10.4.3. Let I be a Borel map from a Polish space E onto a Borel space F satisfying: 1) 1-1({y}) is closed, vu E F; 2) I maps any closed subset of E to a Borel subset of F, or replacing 2) by the following 2') I maps any open subset of E to a Borel subset of F. Then I admits a Borel cross section, i.e., there exists a Borel map 9 from F to E such that 1 0 g(y) = y,vu E F.

Proof. Introduce an equivalent relation

in E : Xl X2, if I(xd = I(X2)' Then by Theorem 10.4.2, there is a Borel subset B of E such that #(/-1({y})n B) = 1, vu E F. Suppose that 9 is a map from F to E such that {g(y)} = 1-1 ({y}) n B, vu E F. Then I 0 g(y) = u, vu E F. Let G be either any closed subset ( in the case of 1) and 2) ) or any open subset ( in case of 1) and 2' ) ) of E. Then I(G) is a Borel subset of F. Since g-l(G) = I(G), it follows from Proposition 10.3.2 that 9 is a Borel map. Q.E.D. I'"<J

I'"<J

Lemma 10.4.4. Let 0 be a locally compact Hausdorff space satisfying the second countability axiom, v be a regular Borel measure on 0, and I be a continuous map from JNOO to O. Then there exists a map 9 from 0' = I(JN OO ) to JNOO and a sequence {Kn } of compact subsets of 0' such that

I 0 g(x) =

X,

Vx E 0', and (0'\

and for each n, 9 is continuous on K n .

Un

K n ) C some v - zero subset,

423

Proof. For any x E 0', 1- 1 ( {X}) is a non-empty closed subset of lN OO. Denote the minimal element of 1-1({x}) by g(x). Then 1 0 g(x) = x, "tx EO'. Clearly, 0' is a Sousline subset of O. By Corollary 10.2.12, the a-finiteness ·of v and the inner regularity of v , it suffices to show that for any compact subset K of 0' , there is a sequence {Kn } of compact subsets such that: tc; C K, g is continuous on tc; "tn, and v(K\ u, K n ) = 0. 1) We say that a subset E of 0 has a Borel kernel relative to u, if there exist Borel subsets F and G of 0 such that FeE, (E\ F) c G, and v( G) = 0. By Corollary 10.2.12, any Sousline subset of 0 has a Borel kernel relative to u. Clearly, if En has a Borel kernel relative to u, "tn, then UnEn has also a Borel kernel relative to v. Moreover, if E, has a Borel kernel relative to u, i = 1,2, then (E 1 \E2 ) has also a Borel kernel relative to u. Indeed, let Fe, G i be Borel subsets of 0 such that F; C E i , (Ei \Fi ) C Gi, and v( G i ) = 0, i = 1,2. Define F = F 1 \(F2 U G 2 ) , and G = G 1 U G 2 • Then F and G are Borel, v(G) = 0, F C (E 1 \E2 ) , and (E 1 \E2 )\ F c G. 2) Let Q = {n = (nA:) E lNOOj#{klnA: f:- O} < oo}. Then Q is a countable dense subset of lN oo • Suppose that V is an open subset of lN OO. H n = (nA:) E V, then IN::., ....n'' c V if k sufficiently large. Let Zl = (nl" .. , nA:, 0, ...) , and Z2 = (nI,"',nA: + 1,0",,), Then n E [ZI,Z2) = {z E lNoolz l < Z < Z2} = ,n". Since Q is countable, it follows that V is a countable union of subsets with a form [ZI, Z2) = [0, Z2) \[0, Zl), where ZI, Z2 E Q. 3) For any Z E Q, we have g-I([O, z)) = 1([0,z)). In fact, if x E g-l([O, z)), then x = 1 0 g(x) E 1([0, z)). Conversely, if n E [0, z), then by the definition of g, go I(n) is the minimal element of 1- 1 ( {/(n)}). Thus, go I(n) < n, and go I(n) E [O,z), i.e., I(n) E g-l([O,z)). 4) For any Z E Q, clearly [0, z) is a Borel subset of lN OO. Since 1 is continuous, it follows from 3) that g-l([O, z))(= 1([0, z))) is a Sousline subset of O. Further by Corollary 10.2.12, g-l([O, z)) has a Borel kernel relative to u. 5) For any open subset V of lN OO and any closed subset F of lN OO, g-l(V) and g-I(F) have Borel kernels relative to u. In fact, from 1) , 2) , and 4) , g-l(V) has a Borel kernel relative to u. Moreover, from g-l(F) = O'\g-l(F') ( where F' = lNOO\F) and 1), g-l(F) has also a Borel kernel relative to u.

lV::.. .

Now let K be a compact subset of 0 with K c 0', {aA:} be a countable dense subset of lN oo , and d be a proper metric on IN°O as in Definition 10.1.6. Put

and

424

Vk, p. Then for any p, we have

Bi,p

n Bi,p = 0,

Vi t= i,

U~=lBk,p = U~lAk,p = K.

Further, define Yp : K

----+

IN OO as follows:

Yp(x) = ak, if x E B kp, Vk. Clearly, yp(x) ----+ y(x) in IN 00 , Vx E K. For any k, p, by the discussion 5) Ak,p has a Borel kernel relative to u, Thus, Bk,p has also a Borel kernel relative to t/. Then for any m, by the regularity of v and the definition of yp we can find a compact subset K~m} C K such that yp is continuous on K~m), and v(K\K~m») < (m. 2P)- 1. Let K(m) = npK;m}. Then Yp is continuous on K(m) , v», and v(K\K(m}) < m- 1 , Vm. Denote the projection from IN°O onto its j-th component by 1f'i. Clearly, (1rjYp)(x) ----+ (1rjY)(x) , Vx E K. By the Egorov theorem, for any e > 0 there is a compact subset K(e). C K(m) such that mJ

(1rjYp)(x)

----+

(1rjY)(x) , uniformly for x E K~J,

and v(K(m)\K~)) < 2- j c. Let K~}

Yp(x)

----+

= niK~).

Then

y(x)(inJN°O), uniformly for x E K~),

and v(K(m)\K!:») < c. Thus, y is continuous on K!:). Further, pick {Kn } = {K~-l}lm,p}. Then tc; C K,y is continous on tc; Vn, and v(K\ u, K n ) = o. Q.E.D. Theorem 10.4.5. Let E, F be two Polish spaces, v be a a-finite measure on all Borel subsets of F, and G be a Sousline subset of E X F. If 1rF is the projection from E X F onto F, then there exists a map y from R = 1rF (G) to E and a Borel subset B of F with B c R such that

(y(y), y) E G,

Vy E R,

and y is a Borel map from B to E, and (R\B)

C

some v-zero subset.

Proof. Since v is a-finite, by Proposition 10.3.15 and 9.3.16 we may assume the F is a locally compact Hausdorff space satisfying the second countability axiom, and v is a regular Borel measure on F. Now G is a Sousline subset of E X F. Then there is a continuous map h from INOO to E X F such that h(IN°O) = G. Let f = 1rF 0 h. Then R = f(IN°O). By Lemma 10.4.4, we have a map TJ from R to INOO and a Borel subset B C R such that fOTJ(Y) = Y, Vy E R,

425

and '1 is Borel on B, and (R\B) C some v-zero subset. Let 1rE be the projection from E x F onto E, and 9 = 1rE 0 h 0 '1. Then 9 : R -----+ E, and 9 is Borel on B. For any y E R, since h 0 '1(y) E G,lfE 0 h 0 '1(y) = g(y),1rF 0 h 0 '1(y) = 1 0 '1(y) = y, it follows that

(g(y), y) = h 0 '1(y) E G. Q.E.D.

Proposition 10.4.6. Let E, F be two standard Borel spaces, I be a Borel map from E onto F, and v be a o -finite measure on all Borel subsets of F. Then I admits a Borel cross section relative to u, i.e., there is a Borel subset Fo of F and a Borel map 9 from (F\Fo) to E such that v(Fo) = 0, and 1 0 g(y) = y, Vy E (F\Fo).

Proof. By Proposition 10.3.4, the Graph G = {(x, I (x)) Ix E E} of I is a Borel subset of E x F. Clearly, 1rF (G) = F. Now from Theorem 10.4.5, we can get the conclusion.

Q.E.D.

Proposition 10.4.7. Let E,F be two standard Borel spaces, I be a map from E onto F, and Jl be a finite measure on all Borel subsets Introduce an equivalent relation in E : XI X2, if I(xd = I(X2)' there is a saturated Borel subset Eo with Jl(Eo) = 0 such that I has a cross section on (E\Eo). I"oJ

Proof. Let

= Jl 0 I-I. Clearly,

I"oJ

Borel of E. Then Borel

is a finite measure on all Borel subsets of F. By Proposition 10.4.6, there is a Borel subset Fo of F and a Borel map 9 from (F\Fo) to E such that v(Fo) = 0 and 1 0 g(y) = y, vu E (F\Fo). Now let Eo = f-I(Fo). Clearly, Eo is a saturated Borel subset of E, and Jl(Eo) = O. Moreover, since I(E\Eo) = F\Fo and g(F\Fo} C (E\Eo), it follows that f has a Borel cross section on (E\Eo). Q.E.D. II

Refereces.

II

[10], [24], [28], [120].

Chapter 11 The Borel Spaces of Von Neumann Algebras

11.1. The standard Borel structure of W{X·) Let E be a topological spce. Denote the collection of all non-empty closed subsets of E by C(E) . Lemma 11.1.1.

Let (E, d) be a compact metric space. For any FI, F2 E

C(E), define

Then (C(E), p) is also a compact metric space. Proof. It is easily verified that p is a metric on C(E). Let {x n } be a countable dense subset of E. For any e > 0, since (E, d) is compact, there is k such that U~=lSd(Xi' e) = E, where Sd(Y, e) = {z E Eld(y, z) < e}, vv E E. We claim that UIC{l ....,k}Sp({xihElJ e) = C(E).

In fact, for any F E C(E), there is I c {I,···, k} such that Sd(Xi' e) n F =I 0, V.,; E I, and UiEISd(Xi, e) => F. Then d(Xi, F) < e, Vi E I. On the other hand, for any y E F, there is j E I such that y E Sd(Xj, e), i.e., d(x;, y) < s, and d(y, {xihEI) < e. Therefore, p({xihElJ F) < e. Now it suffices to show that (C(E),p) is complete. Let {Fn } C C(E) and p(Fn , Fm ) ---t o. Put F =

{x EEl there is a subsequence {nk}, and

}.

x n " E Fn " , Vk, such that x n " ~ x

Since E is compact, it follows that F =I 0. Further, F E C(E). For any e > 0 , there is no such that p(Fn , Fm ) < s , "In, m > no. Fix n(> no). If

427

y E F,,, then from d(y, Fm ) < e, \1m > no , we can find X m E Fm such that d(y, x m ) < e, \1m > no. Since E is compact, there is a convergent subsequence of {xmlm > no}. Suppose that its limit point is x. Then x E F and d(y, x) < e. Thus, d(y, F) < e, \ly E Fn. Conversely, let x E F. Then there is a subsequence {nA:} , and xn" E Fn", \lk, such that xn" -----+ x. Pick k sufficiently large such that d(x n", x) < e, and nA: > no . Since d(x n", Fn) < e, there is y E Fn with d(x n" , y) < e. Further, d(x, y) < 2e. Hence, d(x, Fn) < 2c, \Ix E F. Therefore, p(Fn, F) < 2e, \In > no , and p(Fn, F) -----+ O. Q.E.D. Lemma 11.1.2. Let (P, d) be a compact metric space, and E be a Polish subspace of P. Then (C(E), p) is also a Polish space.

Proof. Denote the closure of E in P by E. Then (E, d) is a compact metric space. By Lemma 11.1.1, (C(E),p) is also a compact metric space. Define a map f : C(E) -----+ C(E) as follows:

f(F) = F,

\IF E C(E).

Clearly, p(f(F1),f(F2 )) = p(F1,F2),\lF1,F2 E C(E), and f(C(E)) = {K E C(E)I(K n E) is dense in K}. Since (C(E},p) and (f(C(E)),p) are isometrically isomorphic, so it suffices to show that f(C(E)) is a Polish subspace of (C(E),p). By Proposition 10.1.3, we need to prove that f(C(E)) is a G 6 - subset of (C(E) , p). Write E = nnVn, where Vn is an open subset of P,\ln. IT K E C(E), and (K n Vn) is dense in K, \In, then we have K E f(C(E)). Indeed, since K is a compact subset of P, K is a Baire space. Now (K n Vn ) is an open dense subset of K, \In, it must be that nn(K n Vn ) = K n E is dense in K, i.e., KEf ( C(E)). Therefore, we get

f(C(E)) = {K E C(E)I(K n Vn) is dense in K, \In}. Let Dn = {K E C(E)I(K n Vn) is not dense in K}, \In. Then f(C(E)) nn(C(E)\Dn } . Now it suffices to prove that each Dn is a FO'---subset (a countable union of closed subsets) of C(E). Fix n. Let K E C(E). IT K = {x} ( some x E E), then K E Dn ¢:::::> K n Vn = 0 <==> x E E\Vn • IT # K > 2, then K E Dn ¢:::::> there exists L E C(E) such that K n Vn C L c K and L t= K. Thus, we have

Dn

=

'7 U 1r"l(Sn \6),

where '7 = {{x }Ix E E\Vn}, 1rl is the projection from C(E) x C(E) onto its first component, 6 = {(K,K)IK E C(E)}, and

s;

= {{K,L)IK,L E C(E),and

K n Vn C L c K}.

It is easily verified that '7 is a closed subset of (C (E), p) . Moreover, 6 is a

closed subset of C(E) x C(E) obviously.

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We claim that Sn is also a closed subset of C(E) X C(E). In fact, suppose that {(Km , L m ) } C s.; and (Km , L m ) --+ (K, L) in C(E) X C(E). Since t.; C K m , \1m, it follows from the proof of Lemma 11.1.1 that L C K. Now if x E K n Vn , then there is a subsequence {mk}, and x m k E K m " , \lk, such that d(x m k , x) --+ O. But Vn is open and x E Vn , so we may assume that x m " E Vn , \lk. Thus, x m k E K m k n Vn C L m k , \lk. Further, by the proof of Lemma 11.1.1, x E L, i.e. K n Vn C L c K. Therefore, (K, L) E s; Now (Sn\6) is a FO'-subset of C(E) X C(E). So we can write Sn \6 = UmG m, where G m is compact in C(E) X C(E), \1m. Therefore,

Dn = '7 U Um1rt(Gm) is a FO' -subset of C(E).

Q.E.D.

Definition 11.1.3. Let E be a Polish space, and C(E) be the collection of all non-empty closed subsets of E. For any open subset U of E, put

u(U) = {F E C(E)IF n U "l0}. Further, we shall denote by P the Borel structure of C(E) generated by {u(U)!U is any open subset of E}.

Theorem 11.1.4. Borel space.

Let E be a Polish space. Then (C(E), P) is a standard

Proof. By Proposition 10.1.4, we may assume that E is a G 6-subset of P = [0,1]00. Clearly, there exists a metric don P such that (P, d) is a compact metric space. From Lemma 11.1.2, (C(E),p) is a Polish space. Now it suffies to show that the Borel structure of C(E) generated by p-top. is equal to P . First, for an open subset U of E, we say that u(U) is an open subset of (C(E), pl. Indeed, let F E u(U). Then there is x E F n U. Now if G E C(E) and p(F, G) is very small, then d(x, G)( < p(F, G)) is also very small, further, GnU "10, i.e., G E u(U). Thus, u(U) is open in (C(E),p). From the preceding paragraph, the Borel structure of C(E) generated by p-top. Contains P. By Theorem 10.3.13, It suffices to prove that P contains a countable separated family. Let {Un} be a countable basis for the topology of E. We need only to prove that {u(Un)}n is separated (for C(E)). If F,G E C(E) and F"I G, then we may assume that there is x E F\G. Clearly, we can find k such that x E Uk and ti, n G = 0. Thus, F n Uk "I 0 and G n Uk = 0, i.e., F E u(U k) and G rf. u(Uk). Therefore, {u(Un)}n is separated ( for C(E)). Q.E.D.

Proposition 11.1.5. Let (E, d) be a separable complete metric space. Then the standard Borel structure P of C(E) is the minimal Borel structure

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such that F ---+ d(x, F) is measurable on C(E), Vx E E. In other words, P is generated by {F E C(E) Id(x, F) < A}, Vx E E and A > O. Proof. First, for any x E E and A > 0, let U = {y E Eld(x, y) < A}. Then it is easy to see that u(U) = {F E C(E) Id(x, F) < A}. Thus, {F E C(E) Id(x, F) < A} E P. Now by Theorem 10.3.13, it suffices to show that the collection of {F E C(E)ld(x, F) < A}(Vx E E, A > 0) contains a countable separated family. Let {x,,} be a countable dense subset of E,Um,n = {x E Eld(x,x n) < m- l } , and (Jm,n = u(Um,n) = {F E C(E)ld(x n, F) < m-1},Vm,n. H F,G E C(E) and F i= G, then we may assume that there is x E F\G. Thus d(x, G) > 2m l if mo sufficiently large. Pick no such that d(x, x no) < mol. Then d(x no' F) < mol , i.e., F E (Jmo,no' On the other hand, since d(x no' G) > d(x, G) - d(x no, x) > mot, it follows that G f/. (Jmo,no' Therefore, {(Jm,n}m,n( C P) is separated ( for

o

C(E)).

Q.E.D. Proposition 11.1.6. Let X be a ( real or complex) separable Banach space, and C(X) be the collection of all closed linear subspaces of X. Then C(X) is a Borel subset of (C(X), Pl. Proof. Let {Vn } be a countable basis for the topology of X. It suffices to show that

where {AA:} is the set of all ( real or complex) rational numbers, and u(Vm )' = C(X)\u(Vm ) , Vm. In fact, if E belongs to the right side of above equality, then for any m, n, i, we have: i) if E n Vm t= 0 and E n Vn t= 0, then En (Vn + Vm) t= 0; ii) if En Vi i= 0, then En (AA:Vi) t= 0,Vk. Thus, for any z, y E E we get En (Vn + Vm) t= 0, En (Ak Vn) t= 0, Vk, m, n and x E Vm, y E Vn. By the closedness of E, we can see that (x + y) E E and Ax E E, VA E Dl ( or a:'), i.e., E E C(X). Conversely, if E E C(X), then for any m,n,i the above properties i) and ii) hold obviously, i.e., E belongs to the right side of above equality.

Q.E.D. Theorem 11.1.7. Let X be a ( real or complex) separable Banach space, C(X) be the collection of all closed linear subspaces of X, and W(X*) be the collection of all w*-closed linear subspaces of X*, where X* is the conjugate space of X. Then: 1) The standard Borel structure of C(X) is generated by

{E E C(X)lllx + Ell < A},

Vx E X, A > 0,

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2) The subsets of W(X*) with the following form

{E* E

W(X*)lllx + E~II

< A},

where El = {y E Xlf(y) = 0, \If E E*}, \Ix E X, A > 0, generate a standard Borel structure of W(X*). 1) It is obvious from Propositions 10.3.15, 11.1.6 and 11.1.5. 2) Notice that E* ~ El(\lE* E W(X*)) is a bijection from W(X*) onto C(X). Then by 1) we can get the conclusion. Q.E.D. Proof.

Proposition 11.1.8. Let H be a separable Hilbert space, and W(H) be the collection of all closed linear subspaces of H. Then the subsets of W (H) with the following form

{E E W(H)llle+

Ell < A},\le E H,A > 0

generate a standard Borel structure of W (H), and E ~ E.l. (\lEE W (H)) is a Borel isomorphism on W (H). Proof. From Theorem 11.1.7 and Proposition 10.3.2, it suffices to show that for any e E H, A > 0, {E E W(H)llle + E.l.II < A} is a Borel subset of W(H). If A > Ilell, then we have {E E W(H)llle + E.l.II < A} = W(H) obviously. Thus, we may assume A < Ileli. For E E W(H), let p be the projection from H onto E. Then we have

lIe + E.l.II = IIpell, lie + Ell = 11(1 - p)ell· Let Il =

(II el1 2 -

A2)1/2. Then {E E W(H) {E E W(H)

Ille + E.l.II < A} Ille + Ell> Il}

W(H)\ nn {E E W(H)

I lie + Ell < ~n + Il}.

Therefore, it is a Borel subset of W(H).

Q.E.D.

References. [34], [177].

11.2. Sequences of Borel choice functions First, we study the process of the Hahn-Banach theorem. Let X be real Banach space, E be a linear subspace of X, f be a linear functional on E with

431

norm < 1, and x E X\E. We want to extend / from E onto (E+[xJ) still with norm < 1, i.e.,

I/(x + w)1 < IIx + wll, Vw

E E.

So we need to pick the value of /(x) satisfying -lIx + ull- /(u) < /(x) < Ilx + vll- /(v),

Vu,v E E.

Then the value of /(x) must satisfy the following inequality: sup{(-llx + ull- /(u))lu E E}

< / (x) < inf{ (II x + v II

- / (v)) IvEE} .

Conversely, if the value of / (x) satisfy the above inequality, then / is a linear functional on E+[x] still with norm < 1. Definition 11.2.1. Let X be a real Banach space, E be a linear subspace of X, and x E X ( maybe x E E) . For any linear functional/on E with norm < 1 , define

L~}(/) and

Mk

=

Z

Since

II/II < 1, it

}( / )

sup{(-llx + ull- /(u))lu E E}.

= inf{(lIx + vII - /(v))lv

E E}.

follows that L~}(/) < M~z}(f).

Lemma 11.2.2. Let X, E, x and f be as in Definiton 11.2.1. 1) IT x E E, then L~}(/) = /(x) = M~z}(/). 2) / can be extended to a linear functional on E + [x] still with norm < 1 if and only if the value of /(x) must satisfy the inequality: L~)(/) < /(x) <

M1

Z } (/).

Proof.

1) Suppose that x E E. Then

I/(x + w)1 < IIx + wll, Vw

E E.

Further, we have

-lIx + ull- /(u) < /(x) < lIx + vII + /(v), Thus ,L~}(/) that

< /(x) < M1z ) ( /

).

L~)(/) > - /( -x) = /(x),

Vu,v E E.

On the other hand, since x E E, it follows and

Ml;)(/) < - /( -x) = /(x).

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Therefore L~)(/) = I(x) = M~z)(/). 2) It is obvious from 1) and the discussion of Hahn-Banch theorem.

Q.E.D.

Lemma 11.2.3. Let X be a real Banach space, E be a linear subspace of X, and x E X, and

8

=

{III

is a linear functional on E, andll/ll

< I}.

Write L~)(.) = L(.) and M~z)(.) = M(·) simply. Then L(.) is a convex function on 8, and L(·) = -M(-·) is continuous in the interior of 8.

Proof.

Let A E [0,1], and

i,» E 8. For any u E E, we have

-llx + ull - (AI

+ (1 - A)g)(U) = A(-llx + ull- I(u)) + (1- A)(-Ilx + ull- g(u)) < AL(/) + (1 - A)L(g). Thus, L(AI + (1 - A)g) < AL(/) + (1 - A)L(g), i.e., L(·) is convex on 8. Now let 10 E 8 and 11/011 < 1-1] for some TJ E (0,1). On V = {I E 8111/11 < TJ}, define F(/) = L(I + 10) - L(/o), We need to show that F(/) is continuous at convex on V, and

F(/) < M(I

I

+ 10) - L(/o) < llxll

VI E V. = 0. Clearly, F(O)

- L(/o),

Put 0: = Ilxll - L(/o). For any e E (0,1) and I E 8 I, ±g-1 I E V, it follows from the convexity of F(·) that

F(/)

=

= 0, F(.)

VI E V. with 11/11 <

is

TJg, since

F({I - g) ·0+ &. &-1 I)

< gF(g-1 I) < go: and

o

+ &)-11+ g(1 + g)-I. (_g-1 I)) < (1 + g)-l F(/) + g(1 + &)-1F( _g-1 I).

= F((1

From the second inequality, we get F(/) > -gF( _g-l I) > -go:. Thus, IF(/)I < go:, VI E 8 with I1I1I < TJg, i.e., F(·) is continuous at O. Q.E.D.

Theorem 11.2.4. Let X be a separable Banach space, and W(X t ) be as in Theorem 11.1.7 ( a standard Borel space). Then there is a sequence {In} of Borel maps from W(X t ) to (Xt,u(X\X)) such that: for any E" E W{X*) and n, In{E*) E (Eth ( i.e., In(E t) E E* and Il/n(E*)!1 < 1); and {In(Et)ln} is w*-dense in (Eth, VEt E W(Xt).

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First, let X be real. Suppose that {xnln = 1,2," .} is a dense subset of X, and fix E· E W(X·). Then {~ = X n + E~ln = 1,2'·"1 is dense in X/ El., where El. = {x E XI/(x) = 0, VI E E·}. Put B o = {O}, B n = [Xi,···, ~], Vn. These are finite dimensional linear subspace of X/El. Moreover, since (X/El)· ~ E·, we shall identify them in the following. For each t = (h,···, t n , · · · ) , where t n E [0,1], Vn, we say that there is a linear functional I tE - on X/ El such that Il/tE -1I < 1 (i.e., ItE - E (E·h) and Proof.

ItE- (X---) It ), n+l = t n+ 1Ln(ltE- ) + (1 - t n+ 1)u; (E-

(1)

where L n ( · ) = L~:+t}(.) and M n ( · ) = M;;:"+l)(.), Vn > 0. We prove this by induction. Assume that such I tE - exists on B n • Put A = tn+1Ln(lf-) + (1 - tn+1)Mn(lf-). Since Ln(lf-) < A < Mn(lf-), it follows from Lemma 11.2.2 that If- can be extended to a linear functional on B n+1 with norm < 1 still and I tE - (xn+d = A. Therefore, there exists a linear with norm < 1 and satisfying (1). functional If- on X/ Define Q = {r = (r1,· .. ,rn,· ..) ITn is rational and E [0,1], Vn; and #{ nlr n f O} < oo}, and fix I E ErI< with 11/11 < 1. For n = 1 and any E: > 0, since I:-(Xl) = (1- 2r d II XI II and I/(Xl)1 < IlxllI, there is a rational number r~O) E [0,1] such that 1(1;:- - I)(Xl)! < E:, Vr = (rn) E Q with rl = r~O). For n = 2 and any E: > 0, by Lemma 11.2.3 there is 1] > 0 with following property: for any g E e; with I(g - I) (xI) I < 1] ( thus Ilg - (/1Bd II is very small, and IlglI < 1), we have

s;

(2) From the preceding paragraph, there is a rational number r~O) E [0,1] such that I(/~- - I)(xdl < 1], Vr = (r~) E Q and rl = r~O). (3) By (2)' we have

ILd/;;-) - L1 (/ )1< E:, IMl(/;;-) - Ml(/)1 < e, (rn ) E Q with rl = riO). Clearly, 1!(/IB2 )l1 < 1. By Lemma 11.2.2, we have t 2 E [0,1] such that

Vr =

(4) Pick a rational number r~O) E [0,1] satisfying

I[r~O) L 1 ( / ;;- )

+ (1 -

r~O»)Ml(I:-)I- [t 2 L 1 ( /

)

+ (1 - t 2)Ml ( /

) ]]

<

s,

(5)

434

Vr = (rn ) E Q with , we get

ri

= ria). We may assume

I(I:·

- I)(Xi) I < e,

TJ

< e. Now by (3), (1), (4), (5)

i = 1,2,

. h rl = r (0) an d r2 = r2(0) . vr = (r n) E Q WIt 1 Repeating this process, for any n and e > there exist rational numbers ria), ... , ria) E [0,1] such that 1(/;;* - I)(Xi) I < c,I < i < n, Vr = (rA:) E Q with rA: = ria), 1 < k < n. Since above I( E E* and 11/11 < 1) is arbitrary, the set {/;;·lr E Q} is w*-dense in (E*h. Now for any t = (tI,···, t n, · · ·) with t n E [0,1], "In, we say E* - - t I tE* is a Borel map from W(X*) to (X*, C1(X*, X)). It suffices to show that E* - - t If· (xn) is a Borel measurable function on W(X*), "In. For n = I,ltE·(Xl) = (1- 2tdliXlii is measurable on W(X*) obviously. Now assume that E* --to ItE·(xA:) is measurable on W(X*), 1 < k < n. Then E* - - t ftE·(u) is measurable on W(X*), Vu E B n. Moreover, by Theorem 11.1.7 E* - - t IIxn+l +ull = lIXn+l +tt+E.i.11 is also measurable on W(X*), Vu E B n , where tt E [x},···, xn] and tt + E.i. = u. Thus, E* --to Ln(ltE*) =

U

°

n

sup{(-llxn+l+ull-ltE·(u))!u E B n} = suP{(-llxn+l+ull-lf·(u))lu = Lrixi, i=l and ri is rational, 1 < i < n} is measurable on W(X*). Further, by (1) E* ----+ If· (Xn+l) is measurable on W(X*). Therefore, the theorem is proved for real case. In the following, let X be complex. Clearly, X can be regarded as a real space, denoted by X r . Then there is a sequence {In} of Borel maps from W(X;) to (X;,C1(X;,Xr )) such that for any E; E W(X;) and n,ln(E;) E (E;h; and {In (E;) In} is w*-dense in (E;h, "IE; E W(X;). Now for any E* E W (X*) and n, define

gn(E*)(x) = In(ReE*)(x) - iln(ReE*)(ix),

Vx E X,

where ReE* = {Re III E E*}(E W(X;)). Clearly, gn(·) is a Borel map from W(X*) to (X*,u(X*,X)), and Ilgn(E*)lI < I,VE* E W(X*). Let x EEl. Since ix E E~ and E.i. = (ReE*)J.' it follows that gn(E*)(x) = O. Thus, gn(E*) E (E*)., VE* E W(X*). Moreover, fix E* E W(X*). For any 9 E (E*)., Yl,· .. , Yrn EX, and c > 0, since Reg E (ReE*h, we can find n such that 1(ln(Re E*) - Re g)(Yj)1 < e and 1(/n(Re E*) - Re g)(iYj) I < s, 1 <j < m. Then l(gn(E*) - g)(y;) I < e,I

(E*)., VE* E W(X*).

< i < m. Therefore, {gn(E*)ln} is w*---dense in Q.E.D.

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Theorem 11.2.5. Let X be a separable Banach space, and (E, B) be a Borel space. Then a map 1/J : (E, B) -----+ W (X*) is Borel if and only if there is a sequence {gn} of Borel maps from (E, B) to (X*, u(X*, X)) such that for each t E E,gn(t) E (1/J(t)h, Vn, and {gn(t)ln} is a w*-dense subset of (1/J(t)h, where (1/J(t))l is the closed unit ball of 1/J(t)(E W(X*)). Proof. Suppose that {In} is as in Theorem 11.2.4. If 1/J is Borel, then {gn = In 0 1/J} satisfies our conditions . Conversely, if {gn} satisfies the conditions, then for any x E X,t E E, we have

Ilx + 1/J(t).L11 =

sup Ign(t)(x) I· n

Thus, t - + Ilx + 1/J(t).L11 is measurable on (E, B), Vx E X. Now by Theorem 11.1.7, 1/J is Borel. Q.E.D.

References. [35], [177}.

11.3. The Borel spaces of Von Neumann algebras Let H be a ( complex) separable Hilbert space. Then X = T (H) is a separable Banach space, and X* = B(H). For any E E W(X*), let

E'" = {a*la E E},

E' = {b E B(H)lab = ba, Va E E}.

Proposition 11.3.1. E - + E* and E - + E' are Borel maps on W(X*), where the standard Borel structure of W(X"') is as in Theorem 10.1.7. Proof. Let ct>(E) = E*,VE E W(X*). Since (E"').L = (E.L)"', it follows from Theorem 11.1.7 that

ct>-l{E E W(X*)lllt + E.Llll < A} {E E W(X*)lllt*

+ E.Llh < A}

is a Borel subset of W(X*), Vt E X, A > 0, where

= T(H). Therefore, E

I . 111

is the trace norm of

E* is a Borel map on W(X*). By Theorem 11.2.4, there is a sequence {a n ( ' ) } of Borel maps from W(X*) to (X"',u(X*,X)) such that for any E E W(X*),{an(E)ln} is a w*-dense subset of (Eh. Then X

-+

E' = {b E X*I ban(E) = an(E)b, Vn}, VE E W(X*).

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Define

M = {(xn)lxn E B(H),Vn, and sUPllxnl1 < oo} n

and

M* = {(tn)jt n E T(H),Vn,and

2: Iltnlh < oo}. n

Clearly, M = L fBB(H) is a W*-algebra, and M. is the predual of M. n

For any E E W (X*) , define a map T E

:

B(H)

TE(b) = (ban(E) - an(E)b),

-+

Vb E B(H).

Then E' = Ker T E = {b E B(H)ITE(b) = O},and Further, define a map T~ : M* - + T (H) as follows T~((tn))(b)

=

M as follows: T E is a-a continuous.

TE(b)((t n))

= L tr((ban(E) - an(E)b)tn) n

Vb E B(H), (t n) E M*. Since (T~)* = T E, it follows that (E').L = ( Ker TE).L = T!M*. Let 8 be the unit ball of B(H) , {b;} be a countable dense subset of (8, a), and {( t!!»)} be a countable dense subset of M*. Then for any t E X, E E W(X*), we have

But !It+T!((tW»)) 11 1 = SUPi Itr(tb i) + Ltr((bian(E)-an(E)bi)t~»)I, and an(·) : n

(B(H),a) is Borel, so E -+ lit + (E').LII is a Borel measurable function on W(X*). Therefore, E ----+ E' is a Borel map on W(X*). Q.E.D.

W(X*)

-+

Theorem 11.3.2. Let H be a separable Hilbert space, X = T(H), and A be the collection of all VN algebras on H. Then A is a Borel subset of W(X*). Consequently, the family of following subsets

{M E

Ailit + M.Llll < .x}, 'It E X,.x > 0

will generate a standard Borel structure of A.

Proof. By Proposition 11.3.1 and 10.3.4, {E E W(X*)IE = E*} and {E E W(X*)IE = E"} are Borel subsets of W(X*). Then A = {E E W(X*)IE = E*} n {E E W(X*)IE = E"} is also a Borel subset of W(X*). Q.E.D. Proposition 11.3.3. Let H be a separable Hilbert space, 8 be the unit ball of B(H) , and A be the collection of all VN algebras on H. Then there

437

is a sequence {an(.)} of Borel maps from A to (8, a) such that for any M E A,{an(M)ln} is a r(M,M.)-dense subset of (Mh.

Proof.

By Theorem 11.2.4 and 11.3.2, there is a sequence {bn ( · ) } of Borel maps from A to (8, a) such that for each MEA, {bn{M) In} is a weakly dense subset of (M)l' Let

{an(')ln}

=

{ :L AA;bk ( · )

AA; is non-negative and rational, } vi; and:L Ak = 1

k

k

Now by Proposition 1.2.8, {a n ( ' ) } satisfies our conditions.

Q.E.D.

Theorem 11.3.4. Let (E, B) be a Borel space, and A be the collection of all VN algebras on a separable Hilbert space H. Then a map t/J : E ----+ A is Borel if and only if there is a sequence {an(.)} of Borel maps from E to (B(H), u(B(H), T(H))) such that for each tEE, the VN algebra t/J(t) is generated by {an(t)ln}.

Proof.

The necessity is obvious from Theorem 11.2.3. Now let such {a n ( · ) } exist. By Theorem 1.6.1 and a proper treatment, we have a sequence {bn ( · ) } of Borel maps from E to (B(H), u) such that for each tEE, {bn(t) In} is a a-dense subset of (t/J(t)h. Then by Theorem 11.2.3, t/J : E ----+ A is Borel. Q.E.D.

Proposition 11.3.5. (M, N) Borel maps from A x A to A.

-------4

M n Nand (M, N)

----+

(M UN)" are

Let {an(')}n~l be a sequence of Borel maps from A to (B(H),a) as in Proposition 11.3.3. For any M, N E A, let

Proof.

b2n(M, N) = an(M),

b2n- 1(M, N) = an(N), \In = 1,2,···.

Then bn(·,·) is a Borel map from A x A to (B(H),a),\ln, and {bn(M,N)ln} = {am(M),an(N)lm, n}, \1M, N E A. Thus, (MUN)" is generated by {bn(M, N) In}, \1M, N EA. Now by Proposition 11.3.4, (M, N) ----+ (M uN)" is Borel from A x A to A. In the process of (M, N) ----+ (M', N') -+ (M' UN')" ----+ (M' UN')'" = M n N, each map is Borel. Therefore, (M, N) - - t (M n N) is also Borel from A x A to A. Q.E.D.

Theorem 11.3.6. Let H be a separable Hilbert space, A be the collection of all VN algebras on H, and 1 be the collection of all factors on H. Then 1 is a Borel subset of A. Consequently, the family of following subsets

{M E

1111t +MJ.lll < A},

\It

E

T(H),A >

0

438

will generate a standard Borel structure of 1.

Proof. Since M - + (M,M') -+ M n M' is a Borel map from A to A, 1 = {M E AIM n M' =
The Borel spaces of Von Neumann algebras was introduced by E.G.

References.

[34], [35], [177].

11.4. Borel subsets of factorial Borel space Let H be a separable Hilbert space, A be the standard Borel space of all VN algebras on H , and 1 be the standard Borel space of all factors on H. Lemma 11.4.1. Let G be the set of all unitary operators on H. Then G is a Polish topological group with respect to strong ( operator ) topology.

Proof. Clearly, G is a topological group with respect to strong topology. Let 8 be the unit ball of B(H). Then 8 is a Polish space with respect to strong topology. If {€A;} is a dense subset of {€ E and only if lIu€kll = Ilu*€kll = 1, 'v'k. Thus G

Hili €II =

I}, then u E G if

= nk{U E SllIu€kll = l}n

nk,n Urn {u

E 8

1

11(€k,u€rn)1 > 1- n}

is a Go -subset of (8, s-top ) . That comes to the conclusion.

Q.E.D.

Proposition 11.4.2. For any MEA, let s(M) = {uMu*lu E G}, where G is as in Lemma 10.4.1. Then s(M) is a Borel subset of A,'v'M E A.

Proof. Fix MEA, and put Go = {u E GluMu* = M}. Define an equivalent relation r - in G : u '" v if v E uG o. By Lemma 11.4.1 and Theorem 10.4.2, there is a Borel subset E of G such that #(E n uG o) = 1, 'v'u E G. Then, s(M) = {uMu*lu E E}. We say that u ----t uMu* is a Borel map from G to A. In fact, if {an} is a countable dense subset of ((Mh,O'(M,M.)), then {an(u)ln} generates uMu·, 'v'u E G, where an(u) = uanu*('v'u E G) is a continuous map from G to (B(H), 0'), 'v'n. Now by Proposition 11.3.4, u - + uMu· is a Borel map from G to A.

439

In particular, u --+ uM u· is an injective Borel map from E to A. Therefore, by Theorem 10.3.12 s(M) is a Borel subset of A. Q.E.D.

Proposition 11.4.3. Let MEA. Then a(M) to M } is a Borel subset of A.

= {N

E

AIN is

* isomorphic

A = A(H) is the collection of all VN algebras on H. We shall denote the collection of all VN algebras on H ® H by A(H ® H). Define a map cl) : A(H) --+ A(H ® H) as follows Proof.

cl)(M)

= M®
Let M, N E A(H). Then M and N are * isomorphic if and only if cl)(M) and cl)(N) are * isomrorphic. Notice that cl)(M)' = M'®B(H) and cl)(N)' = N'®B(H) are properly infinite ( here dim H = 00; if dim H < 00, then a(M) = s(M) is a Borel subset of A ) . By Proposition 6.6.7, cl)(M) and cl)(N) are spatially * isomorphic if cl)(M) and cl)(N) are * isomorphic. Thus, a(M) = cl)-l(s(cl)(M))). From Proposition 11.3.4, it suffices to show that cl) is a Borel map. By Proposition 11.3.3. there is a sequence {a n ( · ) } of Borel maps from A to (S,a), where 8 is the unit ball of B(H) , such that {an(N)ln} generates N, VN E A. Then {a n(·)®l Hln} is a sequence of Borel maps from A to (B(H® H),u) , and {an(N) ® 1H ln } generates cl)(N), VN E A. Now by Proposition 11.3.4, cl) is a Borel map. Q.E.D.

Proposition 11.4.4. Denote the collection of all type (In) factors on H by ~n' Vn. Then ~n is a Borel subset of 1, n = 00,1,2" ... Noticing that all type {In) factors are is obvious from Proposition 11.4.3.

Proof.

* isomorhpic,

the conclusion Q.E.D.

Lemma 11.4.5. Denote the projection from H onto [M'pH] by eM(p). Then (M, p) - + eM(p) is a Borel map from A x P to P, where P is the collection of all projections on H, and with strong (operator) topology P is a Polish space.

Proof.

Clearly, from Theorem 11.1.7 the standard Borel spaces P and W(H) are Borel isomorphic. So we need to show that (M, p) --+ [M'pH] is a Borel map from A x P to W(H) . By Theorem 11.2.5, it suffices to find a sequence {11n(', .)} of Borel maps from A x P to (H, w), where "w" means the weak topology in H, such that {11n(M,p)ln} is a dense subset of [M'pH] , VM E A,p E P. By Proposition 11.3.3, there is a sequence {a n ( · ) } of Borel maps from A to (8,0') such that {an(M) In} is a r(M,M.) -dense subset of (Mh,VM EA. Let

440

{6;} be a countable dense subset of H, and define ~n,Il:(M,p) = an (M')pell:, V n, k, Clearly, ~n,Il:("') is Borel from A x P to (H, w), Vn, k, and [~n.Il:(M,p)ln,k] =

[M'pH] , VM E A,p E P.

Therefore, we can find Borel maps TIn : A X P {Tln(M,p) In} is dense in [M'pH] , VM E A,p E P.

~

(H,w),Vn, such that Q.E.D.

Lemma 11.4.6. Denote the collection of all infinite factors on H by Then ~I is a Sousline subset of 1.

~/'

Proof. A factor M is infinite if and only if there is v E M such that v·v = 1 and vv· -=I 1. By Proposition 11.3.3, there are Borel maps an ( · ) : A ~ (S,u)(n = 1,2,"') such that {an(N)ln} is r -dense in (Nh,VN E A. Notice that

E=

M E 1, v E, S' v·v 1 vv· -r, -/.. l' } -- , ( , ) 1 and an(M')v = van(M'), Vn (1 x {v E Slv·v = l,vv· -=I 1})n {

Mv

nn{(M,v)lan(M')v = van(M')} (1 x {v E slv·v = l,vv· -=I 1})n

nn,i,j{(M,v)I((an(M')v - van(M'))ei, e;) = o} is a Borel subset of 1 X (S,r), where {ei} is a countable dense subset of H. If 1(" is the projection from 1 x S onto 1, then lsi = n E: Therefore, ~, is a Q.E.D. Sousline subset of 1.

Proposition 11.4.7. Denote the collection of all type (Ill) factors on H by ~Il' Then ~Il is a Borel subset of 1.

'1.

Proof. Denote the collection of all finite factors on H by From Proposition 11.4.4 and Lemma 11.4.6, it suffices to prove that 1, is a Sousline subset of 1. Since H is separable, a factor M on H is finite if and only if there exists a faithful normal tradal state on M, i.e, there is a sequence {ell:} of H such that II ell: 11 2 < 00, L((ab - ba) ell:, ell:) = 0, Va, se M, and [a' ell: la' EM', k] is

L Il:

Il:

441

dense in H. Then by Lemma 11.4.5,

E

= {(M (e )) 1M ,Jr:

{

(M

r

E (€A;) E Hoo;[a'6:la' E M',k] = H;} andEJr:((ab- ba)eJr:,eJr:) = O,Va,b EM

«)) I ME 1, (6;) E Hoo,and eM(p) = 1, }n , Jr: where p is the projection from H onto [6: Ik] 00

is a Borel subset of 1

X

Hoo ' where H oo = L $H; a(·) : A ----+ (S,a) is Borel n=l

,Vn, and {an(M)ln} is r-xlense in (Mh,VM E A( see Proposition 11.3.3). Let 1r be the projection from 1 x H oo onto 1. Then 11 = 1rE is a Sousline subset of 1. Q.E.D. Lemma 11.4.8. Denote the collection of all semi-finite factors on H by 'fal. Then 'fal is a Sousline subset of 1.

Proof.

A factor M on H is semi-finite if and only if there exists a finite projection p of M with c(p) = 1, i.e., there is a projection p of M with [MpH] = H, and there exists a sequence {eJr:} of pH with L IleJr:11 2 < 00 such Jr: that L(·eJr:, eJr:) is a faithful trace on (pMp). Jr: Consider a subset Eof 1xPxHoo • (M,p, (eJr:)) E E, ifpeJr: = eJr:, Vk;pan(M') = an(M')p,Vn; [a'eJr:lk,a' E M'l = pH;[MpH] = H, and

L((pan(M)pam(M)p - pam(M)pan(M)p) eJr:, eJr:)

=

0,

Jr: Vn, m where an(·) : A ----+ (8, a) is Borel , Vn, and {an(M) In} is r-xiense in (Mh, VM E A ( see Proposition 11.3.3). By Lemma 11.4.5, the map (M, p) ----+ [MpH] is Borel. Thus , E is Borel . Let 1r be the proj ection from 1 x P x H oo onto 1. Then 'fal = 1rE is a Sousline subset of 1. Q.E.D. Lemma 11.4.9. Let M be a factor on H, and c1> be a * automorphism of M. If there is a non-zero element a of M such that c1>(b)a = ab, Vb E M, then c1> is inner, i.e., there exists a unitary element u of M such that c1>(b) = ubu"; Vb E M.

Proof.

From c1>(b)a = ab, we have a*c1>(b*) = b*a*, Vb E M. In particular, if

442

b is unitary, then we get

= a*~(b*) . ~(b)a = a*a, ~(b)aa*~(b*) = ab . b*a* = aa*.

b*(a*a)b

Thus, a*a and aa" E MnM' = (cl R . Now let u = element of M, and ~(b) = ubu", Vb E M. Lemma 11.4.10.

Ilall-la.

Then u is a unitary Q.E.D.

Let G be the group of all unitary operators on H. Then

E={(M)I ,u

is a Borel subset of f

X

MEf,uEGjuMu*=M,

but.

-4

u . u*is not inner for M

}

G.

Proof. By Proposition 11.3.3, we have Borel maps an ( · ) : A - 4 (8,0), n = 1,2,···, such that {an{M)ln} is r-dense in (Mh, VM E A. Since (M,u) - 4 uan(M)u* is a Borel map from f X G to (8,0), Vn, it follows that

E = {(M,u)IM

_n -

",

m

{(M u) ,

f,U

E

E G,and

uMu* = M}

Iuan(M)u* . ME r, u E G,and } am(M') = am(M') . uan(M)u*

is a Borel subset of f X G. Let d be a proper metric on (8,0) ( see Definition 10.1.1 ) , and consider a subset EU, k, m, n) of f X G. (M, u) E EU, k, m, n), if uMu* = M and satisfies one of following conditions:

1) d{aj(M), 0) < n- l ; 2) d(uak(M'),O) < n- 1 ; 3) d(aj(M),O) > n- 1 , d(uak(M') ,0) > n-t, and d(aj(M),uak(M')) > m- 1 • Noticing that (M, u) ~ (aj(M) , uak(M')) is a Borel map from f X G to (8,0) X (8,0), so E(i,k,m,n) is a Borel subset of f X G. Now it suffices to prove that E = nn u., nj,kE(j, k, m, n).

If . ~ u . u* is an inner * automorphism of M, i.e., there exists a unitary element v of M such that uau* = vav*,Va E M, then v E uM'. If d(v, 0) > 2n- 1 , for any m we can choose j,k such that

d(aj(M), v) < (2mn)-1,d(uak(M'),v) < (2mn)-1. Thus (M, u) tJ. El]; k, m, n). Further, (M, u) tJ. nn U m nj,kE(i, k, m, n). Conversely, let (M,u) E f X G,uMu* = M, and (M,u) tJ. nnUmnj,kEU,k, m, n). Then there is n such that for any m, we have j(m), k(m) and (M, u) tJ. EU(m),k(m),m,n). Thus, d(aj(m) (M), 0) > n-l,d(uak(m)(M'),O) > n- 1 , and d(aj(m)(M), Uak(m) (M')) < m- 1 , Vm. Since the unit balls of M and M' are 0 compact, there exist o-cluster points a,a' of {aj(m)(M)lm},{uak(m)(M')lm} respectively. Hence, we get

a = ua',

d(a,O) > n-t,d(ua',O) > n- 1 •

443

Now for any b E M, we have ubu'a = uba' = ua'b = abo Thus, by Lemma 11.4.9 . ~ u . u· is an inner * automorphism of M, i.e., (M, u) ~ E. Therefore, E = nn u., ni,kEU,k,m,n). Q.E.D.

Lemma 11.4.11. Let G be the group of all unitary operators on H ( G is a Polish topological group with respect to strong operator topology, see Lemma 11.4.1 ) , and Go = {u E Gil is not an eigenvalue of u}. Then

is a Borel subset of G, where {E;} is a countable dense subset of the unit ball of H, {1m} is a sequence of continuous functions on {z E 2- m+1 , then fm(z) = 0, Vm.

Proof. Let u E G, and E E H with E, Vm. Thus , we have 11 - Il/m(u)Eilll =

<

II Ell = 1,

and uE

=

E· Then Im( u) E =

1II/m(u)EII-llfm(u)E;111 lIE - E;II, vi.

Pick jo such that IIE;o - Ell < 1/4. Then II/m(u)E;oll > 3/4,Vm. Hence, if n > in and k > 2 , then

Conversely, let u E Go, and e(.) be the spectral measure of u on {z E
Pm

= e({z

llzl

= 1,11 -

zl

< 2- m}),

Vm.

Then Pm+l < Pm,O < Im+l(U) < Pm < Im(u) , Vm, and Pm ~ 0 ( strongly). For any n, k, choose m sufficiently large such that sup{11PmE;1l11 < j < n} < k- 1 • Then we have

Q.E.D. Lemma 11.4.12. Let X, Z be two Polish spaces, Y be a Borel space, and I be a map from X x Y to Z such that: 1) for each y E Y, 1(·, y) is continuous from X to Z ; 2) for each x E X, I(x,') is Borel from Y to Z. Then I is Borel.

444

Proof.

Let d,6 be proper metrics on X, Z respectively, and {Xk} be a countable dense subset of X. If F is a closed subset of Z, then

f-l(F) =

{(x, y) E X

n, u,

X

Ylf(x, y) E F}

{( )I

d(x, Xk) < n- 1 , and } z, y 6(f(Xk' y), F) < n-1

is a Borel subset of X x Y, where Xk,n = {x E Xld(x, Xk) < n- 1 } , and Yk,n = f(Xk,·)-l({z E ZI6(z,F) < n- 1 } ) . Now by Proposition 10.3.2, f is Borel. Q.E.D. Lemma 11.4.13. Let G be the group of all unitary operators on H, Go = {u E GIl is not an eigenvalue of u}, and f(t, z) = exp] -t(z + l)(z l)-I)),\it E lR,z E (C with Izl = 1 and z f: 1. Then f is a Borel map from lR x Go to G , where the Borel structures of G and Go are generated by strong operator topology.

Proof.

Fix u E Go. Clearly, t ---t f(t, u) is a continuous map from lR to G. Now fix t E lR, and let r = {z E (Clizi = I}. Then g(z) = it(z + l){z _1)-1 is a real valued continuous function on [I' \ {I}). Pick a sequence {gn} of real valued continuous functions on r such that gn(z) ---t g(z), \iz E (r\ {I}). Then ,exp(ign(u)) ---t exp(ig(u)) = f(t,u) ( strongly) , \iu E Go. Let 6 be a proper metric on the Polish space G, and F be a closed subset of G. Since 6(exp(ig n(u)) , f(t, u)) ---t 0, \iu E Go, it follows that

f(t, .)-l(F) = {u E Golf(t, u) E F} =

nk

Un~k {u E G o I6(exp(ign(u )), F )

< k- 1 } .

But exp(ign ( · )) is continuous on G, \in, and Go is a Borel subset of G (Lemma 11.4.11) , thus f(t, .)-I(F) is a Borel subset of Go, and f(t,') is a Borel map from Go to G. Now by Lemma 11.4.12, f is a Borel map from lR X Go to G. Q.E.D. Lemma 11.4.14. Let E be a Sousline subset of 7. Then s(E) = {uMu*'M E E, u E G} is also a Sousline subset of 7, where G is the goup of all unitary operators on H.

By Proposition 11.3.3, we have Borel maps an ( · ) : 7 ---t (8,0'), n = 1,2" .. , such that {an(M) In} is T -dense in (Mh, \iM E 7. Let bn(M, u) = uan (M) u .. ,\in. Then {bn ( " .)} is a sequence of Borel maps from 7 x G to

Proof.

445

(8,0') ,and {bn(M,u)ln} generates uMu*,V(M,u) E 7 X G. By Proposition 11.3.4, 1/J : (M, u) ---+ uMu· is a Borel map from 7 X G to 7. Therefore, by Proposition 10.3.5, s(E) = 1/J(E X G) is a Sousline subset of 7. Q.E.D. Now let D = {z E CL"IO

A(D)

=

< Imz < I},

{f If is complex valued, bounded and continuous on D, } and f is analytic in the interior of D '

C~(D)

be the collection of all complex valued continuous functions on D vanishing at 00. By maximal modulus, COO(D) is a Banach space. Clearly, f(z) ---+ exp] -IRezl)f(z) is an injective map from A(D) into Co(D); and this map transforms {f E A(D) Ilf(z) I < r, Vz E D} to a closed subset of Co(D); further, A(D) can be regarded as a Borel subset of Co(D). Thus, A(D) admits a standard Borel structure induced by the Borel structure of Co(D).

Proposition 11.4.15. Denote the collection of all type (III) factors on H by 7.11' Then is a Borel subset of 7.

s.;

Proof. From Lemma 11.4.8, it suffices to show that 7.11 is a Sousline subset of 7. Pick Eo E H with IIEoll = 1, and consider a subset E of 7 X Go X JR. (M,u,s) E E, if: I)MEo = M'Eo = H; 2) for any t E JR,f(t,u)Eo = Eo; 3) f(t, u)Mf( -t, u) = M, Vt E JR; 4) x ---+ f(s, u)xf( -s, u) is not inner for M, where Go and f are as in Lemma 11.4.3. By Proposition 11.3.3 and 11.2.3, the condition 1) determines a Borel subset of 7. For any rational number r, from the proof of Lemma 11.4.13 f(r,') is a Borel map from Go to G. Thus, the condition 2) determines a Borel subset of Go. By the proof of Lemma 11.4.10, {(M,v)lvMv· = M} is a Borel subset of 7 X G. Since it is enough that the condition 3) holds for all rational numbers, hence the condition 3) determines a Borel subset of 7 X Go. Moreover, the condition 4) determines a Borel subset of 7 X Go X JR from Lemmas 11.4.10 and 11.4.13. Therefore, E is a Borel subset of 7 X Go X JR. From Proposition 11.3.3, we have Borel maps an ( · ) : 7 ~ (8,O'),n = 1,2,,'" such that {an(M)ln} is r -denae in (Mh, VM E 7. For any positive integers i, k, consider a subset E(j, k) of 7 X Go X JR X A(D). (M, u, s, g) E E(j,k), if: 5) (M,u,s) E E; 6) g(t) = p(f(t,u)aj(M)f(-t,u)ak(M)),Vt E JR; 7) g(t + i) = p(ak(M)f(t, u)aj(M)f(-t, u)), Vt E JR, where p(.) = (·eOI Eo). Since it is enough that the condition 6) holds for all rational numbers, hence the condition 6) determines a Borel subset of 7 X Go X A(D). The case of the condition 7) is similar. Thus, E(i, k) is a Borel subset of 7 X Go X JR X A(D). Let 1rl be the projection from 7 X Go X JR X A(D) onto 7 X Go X lR, and 1r2 be the projection from 7 X Go X JR onto 7. Then

Eo = 1r2(nj,k'JrlE(i, k))

446

is a Sousline subset of 7. By Lemma 11.4.14, s(Eo) is also a Sousline subset of 7. If M E 7.11' then from Proposition 6.6.6 there is a cyclic-separating vector e(E H, lIell = 1) for M. Then we have u E G such that eo is cyclic -separating for uMu·. Now it suffices to show that

Eo

=

{M E 7.111 eo is cyclic-separating for M}.

Let M E 7.1U and eo be cyclic -separating for M. Then tp(.) = (·eo, eo) is a faithful normal state on M. Suppose that {Utlt E lR} is the modular automorphism group of M corresponding to tp. Since M is type (III), from Theorem 8.3.6 there is s E Hi such that a, is not inner. By the invariance of rp for {Ut}, for each t E Hi we can define a unitary operator Ut on H such that

Clearly,

UteO = eo,

UtaU-t = Ut(a), Vt E Hi, a E M,

and t --+ Ut is strongly continuous. If U is the Caley transformation of the generator of {Ut} , then U E Go and Ut = f(t, u), Vt E lR. Let gjk be the KMS function corresponding to (tp, {ut},aj(M), ak(M)). Then (M, u,s,gjlc) E E(j, k), vi, k. Thus, M E Eo Conversely, let M E Eo , then there is a U E Go and a s E Hi such that

(M,u,s) E nj,lc1rlE(j,k). Hence, ME 7, eo is cyclic-separating for M, {Ut(') = f(t,u) . f(-t,u)} is an one-parameter strongly continuous * automorphism group of M, and u a ( · ) is not inner. For any a, b E (M) 1, we can find

alc(n)(M) ~ b.

aj(n)(M} .z., a,

For each n, since (M, u, s) E 1rI E (i (n ), k(n)), there is gn E A(D) such that

gn(t) = tp(Ut(a;(n) (M))alc(n) (M)), gn(t

+ i) =

tp(alc(n) (M}ut(a,(n) (M))) ,

Vt E Hi. Noticing that f(t, u)eo = eo and f(t, u) E G, Vt E Hi, from the maximum modulus theorem we have

19n(z) - gm(z)1

--+

0,

uniformly for

Thus, there is 9 E A(D) such that gn(z)

g(t)

=

tp(Ut(a)b),

--+

zED.

g(z), Vz E D. Consequently,

g(t + i} = tp(but( a)),

447

Vt E JR. By Theorem 8.2.10, {Ut(') = f(t, u) . f( -t, u)lt E lR} is the modular automorphism group of M corresponding to cp(.) = ('€o, €o). Since U. is not inner, it follows from Theorem 8.3.6 that M E ;"11' Q.E.D. From above discussions, we obtain the following.

Theorem 11.4.16. of 7.

Notes.

;,... (n = 1,2" . '), ;"00' ;"11' ;"100 and

fru are Borel subsets

Proposition 11.4.15 is due to O. Nielsen. It is a key result for Theorem

11.4.16.

References. [120], [154].

Chapter 12 Reduction Theory

12.1. Measurable fields of Hilbert spaces Let (E, B) be a Borel space. A complex valued function f on E is said to be measurable, if it is B-measurable. H(.) is called a field of Hilbert spaces over E , if for each tEE, H(t) is a Hilbert space. €(.) is called a field of vectors ( relative to H(.)) over E, if

€(t)

E

H(t), Vt

E

E.

Definition 12.1.1. A field H(·) of Hilbert spaces over a Borel space (E, B) is said to be measurable, if there is a sequence {€n (.)} of fields of vectors over E such that: 1) the function (€n(t), €m(t))t is measurable on E, Vn, m, where (,)t is the inner product in H(t),Vt E E; 2) for any t E E,{€n(t)ln} is a total subset of H(t) ( in particular, each H(t) is separable). In this case, a field €(.) of vectors over E is said to be measurable ( with respect to the measurable field H (.) of Hilbert spaces ) , if the function (€(t), €n(t))t is measurable on E, Vn. Denote the collection of all measurable fields of vectors bye. Proposition 12.1.2. Let H(o) be a measurable field of Hilbert spaces over a Borel space (E, B). Then: 1) from any n = 00,0,1,0",

En = {t E EldimH(t)

=

n}

is a Borel subset of E; 2) there exists a sequence {t]n(')} C e with following properties: (a) if tEE with dim H (t) = 00, then {t]n (t) In} is an orthogonal normalized basis of H(t); (b) if tEE with dim H(t) = n < 00, then {t], (t), .. . ,t]n(t)} is an

449

orthogonal normalized basis of H{t), and '7k{t) = 0, Vk > n; (c) €(.) E and only if the function (€(t), '7n{t)), is measurable on E, Vn.

e

if

Proof. Suppose that we have {'71(.), ... , n« (.)} c e with following properties: 1) if tEE with dim H(t) > n , then ('7i{t), '7; (t)), = 6i j, VI < i,j < n; 2) if tEE with dim H(t) = k < n, then {'71(t),···,'7k(t)} is an orthogonal normalized basis of H(t), and '7i(t) = O,Vk n; 4) if €(.) E e, then the function (€(t),'7i(t)), is measurable on E, VI < i < n. For each tEE, let Pn(t) be the projection of H(t) onto ['7i(t)!1 1 , let

F. = {t 1

EEl

(1 - Pn{t)) €n+i(t) 1: 0, and } (1 - Pn{t))€i(t) = 0, i < n + j

and

F oo = {t E EI(1 - Pn(t))€i(t) = 0, Vi} = {t E EldimH(t) < n}. Since II (1- Pn(t))€i(t) II, is a measurable function on E, it follows that {Foo ' F1 , F 2 , ••• } is a Borel partition of E. Let O, '7n+l{t) = { (1- Pn{t) €n+' t 1 - Pn t €n+i t

iftEFoo, if t E F· ,1

l'

. = 1 2 ...

J

"

Clearly, {'7I(·),···' '7n+I(·)} also satisfies the conditions (1)-(4). Repeating this process, we get a sequence {'7n(·)ln = 1,2,···} of e . If €(.) is a field of vectors over E such taht (€ (t), '7i (t)), is measurable on E, Vi, then

(€(t), €n(t)) , = L(€(t), '7i(t))t . ('7i(t) , €n(t)), t

is also measurable on E, Vn, and further, €(.) E e. Thus, the conclusion 2) is proved. Notice that II'7i(t) lit is a measurable function on E, Vi. Therefore,

En = {t E EI'7i(t) 1= 0, i < nj '7j{t) = O,i > n} is a Borel subset of E, Vn.

Q.E.D.

450

Definition 12.1.3. The {f7n(')} is Propostion 12.1.2 is called an orthogonal normalized basis of the measurable field H(·). Moreover, a sequence {~n(')} of measurable fields of vectors is said to be fundamental ,if {~n(t) In} is a total subset of H(t), Vt E E. Proposition 12.1.4. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B). 1) a field E(·) of vectors over E is measurable if and only if (E(t), ~n(t))t is measurable on E, Vn, where {~n} is a fundamental sequence of measurable fields of vectors over E. 2) if E(') is a measurable field of vectors over E, then II E(t) lit is measurable on E. 3) if E('),f7(') are two measurable fields ofveetors over E, then (E(t),77(t))t is measurable on E. 4) let {~m(')} c a, and suppose that for each tEE, there is ~(t) E H(t) such that (~m(t) - ~(t), E)t ---4 0, VE E H(t). Then d,) is also a measurable field of vectors over E.

Proof. E

3) Let {f7n(')} be an orthogonal normalized basis of H(·). If E(·), f7(')

a, then

(E{t),f7{t))t = 2:(e(t),f7n(t))t' (f7n(t),f7(t))t t

is measurable on E. 2) It is obvious from the conclusion 3). 4) Let {En{')} be as in Definition 12.1.1. Then

is measurable on E, Vn. Thus ~(.) E a. 1) The necessity is obvious from the conclusion 3). Now let (e{t), ~n(t))t be measurable on E, \In, and {en(')} and a be as is in Definition 12.1.1. Let

a' = {f7(·)l(f7(t), ~n(t))tis measurable on E, Vn}. Then e(·) E a', and {en(')} c a'. Applying the conclusion 3) to is measurable on E, Vn. Therefore, E(·) E a.

a', (E(t), En(t))t Q.E.D.

Example 1. The constant measurable field of Hilbert spaces. Let (E, B) be a Borel space, n; be a separable Hilbert space, and {En} be a total subset of H a . Define

H(t)

= Ho, En(t) = En' Vt E E,

451

and e = {€(.)[(€(t),€n(t))t = (€(t),€n)O is measurable on E,Vn}. This measurable field of Hilbert spaces is called the constant field corresponding to H«. Clearly, €(.) E e if and only if (€(t),17)o is measurable on E,V'1 E Ho. Consequently, e is independent of the choice of the total subset {€n}.

Example 2. Let A be a separable C·-algebra, and S(A) be its state space. Consider (S(A),o(A·,A)) as a Borel space. For each p E S(A), through the GNS construction we get a Hilbert space Hr If {an} is a countalbe dense subset of A, then {(an)p}n is dense in HpJ and ((an)p, (am)p) = p(a:n,a n) is a continuous function of p on S(A), Vn, m. Let H(p) = Hp, Vp E S(A), and e = {€(·)I(€(p), (am)p)p is measurable on S(A), Vn}. Then we get a measurable field H(·) of Hilbert spaces. Clearly, €(.) E e if and only if (€(p),ap)p is measurable on S(A), Va E A. Consequently, e is independent of the choice of the countable dense subset {an}. Proposition 12.1.5. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B). For n = 00,0,1,"" define En = {t E EldimH(t) = n}. Suppose that H n is a fixed n-dimensional Hilbert space, n = 00,0, 1, .... Then there exists u(.) satisfying: 1) for any tEEn, u(t) is a unitary operator from H(t) onto n.; Vn; 2) €(.) E e if and only if for any nand '1 E n.; (u(t) €(t), '1)n is measurable on En , where (,)n is the inner prodeuct in Hn.

Proof. Let {'1k(')} be an orhogonal normalized basis of the field H(.), n and {'1i ) 11 < k < n} be an orthogonal normalized basis of H n , Vn. Define : U(t)17k(t) = '1 in) ,Vt E En, 1 < k < n, Vn. Then u(.) satisfied the condition 1). Moreover, notice that €(.) E e if and only if (€(t), '1k(t))t is measurable on E, Vk. Thus, u(·) satifies the condition 2) also. Q.E.D. Proposition 12.1.6. Let Ho be a countably infinite dimensional Hilbert space, and H(·) be a measurable field of Hilbert spaces over a Borel space (E, B). Then there exists u(.) such that for each tEE, u(t) is an isometry H(t) into Ho; and t -----+ u(t)H(t) is a Borel map from (E, B) to W(Ho) ( see Proposition 10.1.8 ) . Moreover, €(.) E e if and only if (u(t)€(t), '1)0 is measurable on E, V'1 E H o. Conversely, if H(·) is a field of Hilbert spaces on E, and for each tEE there is an isometry u(t) from H(t) into Ho such that t -----+ u(t)H(t) is a Borel map from (E, B) to W(Ho) J then (H(·), e) is measurable, where e = {€(')I(u(t)€(t), '1)0 is measurable on E, V'1 E Ho}. Let H (.) be a measurable field over (E, B), {17n (.)} be an orthogonal normalized basis of H(.), and {'1n} be an orhtogonal normalized basis of He:

Proof.

452

For any tEE, define

U(t)'7n(t) = '7n if n < dimH(t), u(t)'7n(t) = 0,

if

n > dimH(t).

Then u(t) is an isometry from H(t) into Ho, Vt E E. If Pn is the projection from H o onto ['7.," . ,'7nl , then for any '7 E H o we have

11'7 + u(t)H(t) 110 = 11 (1 - Pn)'7llo,

Vt E Em

where En = {t E EldimH(t) = n}. Thus, 11'7 + u(t)H(t)llo is measurable on E. By Proposition 11.1.8, t ~ u(t)H(t) is a Borel map from E to W(Ho) . Moreover, from

(u(t)€(t), '7)0 = L(€(t),'7n(t))t' (u(t)'7n(t),'7)o n

and

_{O,

(€(t),'7n(t))t -

(u(t)€(t),'7n)O'

n > dimH(t), if n < dimH(t) ,

we can see that €(.) E e if and only if (u(t) €(t), '7)0 is measurable on E for any '7 of H o. Conversely, let u(t) be an isometry from H(t) into Ho, Vt E E, such that t ----i' u(t)H(t) is Borel from E to W(Ho). Denote the projection from Ho onto u(t)H(t) by p(t), Vt E E. Then for each € E Ho, the function II€ + u(t)H(t) 110 = 11(1 - p(t))€llo is measurable on E. Further, (p(t)€,'7)o is measurable on E, V€, '7 E H o. · Suppose that {en} is a countable dense subset of Ho. Let €n(t) = u(t)*p(t)€n,Vt E E and n. Since {€n(t)ln} is dense in H(t), Vt E E, and (€n(t), €m(t))t = (p(t)€n, €m)O is measurable on E, Vn, m, we can get a measurable field H(.) over E with {€n(')}' Notice that (€(t), €n(t))t = (u(t)€(t), €n)O for any field €(.) of vectors over E. Now by the density of {en} in Ho, we can see that e = {€(·)I(u(t)€(t),'7}o is measurable on E, V'7 E H o}. Q.E.D.

Definition 12.1.7. Let (E, B) be a Borel space, v be a measure on B, and H (.) be a measurable field of Hilbert space over E. Let

H = =

Proposition 12.1.8. follows:

L(1) H(t)dv(t) {€(.) Eel JEll €(t) II; dll(t) < oo} Define an inner product in H

(€('), '7{.)} =

L

(€(t), '7(t))t dll(t).

=

J: H(t)dll(t) as

453

Then H is a Hilbert space. Moreover, if En(') -----? E(') in H, then there is a subsequence {nk} such that II Eni; (t ) - E(t)llt ~ 0, a.e.z-. Proof. Let {En(')} be a Cauchy sequence of H. Pick a subsequence {Eni;(')} such that IIEnk+l(') - Eni;(')ll < 00. Put

L k

N

II Eni;+l (t ) -

aN(t) = L

Enk(t)llt,

N -- 1 " 2 ... .

k=l

Clearly, aN(') is a non-negative measurable function on E, and N

(/ aN(t)2dv(t))I/2 < L IJEni;+l(') - Eni;(·)II,VN. k=l

Thus, a(t)

= LIIEni;+l(t) - Eni;(t)l!t k

FEB with v(F) =

°such that a(t) <

E L

2(E , B,v ). Consequently, there is

00, Vt

(j F. Let

E(t) = { En,(t) + ~(EnHl(t) - En.(tl), if t

if t E F.

0,

Then, Eni; (t)

-----?

rt F,

E(t), a.e.z-, and E(') E e. Moreover, since

(f II E( t) lI: dv (t ))1/2

< II Enl(.)II + L II Enk+l (.) - enl;(')II < 00, k~l

it follows that e(·) E H. Noticing that

IJEni;(') - E(·)II <

L HEn;+l(') -

Enj(·)11

-----?

0,

j~k

we have II En(') - e(·) II

-----?

°in H.

Q.E.D.

Proposition 12.1.9. Let (E, B),v,H{·) and H be as above, and {71n(')} be an orthogonal normalized basis of H (.). Then : 1) E(') E H if and only if

L n

JEr I(E(t), 71n(t))tI

2dv(t) <

00;

2) for and E('}, 71{') E H, we have

(E('),71(')) = L n

r (E(t), 71n(t))t . (71n(t), 71(t))tdv(t);

JE

454

I: en('), where en(t) =

3) for any e(·) E H, we have e(·) =

(e(t), 1]n(t))t1]n{t) ,

n

Vt E E, and en(') E H, Vn; 4) if X is a total subset of L 2(E, B, v) , then {(f1]n)(·)lf E X, n} is also a total subset of H. Proof.

Q.E.D.

All conclusions are obvious.

Let H(·)' K(·) be two measurable fields of Hilbert spaces over a Borel space (E, B). Then there is a unique manner such that (H®K)(·) becomes a measurable field of Hilbert spaces over E, and (e®1])(') E 8((H ® K)(·)) , Ve(·) E 8(H(·)) and 1](') E 8(K(.)), where (H ® K)(t) = H (t) ® K (t), ® 1])(t) = t) e 1] (t), Vt E E.

Proposition 12.1.10.

(e

e(

Proof. Let {en(')}, {1]m(')} be fundamental sequences of measurable fields of vectors of the fields H(·), K(·) respectively . Taking {en ® 1]m(') In, m} as fundamental fields of vectors, (H®K){') becomes a measurable field of Hilbert spaces over E, and e ® 1](') E 8((H ® K)(·)), Ve(·) E 8(H(·)) , and 1](') E 8(K(·)). Conversely, if (H ® K)(·) is a measurable field such that (e ® 1])(') is measurable, Ve(·) E 8(H(·)) and 1](') E 8(K(·)), then {(en ® 1]m)(') In, m} is a funcdamental measurable field of vectors. Therefore, the manner is unique.

Q.E.D. Proposition 12.1.11. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B) , H o be a separable Hilbert space, and v be a measure on

L(f) H(t)dv(t) e n; ---+ L(f) (H(t) ® e(t) e 1], Vt E E, e(·) E L(f) H(t)dv(t),

B. Then there is a unique isomorphism <1>

:

Ho)dv(t) such that (<1>(e(·) ® 1]))(t) = and 1] E Ho, where H(·) ® Ho is the tensor product of H{·) and the constant field corresponding to H o. Proof. <1>

is

Naturally define the map

fE(f) (H(t)

<1>.

If suffices to show that the image of

® Ho)dv(t). Let {en(')} be an orthogonal normalized basis of

the field H ('), and {1]m} be an orthogonal normalized basis of u; Then {en(') ® 1]mln,m} is an orhtogonal normalized basis of the field H(.) ® Ho. By Proposition 12.1.9,

will be a total subset of J:(H(t) ® Ho)dv(t). Therefore, the image of <1> is J:(H(t) ® Ho}dv(t). Q.E.D.

455

L2(E,B,v) e Ho = L(j) Hodv(t) = L 2(E,B,v,Ho). In fact, B
get

t

. the conclusion,

JE

References. [28], [36], [119].

12.2. Measurable fields of operators Definition 12.2.1. Let (E,B) be a Borel space, and H(·),K(.) be two measurable fields of Hilbert spaces over E. A field a(.) of operators from H(·) to K(·) is said to be measurable, if for each tEE, a(t) is a bounded linear operator form H(t) to K(t) , and for each E(') E 8(H(·)) we have

(aE){') = a(')E(') E 8(K(·)). Proposition 12.2.2. A field a(.) of operators from H(·) to K(·) is measurable if and only if the function

(a(t) En(t), 7Jm(t))t is measurable on E, Vn, m, where {En(')}, {7Jm(')} are fundamental sequences of measurable fields of vectors of the fields H(·), K(.) respectively. Moreover, in this case , t - + II a (t) II t is a measurable function on E.

Proof.

The necessity is obvious. Conversely, let (a(t) En(t), 7Jm(t))t is measurable on E, Vn, m. Then, a·(·)7Jm(·) E 8(H(.)), \/m, and for each E(') E 8(H(·))

(a(t)E(t),7Jm(t)) = (E(t),a(tr7Jm(t))t is measurable on E, \/m. Thus a(')7J(') E 8(K(·)), and a(.) is measurable over

E. Moreover, if {En(.)}, {7Jm (.)} are the orthogonal normalized bases of H (.), K(·) respectively, then we have

l(a(t ) L anEn(t), L ,Bm7Jm(t))tl

II a(t) II. = sup

l

(~ '::'nI2) 172 • (~ IPm 12) 172

an, Pm are complex rational numbers, \/n, m} .

Therefore, t

-----t

lIa(t) lit is measurable on E.

Q.E.D.

Proposition 12.2.3. Let a(·) be a measurable field of operators from H(.) to K(·), and t -----t Ila(t) lit be eseentially bounded with respect to v, where v

456

is a measure on (E, B). Then (ae)(·) from H =

L H(t)dv(t) ffi

operator a by

to K =

= a(·)e(·) is a bounded linear operator

L K(t)dv(t) ( ffi

and we shall denote this

ffi

/E

a(t)dv(t)). Moreover, if v is semi-finite ( i.e., for any FEB with v(F) > 0, there is G E B with G c F and < v( G) < 00), then

°

Iiall = ess sup Ila(t) lit. Let A = esssuplla(t)llt. Since Ila(t)e(t)llt < Alle(t)lIh a.e.v,\le{·) E H(·), it follows that [e] < A. Now suppose that v is semi-finite. For any c > O,F = {t E Ellla(t)llt > A-c}(E B) is not v-zero. Then there is G E B with G c F and 0< v(G) < 00. IT {en (.)} is an orthogonal normalized basis of the field H (.), { an} is a sequence of complex rational numbers with lanl2 < 00, and f E LOO(E, B, v), then

Proof.

L n

L(anfxGen)(') E H, and n

2

2

Iia L(onfxGen)(') 11 < IIal1 ·11 I)anfxGen){') 11 n

Since

2

•

n

f is arbitrary, it follows that Ila(t) L anXG(t) en(t) lit < Iiall . II L anXG(t) en(t) IIh a.e.z-. n

n

Thus, Ila(t) lIt < lI all, \It E G, a.e.v, and Iiall > A-c. Since e is arbitrary, Iiall > A and Iiall = A = ess sup lIa(t) lIt. Q.E.D.

Example. Let H{·) be a measurable field of Hilbert spaces over (E, B) , and H o be a countably infinite dimensional Hilbert space. By Proposition 12.1.6, we have a measurable field u(·) of isometries from H(.) to the constant field

corresporcding to

tt; ITv is a semi-finite measure on B, then u ffi

is an isometry from H =

/E

L ffi

H(t)dv(t) to

Hodv(t) =

L ffi

u(t)dv(t)

u, ® L 2(E , B, v).

Definition 12.2.4. A bounded linear operator a from H =

K =

L ffi

=

/E

ffi

H(t)dv(t) to

K(t)dv(t) is said to be decomposable, if there is a measurable field

a(·) of operators from H(.) to K(.) such that a = Proposition 12.2.5.

L ffi

a(t)dv(t).

Let v be semi-finite, and an

=

L an(t)dv(t) , ffi

n

=

0,1,2,···. 1) Suppose that an ---+ ao (strongly). Then for any FEB with v(F) < 00, there is a subsequence {an,,} such that an,,(t) ---+ ao(t) (strongly), \It E F, a.e.v.

457

2) If an(t) ---+ an(t) ( strongly) , Vt, a.e.z/ , and sup Ilanl! < 00, then n an ~ ao ( strongly ). 1) Let {em{')} be an orthogonal normalized basis of H{·). Then (XFem)(') E H and Proof.

Ilan{XFem){') - an(XFem)(·)1I

----T

Vm.

0,

By Proposition 12.1.8, there is a subsequence {anA;} such that II anA; (t) em(t) an(t)em(t) lit ~ 0, Vt E F, a.e.z/ , and m, We may assume lIa n(t)l!t < sup Ilamll, Vn, tEE. Thus, anA;(t) ~ ao(t) ( strongly) , Vt E F, a.e.z-. m

2) Let K = sup{llan(t)llh Ilanlllt E E,n}. For any e(·) E H, since In(t) = lIan(t)e(t) - an(t)e(t)ll~ ~ 0, a.e.z- and I/n(t) I < 4K21Ie(t)lI: E LI(E, B,v), it follows from the bounded convergence theorem that 2

Ilane(·) - aoe(·) 11 =

f I/n(t) Idv(t)

----T

0,

i.e. an ~ ao ( strongly ) .

Proposotion 12.2.6.

L an{t)dv{t)}

Q.E.D. Let v be semi-finite. Then there is a sequence

ffi

{an

=

L H(t)dv{t) ffi

of decomposable operators on H =

such

that B(H{t)) is generated by {an(t) In}, Vt E E. Proof. Let E Jc = {t E EldimH(t) = k}, k = 00,0,1,···. By Proposition 12.1.5, there is u(·) such that for each t E E Jc, u(t) is a unitary operator from H(t) onto HJc , where HJc is a k-dimensional Hilbert space, Vkj and e(·) E e if and only if (u(t) e(t), TJh is measurable on E Jc, VTJ E H Jc and k, Pick {b~Jc)ln} C B(HIe ) with Ilb~Jc)11 < I,Vn, such that B(HIe ) is generated by {b~Jc) In}, Vk. Define

an(t) = u(t)*b~le)u(t), Vt E EJc,k, n. Clearly, Ilan(t)11 < 1, Vn, and B(H(t)) is generated by {an(t)ln}, Vt E E. Now it suffices to show that an (.) is measurable for each n. Let {en(·)} be an orthogonal normalized basis of H(.), and {u(t)en{t) = 11 < n < k} be an orthogonal normalized basis of H Jc, Vt E E Jc, k (see Proposition 12.1.5) . Thus, (an(t) ei (t) , e;{t))t is a constant on each E Jc (notice that u(t)en{t) = O,Vn > k,t E EJc),Vi,j, and the field an(·) is measurable, Vn.

ei")

Q.E.D. Definition 12.2.7. Let H(·) be a measurable field of Hilbert spaces on a Borel space (E, B) and v be a measure on B. For any I E LOO (E, B, v) , define

458

a decomposable operator

mf on H =

LG) H(t)d1l(t),

=

lEG) l(t)d1l(t)

i.e., mfE(') = 1(,) E('), VE(') E H. The operator mf

is said to be diagonal, and the collection Z

= {mf II

diagonal operators is called the diagonal algebra on H

E

L 00 (E, B, 11)} of all

= lEG) H(t)d1l(t).

Proposition 12.2.8. Let 11 be semi-finite. Then f ~ mf is a bijection from LOO(E, B, 11) onto Z = {mfll E LOO(E, B, 11)} if and only if 1I(Eo) = 0 , where Eo = {t E EldimH(t) = O}, i.e., H(t) =1= {O}, a.e.v. In this case, we have also Ilmfll = lillI, VI E LOO(E, B,1I).

Proof.

It is obvious from Proposition 12.2.3.

Q.E.D.

Proposition 12.2.9. Let (E, B) be a Borel space, 11 be a a-finite measure on B, H(·) be a measurable field of Hilbert spaces over E, and H(t) =1= {O}, a.e.v . Then the diagonal algebra Z is a commutative VN algebra on H =

LG) H(t)d1l(t) , Z'

I

is a-finite, and

---+

mf is a faithful W*-representation of

LOO(E, B, 11) on H. Let {II} be a net of LOO(E, B,1I), and It ~ 0 with respect to the oo 2 w* -top. in L (E, B, 11). If en(')' Tln(') E H with En (1\ en(') 11 + 11Tln(') 11 2 ) < 00, then we have

Proof.

IL

(En(t), Tln(t))tl < L

n

and

n

I en( t) lit' Il TIn (t) lit E L 1(E, B, 11)

IL(mfIEn(')' Tln('))l n

IJEr Il(t). L(en(t), Tln{t))t d1l (t )I ~ 0 n i.e., mil --+ 0 with respect to u(B(H),T(H)). Thus, from Proposition 12.2.8 f ~ mf is a faithful W*-representation of LOO(E, B, 11). Consequently, Z is a commutative VN algebra on H. Now let E = UnEn, where En E Band 1I(En} < 00, Vn, and {en{')} be an orhtogonal normalized basis of H(·). From Proposition 12.1.9,

will be a total subset of H. Thus Z admits a cyclic sequence of vectors, i.e., Z' is a-finite. Q.E.D.

459

Theorem 12.2.10. Let (E, B) be a Borel space, v be a a-finite measure on B , and H c(·) be a measurable field of Hilbert spaces over E, i = 1,2. Then a

L ffi

bounded linear operator a from HI =

L ffi

Hdt)dv(t) to H2 =

H 2(t)dv(t)

is decomposable if and only if am}l) = m}2)a, TIl E LOO(E, B, v), where m~) is the diagonal operator on Hi corresponding to I, i = 1,2.

Proof.

The necessity is obvious. Now let am}l) = m}2)a, TIl E LOO(E, B, v). From the proof of Proposition 12.2.9, there is a fundamental sequence {En(')} of measurable fields of vectors of H 1 ( · ) such that En(') E HI, \In. Define fJn(·) = aEn (.),\In. Then for any complex rational numbers 0:1, 0:2, ••• with '* {n JO:n =1= oj < 00 and f E LOO(E, B, v), from am}1) = m}2)a we can see that

Since

f is arbitrary, it follows that

II I: O:nfJn(t) lit < lIall ." I: O:nEn(t) lit n

n

TI{ O:n} as above and a.e.v . Thus, there is FEB with v(F) = 0 such that for each t fI. F , we can define a bounded linear operator a(t) from H 1 (t ) to H 2 (t) satisfying a(t)En(t) = fJn(t) ,TIn. Further, let a(t) = 0, TIt E F. Then a(·) is a measurable field of operators from Hd·) to H 2 ( · ) , and Ila(t)llt < lIall, TIt E E. ffi

Put b =

JE

a(t)dv(t). Then bm}l) En(')

\In and I

E

=

m}2)bEn(·)

= m}2)aEn(-) = am}1) En(·),

LOO(E, B,v). But {m}l)En(·)jn, f E LOO(E, B, v)} is a total subset

of H}, so a = b =

ffi

JE

a(t)dv(t).

Q.E.D.

References. [28]' [119], [158].

12.3. Measurable fields of Von Neumann algebras Definition 12.3.1. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B). A field M(.) of VN algebras on H(·) ( i.e., for each tEE, M(t) is a VN algebra on H(t)) is said to be measurable, if there is a

460

sequence {an(')ln} of measurable fields of operators on H(.) such that M(t) is generatecd by {an(t)ln}, Vt E E.

Proposition 12.3.2. Let u; be the constant field over (E, B). A field M(·) of VN algebras on H o is measurable if and only if t ~ M(t) is a Borel map from E to A, where .It is the collection of all VN algebras on H o. It is obvious from Definition 12.3.1 and Proposition 11.3.4.

Proof.

Q.E.D.

Proposition 12.3.3. Let M( '), N(·) be two measurable fields of VN algebras on H(·) . Then M(·)',M(·) n N(·) and (M(·) U N(.))" are also measurable. Proof. Let E" = {t E EldimH(t) = k}, k = 00,0,1,···. Then H(·) can be regarded as a constant field over each E". Further, by Proposotion 12.3.2, Q.E.D. 11.3.1 and 11.3.5, we can get the conclusions.

Proposition 12.3.4.

Let H(.) be a measurable field of Hilbert spaces over

a Borel space (E,B),v be a u-finitie measure on B,H = the diagonal algebra on H, {an =

L ffi

H(t)dv(t),Z be

L an{t)dv(t) In} a L a(t)dv(t) ffi

be a sequence of decom-

posable operators on H, M be the VN algebra generated by Z and {an}, and ffi

a

B(H). Then a E M if and only if = is decomposable, and a(t) E M(t), a.e.v, where M(t) is the VN algebra on H(t) generated by {an(t) In},Vt E E. E

Let a =

Proof.

L a(t)dv(t), a(t) ffi

E

M(t), a.e.z-, and a' EM'. Since M' c Z'

L a'(t)dv(t) ffi

,it follows from Theorem 12.2.10 that a' =

is decomposable.

Noticing that a' commutes with an and a:, Vn, we have a'(t) E M(t)', a.e.v . Thus a'(t)a(t) = a(t)a'(t), a.e.u, Further, a'a = aa', Va' EM', and a E M. Conversely, let a E M. Since M c Z', it follows from Theorem 12.2.10 that ffi

IE

a(t)dv(t) is decomposable. By Proposition 12.2.9 and M c Z', M is a-finite. Let M o be the * algebra generated by Z and {an}. Then M o a =

is strongly dense in M. From Proposition 1.14.4, there is a sequence {bn =

L ffi

bn(t)dv(t)}

C

u; such that bn ~ a (strongly).

Further, by Proposition

12.2.5 and the a-finiteness of v , we have a subsequence {bn ,, } such that

bn "

(t)

~

Therefore, a(t) E M(t), a.e.z/ .

a(t)

(strongly), a.e.u. Q.E.D.

461

Proposition 12.3.5. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B), 11 be a a-finite measure on B, and M(·) be a measurable field of VN algebras on H(·). Then

M = {a

=

lEG) a(t)dll(t)

is a a-finite VN algebra on H =

B(H) la(t)

E

E

M(t), a.e.lI}

LG) H(t)dll(t).

Proof. Let {a n ( · ) } be a sequence of measurabel fields of operators such that M(t) is generated by {an(t) In}, \It E E. We may assume that lI an(t ) lit < 1, 'It E E and n. From Proposition 12.3.4, M is the VN algebra generated by

Z and {an} exactly, where Z is the diagonal algebra on H = and an

= lEG) an(t)dll(t), \In.

LG) H(t)dll(t),

Moreover, from M c Z' and Proposition 12.2.9,

Q.E.D.

M is also a-finite.

Definition 12.3.6. Let H (.) be a measurable field of Hilbert spaces over a Borel space (E, B) , and 11 be a a-finite measure on B. A VN algebra M on

H =

lEG) H(t)dll(t)

is said to be decomposable, if there is a measurable field

M(.) of VN algebras on H(·) such that

M = {a =

kG) a(t)dll(t)

In this case, we shall denote M by

E

B(H) la(t) E M(t), a.e.lI}.

lEG) M(t)dll(t).

Proposition 12.3.7. A VN algebra M on H = LG) H(t)dll(t) is decomposable if and only if M is generated by the diagonal algebra Z and a sequence

LG) an(t)dll(t)} of decomposable operators.

{an =

In this case, M(t) is gener-

ated by {a n(t)!n},a,e.lI, M(·) is unique (a.e,lI), and Z Proof.

Q.E.D.

It is immediate from Proposition 12.3.4 and 12.3,5.

Remark. (H(t))dll(t)

LG) a;'1 tdll(t) and Z' are decomposable VN algebras on H = kG) H(t)dll(t). From Proposition 12.2.6, Z =

=

LG) B

LG) M(t)dll(t),Mn = kG) Mn(t) dll(t), (n = ..) be decomposable VN algebras. Then we have M' = LG) M(t)'dll(t),

Proposition 12.3.8. 1,2"

c Me Z'.

Let M

=

462

(UnMn)" = fE(JJ ( UnMn(t))"dv(t) and nnMn

= L(JJ (nnMn(t))dv(t).

Proof. From Proposition 12.3.3, M(·)' is also measurable, Let N = L(JJ M(t)' dv(t). Clearly, N eM'. Now let a' E M'. Since Z c M c Z',a' = L(JJ a'(t)dv(t) is decomposable. Suppose that M is generated by Z and {an =

L an(t)dv(t)}. m

Then a'(t) commutes with {an(t), an(t)·}n, a.e.u, Thus a'(t) E

M(t)', a.e.z/, a' E N, and M' = N = fE(JJ M(t)'dv(t). For each n, suppose that

u;

is generated by Z and

{a~n)

= fE(JJ

a~n} (t) d

v(t)1 k}, and Mn(t) is generated by {a~n)(t)lk}, Vt E E. Then (UnMn)" is generated by Z and {ain)ln,k}, and (UnMn(t))" is generated by {ain)(t)ln,k}. Thus (UnMn)"

= L(JJ (UnMn(t))"dv(t). Further, nnMn

=

(UnM~)' = fE(JJ (nnMn

(t))dv(t).

Q.E.D.

Proposition 12.3.9.

Let M

= fE(JJ M(t)dv(t).

Then M

n M' =

Z if and

only if M(t) is factorial, a.e.z/.

Proof.

m

It is immediate from M n M' = fE (M(t) n M(t)'}dv(t) and Z =

L(JJ QJ1 t dv (t ).

Q.E.D.

= L(JJ H(t)dv(t} is separable, then a VN algebra M on H is decomposable if and only if Z c M c Z', where Z is the diagonal

Proposition 12.3.10. If H algebra on H.

The necessity is obvious. Now let Z C M c Z'. By Theorem 12.2.10, every operator of M is decomposable. Moreover, since H is separable, M is countably generated. Thus, M is generated by Z and a sequence of decomposable operators, and M is decomposable. Q.E.D.

Proof.

Remark. If (E, B) is a standard Borel space, and v is a u-finite measure on B, then L 2 (E, B, v) is separable. Now by Proposition 12.1.9, H =

L(JJ H(t)dv(t)

is also separable.

Definition 12.3.11. Let H(·), K(·) be two measurable fields of Hilbert spaces over a Borel space (E, B), and M(·),N(·) be measurable fields of VN algebras on H(·), K(.) respectively. For each tEE, let ()(t) be a * homo-

463

morphism from M (t) to N (t). The field () (.) of * homomorphisms is said to be measurable, if for any measurable field a(·) of operators on H(·) with a(t) E M(t),Vt E E,()(·)(a(·)) is a measurable field of operators on K(·). In this case, if v is a a-finite measure on B, then we can define a * homo-

CB

morphism () i.e., ()(a)

=

= fE ()(t)dv(t)

L CB

from M

=

L M(t)dv(t) CB CB

to N

CB

= fE N(t)dv(t),

()(t)(a(t))dv(t) , where a = fE a(t)dv(t) E M.

Proposition 12.3.12. Keep all notations in Definition 12.3.11. 1) If ()(t) is normal, Vt E E , then () is normal. 2) H ()(t) is a * isomorphism from M(t) onto N(t), Vt E E, then () is also a * isomorphism from M onto N.

Proof. 1) From Proposition 1.12.1, it suffices to show that () is completely additive. By Proposition 12.3.5, M is a-finite. Thus, we need to prove that n

where {Pn} is an orthogonal sequence of projections of M, and P = LPn' Let n

P=

L p(t)dv(t) , CB

Pn

=

L Pn(t)dv(t), CB

where p(t),Pn(t) E M(t), Vt E E and n. From PnPm = 0, Vn -:j:. m, we may assume that Pn(t)Pm(t) = 0, Vt E E and n t- rn, Since v is a-finite, and n

{LPi(t)}n is increasing (Vt E E), it follows from Proposition 12.2.5 that i=l

LPn(t) = p(t), a.e. v . Now by the normality of ()(t)(Vt E E) and Proposition n

12.2.5, we get

2) Suppose that M is generated by the diagonal algebra ZH ( on H) and a sequence {an =

L an(t)dv(t)}n CB

of decomposable operators (on H) ,and M(t)

is generated by {an(t)}n(Vt E E). Since ()(ZH) = ZK, where ZK is the diagonal

CB

= fE ()(t)(an(t))dv(t)}n. But N(t) is generated by {() (t)(a n(t)) }n,Vt, thus we have ()(M) = N. More-

algebra on K,()(M) is generated by ZK and {()(a n) over, () is injective obviously.

Q.E.D.

464

References. [28], [35], [36], [119], [121], [158].

12.4. Decomposition of a Hilbert space into a direct integral From Proposition 5.3.14, if Z is an abelian VN algebra on a Hilbert space H and Z admits a cyclic vector, then there is a regular Borel measure v on the spectral space a compact Hausdorff space ) of Z and a unitary operator 2(0, u from H onto L v) such that

°(

C(O) = LOO(O, v),

suppz/ = 0,

VI

.

..-

umJu = mJ,

LOO(O,v), where I ---t mJ is the Gelfand transformation from C(O) = LOO(O,v) onto Z, and TnJ is the multiplicative operator on L 2(0,v) correE

L

e CX'dv(t) , and sponding to I. In version of Hilbert integral, L 2(0, v) = e {mJII E L OO (O,lI)} is the diagonal algebra on CX'dll(t). Thus Z is unitarily e equivalent to the diagonal algebra on CX'dv(t).

L

L

The above case is not very interested (since Z' = Z by Proposition 5.3.15 ) . Now let Z be an abelian VN algebra on a Hilbert space Hand Z admits a cyclic sequence {en} of vectors ( i.e. Z' is u-finite). Put

= 6,

''11

= Z'1b

HI

PI: H

---t

HI,

n-I

n« = en -

L

plcen,

H n = Z'1n,

Pn: H

---t

H n,

Ie:::: I ••

til

....

Clearly, PiPi = 0, Vi :f:. i,

L

Pi = 1; and Pi E Z', Vi.

i

We may assume that

II enll < 1, Vn.

Then

II'1nll < 1, Vn.

Let

'10 =

L 2-~'1n' n

Then '10 is a cyclic vector for Z'. Let 0 be the spectral space of Z, and V« be a regular Borel measure on such that (mJ'1n, '1n) = ~ I(t)dvn(t), VI E C(O),

°

n = 0,1,' . " where I ---t mJ is the Gelfemd transformation from C(O) onto Z. Define 1,1=1,10 , clearly, v = 2- n v n • Then for each n = 1,2"", there is

L

n~1

h n E LI(O, v) with h n > 0 such that V n = h n . u.

465

For each tEO, we construct a Hilbert space H(t) and a sequence {~n(t)ln = 1,2,"'} of vectors of H(t) such that ~n(t) = if hn(t) = 0, and {~n(t)ln with hn(t) > oj is an orthogonal normalized basis of H(t). Since (~n(t), ~m(t))t = OnmXnpphn (t) is measurable on 0, Vn, m, H(·) becomes a measurable field of Hilbert spaces over 0 with a fundamental sequence {!:n(')ln = 1,2,"'} of vector fields. We say that H(t) =1= {O}, a.e.z-. In fact, if there is a Borel subset E of 0 with lI(E) > such that H(t) = {O}, Vt E E, then hn{tl..= 0, Vn > 1 and tEE. Futher, lIn(E) = (h n· lI)(E) = 0, Vn > 1 and lI(E) = 2- nlln{E ) = 0,

°

°

L

n~l

a contradiction. Thus, H(t) =1= {O}, a.e.z/, __ (CfJ __ Let H. = 1 H(t)dll(t), and define u : H ----+ H as follows 0

wnm«

=

1(·)hn(·)1/2~n(·),

Clearly, u is an isometry. If ~(.) E

> 1 and I

Vn

E C(O).

fi satisfies

(/(·)h n(·)1/2!:n(·), d,)) = 0,

> 1 and I E C(O),

Vn

then for each n > 1, we have hn(t)l/2(~n(t),~(t))t = 0, a.e.v , By the definition of {!:n(')} , we get ~(t) = 0, a.e.z- . Thus, u is unitary. Moreover, it is easy to see that umfu· = mb where mf is the diagonal operator on H corresponding to I, VI E C(O). By the proof of Theorem 5.3.1, we have also supp II = 0 and C(O) = LOO(O, z-}. Therefore, we obtain the following.

--

Theorem 12.4.1. Let Z be an abelian VN algebra on a Hilbert space H, Z' be a-finite, and 0 be the spectral space of Z. Then there is a regular Borel measure II on 0 with supp II = 0, a measurable field H(·) of Hilbert spaces .-. (CfJ over 0, and a unitary operator u from H onto H = 10 H(t)dll(t) , such that

H(t)

=1=

{O},a.e.lIj

umqu"

C(O) = LOO(O,lI);

= mf, VI

E

LCO{n, lI),

--

where f ----+ mf is the Gelfand trandformation from C(O) onto Z, mf is the .....
Theorem 12.4.2. Let H be a separable Hilbert space, M, Z be VN algebras on H, and Z C M n M'. Then there is a finite Borel measure on JR, a measurable field H(·) of Hilbert spaces over JR , a measurable field M(·) of .-. (CfJ VN algebra on H(·) .and a unitary operator u from H onto H = 11l H(t)dll(t), such that

«z« = Z, «u« =

I:

M(t)dll(t) ,

466

where Z is the diagonal algebra on is factorial, a.e.z/ .

H.

Moreover, if Z = M

n M'

, then M(t)

Proof. From Theorem 5.3.7, Z can be generated by a self-adjoint operator a. Let A be the C*-algbera generated by {1, a} . Then A is weakly dense in Z, and the spectral space of A is u(a). By the proof of Theorem 12.4.1, _ (e we have also v, H(.) and u : H ----+ H = 1 H(t)dv(t), such that umfu* =

mf, Vf

E C(O), here 0

=

u(a), f

0

----+

mf, is the

* isomorphism from

C(O)

onto A. From Lemma 5.4.4 and the proof of Theorem 12.4.1, we can see that v

1

>-

Ve, Ve E H, where ve is defined by (mfe, e) = f(t)dve(t), Vf E C(O). Since supp v = 0, C(O) can be embedded into LOO(O, v). Also by v >- ve, Ve E H, f ~ mf(: C(O) ~ A) is u(LOO, Ll)-weakly continuous. Thus, the map f ~ mf can be extended to a * isomorphism from LOO(O, v) onto Z. Further, we have uZu* = Z. c Z'. Now since _ From Z C M n M',Z c M c Z', we have Z C «u« (e H = uH is separable, by Proposition 12.3.10 uMu* =

1

M(t)dv(t) is decom-

0

posable. Moreover, u, H(·) and M(·) can be trivially extended from 0 = u(a) onto JR. That comes to the conclusion. Finally, if Z = MnM', then Z = (uMu*)n(uMu*)'. By Proposition 12.3.9, M(t) is factorial, a.e.z/ . Q.E.D.

Theorem 12.4.3. Suppose that 0 is a locally compact Hausdorff space, and o is a countable union of its compact subsets. Let Vb 1.12 be two regular Borel measures on 0, and HI (.) , H 2 (.) be two non-zero measurable fields of Hilbert spaces over 0 . If there is a unitary operator u from HI H 2 = foe H2(t)dv2(t) such that um}l)u*

= foe

Hdt)dvl(t) onto

= m}2), Vf E cgo(n), where m}) is the

diagonal operator on Hi corresponding to f, i = 1,2, then VI 1.12 , and there exists a measurable field v(·) of operators from H 1 ( · ) to H 2(·) such that v(t) is "'-J

unitary from H 1(t) onto H 2(t) , a.e.zq , and u

= wv,

where v

= foe

v(t)dvdt)

, and w is the canonical isomorphism from foe H 2(t )dvd t ) onto H 2 , i.e., if 1.12 = p . VI , then

Proof. Let K be a compact subset with vl(K) = O. Since H 2 ( · ) is non-zero, we can pick a measurable field '1(.) of vectors of H 2 ( · ) with 11'1(t)llt = 1, Vt E O.

467

Suppose that U is an open neighborhood of K and the closure U of U is compact. Then (XUTI)(') E H 2 • Put e{·) = u*{XufJ('))(E HI)' For any e > 0, since vI{K) = 0, we can pick an open subset V such that

K eVe U, Now let I E cgo(n) with we have

and

[lIe{t)lI~dvI(t) < e.

°< I < 1; I{t) = 1, 'It E K; I{t) = 0, 'It fI. V. Then

f II (/fJ)(t) II;

v2(K) < / 1 2(t) dv2{t) =

dv2(t )

= Ilu*m}2}(xufJ)(')lliII = II m}l)e (·) II iII < e. it follows that 1.12 (K) = 0. Thus, 1.12 -< VI.

Since e is arbitrary, VI -< 1.12. Hence VI ,-y 1.12.

Similarly,

H2 = ~(IJ H 2{t )dv l {t). Then Tnf = vm}l)v*, where v = w*u, and fflf is the diagonal operator on H2 corresponding to I,VI E C~(O). Since Cgo{O) is w*-dense in L oo (0, v.) , by Theorem 12.2.10 v = L(IJ V(t)dVI(t) is decomposable. Moreover, v is unitary from HI onto H2 . Thus, v{t) is unitary Now let

from H.(t) onto H 2(t), a.e.z-r.

Q.E.D.

Lemma 12.4.4. Let (Ei , Bi ) be a standard Borel space, and Vi be a a-finite measure on Bi , i = 1,2. If there exists a * isomorphism 1r from L OO (E., B., v.) onto L OO(E2, B2, 1.12), then there is a Borel subset F; of Ei , i = 1,2, and a Borel isomorphism ~ from (E 2\F2) onto (EI\FI) such that

v.(FI) = v2(F2) = 0, and for any 9 E L OO(Et,B bJLI),1r{g)(t) =

Vl,-y 1.12

0

~-1,

g(~(t)),Vt E

(E 2 \ F 2), a.e,v2'

Proof. From Theorem 10.3.16, we may assume that E 1 = E 2 = [0,1], B1 and B2 are the collectionof all Borel subsets of [0, 1] .and 1.11 and 1.12 are two probability measures on [0,1]. Let 11(t) = t(E LOO(E.,v.)) and 12 = 1r(ft)(E LOO(E2 , 1.12))' Clearly, we may assume that 0 < 12(t) < 1, 'It E E 2 • Then ~(t) = f2(t) is a Borel map from E 2 to E 1 • The function 1 ( E L I (E 2, 1.12)) determines a faithful normal state W2 on LOO(E2,V2), i.e., 1

w2(h) = 10 h(t)dv2(t),

Vh E

tr (E 2 , 1.12)'

Then WI = W2 01r is also a faithful normal state on LOO(E., v.). Thus there is unique I E LI(Eb VI) such that wI(g) Also we may assume that I(t)

=

10

> 0, 'It EEl.

1

I(t)g(t)dv.(t), Vg E LOO(E., VI)'

468

If p(.) is a polynomial , then we have

1r(p(ft))(t)

=

p(~(t)),

P{f2)(t) =

Vt E E 2.

For any 9 E C[O, 1],pick a sequence {Pn} of polynomials such that Pn (t) -+- g(t), uniformly for t E [O,IJ. Then in L oo(E2, V2), we have 1r(Pn(fd) -+- 1r(g). Thus 1r(g)(t) = g(~(t)), a.e'V2' Further,

10

1

g(t)f(t) dvl(t)

= wdg) = W2(1r(g)) =

i

1

9(~ (t)) dV2 (t),

Vg E C[O, 1]. SO ~(V2) = f . vb where ~(V2) = V2 0 ~-1. For 9 E L oo(E1 , vd, pick gn E C[O, I} such that gn ~ g. Then

i

1

11r(gn)(t) - 1r(g)(t)1 2dv2(t) = W2(1r((g - gn)·(g - gn)))

Wl((gn - gt(gn - g)) =

10

1

f(t)lgn(t) - g(t)1 2dvl(t)

-+-

O.

Since f(t) > 0, Vt E E, we can find a subsequence {gnA;} such that

gnA;(t)

g(t),

-+-

a.e.v},

and

1r(gnA;)(t) -+-1r(g)(t), a.e,v2'

From 1r(gnA;)(t) = gnA:(~(t)),a.e,v2,Vk, it follows that gnA;(~(t)) -+-1r(g)(t)'a.e. V2. On the other hand, gnA: (t) -+- g(t), a.e'Vl, ~(V2) = f . VI and gnA: (~(t)) -+g(~(t)), a.e'V2, so we get

Vg E L oo(E1,VI)' Replacing 1r by 1r~I(: Loo (E 2, V2) -+- L OO(E1 , VI)), there is a Borel map '11 from E 1 to E 2 such that W(vd = VI 0 '11-1 -< V2, and ~-I(h)(t) = h(W(t)),a.e.lIt, Vh E L oo(E2,V2)' 1r(g)(t) =

g(~(t)),a.e,v2'

Thus we have

(g

0

~)(W(t))

= 1r- 1(1r(g))(t) =

(h 0

W)(~(t))

g(t),

a.e,vb Vg E Loo(Et, vd;

1r(1r- I

= (h))(t) = h(t), a.e'V2, Vh E Loo(E2, V2)'

In particular, from t E LOO (E}, vd n Loo(E2, V2) we get ~

0

W(t) = t,

a.e,vl;

'11

0

~(t)

= t,

a.e,v2'

So there is F~ E 8 2 with v2(Fi) = 0 such that'll 0 ~(t) = t, Vt E (E 2\F~). Let Fi = W-l(F~). Then vl(Fi) = since VI 0 q,-l -< V2' It is easy to see that

°

(1)

469

and eI' is injective on (E 2 \F~). Further, pick F;' E B1 with F;' vl(F;') = 0, such that

c (E 1\FD and (2)

where F1

= F; U Fi'. Clearly, vdFd = O. Let

F~'

= eI'-I(F;') n (E2\F~).

Then

v2(F~') = 0 since 1.12 0 eI'-1 -< VI' Further, v2(F2) = 0, where F2 = F~ U F~' = F~ U eI'-l(F{'). Then, eI' maps (E 2\F2) into (E 1 \F1 ) injectively. Now let t E (E 1\F1 ) . By (1) , we have W(t) E (E 2\F~). If W(t) E F~' c

eI'-I(F;'), then eI' 0 w(t) E F;'. But by (2) , eI' 0 w(t) = t t/. FI, a contradiction. Thus W(t) t/. F~', and W(t) E (E 2\F2). Further, by (2) we get eI'(E2\F2) = (E1 \F1 ) . From Theorem 10.3.12 and Proposition 10.3.15, eI' is a Borel isomorphism from (E 2\F2) onto (E 1\Fd, and its inverse is W. Now by 1.12 0 eI'-l -< 1.1.,1.11 0 '11- 1 -< 1.12, we can see that VI 1.12 0 eI'-1 on (E 1 \Ft ) . Q.E.D. #"oJ

V,

Theorem 12.4.5. Let (E" B,) be a standard Borel space, be a a-finite measure on B" Hi ( · ) be a non-zero measurable field of Hilbert spaces over E i , M, (.) be a measurable field of VN algebras on H, (.), and Z, be the diagonal

algebra on Hi =

m

1 Hi(t)dv,(t), i = 1,2. If there is a lEi

unitary operator u from

Hi onto H 2 such that

where M,

m

= , M,(t)dvi(t), i = 1,2, then there is a Borel subset F; of E, with lE. I

v,(Fi ) = 0, i = 1,2, a Borel isomorphism eI' from (E 2 \F2) onto (E 1\Fd, and a measurable field u(·) of unitary operators from H'(·) to H2 (eI'- I(.)) , such that 1) u(t)M1(t)u(t)· = M 2(eI'- 1(t)) , Vt E (E 1\Fdi 2) eI' (1.12) = 1.12 0 eI'-1 1.11; #"oJ

3) u = I

m

lE1\F1

(del' (1.12 )(t))1/2 U(t) dVl(t). dVI

Proof. By Proposition 12.2.8 and UZIU· = Z2, there is a * isomorphism 1r OO(E from L 1 , B.,vI) onto LOO(E2, B2, 1.12). From Lemma 12.4.4, we have Fi E Bi with v,(Fi) = 0, i = 1,2, and a Borel isomorphism eI' from (E 2 \F2 ) onto (E 1 \ F1) , such that eI'(V2) "'" VI , and 1r(g)(t) = g(eI'(t)),Vt E (E 2\F2), a.e. 1.12, and g E L OO(E.,B.,Vl)' Replacing E bE2,VI by (E 1\Ftl,(E2\F2),eI'(V2) respectively, with the Borel isomorphism we can identify (E b Bb vtl with (E2 , B2 ,V2)' Now our case becomes the following. Let (E, B) be a standard Borel space, V be a a-finite measure on B, H, (.) be a measurable field of Hilbert

spaces over E, i = 1,2, and u be a unitary operator from HI

=

L H1(t)dv(t) m

470

onto H 2

= L(1J H 2(t)d1l(t) , such that' uM1u*

where M, to

= M 2, um}l)u* = m}2), VI E

= L(1J M i(t)d1l(t) , m~)

I, i = 1,2. By

L OO (E , B, 1I ),

is the diagonal operator on Hi corresponding

Theorem 12.2.10, u

= L(1J u(t)d1l(t) , a.e.z-. From Proposition

12.3.7, there is a sequence {an = L(1J a n(t )d1l(t )In} of decomposable operators on HI such that M l is generated by Zl and {anln}, and M1(t) is generated by {an(t)ln}, a.e.v. Then M 2 is generated by Z2 and {uanu*ln}, and M 2(t) is generated by {u(t)a n(t)u(t)*ln},a.e.1I. Therefore, M 2(t) = u(t)Ml(t)u(t)*,a.e.1I.

Q.E.D.

References.

[28], [120], [158].

12.5. The relations between a decomposable Von Neumann algebra and its components Let (E, B) be a standard Borel space,

11

be a a-finite measure on B, H(·) be

a measruable field of Hilbert spaces over E, H =

L(1J M(t)d1l(t) be a decomposable VN algebra on

L (1J

H (t) d1l (t), and M =

H. In this section, we shall

discuss the relations between M and M(t)'s. Proposition 12.5.1. Let p = L(1J p(t)d1l(t),p'

=

L p'(t)d1l(t) ffi

be projections

of M,M' respectively. Then ffi

u, = fE and c(p)

M(t)p(t)d1l(t),

u; = JEff) M(t)pl(t)d1l(t),

= fE(1J c(p(t))d1l(t).

Proof.

From Proposition 12.3.7, we can get the expressions of M p and Mpl. Now suppose that M is generated by the diagonal algebra Z and a se-

L an {t)d1l (t) In} ffi

quence {an =

of decomposable operators, and M(t) is gen-

erated by {an(t)ln},Vt E E. Through a suitable treatment, we may assume that {an(t)ln} is strongly dense in M(t), Vt E E. Let {em{·)lm} be a fundamental sequence of measurable fields of vectors. Then {an(t)p(t) em(t) In, m}

471

is a total subset of c(p(t))H(t), Vt E E. By the method in Proposition 12.1.2,

we can construct an orhtogonal normalized basis {'7A:(')} of c(p(·))H(.) from {an(·)p(·)em(·)ln,m}. Clearly, '7A:(') is measurable, Vk. Then

(c(p(t))€n(t), em (t))t = :L(en(t),'7A:(t))t' ('7A:(t), em(t))t A: is measurable on E, \In, rn, Thus, the field c(p(·)) of operators is measurable.

L CB

Put z =

c(p(t))dv(t). Clearly, zanP€(') = anpe(·),

> c(p). Now

e(·) E H.

L CB

q(t)dv(t), where q(t) is a central projection of M(t), Vt E E. Since c(p)anP = anP, \In, it follows that q(t)an(t)p(t) = an(t)p(t) , a.e,u, Vn, i.e., q(t) > c(p(t)) , a.e.t/. Therefore, c(p) > z, and c(p) = So z

write c(p) =

Vn and

L CB

c(p(t))dv(t).

Q.E.D.

Proposition 12.5.2. If M is discrete, then M(t) is also discrete, a.e.z/. Proof. By Theorem 6.7.1, there is an abelian projection p of M with c(p) = I H • Further from Proposition 12.5.1, p(t) is also an abelian projection of M(t) with c(p(t)) = I H (t), a.e.z/, Therefore, M(t) is discrete, a.e.z-, Q.E.D. If M is properly infinite, then M(t) is also properly

Proposition 12.5.3. infinite, a.e.z-, Proof.

It is immediate from Theorem 6.4.4.

Q.E.D.

Proposition 12.5.4. If M(t) is finite, a.e.v, then M is also finite. Proof.

It is immediate from the definition of finite VN algebras.

Q.E.D.

If M is continuous, then M{t) is also continuous, a.e.

Proposition 12.5.5.

v. Proof. Put EA: = {t E EldimH(t) = k}, and let ZA: be the diagonal operator corresponding to XE." Vk. Clearly, ZA: is a central projection of M, M ZA: is con-

r M(t)dv(t), Vk. Thus, we may assume that H(·) = Ho JEk CB

tinuous, and MZA: =

is a constant field over E.

Suppose that M, M' are generated by the diagonal algebra Z and {an

Lan(t)dl.l{t)ln},{a~ La~(t)dv(t)ln} CB

CB

=

=

respectively, and M(t),M(t)' are

472

generated by {an(t)ln},{a~(t)ln} respectively, Vt E E. We may assume that lIanll, lIa~ll, Ilan(t)II and lIa~(t)1I < 1, Vn and tEE, and {an(t)ln}* = {an(t)ln}, {a~ (t) In}· = {a~ (t) In}, Vt E E. Let S be the unit ball of B(Ho). Clearly, S is a Polish space with respect to the strong operator topology. Consider a subset G of S x E. (a, t) E G, if: 1) aa~(t) = a~(t)a, Vn; 2) a is a non-zero projection; 3) aan(t)aam(t)a = aam(t)aan(t)a, Vn, m. Noticing Proposition 10.3.14, G is a Borel subset of S x E. Let 1f be the projection from S x E onto E. Then from Theorem 1004.5 there is a Borel subset F C 1fG, and a Borel map p(.) from F to S such that (p(t), t) E G, Vt E F, and (1rG\F) C some v-zero subset. e Let p(t) = 0, Vt E E\F, and p = p(t)dv(t). Then p is an abelian

L

But M is continuous, so p = 0 and v(F) = O. Thus , 1f(G) = {t E EIM(t) is not continouos } is contained in some v -zero subset, i.e., M(t) is continuous, a.e.v. Q.E.D. projection of M.

Proposition 12.5.6. infinite, a.e.v.

If M is purely infinite, then M(t) is also purely

Proof.

With the same reason as in Proposition 12.5.5, we may assume that H(-) = Ho is a constant field. Keep the notations of {an,an(t),a~,a~(t)ln},S and etc. in Proposition 12.5.5. Further, let (Ho)oo = ffiHo. Consider a

L n

subset G of S x (Hol oo x E. (a, ('7k), t) E G, if: 1) aa~(t) = a~(t)a, Vn; 2) a is a non-zero projection; 3) a'7k = n», Vk; 4) L II'7kIl 2 = 1; 5) for any finite sets k

A1' A 2 of positive integers,

L((aII nEJ\l an(t)aIImEJ\2am(t)a - aII m E J\ 2 am(t) aIInEJ\l an(t)a) m, '7k) =

o.

k

Clearly, G is Borel subset. Let 1f be the projection from S x (Ho)oo x E onto E. From Theorem 1004.5, there is a Borel subset F C 1rG, and Borel maps p('), ('7k(')) from F to S, (Ho)oo, such that (1rG\F) C some v-zero subset, and for any t E F, (p(t), (tlk(t)) ,t) E G. Define p(t)

=

0, tlk{t)

=

0, Vk and t

ti F.

Then p

=

fEe p(t)dv(t)

is a

projection of M. We may assume that v{E) < 00. Then ~k = tlk{') E H,p~k = ek, Vk, and lIekl1 2 = v(F). By the construction, ~k, ~k) is a normal

L k

L(" k

trace on M p • Since M is purely infinite, it follows that lI(F) = 0, i.e., 1f(G) C some v-zero subset. Moreover, it is easy to see that 1rG = {t E EIM(t) is not purely infinite }. Therefore, M(t) is purely infinte, a.e.z/, Q.E.D.

Proposition 12.5.7. If M is finite, then M(t) is also finite, a.e.z/,

473

Keep the notations: H(·) = Ho,an,an(t),a~,a~(t),S and etc. as in Proposition 12.5.5. Consider a subset G of S x E. [v, t) E G, if :1 ) va~(t) = a~(t)v, 'In; 2) v·v = 1, vv· =I 1. Clearly, G is a Borel subset of S x E. From Theorem 1004.5, there is a Borel subset F C 1rG, where 1r is the projection of S x E onto E , and a Borel map v(·) : F ---t S, such that (1rG\F) C some v-zero subset, and (v(t),t) E G,Vt E F.

Proof.

= 0, 'It (j. F, and v =

L fJ3

v(t)dv(t). Then v·v = p is the diagonal operator corresponding to XF' Since M p is finite, and v(t)v(t)* f. 1, 'It E F, it follows that v(F) = O. Therefore, M(t) is finite. a.e.v. Q.E.D. Define v(t)

If M is semi-finite, then M(t) is also semi-finite,

Proposition 12.5.8. a.e.v.

c(p) = 1. Write

=

L

p(t)dv (t). From Propositon 12.5.1 and 12.5.7, p(t) is a finite projection of M(t), and c(p(t)) = It, a.e.z/, Therefore, M(t) is semi-finite, a.e.z/. Q.E.D. Proof.

M contains a finite projection

fJ3

p with

p

Theorem 12.5.9. Let (E, B) be a standard Borel space, 1I be au-finite measure on B, H(·) be a measurable field of Hilbert spaces over E, H =

L H(t)dv(t), fJ3

and M

L

=

L M(t)dv(t) fJ3

be a decomposable VN algebra on

fJ3

H. If z = z(t)dv(t) is the maximal central projection of M such that Mz is finite, or semi-finite, or discrete, then z(t) is also the maximal central projection of M(t) such that M(t)z(t) is finite, or semi-finite, or discrete, a.e.v.

Proof.

It is immediate from Mz

=

L fJ3

M(t)z(t)dv(t), M(1 - z) =

L fJ3

M(t)

(It - z(t))dv(t) , and Propositions 12.5.3, 12.5.7, 12.5.8, 12.5.6, 12.5.2, 12.5.5 . Q.E.D. Theorem 12.5.10. Let (E, B) be a standard Borel space, v be a (1finite measure on B,H(') be a measurable field of Hilbert spaces over E,H =

LCD H(t)dv(t),

and M =

LCD M(t)dv(t)

be a decomposable VN algebra on H.

Then M is finite, semi-finite, properly infinite, purely infinite, discrete, or continuous, if and only if , M(t) is finite, semi-finite, properly infinite, purely infinite , discrete, or continuous, a.e.v,

Proof.

It is immediate from above discussions.

References.

[21], [28], [119], [154].

Q.E.D.

474

12.6. The constant fields of operators and Von Neumann algebras Lemma 12.6.1. Let A be a separable C"'-algebra with an identity, and H o be a separable Hilbert space. Define

Rep(A, H o} = {1r

11r is a

nondeg:~e~a~: ~epresentation }

and give Rep(A, H o) a topology as follows: 1rl --+ 1r, if II1rl(a)~ - 1r(a)~11 0, Va E A, ~ E H o. Then Rep(A, H o) is a Polish space.

--+

Proof. Let {an}, {~m} be dense subsets of the unit balls of A, H o respectively. For any 1rl, 11"2 E Rep(A, H o), define

n,m Now it suffices to show that ( Rep(A, H o), d) is separable. Let J

= {(n,m)!n,m= 1,2,···},

E

=

{I : J

For any fEE, define

--+

IIIII

Hal L2-(n+m)ll/(n,m)11 < co}. n,m

= L2-(n+m)llf(n,m)ll. Clearly, (E,

n,m

II . II) .IS

a

separable Banach space. Moreover, for any 1r E Rep(A,Ho), let

f'll'"(n,m) = Then 1r

I«

1r(an)~m,Vn,m.

is an isometric map from Rep(A, H o) to (E, Rep(A, H o), d) is separable. --+

II . II).

Therefore, (

Q.E.D.

Lemma 12.6.2. Let (E, B) be a Borel space. Then a map (f) : (E, B) --+ Rep(A, H o) is Borel if and only if t --+ ((f)(t)(a)~, TJ) is measurable on E, Va E A,~, TJ E H o. Here Rep(A, H o) is as in Lemma 12.6.1.

It suffices to prove the sufficiency. By Lemma 12.6.1, Rep(A, H o} admits a countable dense subset {1r n } . Now we need only to prove that for any n,m,(f)-I({1rld(1r,1rn) < m- 1 } ) E B. But it is immediate from the sufficiency Proof.

475

and ~-l({1rld(1r,1rn) < m- 1 } )

EI L2-(H;)II(1rn(ai) - ~(t)(ai))~;11 < m- I } i,j {t E EI L2-(H;) sup 1((1rn (a.) - ~(t)(ai))~;' 6;)1 • . Ie

{t E

< m- I } .

."

Q.E.D. Let (E, B) be a Borel space, H(·) be a measurable field of Hilbert spaces over E, 1\ be any index set , and a,(.) be a measurable field of operators on H(·), Vl E 1\. Suppose that Hi, is a separable Hilbert space, and for each tEE there is a unitary operator u(t) from H(t) onto H« such that u(t)a,(t)u(t)· = b where b, E B(Ho) , Vl E 1\. Then there is a measurable " field v(·) of unitary operators from H(·) to the constant field Ho such that v (t) ai (t)v (t)· = b" Vt E E, l E 1\.

Theorem 12.6.3.

Proof.

We may assume that H (.) is the constant field H o . Then for any eo,71 E Ho,l E I\,t --+ (u(t)*b ,u(t)e,7]) is measurable on E. Let M be the VN algebra generated by {bzll E I\} , and M o be the * subalgebra of B(Ho) generated by {bzll E A}. For any b E M, since Ho is separable, there is a sequence {bn } of M o such that bn --+ b ( strongly ) . Thus, we can see that t --+ (u(t)·bu(t), 71) is measurable on E, Ve, 71 E Ho , bE M. Clearly, M is countably generated. Hence, there is a separable C· -algebra A on H o such that 1 E A c M, and A is strongly dense in M. Let G be the group of all unitary operators on' H o. With the strong operator topology, G is Polish ( see Lemma 11.4.1 ) . Put

e,

Go = {u E Glu·au = a, Va E A}. Then Go is a closed subgroup of G. By Theorem 10.4.2, there is a Borel subset F of G such that #(F n Gou) = 1, Vu E G. For any u E G, define a nondegenerate * representation 1ru of A on H o:

1ru ( a) = u· au,

Va E A.

Clearly, u --+ 1ru is a continuous map from G to Rep(A, H o). Denote this map by W. Obviously, W is injective on F. From Lemma 12.6.1, W is a Borel isomorphism from F onto W(F) = W(G). Define (w IF) -I = (b. For any tEE, let v(t) = (b 0 w(u(t)). Then we get

E

v(.).

F

.q w(F) = w( G)

~q F.

476

From the preceding paragraph, the function

t

-40

(\lI 0 v(t)( a)

e,

1])

= (v(t) *av(t)

e,

1])

= (u(t)*au(t)e, 1])

is measurable on E,Va E A(C M),e,1] E H o . By Lemma 12.6.2, the map \lIov(·) is Borel from E to Rep(A, H o). Thus, v(.) =
v(ttav(t)

=

u(t)*au(t),

'Va E A,t E E.

Since A is strongly dense in M, we obtain v(t)*b,v(t) = a,(t), Vt E E,l E /\.

Q.E.D. Corollary 12.6.4.

Keep the assumptions of Theorem 12.6.3, and let v be

a measure on B. Then there is a unitary operator v from H onto L 2(E, B, v) ® Ho such that va,v* = 1 ® b" where at

=

=

fJJ

IE

L

H(t)dv(t)

fJJ

a,(t)dv(t), Vi,

and "1» is the identity operator on L2 (E, 8, v). Theorem 12.6.5. Let (E, B) be a Borel space, H(·) be a measurable field of Hilbert spaces over E, and M(·) be a measurable field of VN algebras on H(.). Suppose that H o be a separable Hilbert space, and for any tEE there is a unitary operator u(t) from H(t) onto Ho such that u(t)M(t)u(t)* = M o, where M o is a fixed VN algebra on H o. Then there is a measurable field v(·) of unitary operators from H(·) to the constant field H« such that v(t)M(t)v(t)* =

Mo,Vt E E. We may assume that H(·) = Ho, and let G be the Polish group of all unitary operators on H o. Let Go = {u E Glu* Mou = Mo}. Then there is a Borel subset F of G such that #(F n Gou) = 1, 'Vu E G. Define a map \lI : G -40 A, \lI(u) = u* Mou, where A is the Borel space of all VN algebras on H«. From the proof of Proposition 11.4.2, \lI is Borel. Clearly, \lI is injective on F. Thus, \lI is a Borel isomorphism from F onto \l1(F) = \l1(G). Define (\lIIF)-l =
Proof.

E ~F

!U \lI(F) =

\lI(G) ~ F.

Notice that \lI 0 v(·) = \lI(u(.)) = u(·)*Mou(·) = M(·) is a Borel map from E to A ( see Proposition 12.3.2. ) Thus , v(·) =
Q.E.D.

477

Corollary 12.6.6. Keep the assumptions of Theorem 12.6.5, and let v be a a-finite measure on B. Then there is a unitary operator from H = (ffi

JE H(t)dv(t)

r

JE M(t)dv(t), Proof.

«u»: =

onto L2(E, B, v) ® tt; such that

«u»:

~

and Z is the multiplicative algebra on L 2(E, 8, v).

Pick v{·) as in Theorem 12.6.5, and let v =

L v(t)dv(t). ffi

Then

Q.E.D.

Z®Mo.

References.

~

= Z®Mo, where M =

[28], [35], [120], [173].

12.7. Borel subsets of the Borel space of Von Neumann algebras Let H be a separable Hilbert space, and A, 1 be the collections of all VN algebras, factors on H respectively. By Theorem 11.3.2, 11.3.6, A and 1 are two standard Borel spaces.

Proposition 12.7.1. Let AI n be the collection of all type (In) VN algebras on H. Then Al n is a Borel subset of A, n = 00,1,2,···. By Proposition 6.7.8, any type (In) VN algebra is spatially * isomorphic to ZrsB{Hn ) , where Z is an abelian VN algebra, and dim H n = n. Now from corollary 5.3.9 and Proposition 11.4.3, AI n is a Borel subset of A.

Proof.

Q.E.D. Proposition 12.7.2. Let Ax be the collection of all type (I) VN algebras on H. Then A is a Borel subset of A. I

Proof. As the proof of Proposition 11.4.3, define a Borel map (l) from A = A(H) to A(H®H): (l)(M) = M@(CI H,VM E A(H). Then MEAl if and only if (l)(M)' = M'rsB(H} is type (I oo ) ( here we assume that dim H = 00. H dim H < 00 , then AI = A, and the conclusion is trivial). Let E be the collection of all type (100) VN algebras on H ® H. Then E is a Borel subset of A(H ® H) by Proposition 12.7.1. From Proposition 11.3.1, E' = {N'jN E E} is also a Borel subset of A(H ® H) . Then A = (l) -1 (E') is a Borel subset of A = A(H). I

Q.E.D. Proposition 12.7.3. Let A, be the collection of all finite VN algebras on H. Then A, is a Borel subset of A.

478

Proof. First, by a similar proof as Lemma 11.4.6, (A\ A I) is Sousline. Then from a similar proof as Proposition 11.4.7, AI is also Sousline. Therefore, AI is a Borel subset of A. Q.E.D.

Lemma 12.7.4. Let A$I be the collection of all semi-finite VN algebras on H. Then A61 is a Sousline subset of A. Proof.

It is immediate from a same proof as Lemma 11.4.8.

Q.E.D.

Lemma 12.7.5. Let Am be the collection of all type (III) VN algebras on H. Then (A\A is a Sousline subset of A. III )

Proof. A VN algebra M on H is not type (III) if and only if M contains a non-zero finite projection. Consider a subset E of Ax P X H oo , where P is the collection of all projections on H, and tt; = EBH. iu,« (6:)) E E,

L n

if: p is a non-zero projection of M; p6, =

6:, Vk; L('6:, ~I:) is a trace on

I: pMp; { a' 6.1 a' E M', k} is a total subset of pH. By a similar proof as Lemma 11.4.8, E is a Borel subset. Therefore, (A\A m ) = 1r E is a Sousline subset of A, where 1r is the projection from A X P x H oo onto A. Q.E.D.

Lemma 12.7.6. Let (E, B) be a standard Borel space, and M(E) be the collection of all non-zero finite measures on B. Give M(E) the minimal Borel structure such that for any bounded measurable function f on E, v ----+ v(f) = f dv is a measurable function on M (E). Then M (E) is also a standard Borel

L

space, and its Borel structure is generated by following subsets:

{v E M(E)lv(F) < A},

VF E B, A >

o.

By Theorem 10.3.16, we may assume that E = [0,1]. Then M(E) = C(E)~ \{O}, where C(E)~ is the collection of all continuous positive linear functionals on C(E) . Notice that for any bounded measurable function f on E, there is a sequence {fn} of simple functions on E such that fn(t) -+ I(t), uniformly for tEE. Thus, the Borel sstructure of M(E) is generated by {v E M(E)lv(F) < A}, VF E B, and A > 0 exactly. Since C(E) is a separable Banach space, (C(E)*,w*-top. ) is a standard Borel space. Hence, M(E) = C(E)~ \ {a} is also a standard Borel space with respect to the w*-top. in C(E)*. For any I E C(E),v ----+ v(f) is w*continuous on M(E). Further, v ----+ v(F) is measurable on the Borel space Proof.

479

(M(E), w*-top.) ,\IF E B. Thus the Borel structure P of M(E) generated by {v E M(E)jv(F) < -X}(\lF E B and -X > 0) contains the standard Borel structure generated by w* -topology. Now by Theorem 10.3.13, it suffices to show that P contains a separated countable family. Let {r n } be all rational numbers in [0,1], and {t k } be all rational numbers in (0, +00). Put

Then it is easy to see that {Qi;kli,j,k} will be a separated family for M(E).

Q.E.D. Lemma 12.7.7. Let (E, B) be a standard Borel space, M(E) be as in Lemma 12.7.6, H(·) be a measurable field of Hilbert spaces over E, e(·) be a bounded measurable field of vectors of H (.) , a(.) be a uniformly bounded measurable field of operators on H(·), and M(·) be a measurable field of VN algebras on H (.). Then:

1) By the a fundament family {v

L

-+

ffJ

'1( t) dv (t) 1'1 (.) is any bounded

measurable field of vectors of H(·)} of vector fields ( where e the element '1(.) of H(t)dv(t)),

L

v

-+

K(v) =

L e

'1 (t )dV(t ) is

L ffJ

H(t)dv(t)

is a measurable field of Hilbert spaces over M (E) ;

L ffJ

2) v -+ e(t)dv(t) is a measurable field of vectors of K(·), where dv(t) is the element e(·) of K(v), \Iv; 3) v -+ on K(')j

4) v

t

-+

e

a(t)dv(t) is a uniformly bounded measurable field of operators

L M(t)dv(t) e

L e(t)

is a measurable field of VN algebras on K(·).

Proof. 1) We may assume that E = [0,1]. Let {en(')} be an orthogonal normalized basis of H(·), and {1m} be a countable dense subset of G(E). ffJ

By Proposition 12.1.9, for each v E M(E){L (/m
lEe H(t)dv(t). Then from Lemma 12.7.6, v

L H(t)dv(t) ffJ

-+

K(v) =

is a measurabel field of Hilbert spaces over M (E). Now if '1 (.) is a bounded measurable field of vectors of H(·), then t -+ ('1(t),/m(t)en(t)), is bounded

480

measurable on E. Further, from Lemma 12.7.6 V

~ V ( (1J (.), f m

( •)

en (.)).)

(L 1J(t)dv(t), L fm(t)en(t)dv(t)) 6J

6J

L 6J

(1J(t), f m(t) en (t))tdv( t)

is measurable on M(E), '1m, n. Thus, v

~

L 1J(t)dv(t) 6J

is a measurable field

of vectors of K(·). 2) It is immediate from 1). 3) Since a(·)fm(·)en(·) is still a bounded measurable field of vectors of H(·) , '1m, n, from Propositon 12.2.2 v

~

L a(t)dv(t) 6J

is a measurable field of

operators on K(·) . Moreover, II (6J a(t)dv(t) II < sup Ila(t)llh 'Iv E M(E).

JE

t

4) Let {a n ( · ) } be a sequence of measurable fields of operators on H(·) such that M(t) is generated by {an(t)ln}, Vt E E. We may assume that 6J Ilan(t) lit < 1, 'It E E and n. Then for each u E M(E), M(t)dv(t) is gen-

{L ~L 6J

erated by

L 6J

fm(t)l tdv(t),

L

an(t)dv(t) 1m, n}. Now from 3) and Definition

6J

12.3.1, v

M(t)dv(t) is a measurable field of VN algebras on K(·). Q.E.D.

Lemma 12.7.8. Let M = M(.'1;n X N) = {vlv is a non-zero finite measure on .'1;u X N}, where .'1;n is the collection of all type (III) factors on H, and is a standard Borel space from Theorem 11.4.16. Suppose that {Zn} is a sequence of pairwise non * isomorphic abelian VN algebras on H such that each abelian VN algebra on H must be * isomorphic to some Zn (see Corollary 5.3.9). Then there exists a Borel map B (.) from M to A such that for each v E .M, B (v) is

spatially

* isomorphic to

(6J

JJin xIV

(A~Zn)dv(A, n).

Proof. From the proof of Proposition 11.4.3, A ~ A~IH is a Borel map from A = A(H) to A(H ® H) . Clearly, n ~ a:lH~Zn is a Borel map from N to A(H®H). Then by Proposition 11.3.5, (A,n) ~ A~Zn is a Borel map from

N to A(H ® H). Now from Lemma 12.7.7, v ~ (6J J, m xlV (A~Zn)dv(A,n) is a measurable field of VN algebra on K(·) , where .'1;n

X

K(v)

=

(6J

JJinxN

(H ® H)dv(A, n)

=

H ® H e L 2 (.'1;u X N, v),

481

Vv EM. Since H ®H ®L 2 (1,.11 X IN, v) and H are countably infinite dimensional, from Proposition 12.3.2 there is a Borel map B(·) from M to .A such that B(v) ffJ is spatially * isomorphic to r (A®Zn)dv(A, n), Vv E M.

J7mxN

Q.E.D. Now let M be a fixed type (III) VN algebra on H. By Theorem 12.4.2, there is a finite Borel measure v on JR , a measurable field H(·) of Hilbert spaces over JR and a measurable field M (.) of factors on H ('), such that :

M is spatially

* isomorphic

to

f:

n M' to the diagonal algebra on

M(t)dv(t); and this

f:

* isomorphism maps

H(t)dv(t). From Theorem 12.5.10, we may assume that M(t) is type (III) and dim H(t) = 00, Vt E JR. Further, we may assume that H(·) is the constant field H over JR. Then M(·) is a Borel map from JR to 1 ( see Proposition 12.3.2). M

Lemma 12.7.9.

We may assume that M(lR) is a Borel subset of 1.

Proof. Clearly, M(JR) is a Sousline subset of 1. Let J.t = vc M::', Clearly, J.t is a finite measure on F, By Corollary 10.2.11, there are Borel subsets E, F of .A such that E c M(JR), (M(JR)\E) c F, and J.t(F) = o. Put lRo = M- 1(1\E). Then v(lRo) = 0 and M(lR\lRo) = E. Now change the definition of M(·) on lRo such that the new M(·) on lRo are a fixed type (III) factor on H. Then M(JR) is a Borel subset of 1. Q.E.D. Lemma 12.7.10. Let M(lR) be a Borel subset of 1 , and introduce an equivalent relatjon r - in JR : ft _ t 2 if M(td = M(t 2 ) , and give JR/ _ the quotient Borel structure. Then JR / _ is also a standard Borel space, and
482

JR/

is a standard Borel space. The rest conclusion can be obtained from Proposition 1004.7. Q.E.D. ""'-I

Lemma 12.7.11. Let M(JR) be a Borel subset of 7. Then there is a Borel map 'l ~ V; from lR/ ~( see Lemma 12.7.10 ) to M(lRh such that for any bounded measurable function f on lR and any ii-integrable function h on lR/ where v = v 0 1r- 1 , ""'-I,

IB (h

0 1r)

(t)f(t)dv(t)

f h(t}dil(t) f f( s)dvrls), lRI'" lR and supp V; C

1r-

1

({ t}), a.e.v.

Proof.

A rational semiclosed interval means a subset of JR with form [a, b), where a and b are rational or ±oo ( for the conveninece, [a, b) = 0 if b < a, and [a,b) = (-oo,b) if a = -00). Let E be the collection of all disjoint finite unions of rational semiclosed interval. Then E is a countable Bool algebra, and any disjoint union of elements of E must be finite. Thus, any finitely additive finite measure on E must be countably additive, and can be uniquely extended to a finite Borel measure on lR. For any E E E, v(En1r- 1 ( . )) is a finite measure on lR/ and is absolutely 1 continuous with respect to v = v 0 1r- . Thus, there is 0 < gE E L1(JR/ ~,v) such that v(En1r- 1(.)) = gE·iI(.). Since E is countable, then there is a v-zero subset Eo of lR/ ~ such that for each t E (lR/ ~)\Eo, E ~ gE(t) is a finitely additive probability measure on E. Further, g.(t) can be uniquely extended to a probability measure on lR. Thus, we get a map 'l ~ V; from lR/ to M(lRh such that v;{E) = gE(t), 'It E (lR/ .....,)\Eo and E E E, and is a fixed probability measure on lR, 'It E Eo. Since the Borel structure of lR is generated by E, it follows from Lemma 12.7.6 that 'l ~ Vi'is a Borel map from lR/ ....., to M(lRh. And for any Borel subset E of lR and any Borel subset F of lR/ ....." we have ""'-I

,

""'-I

vr

(1) Further, for any bounded measurable function function h on lR/ ....., we get

Ill. (h f

llll'"

0 1r)

f on lR and any v-integrable

(t) f(t)dv(t)

h(t)dv(t) f f(s)dvrl s).

lR

483

For any Borel subsets E, F of JIll .-, by (1) we have

kXi(i)df/(t) = f/(E n F)

Thus for fixed

E,

=

v(1r- 1(E) n 1r- 1 (F ))

=

kl-f{1r-

1(E))df/(t).

we get

(2) Since JIll .- is a standard Borel space, we can find a countable family f: of Borel subsets of JIll .- such that for any t E JIll .-, there is {En} C f: with E1 ~ ••• ~ En ~ ... and nnEn = it}. Now from (2) , we can see that 1 = l-f{1r- 1 ({t})), a.e.D, i.e., supp Vi'C 1r- 1 ({t}), a.e.D. Q.E.D.

f: f:

Lemma 12.1.12.

Let

t

---+

Vi' be as in Lemma 12.7.11.

Then

t

---+

M(s)dv;{s) is a measurable field ofVN algebras on L(·), where t ---+ L(t) =

H~L2(JIl, vr) is a measurable field of Hilbert spaces over JIll .-, is spatially * isomophic to

H dl-f{s)

and M

=

l" dfi(t) l" M(s)dl-f{s). JEt/'" JIl Pick a sequence {In} of bounded measurable functions on JIl such that {In} is dense in L 2 (JIl, v) and is w· -dense in Loo (JR., J1.), VJ1. E .M (JIl). Let {Em} be an orthogonal normalized basis of H, and {a n ( · ) } be a sequence of measurable fields of operators on the constant field H such that M (t) is generated by {an(t) In}, Vt E JIl. Since ---+ Vi' is Borel, by Lemma 12.7.7, Proof.

,."

t

fm

12.7.6, {t ---+ JEt (InEm)(t)dl-f{t) In, m} is a fundamental family of measurable fields of vectors of L(·) , and

t ---+

f:

an(t)dl-f{t),

t ---+

f:

Im(t) I Hdv;{t)

are measurable fields of operators on L(.), Vn, m ( here we may assume that

JIl, n), and for each t'" E JIll .-, Jfm M(s)dl-f{s) is generated Il fm fm ,." fm by {JIl an(s)dl-f{s) , JIl Im(s)IHdl-f{s)ln,m}. Thus, t ---+ JIl M(s)dl-f{s) is a

Ilan(t)11 <

1, Vt E

measurable field of VN algebras on L(·). By Proposition 12.1.9,

E )( )d....J ) I Vn, m, and h is } {t ---+ h(l\ Jfm(~ Il n m S vf\s bounded measurable on JIll """ l<)

484

is a total subset of (fB dv(t) {fB H dV; = {fB (H

JRI...-

JR

JElI...-

e L 2(lR, vr))di/(t).

Define

Vn, m, h. From Lemma 12.7.11, u can be uniquely extended to a unitary operator from {fB (H ® L 2 (lR, vr))dil{t) onto

JR~

u maps the operators

f:

Im(s) IHdvr(s) to

JRI...-

® L 2 (lR, v). Clearly,

t~ h(t) l tdv(t), (fB di/(l) (fB an(s)dv;(s), (fB di/(t)

f:

JRI'"

f:

JRI'"

h(1r(t))IHdv(t),

spectively, Vn, m, h. Therefore, u (fB

I" H dv = H JR JR

f:

.

an(t)dv(t),

JRI'"

Im(t)IHdv(t) re-

di/(t) {fB M(s)dvr(s)u* = (fB M(t)dv(t). JR

JR

Q.E.D.

Lemma 12.7.13. There exists JL E M such that M is ,where M,B(.) are defined as in Lemma 12.7.8. Proof.

Let t

---+

* isomorphic to B(JL)

V; be as in Lemma 12.7.11, and {In} be as in the proof {fB

;v

of Lemma 12.7.12. Then with {(t

---+

J In(s)dvr(s))!n}

as a fundamental

R

sequence of measurable fields of vectors, (t ---+ L2 (lR, Lot)) is a measurable field of Hilbert spaces over JR/ and (t ---+ Zr) is a measurable field of 2 VN algebras on (t ---+ L (lR, vr)), where Zr is the multiplicative algebra on L 2 (lR, V;)), Vt E lR/ Since L 2 (lR, vr) is separable, there exists unique positive integer n(l) such that Zr is * isomorphic to Zn{t)' Vt E lR/ where {Zn} is defined as in Lemma 12.7.8. We say that (t ---+ n(t)) is a Borel map from lR/ to IN . By 2(lR, Proposition 12.1.2, {tldimL ve) = k} is a Borel subset of lR/ Vk. Thus 2 2 , we may assume that dimL (lR, vr) is a constant, Vt. Then (t ---+ L (lR, vr)) is unitarily isomorphic to a constant field Kover lR/ and (t ---+ Zr) is spatially * isomorphic to (t ---+ N(t)), where N(·) is a Borel map from lR/ ,..., to A(K). For fixed No E A(K), a(No) is a Borel subset of A(K) ( see Proposition 11.4.3 ). Then N- 1(a(No)) is a Borel subset of lR/ ,..., . Further (t ---+ n(t)) is Borel. We can also construct a measurable field W(·) of * isomorphisms such that W(t) is a * isomorphism from Z; onto Zn{t)' 'it E JIl/ In fact, from the preceding paragraph, (t ---+ Zr) is spatially * isomorphic to (t ---+ N(t)), where N(.) is a Borel map from lR/ to A(K). Since the function I(E L 2(lR, V;)) is cyclic and separating for Zr, N(t) admits also a cyclic-separating vector in .-.J,

.-.J

•

.-.J,

.-.J

.-.J,

.-.J,

.-.J

.-.J

•

485

K, TIt E JRI - . By Corollary 5.3.9, we may assume that all N(t}(t E JRI -) are * isomorphic. Fix a to E JRI Then by Theorem 1.13.5, N(t} is spatially * isomorphic to N(to) ,Vt E JRI - . Further, from Theorem 12.6.5 we can get the field \II(.). By Lemma 12.7.10, cP is a Borel isomorphism from JRI - onto M(lR) . Let k(A) = no CP-l(A)' \fA E M(JR), and k(A) = 1, VA E (A\M(JR)). Then k(·) is a Borel map from A to IN. Let fj = i/ 0 cp-l = V 0 M::', Then f) is a finite measure on A, and supp fi c M(JR). In the following, the symbol " ~" means the * isomorphism r«,

between VN algebras. Since cP is a Borel isomorphism, by Proposition 12.3.12 and Lemma 12.7.10 we have

(rB (A~Zk(A))dfi(A) =

rrB

lA

I"

cP

lM(E)

0

lM{E)

(AlZlZk(A))dfi(A)

CP-l(A)lZlZko~o~-I(A)dv 0 CP-l(A)

(rB cp(t) lZlZn(t)dv(t) '" t" (M(a(t))lZlZr)dv(t). lE/,.., lEI""

-

From Lemma 12.7.11, supp '1'C 1("-I({t}), a.e.i/. Clearly, M(t) = M(a(t)),\ft E 1("-1 Thus ,

«r»

t" M(s)dv;(s)

1JR

=

t" «(i}) M(a(t))dv;(s)

1

'It-I

= M(a(t})lZlq-, a.e.D. Now by Lemma 12.7.12, M is

* isomorphic to £rB (AlZlZk(A))dfi(A).

Noticing that supp fi C M(JR) C 1.11' we get

£rB (AlZlZk(A))dfi(A) ~$ where Ai

= {A

£:

(A®Zj)dv(A)

E Alk(A) =

i}, Vi·

= ~$

(n"" (A®Zj)dv(A),

Further, pick J.L E .M such that J.L

{i}, Vj. Then, M is * isomorphic to Again by Lemma 12.7.8, M is * isomorphic to B(J.L).

on (Ai

n 1.11)

X

=

f)

ffi

r

lJ[lIxN

(AlZlZn) dJ.L(A , n). Q.E.D.

Lemma 12.7.14. Let E be a Sousline subset of A. Then a(E) = {N E AIN is * isomorphic to some element of E} is also a Sousline subset of A. Define CP(N) = N®Q:1H' Then cP is a Borel isomorphism from A = A(H) into A(H ~ H). From the proof of Proposition 11.4.3, we have

Proof.

a(E)

=

cp-l(s(CP(E))) = cp-l(s(CP(E)) n cp(A)).

486

From Lemma 11.4.14, s(c)(E)) is a Sousline subset of A(H ® H). Then there is a Polish space P and a Borel map f from P to A(H ® H) such that f(P) = s(c)(E)) n c)(A). Now C)-I 0 f is a Borel map from P to A. Thus, by Proposition 10.3.5 a(E) = C)-I 0 f(P) is a Sousline subset of A. Q.E.D.

Proposition 12.7.15.

Am is a Borel subset of A.

By Lemma 12.7.5, it suffices to show that AlII is Sousline. From Lemma 12.7.8, B(M) is a Sousline subset of A. By Theorem 12.5.10, we have B(M) c Am. Again by Theorem 12.7.13, AlII = a(B(M)). Now from Lemma 12.7.14, Am is a Sousline subset of A. Q.E.D.

Proof.

Theorem 12.7.16. Af ( the collection of all finite VN algebras on H) ,A,f( the collection of all semi-finite VN algebras on H) , Api( the collection of all properly infinite VN algebras on H) ,Alit ( the collection of all type (Ik ) VN algebras on H), k = 00,1,2", . ,AI ( the colletion of all type (I) VN algebras on H) , Ac ( the collection of all continuous VN algebras on H), AII l ( the collection of all type (III) VN algebras on H), Alloo ( the collection of all type (II 00) VN algebras on H) , An ( the collection of all type (II) VN algebras on H ) , and Am ( the collection of all type (III) VN algebras on H) are Borel subsets of A.

Proof. From Propositions 12.7.1, 12.7.2, 12.7.3 and 12.7.15, Alit' AI) Af and AIII are Borel subsets of A. For the case A,j, by Lemma 12.7.4 it suffices to prove that (A\A,f) is a Sousline subset of A. Notice that M E (A\A,f) if and only if M = M I EEl M 2 and M 2 is type (III), i.e., M is

* isomorphic to a VN algebra

(MI

M2) on

H EEl H , where Mt, M 2 E A and M 2 is type (III) . From Proposition 11.3.3 and 11.3.4, (Mt, M 2 )

~

(M1

M

)

is a Borel map from A X A to A(H EEl H)'

2

and this map is injective obviously. Then by Proposition 12.7.15,

E = {

(M

1

M

) 2

I M 1,M2

E A, and M 2 is type (III) }

is a Borel subset of A(H EEl H) . Let v be a unitary operator form H EEl H onto H. Similarly, from Proposition 11.3.3 and 11.3.4 v . v· is a Borel isomorphism from A(HEElH) to A. Then by Lemma 12.7.14, (A\A,f = a(vEv*) is a Sousline subset of A. Notice that M ~ Api if and only if M = M 1 EEl M 2 and M 2 is finite. Then by the same method as in the preceding paragraph, we can see that (A\A pi) is Sousline. Moreover, M E Api if and only if there is a projcetion p of M such that 1 ,.... P ,.... (1 - pl. Let S be th unit ball of B(H), and consider a

487

subset E of Ax S X S. (M, Vb V2) E E if: 1) an(M')vi = vian(M'), \In, and i = 1,2; 2) vivj = 1,j = 1,2; 3) V;Vl • V;V2 = 0; 4) V;Vl + V;V2 = 1, where {a n ( · ) : A --+ Sin} is as in Proposition 11.3.3. Clearly, E is a Borel subset of A X S X S . Then Api is Sousline. Further, Api is a Borel subset of A. By Theorem 6.8.4 and the same method, we can see that An is Sousline. Moreover, M fj. An if and only if M = M 1 (£l M 2 and M 2 is either type (I) or type (III) . Then we can prove that (A\A n ) is Sousline. Thus, Au is a Borel subset of A. Further, AU I = An n AI and Anoo = An n Api are Borel subsets of A. Finally, since Ac = Au U Am U {M E AIM = M 2 (£l M s , and M 2,Ms are type (II) , (III) respectively} , Ac is Sousline. Moreover, M fj. Ac if and only if M = M 1 (£l M 2 , and M 1 is discrete. Thus, (A\A c ) is Sousline , and Ac is a Borel subset of A. Q.E.D.

Notes. Lemma 12.7.5 was proved by J.T.Schwartz, and the hard result, Propositon 12.7.15, is due to a.Nielsen. References.

[36], [120], [121], [154], [177].

12.8. Borel subsets of the state space of a separable C* -algebra Let A be a separable C·-algebra with an identity, and S (A) be its state space. By a(A*,A),S(A) is a compact Polish space. In fact, let {an} be a countable n dense subset of the unit ball of A, define d(ip,t/J) = l(ip - t/J)(an)I,Vip,

L2n

t/J E S(A). Then d is a proper metric on (S(A),a(A*, A)). For any n = 00,1,2,' .. , let H n be n-dimensional Hilbert space, and R n be the collection of all nondegenerate * representations of A on H n • In Rn , the topology is defined as follows: 1r1 --+ 1r if II (1rI(a) -1r(a)) Ell --+ 0, Va E A, E H n . By Lemma 12.6.1, R n is a Polish space, 'In.

e

Lemma 12.8.1. (): 1r -----t 1r(A)" is a Borel map from is the collection of all VN algebras on H n , 'In.

Rn to An' where An

Proof. Let {ak} be a countable dense subset of the unit ball of A, and define aA:(') : Rn --+ (B(Hn),a), ak(1r) = 1r(ak), V1r E R n and k . For any E,,, E H n , (ak(')E,,,) is continuous on R n obviously. Thus, {ak(')} is a sequence of Borel maps from R n to (B (Hn) , 0"). Moreover, for each 1r E R n, 1r ( A)" is generated by {ak(1r)jk}. Now by Proposition 11.3.4, () is a Borel map from Rn to An. Q.E.D.

488

Proposition 12.8.2. R~) = {1r E RnI1r(A)" is a type (t) VN algebra on H n } is a Borel subset of Rn, Vn, where (t) is factorial, finite, semi-finite, properly infinite, (Ik)(k = 00,1,2,00'), (I), (II I), (11 00 ) , (II), (III), (c) (continuous ) , or (Irr) ( irreducible). Except the case of (t) = (Irr), the conclusions are obvious from Theorem 11.3.6, Theorem 12.7.16 and Lemma 12.8.1. Now let (t) = (Irr). Notice that 1r E Rit ) if and only if 1r(A)" = B(Hn ) , i.e., the unit ball of 1r(A) is strongly dense in S, where S is the unit ball of B(Hn ) . Let be a proper metric on the Polish space (S, strong top.) ,{bk } be a countable dense subset of (S,O), and {ak} be a countable dense subset of the unit ball of A. Then

Proof.

°

R~) = nk,p U m {1r E Rnlo(1r(a m ) , bk )

< p-I} Q.E.D.

Now for n = 00, 1, 2, ... , put

E)I1r ERn, EErn, and Eis cyclic r n = {E E HnlllEII = I}. En = {(1r,

where

Lemma 12.8.3. En is a Gs-subset of R n with respect to the relative topology, Vn.

X

for 1r(A)},

r n' Thus

En is a Polish space

Let {ak} be a countable dense subset of A, and {Ek} be a countable dense subset of H n • Then

Proof.

En = ni,;

u, {(1r, E)

E

u; x r nlll1r(ak)E - E;II < i-I} Q.E.D.

Put E = U{Enln = 00,1,2," .}. Then E is also a Polish space, and each En is an open and closed subset of E. Further, define a map w : E -+ S(A), w(1r, E)(a) = (1r(a) E, E), V(1r, E) E E, a E A. Lemma 12.8.4.

W is continuous from E onto S(A).

The continuity is obvious. Now let cp E S{A). Then we have the cyclic * representation {1rtp,Htp,ltp} of A. If dim Hlp = n, and U is a unitary operator from Hlp onto H n , and let 1r = U1rtpu* E R m then

Proof.

cp(a) = (1rtp(a) lip' lip) = (1r(a)E, E)

= W(1r, E)(a) , Va

E A

489

where

E=

ul~ Ern' Thus,

Q.E.D.

'1'(E) = S(A).

Lemma 12.8.5. Let /\ C S(A). IT '1'-1(/\) is a Sousline or Borel subset of E, then /\ is also a Sousline or Borel subset of S(A) .

Proof.

By Lemma 12.8.4, '1' maps a Sousline subset of E to a Sousline subset of S(A). So , if '11- 1 ( /\ ) is a sousline subset of E, then /\ = W('I1- 1 ( /\ )) is also a Sousline subset of S(A). Now let w- 1 ( /\ ) is a Borel subset of E. Then (E\ '11- 1 ( /\ )) is also a Borel subset of E. From the preceding paragraph, /\ and W(E\W- 1 ( /\ )) = S(A)\/\ are two Sousline subsets of S(A). Therefore, 1\ is a Borel subset of S(A).

Q.E.D. Theorem 12.8.6. S(A)(t) = {cp E S(A)I1r~(A)" is a type (t) VN algebra on H~} is a Borel subset of S(A), where {1r~,H~} is the * representation of A generated by cp, \lcp E S(A), and (t) is factorial, finite, semi-finite, properly infinite, (IA;)(k = 00,1,2," '), (I) , (Ill) , ( 1100 ) , (II), (III) , (c) or (Irr). For n = 00,1,2"", by Proposition 12.8.2 (R~) X r n) is a Borel subset of (Rn X r n)' Thus , Un((R~) X r n) n En) is a Borel subset of E, \I(t). IT (1r, E) E (R~) X r n) nEn, then 1r is a type (t) * representation of A on H n, and is cyclic for 1r(A). Clearly, the cyclic * representation {1r,Hn of A is unitarily equivalent to the * representation of A generated by cp = '11(1r, e). Thus '11(1r, e) E S(A)(t). Conversely, if cp E S(A)(t), then by Lemma 12.8.4 there is (1r, E) E En ( for some n) such that '11(1r, e) = cp. Since (1r(a)E, E) = (1r~(a)l~,l~),\la E A, it follows that {1r,Hn, e} ,-.,J {1r~,H~,l~}. Hence, 1r is a type (t) * representation of A on H n , and E is cyclic for 1r(A), i.e, (1r, E) E (R~) X r n) n En. Therefore,

Proof.

e

,e}

W- 1(S(A)(t)) = Un((R~)

X

r n) n En)

is a Borel subset of E. Now by Lemma 12.8.5, S(A) (t) is a Borel subset of

S(A), \I(t).

Q.E.D.

Remark.

Since S(A) is a metrizable compact convex subset of (A 1ft, a(A", A)), it follows from the Choquet theory (see Section 14.1) that S(A)(Irr) = ExS(A) = P(A} (the pure state space of A) is a Grsubset of S(A) indeed.

Notes.

Theorem 12.8.6 is due to J. Feldman, J.T. Schwartz, and etc.

References. [45], [120], [148], [154].

Chapter 13 Type I C*-Algebras

13.1. The spectrum of a C*-algebra Let A be a C· -algebra, and Prim(A) be the collection of all primitive ideals of A ( see Definiition 2.8.1.). For any T C Prim(A), define I(T) = n{JIJ E T}, and T = {J E Prim(A) IJ :::> I(T)}.

Lemma 13.1.1. have that

For the bar "-" operation on the subsets of Prim(A), we

0" = 0,

vr, Tl' T2 C Proof.

T

C

T,

T

= T,

and

T 1 U T2 = T 1 U T2 ,

Prim(A).

For any T

1(1)

C

Prim( A) , since

C I(T)

c n{J

E Prim(A)IJ ~ I(T)} =

it follows that T = T. For any Ti, T 2 C Prim(A), put Ii = I(Ti ) , i 11 n 12 • Then by Lemma 2.8.7 we have that

J E T1 U T2

Thus, T 1 u 12 = T 1 U T 2 •

~

II

=

1(1),

1,2. Clearly, I(T1 U T 2 ) =

n 12

{=>

J

-¢::::=:>

either J :::> 11

or

-¢::::=:>

either J E T 1

or

J:::> 12

J E T2 • Q.E.D.

It follows from above lemma that there is a unique topology in Prim(A) such that for each T C Prim(A), T is the closure of T with respect to this topology.

491

Definition 13.1.2. Prim(A).

This topology

IS

called the Jacobson topology

in

Proposition 13.1.3. Let A be a C*-algebra. 1) A subset T of Prim(A) is closed <==::}- there is a subset E of A such that

T

=

{J E Prim(A)IJ

~

E}

there is a closed two-sided ideal I of A such that T = {J E Prim( A) IJ ~ I}. Moreover, the ideal I is indeed generated by E, and is uniquely determinded by the closed subset T. 2) A subset W of Prim(A) is open <=> there is a subset E of A such that <==::}-

W <==::}-

= {J

E Prim(A)IJ

1J

E}

there is a closed two-sided ideal I of A such that W = {J E Prim (A) IJ

1J

I}. 3) Let J E Prim(A) . Then the subset {J} of Prim(A) is closed if and only if J is a maximal closed two-sided ideal of A.

4) The space Prim(A) is a To-space. Proof. 1) Picking E = I = I(T) = n{ JIJ E T}, the necessity is obvious. Now let E c A and T = {J E Prim(A)IJ ~ E}. Clearly, E c I(T). Then

T = {J E Prim(A)IJ ~ I(T)}

c {J

E Prim(A)IJ ~ E}

= T c T.

So T = T is closed. Moreover, by Proposition 2.8.2, I and the closed two-sided ideal generated by E are indeed equal to I(T). 2) It is obvious from the conclusion 1). 3) Let J E Prim(A) and J be maximal. Since I({J}) = J, so {J} = {J} is closed. Conversely, let {J} = {J}, and I be a closed two-sided ideal of A with I ~ J. By Proposition 2.8.2, there is J' E Prim(A) with J' ~ I. Then JI E {J} = {J} ,and J' = 1= J. So J is maximal. 4) Let J I1 J 2 E Prim(A) and J 1 # J 2 • We may assume that J 1
Let A be a C*-algebra, A be the collection of all unitary equivalent classes of non-zero irreducible * representations of A, and ker be the nature map from A onto Prim(A). Then there is a topology in A induced by the Jacobson

492

topology in Prim(A) , i.e., a subset fj of open subset U of Prim(A) such that

..4 is said

to be open, if there is an

(j = ker -1(U) = {11" E Alker1l" E U}. A.

Definition 13.1.4. The spectrum of a C·-algebra A is the set A endowed with the inverse image of the Jacobson topology under the natural map ker : A ---t Prim( A). Clearly, now the map ker : A ---t Prim(A) is continuous and open. Proposition 13.1.5. The map I ~ AI = {11" E Alker1l" 1J I} is a bijection from the collection of all closed two-sided ideals of A onto the collection of all ...... open subsets of A. Moreover, for any closed two-sided ideals I, II and 12 of A, we have that AI '" i{ homeomorphism ), A(lt+I2) = All

Consequently, II n 12

Proof.

= {O}

U

AI 2 ,

-¢;::;::;>

All n

AI

A(lt n I2) = All n A I 2 .

..412 = 0.

Let I be a closed two-sided ideal of A. It is clear that

AI = ker- I( {J SO

and

is an open subset of A.

A.

E Prim(A) IJ

1J I}).

......

A.

Now let U be an open subset of A. By Definition 13.1.4, we have U ker-I(U), where U is an open subset of Prim(A). Further, by Proposition 13.1.3 there is unique closed two-sided ideal I of A such that Prim(A)\U = {J E Prim(A)IJ

:J

I}.

Hence, there is unique closed two-sided ideal I of A such that

D=

{7r E Alker7r 1J I} =

The rest conclusions are easy.

A? Q.E.D.

Proposition 13.1.6. Let A be a C·-algebra. Then the following statements are equivalent: 1) A is a To-space; 2) Two irreducibe * representations of A with the same kernel are unitarily equivalent, i.e., the map ker : A ---t Prim(A) is injective; 3) 7r ---t kerx is a homeomorphism from ..4 onto Prim{A).

493

Proof. 2) =::::? 3) . It is obvious since ker : A ~ Prim(A) is continuous and open. 3) =::::? 1). It is immediate from Proposition 13.1.3. 1) =::::? 2) .Let A be a To -space, and 7r17 7r2 be two irreducible * representations of A with ker7rl = ker7r2. If 7rl and 7r2 are....n ot ullitarily equivalent;., then we may assume that there is an open subset U of A such that 7r1 E U and 7r2 ¢ U. Suppose that U = ker- 1(U), where U is an open subset of Prim(A), and put J = ker7rl = ker7r2' Then we get J = ker7rl E U and' J = ker7r2 ¢ U, a contradiction. Therefore, 7r1 and 7r2 are unitarily equivalent.

Q.E.D. Definition 13.1.7. A topological space X is said to be Baire , if {Xn } is a sequence of open dense subsets of X, then nnXn is still dense in X. For example, a completely metric space is a Baire space, consequently, it must be second category.

Lemma 13.1.8.

Let E be a Hausdorff locally convex linear space, K be a compact convex subset of E, and P be the set of extreme points of K. Then P is a Baire space with respect to the relative topology.

Proof.

We may assume that E is real. For each

Uja = {x

E

KI/(x) < a},

Fja = {x

I

E

E E* and a E lR, let

KI/(x) < a}.

And for x E P, put

t; = {(I, a) I! E E*, a E lR, I(x) < a}. We claim that

{x} = n{Fjal(/, a) E L z }, In fact, if

(I, a)

"Ix E P.

E L z , then x E Uja' Thus we have

On the other hand, if y E E and y f. x, then there is some 10 E E* and some ao E lR such that 10(Y) > an > lo(x). So x E Ujoao and Y ¢ Fjoao' Therefore,

{x} = n{Fjal(/, a) E L z },

"Ix E P.

Now let x E P, and we prove that {Fjal(/, a) E L z } is a basis for the neighborhood system of x in K. Indeed, let U be an open subset of K, and x E U. Since {x} = n{Fjal(/,a) E L z } C U, it follows that (K\U) C U{(K\Fja) 1(/, a) E L z}. By the compactness of (K\U), there exist (11' al),' ", (In, an) E L; such that (K\U) C U~=l(K\Fi), where Fi = Fft,ai71 n~=lFi. Let Ut = Uft,aj' K, = K\Ui , 1 < i < n. Then F Co U~=lKi is a compact subset

=

494

of K. Clearly, x E U, and x f!. K i, VI < i < n. So by x E P, we have x Then we can find IE E* and a E JR such that

f!. F.

I(x) < a < inf{/(Y) Iy E F}. Hence ,(/,0:) E t; and Ffa n F = 0 . Further, (Ula and Ur« CUi, I < i < n. Then we have that

x

E

Ufa

C

Fl a C niUi = niFi

C

n K i)

C

(F fa n F) = 0,

U.

Hence, {Ffal(/, 0:) E L z } is a basis for the neighborhood system of x in K. Now let {Vn } be a sequence of open dense subsets of P, and V be any non-empty open subset of P. We need to prove that n, Vn n V =f. 0. In fact, we have open subsets U, Ub " ' , Um · · · of K such that V = Un P, Vn = Un n P, Vn. We may assume that each U« is dense in K, and

UI

:)

U2

:) ••• ,

VI:) V2

:) ••••

Since V C P, and {Ffal(/, Q) E L z } is a basis for the neighborhood system of x in K, Vx E P, we can suppose that U = Uh,erl for some II E E* and 0:1 E JR. Pick X2 E V n V2 • Since X2 E Uh ,er 1 n U2 , it follows from the preceding paragraph that there is some (/2,0:2) E L Z2 such that X2 E Uh, er 2 C Fh, er 2 C Uh,erl nU2 • Again Pick Xs E Vsn(Uh,a2np), similarly we can find (Is, 0:3) E L Z3 such that Xs E U/a,a3 C F/a,a3 C Uh,a2 nUs, .... Generally, we have {In} C E* and {O:n} C JR such that

Ffn+l,an+l

C

U/n,a n n Un+1 l

U/n,er n n P f=. 0, Vn.

Now {F/n,er,J is a decreasing sequence of non-empty compact subsets of K. By the compactness of K, F - nnFfn.an f=. 0. Clearly, F is compact and convex, and F c nnUn' Noticing that (K\F/ er) = {x E KI/(x) > a} is convex (VI E E., Q E lR) , and the sequence {(K\Ffn.er..lln} is increasing, we can see that (K\F) is also convex. Since FnP c unnnUnnP = VnnnVn, it suffices to show that FnP =f. 0. Indeed, let x be an extreme point of F. If x is extremal in K, we are done. If not , let 6 be a line passing through x and such that x is an interior point of the segment K n 6, we then can show that one of the end-points of F n 6 is an extreme point of K since F and (K\F) are convex. Q.E.D. Let X be a Haire space, Y be a topological space, and T be a continuous open map from X onto Y. Then Y is also a Baire space.

Lemma 13.1.9.

Proof. Let {Vn } be a sequence of open dense subsets of Y, and V be any open subset of Y. We need to show that n, Vn n V f=. 0. Let U = T- 1(V), U« = T- 1(Vn ) , Vn. H W is any open subset of X. Then TW is also open. Hence, TW n Vn f=. 0, \In. Let x E Wand Tx E Vn. Then x E

495

WnT-l(Vn) = WnUn. So WnUn f. 0, and U« is an open dense subset of X, Vn. Now Un nnUn f. 0 since X is Baire. Therefore, V n n,Vn = T(U n nnUn) =1= 0.

Q.E.D.

Lemma 13.1.10. Let A be a C·-algebra, {{7r"H, } ll E I\} be a family of nondegenerate * representations of A, I = n{ker7r,jl E I\}, and P be a state on A with plI = O. Then P belongs to the a(A*, A) -closure of following subset of A* : CO{(7rI(')6, 6)16 E H, and lle,ll = 1,1 E I\}

Proof. Let 7r = $'EA7rh H = $fEAH,. Then {11", H} is a faithful * representation of AI I. By plI = 0, P can be regarded as a state on AI I. Now from Lemma 16.3.6 (it is easy and elementary), p is a u(A·, A)-limit of states which belong to the following subset: Co{ (7r (.) e, e) leE H, II ell = I}.

Q.E.D.

That comes to the conclusion.

Let A be a C*-algebra, P(A) be the pure state space of A, and PI, P2 E P(A).Pl and P2 are said to be unitarily equivalent, and denoted by Pl ,...., P2, if there exists a unitary element u E (A -Hv) such that Pl (a) = P2 (u· au), Va E A.

u..

Proposition 13.1.11. Let A be a C·-algebra, PbP2 E P(A), and {1I"b ell, {7r2' H 2, e2} be the irreducible cyclic * representations of A generated by Pb P2 respectively. Then we have that

Pl ,...., P2

-<==> { 7r b H l } ,...., {7r2' H2} -<==> there is 11 E H l with P2 (a)

=

(7r 1 ( a)1], 1]) ,

111111 = 1 such that

Va EA.

Proof. Let Pl(a) = P2(u·au), Va E A, where u is a unitary element of (A-Hv). Define U7rt{a) 6 = 7r2{au)6, Va E A. Then U can be uniquely extended to a unitary operator from H l onto H 2 and U1I"l(a)U· = 7r2(a), Va E A.

,

Therefore, {1I"b H l } ~ { 7r 2' H 2 } . Conversely, let U be a unitary operator from H l onto H 2 such that U7rl{a)U· = 11"2 (a) , Va E A. Then we have that P2(a) = (7rl{a)1], 1]),Va E A , where 1] = U" 6 E HI, and II'711 = 11611 = 1. Further, we can find a unitary operator

496

Von HI such that V6 = n. By Theorem 2.6.5, there exists a unitary element u of (A-kX') such that 7r1(U) 6 = fJ. Therefore, we get

P2(a)

= (7rdU"'au)€h €I) = pdu"'au),

Va E A.

Q.E.D. Let A be a C*-algebra. Then (P(A),u(A"', A)), A and Prim(A) are Baire spaces.

Theorem 13.1.12.

Proof. Let K = {I E A*II > 0, IIIII < I}. Clearly, K is a compact convex subset of (A\ u(A*, A)). By Lemma 13.18, (ExK, u(A"', A)) is a Baire space. From Proposition 2.5.5, we have that ExK = P(A) U {O}. Now it is easy to see that (P(A), u(A., A)) is a Baire space. ..... By the GNS construction, P ~ 7r p is a surjective map from P(A) to A. We claim that the map p -+ 7r P is continuous. Indeed, by Proposition 13.1.5 any open subset of A has the form of AI, where I is some closed two-sided ideal of A. Then inverse image of AI under that map is as follows:

U = {p E P(A) l7r p E

A?

~

7rp

iI

=1= O}

Let P E U. H for any u(A*, A)-neighborhood U(p, F, 1) = {cp E P(A) IIp(x) cp(x) I < 1, Vx E F} of p in P(A), where F is a finite subset of A, there is PF E U(p,F, 1)\U. Then we have that PF ----t pin u(A" ,A) and 7r pP II = 0, VF. Consequently, PF(axb) = 0, \Ix E I, a, s e A, and F. Thus we obtain that p(axb) = 0, Vx E I, a, b E A, and 7rp iI = 0, a contradiction. Therefore, U is open, and the map p --) 7rp is continuous. ..... Now let U be an open subset of (P(A),u(A*,A)), and V = {7r E AI there is p E U such that 7r """ 1r p } . For each subset E of P(A), put

E=

{cp E P(A)lthere is pEE such that cp ,...;

Clearly , V = {7r E AI there is p E element u of (A -Hr), let

if

pl.

such that 7r ,...; 7r p}. For each unitary

u(U) = {p(u* . u)lp E U}. Clearly, u(U) is open. By Proposition 13.1.11,

fJ =

U{u(U)lu is a unitary element of (A-Hr)}

is also open. Let F = P(A)\U. Then F is closed and F = F. Let I = n{ker7rp lp E F}. H cp E P(A) and ker 7rrp :> I(-<=:> cplI = 0), then by Lemma 13.1.10 and F = F we have that cp E CoY. Since F = FC' and cp is also an

497

extreme point of CoF(7, it follows from the Krein-Milmann theorem ( see [89, Theorem 15.2 ] ) that cp E F. This means that

{7r E AI

there is p E F such that

7r

f',J

7r p}

is a closed subset of A. Therefore, V is an open subset of A, and p -----t 7rp is an open map. Now by Lemma 13.1.9, A is a Baire space. It is well-known that ker : A -----t Prim( A) is a continuous and open surjection. So Prim (A) is also a Baire space from Lemma 13.1.9. Q.E.D.

Proposition 13.1.13. Let A be a C·-algebra, and x E A. Then 111r(x) II is a lower semicontinuous function on A. Proof.

7r

-----t

For any k > 0, we need to show that E

= =

{7r E AIII7r(x)11 < k} = {7r E AIII7r(x·x)11 < k 2 } {7r E Ala(7r(x·x)) C [-k 2 , k 2 ]}

is a closed subset of A. Let a = x·x,L = [-k 2,k 2 j, and 1= n7r'[7r' E E}. Suppose that 7r E E, and there is some). E JR with), E a(7r(a))\L. Pick a continuous function f on JR such that flL = 0 and f().) =I- O. Then we have that 7r'(f(a)) = f(7r'(a)) = 0, V7r' E E, i.e., f(a) E I; and 7r(f(a)) = f(7r(a)) =I- o. Let T = {J E Prim(A)IJ => I}. Then T is a closed subset of Prim(A), and ker- 1(T) is a closed subset of A. Clearly, E c ker- 1(T). Since 7r E E, it follows that 7r E ker- 1(T) and kern => I. Hence, we get f(a) E I c kerx and 7r(f(a)) = 0, a contradiction. Therefore, E must be closed. Q.E.D.

en

Lemma 13.1.14. Let H be a Hilbert space, 6,"" E H, and e > O. Then there exists 0 > 0 with the following property: if 171, •.. ,17n E H such that 1(17i,17i) - ('7i,17i)1 < 0, 1
eill < e,1 0 there is 0 = o(e,e) E (o.e) such that if 17 E H and 111'711 2 - Ilel1 21 < 0, then we can find a unitary operator U on H with IIU'7 - ell < e. Now we assume that for any 6,"" E Hand e > 0 there exists o = o( 6,' .. ,en-l, e) E (0, e) with the following property: if 171,' .. ,'7n-1 E H and 1('7.,17i) - (ei, ei) I < 0,1 < t.i < n - 1 , then there is a unitary operator U on H such that lIU'7i - eill < e,l < i < n-1.

e

en-1

498

Let 6,"" en E Hand e E (0,1). Write H = H' ffi H", where H' = [eb" .,en-I] and H" = (H').l, and ler P, Q be the projections from H onto H',H" respectively, and K = max{lleill11 there exists 6' = 6'(e) E (0, e) with following property: if 11 E H' and 1(11, ei) I < 6',1< i < n -1, then liT/II < e. Pick e'; ttl E (0, t) such that

°

e"(1 + K + (1 + K 2)1/2) < 6(Qcn, e/2), e'(1 + (1 + K 2)I/2) < 6'(e"). Let 6 = 6( el" .. , en-It e'), and 11h ... .n« E H with 1(11i' T/,.) - (€i' e,.)! < 6, 1 < i,i < n. Clearly, II11ill = (1 + K 2)l /2, 1 < i < n. And by the preceding paragraph, we can find a unitary operator U on H such that

II U 11i - €; I < e' < s,

l
Let e~

= Pen,

€~ = QCn,

11~

= PUT/n, 11: = QUT/n.

Noticing that

- T/~, €i) I = I(P(cn - U11n) , Ci) I < I(en, Ci) - (U11n, U11i)! + !(U11n, U11i < 6+ s'{I + K2)l/2 < e'(1 + (1 + K 2)1/2) < 6'(e"), 1 < i < n - 1, II €~ - T/~ 11 < e". Further, I(c~

we have that

ei)]

Ille~112 -ll11~1121

< (II enll + liT/nil) . II c~ - T/~II < e"(K + (1 + K 2)l/2). Thus , we have

Ille:1I 2 -1111:11 2 < IIIU11nlI2 -IICnIl 21 + Illc~112 -11T/~1121 < 6+ e"(K + (1 + K 2)l/2) < g"(1 + K + (1 + K 2)l/2) < 6{C:,e/2). 1

Now from the preceding paragraph, there is a unitary operator V on H" such that IIV11~ - c:11 < e/2. Let V = I on H'. Then V is unitary on H, and

499

Moreover, IIVU1]n -

<

c',2

enl1 2=

111]~

-

e~112

+ Ilv1]: - e:11 2

+ c2 /4 < c2 , Q.E.D.

i.e., IIVU1]n - enll < c.

Lemma 13.1.15. Let A be a C· -algebra, H be a Hilbert space such that every irreducible * representation of A can be realized in a closed subspace of H, and {7ro, H o} be an irreducible * representation of A, where H o is a closed subspace of H. Then for any al, ... ,ap E A, 6, ... , E H o and e > 0, there is a neighborhood V of 7ro in A such that for any rEV we can find an irreducible * representation {7r,H'Ir} of A with 1r""" t .H; cHand

en

VI

< i < p, 1 < i < n,

Proof. i < p.

where P1(' is the projection from H onto H1("

We may assume that A has an identity 1, and al = 1, lIaill < 1,1

1) Suppose that n

=

1 , and

II ell =

1, where

<

e = 6. Consider the open

neighborhood

W = U((7ro(·)e, e), a;a;, 1 < i,j < p,

cd

of (7ro(·)e, e) in P(A), where Cl = o( 7rO(al)e, ... , 7ro( ap)

e, c/2}

and 0(" .) is as in Lemma 12.1.14. From the proof of Theorem 13.1.12, V

= {r

E Althere is pEW such that 7r p

::::::

r}

.....

r, Suppose that {7r p, H p , p } is the irreducible cyclic * representation of A generated by p and C H. Then we have is an open subset of A. For fixed rEV, let pEW be such that 7r p

u,

e

1(7r p(a;a;)ep l

e

p) -

(

7ro(a;a; )e, e)1 <

:::

Cb

i.e.,

1(1rp (a;)ep ,7r p (ai )ep ) VI

< i.i <

-

(7ro(a;)e,7ro(a;)e)1 < CI,

p, By Lemma 13.1.14, there is a unitary operator

that In particular,

U on H such

500

Thus , we have

IIU7rp(a·dPpU-1€ - 7ro(at)€11 IIU7rp(~)PpU-l€

<

- U7rp(at)€pll + IIU7rp(at)€p - 7rO(a.)€lI

< Ilaill·IIPpU-1€ - €pll + c/2 < s,

1

< i < p,

where Pp is the projection from H onto HpNow let 7r(.) = U7r p(·)U- 1 , and H1r = UHpo Then 7r ,. . " 7r p ,. . " T. Since P" = UPpU- 1 , where P" is the projection from H onto H", it follows that

7r(a.)P1r€ = U7r p(ai)U- 1 • UPpU-1€ = U7r p(ai)PpU- 1€,

1 < i < p.

Thus , we obtain that

2) For any n , by Theorem 2.6.5 we can find bl,··· ,bn E A such that

7ro(bi)€ = Ci,

1

< i < n,

where € = 6. From the preceding paragraph 1) , there is a neighborhood V of 7ro in A'" with the following property: for any T E V we can find an irreducible * representation {7r, H,,} such that 7r ,. . " T, H1r C H, and

VI < i < p, 1 < j < n. Then , we have

117r( at}P1r C; - 7ro (ai) €; II <

117r(~)P,,(€j

- 7ro(bj) C) 11

+ 11 7r( ai) P1r7rO (b;) C- 7r( ai)P,,7r(bj )P"cll +117r(a.b;)P,,€ - 7ro(aibj)€11 + II 7ro(aibi)€ - 7ro(ai)€;11

< lIaill·c/2+c/2
Q.E.D.

Let A be a C·-algebra, and x E A+. Then 7r '" tr7r(x) is a lower semicontinuous function on A.

Proposition 13.1.16.

~

Proof. Pick a Hilbert space H such that every irreducible * representation of A can be realized in a closed subspace of H. Let {7ro, H o} be an irreducible

501

* representation of A and H o C

H. If tr1ro(x) > a, then there is an orthogonal normalized family {eh' .. ,en} C H o such that n

L( 1ro(x)e"

;=1

e;) - a = f3 > o. ......

By Lemma 13.1.15, there is a neighborhood V of 1ro in A such that for each T E V we can find an irreducible * representation {1r, H 1r } with 1r ~ T, H 1r C H, and Then we have

Therefore, for each a >0, {1r E Altr1r(xl > a} is an open subset of A , i.e., 1r ~ tr1r(x) is lower semicontinouos on A. Q.E.D.

References.

[27], [46], [73].

13.2. Elementary C*-algebras and CCR ( liminary ) algebras Definition 13.2.1. A C*-algebra A is said to be elementary ,if A is isomorphic to C(H), where H is some Hilbert space.

*

Let A = C(H) be an elementary C*-algebra. 1) Each positive linear functional on A has the following form: A.(·e., e.)

Proposition 13.2.2.

L i

,where {ei} is an orthogonal normalized sequence of H; Ai 00.

> 0, Vi; and L Ai < i

2) Each pure state on A must be the following form: (·e, e), where e E H and Ilell = 1. 3) # A = 1. 4) A is simple , i.e., if I is a closed two-sided ideal of A , then I must be either {o} or A. 5) If B is a C*-subalgebra of A , and B is irreducible on H, then B = A.

502

6) If B = C(H') is

* isomorphic to A, then there exists a unitary operator

U from H onto H' such that U* BU = A.

Proof. 1) It is immediate from C(H)* = T(H). 2) The conclusion is obvious by 1). 3) Let p(.) = (·e, e),a(.) = (·TJ,TJ) be two pure states on A, where e,TJ E H and II ell = IITJ II = 1. Since A is irreducible on H, from Theorem 2.6.5 there is a unitary operator u E (A+ I}. By 3) , Prim(A) = {Ole Therefore, 1= {Ole 6) Let ~ be a * isomorphism from A = C(H) onto B = C(H') . Then {id, H} and {~, H'} are two irreducible * representations of A. By 3) , ~ must be spatial. 5) Let B be a C*-subalgebra of A = C(H), and B be irreducible on H. Then B is weakly dense in B(H). If t E T(H) = C(H)* is such that tr(tb) = 0, Vb E B, then we also have tr(tb) = 0, Vb E B(H). Since T(H)* = B(H) , it Q.E.D. follows that t = o. Therefore, B = A. I"V

Now we consider C*-algebras of compact operators, and the C"-subalgebras of an elementary C"-algebra.

Proposition 13.2.3. Let A be a C*-subalgebra of C(H), where H is some Hilbert space. 1) If p is a non-zero projection of A, then the rank of p is finite, i.e., dim pH < 00 j and p is an orthogonal sum of minimal projections of A. Moreover, p is minimal in A if and only if pAp =
e

Proof.

Since each slef-adjoint element of A has the form of

2:: AiPi, where i

{Pi} is an orhtogonal sequence of projections on H j Ai =f. A,., Vi =f. i j and Ai ---t 0, it follows that A = [Proj(A)]. Now if p E Proj (A), then we also have pAp = [Proj (pAp)]. Hence , p is minimal in A if and only if pAp =
503

where pt· sp = >"P and >.. E (C ( since P is minimal) . Thus b = (be, e)l K , i.e., A is irreducible on K. Further, from Proposition 13.2.2 we have AIK = C(K).

Q.E.D. Let {Ad' E Il} be a family of C·-algebras. lim e, I

=

0, where al E AI,

VI E Il, means that for any e > 0 there is a finite subset Fe of Il such that Iladl < s, VI(/. Fe. The following set

{(a')IEElal E AI, VI E Ilj and lim a, = O}, I

denoted by

L

Al, is called the direct sum of {Adl E Il}. Clearly, it can

lEI

become a C·-algebra with respect to the norm

II(a,)11 = sup Ilalll

and natural

I

operations.

Proposition 13.2.4. Any C·-algebra of compact operators is to the direct sum of a family of elementary C·-algebras.

* isomorphic

Proof. Let A be a nondegenerate C*-subalgebra of C(H) on a Hilbert space H. By Proposition 13.2.3 and the Zorn lemma, we can write H = mH"

L

lEA

where AH, C Hi, and AIH, = C(H,), VI E 1\. For each I E 1\, let 1r1(a) = aIH Va E A. Then {1rh HI} is an irreducible * representation of A. Pick a subset "Il of 1\ such that = for any different I, I' E Il, the representations {1rl, H,} and {1rI', H,,} are not unitarily equivalent; and for any I E 1\ there is l' E Il such that {1rI' H,} ::: {1fp, HI' }. Since a E A is compact, it follows that lim1fl(a) = O. Then by the property lEE of Il we can see that a ----+ (1rI(a))'EE is a * isomorphism from A into C(Ht ). lEE Now it suffices to show that

L

{( 1rI(a))'EEla E A} =

L C(H,).

lEE

For each I E Il, let PI be the projection from H onto [A'Hz]. Clearly, PI E A' n A" . Pick a net {tal C A such that Iitall < Ilpzll, Va, and t a ~ Pl( strongly) . Then for each projection P of A, we have I){ta - PI)pll ~ 0 since dim pH < 00. Hence, PIP E A, Vp E Proj(A). By Proposition 13.2.3, we get PIA C A and ApI C A, Vl E Il. Let q, be the projection from H onto H" Vl. Then ql E A', VI E Il. By Propositions 2.10.7, 2.10.4 and 1.5.9, we have qIA'q" = {O}, Vl,l' EIland I ;:j:. I'. Thus, A'H, 1- A'HI', or PIPI' = 0, VI, l' EIland I ;:j:. l' . Now let (b,),EE E C(H,). For each I E Il, pick a, E A such that 1r1(al) = b, lEE and lIadl < 2l1bdl. Then, lim Ilazil = O. By the properties of {pzlp E Il}, we can lEE

L

504

define an element a =

L

a,P, of A. Clearly, 1rl(a)

=

alpl

lEE b" Vi E JI, i.e., (1rl(a))'EE = (b,hEE.

=

a,lp,

=

7I'"l(a,) = Q.E.D.

Definition 13.2.5. A C·-algebra A is said to be CCR (or liminary ) , if for each irreducible * representation {7I'",H} of A and a E A,7I'"(a) E C(H). From Propositions 13.2.2, 13.2.4 and 2.3.10, we can see that any elementary C·-algebra, any C·-algebra of compact operators, and any abelian C·-algebra are CCR.

Lemma 13.2.6. Let A be a C·-algebra, and {1r, H}, {1r' ,H'} be two nonzero irreducible * representations of A. 1) If 1r(A) n C(H) "# {O}, then 1r(A) :::> C(H). 2) If 1r( A) :::> C(H), 1r' (A) :::> C(H'), and kerx = kerer', then {1r, H} ,...; {1r', H'}. 3) If 1r(A) c C(H), then 1r(A) = C(H), and kerx is a maximal closed two-sided ideal of A. Proof. 1) Let I = {x E AI1r(x) E C(H)}. Clearly, I is a closed two-sided ideal of A, and 1r11 "# {O}. Thus, 1r(I) is also irreducible on H. By Proposition 13.2.2. 5) , we have that 1r(A) :::> 1r(I) = C(H). 2) Let I be as in 1). Then (I + kerx] is also a closed two-sided ideal of A, and (I + kersr] "# kersr. Thus, 1r'(1 + kerrr] = 1r'(I) "# {O}, and 71'"'(1) is irreducible on H' . Define a * representation, {p, H'} of C(H) as follows:

p(t)

=

1r'(1r- 1 (t)) ,

Vt E C(H).

Since p(C(H)) = 1r'(I), {p, H'} is irreducible. By Proposition 13.2.2, there is a unitary operator U from H onto H' such that

UtU·

=

p(t)

=

1r'(1r- 1(t)) ,

Vt E C(H).

Then we have that U1r(x)U· = 1r'(x), 'Ix E I. Let {u a } be an approximate identity for I. Then 1r( u a ) ~ IH( strongly) and 1r'(u a ) ...-+ IH' (strongly) . Since U1r(a)U- 1,U1r(u a)U- 1 = U1r(au a)U- 1 = 1r'(au a )) = 1r' (a) 1r'(u a ), Vex, it follows that U1r(a)U- 1 = 1r'(a) , Va E A, i.e, {1r, H} :: {1r', H'}. 3) From 1) we have 1r(A) = C(H). Now let {p, K} be an irreduable * representation of A such that kerp :::> kerer. Then {p, K} can be regarded as a * representation of A/ker1r ,...; C(H). Since C(H) is simple, p is a faithful * representation of A/ker1r, i.e., kerer = kerp. By Theorem 2.7.6, kersr must be maximal. Q.E.D.

505

Let A be a CCR algebra. Then: 1) Prim(A) = {JIJ is a maximal closed two-sided ideal of A}; 2) for each J E Prim(A) , AI J is a elementary C*~algebra; 3) A is a T1-space; 4) ker: A ~ Prim(A) is a homeomorphism.

Proposition 13.2.7.

Proof. By Theorem 2.7.6, each maximal closed two-sided ideal of A must be primitive. Conversely, if {n, H} is an irreducible * representation of A, then by Lemma 13.2.6 we have 1r(A) = C(H), and kersr is a maximal closed two-sided ideal of A. Thus, the conclusions 1) and 2) are obvious. If 7r and 7r' are two irreducible * representations of A and kerzr = kerer' , then by Lemma 13.2.6 we have 1f ,...., 7r' . From Proposition 13.1.6, ker : A ~ Prim(A) is a homeomorphism. Moreover, from 1) Prim (A) is a T1-space Q.E.D. obviously. Proposition 13.2.8. Let A be a CCR algebra, B be a C· -subalgebra of A, and I be a closed two-sided ideal of A. Then B and AI I are also CCR. Let {1f, H} be an irreducible * representation of AI I , and define 1r(a) = 1f(a), where a = a + I, Va E A. Then {1r, H} is an irreducible * representation of A, and 1f(a) = 1r(a) E C(H), va E AI I and a E a. SO AI I is CCR. Let p be a pure state on B, and {7rp, Hp} be the irreducible * representation of B generated by p. p can be extended to a pure state on A, which is still denoted by p, Let {1f p, H p} be the irreducible * representation of A generated by p. Then we have that H, C H p,1f p(b)Hp C H p and 1f p(b) IH p = 1f p(b), Vb E B. Since 1f p{b) E C{H p), it follows that 7r'(b) E C(H p), Vb E B. Therefore, B is also CCR. Q.E.D. Proof.

Proposition 13.2.9. Let A be a CCR algebra, {7rt, HI} and {1r2' H 2 } be two irreducible * representations of A which are not unitarily equivalent, and t, E C(Hi),i = 1,2. Then there exists a E A such that 1fi(a) = ti,i = 1,2. Proof. Let I, = kerer., i = 1,2. By Proposition 13.2.7, we have II i= 12 • Thus , II + 12 i= II. But II is maximal, it follows that II + 12 = A. Since 1ri(A) = C(Hi ), we can find ai E A such that 7ri(ai) = ti,i = 1,2. Now write

where aij E I j , 1

< i,j < 2, and let

a

= au

+ a2l.

Then we obtain that

506

and Q.E.D. Proposition 13.2.10.

I

= {x

Let A be a C"-algebra. Then E

AI for any 11" E..4,

1r(x) E C(HlI")}

is the largest CCR closed two-sided ideal of A.

Proof. Clearly, I is a closed two-sided ideal of A. Let {p, H} be any irreducible * representation of I. Then {p, H} can be extended to an irreducible * repsentation {7r, H} of A. Hence, p(x) = 7r(x) E C(H), \/x E I, and I is CCR. Now let J be a CCR closed two-sided ideal of A, and {1r, H} be any irreducible * representation of A. If 7rIJ = 0, then 1I"(J) = {o} c C(H) obviously. If 1rIJ i- 0, then {1rIJ, H} is also an irreduible * representation of J. Since J is CCR, it follows that 1I"(x) E C(H), \/x E J. By the definition of I, we have that J C I. Q.E.D. Notes. CCR (completely eoniinouous representations ) algebras were introduced by 1. Kaplansky. References. [10], [27], [84].

13.3. GCR ( postliminary) algebras and NGCR ( antiliminary ) algebras Definition 13.3.1. A C" -algebra A is said to be GOR ( or postliminary ) , if for any closed two-sided ideal I of A, AI I contains a non-zero CCR closed two-sided ideal. Clearly, a CCR algebra must be GCR{a GCR algebra contains a non-zero CCR closed two-sided ideal. A C"-algebra A is said to be NGCR (or antiliminary ) , if A contains no non-zero CCR closed two-sided ideal. Clearly, a NGCR algebra also contains no non-zero GCR closed two-sided ideal. Definition 13.3.2. Let A be a C" -algebra. A strictly increasing family {Ia } of closed two-sided ideals of A indexed by a segment {o < 0: < tJ} of the

507

ordinals is called a composition series for A, if 10 limit ordinal ,( < f3) we have

= {O}, Ip = A; and for each

(norm closure).

Proposition 13.3.3. Let A be a C* -algebra. Then there exists unique strictly increasing family {lalO < a < f3} of closed two-sided ideals of A satisfying: 1) 10 = {O}; and Allp is NGCR ; 2) for each limit ordinal ,( < f3) we have

IT = (Ua<..,la)- ; 3) if a < f3, then la+l/1a is the largest ( non-zero) CCR closed two-sided ideal of Alla'

Proof. The family {la} can be constructed by induction: if a is a limit ordinal, the condition 2) defines la; otherwise, we have a = a' + 1 , and pick L; such that lalla' is the maximal ( non-zero) CCR closed two-sided ideal of AIla' ( if AIla' is NGCR, then let f3 = d). Q.E.D.

Proposition 13.3.4. Let A be a C*-algebra. Then the following statements are equivalent: 1) A is GCR; 2) There is a composition series {lalO < a < f3} for A such that la+l/1a is GCR, Va < f3; 3) There is a composition series {la 10 < a < f3} for A such that la+l1la is CCR , Va < f3. Proof. 1) ~ 3) . Pick {lalO < a < f3} as in Proposition 13.3.3. Then Allp is NGCR. But A is GCR, so 1{3 must be equal to A, i.e., {lalO < a < f3} is a composition series for A. Moreover, by Proposition 12.3.3, la+llIa is CCR, Va < 13. 3) ~ 2) . It is obvious. 2) =:} 1) . Let J be a proper closed two-sided ideal of A. It suffices to show that AI J contains a non-zero CCR closed two-sided ideal. Suppose that a is the minimal ordinal such that la ¢.. J. Then la' C J, Va' < a. If a is a limit ordinal, then we have la = (Ua'
508

Proposition 13.3.5. Let A be a GCR algebra, B be a C·-subalgebra of A, and J be a closed two-sided ideal of A. Then Band A/ J are also GCR. Proof. By Definition 13.3.1, A/J is GCR obviously. Now let {IaIO < a < p} be a composition series for A such that I a +1/Ia is CCR, Va < p. For any a( < P), [(Ia+ 1 n B) + 101]/101 is a C·-subalgebra of I a+1/Ia. Since I a +1/Ia is CCR, it follows from Proposition 13.2.8 that [(/01+ 1 n B) + 101]/101 is also CCR. But (Ia +1 n B)/(Ia n B) is * isomorphic to [(/01+ 1 n B) + 1011/ t; so (I a + 1 n B) /(1 01 n B) is CCR , Va < p. Clearly, 10 n B = {a} , and I p n B = An B = B. From Proposition 13.3.4, it suffices to show that {Ia n BIO < a < P} is a composition series for B. By Definition 13.3.2 we need to prove that

where I is a limit ordinal and I < p. Clearly, the left side contains the right side. Conversely, let x E L, n B, and e > 0. Since 1"1 = (Ua<-yla))-, there is a < I such that dist (la, x) < e. But B /(IanB) is * isomorphic to (B +Ia)/ la, and IIxll < e , where x = x + I a is an element of (B + I a ) / la, hence we have dist (Ia n B,x) < c. Therefore, 1"1 n B = (Ua<-y(Ia n B))- . Q.E.D.

Proposition 13.3.6. Let A be a C·-algebra, and {Ia 10 < a < P} be as in Proposition 13.3.3. Then I p is the largest GCR closed two-sided ideal of A, and I p is also the smallest closed two-sided ideal of A such that A/lp is NGCR. Proof. Clearly, {IaIO'< a < P} is a composition series for I p. From Proposition 13.3.3 and 13.3.4, I p is GCR. Let J be any GCR closed two-sided ideal of A. Then J /(J n I~) is also GCR. Since J /(J n I~) is * isomorphic to (J + Ip)/Ip, (J + I~)/Ip is a GCR closed two-sided ideal oIA/lp. But A/lp is NGCR, it must be J + I~ = I p, i.e., J c I p. Now let I be a closed two-sided ideal of A such that A/lis NGCR. Since Ip/(Ip n I) is GCR, and is * isomorphic to (Ip + 1)/1, it follows that (Ip + 1)/I = {a}, i.e. I~ c I. Q.E.D.

Proposition 13.3.7. Let A be a GCR algebra. Then: 1) for any irreducible * representation {1f,H} of A, we have 1f(A):> C(H); 2) ker: A -----+ Prim(A) is a homeomophism; 3) A is a To-space. Proof. J / ker

1f

1) Let J be a closed two-sided ideal of A, and ker 1r C J such that is a non-zero CCR closed two-sided ideal of A/ ker 1r. Then {1r, H}

509

can be regarded as an irreducible

1r(A) :> 1r(J)

* representation of J I kerrr. Hence, we have =

7f(Jlker1r)

= C(H).

2) and 3) are obvious from 1) , Lemma 13.2.6 and Proposition 13.1.6.

Q.E.D. Lemma 13.3.8. Let X be a Baire space, and 1 be a non-negative lower semicontinuous function on X. Then there exists Xo E X such that 1 is continous at xo. Proof. Replacing For each x E X, let

1 by 1(1 + /)-1, we may assume that I(x) < 1, \Ix E X.

w(x) = inf {sup I(y) - inf I(y) I V is a neighborhood of x} . I/EV

Clearly,

1 is continuous

I/EV

at x

r;

-¢::::::}

=

w(x)

= O. For each

n , put

1

{x E X]w(x) > -}. n

We claim that Fn is closed, \In. In fact, if x E Fn, then for any open neighborhood V of x there exists z E F n n V. Since V is also a neighborhood of z, it follows that [ SUP I/EV

I(Y) - inf I(Y)] I/EV

> inf { [sup I(Y) - inf I(Y)] I/EW

= w(z) >

I/EW

IW is open, and z E W}

~.

Thus, w(x) > lIn, and x'E F n . Suppose that the interior U of Fn is non-empty for some n, and a = sup{/(x) Ix E U}. Pick Xo E U such that I(xo) > a - 2~' Since 1 is lower semicontinuous, there is a neighborhood V( C U) of Xo such that I(Y} > a - 2~' vv E V. Of course, I(Y} < a, Vy E V. Thus, we have that w(xo} < 2~' This contradicts the fact of Xo E U c Fn. Therefore, (X\Fn) is open and dense in X, \In. Since X is Baire, we can pick Xo E nn(X\Fn}. Clearly, w(xo} = 0, i.e., 1 is continuous at xo. Q.E.D.

Lemma 13.3.9.

Let A be a CCR algebra. Then

J = {x E Althe rank of 7f(x} is finite.Vsr E is a

* dense two-sided ideal of A.

A}

510

Clearly, J is a * two-sided ideal of A. Now it suffices to show that h E J for any h: = h E A with Ilhll < 1. Pick a sequence {fn} of continuous functions on lR such that fn(t) = if It I < l/n; and fn(t) ~ t uniformly for '" since 1r(h) is a completely continouos symmetric t E [-1,1]. For any 1r E A, operator, the rank of 1r(fn(h)) = fn{1r(h)) is finite, Vn. Hence, fn{h) E J, Vn. Clearly, fn(h) ~ h. Therefore, we have hE J. Q.E.D.

Proof.

°

Lemma 13.3.10. Let A be a C·-algebra, {1ro, H o} be an irreducible * representation of A, and x E A+. Suppose that the function 1r ~ tr1r(x)(V1r E A) is finite and continuous at 1ro. 1) If y E A+ and y < x, then tr1r(Y) is also continuous at 1fo. 2) Let 1fo(x) =1= 0. Then there is some Z E A+ and a neighborhood V of 1fo in A such that the rank of 1f(z) is one, V1f E V.

Proof. 1) Let Z = x - y(E A+). By Proposition 13.1.16, tr1r{x),tr1r{Y) and tr1r(z) are non-negative lower semicontinuous functions on A. Put a = tr1fo(X)(E [O,+oo)),al = tr1fO(y),a2 = tr1fo(z). Then al,a2 > and a = al +a2. For any e > 0, since tr1f(x) is continuous at 1ro, there is a neighborhood

°

VI of 1ro in

A such that a + e/3 > tr1r(x) > a -

e,

V1r E VI.

Clearly

and

'" e {1f E AI tr1f(z) > a2 - -} 3 are open subsets containing 1ro. Then V = VI n V2 n Vg is a neighborhood of 1ro in A. If 1r E V is such that tr1r(Y) > al + ~e, then Vg

tr1f{x)

=

= tr1f(y) + tr1f(z) >

al + a2 + e/3

= a + e/3,

a contradiction. Thus, we have that

2 V1f E V. 3 Since e is arbitrary, tr1f(Y) is continuous at 1ro. 2) We may assume that II1ro(x) II = 1. Since the operator 1fo(x) is trace class, 1 is an eigenvalue of 1fo(x), Further, there is an one rank projection p on H o such that 1fo(x)p = p. From Lemma 13.2.6, we have C(Ho) C 1ro(A). So we al

can find

ZI

E

2

+ -e > tr1f(Y) > al 3

A+ such that

1rO(ZI)

= p. t,

f(t) =

0,

{ 1,

- -e,

Let if t E [0, 1J, if t < 0, if t > 1,

511

Then Z2 E A+, IIz211 < 1, and 1rO(Z2) = f{p) = p. Further, let Z3 = XI/2Z2XI/2. Then 0 < Z3 < x, and 1fO{Z3) = 1rO{x)l/2p1rO(X)I/2 = p. From the preceding paragraph, tr1r(zs) is continuous at 1ro. So there is a neighborhood VI of 1ro in A such that tr1f(zs) < 5/4, V1r E VI' By Proposition 13.1.13, we also have a neighborhood V2 of 1ro in A such that 111r(zs) II > 3/4, V1r E V2. Let V = VI n V2. Then and Z2

= /(ZI)'

111r(zs)ll > 4/3,

and

tnr(zs) < 5/4,

V1f E V.

Since 1r(zs) is non-negative and trace class, 1f(zs) has only one eigenvalue A,... with multiplicity 1 and A,... > 3/4, and other eigenvalues of 1r(zs) belong to [0,1/2), V1r E V. Now let z = g(zs), where

t) _ ( 9 -

{O,1,

if t < 1/2, ift>3/4,

and 9 is continuous on JR. Then 0 < Z < zs, and 1f(z) = g(1f(zs)) is the spectral projection of one rank corresponding to the eigenvalue A,... of 1r(Zs) , V1f E V. Q.E.D.

Proposition 13.3.11. Let A be a C·-algebra which is not NGCR. Then there exists a ::j:. x E A+ such that the rank of 1f(x) is either 0 or 1 , V1r E A. Proof. Since A contains a non-zero CCR closed two-sided ideal, by Lemma 13.3.9 we can find 0 ::j:. y E A+ such that the rank of 1f(Y) is finite, V1r E A. Now tr1r(Y) is non-negative finite lower semicontinuous function on A. By y =I 0, {1f E Ajtr1r(y) > O} = U is a non-empty open subset of A. From Proposition 13.1.5 and Theorem 13.1.12, U is also a Baire space. From Lemma 13.3.8, tr1r(Y) will be continuous at some 1ro E U. Now by Lemma 13.3.10, there is some Z E A+ and some neighborhood Al of 1fo, where I is a closed two-sided ideal of A, such that the rank of 1r{z) is one, V1r E AI. We claim that Iz =I {O} . Otherwise, let J = [AzA]. Then J n I = [IJ] = {O}. From Proposition 13.1.5, it follows that AJ n Al = 0. Thus, we get 1r(z) = 0, V1f E Al , a contradiction. Now let wz =I 0 for some wEI, and x = Z·W·WZ. Then a =I x E A+ n I, and clearly the rank of 1f(x) is either 0 or 1 , V1f E AI. Further, for any 1f E A the rank of 1r(x) is also either 0 or 1 . Q.E.D.

Proposition 13.3.12.

Let A be a C·-algebra. Then A is NGCR if and only if A satiesfies the Glimm condition, i.e., for any 0 ::j:. x E A+, there is an irreducible * representation {1r, H} of A such that dim 1f(x)H > 2.

Proof. The sufficiency is obvious from Proposition 13.3.11. Now let A be NGCR. If there exists 0 =I x E A+ such that the rank of 1r(x) is either 0 or 1

512

, V1r E

A:, let

J = [AxA], then J is a non-zero CCR closed two-sided ideal of A. This is impossible since A is NGCR. Therefore, A must satisfy the Glimm condition. Q.E.D. Reference.

[23], [47], [55], [84].

13.4. The existence of type (III) factorial tions of a NGCR algebra

* representa-

Proposition 13.4.1. Let A be a NGCR algebra. 1) For any 0 -=I- h = h * E A, there is an irreducible * representation {1r, H} of A such that dim 1r(h)H > 2. 2) For any 0 -=I- a E A and a*a = aa*, there is an irreducible * representation {1r, H} of A such that dim 1r(a)H > 2. 3) H A has no identity, (A-to:) is also NGCR.

Proof. 1) Let h = h+-h_, where h+ and h.: E A+, and h.; . h : = O. We may assume that h + -=I- O. Then by Proposition 13.3.12, there is an irreducible * representation {1r, H} such that dim 1r (h+) H > 2. Pick TJl, TJ2, 6, 6 E H such that (ei, e;) = Oi;' and 1r(h+)TJi = ei, 1 < i,j < 2. If >",J-L E (C are such that 1r(h)TJ = 0, where TJ = >"TJl + J-LTJ2, then 0 = 1r(h+)1r(h)TJ = 1r(h+)2 TJ , 0 = (1r(h+)2 TJ, TJ) = 1I 1r (h+ )TJ II 2, i.e., >"6 + J-L6 = 0, so >.. = J-L = O. Thus, dim 1r(h)H > dim[1r(h)TJili = 1,2] = 2. 2) Write a = hI + ih 2 , where hi = hi E A, i = 1,2, and h 1h2 = h 2h 1 • We may assume that hi -=I- O. By 1), there is an irreducible * representation {1r , H} of A such that dim 1r(ht}H > 2. Pick TJ1, 112,6, e2 E H such that (ei, e;) = Oi;,

and

If >",J-L E (C are such that 1r(a)TJ

= 0,

1r(h 1)TJi = ei, where TJ

1

< i.i < 2.

= >"TJ1 +J-LTJ2,

then

o = (1r(a)TJ, 1r(h 1)TJ) = 111r(h 1)TJ I12 + i{1r(h 2)TJ, 1r(h1)TJ). Clearly, (1r(h l)TJ,1r(h 2)TJ) E JR. So we have 0 = 1r(ht}11 = >..el + J-Le2' and >.. = J-L = O. Thus, dim 1r(a)H >dim[1r(a)TJili = 1,2] = 2. 3) Let I be a non-zero CCR closed two-sided ideal of (A-t-(C). Clearly, it must be that A n I = {o). Thus, we have I = [e - 1] for some e E A. Since AI = I A = (o), e is an identity of A, a contradiction. Therefore, (A-to:) is also NGCR. Q.E.D.

513

Lemma 13.4.2. Let A be a NGCR algebra with an identity 1, d E A+ with Ildll = 1, and t E (0,1]. Then we can find w,w',d' E A such that 1) Ilwll = Ilw'lI = 1Id'11 = 1, W > 0, d' > 0, and w"w = OJ 2) It(d)w = w, It{d)w' = w'; 3) w2 tl = d', w"w'd' = d', where if r < 1 - t, 0, It (r) = affine, if 1 - t < r < 1 - ~, (0 < t < 1) { 1, if r > 1 - ~ Let s = tiS, pick u, c E A with lIull < 1,0 < c < 1, and put do = 12~(d)c/2~(d), d1 = 141J(d) - do. Clearly, 0 < do < 1, -1 < d1 < 1. Since 141J/21J = 121J' if follows that 141J(d)do = do = 141J(d). Then {1,d o,146(d)} can generate an abelian C*-subalgebra B of A. Let B ,..., C(n). By /46(d)do = do, we have p(f.b(d)) = 1, p(dd = 1 - p(do),

Proof.

Vp E nand p(do) =j:. O. Hence, if 9 : 1R ~ 1R is continuous and 9 vanishes on [0, !], then by 0 < p(do) < 1 we have p(g(do)g(dd) = g(p(do))g(p(d1)) = 0, Vp E n. It follows that g(ddg(do) = 0, in particular, /21J(dd/21J(do) = O. Let v = IIJ(d1)ullJ(do). Then we have

Hvll = and

v*v = IIJ(do)u* IIJ(dd 2UIIJ(do).

1,

12~(do)v*v =

v*v = V*V/21J(d o),

e" /2~(dl)

=

v",

v*(v·v) = V*/21J(dd/21J(d o)v*v = O. Furthermore, 181J(d)do = do and 181J(d)d1 = dI, hence 181J(d)p(do) = p(do) ,

181J(d)p(d 1 ) = p(d1)

if p is a polynomial with no constant term, hence also if p is a continuous function vanishing at 0 . In particular 181J(d)v = v, and /81J(d)v· = e". Finally, put

d' = /!(v*v), 4.

W

= I! (V*V)1/2, 2

W'

= vk(v*v),

where k : 1R ~ 1R is the function which is equal to (I! (t)t- 1)l/2 if t =j:. 0, and '2 to 0 if t = O. Clearly, 0 < d' < 1,0 < w < 1. Since v*(v*v) = 0 and 11/2(0) = 0, it follows that w'*w = O. Since 188(d)v = v, 181J(d)v* = v*, we have

/81J(d)v*v

= v*v,

18IJ(d)w

= w,

181J(d)w'

= w'.

That comes to 2) . Further,

w2 = h/2(V*V),

w"w'

= k2(v*v)v*v =

/1/2(V*V)

514

and w

2d'

= /t/2(V"'V)/t/4(V"'V) = d'

and similarly w""w' = d'. That comes to 3) . If we can choose u, c such that Ild'lI > 1, then it must be Ild'll = 1. Since w 2 > d' and 1 > w""w' > d', it follows that Ilwl! = Ilw'll = 1. That comes to 1) , and the proof will be completed. Now it suffices to find u, c such that IId'll > 1. By the Glimm condition, there is an irreducible * representation {1I",H} of A such that dim 1I"(/,,(d))H > 2. Pick e,TJ E 1I"(/,,(d))H with Ilell = IITJII = 1 and (e,TJ) = o. By Theorem 2.6.5, there exists h'" = h E A such that

1I"(h)e = e,

1I"(h)TJ = O.

Let 9 : JR ---+ JR be the function which is equal to 0 if r < 0, to 1 if r > 1, and to affine on [0,1]. Pick c = g(h). Then 0 < c < 1 and 1I"(c)e = e,1I"(c)TJ = o. By Theorem 2.6.5, there is also a unitary element U of A such that

1I"(U)e =TJ. Since

/21J/"

=

/IJ and e,TJ E 1I"(/,,(d))H, it follows that 1I"(/2,,(d))e = e,

1I"(/2,,(d))11 = TJ·

Similarly, 1I"(/4,,(d))TJ = TJ. Hence, we have

1I"(do)e

= 1I"(/2,,(d))1I"(c)1I"(/2,,(d))e = e,

1I"(do)11

= 0,

= 11"(/4" (d))11 - 11"( do)TJ = 11, 1I"(v"'v)e = 1I"(/,,(do))1I"(u)*1I" (/ ,, (dt ))211" (u)1I" (/ IJ (don e = e, 11"( d1)TJ

= 1I"(ft/4(V"'V))e = /1/4(I)e = e· lid' I > II 11"(d') II > 1. 1I"(d')e

Therefore,

Q.E.D.

Proposition 13.4.3. Let A be a NGCR algebra with an identity 1. Then there exist non-zero elements v(ah"" an) and b(n) in the unit ball of A, where at,"', an E {O, I}, n = 0,1,2,"', with the following properties: 1) if j < k and (at,"" ail t- UJ1 , " ' , (Ji), then

v(at,"',aj)"'v({Jl,''',{Jk) = 0; 2) if k > 1, then v( at,· .. , ak) = v( at, ... , ak-l)v(Ok-b ak); 3) if j < k , then

4) v(0) = 1, V(Ok) > 0;

515

5} v(a1"'" an)*V(ab"" an)b(n}

= b(n), b(n) >

0, and IIb(n} II

=

1, n

=

0,1,2, .... Proof. For n = 0 , put v(0) = b(O) = 1. Now suppose that non-zero elements v(a1' ... , a;) in the unit ball of A and b(j) of norm 1 in A+ have been constructed for j < n and they satisfy these properties. Using Lemma 13.4.2 to d = b(n}, we get w, w' and d', then let

v(On+tl

= w,

V(On, 1) = w',

and

b(n + 1}

= d'.

By Lemma 13.4.2, we have

v(ab"" an)*v(ab···' an}v(On' an+d v( ab· .. ,an) *v( at,' .. ,an} ft (b(n))v (On' an+d ft(b(n))v(On' a n+l)

=

v(On' an+tl.

= v(On)*v(On)b(n} = v(On)2b(n), it follows that v(On)2b(n} = b(n) v(On)2,b(n)v(On) = v(On)b(n), and b(n)2 = (v(On)b(n))2. Hence, b(n)

Since b(n)

v(On)b(n) and v(On)v(On, an+d Then for j

= v(On)ft(b(n))v(Om an+d = v(Om a n+1)'

< n we get v(al,···, aj)+v(ab···' aj)v(On, an+d v(al,···, a;)*v(al,"" aj)V(On)V(On, an+l) v(On)v(On, an+d = v(On' an+d.

Thus, v(On, an+d satisfies the conditions 2) , 3) , 4) . Again by Lemma 13.4.2, v(On,an+1) and b(n+ 1) satisfy the condition 5) ,and v(OmO)*v(On, l ) = 0.1f j < nand (ab···' ail 1= (OJ), then

v(al,···, aj)*v(On, a n+1) = v(aI,···, aj)*V(On)V(Onl an+d = O. So v(On' an+tl and b(n + 1) satisfy all conditions 1) - 5). Now let v(al,· .. ,an+l) = v(al,· .. ,an)v(On' a n+l). Clearly, the conditions 2), 3), 4) hold for (n + 1). If (all' . " an+tl #- (fJb··· ,Pn+tl, then

v(al, .. " an+l)*v(Pl'···' Pn+l) V(On' an+tl*v (ab ... , a n)*v(fJl' ... , Pn) v(On' Pn+d 0, {

if (at, ... , an)

#- (fJI,' .. ,Pn)

v(Om an+l)+v( ab· .. ,an)~vCal," . ,an)v(O", Pn+l), otherwise, 0, If (ab ... ,a~) #- tPb ... , Pn) { v(On' a n+l)+V(On, .Bn+d, otherwise

o.

516

When j < n and (ah" . , a;) =I- (Pb ... , f3;), it is obvious that

= v(al, ... , a; )*v(fJl, ... , fJn)V(On, bn +l ) = O. Thus, the condition 1) holds for (n + 1). Finally, if (al,"" an) =I- (On), then v(at,···, an+I)*v(al,"', an+l)b(n + 1) v(On, an+ltv(al,"" antv(ab"', an)v(On, an+t)b(n + 1) v(On, an+d*v(On, an+db(n + 1) = b(n + 1). So the condition 5) holds also for (n + 1). v(ai, ... , a;)*v (Pl, ... ,fJn+l)

Q.E.D.

By induction, we can complete the proof.

Denote the 2n X 2n matrix algebra by B n • Then B n contains an orthogonal family {p(al,"" an)la; = 0 or 1,1 < i < n} of minimal projections. Also there are partial isometries w( at, ... ,an) such that

w(at, . . . , an)*w(at,' .. , an) = p(On), w(al,"" an)w(al"'" an)· = p(a~, ... , an), { w(On) = p(On) , Va; E {a, I}, 1 < l < n. Clearly, if (ab ... , an) =I- (131,"', fJn), then w· (ah ... , an).w(1317 .•. ,fJn) = O. Hence n

w(al,"" an)*w(fJt," . ,fJn)

=

II 6 .,.8.p(On), a

i=l and

w( at, ... , a n)w(f3b ... ,fJntw( a~, ... , a~)w(fJ~, ... , P~)*

(.=tfI 6.8i,a~) (fI 6.8.,a~) 1=1

Vai,fJi,a~,fJ: E {0,1},1

w( ab' .. , an)p(On)w(fJi,' .. , P~)* w(al,' .. , an)w(fJ~, ... , fJ~)*,

< i < n.

Now let A be a NGCR algebra on some Hilbert space H, and 1 = IH E A. Pick v(at,'" ,an),b(n) as in Proposition 13.4.3, and put

en =

L

v(al, ... ,an)v(al,···,an)*,

a.E{O,l} l~i~..

and H(n) = [e(n)H], Vn. We say that H(n) => H(n+l), Vn. In fact, if e(n)e = 0 for some e E H, then v(at,"" a n)* e = 0, Vat,"" an' Hence, v(al, ... , an+d* e = v(On, an+d*v(at, ... a n)* e = 0, Vat, ... ,an+t, i.e., e(n + l)e = 0.

517

Let p(n) be the projection from H onto H(n+l). Then we have p(n) > p(n+ 1),'tn. For fixed n, since v(ab···,an)H ..1 v(f3b"',f3n)H for (ab"',an)-=I(f3l, ... , f3n), it follows that

H(n) = p(n)H =

2:

$[v(al,"" an)H], 'tn.

Qi E {o, I },

19:$..

Noticing that

v( ab' .. , a n)v(f3b" . , f3n)*V(Il" .. ,In+l) v( at,' .. , a n)[v(J31" .. , f3ntv( 111 ... 'In) ]v(Om In+d

(fI 6f3im)

v(at, ... ,an,In+t}, 'tai,f3i,Ii,

1=1

we have v(al,···,a n)v(f3l"",Pn}*H(n+l)

C

H(n+ l),'tai,f3i, i.e.,

v(at,"" a n)v(f3h'" ,f3n}*p(n + I} = p(n + l)v(al"'" an)v(Pb'" ,f3n)*, 'tai,J3i E {O, I}, 1 < i < n. Since v(at,

, an)V(f3l,

, f3n)* V(a~,

, a~) V(J3~,

, f3~) *v (11,

, In+ d

V(at,

, an)v (f3l,

,f3n)*v(a~,

, a~)v (J3L

,J3~)*v( II,

, In)

'V(On,In+tl

(fI 6a~'f3i) (fI 6a~'f3i) 6f3:,-1i •

1=1

v( at,' .. ,an)v(On,In+l)

v( ab ... , an}) v(P~, ... ,P~) *v (111 ... , In+l),

1=1

it follows that

v(al,"" a n)v(f31"", f3n)*p(n + 1) . v(a~,···, a~)v(f3~,"" .p(n + 1)

(g

6a

:,p,) v(

at> ••• ,

is a C* -algebra on H, and

an)v(,8[, ... ,,8~)'p(n + 1),

f3~}*

518

(\lcxt,f3. E {O, I}, 1 < i < n) is a * homorphism from B n onto D n. Since v(ah"', lXn+tl* . [v (a}, ... , an)v(at,"" an)*p(n + 1)] . v(ah"" an+l) .b(n + 1) ·b(n + 1) v(at,"" an+l))*v(at,"" a n+l)b(n + 1) = b(n + 1) =I- 0, it follows that v(at,"" an)v(at,"" an)*p(n + 1) =I- 0 and ~n(Bn) =I- {O}. Now by the simplicity of B m ~n is a * isomorphism. Moreover,

[v( at,' .. ,an-I, 0) V(f3t, ... ,f3n-t, 0)*

+ v( at, ... ,an-I, l)v(f3t," . ,

f3n-b 1)*]v(I'h"" I'n+l) n-l

II 6,8Jo.'Y.[60.'Yn v (a t, "

' , an-bO,I'n+tl

+ 61.'Yn V(a b · · · ' an-b 1, I'n+l)l

1=1

n-l

II 6,8i.'Yi v (a b · · · ' an-bl'n,I'n+l)

i=1

v(al,' .. , an-d, v(fJl,' .. , ,Bn-t}*V(I't,' . " I'n-dV(On-b I'n) . v(On, I'n+l) v( at, .. " an-dv(,Bh' .. ,f3n-t)*v("tt-: .. 'In+t), hence t

L

v( ab" ., an-t, i)V(,Bb' ", ,Bn-t, i)*p(n + 1).

i=O

Generally, from p(t) > p(t + 1), \It, we can see that

v(at,"" an)v(,Bt,'" ,,Bn)*p(n + s)

L

v(at,"" an, it,' .. ,is-d

(1)

ijE{O,l} l~j~.-l

·v(,Bl,··· ,,Bm it,"', i,,_t)·p(n + s) Therefore, for any n < r, ai,f3i E {a, I}, 1 < i < n, H(r) is invariant under v(ab"', an)v(,Bb'" ,,Bn)·, i.e., p(r) and v(ab"" an)v(,Bb"" ,Bn)'" commute. Now let B(n) be the C*-subalgebra of A generated by {I, v( ab" . ,an)v(f3b "',fJn)*laa,fJi E {O,I},1 n, and D n = B(n)p(n + 1) is * isomorphic to B n.

519

Further, let A(n) be the C'" -subalgebra of A generated by {B(i)li From (1) , we have

< n}.

A(n)p(n + 1) = B(n)p(n + 1) = D n. Let I(n) = {x E A(n)lxp(n + 1) = O}. Then I(n) is a closed two-sided ideal of A(n), and A(n)jI(n) is * isomorphic to B n ( the 2 n x 2 n matrix algebra) . Clearly, A(n) C A(n+l), and I(n) C I(n+l). But A(n}jI(n) is simple, so that A(n) n I(n + 1) = I(n). Now let B be the C"'-sebalgebra of A generated by UnA(n), and I be the closure of UnI(n). Then I is a closed two-sided ideal of B. Moreover, since A(n)jI(n) is simple, A(n) n I = I(n}, \In. Consider the quotient algebra BjI. By A(n)jI(n) = A(n)j(A(n) n I) ,..., (A(n)+I)j I, B j I = Un(A(n) + I)j I is an (UHF) algebra of type (2°,2 1 " . " 2n, ...). Therefore, we have the following. Proposition 13.4.4. Let A be a NGCR algeba with an identity 1. Then there is a separable C"'-subalgebra B of A with 1 E B and a closed two-sided ideal I of B such that BjI is an (UHF) algebra of type {2 n}.

Lemma 13.4.5. Let X, Y", be two Banach spaces, Y = (Y",)"', and B(X, Y) be the Banach space of all bounded linear operators from X to Y. Then through the following way

g(x ® I) \Ix E X,I E l:,T E B(X,Y),g E

=

(Tx, I)

(ry - (X®Y.))"',

where ry(.) is the largest cross norm on X ® Y", ( see Proposition 3.1.2 ) , and rt- (X ® Y",) is the tensor product of X and l: with respect to ,(·),B(X, Y) is isometrically isomorphic to (, - (X ® Y",))"'.

Proof.

For any T E B(X, Y), define

g(u) = L(Txi' Ii),

i

where u = LXi®/i E X®Y",. Clearly, i

Ig(u)1 < IITII L II xill·ll/ill. Further, we i

have Ig(u)1 < ry(u)IITII, \lu E X®Y",. So 9 can be uniquely extended to a linear functional on "t : (X ® Y",), and Ilgll < IITII. For any e > 0, pick x E X, lEY. with Ilxll = 11/11 = 1 such that I(Tx, I) - IITIII < c. Since ,(x ® I) = 1, it follows that IlglI > Ig(x®/)1 = I(Tx,/)1 > IITII-c. Thus ,we have Ilgll = IITII· Conversely, let 9 E (, - (X ® Y",))"'. Since Ig(x ® 1)1 < Ilgll '11xll ,11/11, \Ix E X, lEY"" there is T E B(X, Y) such that g(x ® I) = tt», I). Q.E.D.

Lemma 13.4.6. Let M be a hyperfinite VN algebra on a Hilbert space H, l.e., M = (UpMp)" , where 1 E M 1 C ... c M p C ... c M, and for each p, M p

520

is a matrix algebra. Then there exists a projection of norm one from B(H) onto M'. Proof. For any x E B (H) I denote by C (x) the weak closure of C o{u* xu Iu E M and is unitary}. We say that C(x) n M' =j:. 0. In fact, for any p, Up = {u E Mplu is unitary} is a compact group. So there is an invariant Haar measure J.L on Up with J.L(Up) = 1. Let xp = JuI' u*xudJ.L(u). Since v·xpv = xp, Vv E Up, it follows that xp E C(x) nM;. Thus, C(x) nM; =j:. 0, Vp. Now {C(x) nM;lp} is a decreasing sequence of non-empty weakly compact subsets of B(H). Therefore

Vx E B(H). Denote by B(B(H)) the Banach space of all bounded linear operators on B{H). By Lemma 13.4.5, B(B(H)) = (, - (B(H) ® T(H))*. Thus, any bounded ball of B(B(H)) is w*-eompact. Let U be the set of all unitary elements of M, and for any u E U define TU E B(B{H)) as follows:

TUx = u*xu,

Vx E B(H).

Clearly, J = -=C,. . . o. .,. .,{T=u---.l-u-E---::":U.. . -'l}w· is a w*-eompact convex subset of B{B(H)). Introduce a partial order "<" into J : T 1 < T2 if

-::---,---=-w"

Let {1l11 E I\} be a total ordered non-empty subset of J, and J, = {11,ll' > I} , VI E 1\. Then {Jill E I\} is a family of w*-compact subsets of J, and {J,II E I\} has the property of finite intersection. Hence, there is T E nzJi. Clearly, for any x C B(H), Tx E {ll,xl/' > I} w, VI E 1\. Since 11,x E C(11 1x) C C(l1x), VI' > I, it follows that Tx E C(l1x), and C{Tx) C C(l1x) , VI E 1\, x E B(H). Thus, T > 11, VI E 1\. Now by the Zorn Lemma, J admits a maximal element To at least. We claim that C(Tox) = {Tox}, Vx E B(H). In fact, fix x E B(H), and pick a' E C(Tox) n M' . Then there is a net {fa(')} of functions on U, where fa{') > 0, and fa(u) = 1, Va., such that

L

uEU

a' = w-lim a

Clearly {Ta =

L

L

fa(u)u*(Tox)u.

uEU

fa (u)TUTo} C J. Since J is w*-compact subset of B(B(H)},

uEU

{Tala.} admits a w*-cluster point T1(E J). In particular, T1 E Co{TuToru E U}w·. Thus, we have T1y E Co{TuToylu E U}W

=

C(ToY),

521

and C(T1y) C C(ToY), vu E B(H). That means T1 > To. But To is maximal, so we have T1 = To. On the other hand, T1 is a w*-cluster point of {Tala}. It follows that a' = T1x EM'. Therefore, C(Tox) = C(T1x) = {a'l = {Tax} c

C(x), Vx E B(H). Now we have a linear map E(x) = Tox(Vx E B(H)) from B(H) to M'. Clearly, IIEII < 1, and E(a') = ai, Va' EM'. Therefore, E is a projection of norm one from B(H) onto M'. Q.E.D. Remark. A VN algebra M on H has the property ( P ), if C(x) n M' =j:. 0, Vx E B(H). From the proof of Lemma 13.4.6, we can see that: there is a projection of norm one from B(H) onto M' if M has the property (P); and any hyperfinite VN algebra has the property (P).

Lemma 13.4.7. Let A be a C"'-algebra with an identity 1, B be a C"subalgebra of A with 1 E B, and M be a type (III) factor on a separable Hilbert space H. If there is a linear map P from A to M satisfying : i) P(a) > 0, Va E A+; ii) P(b1ab2 ) = P(bdP(a)P(b2), VbI,b 2 E B,a E A; iii) P(B) is weakly dense in M, then A admits a type (III) factorial * representation. Proof. 1) Denote by 0 the set of all linear maps Q from A to M satisfying: Q(a) > 0, Va E A+; Q(b 1ab2) = Q(b1)Q(a)Q(b2), Vb b b2 E B, a E A; and Q(b) = P(b), Vb E B. We claim that 0 is a compact convex subset of (B(A,M),u(B(A,M)" - (A ® M,,))) ( see Lemma 13.4.5 ) , and Q(x"'x) > Q(x)*Q(x), Vx E A. In fact, if Q E 0, then we have -llhlllH < Q(h) < IlhIJIH, Vh* = h E A. Hence, 0 is a bounded subset of B(A,M). By Q = P on B, VQ E 0,0 is also convex. Morevoer, it is easily verified that 0 is w* -closed. Thus, 0 is a w*-compact convex subset of B(A,M). For any Q E 0, and x E A, by the Kaplansky density theorem there is a net {Cl} C P(B) such that Cl -----+ Q(x) ( * strongly) , and Ilcdl < IIQ(x)II,Vl. For any
0, it follows from the Schwartz inequality that

(Q(x)"'Q(x),
lim(x"'b" Q"'
< Q*
(Q(x"'X),
F(c;cl,
= (Q(x*X),
where b, E Band P(b,)

= ci, Vl. Hence, Q(x"'x) > Q(x)"'Q(x), VQ

E 0, x E A.

522

2) Since H is separable, there is a faithful normal state lp on M. For any Q E 0, let lpQ(a) = lp(Q(a)), Va E A. Then lpQ is a state on A since Q(l) = P(l) = 1H • Define

e = {lpQIQ EO}. Then by 1) e is a w·-compact convex subset of the state space on A. Now fix Qo E 0 such that lpo = lpQo is an extreme point of e. Let {1fo, H o, eo} be the cyclic * representation of A generated by lpo, and N = 1fo(A)"( a VN algebra on H o). If 1fo(x) = 0 for some x E A , then we have lp(Qo(x*x)) = lpo(x·x) = O. Since Qo(x·x) > 0 and lp is faithful, it follows that Qo(x*x) = O. By 1) , Qo(x)*Qo(x) < Qo(x·x), hence Qo(x) = O. Now for any fixed f EM., we can define a linear functional F on 1l'"o(A) as follows:

F(1fo(x)) = f(Qo(x)),

Vx E A.

We say that F is strongly continuous on the unit ball of 1fo(A). In face, let {a,l be a net of A such that \I 1fO(a,) II < 1, Vl, and 1fo(a,) ----+ 0 ( strongly ). Then lpo(aia,) = lp(QO(aia,)) > lp(QO(a,)*Qo(al)) ----+ O. We may assume that Iladl < 2, Vl. Then {IIQo(a,)llll} is bounded. Further, since lp is faithful and normal, so we have QO(a,) ---+ 0 with respect to s-top. of M. Hence

F(1fo(a,)) = f(QO(a,)) ----+ O. For any a E N = 1fo(A)", we can find a net {all C 1fo(A) such that a, ----+ a( strongly ) . Then (a, - a,,) -------+ 0 ( strongly ) , and from the preceding paragraph {F(a,)} is a Cauchy net of numbers. Hence, we can define F(a) = lim F(a,), I

and this definition is independent of the choice of {al} obviously. In such way, F is extended to a linear functional on N, still denoted by F. We claim that this extension F is strongly continuous on the unit ball of N. In fact, for any c > 0 by the strong continuity of F on the unit ball of 1fo(A) we can find

such that IF(a) I < s, Va E V, where ell"', 6~ E H o. Let

U

= U(O, 6,""

ek, 6) =

{a E

Niliall < 1, Ilaeill < 6,Vi}.

Clearly, U is a strong neighborhood of 0 in N. For any a E U, there is a net {a,l C 1fo(A) such that a, -------+ a ( strongly) , and Iladl < 1, VI. We may

523

assume that Ila,eill < 6,Vl,i, i.e., {all C V. By the definition of F on N, we get

IF(a) I =

liF IF(a,) I <e,

Va E U.

Therefore, F E N*. Moreover,

IIFII = sup{IF(1fo(x)) II x E A < sup{lf(Qo(x)) II x E A

and

lI 1fo(X)II <

and

IIxll < 2}

I}

< 211f1l11Qoll· If define C,l)(f) = F, then C,l) is a bounded linear map from M. to N •. Further, C,l)* is bounded and u(N, N+)-u(M, M.) continuous from N to M. 3) The map C,l). : N --+ M has the following properties:

I) C,l)* (7fo(x)) = Qo(x), Vx E A; II) C,l)*(a) > 0, Va E N+; III) C,l)*(b 1ab2 ) = C,l)*(bdC,l)*(a)C,l)*(b 2), Va E N, i., b2 E 1fo(B)"j IV) C,l)* (a*a) > e- (a*)C,l)* (a), Va E N. In fact, for any f E M* and x E A, we have

(C,l)* (1fo(x)), f) = ( 1fo(x), C,l)(f))

= f(Qo(x))

= (Qo(x), f).

Hence, C,l)*(1fo(x)) = Qo(x), Vx E A. For any a E N+, we can pick a net {Xl} of A such that 1fO(Xl) --------t al/2( * strongly) ,and II 1fo(Xl)II < Ilal / 2 1 1,VI. Then 1fo(xix,) --------t a( strongly) . Hence,

By the positivity of Qo, we have C,l)*(a) For any Yl,Y2 E B,x E A, we have

> 0, Va E N+.

C,l)* (1fo(yt}1fo(X)7f0(Y2)) = QO(YlXY2) = Qo(yt} Qo(x) Q(Y2) = C,l)* (1fo(yd) C,l)" (1fo (x)) C,l)+ (1fo(Y2))

obviously. Then by the u-continuity of C,l)*, we get C,l)+(b lab2 ) = C,l)*(b1)C,l)*(a)C,l)* (b 2 ) , Va E N, bl , b2 E 1fo(B)". For any a E N, since P(B) = Qo(B) is a weakly dense * subalgebra of M, there is a net {Xl} C B such that C,l)*(1fO(Xl)) = Qo(x,) --------t C,l)*(a) ( strongly) , and IIQo(xl)1I < 1IC,l)*(a) II, VI. Clearly, C,l)((M*)+) C (N*)+. Then for any ,p E

524

(M*)+, we have

o «

C)*(a*)C)*(a), ,p) = lim(C)*(1fo(I))C)*(a*)C)*(1fO(XI))',p)

= lim(C)* (a* 1fo(XI)) , ,p) = lim(a*1fo(XI) , C)(,p)) I < (C)* (a* a), ,p) 1/2lim(C)*(7fo( x; Xl)),,p) 1/2 = (C)*(a*a), ,p) 1/2lim(C)* (1fo(xi)) C)* (1fO(XI)) , ,p)1/2 = (C)* (a*a), =

,p)1/2lim(QO(XI)*QO(XI) , ,p)1/2

(C)*(a*a),?jJ)1/2. (C)*(a*)C)*(a),?jJ)1/2.

Hence (C)*(a*)C)*(a),,p) = (C)*(a*a),?jJ),V,p E (M*)+, and C)*(a*)C)*(a) (a'" a) , Va E N. 4) Now we prove that N is a factor. In fact, let z be a central projection of N. Since

< C)*

C)*(z)Qo(b) = C)*(z7fo(b)) = C)*(7fo(b)z) = Qo(b)c)*(z), Vb E B, and Qo(B) = P(B) is weakly dense in M, C)* (z) is also a central element of M. But M is a factor, thus C)*(z) = >'(z)I H, where >.(z) E [0,1]. If >.(z) E (0,1), we can define

Qdx) = >.(z)-lC)*(7fo(x)z), { Q2(X) = (1- >'(z))-lC)*(1fo(x)(l- z)), "Ix E A. By 3) , Qb Q% E 0, and >'(Z)Ql + (1 - >.(z))Q% = Qo. Then fPo = >'(Z)fPQl + (1 - >'(Z))fPQ2' But fPo is an extreme point of e( see 2) ) , so we have fPo = fPQ 1 = fPQ2' i.e., >'(Z)-l(7fO(X)Z, C)(fP))

= (1 - >.(z))-l(1fo(x)(1 - z), C)(fP)),

"Ix E A. Pick a net {Xl} C A such that 1fO(XI) 1, VI. Then we get

1 = >.(z)-l(C)*(z),fP)

---+

z (strongly) , and I 7fo(xl) II <

= >.(z)-l(z,C)(fP)) = (1 - >.(z))-l(z(1 - z), C)(fP)) = 0,

a contradiction. Hence , >.( z) is either 0 or 1 . If >.(z) = 0, pick {Xl} c A as above, then we have

(zeo, eo) = lim( 7fO(XI) eo, eo) = limfPO(XI) = limfP(QO(XI))

=

lim(C)"'(7fO(XI)),fP) = (z, C)(fP))

= >.(z) = O.

Hence, z1fo(A) eo = 1fo(A)zeo = {O}, i.e., z = O. If >.(z) = 1, similary we have z = 1. Therefore, N is a factor.

525

5) It suffices to show that N is not semi-finite. Then by 4) , N = 1fo(A)" is a factor of type (III) , and {1fo, H o} is a type (III) factorial * representation of A. Now suppose that N is semi-finite. Let E' be the projection from H o onto [11"0 (B) eo]. If 1fo(b)E' = 0 for some b E B, then 1I"0(b)E'€0 = 1fo(b)eo = 0, and 0 = 111I"0(b)eoI1 2 = rpo(b*b)). Since rp is faithful and Qo(b*b) > Qo(b)*Qo(b) , it follows that Qo(b) = o. Conversely, if Qo(b) = 0 for some b E B, then for any c E B we have 111I"0(b)1fo(c) eo 11 2 = rp( Qo(c* b* bc)) = rp(Qo(c*b*)Qo(b)Qo(c))

= o.

Hence, 1fo(b)E' = O. Therefore, 1fo(b)E' ---+ Qo(b) = P(b) is a * isomorphic from 1fo(B)E' into M. We say that the above * isomorphism is s-s continuous on the unit ball of 1fo(B)E'. In fact, let {b,} be a net of B such that 1I"0(b,)E' ---+ 0 ( strongly) , and lI1fo(b,)E'lI < 1, VI. Since b ---+ 1fo(b)E' is a * homomorphism from B to lI"o(B)E', we may assume that Ilbtll < 2, VI. Now if a is a w-cluster point of {Qo(b;b ,) = Qo(b,)*Qo(b,)!I}, then

rp(a)

= limrp(Qo(b,)*Qo(b,))

= limll1fo(b,)eoI12 = O.

Since rp is faithful, it follows that a = O. Hence, Qo(b,) ---+ O( strongly). Conversely, if {llb,III'} is bounded, and Qo(b,) ---+ 0 ( strongly) , then lI1fo(b,)1I"0(c)€oIl2 = rp(Qo(c*)Qo(b,)*Qo(b,)Qo(c)) ---+ o. Hence 1I"0(b,)E' ---+ 0 ( strongly). Therefore, the * isomorphism 1fo(b)E' ---+ Qo(b)(Vb E B) can be extended to a * isomorphism r from 1I"0(B)"E' onto M, and 1fo(B)"E' is a type (III) factor. Let F' be the central cover of E' in 11"0 (B)' . Then'll: xF' ---+ xE' is a * isomorphism from 1I"0(B)"F' onto 1fo(B)"E', and row is a * isomorphism from 11"0 (B)" F' onto 11"0 (B)" E', and row is a * isomrophism from 11"0 (B)"F' onto

M. Since F' > _ E' , E'c = ~o

C

~o

( ~ *( F') , rp)

and

= lim(1f0 (b,) , ~ (rp)) = lim rp (Q0 ( b,)) = lim (11"0 (b ,) eo,

eo) = (F' eo, eo) = 1,

where {b,} C B such that 1fo(b,) ---+ F' ( strongly) , so ~·(F') =I- o. By the semi-finiteness of Nand F' E 1fo(B)" c N, we have F' = sup{e E Nle is a finite projection, and e < F'}. Moreover, ~* is a-a continuous, so there is a finite projection e of N with e < F' such that ~* (e) =I- o. Further, we can find a number A > 0 and a non-zero projection p of M such that ~. (e) > Ap.

526

For any a E M, pick a net {b,} of B such that Qo(b,) ~ a ( strongly ). Then w- 1 0 r-1(Qo(b,)) ~ w- 1 0 r-1(a) ( strongly) , and ~*(\I1-1 0 r~I(Qo(b,))) ~ ~*(\I1-1 0 r-1(a)) ( weakly) . On the other hand, by the definition of I', W, and the properties of ~* , we have ~*(W-l

0

r-1(Qo(b,)))

= ~*(7fO(b,)F') = ~*(1fo(b,))~*(F') = QO(b,)~*(F')

-+ a~*(F')

(strongly).

Thus, we obtain that ~*(W-l

0

r-1(a))

=

a~*(F'),

~*(W-l

r-1(a*)) = ~*(F')a*, Va E M.

0

Now let {a,l be a net of Mp with Ilalll < 1, Vl, and a, ~ O( strongly ) . Clearly, w- 1 0 r-1(a,)e ~ O( strongly) . Since e is finite, it follows from Propsition 6.5.16 that eW- 1 0 r-1(ai) ~ O( strongly) . By ~*(a*a) > ~"'(a*)~*(a),Va EN, and the a - a continuity of ~*, we also have ~*(eW-l 0 r- 1(an) ~ O( strongly) . Since w- 1 0 r-1(M) C 'Ko(B)" F' C 7fo(B)", e < F', and {a,} C M p, it follows that {p~"'(e)p

+ (1 -

p)}-l{p~*(e)p

= {p~*(e)p

+ (1 -

p)}p~*(e)a~

ai =

+ (1 -

p)}a;

= {p~*(e)p + (1- p)}p~*(eF')a~ = {p~*(e)p + (1 = {p~*(e)p + (1 = {p~*(e)p

+ (1 -

p)}p~*(e)~*(F')ai p)}p~*(e)~*(W-l p)}p~*(e\l1-1

0

0

r-1(aj))

r-1(an) -+ O(strongly).

This means that the * operation is strongly continuous on the unit ball of M p, That contradicts the facts: p =1= 0, and M is type (III) ( see Proposition 6.6.3). Therefore, N is type (III) . Q.E.D.

Proposition 13.4.8. Let A be a NGCR algebra. Then A admits a type (III) factorial * representation. Proof. If A has no identity, and {s, H} is a type (III) factoral * representation of (A+a:) , then by Theorem 1.3.9 there is a projection Po E 1f(A)'n1f(A)" such that 7f(A)w = 7f(A)"po. Hence, {1f(')Po,P oH} is a type (III) factorical * representation of A. Moreover, (A-HD) is also NGCR. Thus, we may assume that A has an identity 1. From Proposition 13.4.4, there is a separable C*-subalgebra B of A with 1 E B and a closed two-sided ideal I of B such that B / I is a (UHF) algebra of type {2 n } .

527

Write B / I = ao - l8l nMJn), where MJn) = M 2 is the 2 Vn. Fix A E (O,!), pick a state lP on M 2 as follows:

X

2 matrix algebra,

and let ,p = l8l nlPn, where lPn = lP, Vn. By Theorem 9.5.11, The cyclic representation {1r, H, e} of B / I generated by ,p is a type (III) factiorial representation. From Proposition 3.8.7, we have

1r = l8ln7rn,

H = I8l!Hn ,

* *

e= l8lnen,

where {1r n , H n , en} is the cyclic * representation of MJn) generated by lPn, Vn. Clearly, 1rn (M Jn») ~ M 2 , B(Hn ) ~ M.. , and 1rn (M Jn»), ~ M 2 , Vn. Further, by Proposition 3.8.6 7r(B / I)' is generated by

{(1rn (M Jn»)' l8ll8l m#n1 m )ln} . Hence, 1r(B/ I)' is hyperfinite. Now from Lemma 13.4.6, there is a projection E of norm one from B(H) onto 7r(B/ I)". Summing the above discussion, we may assume the following: there is a state cp on B, and a projection E of norm one from B(H) onto the type (III) factor M = 7r(B)", where {1r,H} is the * representation of B generated lP, and H is separable. Pick a state ep on A such that eplB = cp, and let {7r, H} be the * representation of A generated by cp. Then we have H C H, 7r(b)H C H, and 1r(b) IH = 1r(b) , Vb E B. Let p be the projection from H onto H, and define P : A ---+ M as follows:

P{ a) = E(p1r(a)p),

Va E

A.

By Theorem 4.1.5, {A,B,P,M = 1r(B)",H} satisfies the conditions of Lemma 13.4.7. Therefore, A admits a type (III) factorial * representation. Q.E.D.

Remark. Let H be a separable infinite dimensional Hilbert space. Clearly, B(H) is not CCR and C(H) is a CCR closed tow-sided ideal of B(H). By Proposition 1.1.2 and 13.3.6, the Calkin algebra A = B(H)/C(H) must be NGCR. Thus by Proposition 13.4.8, the Calkin algebra A admits a type (III) factorial * representation. Notes. Quasi-matrix systems (Proposition 13.4.3) were introduced by J. Glimm. Proposition 13.4.8 is due to S.Sakai. The property (P) (see the Remark under Lemma 13.4.6) was introduced by J:T.Schwartz.

References. [55], [150], [153].

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13.5. Type I C*-algebras Definition 13.5.1. A C·-algebra A is said to be type I ,if for any nondegenerate * representation {a, H} of A,1r(A)" is a type (I) VN algebra on H. Proposition 13.5.2. If A is a GCR algebra, then A is type I. Proof. Let {s, H} be a nondegenerate * representation of A, and {PI} be a maximal orthogonal family of non-zero central projections of 1r(A)' such that 1r(A)"pl is a type (I) VN algebra on PIH, Vl. By Proposition 6.7.2, it suffices to show that PI = 1. Suppose that

L I

P=

1 -

LPI is not zero. Then {1rp,Hp } = {7fIPH,pH} is a non-zero

*

I

representation of A. Since 1rp(A) is * isomorphic to AI ker7fp and A is GCR, 7fp(A) is GCR. By Proposition 13.3.11, there is 0 =I a E 1rp (A)+ such that dim 1r'{a)H' < 1 for any irreducible * representation {1r', H'} of 7fp(A). Thus , a1rp(A)a is commutative. Further, aMa is also commutative, where M =

1rp {A)" ( on

u,

1

L

oo

= pH). Let a =

Ade>. be the spectral decomposition of a,

00

and Ye = A-Ide>.. If c(> 0) is small enough, then I = aYe = 1 - e, is a non-zero projection of M. Clearly, lall . j a21 = a(YeaIYe)a. a(Ye a2Ye)a = a{Ye a2Ye)a . a(Yea2Ye)a = la2/·jal/,Val,a2 E M. So IMj is commutative. Let z be the central cover of j in M. Then M; and Mj are * isomorphic. But (M})' = 1MI is commutative, from Theorem 6.7.1 M} is type (I) . Then M; and (M;)' = Mz are also type (I). Clearly, M z = 7fp(A)" z = 7f(A)"pz = 1r(A)"z, z =I 0, and ZPI = 0, Vl. This is impossible since the family {PI} is maximal. Therefore,

L

PI = 1.

Q.E.D.

I

Definition 13.5.3. A C·-algebra A is said to be smooth, if for any nonzero irreducible * representation {1r, H} of A, we have 7f(A) n C (H) t= [O}, By Lemma 13.2.6, this condition is equivalent to C(H) C 1r(A). Proposition 13.5.4. If A is a smooth C·-algebra, then A is GeR. Proof. Clearly, (A+a:) is also smooth. Moreover, by Proposition 13.3.15, A is GCR if (A+a:) is GCR. Thus, we may assume that A has identity 1 .

529

If A is not GCR, then by Proposition 13.3.6 there is a closed two-sided ideal I of A such that A/lis NGCR. Further, from Proposition 13.4.4 there is a C* -subalgebra B of A with 1 E B and a closed two-sided ideal J of B such that B/J is a (UHF) algebra of type {2 n } . By the Remark under this Proposition, B / J is simple ( i.e., B / J contains no non-zero proper closed two-sided ideal ) , so J is the largest proper closed two-sided ideal of B. Fix a pure state w on Band wlJ = {O}. Let {1fw , H w } be the irreducible * representation of B generated by w. Then ker7fw = J, and {7f w , H w } can be regarded as a faithful * representation of B / J. Since B / J is infinite dimensional, dim H.; = 00. Let £ = {p E P(A) IplB = w}. For any p E £, let {1f p , H p } be the irreducible * representation of A. Clearly, Hi; c H; , so dim H p = 00, and the identity operator 1f p (l ) = I p on H p is not contained in C(Hp ) . Since A is smooth, it follows that 1f p (A) :J C(Hp ) . Let I(p) = 1f;l(C(Hp )) . Then I(p) is a closed two-sided ideal of A, and 1 ff- I(p). Thus IlP) 1J B, and B n I(p) c J. Introduce a partial order" <" in E : PI < P2 if I(PI) c I(P2)' We say that £ has no maximal element with respect to this partial order. In fact , for any pEe, since wlJ = {a} and I(p) nBc J, so w can be regarded as a pure state on B /(B n I(p)). But B /(B n I(p)) ~ (B + I(p))/I(p) C A/I(p) , hence there is a pure state o E e such that uII(p) = O. Since 1fu (I (p)) = [O} and I(u) = 1f;I(C(Hu )) , it follows that I(p) C I(u) and I(p) I(u), i.e., p < a and p =I u. On the other hand, let {PI} be a non-empty totally ordered subset of c, and I = U,I(P,). Clearly, 1 ff- I, and I is a proper closed two-sided ideal of A, and B n I c J. Then w can be regarded as a pure state on B/B n I. Again by B/(B n I) ,..., (B + 1)/1 c A/I, there is pEe such that plI = {Ole Then I(p) :J I:J I(PI), VI, i.e., p > Ph Vl. By the Zorn lemma, contains a maximal element at least. This contradicts the conclusion of the preceding paragraph. Therefore, A is GCR. Q.E.D.

t=

e

Remark. Let A = UnAn be a (UHF) algebra, where 1 E Al C ... C An C ... , and each An is a matrix algebra. If I is a proper closed two-sided ideal of A, then there is no such that I nAn = {O}, 'tin > no. Consider the quotient * homomorphism -t> : A -------+ A/I. Then -t> IAn is isometrical, Vn > no. Hence, -t> is isometrical on whole A, and I must be zero. Therefore, any (UHF) algebra is simple. There is a more general result, see Lemma 15.4.1.

Theorem 13.5.5. are equivalent: 1) A is type I; 2) A is GCR; 3) A is smooth;

Let A be a C·-algebra. Then the following statements

530

4) Each factorial * representation of A is type (I) ; 5) A has no type (III) factorial * representation. Proof. From Propositions 13.5.4 and 13.5.2, we have 3) ==> 2) ==> 1) . Moreover, 1) ==> 4) ==> 5) are obvious. 5) ==> 2) . If A is not GCR, then by Propositions 13.3.6 and 13.4.8 A admits a type (III) factorial * representation. This contradicts the condition

5) . Finally, 2) ==> 3) is obvious from Proposition 13.3.7.

Q.E.D.

Lemma 13.5.6. (i) Let A be a C·-algebra of type I . Then A = A+a: is also a C·-algebra of type I. (ii) Let Ai be a C·-algebra with an identity Ii, i = 1,2, and Al be type I. If 0:(') is a C· -norm on Al ® A 2 , and 1f is an irreducible * representation of 0:(AI ®A 2 ), then we have 7r '" 1fl ®1f2, where 1fi is an irreducible * representation of Ai, i = 1, 2. (iii) Let A be a C·-algebra of type I, B be a C·-algebra, and 1f be an irreducible * representation of o:o-(A ® B). Then we have 7r = 1fA ® 1fB, where 7rA,1fB are irreducilbe * representations of A, B respectively. (iv) Let AI, A 2 be two CCR algebras. Then O:o-(A 1 ® A 2) is also CCR. (v) Let A be a C·-algebra of type I, B be a C·-algebra, and oJ be a closed two-sided ideal of B. Then we have

O:o-(A e B) /o:o-(A ® J) ,..., O:o-(A ® B / J). (i) By Proposition 13.3.4, there is a composition series {1er I 0 < 0: < fJ} for A such that I er + 1 / t; is CCR Vo: < {j. Let 113+1 = A. Then {Ier I 0 < 0: < fJ + I} is a composition series for ..4, and 1er+1 / t; is CCR, Vo: < fJ + 1. Hence A is GCR by Proposition 13.3.4. (ii) Clearly, 1f(A1 ® 12 ) C 7r(1 1 ® A 2)', 1f(1 1 ® A 2 ) C 1f(A 1 ® 12 )' , and 7r(A1 ® A 2 ) " = B(H). Then Proof.

(C

=

7r(A 1 ® A 2 ) "

n 7r(A1 ® A 2 ) '

(7r(A 1 ® 12 ) U 7r(1 1 ® A 2 )) " n 1f(A 1 ® 12 ) ' n 1f(1 1 ® A 2 ) ' ::J 7r(A 1 ® 1 2 ) "

n 7r(A 1 ® 12 )' .

Hence {7r I Al ® 12 , H} is a factorial * representation of A}, and M = 1f(A 1 ® 12 ) " is a type (I) factor on H. Let {PI II E A.} be an orthogonal family of minimal projections of M such that LPI = IH' Clearly, PI'" Pl' in M,Vl,l' E lEA

A.. By Theorem 1.5.6, we can write that {M,H} = {B(Ht}®pMp,H1®H2 } =

531

{B(HI)~lH2'

dimHI =

HI ®H2}, where p is a minimal projection of M, H 2 = pH, and

'A. Hence, we have

and {1ft, HI} is an irreducible 11"(1 1 ® A 2) C 1f(A I ® 12 ) ' = M'

1l"(1 1 ® a2)

* representation

of AI' On the other hand, = C1JI H1 ® B(H2), so

=

IHI

® 1f2(a2) ,

'Va2 E A 2.

Further, by B(H1)®B(H2) = B(H) = 1f(A l ® A 2)" = 1l"1(Ad"®1l"2(A2)", we can see that 1f2(A2) = B(H2), and {11"2' H 2} is an irreducible * representation of A 2 , and 1l" = 1fl ® 1f2' (iii) Clearly, ao-(A ® B) ,., . ,., . is a closed two-sided ideal of ao-(A ® .8), where A = A-HlJ, and B = B+C1J. Then 1f can be extended to an irreducible * representation 1r of ao-(A ® B). By (i) and (ii), !Ve""have 1r = 1rA e 1rB, where 1rA, iB are irreducible * representations of A, B respectively. If let 1I"A = 1rAIA and 1I"B = ?rBIB, then 1l"A,1fB are irreducible * representations of A, B respectively, and 1f = 1l"A ® 1l"B. (iv) Let 1f be any irreducible * representation of ao-(A 1 ® A 2). By (iii), 1f = 11"1 ® 1f2, where 1fi is an irreducible * representation of Ai, i = 1,2. Since Ai is CCR, it follows that 1l"i(Ai) C C(H,), i = 1,2. Then 1l"(A l ® A 2) C C(H1) ® C(H2 ) c C(H). Therefore, ao-(A 1 ® A 2) is also CCR. (v) Let 1fA = EB{1l"p I p is any pure state on A} and

1l"B = EB{1fu

Io

is any pure state on B, and ulJ = O}.

Clearly, we have

On the other hand, if v is a pure state on ao-{A ® B) and vlao-(A ® J) = 0, then by (iii) we can write 1fv = 1l"p ® 1fu , where p is a pure state on A, and o is a pure state on B with ulJ = O. Therefore,

ao-(A ® B/J) ~ (1l"A ® 1fB)(ao-(A ® B)) '"-J

'"-J

EB

{1f v

Iv is a pure state on ao-(A ® B), } and vlao(A ® J) = 0

ao-(A ® B)jao-(A ® J).

Q.E.D. Proposition 13.5.7.

532

(i) Let A be a C"-algebra of type I, B be a C"-subalgebra of A, and J be a closed two-sided ideal of A. Then B and A/ J are type I C" -algebras. (ii) Let A be a C"-algebra of type I. Then A is a nuclear C"-algebra. n

(iii) Let At,·' . ,An be C"-algebras of type I. Then ao-tg) A; is also a C·i=1

algebra of type I.

(i) By Proposition 13.3.5 and Theorem 13.5.5 it is obvious. (ii) Let B be any C"-algebra, and a(·) be a C"-norm on A ® B. We want to prove that a(.) = ao(') on A ® B. By Lemma 3.9.4 and' Proposition 3.9.5, we may assume that A and B are unital. Then by the (ii) of Lemma 13.5.6, we have Proof.

""(u) = Lr.

<

sup

{II 11" (u )11111" is any

irreducible * representatiOn} of a-(A ® B)

{ 1 ?rA l1l"B are irreducible * } sup 1I11"A(®)1I"B(u) II representations of A, B respectively ao(u),

\iu E A ® B.

Therefore, A is nuclear. (iii) First step. Let A be a CCR algebra, and B be a GCR algebra. We claim that ao-( A ® B) is GCR. In fact, let {I p I 0 < P < I'} be a composition series for B such that I p+ 1/Ip is CCR, \ip < 1'. Then by the (iv), (v) of Lemma 13.5.6,

ao-(A ® I p+ 1/Ip ) ~ ao-(A ® Ip+d/ ao-(A ® I p) is CCR, \ip < 1'. Clearly, {ao-( A ® Ip) I 0 < P < I'} is a composition series for ao-(A ® B). Hence, ao-(A ® B) is GCR. Second step. Let A, B be GCR. We show that ao-(A ® B) is GCR. In fact, let {I p I 0 < P < I'} be a composition serise for A such that I p+ 1/Ip is CCR, \ip < 1'. Then by the First step, ao-(Ip+ 1/Ip ® B) is GCR, \ip < 1'. Clearly, {ao-(Ip ® B) 10 < P < I'} is a composition series for ao-(A ® B), and ao-(Ip+ 1 ® B)/ ao-(Ip ® B) ao(Ip+ 1/Ip ® B) is GCR, \ip < 1'. Now by Proposition 13.3.4, ao-(A ® B) is GCR. I".J

n

Now let At, ... ,An be GCR (type I) algebras. We prove that ao-tg) Ai is i=1

GCR (type I). The second step is the case of n = 2 exactly. We assume that the conclusion n

is true for (n -1). Since min-tg) Ai i=1

n

= min-(A 1 ® min - tg) Ai), it follows from n

i=2

the second step and the induction that ao-tg) Ai is type I. i=l

Q.E.D.

533

Notes. Proposition 13.5.2 is due to I. Kaplansky. Theorem 13.5.5 was proved by J. Glimm for separable C·-algebras. Then S.Sakai pointed out that this theorem holds in the absence of separability. Proposition 13.5.7 is due to W. Wulfoshn.

References.

[27], [55], [84}, [85], [150}, [198].

13.6. Separable type I C*-algebras Definition 13.6.1. Let H(·) be a measurable field of Hibert spaces over a Borel space (E, B) ,and A be a separable C*-algebra. Suppose that {1r(t), H(t)} is a * representation of A, Vt E E. The field {1r(·),H(·)} of * representations of A over E is said to be measurable, if for any a E A, t ----+ 1r(t)(a) is a measurable field of operators on H (.). Clearly, 111r(t)(a)I! < lIall,Vt E E,a E A. Thus, if the field {1r(·),H(·)} of * representations of A is measurable, and v be a u-finite measure on B, then e e 1r(t)(a)dv(t) is a bounded linear operator on H(t)dv(t) ( see Proposi-

L

L

tion 12.2.3 ) , Va E A. Hence, we get a

1r(a) =

L e

1r(t)(a)dv(t),

* representation {1r,H}

Va E A;

H=

of A, where

L e

H(t)dv(t).

Definition 13.6.2. The above * representation 1r is called the direct integral of the measurable field 1r(') with respect to the measure v and denoted by e

1r =

L 1r(t)dv(t).

Proposition 13.6.3. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B), v be a u-finite measure on B, and A be a separable C* -algebra. 1) Suppose that {1r(·),H(·)} is a measurable field of * representations of e A, and 1r = 1r(t)dv(t). Then 1r is nondegenerate if and only if 1r(t) is nondegenerate, a.e.z/. 2) Suppose that {1rl ('), H(·)}, {1r2('), H(·)} are two measurable fields of * e representations of A, and 1ri = 1ri(t)dv(t), i = 1,2. Then 1rl = 1r2 if and only if 1rl(t) = 1r2(t), a.e.s-.

L

L

Proof.

1) Let Z be the diagonal algebra on H =

L H(t)dv(t) ( e

Definition

534

12.2.7 } . Since 1r(a) is decomposable, it follows from Theorem 12.2.10 that 1r(a) E Z', Va E A. So Z C 1r(A)'. Let p be the projection from H onto the essential subspace of 1r ( i.e., pH = [1r(A}H]). Clearly, p E 1r(A)" c Z'.

L 63

Thus, p is decomposable, and we can write p =

p(t)dv(t), where p(.) is a

measurable field of projections on H(·). Since A is separable, there exists an approximate identity {dnln = 1,2,·· .} for A. By Proposition 12.2.5, and the semi-finitness of v ; and 1r(dn) --+- P ( strongly) , we have 1r(t)(dn} ---+ p(t), a.e.v ( replacing {d n } by a subsequence if necessary) . Thus, p(t) is the projection from H(t) onto the essential subspace of 1r(t), a.e.z-. Therefore, 1f is nondegenerate ( i.e., p = IH) if and only if 1f(t) is nondegenerate, a.e.v, 2) The sufficiency is obvious. Now let 1rt = 1r2, and {an} be a countable dense subset of A. Then there is a Borel subset N of E such that

v(N) = 0, and 1rdt)(an) =

1r2

(t)(a n), Vt ft N,n.

Since {an} is dense in A, it follows that 1rdt)(a) = 1r2(t)(a) ,Vt i.e., 1rt(t) = 1r2(t), a.e.z-.'

ft N, a

E

A,

Q.E.D.

Proposition 13.6.4. Let H (.) be a measurable field of Hilbert spaces over a Borel space (E, B), v be a a-finite meausre on B, A be a separable C·-algebra, and {1r (.) , H (.)} be a measurable field of * representations of A. Suppose that there is a * representation {1ro,Ho} of A such that ~1r(t),H(t)} -

{1ro, Hol, Vt E E. Then there is a unitary operator u from H onto L 2(E, B,v) ® u; such that u1r(a)u* =

L 1r(t)dv(t), 63

where

1r

=

1®

1ro(a) ,

=

L

H(t)dv(t)

Va E A,

"I" is the identity operator on L 2(E, B,v).

Proof.

Since each H(t) is separable, Ho is also separable. Now by Theorem 12.6.3 and Corollary 12.6.4, the conclusion is obvious. Q.E.D. Let H(·) be a measurable field of Hilbert spaces over a Borel space (E, B),v be a a-finite measure on B, and A be a separable

Proposition 13.6.5. C"'-algebra. If

1r(A)

C

1r

is a

* representation

of A on H =

L H(t)dv(t) L 1r(t)dv(t). 63

such that

Z' , where Z is the diagonal algebra on H, then there is a measurable

field {1r(o), H(·)} of

* representations of A such that 1r =

63

Since A is separable, then we can construct a countable dense * subalgebra B of A over the field of complex rational numbers. By 1r(A} C Z', 1r( a) is decomposable, Va E A. Since B is countable, we can find a Borel

Proof.

535

subset N of E such that

lI(N)

= 0,

Ta(t)*

=

Ta(t)Tb(t)

Ta,,(t),

Tc(t)

=

Tab(t) ,

=

.\Ta(t) + ILTb(t) , Vt rf- N,

where a, b E B, c = .\a + ILb,.\, IL are complex rational numbers, and 1r(x)

L Tz(t)dll{T) , II Tz(t)ll < Ilxll,

=

ffJ

t

Vt E E, x E B. Since B is dense in A, for each

rf- N we can get a * representation {1r(t) , H(t)} of A such that 1r(t)(b)

Tb{t), Vb E B. Further, let 1r(t) is measurable on E, and 1r

=

=

0, Vt E N. Now it is easy to see that t ----+ 1r{t)

=

L 1r(t)dll(t). ffJ

Q.E.D.

Proposition 13.6.6. Let H be a separable Hilbert space, A be a separable C·-algebra, 1r be a * representation of A on H, and Z be an abelian VN algebra on H with Z C 1r(A)'. If n is the spectral space of Z ( a compact Hausdorff space) , then there is a measurable field H(·) of Hilbert spaces over n, a regular Borel measure

n, a unitary operator u from H onto £fB H(t)dll(t), and a {1r(·),H(·)} of * representations of A, such that uZu* is the

11 on

measurable field

L H(t)dll(t) , ffJ

diagonal algebra on

fB

and U1rU·

=

IE

1r(t)dll(t).

It is immediate from Theorem 12.4.1 and Proposition 13.6.5.

Proof.

Q.E.D.

Proposition 13.6.7. Let H(.) be a measurable field of Hilbert spaces over a Borel space (E, B), 11 be a a-finite measure on B, A be a separable C· -algebra, and {1r(·),H(.)} be a measurable field of * representations of A. Suppose that Z C 1r(A)', where Z is the diagonal algebra on H =

L 1r(t)dll(t).

L ffJ

H(t)dll(t), and

ffJ

1r

Then Z is a maximal abelian VN subalgebra of 1r(A}' if and only if {1r(t),H(t)} is irreducible for A, a.e.z>, =

Let M = 1r(A)". Clearly, Z is a maximal abelian VN subalgebra of M' -¢:::::::> Z' n M' = Z {::::::> Z' is generated by Z and M. Now let N = (Z U 1r(A))". Then Z is a maximal abelian VN subalgebra of M' if and only if N = Z'. Let {arJ be a countalbe dense subset of A. Then N is generated by Z and Proof.

{1r( an) =

L ffJ

1r(t)(an)dll(t) In}. By Proposition 12.3.7, N is decomposable, i.e., N

=

IE

ffJ

N (t)dll (t),

where N(t) is a VN algebra on H(t) generated by {1r(t)(a n) In}, Vt E E.

536

It is well-known that Z' =

Lf:B B{H{t))dv{t).

Therefore, Z is a maxi-

mal abelian VN subalgebra of 1r(A)' if and only if N(t) = {1r(t)(a n )ln }" = 1r(t)(A)" = B(H{t)), a.e.z/, i.e., {1r(t), H(t)} is irreducible for A, a.e.z/,

Q.E.D. Proposition 13.6.8. Let H be a separable Hiblert space, A be a separable C·-algebra,1r be a * representation of A on H, and Z be a maximal abelian VN subslagebra of 1r(A)'. Then there is a compact Hausdorff space 0, a regular Borel measure v on 0, a measurable field H(·) of Hilbert spaces over 0, a unitary operator u from H onto

L H(t)dv(t), e

and a measurable filed

{1r(·),H(·)} of * representations of A, such that uZu· is the diagonal algebra on H(t)dv(t), U1ru· = 1r(t)dv(t), and {1r(t) , H(t)} is irreducible for A,

Le

Ire

a.e.il! Proof.

It is immediate from Propositions 13.6.6 and 13.6.7.

Q.E.D.

Theorem 13.6.9. Let A be a separable C·-algebra. Then the following statements are equivalent: 1) A is type I ; 2) ker: A -----+ Prim(A) is a homeomorphism; 3) A is a To-space. Proof. By Propositions 13.1.6 and 13.3.7, it suffices to show that the statement 2) implies the statement 1). Now let ker : A -----+ Prim(A) be a homeomorphism, and {1r, H} be a nondegenerate factorial * representation of A. We need to prove 1r(A)" is type (I) . Pick a non-zero vector € of H, and let p' be the projection from H onto 11r(A)€].· Clearly, p' E 1r(A)'. Since p' f:. 0 and 1r(A)' is a factor, the central cover of pi in 1r(A)' is In . Hence, a -----+ ap' is a * isomorphism from 1r(A)" onto 1r(A)"p'. Clearly p' H = [1r(A) €] is separable. Thus, we may assume that H is separable. Fix a maximal abelian VN subalgebra Z of 1r(A)'. By Proposition 13.6.8, we may assume that:

H=

e

L H(t)dv(t) ,

L e

1r =

Lf:B 1r(t)dv(t),

Z is the diagonal algebra on H(t)dv(t), and {1r(t),H(t)} is irreducible for A, Vt E 0, where is compact, and v is a regular Borel measure on n.

°

537

For any Borel subset B of 0, denote by p~ the diagonal operator corresponding to XB. Clearly, p~ E Z C 11" (A)', and p~ =I 0 if lI(B) > O. Since 1I"(A)' is a factor, the central cover of p~ in 1I"(A)' is I H • Thus, a --+- ap~ is a * isomorphism from 1I"(A)" onto 1I"(A)"p~. In particular, we have

111I"(x) II

= Ilp~1I"(x) II,

Fix 0 =I x E A. By Proposition 12.2.2, t function on 0. Thus for any n

Vx E A. ---+

111£"(t)(x) II is a measurable

is a Borel subset of fl. Denote by P~n the diagonal operator corresponding to XB",n'

Then

From the preceding paragraph, it must be that lI(B zn) = 0, Vx E A, n. Thus, we obtain that II 11" (t)(x) II = II 11" (x) II, Vt ~ B z, where B; = unB zn, and lI(B z) = 0, x E A. Suppose that D is a countable dense subset of A. Then we have

II 11" (t)(x) II = 111I"(x) II, Vx E A,t ~ B where B = UzEDBz, and lI(B) = O. Consequently, ker 1I"(t) = ker 11", Vt ~ B. Since {1I"(t) , H(t)} is irreducible, Vt E 0, by the condition 2) we may assume that {1I"(t),H(t)} ,. . . {1I"o,Ho},Vt E 0, where {1I"o,Ho} is some irreducible * representation of A. Now by Proposition 13.6.4, the * representation {11", H} is unitarily equivalent to {I ® 1£"0, L 2(0 , 1I) ® H o}. Clearly, 1I"0(A)" = B(Ho) is type (I) factor. Therefore, 1I"(A)" is type (I) . Q.E.D.

Notes.

Theorem 13.6.9 is due to J. Glimm. But we still don't know whether this theorem holds in the absence of separability.

References.

[55].

Chapter 14 Decomposition Theory

14.1. Choguet theory of boundary integrals on compact convex sets Let E be a ( real) locally convex ( Hausdorff topological linear ) space, X be a nonempty compact convex subset of E, and ExX be the set of all extreme points of X. A probability measure on X means a non-negative regular Borel measure and its total mass is 1.

Definition 14.1.1. Let J.l be a probability measure on X. A point x of X is said to be represented by J.L, if f(x) = fdJ.L, vt E E"'. Clearly, such x is one at most. We also say that J.L is a representing measure of x , and x is the barsrcenter or resultant of J.L , denoted by r(J.L) = x.

Ix

Proposition 14.1.2. existential.

The barycenter of a probability measure J.L on X is

For any f E E*, let HI closed. We need to show that Proof.

= {y

E EI/(y)

n{Hf n XII E E"'}

=

Ix fdJ.L}.

Clearly,

n,

is

-# 0.

Since X is compact, it suffices to prove that for any finite set 11,"', In of E"', n{H/i n XII < i <} f:. 0. Define T : E ---+ JRn by Ty = (fI(y),···, In(Y)). Then TX is compact and convex in JRn. Let suffices to show that such that

0:

E

T X. If 0:

~

0:

=

(Ix 11dJ.L,···, Ix IndJ.L) , n

T X, then there is

it

P=

(PI,' .. ,Pn) E lR

539

Let

f

=

I;, «s: Then Ix fdJ.L •

since J.L(X) = 1. Therefore,

IX

> sup{f(x) Ix

E X}. This is a contradiction

E T X.

Q.E.D.

Proposition 14.1.3. Let x E X. Then x E ExX representing measure of x.

{::::=}

Oz is the unique

Proof. If x = ..\XI + (1 - ..\)X2, where XI, X2 E X with Xl -I X2, and)" E (0,1), then v = )..OXI + (1 - )..)oz~ is also a representing measure of x, and v -I oz· Conversely, let X E ExX, and J.L be a representing measure of x. Suppose that F is a closed subset of X with J.L(F) > and x (j. F. For each y E F,

°

there is a closed convex neighborhood V" of y such that x (j. V". Since F is compact, there is a finite set Yb"', Yn of F such that U?=I Vi ::::> F, where Vi = V"., 1 and x (j. K. By Proposition 14.1.1 and the uniqueness of the baryeenter of J.L, it must be J.L(K) < 1. Now for each Borel subset B of X, define

°

J.LI(B) = r-IJ.L(B n K),

J.L2(B) = (1 - r)-IJ.L(B\K),

where r = J.L(K). Then J.LI and J.L2 are two probability measure on X. Let Xl, X2 be the barycenters of J.LbJ.L2 respectively. Then X = rXI +(1-r)x2' and X -I Xl' This is impossible since X E ExX. Therefore, supp J.L = {x} and J.L = Oz. Q.E.D. Proposition 14.1.4.

Proof.

If X is metrizable, then ExX is a G 6 subset of X.

Let d be a proper metric on X, and let 1 E; = {x E Xix = -(y + z), where Y,z E X and d(y,z) 2

> n- l }

for each n. It is easily checked that each Fn is closed, and a point X of X is not extreme if and only if it is in some Fn. Thus, the complement of ExX is F6 , and ExX is G6' Q.E.D.

Remark.

If X is not metrizable, then ExX need not be a Borel subset of

X. A real function f on X is said to be convex if f()..x + (1- ..\)y) < )..f(x) + (1- ..\)f(y),Vx,y E X,D < ).. < 1;f is said to be affine, if f(..\x + (1..\)y) = )..f(x) + (1- )..)f(y), Vx,y E X,D < ..\ < 1. Denote by Cr(X) the space of all real continuous functions on X, and A(X) = {f E Cr(X)lf is affine },M(X) = {(fix) + rlr E JR, f E E*}, P(X) = {f E Cr(X)lf is convex }. Clearly, P(X) n (-P(X)) = A(X) ::::> M(X).

540

Proposition 14.1.5. A(X) is closed in Cr(X); M(X) is dense in A(X); and (P(X) - P(X)) is dense in Cr(X).

Proof. For any g E A(X) and e > 0, consider the following subsets of Ex JR: J1

= {(x,r)lx E

X,r

= g(x)},

J 2 = {(x,r)lx E X,r

= g(x) + s}.

Clearly, J 1 and J 2 are two disjoint nonempty compact convex subsets of E X JR. By the separation theorem, there is some L E (E X JR). and some A E JR such that sup{L(·)J· E J 1 } < A < sup{L(·)I· E J 2 } . If (x, r) E J 1 , then (x, r

+ c) E J 2 • Hence

L(x, r) < L(x, r + c)

=

L(x, r) + cL(O, 1),

and L(O,I) > 0. Define /(x) = L(x,O), Vx E E. Then / E E*, and m(.) L(O, 1)-I(A - /(.)) E M(X). For any x E X, we have

L(x,g(x)) = L(x,O)

=

+ g(x)L(O, 1)

< A = L(x, m(x)) = L(x,O) + m(x)L(O, 1)

< L(x,g(x) + c) = L(x,O) + (g(x) + c)L(O, 1), i.e., g(x) < m(x) < g(x) + s, Vx E X. Hence, M(X) is dense in A(X). For any It, /2, gl, g2 E P(X), SUP(/l - gl, /2 - g2) = [sup(1t + g2, /2 + gl) (gl + g2)] E (P(X) - P(X)). So (P{X) - P(X)) is a vector lattice. Now by the Stone-Weierstrass theorem, (P(X) - P(X)) is dense in C,.(X). Q.E.D. Definition 14.1.6. Let J.L, v be two probability measures on X. We say that J.L is bigger than v in the sense of choquet-Meyer and denoted by v -< J.L ( C. M. ) , if

Ix

/dJ.L

>

Ix

[du,

V/

E

P{X).

A probability measure J.L on X is maximal (C. M. ) , if v some prabability measure v on X, then it must be v = J.L.

~ J.L (

C. M. ) for

Proposition 14.1.7. (i) If v -< J.L ( C. M. ) and J.L -< v ( C.M. ), then it must be J.L = v; (ii) If·v -< p, ( C.M. ) , then the barycenters of p, and v are the same; (iii) For any probability measure v on X, there exists a maximal ( C.M. ) probability measure J.L on X such that v -< J.L ( C.M. ).

541

f

f

Proof. (i) By the assumption, we have IdJ.L = [du, VI E P(X). Since (P(X) - P(X)) is dense in Cr(X), it follows that J.L = v,

(ii) H v -< J.L ( C. M. ) , then for any lEE· we have

f I dJ.L = I I dv

since

±/lx E P(X).

Therefore, the barycenters of J.L and v are the same. (iii) Let v be a probability measure on X, and let

.c - { IJ.L is a probability measure on X, } -

and J.L >- v( C.M.)

J.L

.

.c

With" -< ( C.M. ) ", is a partially ordered nonempty set obviously. Now let {J.Ldl E A} be any totally ordered nonempty subset of L, where A is a totally ordered index set such that J.L1 -< J.L1' ( C.M. ) if l, l' E A and I < l'. By the weak * compactness, there is a su bnet {A a Ia E ll} of {PI} and a probability measure J.L on X such that

where II is a directed index set. Since {Aala E ll} is a subnet of {pdl E A}, there is a map d : 1l ~ A with following properties: (i) for each a E ll, Aa = J.Ld(a); (ii) for each lEA, we can find a, E II such that d(a) > l, Va > a" Now for any I E A, pick as in (ii) . Then Aa = J.Ld(a) >- J.L1 ( C.M. ) , Va > al. Hence, we have J.L >- J.L1( C.M. ) ,Vl E A, i.e., J.L is a upper bound of {J.Lz!l E A} in L, By the Zorn lenuna, admits a maximal element J.L at least, and this J.L is what we want to find . Q.E.D.

a,

.c

Theorem 14.1.8. Let P be a maximal (C.M. ) probability measure on X. Then J.L is pseudoconcenirated on ExX in the sense that J.L(B) = 0 for each Baire subset B of X disjoint from ExX. In particular, supp J.L C ExX, and if ExX is a Baire subset of X ( for example, X is metrizable), then J.L is concentrated on ExX, i.e., J.L(ExX) = 1. Moreover, for any x EX, there is a probability measure J.L on X such that : J.L is a representing measure of x; and J.L is pseudoconcentrated on ExX in the above sence. Proof.

See [128].

Q.E.D.

Remark. Generally, we can't require that J.L(B) = 0 for each Borel subset B of X disjoint from ExX.

Definition 14.1.9. Assume that X is contained in a closed hyperplane of E which misses the origin ( There is no generality lost in making this assumption,

542

since we may embed E as the hyperplane E X {1} of E X lRj the image X X {1} of X is affinely homeomorphic with X), and let P = {axla > 0, x EX}. X is called a simplex (in the sense of Choquet), if (P - P) with the positive cone P is a vector lattice ( i.e., if each pair x, y in (P - P) has a least upper bound sup{x, y} in (P - P)) . Such definition of a simplex coincides with usual one in case X is finite dimensional.

Theorem 14.1.10. The following statements are equivalent: (i) X is a simplex ( in the sense of Choquet); (ii) For each x E X, there is unique maximal (C.M.) probability measure J.L on X such that J.L is a representing measure of x. Proof.

Q.E.D.

See [128J.

Notes. Theorem 14.1.8 is due to G. Choquet, P.A. Meyer, E.Bishop and K. de Leeuw, and its metrizable case is due to G.Choquet. Theorem 14.1.10 is due to G.Choquet and P.A.Meyer. Proposition 14.1.3 is due to H.Bauer.

References. [128].

14.2. The C-measure and C-isomorphism of a state In this section, let A be a C"-algebra with an identity 1 , S = S(A) be the state space of A, cp be a fixed state on A, and {1rIp' Hlp, lip} be the cyclic * representation of A generated by cp. Clearly, S is a compact convex subset of (A*, q(A\ A)). For any a E A, define a(p) = p(a), Vp E S. It is obvious that a -----+ ii(.) is a positive linear map from A into C(S). By Corollary 2.3.14, E = {li(')la E A} is a closed * linear subspace of C(S) containing the constant function 1 . Now cp can be regarded as a state on E. Then by Propostion 2.3.11 cp can be extended to a state on C(S). Hence, there is a probability measure v on S such that cp(a)

Definition 14.2.1.

l a(p)dv(p) ,

Va E A.

(1)

Let

n( ) = cp

=

{v Ivis asuch probability measure on S } . that (1) holds.

Clearly, for each v E n(cp), the barycenter ( or resultant) of v is cp, i.e., r(v) = cp; and v is a representing measure of the point CP(E S).

543

Moreover, by the above discussion, n(~) #- 0. Hence, n(~) is a non--empty compact convex subset of (C (S) *, a( C (S)*, C (S))) .

Proposition 14.2.2. 1) IT C is an abelian VN algebra on Hip and C c 1rip(A)', P is the projection from Hip onto Clip' then: c ---+ cp is a * isomorphism from C onto C p; and C p is a maximal abelian a-finite VN algebra on pHip' i.e., Cp = (Cp)' = pC'p; the central cover of p in C' is 1 = lH.. .i.e., c(p) = 1; and p is a maximal abelian projection of C';

p1rip(a)p1rip(b)p

= p1rip(b)p1rip(a)p, Va, bE A,

i.e., p1rip(A)p is abelian; and p1rrp(A)p

C

pC'p = Cp; and

C = {1rip(A),p}'. 2) Suppose that p is a projection on Hip such that plip = lip and p1rip(A)p is abelian. Then pHip = Clip' C is abelian, and C C 1r rp(A)', where C =

{1rip(A),p}'. 3) There is a bijection between the collection

{

c IC

is an abelian VN algebra on Hip, } and C C 1r ip(A)'

and the collection p is a projection on H ipsuch that {p IPi.. = lip and p1rip(A)p is abelian such that pHip

= Clip

and C

}

= {1rip(A),p}'.

Proof. 1) By c(p)Hip = [C'pHip] = [C'Clip] ::J 1rip(A)lip = Hip, we can see that c(p) = 1. Since C p is abelian and admits a cyclic vector lip on pHip' thus C p is maximal abelian and a-finite from Proposition 5.3.15. IT q is an abelian projection of C' and q > p, then by Proposition 1.5.8 we have p = c(p)q = q, i.e., p is maximal abelian in C'. Since p1rip(A)p C pC'p = Cp, it follows that p1rrp(a)p1rip(b)p = p1rip(b)p1rip(a) p, Va,b E A. Now p1rip(A)p is abelian and admits a cyclic vector lip on pHip' so by Proposition 5.3.15 we have

(p1rip(A)p)"

= (p1rip(A)p)'

= (pNp)'

= N'p,

where N = {1rip(A),p}". Moreover, from Cp = (Cp)' = pC'p ::J p1rip(A)p, we get (P1r rp(A)p)' ::J (Cp)' = Cp ::J (P1rip (A)p)".

544

Hence, Cp = Ntp. Noticing that [NpH",l :> 1r",(A)I", = H"" the central cover of p in N is also 1 . Thus, c -----+ cp(Vc E C) and -----+ x'p(Vx' EN') are * isomorphisms from C onto C p and from N' onto N' p respectively. Now by C eN', we must have C = N' = {1r",(A),p}'. 2) Let N = {1r",(A),p}". Then the central cover of p in N is 1 , and x' -----+ x'p is a * isomorphism from N' onto N'p. Since pNp = (p1r",(A)p)" is abelian and admits a cyclic vector 1", on pH"" it follows that pNp = (pNp)' = N'p . Hence, N'p is abelian. Further, C = N' is abelian, and C C 1r",(A)', and Cl", = N'pl", = pNpl", = pH",. 3) It is obvious from 1) and 2). Q.E.D.

x

Now fix an abelian VN algebra C C 1r",(A)', and let p be the projection from H", onto CI",.

Lemma 14.2.3. that

There exists unique

* homomorphism 1\ : C(S)

I\(a)p = p1r",(a)p,

-----+

Va E A,

C such

(2)

i.e., the following diagram is commutative:

C(S)

.c;

C

i A Moreover,

1\ C(S)

-----+

-----+

Cp

i id v«'" (A) p

is strongly dense in C, and

l\(a 1 ••• an)p

=

p1r",(adp· .. p1r",(an)p,

Vab' .. , an E A.

Proof. Let B be the abelian C"-algebra generated by p1r'"(A) p, Then B C Cp, and B ,...; C(T), where T is the spectral space of B. For any t E T, define 6(t)(a) = (P1r",(a) p)(t) , Va E A. Clearly, 6(·) is a continuous map from T to S. Further, for any at,' •• ,an E A and any polynomial P of n-variables, define

Since

liP (P1r",(al)p, "', P1r",(an)p) 11 sup tET

sup tET

<

IP{ (P1r'" (al)p)(t), ... , (P1r'" (an)p)(t)) I IP(a1 (6(t )), " ' , an(6(t ))) I

sup IP(al (p), . . . ,an(p)) I = pES

IIP(a1 , ' .. ,an) II,

it follows from the Stone-Weierstrass theorem that 1\ can be extended to a * homomorphism from C(S) to B c Cp, By C ,...; Cp, we can get a *

545

homomorphism

1\ :

C (S)

-----t

C such that

Again by the Stone-Weierstrass theorem, such any aI, ... , an E A we have

1\

must be unique. Clearly, for

I\(li:}, .. . ,li:n)p = 1\ {adp ... 1\ (li:n)p

= p7r~(adp··· P7r~(An)P since p E C'. Finally, since Cp = pC'p = p{7r~(A),p}"p = (p7r~(A)p)", I\C(S) = B is strongly dense in Cp, Therefore, I\C(S) is strongly dense in C. Q.E.D. Now (1\(')I~, 1~) is a state on C(S). Hence, there is unique probability measure p, on S such that

(1\(/)1~, IV') =

Is I (p)dp,(p) ,

VI E C(S).

In particular, by Lemma 14.2.3 we have

Val,'" ,an E A, and p, E O(cp) ( see Definition 13.2.1). Clearly, the measure p, satisfying (3) is unique from the Stone-Weierstrass theorem.

Definition 14.2.4. The unique P,(E O(cp)) determined by (3) is called the C -rnecsure of the state cp. Now we point out that 1\ can be extended to a * isomorphism from LOO{S, p,) onto C. First, for I E C(S) we say that

I\{/)

= 0 -<==> I = 0

e.e.u,

This is immediate from the following equality:

(1\ (/)p7r V'(ad p . . . p7rV' (an)pl~, lcp) (1\(/) 1\ (ad··· 1\ (an) IV', 1~)

! I{p)al{p) ... an{p)dp,(p) , Val, ... ,an E 1\, and by the Stone-Weierstrass theorem.

546

Secondly, if {/z} is a net of C(S) with O(o(Loo, L I )) , then we have

IIllll <

1, VI, such that

I,

-----+

(A(h)pw~(a)1~,w~(b)1~)

(A(I,) . pw~(a)p1~,pw~(b)p1~)

! I, (p )a(p)b(p)dJL(p) Va, bE A. Hence, A(lt)p and A(I,)

---t

0,

0 ( weakly). Then, A can be extended to a normal * homomorphism from Loo(S,JL) to C, still denoted by A, such that

(1\(/)1~, 1~) =

---t

Is l(p)dJL(p), VI

E LOO(S,JL).

On the other hand, by (pw~(A)p)" = Cp ( see that proof of Proposition 14.2.2) and A(C(S))p ~ pw~(A)p, we have A(Loo(S,JL))p = Cp and I\(Loo(S,JL)) = C. Further, if I E LOO (S, JL) is such that A(/) = 0, then

! I(p)al(p)", Cin(p)dJL(p) (A(/) 1\ (at,' . " an) 1~, 1~) = 0, Val,' ..

,an E A. Hence, I = 0, a.e.JL, and

A is a

* isomorphism from Loo(S, p,)

onto C. Denote by r the inverse map of A. Then T is a Loo(S,JL), and

* isomorphism from C

onto

(4) Va E

A,x E C.

Lemma 14.2.5. Let v E Loo(S, v) such that

Va E

A,x E C. Then

Proof.

v =

n(~),

and r 1 be a * isomorphism from C onto

JL is the C-measure, and r l = r.

For any a E A there is x E C such that

r IX = a.

Then

(xpy1~, z1~) = (xz*y1~, 1~)

! (rlxz*y)(p)dv(p) ! (rlz*y)(p)a(p)dv(p) =

(z*yw~(a)1~, 1~) = (pw~(a)py1~, z1~),

547

vv, z

E C. Since C1rp

= pHrp, it follows that xp

Then by p7rrp(a)p

= I\(a)p

we have x

r 1(Xl ... x n ) where rLi E A,

x.

= p7rrp(a)p.

=

E C, and fIX. =

= I\(a),

and I'z

= a = r 1 x.

Further,

a 1... an = r(X1," . , x n ) ,

llt, 1 < i < n. Hence,

Is aI(p) ... an(p)dv(p) Lft{XI ... xn)(p)dv(p) =

(x1 ... xn1rp , l rp ) = ((XlP) ... (x nP)lrp,lrp) (P7r p (a1)p ... pll" rp (an)p1rp, 1rp) , Val,

r 1(X1

, an E A. By Definition 14.2.4, v = p, is the C-measure. Also by x n) = a 1 ... an = r(Xl ... x n) we have

r 1- 1 ( ""a1' .. "") an

=

"" ... ...... r- I ( al an) ,

Since C(S) is w*-dense in LOO(S, p,) , it follows from the Stone-Weierstrass theorem that r l = f. Q.E.D.

Definition 14.2.6. The unique * isomorphism r from C onto LOO(S,Jl) determined by (4) is called the C -isomorphism of the state cp , where p, is the C-measure of cpo From the above discussion, we have the following.

Theorem 14.2.1. Let A be a C·-algebra with an identity 1, S = S(A) be the state space of A (a compact convex subset of (A*,cr(A*,A))),cp E S, {lI"rp, Hrp, 1rp} be the cyclic * representation of A generated by cp, and C be an abelian VN algebra on Hrp with C C 7rrp(A)'. Then there is a unique probability measure Jl on S (the C-measure of cp) and a unique * isomorphism r from C onto LOO(S,Jl) ( the C-isomorphism of cp) such that p, E O(cp),

Is a

1 .••

o'ndJl

= (p7rrp(al)p", p7rrp (an )p1rp , 1rp)

and (X7r rp(a)lrp, lrp) =

Is (rx)(p)a(p)dJl(p),

Va, at,"', an E A, where O(cp) is as in Definition 14.2.1, and p is the projection from u; onto C1rp.

548

Let n be a compact convex subset of a locally convex topological linear space E, and p, be a probability measure on o. Then

Lemma 14.2.8.

p, = u(C(O)*, C(O))-lim P,{h-},

{hi}

.

i.e.,

rgdp, = in

where

lim

r gdP,{h-},

{hi} in

°< hi E LOO(O,p,),Vi,Lh;(t)

J

Vg E C(O),

J

= 1,

a.e.p,;p,{hi} = La;Oti' for each

i

i; a; = ~ h;dp"

j

t j E 0 such that aif(t;) =

~ f(t)hj(t)dp,(t) , "If

E

E·

(notice that if aj > 0, then ajlhjp, is a probability measure on O. Hence, from Proposition 14.1.2, there is unique t; E

n such

that f(t;)

= ail ~ f(t)hj(t)d

p,(t), "If E E·); {h~}k=l < {hj}j=l if there is a partition {Ib · · · , 1m } of {I, ... , n} such that h~ = L hi' 1 < k < m ( Clearly for any {h~} and {hj}, {h~hj} > JEI..

{h~} and {hj}. So that {{hj}IO

< h j E LOO(O,p,),Vi, and Lh; = l,a.e.p,} is i

a directed set with respect to " <"). Proof. First, we claim that P,{hD -< P,{hj} ( C. M. ) if {h~}k=l < {h; }j=l· In fact, let P,{hD = L a~Ot~, P,{hj} = L a;8tj, and {It,· .. ,1m } be a partik

tion of {I,··· n} such that

L

! fhjdp,

JEI..

;

h~ = L hi' 1 < k < m. Since a~f(t~) =

! fh~dp,

=

JEI..

= L

a;f(t j ) , "If E E", it follows that

JEI..

L

a~t~ =

ajt;

in E,

1

< k < m.

iEI"

Then for any convex function 9 E Cr(O) we have

! 9dp,{h~}

=L k

a~g(t~) = a~>O L a~g (L

a: t i )

fEI.. ak

< L a~ L a: g(tj) = L ajg(t j ) = a~>O

( notice that if a~ =

C.M. ).

JEI..

ak

f gdp,{hi}

j

°then it must be aj = 0, Vi

Elk), i.e., P,{hD

-< ""{hi} (

549

Now we prove that J1, = O'-limJ1,{h-}' {hj}

J

In fact, for any g E C(n) and e > 0, we can find a family {Vb'··' Vn } of closed convex subsets of n such that Uj=l Int(V;) = nand

Ig(s) - g(t) I < s, if s, t E

Vi

for some j.

For each j E {I",., n}, pick hi E C(n) such that 0 < h; < 1, supp h; C Int (V;), and

I; h; -

! h;dJ1,

and t; E

Vf E E\

Vj.

1. Suppose that cr.; =

n such that

J

! fh jdJ1"

cr.if(t;) =

If for some j, cr.; > 0 and t f (j. V;, then there is fEE'" such that Ref(t;) > sup Ref(t). But tEV,.

Ref(ti)

=

;.! Ref(t)h;(t)dJ1,(t) = ~ r Ref· h;dJ1, Cti ]Vj

J

< cr.;I sup Ref(t) / hjdtt = sup Ref(t), tEV,.

tEV,.

so we get a contradiction. Hence, t j E

Vi

if

Cti

>

o. Further,

1/ gdJ1, - ! gdJ1,{h,.}!

I! gdJ1, - I; ! hidJ1,g(ti) I J

I;! /g(t) -

<

g(t j ) /hj(t)dJ1,(t) < c.

J

Now if {h~H~l > {hj}i=l' then there is a partition {I.,···, In} of {I"", m} such that h; = h~, 1 

0, Vi E

a.e.J1, on n\Int(V;).

t.; where Ct~ =

! h~dJ1"

Vf E E"'. Then

!

I <

gdJ1, -

! gdJ1,{hH I

L! Ig(t) - g(Sl) Ihat)dll(t) I

LL ;

lEI,.

a

1Ig(t) - g(SI) Ih t)dJ1,(t) < e. Vj

and

cr.~f(8,)

550

Therefore,

Vg E C(n).

JL(t) = lim JL{hj} (g) ,

Q.E.D. Proposition 14.2.9. Let A, S = S(A), tp E S, {1rcp, u.; lcp}, C C 11"'1' (A)' and pH'I' = Clcp be the same as in Theorem 14.2.7. Suppose that JL is the C -measure of cp. Then

JL = a (C (Sf, C (S))- lim JL{B .} , J {Bj}

.i.e.,

lim r gdJL{B'}' isr gdJL= {Bj} is

VgE C(S),

J

where

e, E C+, Vj, L n, =

1; JL{Bj} =

i

L a.jOpi' for each i, a.i =

(Bj l cp , 1'1')' Pi E

;

S such that

a.; Pi(a) = (11" 'I' ( a) B; 1'1" 1 cp) ,

Va E A;

{B~}r=l

< {Bj};=l if there is a partition {II,"', 1m } of {I"", n} such that I: B;, 1 < k < m. Morevoer, if {B~} < {Bi } , then JL{BD --< JL{Bj} --< JL (

B~ =

iElk

C.M.) .

Proof.

Let

r be the C-isomorphism of cp from C (x1I"cp(a)lcp,lcp)

=

onto LOO(S,JL) . Then

£

et(p)(rx)(p)dp,(p),

Va E A, z E C. For any {Bi}, let hi = r B" Vj. Now using Lemma 14.2.8 to E = (A*,cr(A",A)),n S and the C-measure JL of cp, we can get the conclusion.

Q.E.D. Let A be a C·-algebra with an identity 1, S = S(A) be the state space of A, cp E S, {11" '1" Hcp, Ill'} be the cyclic * representation of A generated by cpo Suppose that Ci is an abelian VN algebra on Hcp with Ci C 1I"cp(A)', and JLi is the Ci -measure of cp on S, i = 1,2. Then the following conditions are equivalent: (i) JLI --< JL2 ( C.M.), i.e., for any convex function g E Cr(S),

Theorem 14.2.10.

Is gdJLI < Is gdJL2; (ii)

551

(iii)

c, C

C2 Consequently, if 0

JLl

= JL2, then C l = C 2

0

(i) ==> (ii) . It is obvious since a( .)2 is convex on S, Va* = a E A. (ii) ==> (iii) . Suppose that the inequality (ii) holds, Let Pi be the projection from H4p onto C,l4p' I', be the C,-isomorphism of rp from C. onto Loo(S,JLi)' and Ai = fi l , i = 1,2. Notice that Ai(a)Pi = Pi1r4p(a)Pi, Va E A, i = 1,2. For any a = b+ ic.b" = b,c* = c E A, we have

Proof.

(Pl~~(a)I~'~4p(a)I4p) = (Pl1r4p(a*)pl1r4p(a)Pll4p,I~)

=

(Al(a*a)I~, 14p) = L[b(P)2 + c(p)2]dJLl(P)

< t[b(p)2 + c(p) 2]dJL2 (p) = (A 2 (a*a)I4p ' 14p) = (P21r ~ (a) 1 ~' 1r 4p (a) 14p) .

Hence we get PI < P2 since 14p is cyclic for 1r~(A), which means that C 114p C C 2 14p o Let U be the isometry of L 2(S,JL2) onto C21cp given by U I = t\2(/)1~, VI E Loo(S,JL2)' For any x E Ct, by C 1 1lp C C 214p there exists I E L 2(S,JL2) such that U I = xl~o For any n = 1,2,0", let En = {p E SII/(p)! < n} and en = 1\2 (XE,J. Then we have en x l 4p

= 1\2(XEn)UI = 1\2 (XEn) lim U (XE... I) f7&

= 1\2(XEn) limI\2(XE... /)I4p = m

1\2(XEnl)1~

since XEml ----+ I in L 2(S,JL2)' But 14p is separating for 1r4p(A)', so we get enx = 1\2(XEn/) E C2, Vn. Moreover, since XE n --+ 1 in O'(Loo, L 1 ) and 1\2 is normal, it follows that en ---t 1 ( weakly). Hence, x E C 2 , and C 1 C C 2 • (iii) ==> (i). Suppose that C 1 C C 2 • By Proposition 14.2.9, we have JLl

= limJL{B j }

VB; E (C1)+ C (C 2)+, and

and

JL{Bi} -< JL2(C.M.),

L e, = 1. Therefore, JLl -<

li-2 (

C.M.).

Q.E.D.

i

Corollary 14.2.11. If 1r~(A)' is abelian, let C = 1r~(A)', then the Cmeasure Ii- of rp is maximal (C.M.) on S = S(A). Consequently, Ii- is pseudoconcentrated on the pure state space P(A) in the sense that JL(E) = 0 for every Baire subset E of S(A) disjoint from P(A). Moreover, if A is separable, then JL is concentrated on P (A).

552

Proof. Let v be a probability measure on S, and v :>- #L ( C.M. ). By Lemma 14.2.8, it suffices to show that #L :>- V{hj} (C.M. ) , VO < hi E LOO(S, v), and h; = 1, a.e.v.

E ;

Let V{hj}

= ~(Xic5pj' where

,

(Xi

=

!

hidv,Pi E S, and (XiPi(a)

=

!

ahidv, V

a E A. Since v :>- #L( C.M.), v :>- V{hj}( C.M. ) and #L E n(lp), it follows that t/,

V{hj} E n(lp). Hence, / ad#L = lp(a) =

lp > (XiPi > 0, vi- Hence, for each

Clearly,

L

B;

= 1,

and V{h j }

i.

~ (Xi Pi(a), Va

,

E A. Consequently,

there is B; E 1r/p(A)~ = C+ such that

= #L{Bj}'

Now by Proposition 14.2.9, we obtain

;

that

Q.E.D. Remark. Applying Choquet theory (Section 14.1) to S(A), and by n(lp) =1= 0( see Definition 14.2.1), there is a maximal ( C.M. ) probability measure #L on S(A) such that

lp{a) = (

}S(A)

a(p)d#L(P) ,

Va E A.

If 7r/p(A)' is abelian, then by Corollary 14.2.11 we can pick that #L is the Cmeasure of lp, where C = 1r/p(A)'. #L is pseudoconcentrated on the pure state space P(A). So lp seems to be an "integral" of pure states. In particular, if A is separable, then we have

lp(a) = (

}P(A)

a(p)d#L(P),

Va E A.

Proposition 14.2.12. Let #L E n(lp) ( see Definition 14.2.1 ) . Then #L is the C-measure of lp for some abelian VN algebra C C 1r/p(A)' if and only if for each Borel subset E of S there is a projection PE of 7r/p(A)' such that

L

XE(p)a(p)d#L(P) = (PE1r/p{a)l/p' 1/p), Va E A.

Proof. The necessity is obvious from Theorem 14.2.7. Conversely, for any f E LOO(S,#L) with 0 < f < 1 define

lpf(a) = / fad#L,

Va E A.

553

Clearly, 0 < CP, < cp. Thus, there exists unique xI E 1r1p{A)' such that 0 < xI < 1 and

f ladlL = (xl1rlp{a) lip' lip)'

Va EA.

Then we can get a bounded positive linear map I ----io xI from Loo{8,1L) to 1r1p{A)'. By the condition, xI = PE is a projection if I = XE·for some Borel subset E of 8. Let EI, E 2 be two Borel subsets of 8, and E 1 n E 2 = 0. Since XE l + XE2 = XE 1uE2' it follows that PEl + PE2 = P ElUE2' By the positity, we have PE 1UE2 > PEt' i = 1,2. Hence, PE lPE2 = (P El UE2 - PE 2)PE2 = o. Now for any Borel subsets Ei, E 2 of 8, by PEt = PE lnE2 + PE,\E 1nE2, i = 1,2, we have PE lPE2 = P El n E2' Hence, for any simple functions I, g on 8 we obtain that x,xg = X'g' Further, I ----io x, is a * homomorphism from Loo{8, IL) to

1r1p(A)'.

If x, = 0 for some real IE L oo(8,1L), then xh = xh, where 0 < 117/2 E L OO(8,J.t) , I = II - hand 11h = O. Hence, xI? = xhxh = xhh = O. Consequently,

f l;adlL =

So that

II

(x/{1rIp(a) lip' lip)

= O. Similarly, h = O. Therefore, I

L oo(8,1L) to 1r1p(A)'. Now let C = {xIII E L oo(8,1L)}, and

fXI

= 0, ----io

Va E

x,

A.

is a * isomorphism from

= I, VI

E LOO(8,1L). Since for

any x E C,a E A,

it follows from Lemma 14.2.5 that IL is the C-measure of cp.

Q.E.D.

Notes. The construction of C-measure and C-isomorphism was presented by D.Ruelle and S.Sakai. Theorem 14.2.10 is due to D.Ruelle. References.

[138], [140], [148J.

14.3. Extremal decomposition and central decomposi-

tion Theorem 14.3.1. (Extremal decomposition ) Let A be a C·-algebra with an identity 1,8 = 8(A) be its state space, cp E 8, {1rIp' Hlp, lip} be the cyclic * representation of A generated by cp. If C is a maximal abelian VN subalgebra of 1r1p(A)', then the C-measure IL of cp is pseudoconcentrated on the pure state

554

space P( A) in the sense that p.(E) = 0 for each Baire subset E of S disjoint from P(A). In particular, if A is separable, then p. is concentrated on P(A).

Proof. Let B be the C*-algebra on Hrp generated by 1rrp(A) and C. Clearly, B' = C. Let ~(.) = (·11p' 1rp), V· E B, and {7rV;' HV;' IV;} be the cyclic * representation of B generated by ~. Clearly, there is a unitary operator U from Hrp onto H~ such that U1rp = lV;' U*1rV;(x)U = X, Vx E B. Hence, 1rV;(B)' = (UBU*)' = UCU* is abelian. By Corollary 14.2.10, the 1rV;(B)'-measure ji, of rp is a maximal (C.M.) probability measure on S, where S is the state space of B. Let Srp be the state space of 1rIp (A). We have a natural continuous map r from S onto Sip' i.e., r(p) = pl1rrp(A) , Vp E S. Then p.rp = ji,or- 1 is a probability measure on S rp' For each Prp E Srp, Prp 01rrp is a state on A. Hence, Srp can be regarded as a compact convex subset of S. For each Borel subset E of S, define

Then p. is a probability measure on S. 1) P. is the C-measure of cpo In fact, for any ab ... ,an E A, by the definitions of p. and p.rp we have

l

tll(P) ... iin(p)dp.(p)

r Xl(Prp)'"

Js",

xn(plp)dp.rp(prp)

hX1(r(p))'" xn(r(p))djL(P) , where Xi = 1rrp(ai) , 1 < i < n. Since r(p) = ,o11rrp(A) and Xi E 7rrp(A), it follows that xi(r(p)) = r(p)(xi) = ,o(Xi) = Xi(P) , V,o E S, 1 < i < n. Hence,

l (ll(P) ... (In(p)dp.(p)

= !SX1(P) ... xn(P)dji,(p).

Let 11 be the projection from H~ onto 1rV;(B)'IV;' Since ji, is the 1r~(B)'-measure of~, and Xi E B, Vi, we have

Is xt{P) ... xn(P)dji,(p) =

(P1rV;(Xl)P' .. P7rV;(xn)jH~,1~).

Noticing that U1rp = IV;; U*1r~(Xi)U = Xi, Vi; and U*jJU = p, where p is the projection from Hrp onto C1rp, we obtain that

Ltll(P) ... tln(p)dp.(p)

= (PXIP" . px nPlrp, 1rp)

= (P1r rp( all P: .. pt:rp (an)p1rp, 1rp).

555

Now by Definition 14.2.4, JL is the C-measure of cpo 2) H p is a pure state on B, then r(p) is a pure state on 1rrp(A). In fact, let {1r, H, €} be the irreducible cyclic * representation of B generated by p. Since C is abelian and C C 1rrp(A)', it follows that either 1r(q) = 1H or 1r(q) = 0 for each projection q of C. So there is a non-zero multiplicative linear functional A on C such that 1r(c) = A(c)lH,\lc E C. Then {1r,H} is irreducible for 1rrp(A). Since € is cyclic for 1rrp(A), and r(p))(x) = p(x) = (1r(x)€, e), \Ix E 1rrp(A) , hence the * representation {1r, H} of 1rrp(A) is unitarily equivalent to the * representation of 1rrp (A) generated by r (P). Therefore, r (p) is pure on 1rrp(A). 3) JLrp is pseudoconcentrated on ExSrp. In face, let Erp be a Baire subset of Srp disjoint from ExSrp. By 2 ) , r-1(Erp) is a Baire subset of S disjoint from ExS. Since iJ, is maximal (C.M.), it follows that JLrp(Erp) = iL(r- 1 (Erp)) = O. Finally, we prove that JL is pseudoconcentrated on P(A) = ExS. Let E be a Baire subset of S disjoint from ExS. Then E n Srp is a Baire subset of Srp. By Proposition 2.4.11, ees; C ExS. Hence, (EnSrp)nExSrp = 0. Now by 3) we obtain that

Q.E.D. Theorem 14.3.2. (Central decomposition ) Let A be a C*-algebra with an identity 1, S = S(A) be its state space, cp E S, {1rrp, Hrp, 1rp} be the cyclic * representation of A generated by cp, and Z = 1rrp(A)" n 1rrp(A)'. Then the Z-measure JL of cp is pseudoconcentrated on the factorial state space 1 in the sense that JL( E) = 0 for any Baire subset E of S disjoint from ,. In particular, if A is separable, then JL is concentrated on 1.

Proof. Let B be the C*-algebra on Hrp generated by 1rrp(A) and 1rrp(A)'. Then B' = Z. Consider the state ~(.) = (·lrp, 1rp) on B. Similar to the proof of Theorem 14.3.1, the 1r~(B)'-measure j1 of ep is maximal (C.M.) on S, where S is the state space of B. Let Srp be the state space of 1rrp(A), and r(p) = pl1rrp(A), \lp E S. Then JLrp = j1 0 r- 1 is a probability measure on Srp. Srp can be regarded as a compact convex subset of S, and JL(E) = JLrp(E n Srp)(\1 Borel E C S) is a probability measure on S. Similar to the proof of Theorem 14.3.1, JL is the Z-measure of cpo

Let

p be a

pure state on B, {1r, H, €} be the irreduable cyclic

* representa-

556

tion of B generated by

p, and a

= 1r1p(A). Then

1r(a)" n 1r(a)' = {1r(a) C

{1r(a)

U 1r( a)'}' U

1r(a')}' =

(]J1H.

So 1r(a)" is a factor on H. Let q be the projection from H onto 1r(a)e. Clearly, q E 1r(o:)',1r(a)"q is a factor on qH, and {q1r,qH, e} is a factorial cyclic * representation of a. Since (q1r(x)e, e) = (1r(x)e, e) = p(x) = r(p)(x), Vx E 0:, the * representation {q1r, qH} of a is unitarily equivalent to the * representation of 0: generated by r(p). Therefore, r(p) is a factorial state on 1r1p(A). Let ,ip be the factorial state space on 1r1p(A), and Elp be a Baire subset of Sip disjoint from lip' From the preceding paragraph, r- 1(EIp) n ExS = 0. Since iJ, is maximal (C.M.) , it follows that J1.1p(EIp) = iJ,(r- 1(EIp)) = O. Finally, let E be a Baire subset of S disjoint from 1. By lnSrp = IIp,EnSIp is a Baire subset of Sip disjoint from lip' Therefore, J1.(E) = J1.1p(E n Sip) = 0, i.e., J1. is pseudoconcentrated on 1. Q.E.D. The Z(= 1r1p(A)" n 1r1p(A)')-measure is also called the central measure of cp. Now we give an equivalent definition of central measure.

Definition 14.3.3. JI, E n(cp) ( see Definition 14.2.1 )) is called the central measure of cp, if there is a normal * homomorphism W from the center Z" of A'" onto L 00 ( S, JI,) such that

cp(za) =

Is W(z) (p)a:(p)dJl,{p) ,

Vz E Z ... , a E A ( notice that cp can be uniquely extended to a normal state on AU). The following proposition will show that W is unique, and this W is called the central homomorphism of cp.

Proposition 14.3.4. The JI, and W in Definition 14.3.3 are existential and unique, and JI, is the Z-measure of cp exactly, where Z = 1r1p(A)" n 1rIp{A)'.

Proof. Let JI, be the Z-measure of cp, and f be the Z-isomorphism from Z onto LOO(S,JI,), where Z = 1r1p(A)" n 1r1p(A)'. Then we have (x1rIp{a) lip' lip) =

Is (fx)(p)a(p)dJl,(p),

Va E A,x E Z (s~e Theorem 14.2.7). By Theorem 4.2.7, the * representation {1rIp' Hlp} of A can be uniquely extended to a W*-representation {1r;,HIp} of A"', and 1r:(A**) = 1rrp(A)". For each z E Z .. , define

w(z) = f(1r;(z)).

557

Clearly, W is a normal * homomorphism from Z"" to LOO (S, p.). Let z be the central projection of A"" such that A .... z = ker Then is a * isomorphism from A.... (l-z) onto 1r:(A**) = 1r1p(A)". Then center of A**(l-z) is Z**(l-z)( see Proposition 1.3.8. ). Hence, Z = 1r:(Z.... (l - z)) = 7r:(Z**), and W is surjective. Moreover, for z E Z .... , a E A, let x = 1r:(z). Then x E Z and

1r:.

~(za) =

=

1r:

(1r:(za) lip' lip) = (x1rlp(a) lip' lip)

L

(fx)(p) a(p) dJ1,(p)

=

L

w(z)(p)a(p)dp.(p).

Now suppose that p.' and W' satisfy the conditions in Definition 14.3.3. We need to show that p.' is equal to the Z-measure p. of ~, and W' is equal to the above W. Let z E (z .... n ker 7r:)+. Then by 0=

(1r;(z) lip' lip) =

~(z)

=

Lw'(z)(p)dJ1,'(p)

and w'{z) > 0, we have w'(z) = o. So we can define a from Z onto Loo(S, p.') as follows:

f' (1r;(z))

=

\lI' (z),

* homomorphism I"

Vz E Z ...

H w'(z) = 0 for some z E Z**, then by Definition 14.3.3 we have

Va E A. Since lip is cyclic for 7rcp(A), it follows that 1r:(z) = O. Hence I" is a * isomorphism. Moreover, by (x1rlp(a) lip' lip) . (1r:(z)1rIp(a)llp' lip) = ~(za) =

f w'(z)(p)a(p)dJ1,'{p)

=

! (f'x)(p)a(p)dp.'(p),

Vx E Z, a E A, where z E Z**

and 1r:{z) = x, and by Lemma 14.2.5, J1,' is the Z- measure p. of the Z-isomorphism I' of~. Further, \lI' = W.

~

and I" is Q.E.D.

Now we consdier a geometrical characterization of the central measure in n(~).

Definition 14.3.5. Let A be a C*-algebra with an identity 1 , and S = S(A) be the state space of A. Let be a W" -representation of AU. The support of 1r, denoted by s(1r), is the central projection of A" such that ker 1r = A" (1 - S (1r)) . Let ~l, ~2 be two positive linear functionals on A. ~1 and ~2 are said to be disjoint, if s(1rf) . s(1r2') = 0, where 1r;' is the W"-representation of AU generated by ~i, i = 1,2.

1r

558

Now let II be a probability measure on S. For each Borel subset E of S, we can define a positive functional vs on A as follows:

IIE(a) =

L

a(p)dv(p) ,

Va E A.

v is said to be semi-central ,if for any Borel subset E of S, £IE and VS\E are disjoint. Fix cp E S. Let

Oc(p) = {v E O(cp) Iv is semi-central }, where O(cp) is the same as in Definition 14.2.1.

Theorem 14.3.6. Let Z = 1rcp(A)" n 1rcp(A)'. Then the Z-measure ( central measure) of cp is semi-central, and there is a bijection between Oc (cp) and the collection of all abelian VN subalgebras of Z, i.e., for each v E Oc(cp) there is ( unique) abelian VN subalgebra C of Z such that v is the C-measure of cp; conversely, if C is an abelian VN subalgebra of Z, then C-measure of cp is semi-central. Proof. Let C be a VN subalgebra of Z, v and r be the C-measure and C-isomorphism of cp respectively, and E be a Borel subset of S. Let P1 = r- 1XE, P2 = f- 1xs\E. Then PbP2 are projections of C,P1P2 0,P1 + P2 = 1, and by Theorem 14.2.6, £IE (a)

=! =

VS\E(a)

XE(p)a(p)dv(p)

=!

(fpd(p)a(p)dv(p)

(P11rcp(a)lcp, 1cp),

= (P21rcp(a)lcp, 1cp),

Va E A.

Since 1r:(Z"'*) = Z => C, so we can find projections Z1, Z2 of Z1Z2 = 0, Z1 + Z2 = 1, and 1r:(Zi) = Pi, i = 1,2. Then

z*·

such that

Hence, VE(Z2) = VS\E(Z1) = O. Let 7rE,1rS\E be the W*-representations of AU generated by £IE, VS\E respectively. Since Z1, Z2 E Z .., it follows that ZI E ker 1rE,Z2 E ker 1rS\E. Further, ZI < (1- S(1rE)),Z2 < (1- S(1r.\E)), and S(1rE) . S (7r S\E) = O. Therefore, v is semi-central. Now let v E Oc(cp). By the proof of Proposition 14.2.12, there is a bounded positive linear map f ~ xf from Loo(S, v) to 1rcp(A)' such that

f

fadv = (xf1rcp(a)lcp, 1cp),

Vf E tr (S, v), a E A.

559

Let E be a Borel subset of S, and f = XE' By Proposition 14.2.12, it suffices to show that x I is a projection of Z. Let g = XS\E . Then

Ladv (

lS\E

= vf(a) = (xf1rlp(a) lip' lip)'

adv = vg(a) = (xg1rlp(a) lip, lip)'

Va E A.

Further, let {1rI,Hf,lf} and {1rg , H g,l g } be the cyclic W*-representations of A.... generated by vf and vg respectively. Since v E nc(lp), it follows that s(1rf)' s(1rg) = O. Write z = s(1rf),z' = s(1rg). Then z,z' is the central projections of A", and z + z, < 1. Since 1rf(z) = I, ( the identity operator on H,) and 0 < xf < 1, for any a E A+ we have

(1r;(za) lip' lip) > (1r;(za)x f 11p' lip) =

vf(za)

=

(1r/(a) 1" If)

=

(1rf(za)l" If) =

vl(a)

=

(xf1rlp(a) lip' lip)'

Hence, ((1r;(z) - Xf) 1r1p (a) lip' lip) > 0, Va E A+. Further, 1r;(z) > xf' Similarly, 1r:(z') > Xg' Since z + z' < 1, and xf + Xg = XI+g = 1, it follows that 1 - 1r:(z) > 1r:(z') > Xg = 1 - Xf, and Xf > 1r;(z).

Q.E.D.

Hence, xf = 1r;(z) is a projection of Z since 1r:(Z"'*) = Z.

Corollary 14.3.7. Let p, E n(lp). Then p, is the central measure of lp if and only if p, E nc(lp) and p, is the largest (C.M.) measure of Oc(lp).

Proof.

It is immediate from Theorems 14.2.10 and 14.3.6.

Q.E.D.

Notes. The formulation of the central decomposition of a state was first given by S.Sakai. Theorem 14.3.2 in non separable cases is due to W. Wils. References.

[148], [197].

14.4 Ergodic decomposition and tracial decomposition Let (A, G, a) be a dynamical system, where A is a C·-algebra with an identity 1 , G is a group, and a is a homomorphism from G to the * automorphism group Aut(A) of A. Further, let S = S(A) be the state space of A, and

SG

= {lp E Sllp(a,,(a))

= lp(a), Vs E G, a E

A},

560

i.e., 8 G is the set of all G-invariant states. Clearly, 8 G is a closed ( compact ) convex subset of (8, u(A..., A)). Fix cp E 8 G , let {1fIp,HIp, lip} be the cyclic * representation of A generated by cp , and define ulp(s)1fIp(a)11p = 1f1p(aa-1(a))IIp' Va E A,s E G. Then UIp(s) can be uniquely extended to a unitary operator on Hlp, denoted by ucp(s) still, Vs E G. Clearly, s ~ ulp(s) is a unitary

representation of G on Hlp, and {1fcp, ulp' Hlp} is a covariant representation of (A, G, a), i.e., UIp(s)1fcp(a)ulp(s)* = 1fcp(aa-1(a)), Vs E G, a E A. Moreover, ulp(s)lcp = lcp,Vs E G. Let Ecp = {e E Hcplulp(s)e = Vs E G}, and Pip be the projection from Hlp onto Elp. Clearly, Elp is a closed linear subspace of Hcp, and lcp E Elp.

e,

Proposition 14.4.1. With the above notations, we have that: (i) Plpulp(s) = ulp(s)PIp = Pip, Vs E G, and Pcp E Coulp(G)", where" - a" means the strong closure; (ii) {1fIp(A) , Pcp}' = {1fIp(A),ulp(G)}'; (iii) Let MIp = {1fIp(A) , Pcp}" = {1fIp(A) , ulp(G)}". Then the central cover of Pip in MIp is I , and x' ~ x'PIp is a * isomorphism from M~ onto M~plp; (iv) Let Nip = (plp1fIp(A)plp)" ( a VN algebra on plpHcp = EIp). Then N~ = (plp1fIp(A)pcp)' = M~plp. Proof. (i) We need to show that for any '711· .. ,'7n E Hlp and e > 0, U{PIp' '711 "','7n,c) n Coulp(G) =I 0, where U(PIp,'7b···,'7n,c) = {x E B(HIp)III(plpx)'7ill < s, I < i < n} is a strong neighborhood of Pip in B(HIp). Let Hlp = s; ED E;. Since ulp(s)EIp C EIp' it follows that ulp(s)E; C E;, Vs E G. For any '7 E E;, let r'1 = Col ulp{s)'7ls E G}. r, is a closed convex subset of Hlp. Hence, there is unique '70 E r" such that 11'7011 = min{11 elll e E r,,}. Since ulp(s)r'1 C r, and Il UIp (s)'70 11 = 11'7011, it follows from the uniqueness of '70 that ulp(s)'7o = '70, Vs E G. Hence, '70 E Elp n E~ = {O}. From this fact, A~I) = I such that for '71 we can find A~I) > 0, S;I) E G, vi, and ;

I:

II I: A)I)UIp(s?»)(1 -

Pip) '71 II < c.

i

For '72 , similarly there are A~2) > 0, s~2)

E

G, Vk, and

L A~2) = I such that Ie

II L Ar)UIp(s~2») L A}l)UIp(S}l»)(l Ie

j

Pip) '72 II < c.

561

And we also have

11:E Al2)ucp(sl2») 2: A}l)Ucp(S}l»)(1 -

pcp)'lll1 < c.

i

Ie

.... Generally, we can find x E Coucp(G) such that

i.e., U(pcp, '71,' . " n«, c) n COUcp(G) =I- 0. (ii) Since Pcp E ucp(G)", it follows that {1rcp(A),ucp(G)}' C {1rcp(A),pcp}'. Conversely, let x E {1rcp(A),pcp}'. Then for any S E G, a E A we have

xUcp(s)1rcp(a)lcp = X1rcp(O:6-1(a))lcp

=

1rcp(O:6-1(a))xlcp

= ucp(s)1rcp(a)u:xpcplcp =

ucp(s)1rcp(a)ucp(s)+pcpxlcp

= ucp(s)1rcp(a)pcpxlcp = ucp(s)x1rcp(a)lcp. Hence, ucp(s)x = xUcp(s), \Is E G, and x E {1rcp(A) , ucp(G)}'. (iii) By [McppcpHcp] :> 1rcp(A)lcp = Hcp, the central cover of Pcp in Mcp is 1 . (iv) M~pcp = (pcpMcppcp)' = (pcp1rcp(A)pcp)' = N~. Q.E.D. Theorem 14.4.2. pcp1rcp(A)pcp is commutative if and only if for any at, a2 E A, E Ecp, e > 0, there are Ai > 0, s, E G, and Ai = 1 such that

e

L i

I(e, [:E Ai 1rcp (O:6i(a tl ), 1rcp(a2)]€)1 < e, i

where [x, y] = xy - yx, \Ix, y E B(Hcp).

Proof. For any at, a2 E A with lI a l ll < 1,ll a 211 < 1, and 6,6 E pcpHcp with 11611 < 1,1I61L5 1,_and 8 > 0, by Proposition 14.4.1 (i) we can pick Ai > 0, Si E G, and 1.: Ai - 1 such that i

11(2: AiUcp(Si) -

pcp)1rcp(a2) 611 < 8/2

i

and

II (2: AiUcp (Si) i

- Pcp)1rcp (ai) 611 < 8/2.

562

Then by u11'(s) €i

= u11'(S)P11'€i = €i, i =

1,2, Vs E G, we have

1(6, (7r11'(adp11'7r11'{a2) - 7r11' (a2)p11'7r11' (AI)) 6) - (€l' [L Ai 7r11' (a 8i (al)) , 7r '(J (a2)] 6) I i

I( 6, 7r11'(at}p11'7r11' (a2) 6) -

<

(6, L Ai U11'(Si)*7r11'{ al)u11'(si)7r11'( a2) 6) I i

+1 (€b 7r11' (a2)p11'7r 11'( al) €2) - (€b L Ai 7r '{J (a2)U11'( Si) + 7r11' (al) U11' (Si) 6) I

II (p11' -

<

i

L Ai u'{J (Si)) 7r11' (a2) 611 i

+1 (P11'7r11'(a;) €b 7r 11'{ad 6) - (L Ai U11'( Si) 7r11' (a;) 6, 7r11' (al) 6) I i

< II (P'{J - L Ai U11'( si))7r11'(a2) 611 + II (p11' - L Ai U11'( Si))7r11'( a;) 611 < 6. i

Let P'{J7r11'{A)p11' be commutative. For any € E E11" at, a2 E A and e > 0, put 6 = 6 = €, and 6 = Il alllll:21111€11 2 ' Then by the above inequality there are

Ai > O,Si

E G and

LAi

•

= 1 such that

I(€, [L Ai 7r'{J(a8 ; { a d ) , 7r11'{a2)]€) I <

e

i

since (€, (7r'{J{al)p11'7r11'{a2) - 7r11'(a2)p11'7r11'(ad)€) = 0. Conversely, let at, a2 E A, € E E11' and e > 0. By Proposition 14.4.1 pick Jli > 0, tj E G and L Jlj = 1 such that

(i) ,

i

II(LJljU11'(ti) - p11')7r11'(a2)€11 = II LJlju'{J(tj)(l- p11')7r'{J(a2)€11 < e/2 j

and

II(LJljU11'(t i) - p'{J)7r11'(a;)€1I = II LJli u'{J(t j)(l- P11') 7r11' (a;) til < e/2. i

j

By the assumption, for and

L

L

Jli atj ( ad, a2, t and

e

> 0, there are Ai > 0, Si

j

Ai = 1 such that

i

I(t, [L Ai 7r11'(L Jlja8i t j (al)) , 7r'(J( a2)] t) I < e. i

i

Since

IHL AiJli u11'(Sitj) - p11')x€11 = II L AiJli u11'(Sitj)(1 - p11')xtll ij

-

ij

II LAiU'{J(Si) LJli U11'(tj)(l- p11')xtll < e/2, i

j

E G

563

where x = 1I'"Ip(a2) or 1I'"Ip(a;), it follows that

I(€, (1I'"Ip(at}plp1l'"Ip(a2) - 1I'"Ip(a2)plp1l'"Ip(al))€)

- (€, [L AiJLj1l'"Ip (a",tA al)), 11"Ip (a2) 1€) I <

I(e,

ij 11"Ip( al) Pip 11"Ip (a2) e)

AiJLjUIp(Sit;) *11"Ip( at}uIp( Sit;) 11"Ip(a2) €) I ij + I(e, 11" Ip(a2)plp1l"Ip(ad €) - (€, L AiJLj1l"Ip (a2) Ulp (Si t ;)* 11"Ip (al) UIp(Sit;) €) I

ij AiJLjUIp (Sit j))1I"Ip (a2)€11

L i; +11€llllalllll(pIp - L

< 1I€llllalllll(pIp -

i;

Hence

- (€, L

AiJLjUIp(Si t;))1I"Ip(a;)€11 < cll€lllIalll·

I(€, (11"Ip (at}plp1l"Ip( a2) + cll€llllalll. A, e > 0 and € E Elp <

Since aI, a2 E

11"Ip (a2) Pip 11" Ip

(al)) e) I

e

are arbitrary, plp1l"Ip(A)pIp is commutative.

Q.E.D. Corollary 14.4.3.

If for any aI, a2 E A,

€E

EIp, we have

Theorem 14.4.4. Let plp1l"Ip(A)pIp be commutative, and C = {1I"Ip(A),pIp}'. Then C is abelian, C C 1I"Ip(A)' and Elp = Clip' Suppose that JL is the Cmeasure of cp. Then supp JL C SG, and v -< JL ( C.M.) for each probability v on SG with a(p)dv(p) = cp(a), Va E A. Consequently, J.L is pseudoconcentrated

r

lSG

on ExSG in the sense that JL(E) = 0 for each Baire subset E of SG disjoint from ExSG. Moreover, if A is separable, then JL(ExSG) = 1.

Proof. By the 2) of Proposition 14.2.2, C is abelian, C C 1I"Ip(A)' and plpHIp = Elp = Clip' . From Proposition 14.2.8, JL = (J - limJL{B j } , where B; E C+, vi, L B; = ;

l;JL{Bj}

= LA;Dpj -< JL

Ajp;(a) have

=

= (B;llp,llp)'p; E

S such that

> 0, then for each S E

G, a E A we

(C.M.), for each j,A;

j

(1I"Ip(a)Bjllp' lip)' Va E A. If Aj Pj(a,,(a))

= Ajl(1I"Ip(a)uIp(s)B;plpllp' UIp(S) Pip lip)

= Ajl(1I"Ip (a)uIp (s)plpB;llp' uIp(s)plpllp) = p;(a).

564

Hence, Pi E SG, vi, and supp

It C SG'

Now let v be any probability measure on SG with f

1sG

a(p)dv(p) = cp(a) , Va

EA. By Lemma 14.2.7, v = (J-limlt{hj}' where 0 < hj E LOO(SG, v), Vj,'Lhj = j

l(a.e.v);v{ho} = L AiDp" for eachj,Aj = f hidv,pj E SG such that A;pj(a) = 1 1 1s i

G

£G a(p) hi (p)dv(p) , Va E A. Now it suffices to show that V{hj} -< v (C.M.) on SG'

Since

~ A;p;(a) = / a(p)dv(p) = cp(a), Va E A, cp > A;Pi(> 0)

,

on A, vi.

Then for each i, there is B j E 7rrp(A)~ such that

Ajpj(a) = (7rrp (a)Bj1rp, 1rp),

Va E A.

By Pj E SG and 7rrp(A)lrp = Hrp we can see that

B j1rp = urp(s)Bj1rp,

Vs E G,and J'.

On the other hand,

7rrp(a)urp(s)Bjurp(st = Urp (s) 7rrp(a, (a))Bjurp(s) * urp( s) B j 7r rp( a, (a) )urp( s) *

Urp(s) Bjurp (s) *7rrp(a) ,

Va E A.

Hence, urp(s )Bjurp(s)* E 7r rp(A)'. Further

urp(s)Bjurp(s)*lrp = urp(s)Bjlrp

= B j1rp.

But 1rp is separating for 7rrp(A)', so we get urp(s)Bjurp(s)* = B j , Vs E G, i.e., B j E urp(G)'. Further, by Proposition 14.4.1, B j E 7rrp(A)' n urp(G)' = {7rrp(A),urp(G)}' = C,Vi- Clearly, B, > O'LBj = 1, and V{hj} = It{B j } . j

Therefore,

It ~ It{Bj} =

V{hj} ( C.M. ) , V{h j

},

and

It ~

v( C.M.) on SG.

Q.E.D. Definition 14.4.5. ep E SG is said to be ergodic, if cp is an extreme point of SG, i.e., cp E ExSG. Propoaition 14.4.6. {7rrp(A),prp}' = (CIH'P' Moreover, if dim Erp

Let cp E SG' Then cp is ergodic if and only if

= 1, then cp is ergodic.

565

Let 'P be ergodic, and h E {7rIp(A) , Pip}' = {7rIp(A),ulp(G)}' with < h < 1. Then 'Ph is G-invariant , and < 'Ph < 'P, where 'Ph(a) = (7rIp(a)h11p' lip)' 'Va E A. Since 'P E ExSG' it follows that 'Ph = A'P for some A E [0,1]. Now by ((h - A) 7r1p (a)lip' lip) = 0, 'Va E A, we get h = A. Therefore, {7rIp(A),plp}' = (C1B". Now let {7rIp(A),plp}' = {7rIp(A),ulp(G)}' = ~lB". If 'P is not ergodic, then there is some p E SG and some A E (0,1) such that 'P > AP and 'P =1= p. Further, we can find h E (7rIp(A)'\~lB,,)+ such that Proof.

°

°

p(a) = (7rIp(a)h11p' lip)'

'Va E A.

By p E SG, ulp(s)h11p = h11p' 'Vs E G. On the other hand,

Ulp (s)hulp (s)·1l"1p (a)

=

ulp(s)h1l"lp(a 8(a))uIp{s)·

=

7rlp(a)uIp(s)hulp(s)""

'Va E A,s E G,

i.e., uIp(s)hulp(st E 7rlp(A)'. Further, since uIp(s)huIp(s)·llp = ulp(s)h11p hlp''Vs E G , and lip is separating for 7rlp(A)', it follows that h E uIp(G)'. Hence, we get h; E 7rlp(A)'nuIp(G)' = {7rlp(A) , uIp(G)}' = ~lB", a contradiction. Therefore, 'P is ergodic. Finally, let dim Elp = 1. Then by Proposition 14.4.1, {7rIp(A) ,PIp}'Plp = (plp7rlp(A)pIp)' = CPlp' Moreover, x' ~ x'PIp is a * isomorphism from {7rIp(A),plp}' onto {7rlp(A),plp}'plp ( see Proposition 14.4.1). Therefore, {7rlp(A) ,Pip}' = ~lB", and 'P is ergoduic. Q.E.D. Definition 14.4.7. The system (A, G, a) is said to be G-abelian, if for any 'P E SG, Plp7rIp(A)plp is commutative. Proposition 14.4.8. Let (A, G, a) be G-abelian, and 'P E SG. (i) (PIp7r Ip(A)plp)" = (PIp1l"Ip(A)plp)' = {7rIp(A) ,Plp}'PIp is a maximal abelian VN algebra on plpHIp = EIp; (ii) {1l"Ip(A),plp}' = {1l"Ip(A),uIp(G)}' is abelian; (iii) 'P is ergodic <=:> {7rlp(A) ,Pip}' = (C1B" <=:>dimEIp = 1. Proof. (i) Since (plp1l"Ip(A)plp)" is an abelian VN algebra on plpHIp, and it admits a cyclic vector lip' hence (PIp7rIp(A)plp)" = (PIp7rIp(A)plp)' is maximal abelian on plpHIp' (ii) By Proposition 14.4.1, (plp7rIp(A)plp)' = {7rIp(A),plp}'plp is abelian, and x' ~ X'PIp is a * isomorphism from {7rIp(A) , Pip}' onto {7rIp(A),plp}'plp' Thus, {7rIp(A),plp}' = {7rIp(A) , ulp(G)}' is abelian. (iii) Let 'P be ergodic. By Proposition 14.4.6, we have {7rIp(A),plp}' = ~lH". Then from (i) ,

566

Therefore, dim E'P

=

1. Now by Proposition 14.4.6, the conclusion is obvious.

Q.E.D. Theorem 14.4.9. The system (A, G, a) is G-abelian if and only if for any lp E SG, at = D.i E A, i = 1, 2,

Proof. Th necessity is obvious from Theorem 14.4.2. Conversely, it suffices to show that

(€, (7r'P (al)p'P 7r'P (a2) - 7r0'P(a2)p'P7r'P(al))€)

= 0,

Vlp E SG; a; = D.i E A with 11D.i II < 1, i = 1,2; and € E P'PH'P = E'P with l1{11 = 1. For any e > 0, by Proposition 14.4.1 there are Ai > 0,5i E G and L x, = 1 such that i

lI(L Aiu'P(si) - P'P) 7r'P(ad€11 < e/2. i

Let a~ = L Ai a 6,-:-1 (al), then by u'P(t)p'P = p'P(Vt E G) we have i

Vs E G. By the sufficient condition, there are #J-j > 0, t j E G, and

L #J-j = ;

such that 11JI([L#J-jatj(a~),a2])1 < e, i

where 1JI(.) = ( 7r'P (·)€, €)(E SG). Then

I(€, (7r'P( adp'P 7r'P( a2) - 7r'P(a2)p'P 7r'P (al)) €) I

< I(€, [L#J-j7r'P(atj(a~)), 7r'P(a2)]€) I ;

+1( 7r'P (a2)€, p'P 7r'P (al)€ -

L#J-j7r'P(atj(a~))€)1 i

+1 (p'P 7r'P( all €- L #J-j 7r'P( atj (a~)) €, 7r 'P(a2) €) I j

< 2117r'P(a2)€II·llp'P7r'P(ad€ - L#J-j7r'P(ati(a~))€11 j

+I(€, [L #J-j7r'P (atj (a~)), 7r'P (a2)] €) I i

< e + 11/J([L J.L;atA a~), a2]) I < 2e. j

1

567

Since c( > 0) is arbitrary, it follows that

Q.E.D. Remark. Clearly, if A is abelian, then (A, G, a) is G-abelian. Moreover, if for any a; = at E A, i = 1,2 and cp E SG, inf{lep([a.(al), a2]) lis E G} = 0, then by Theorem 14.4.9 (A, G, a) is G-abelian. Theorem 14.4.10. (Ergodic decomposition) Let (A, G, a) be a G-abelian system. Then SG is a simplex ( in the sense of Choquet). Therefore, for any ep E SG there is a unique probability measure J.L on SG such that

Val, ... ,an E A, and v < J.L ( C.M. ) for each probability measure v on SG with ( a(p)dv(p) = ep(a) , Va E A. Consequently, J.L is pseudoconcentrated on

lSa

ExSG in the sense that J.L(E) = 0 for each Baire subset E of SG disjoint from ExSG. Moreover, if A is separable, then J.L(ExSG) = 1.

For any ep E SG, let J.L be the C-measure of cp, where C = {7rlp(A),plp}'. By Theorem 14.4.4, J.L is the unique largest ( C.M. ) probability measure on SG such that J.L >- v ( C.M. ) for each v as above. Now by Theorem 14.4.10, SG is a simplex, and J.L is pseudoconcentrated on ExSG. Moreover, since J.L is the C-measure of ep and supp J.L C SG, it follows from Theorem 14.2.6 that Proof.

( al(p)'" lSa

an(p)dJ.L(p) = (plp1l"Ip(al)pIp'" Plp1l"Ip(an)plp11p, lip)'

Q.E.D. Definition 14.4.11. Let A be a C· -algebra with an identity 1 , G be the group of all unitary elements of A, and S = S(A) be the state space of A. For each v E G, define av(a) = vav" Va E A. Then (A, G, a) is a dynamical system. Clearly, SG is the tracial state space T = T (A) of A, l.e.,

T = T (A) = {ep Proposition 14.4.12. Then it is G-abelian. Proof.

E

Slcp(ab) = cp(ba), Va, b E A}.

Let (A, G, a) be the same as in Definition 14.4.11.

It is immediate from Theorem 14.4.9.

Q.E.D.

568

Theorem 14.4.13. (Tracial decomposition ) Let A be a C·-algebra with an identity, and T = T (A) be its tradal state space ( a compact convex subset of (A*,u(A*,A))). Then T is a simplex ( in the sense of Choquet). Therefore, for any cp E T there is unique probability measure Jl on T such that

1

al(p)" . an (p)dJl(p) = (plp1rlp(at}pIp'" Plp1rlp (an)plp lip' lip), ~

Val, ... ,an E A, and Jl

1

v( C.M. ) for each pprobability measure II on T with

a(p)dll(p) = cp(a) , Va E A. Moreover, Jl is indeed the central measure of cp, and Jl is pseudoconcentrated on ExT = Tn l' in the sense that Jl(E) = for any Baire subset E of T disjoint from ExT, where l' = 1'(A) is the factorial state space of A. In particular, if A is separable, then Jl( T n 1') = 1.

°

Proof. By Proposition 14.4.12 and Theorem 14.4.10, it suffices to show that ExT = T n 1', and {1rlp(A) , Pip}' = 1rlp(A)" n 1rlp(A)' for each cp E T. Let cp E T, Z = 1rlp(A)" n 1rlp(A)'. For any x E Z, pick a net {ad c A such that 7rlp(a,) ~ x( strongly). Then for any unitary element v of A we have

uIp(v)xl1p = lim uIp(v)1rlp(al)uIp(v)*I1p

= lim 7rlp(O:tJ-1(a,))I1p = lim 7rlp(v·a,v) lip

= 1rlp(v)*X1rlp(v) lip = xl1p' Hence, ZI1p

c Elp' Conversely, for any a E A,

where A;l} > 0,

L

A}'J = 1, Vi, V)') E G ( the gropu of unitary elements of A)

j

,L

such that Pip = s-lim . A)'JUIp(VY J) (see Proposition 14.4.1 ) . We may assume J that A;l} 1rlp(v;'J* avJ')) .z, x E 1rlp{A)"

L i

( replacing

{L AYJ 1rlp(V;'J* av)'))}, by a subnet if necessary). j

Then

569

Vb E A, and ptp7rtp(a)ltp = x1tp. Pick {a,l strongly). Then by ptp7rtp(a) 1'1' E Ell' we have

C

A such that 7rtp(a,)

~

x (

xlII' = ulp(v)x1lp = lim Ulp(V)7rtp (al) utp (v)* 1'1' =

lim 7rlp(v*a,v) 1'1'

= 7rlp(v)*x7rtp(v)llp'

Vv E G. Since rp E T, it is easy to see that 1'1' is also separating for 7rtp(A)". Hence, z =

7rlp ( V

)* X7r 'I' ( V ) ,

Vv E G, and z E Z. Therefore, plp7rlp(a)llp E Zllp' Va E A, and Ell' = Zllp' By Proposition 14.2.2, we have

Now let cp E ExT. By Proposition 14.4.8, dim Ell' = 1 . From the preceding paragraph, a:plp = Etp = Zltp, and zllp = A.r:1tp, where A.r: E a:, Vz E Z. Further,

z7rtp(a)llp = 7rlp(a)zltp

= A.r:7rlp(a) 1'1"

Va E A,

i.e. z = A.r: 1H f> , Vz E Z. Therefore, 7rtp(A)" is a factor, and rp E T n f. Conversely, let rp E T n 1. Then Ell' = Zllp = a:1lp' and dim Etp = 1. From Proposition 14.4.8, cp E ExT. Therefore, ExT = T n 1. Q.E.D.

Notes. Theorem 14.4.10 is due to O. Lanford and D. Ruelle. Theorem 14.4.9 is due to D.Ruelle. Proposition 14.4.8 is due to G.G. Emch.

References. [44], [98], [139].

Chapter 15 (AF)-Algebras

15.1. The definition of (AF)-algebras Definition 15.1.1. A C"-algebra A is said to be approximately finite dimensional, or (AF) simply, if there is an increasing sequence {An} of finite dimensional * subalgebras of A such that unA n is dense in A, i.e., unA n = A. Proposition 15.1.2. Let A = UnA n be an (AF)-algebra. Then A has an identity 1 if and only if there exists no such that In = 1, Vn > no, where In is the identity of An' Vn. Proof. Suppose that A has an identity 1. If there is a subsequence {nk} with InA: =I- 1, then In =I- 1, Vn, since {An} is increasing. On the other hand by UnA n = A, we have no and x E A no such that Ilx - 111 < 1. We may assume that A C B(H), and 1 is the identity operator on H. Pick E (1 - I no )H with II ell = 1. Then we get

e

1 > 111 - z] >

lie - xell = lIe - xln oell = Ilell = 1,

a contradiction. Therefore, In must be equal to 1 for all enough large n.

Q.E.D. Lemma 15.1.3. For any e E (0, ~), there exists 1 = 1(g) > 0 with the following property: if A is a C*-algebra on a Hilbert space H, p is a projection on H, and a E A with Iia - pll < 1, then we have a projection q of A with lip - qll < e. Proof. We may assume that a" minimal value of the function IA 2 -

Let 6 E (0, i), and m(> 0) be the on the following set:

= a.

AI

[-2,2]\[(-6,6)

U

(1- 6,1 + 6)].

571

Now pick "1 = "1(e)

>

°such that "12 + 3"1 <

min{~e, ,;}. Notice that

max{I>.2 - >'1 I >. E a(a)} = IIa 2 - all < IIa 2 - ap - pa + pI! + IIp(a - p)1I + II(a - p)plI

+ lip -

all

< II(a - p)211 + 311 a - pll < "1 2 + 3"1 < min{~e,~} and

1>.2 - >'1 > 1 1>.2 -

if

AI > m

1>'1

if

> 2,

>. E (-2,2]\[(-6,6) U (1- 6,1

+ 6)].

Thus, a(a) c (-6,6) U (1 - 6,1 + 6). Pick a continuous function f on JR such that f(A) = if A E (-6,6) and f(A) = 1 if A E (1- 6,1 + 6). Then q = f(a) is a projection of A, and lip - qll < lip - all + lIa - qll < "f + 6 < e. Q.E.D.

°

Lemma 15.1.4. Let e' > 0, and n be a positive integer. Then there exists 61 = 61(e, n) > with the following property: if A is a C·-algebra, and PI, ... .v« are projections of A satisfying IIPiP; 11 < 6b VI < i f. j < n, then we have projections qb" . .a« of A satisfying qiq; = 0, Ilpi - qill < e, '11 0, let 61 (s, n

. {"1(e) } , + 1) = min - , 61 ("1(e) - - , n) 6n 6n

where "1(c:) is as in Lemma 15.1.3, and we may assume that e E (o,~) and "1(e) < c. H A is a C·-algebra, and PI,'" ,Pn+l are projections of A satisfying IIPiP;1I < Dl(e, n+l), '11 < if. j < n+l, then we have IIPiPill < 61(~' n), '11 < i i= j < n. By the inductive assumption, there are projections ql,' .. ,qn of A with qiqj = 0, '11 < i i= j < n, and II P. - qi II

< "1 (e) / 6n,

1
< n.

n

Let q =

L

qt· Then

'=1

IIPn+l - (1 - q)Pn+l (1 - q) II n

< 31lPn+lqll <

3

L IIPn+lq,11

,=1

< 3(f: IIPn+1Pill + "f(e)) < 3n61(e,n + 1) + "1(e) < "f(e). i=1

6

2

572

If B is the abelian C"'-subalgebra of A generated by {qb"', qn, (1- q)Pn+dlq)}, then from Lemma 15.1.3 there is a projection qn+l of B such that IIPn+l qn+ll1 < e. Clearly, qn+lqi is still a projection, and

1 < i < n. So qn+lqi = 0,1 < i < n, and {qI,"', qn+l} is what we want to find.

Q.E.D. Lemma 15.1.5. Let A be a C"'-algebra, {PI,'" ,Pn} and {qI,"', qn} be two orthogonal families of projections of A, and Ilpi - qi I < 1,1 < i < n. Then there exists a partial isometry w of A such that .

=

(Piwqi) '" . (Piwqd

qi,

--~ L..i qi,

and

uno"

=

LPi'

i=1

i=1

Proof. Pick S E (0,1) such that continuous function on 1R with:

f(A)=O

I-Sj

and f is affine on [~(1 - S), 1 - S]. For 1 < i < n, let B, be the abelian C"'-subalgebra of A generated by {Pi, PiqiPd, and 0i be its spectral space, i.e., B i '" C ( n,). Fix i E {I,,,,, n}. Since p(Pi) = 1 and IlpiqiPi - Pill < IIPi - qi II < «5, it follows that P(PiqiPd E (1- S, 1], Vp E ni , and hence from the definition of the function f, we have that

= f(PiqiPi) . PiqiPi. XiPiqiPiXi = PiXiqiXiPi since B,

Put Xi = f(PiqiPi) 1/2. Then Pi = PiqiPi X; = is commutative. Changing Pi and qi, we can see that qi = 'y;qiPiqi, where Yi = f(qiPiqi)' Now let Wi = Pixiqi. Then

Thus, Pi

W;Wi

=

qiXiPixiqi = y;qiPiqi . Pi X; qi y;qi(PiqiPixnqi

=

y;qiPiqi

= qi,

n

1

 0 with the following property: if {PI,"" Pn} and {q1,' .. ,qn} are two orthogonal families of projections of a C~-algebra A, and Ilpi - qill < 02,1 < i < n, then we can find a partial isometry w of A such that

(PiWqi)* . (PiWqi) = qi, IIPi - Piwqill < s,

(PiWqi)' (PiWqi)·

=

Pi,

1 < i < n,

and w·w = Ei=l qi, ww· =. Ei=l Pi. Moreover, if A has an identity 1 and Ei=l Pi = 1, then w can be chosen such that 111 - w II < g. Take 02(g, n) Then we have Proof.

IIPi -

ui;

=

II =

4~n'

and keep all notations of Lemma 15.1.5 (0 = O2),

Ilpi (pi - Xiqi) II

+ IHpi - Xi)Pill < II Xi ll · ll qi -Pill + IIXi -pill, < Ilxiqi - XiPill

1

Since P(PiqiPi) E (1 - O2 , 1], it follows that P(Xi) E [1, (1 - O2 )- 4), Vp E 0i' Then Ilxili < (1 - O2 ) - 1 , and Ilpi - xiII < (1 - O2 )- 1 - 1 = (1 - O2 )- 162 , Thus, we get

If A has an identity 1 and Ei=l Pi we have n

111 - wi! < L Ilpi -

= 1, then from the preceding paragraph

will < 2n62(1 - 62)- 1 <

g.

i=l

Q.E.D. Lemma 15.1.7. For any e E {O, I}, there exists 63 = 63 {g) > 0 with the following property: if A is a C·-algebra on a Hilbert space H, Pi and P2 are two projections of A, ql and q2 are two projections on H, a E A and v E B(H) such that IIPi - qill < 63 , i = 1,2, v·v = qll

vv· = q2,

IIv - all < ha, then there is u E A with u·u = PI, uu· = P2 and lIu - vII < and

g.

Proof. Let 63 = 63 (g) = 1;8' B be the abelian C·-subalgebra of A generated by {P2' p2aa*p2}, and n be the spectral space of B, i.e., B ,.... C(O). Noticing

574

that

IIp2 -

IIp2 - aa"'ll < IIp2 - q211 + IIq2 - aa*1I < IIp2 - q211 + Ilvv'" - va*1I + Ilva'" - aa"'ll < 40s < 1,

P2 aa"'P211 <

we get Ip(p2 - P2 aa"'P2) I < 46s

< 1,

Vp E O.

(1)

Pick x E B+ such that

(2) i.e.,

(3) From (1), we have

00

I: P(P2 -

P2 aa'" P2) n ,

VpEn.

n=O

Further, by (1) (2) we can see that

IIx2- P211 = max{11 <

max

f

pEn n=1

p(x 2 ) II pEn}

Ip(P2 - P2aa"'P2) jn <

s 40 0 < 80s, 1- 4 s

(4)

(5) and

Ilx - P211

=

+ P2)-1 . Ip(x 2 P211 < 80s.

max[p(x pEn

< Il x2 -

P2) I]

(6)

Let w = Xp2a. Then ww'" = P2 by (3). Thus, w"'w is a projection of A. Further, by ql = V"'q2v and (4)' (5), we have Ilw*w -

P2!1 <

+ Ilql - Pili +llv"'P2x2P2a - V"'P2x2P2VII + Ilv"'P2 X2P2 V - qlll < Iia - vii' Iiall . II x211 + II x211 • Ila - vii + IIp2 X2p2 - P211 +llp2 - q211 + llpl - qlll < 20s (1 + Os) + 20s + 80s + 20s Ila"'P2 x 2p2 a - V"'P2 x2p2 all

< 166s = 02(2,1),

575

where ~(i, 1) = 1 (see the proof of Lemma 15.1.6). Now from Lemma 15.1.6 (for W·W,PI, i and n = 1), there is a partial isometry WI of A such that = PI, wlwi =

W~WI {

W ..W

(7)

IIwl - w·wll < c:/2

Let u = WWl' Then u"'u = PI, uu'" q2V and (5), (6), (7), we get

=

P2.

Finally, by W

=

XP2a, W

= ww·w, v

lI u - vii < Ilwwl - wll + Ilw - vII < Ilwl - w"'wll + II v - P2 vll + IIp2 v - xP2 v II + lI XP2V < II wl - w·wll + IIq2 - P211 + IIp2 - xII + Ilxll . Ilv - all ~

<

=

xP2 a li

+ 83 + 8~ + 283 < s. Q.E.D.

Lemma 15.1.8. For any e > 0 and positive integer n, there exists 84 = 84 (c:, n) > 0 with the following property; Let A be a C"'-algebra on a Hilbert space H, and {eJ;) ]1 < i,i < nk, 1 < k < m} be a matrix unit on H, i.e., f i(k) j

E B (H) ,

Vi,i, k , i' ,/, k', where nl + ... + n m = n. If there is a subset {a~:)} of A such that lIe~J) - a~J) II < 84 , Vi,i, k, then we have a matrix unit {q~)} of A such that

Ile~;)

Moreover, if

L e~:)

- q1:) II < s ,

VI

< i,i <

= IH, then IH E A and

nk, 1

< k < m.

{q};)} can be chosen such that

i,k

Lq};) =

IH.

i,k

Proof. First, let 84 < 1(C:I), here 1(') is as in Lemma 15.1.3, and Cl will be chosen later. Then we have projection {pJ~)} of A such that Ile~~) - P~:) II < cI,Vi,k. So IIpJ:)p);1I < 2C:I,V(i,k) i- (i,l). Pick el < 8l (C:2, n ), here 81 ( , , ' ) is as in Lemma 15.1.4, and e2 will be determined in the following. Then there is k an orthogonal family {q1 ) } of projections of A such that IIq};) -P~:) II < e2, Vi, k. Hence, IIq~:) - e~:) II < el + C:2, Vi, k. Notice that (k) .. (k) _ eli

eli

-

(k)

eit,

(k) (k)'" _ eli eli

-

(k) ell ,

IIq!;) - eJ:) II < el + e2, Ilql~) - et~)11 < el + C:2, Ila~~) - e~~)11 < 84 < 'Y(c:d < el + C:2, Vi,k.

576

Now pick Cl + C2 < 6s(cs), here 8s (·) is as in Lemma 15.1.7, and cs will be chosen later. Then there exists a subset {q~:)} of A such that

Vi, k. Let q~) = qi~)*qi~). Then {q~)} is a matrix unit of A. When small enough, we get

(CI

+ C2 + cs) is

Vi,i,k. Moreover, if

L

e~;) = 1H, we may assume that nc < 1, then

i,k

II L

q~ile)

-

IHII < nc < 1.

i,1e

Thus,

~

(Ie)

L...., qi. i,k

Q.E.D.

= IH·

Lemma 15.1.9. Let A be a C*-algebra with an identity 1. Then A is (AF) if and only if the following conditions are satisfied: 1) A is separable; 2) for any finite subset {at,···, an} of A and e > 0, there exists a finite dimensional * subalgebra B of A and a subset {bl , · · · , bn } of B such that II ai - b. II < e, 1 < i < n. Moreover, if A is (AF) and D is a finite dimensional * subalgebra of A , then we can find an increasing sequence {An} of finite dimensional * subalgebras of A such that DeAl; 1 E An' Vn; and UnA n = A. Proof. The necessity is obvious. Now let A satisfy the conditions 1) and 2), let {x n } be a countable dense subset of {x E A llixil < 1/2} with Xl = 0, and D be either (o} or the given finite dimensional * subalgebra of A. Suppose that Al = D+(C. Then 1 E AI, and there is a~l) E Al such that Ila~l)-xIII < 2- 1 (for example, a~l) = 0). Now assume that we have finite dimemsional * subalgebras Al c··· c An' such that for each I E {I,··· ,n}, there are ai'),···,af') E A, with lIa~') - Xiii < 2- 1,1 < i < I. Let {e~;) I 1 < i,i < nk,I < k < m} be a matrix unit of An' and n~

+ ... + n:n

= dim An. Clearly,

L

i,1e

e~~) = 1.

By the condition 2), there is a finite dimensional * subalgebra B of A, and {b~:),bl < i,j < nk,I < k < m,I 0 will be determined in the following. Then by Lemma 15.1.8, we have a matrix unit {INc)} of B with ~ L."

i,k

(k) - Iii(k)11 < Iii(k) = 1 such that II eli

. . k.

C, Vl,),

( I

Let c <

6" ( 11, ) m , where

11

>0

will be chosen later. Then by Lemma 15.1.6, there is a partial isometry w of A such that (k) f(k»). . ((k) f(k») - f(k) ell w 11 ell w 11 11, -- elk) (k)w f 11 (k») . (e(k)wf(k»)* (1) (e11 11 11 11,

lIe~~) - e~~)wll~)11 < 11,

Put u = ~ e;~)wIJ:).

1 < k < m,

Then u is a unitary element of A, and U/i~)U· =

,.

e!;) , Vi,j, k, Suppose that A n+ 1 = u.Bu": Then An Then Ila!n+l) - xiII < II bi -

C

A n+ 1 • Let a!n+l) = ubiu·.

Xiii + Ilubiu* - bill

L [e}~)wfJ~)biIAm)w·e~7') - IJ;)bill~m)]1I j,k,l,m < e + (dimA n)2 sup 1111;)bill~m) - e}~)wlj~)b,l,\m)w*e~7')II, i,1r.,l,m

< c + II

!.

n + 1.

We may assume that c < Since II xill < it follows that Ilb,11 < 1,1 < i < n + 1. Thus, 1

11/ij(k) bo./,u(m) _

! and lib, - xiii < !'

I(k)b./,(m) * (m) II ej(k) 1 w il , 11 W eu

< IIIJ:)bill~m) - IJ;)bifAm)w·e~7')1I - e}~)wli:»)bi/Am)w·e~7')1I ft) k 1 m. < 2 sup II f u(t) - ed(t) w 1Is II , u· vJ",

+/1 (fJ:) !,t

By III!:) - e~~ II < s, lIe~tJ - II:) II < e, Vs, t, and (1), we have

111 ft) !!

e(t)wj(t)ll < d h

<

c

+ Ile(t)e(t)e(t) - e(t)e(t)wj(t)j(t)11 !llll!!ill 11 I!

2c

+ Ile~tlli:) - e~1wl1~) 11:)11 < 2c + 11, Vs,t.

Now suppose that c and 11 are small enough. Then we get Ila~n+l) - xiII < 2- n - 1 , 1 I} can be found. Q.E.D. Lemma 15.1.10.

(A-Hr) is (AF).

Let A be a C*-algebra. Then A is (AF) if and only if

578

Proof. The necessity is obvious. Conversely, let (A+a') is (AF). Then by Proposition 15.1.2, there is an increasing sequence {B n } of finite dimensional * subalgebras of (A+a'), such that 1 E B n, "In, and unBn = A-t
Theorem 15.1.11. Let A be a C·-algebra. Then A is (AF) if and only if the following conditions are satisfied: 1) A is separable; 2) for any finite subset {ab"', an} of A, and E: > 0, there exists a finite dimensional * subalgebra B of A and a subset {bb"', bn } of B such that I a, - bi II < E:, 1 < i < n. Moreover, if A is (AF) and D is a finite dimensional * subalgebra of A, then we can find an increasing sequence {An} of finite dimensional * subalgebras of A such that DeAl, and unA n = A. Proof.

It is immediate from Lemmas 15.1.9 and 15.1.10.

Q.E.D.

Proposition 15.1.12. Let A be an (AF)-algebra, and p be a projection of A. Then pAp is also (AF). Proof. For any Xl,"', X n E pAp and e > 0, by Theorem 15.1.11 there is a finite dimensional * subalgebra B of A and a, Yl, ... ,Yn E B such that Ilxi - Yill < E:,1 < i < n, and Ila - pil < ')'(62 (e: , 1)), where ,),(.) and 62 ( " , ) are as in Lemmas 15.1.3 and 15.1.6 respectively. Then there exists a projection q of B with lip - qll < 62 (E: , 1). Further, there is a partial isometry w of A such that w·w = p, and lip - wll < e.

Let C w· Bw. Clearly, C is a finite dimensional Moreover, we have

IIXi -

* subalgebra

of pAp.

Ilxi - PYiPIl + IlpYiP - w·Yiwll < Ilxi - Yill + Il(p - w·)YiPII + Ilw·Yi(p - w)11 < e + 2e:IIYill < e + 2e:(lIxill + s},

w·Yiwll <

1 < i < n. Therefore, by Theorem 15.1.11 pAp is an (AF)-algebra.

Q.E.D.

Theorem 15.1.13. Let A be a C·-algebra with an identity 1. Then A is a (UH F)-algebra if and only if the following conditions are satisfied: 1) A is separable,

579

2} for any a},···, an E A and e > 0, there is a finite dimensional subfactor B of A with 1 E B and b},· .. ,bn E B such that II a, - b,II < s, 1 < i < n. Moreover, if A is (UHF) and D is a finite dimensional subfactor of A with 1 ED, then there exists an increasing sequence {An} of finite dimensional subfactors of A such that Al = D,1 E An' "In, and A = unA n. Proof.

It is similar to the proof of Lemma 15.1.9.

Q.E.D.

Notes. The theory of (AF)-algebra was introducted by O. Bratteli. It is a generalization of the theory of Glimm (UHF) algebras.

References. [15], [26], [54].

15.2. Dimensions and isomorphic theorem Consider a * algebra A over (C. pEA is called a projection, if p. = P = p 2 • And we denote the set of all projections of A by Proj(A). Moreover, we assume that if a E A and a*a = 0, then a = 0.

Definition 15.2.1.

P, q E Proj(A) are equivalent, denoted by P q, if there exists v E A such that v·v = P and vv* = q. In this case, we have up = v and v'« = v*. Indeed, since (v-vpt(v-vp) = v·v - v·vp - pv·v + pv·vp = 0, it follows that v - vp = 0, i.e., v = up, Similarly, v'« = v*. Thus, is an equivalent relation on Proj(A). For any p E Proj (A), denote its equivalent class by p. "-J

"-J

Definition 15.2.2. Denote all equivalent classes of Proj (A) by E(A), i.e., E(A) =Proj (A)I The canonical map d(·) from Proj(A) onto E(A) is called "-J.

dimensions.

Elements PI,'" ,Pn E E(A) is said to be additive, if there exists a Pi E PI, 1 < i < n, such that PiPi = 0, Vi i= j. In this case, we define PI + ... ,+Pn = (PI + ... Pn)"'. We claim that the partial addition on E(A) is well-defined. In fact, if we have another qi E Pi, 1 < i < n, such that qiqi = 0, Vi f- j. For i E {I" .. n}, there is Vi E A with ViVi = Pi, v,vi = qi. Let v = VI + ... + v n. Since

Definition 15.2.3.

Vi I- j, it follows that v*v = PI Pn)'" = (qt + ... + qn)"'.

+ ... + Pn, vv* =

qt

+ ... + qn, i.e.,

(PI

+ ... +

580

Definition 15.2.4. Let A, B be two * algebra as above. E(A) and E(B) are said to be isomorphic, if there is a bijective map '11 from E(A) onto E(B) such that {PI,' . " 13'n}( c E(A)) is additive if and only if {W(13'l),' .. , W(Pn)}( c E(B)) is additive; and W(PI + ... + Pn) = '11(13'1) ... + W(13'n). ..--.... Clearly, if ~ is a * isomorphism from A onto B, let '11 (P) = ~ (p), then '11 is an isomorphism from E(A) onto E(B). Definition 15.2.5. A * algebra A is called a (LF) algebra, if A = UnA n with Al C ... C An C "', and every An is a finite dimensional C·-algebra. In particular, any (AF)-algebra contains a dense (LF) * subalgebra.

Lemma 15.2.6. Let A = UnA n be a (LF)-algebra, Band C be two finite dimensional C·-subalgebras of A, and ~ be a * isomorphism from B onto C, such that p '" c)(p) , Vp EProj(B). Then there exists a unitary element u of A (if A has an identity) or (A+O;) (if A has no identity) such that ~(b) =

ubu",

Vb E B.

Proof. Let {e~;) I 1 < i,j < nA:,1 < k < m} be a matrix unit of B, and n~ + ... + n~ =dimB. Then {fi~) = ~(e~;»)} is also a matrix unit and a basis of C. By the assumption, for each k E {I", " m} there is VA: E A such that •

vA:VA:

(A:) ,VA:vA: • = = en

I(k) 11'

Clearly, w·w = e, ww·

L et

= ~(e), and wbw· = ~(b), Vb E B, where e =

the identity of B, c)(e) =

L, fJ:)

L: e)~)

is

i,A:

is the identity of C.

i,A:

Let n be large enough such that w, ui" E An. Then e '" By Proposition 6.3.2, we have w' E An such that

w'·w' = 1n - , e w'w'· = 1n -

~(e)

relative to An.

~(e) ,

where In is the identity of An. If 1 is the identity of A (if A has an identity) or (A+a') (if A has no identity), then u = w + w' + (1 - In) is a unitary element of A or (A +0;), and we have

ubu· = wbw· =

~(b),

Vb E B.

Q.E.D.

581

Lemma 15.2.7. Let Ai be a * algebra, E. = E(A.),i = 1,2 , and \II be an isomorphism from E 2 onto E I • If B 2 is a finite dimensional C"-subalgebra of A 2 , then there exists a finite dimensional C"-subalgebra B 1 of Al and a * isomorphism ~ from B I onto B 2 such that \II(~lP)) = p, Vp EProj (B I ) . Let {IBc) 11 < i.i < nk, 1 < k < m} be a matrix unit and a basis of B 2 • By the property of \II, there is an orthogonal family {e~:)} of projections of Al such that (k») -_ eW . \II ( Iii V~, k, ii, Proof.

Since li~k)

.- li~),

it follows that e~~) (k). (k) _

(k)

eil

en,

en

-

.-

e~~) and there is e~~) E Al such that

(k) (k)" _ eil eit

-

(k)

w·

k

VI,.

eii '

(k) (k) (k). ""'( (k}) - r{k} w· • k (k) I . . k] L et eij en eil ,'.l" eii - Jij , vI,), ,an d B I -- [eii ",),. Then B 1 is a finite dimensional c·-subalgebra of A., and ~ is a * isomorphism from B I onto B 2 • Now if p is a projection of B}, then there is a subset A. of { (i, k) 11 e~:}. By the additivity of

L

(i,k}EA

'l1, we get

\II(~1P)) =

L---fi~k»)

\If(

(i,k)EA

L

\II{jJf») =

{i,k)EA

L

e~:)

=

p.

{i,k)EA

Q.E.D. Theorem 15.2.8. Let A = UnA n and A' = UnA~ be two (LF)-algebras. If E = E(A) and E' = E(A') are isomorphic, then A and A' are * isomorphic. Proof. Let's try to find two subsequences {mk}, {nk} of positive integers, and a sequences {B k } of finite dimensional C*-subalgebras of A, and a * isomorphism \Ilk from B k onto A~I: for each k, such that

Ami C B I C A m2 C B 2 C '" C AmI: C B k C " ' ,

\II k+IIBk =

\Ilk,

Vk,

and the following diagram is commutative:

BI

~

!\Ill A'n!

B2

~

!\Il 2

~

A'n2

~

where "~" represents the embedding map. Then we can see that A and A' are * isomorphic.

582

The process and principle are as follows: Al =

Ami

B1

C-.....t

C-.....t

C-.....t

!

UI!

C1 F1! B 'I

A'nl

C-.....t

Ama

C'3

!

B~

'-+

Now we begin the proof. Let W be an isomorphism from E onto E' and ml = 1. By Lemma 15.2.7, there exists a finite dimensional C· -subalgebra B~ of A' and a * isomorphism ~l from Ami onto B~ such that

(1) Pick nl such that B~ C A~l' Similarly by Lemma 15.2.7, we can find a subalgebra C 1 of A and a * isomorphism FI from C 1 onto A~l such that

w(qtl

---

Vql E Proj (CI ) .

= FI(ql),

*

(2)

Then the following diagram is commutative:

and by (1), (2), we have Fit

-

-...... 0

~t{Pt) = W-I(~I(PI)) =

Using Lemma 15.2.6 to A,AmllFIof (A+a') such that U·lau~ = F I-

I

1

0

0

PI,

VPl E Proj (Ami)'

~1(Aml)'

~t( a),

there is a unitary element Ut

Va E Ami'

Let B 1 = UiCIUt. Then B I :::J uHF;-t 0 ~1(AmJ)UI = Ami' Thus, Wd·) = FI(UI' is a * isomorphism from B I onto A~l' and by (2) we have

un

(3) Pick m2(> ml) such that B 1 C A m 2 • From Lemma 15.2.7, there is a subalgebra C~ of A' and a * isomorphism G 2 from A m 2 onto C~ such that

*

(4) Then we have the following commutative diagram:

BI WI! A' nl

'-+ G2

0-'"j" I

Am 2 G 2!

C 2'

583

and by (3), (4), ,.....,.

G2

0

w1

,.....,.......,

1

(pD

= W(W11 (pi) ) = pL

Using Lemma 15.2.6 to A', A~l ' G 2 of (A' such that

+On

0

E Proj

(A~.).

'11 11 (A~J, there is a unitary element u~

,,*2 = G 2 0 'T.-l(') 1 a ,

u I2 a u

vp~

Va'E A'n l '

'J!'

Let B~ = U~C2U';. Then B~ :J u';(G 2 0 Wll(A'nJ)u~ = A 'nl. Thus we have the following commutative diagram:

B1

Co......+-

WI! A'nl

Co......+-

A m2 ~2! B~,

where ~2(a) = u';G2(a)u~, Va E A m2, and by (4),

~;(P2) = G;(P2) = W(P2),

VP2 E Proj (Am~J.

(5)

Pick n2(> nl) such that B~ C A~2' From Lemma 15.2.7, there is a subalgebra C 2 of A and a * isomorphism F 2 from C 2 onto A~2 such that

*

(6) Similar to the preceding paragraph, we have a unitary element u~ of (A' +a') such that F2- 1 0 ~2(a) = U2au;, Va E A m2. Let B 2 = U;C2U2' Then B 2 :J A m2, and '11 2 ( , ) = F2 (U2 ' u;) is a * isomorphism from B 2 onto A'n2' and by (6),

(7) Moreover, we have the following commutative diagram: A ml

Co......+-.

~l!

B'1

BI

Co......+-

A'nl

Co......+-

~2!

WI! Co......+-

A m2

C0......+-

B'2

Going on this way, we can complete the proof.

B2

W2! Co......+-

A~2'

Q.E.D.

Theorem 15.2.9. Two (AF)-algebras A = UnA n and B = UnB n are * isomorphic if and only if the (LF)-algebras UnA n and UnBn are * isomorphic. The sufficiency is obvious. Now let A and B be * isomorphic. By Theorem 15.2.8, it suffices to show that E = E(A) and E ' = E (UnA n) are isomorphic.

Proof.

584

First, E' can be embedded into E naturally. In fact, let P, q EProj ( UnA n) , and v E A with v·v = P, vv· = q. Pick n and a E An such that P, q E An' and Iia - vii < 83 (! ), where 83 ( , ) is as in Lemma 15.1.7. Then there is u E An such that P = u·u and q = uu". So P and q are also equivalent in UnA n. Moreover, the above embedding is also surjective. In fact, let P be any projection of A. By Lemmas 15.1.3 and 15.1.6, we can find a projection q of UnA n, such that P - q. Clearly, the above embedding keeps the partial addition. Conversely, let {Ph"', Pm} CProj (A) with PiPj = 0, Vi 1= i. From Lemmas 15.1.3, 15.1.4 and 15.1.6, there is an orthogonal family {q., ... , qm} of Proj( UnA n) such that PI - q" 1 < i < n. Therefore, E and E' are isomorphic. Q.E.D.

Theorem 15.2.10. Two (AF)-algebras A = UnA n and B = UnB n are * isomorphic if and only if there exists a subsequence {AnA;} of {An} and a * subalgebra B~ of AnA; for each k such that: 1) B~ c ... c B~ c "', and there is a * isomorphism 4.) from UnB~ onto UnB k with 4.)(B~) = B k , Vkj 2) for each n, An is contained in some B~. Proof. The sufficiency is obvious from Theorem 15.2.9. Now let A and B be * isomorphic. By Theorem 15.2.9, we have a * isomorphism 4.) from UnA n onto unBn. For any k, let B1 = ~-l(Bk)' Clearly, B~ C ... c B~ C "', and for each k, B~ C A n le for some nk. We may assume that nl < n2 < .... Finally, for each n, since U;B;. = 4.)-I( UjB j } = UjA j, it follows that An C B~ for some k,

Q.E.D. Definition 15.2.11. Let {Pn} be a sequence of positive integers with PnIPn+., 'In. A sequence {rn} of prime numbers is said to be determined by {Pn}, if there is a sequence {81 < 82 < ...} of positive integers such that 42

41

II ri = i=l

mI,

"3

.

II ri = m2, II r, = i>41

i>"2

m '" ...

,

Theorem 15.2.12. Let Ai be a (U H F)-algebra of type {p~)}, and {r~)} be a sequence of prime numbers determined by {p~)}, i = 1,2. Then Al and A 2 are * isomorphic if and only if for any prime number r, the times of r appeared in {r~l}} and {r~)} are the same. Proof. By Proposition 3.8.3, it suffices to prove the necessity. Let Al and A 2 be * isomorphic. From Theorem 15.2.10, we can find two sequences

585

{k l < k 2 < ... J, {nkl < nk2 < .. '}, a subfactor Bl~) of Al for each 1

i,

and a

*

isomrophism ~ from uiBl~) onto UiA~2.', such that 1 J

and

';r,.(B(I)) *" kj where 1·$ E A (ti ) C ...

=

A(2)

kj'

c ACi) c ... c A· A· n'"

vs :

v J,

= U n ACi) and ACi) is pCi) X pCi) n' n n n

matrix algebra, Vn, i = 1,2. Now by Proposition 3.8.3, we have p~~) Ip~12j and

p~2.lp~~), Vi· That comes to the conclusion.

Q.E.D.

1

Notes. C·-algebraic dimension theory was first explicitly strudied by J. Diximer, who used it to classify the matroid algebras. G.A. Elliott then extended this theory to the (AF)-algebras. Theorem 15.2.10 is due to G.A. Elliott. And Theorem 15.2.12 is due to J. Glimm. References. [15], [26], [42J, [54].

15.3. The Bratteli diagrams of (AF)-algebras Let A = unA n be an (AF)-algebra. Then we can see A as the inductive limite of the increasing sequence {An} of finite dimensional C·-algebra. In fact, let ~n be the embedding map from An into A n+ b and for any m > n, ~mn = ~m-l 0 ••• 0 ~n. By Corollary 3.7.4, A is * isomorphic to lim{A n , ~mn 1m> --+ n} = 1ig1{An , ~n}. Conversely, let {An} be a sequence of finite dimensional n

C·-algebras, and for each n, suppose that ~n is a * isomorphism from An into A n+ l . Then the inductive limit {An, ~nln} is an (AF)-algebra. So it is important to consider the * isomorphism from a finite dimensional C·-algebra into another finite dimensional C·-algebra. Let A = EDj::::lA j , B = ED~lBi be two finte dimensional C·-algebras, where Ai' B, are * isomrophic to matrix algebras, Vi, i, and let ~ be a * isomorphism from A into B. If p is a minimal (i.e. rank one) projection of Ai' then ~(P)Zi is a projection of Bi, where Zj is a central projection of B with B, = BZj. Suppose that the rank of ~(P)Zi in B, is sii (a nonnegative integer), i.e., ~(p)Zj is a sum of an orthogonal family A of minimal projections of Bi, and •A = Sij. We claim that the non-negative integer Sij is independent of the choice of minimal projection p of Ai' In fact, if q is another minimal proj ection of Ai' then there

586

is v E A j such that v·v

= p and vv· = q.

Thus we have

i.e. the ranks of CP(P)Zi and CP(q)Zi in B, are the same: Bij' Therefore, cP determines a unique m X n embedding matrix (Bij) l~i~m,l~j~n of non-negative integers. Clearly, we have n

LBij(dimAi)I/2 < (dimBi)I/2,

1
(1)

;=1

and (1) becomes an equality for any i if and only if CP(I A ) = lB. Moreover, since cp(p) =1= for any minimal projection p of A j , it follows that

°

m

L Bi; > 0,

1<

i < m.

(2)

i=1

Conversely, if a m X n matrix (Bi;) of non-negative integers satisfies (1) and (2), then we can construct a * isomorphism cP from A into B such that the embedding matrix determined by cP as above is (Bij) exactly. In fact, let {e~{) 11 < B, t < (dim A iP/2}, {!~:) 11 < B, t < (dim B iP/2} be matrix units of A j , B, respectively, 1 < j < n, 1 0,

;-1

where k = Bij, T = L Bil(dim Al)I/2, VI < s, t < (dim A j)I/2,1 < j < n,1 < l=l

i < m. Then this cP satisfies our condition.

Lemma 15.3.1. Let CP, W be two * isomorphisms from A into B, where A and B are two finite dimensional C·-algebras. If the embedding matrices determined by CP, Ware the same, then there exists a unitary element u of B such that

Replacing CP, 'If by CPi(') = CP(')Zi, wi ( · ) = W(')Zi respectively (1 < i < m), we may assume that B = B(H), where H is a finite dimensional Hilbert space (dim H =(dim B i )I/2). For i E {I,.·· ,n}, let {e~{) 11 < B,t < (dimA j )l/2} be a matrix unit of A j • By the assumption, we have dimCP(e~'{)H = dim w(e~{»)H. Let {eP),···, j)} and {TJlj)"",TJi be orthogonal normalized bases of CP(e~1)H and 'If(e~{»)H respectively. Since e(j) = e(i)* e(il and e(j) = e(i)e(i)* {CP(e(j») c(j) I 1 < 1 < k} 11 d d "" II d , II ~l -Proof.

elil}

587

and {\It(e~{»)l1!i) 11 < 1 < k} are orthogonal normalized bases of ~(e~~»)H and \It(e~~»)H respectively, 1
0."

'T' (e(i»),,(i) = .T.(e(;») cU) d . I ".t." d ~"

U '!r

\-Is 1 J'

v".

I

Noticing that

u·~(e~{»)u\It(e~{»)l1l(;)=

Ortw(e~{»)l1fi) \It (e(i) ) \It (e(j»)" (j) .t

"Vs,t,r,l,i, we have

u·~(a)u =

rl

'1' , Q.E.D.

\It{a),Va E A.

From the above Lemma, a * isomorphism ~ form A into B is determined completely by the embedding matrix (Sij)' So it is reasonable to write ~ = (B.j). More intuitively, we use the following diagram to describe ~: ·11

BI

·ij

B·•

Al

~

Ai

----+

An

• m,\

~ =

(Bij) l~i~m • l~j~ ..

Em

Lemma 15.3.2. Let Ai, B, be two finite dimensional C·-algebras, (Ji be a * isomorphism from Ai onto Bi, i = 1, 2,~, \It be * isomporphisms from Al into A 2 , B 1 into B 2 respectively, and the embedding matrices determined by ~,\It be the same. Then there exists a * isomorphism O 2 from A 2 onto B 2 such that the following diagram is commutative:

Proof. First, notice the following two facts: 1) Let Ct, C 2 , O« be finite dimensional C·-algebras, ~b ~2 be * isomorphisms from C I into C 2,C2 into C g respectively, and (B~~»),(B~») be the embedding matrices determined by ~h ~2 respectively. Then the embedding matrix determined by ~2 0 ~l is (B~~}) . (B~:»). 2) Let CI , C 2 be two finite dimensional C"-algebras, and (J be a * isomorphism from C 1 onto C 2 • Then the embedding matrix determined by (J is the unit matrix. Their proofs are easy.

588

Now consider * isomorphisms ~,82"1 0 lIt 0 fit from Al into A 2 • From above facts, their embedding matrices are the same. So by Lemma 15.3.1, there is a unitary element u of A 2 such that u·~(·)u =

Define

82 ( , )

= 82 (u·

8i1 0 'If 0 8d'),

V· E AI.

. u), V· E A 2 , then we get lIt

0

81 =

82 0

~.

Q.E.D. Definition 15.3.3. Let a be an (AF)-algebra, and {An} be an increasing sequence of finitie dimensional C· -subalgebras of A with unA n = A. A Bratteli diagram of A corresponding to {An} is

= D = (D,d,U), = {(n, m)1m = 1,·· . ,r(n), n = 1,2,· . '}(r(n) D(A, {An})

where D is the number of minimal central projections of An); d : D ---4 IN, d(n, m) = dim (A nZ!: )P /2 (where {z1n), ... , z~(l)} is the set of all minimal central projections of An), V(n, m) E D;U = {~h···,~n,···} (where ~n is the embedding matrix from An into A n + I , Vn).

Example 1. The Bratteli diagram of a (U H F)-algebra of type {Pn}. In this case, r(n) = 1, d(n,l) = Pn, ~n = (m n) , where m n = p;;l Pn+ b Vn, I.e., PI

P2

Ps

Pn

Pn+1

Example 2. K

=

Let H be a separable infinite dimensional Hilbert space, C (H) be the set of all compact linear operators on H, and A = K-t
If {en} is an orthogonal normalized basis of H, and Pn, qn, rn are the projections from H onto [e1,· .. , en}, [6,' .. ,en]l., [en} respectively, Vn, then we can write A = UnAn, where

An

= Pn B(H)pn+a:l H = PnB(H)Pn ffi (Cqn, Vn.

Clearly, any minimal projection of PnB(H)Pn is still a minimal projection of Pn+lB(H)Pn+l, qn = qn+l +rn+h and rn+l is a minimal projection of Pn+1B(H) 'Pn+},'tn. So D(A, {An}) is as follows

r(n)

= 2,d(n,l) = n,d(n,2)

=

1,~n = (~ ~),

589

Vn. Example 9. The GICAR (gauge invariant canonical anticommutation relation ) algebra. It has a diagram with r(n) = n + 1, d(n, m) = C;:" and 1 1

cP n =

1 1

0

(n

x (n

+ 1),

1 1

0

Vn

+ 2)

= 0,1,2' .. ,m = 0,1, ... ,n, i.e.,

the Pascal's triangle.

Let A = UnA n, B = UnB n be two (AF)-algebras, and V(A,{A n}) = V(B, {Bn}). Then A and B are * isomorphic.

Proposition 15.3.4.

Let cP n l \lin be the * isomorphisms from An into A n+b B n into B n+l respectively. By the assumption and Lemma 15.3.2, for each n we can construct a * isomorphism On from An onto B n such that the following diagram is commutative: Proo].

Al

~

01 ! B1

-.!.4

A2 O2 !

B2

~ ~

As Os !

n,

----+

----+

----+

----+

An On !

s;

~ W

n

)

A n+ 1 On+l !

---+ •.•

B n +1

---+ ••.•

Thus we get a * isomorphism 0 from UnA n onto UnB n such that OIAn = On, \In. Therefore, A and Bare * isomorphic. Q.E.D. Clearly, any (AF)-algebra has a Bratteli diagram at least. Conversely, let V = {D,d,u} be a diagram, and each cPn(E U) satisfy (1), (2), i.e., r(n)

d(n

+ 1, i) > 2: s~i)d(n,j), ;=1

and

1

 0,

1<

i < r(n),

i=1

where cP n

= (s~i)h~i~r(n+l),I~j~r(n), Vn. Then we can construct an (AF)-algebra

A = UnA n such that V(A,{A n}) = V. Moreover, from Proposition 15.3.4 such

(AF)-algebra A is unique up to * isomorphism. The Bratteli diagrams of an (AF)-aglebra depends on not only the algebra itself but also the choice of the dense increasing sequence of finite dimensional C*-subalgebras. So for * isomorphic (AF)-algebras, their Bratteli diagrams may be very different.

590

Notes. The Bratteli diagrams of (AF)-algebras was introduced by O. Bratteli, It is an important tool for studying the constructions of (AF)-algebras. Let A, B be two finite dimensional C· -algebras, and ~ be a * isomoprhism from A into B. Then the embedding matrix is exactly the homomorphism Ko(~) : Ko(A) ---+ Ko(B) from the point of view of K-theory.

References. [11], [15], [99J.

15.4. Ideals of (AF)-algebras Lemma 15.4.1. Let A = ~ be an (AF)-algebra, and J be a closed two-sided ideal of A. Then J = Un(J nAn).

Let I n = J n A,\) Vn. Clearly, I n is a two-sided ideal of An, Vn, and UnJn C J. Let x E A \UnJn, and e = inf{llx - yilly E UnJn}. Clearly, e > o. Pick X n E An, Vn, such that X n ---+ z. Then we have no with Ilxn -xII < e/2, \In > no. So Ilxn - yll > Ilx - yll -llxn - xli> e/2, Proof.

Vy E I n , and n > no. Let a ---+ a be the canonical map from A onto Ani I n can be naturally embedded into AI J, it follows that

IIXnl1 = and Ilxll > e/2, x

fi.

inf{llxn- yilly E I n }

> e/2,

J. Therefore, we have J = UnJn

Vn

AI J. Since

> no, Q.E.D.

Definition 15.4.2. Let [) = {D, d, U} be a diagram of an (AF)-algebra, D = unDn,D n = {(n,m)ll < m < r(n)},Vn,U = {~n = (s~;))ln}. A point (n+ 1, i) is called a descendant of a point (n,J·), if s~;) > o. In general, a point y E Dm. is called a descendant of a point x E D n , which is denoted by x ---+ y, if m > n, and there exist points Xle E Die, n < k < m, such that X n = z, Xm. = y, and Xle+l is a descendant of Xle, n < k < m - 1. Let x = (n, i), y = (m, i). Clearly, x ---+ y if any only if the (i, i)-element of the matrix (~m.-l ... ~n) is not zero. Definition 15.4.3. Let [) = {D, d, U} be a diagram of an (AF)-algebra. A subset E of D is called an ideal, if: 1) any descendant of x belongs to E, Vx E E; 2) suppose that x E D n and {y E D n +1 ly is a descendant of x} c E, then x E E.

591

Lemma 15.4.4. Let A = unA n be an (AF)-algebra, and D(A,{A n}) = D = {D, d, U} be the corresponding diagram. If J is a two-sided ideal of UnA n, then there exists an ideal subset E of D such that

(1)

J = Un EB {An,kl(n,k) E E},

where each A n,k is a matrix algebra, and An = EB~t'fAnlk' \In. Conversely, if E is an ideal subset of D, then Un EB {A n,kI(n, k) E E} determines a two sided ideal J of UnA n, and J n An = EB{An,kl(n, k) E E}, \In. Proof. Let J be a two-sided ideal of UnA n. Since J = Un(J n An), there is a subset E of D such that (1) holds. Now we must prove that E is an ideal subset of D. Let (n, k) E E, and (n, k) --+ (n + 1, l). Clearly, if p is a minimal projection of An,k' then pz =1= 0, where z is the minimal central projection of A n+1 with An+1z = An+1,l' Since pz E J and pz E Anz C A n +1ll , it follows that J n An+1,1 =1= {O}. But An+1,1 is a matrix algebra, hence An+1,l C J, i.e., (n + 1, l) E

E. Now let (n,k) ED, and {(n

+ l,i)l(n,k)

An,k C EB{An+1,il(n, k) C EB{An+1,il(n

--+ --+

+ l,i)

(n

+ l,i)} C

(n

+ l,i)}

E. Then

E E} C J

and (n, k) E E. Therefore, E is an ideal subset. Conversely, let E be an ideal subset of D, and J = Un ED {An,kl(n,k) E E}. Put I n = EB{An,kl(n,k) E E}, \In. If (n,k) E E, then An,k C EB{An+1,il(n, k) --+ (n + l,i)} C I n+ 1 • So I n C I n+h \In, and J = unJn is a two-sided ideal of UnA n. Moreover, if An,k C J , then there is m(> n) such that An,k C J m . Thus {(m, r) I(n, k) --+ (m, r)} C E. Since E is an ideal subset, it follows that (n, k) E E. Therefore, J n An = I n, \In. Q.E.D. Theorem 15.4.5. Let A = UnA n be an (AF)-algebra, and D(A, {An}) = {D, d, U} be the corresponding diagram. Then there are bijections between the following collections: 1) the collection of all closed two-sided ideals of A; 2) the collection of all two-sided ideals of unA n; 3) the collection of all ideal subsets of D. Proof. By Lemma 15.4.4, there is a bijection between the collections of 2) and 3). From Lemma 15.4.1, J --+ J is a map from the collection 2) onto the collection 1). Now let J 1 , J 2 be two different two-sided ideals of UnA n. We

592

must prove that J 1 1= J 2 • By Lemma 15.4.4, we may assume that there is An.A: C J 1 , but An.A: n J 2 = (o). Thus z ¢ J 2 , where z is the minimal central projection of An such that Anz = An,A:' Further, for any m > n, z + (J2 n Am) is a non-zero projection of A m/ (J2 n Am), i.e., inf{lIz -

yilly E Am n J 2 }

= 1,

'Vm > n.

Since J 2 = Um~n(J2 n Am), it follows that

inf{llz and z ¢ J 2 • Therefore, J 1

yilly E J 2 } = 1,

1= J 2'

Q.E.D.

Remark. Let A = UnAn and B - UnBn be two (AF)-algebras, and An = B n, 'Vn. But they can pick different Bratteli diagrams such that they have different sets of two-sided ideals. Hence, the structure of an (AF)-algebra A = UnAn depends on not only each An but also each embedding way from An into A n+1. Proposotion 15.4.6. Let A = UnAn be an (AF)-algebra, and [) = {D,d,U} be the corresponding diagram. H J is a closed two-sided ideal of A, then J and A/J are also (AF)-algebras, and they have diagrams:

{E, diE, UIE},

{D\E, dl(D\E), UI(D\E)}

respectively, where E is the ideal subset of D corresponding to J. Moreover, if U = {Un = (S~;)h~i~r(n+l),l~j~r{n) In}, then

UIE = {Vn . (s~;») {n+l,i)EE,(n,j)EE In} and

Proof. From Lemma 16.4.1 and Lemma 16.4.4, clearly J is an (AF)-algebra, and has a diagram {E,dl§,UIE}. Since A/J = Un(An/J), A/J is also an (AF)-algebra. Notice that for each n, An/J = EB{An,k/JI(n, k) ~ E}

c An+1/J = ffi{An+1,i/JI(n + l,i) ¢ E}, and An,A:/J ~ An,A:,An+1,i/J - An+1,j,'V(n,k) and (n + l,i) ¢ E. Thus, Q.E.D. A/J = Un(An/J) has a diagram {D\E,dl(D\E),UI(D\E)}. Definition 15.4.1. Let [) = {D, d, U} be a diagram of an (AF)-algebra. An ideal subset E of D is said to be prime , if it follows from x, y ¢ E that there is z ¢ E such that x -+ Z, Y -+ z.

593

Theorem 15.4.8. Let A = UnA n be an (AF)-algebra, D = {D, d, U} be its diagram, and J be a closed two-sided ideal of A. Then the following statements are equivalent: 1) J is primitive; 2) J is prime: 3) The ideal subset E of D corresponding to J is prime. Proof.

1) ==> 2). it is obvsious from Proposition 2.8.8.

Replacing A by AI J and from Proposiiton 15.4.6, we may assume that J = [O] and E = 0. 2) ==> 3). For any points x = (n, k), y = (m, I) E D, let J 1 = Up>n ffi {Aprlx -. (p, r)}, J 2 = Up>m ffi {Aprly -. (p, r)}.

By the condition 2), we have J 1 n J 2 f:. {O}. Further, J 1 n J 2 = J 1 n J 2 n (UnA n) f:. {O} from Lemma 15.4.1. Thus there exists (p, r) E D such that Apr C J 1nJ2 • By the definition of J}, we can find (Pb rl) E D with x -. (Ph rl) and (p, r) -. (PI, rl). Thus, Ap1,rl C J 1 n J 2 • Again by the definition of J 2 , there is (P2,r2) ED with y -. (P2,r2) and (Pt,rl) -. (P2,r2). Let z = (P2,r2)' Then x -. z and y -. z. 3) ==> 1). Since any finite subset of D has a common descendant, we can find a subsequence {nl:} and a function j(.) such that for each k,j(k) E {I,' .. , r(nl:+l)} and

(nl:,i)

-+

(nl:+bi(k)),

1
Now we may assume that

(n, i)

-+

(n + 1,1),

VI

 Pn+b Vn. If write PnaPn = Pn(a)Pn, Va E An, then Pn is a pure state on An, Vn. Since

Va E An, it follows that Pn+llAn = P, Vn. Hence there is a state P on A such that plAn = Pn, Vn. Moreover, it is easily verified that P is pure. Now it suffices to show that ker 1r p = {O}, where 1r p is the irreducible * representation of A generated by p. From Lemma 15.4.1, it is equivalent to prove that ker 1r p n An = {O}, Vn. Let z be a minimal central projection, and Anz = AnI:. Since (n, k) -. (n + 1,1), it follows that Pn+l -< P relative to A n + b where P is

594

a minimal projection of A nk . Thus, there is v E A n+1 such that v·v and vv· < P < z: Further, we have 1

= Pn+b

= Pn+l(Pn+l) = p(v·v) = p(v*zv) = (1t"p(Z)1t"p(v)Ep, 1t"p(v) Ep),

and 1t"p(z)

1= O,z fi ker

1t"po Therefore, ker 1t"p n An = {O}, tin.

Q.E.D.

Notes. From Proposition 15.4.6, we have the extension problem for (AF)algebras. This problem was first studied by G. Elliott, and may be stated as follows. Given a C·-algebra A and a closed two-sided J of A such that J and AI J are both (AF)-algebras, does it follow that A is itself an (A F)-algebra? Elliott was able to prove that this would be the case if one could show that any projection of AIJ is the image of projection of A. Then L. Brown solved affirmatively this problem in terms of K -theory.

Referneces. [15], [40), [99)

15.5. Dimension groups Definition 15.5.1. (G, P) is called an ordered group, if G is an abelian group, and P is a subset of G satisfying: (1) P + PCP; (2) P n (-P) = {O}; (3) P - P = G; (4) unperforated, i.e., if a E G and na E P for some n E IN, then a E P. P is also called the positive part of the ordered group G, and denoted by G+ sometimes. We shall write a > 0 if a E P; and a > b if (a - b) E P. Definition 15.5.2. An element u of an ordered group (G, P) is called an order unit ,if any a E P, there exists n E IN such that a < nu. Since P is unperforated, it follows that u E P. An homomorphism P from an ordered group (G, P) to JR is called a state relative to an order unit u, if P is positive ( denoted by P > 0), i.e., p( a) > 0, Va E P; and p(u) = 1. We shall denote by Su(G) the set of all states relative to u. Clearly, JRG = xGJR with product topology is a locally convex Hausdorff topologyical linear space. With the embedding: P --+ (p(a))aEG, Su(G) is a closed convex subset of JRG obviously. We claim that Su(G) is also a compact subset of JRG. In fact, for any a E G there is n a E IN such that -nau < a < nau. Thus, we have Su(G) C xaEG[-n a, n a). Clearly, xaEG[-na,n a] is a compact subset of JRG. Therefore, Su(G) IS a compact convex subset of JRG.

595

Moreover, if v E P is another order unit, then p homeomorphism from Su(G) onto Sv(G) obviously.

-?

p(V)-l p is an affine

Lemma 15.5.3. Let (G, P) be an ordered group, U(E P) be an order unit, H be a subgroup of G, and U E H . Then (H, H+ = P n H) is also an ordered group with an order unit u. Moreover, if p E Su(H) , and a E G\H, then p can extended to a state on (H on (G,P, u). Proof.

+ ~a).

Consequtly, p can be extended to a state

It suffices to define p(a) = A such that

p(h + mal

= p(h) + mA > °

(1)

for any h E Hand m E ~ with (h + mal E P. H h + na > O,h' - n'a > 0, where h,h' E H,n,n' E IN, then

nh' -> nn'a -> -n'h and -p(h)/n < p(h')/n' . Since u E H, we can pick

, A

I

° '

[ {-P(h) h E H, n E IN, } E sup n (h + na) >

inf {P(h)' I h' E h, n' E IN, }]

n'

1

(h' - n'a) >

° . Q.E.D.

Clearly, such A satisfies the condition (1).

Definition 15.5.4. Let (G, P) be an ordered group. A subgroup J of G is called an order ideal, if J = J+ - J+, where J+ = J n P, and for any a, bE P with a < band bE J+ we have also a E J+. An order ideal J is said to be prime, if J 1 and J 2 are two order ideals with J = J 1 n J 2 , then either J = J 1 or J = J 2 • Definition 15.5.5.

A group G with the following form

G

= lim{ ~r(n) , cI»n} ----.

is called a dimension group, where cI»n =

(sJi»)

is a r(n

+ 1)

x r(n) matrix

r(n+l)

of non-negative integers with

L sJi) > 0,1 < j

< r(n), "tn. In detail, every

l=1

element of G has the following form:

cI»noo(t n)

= (0,··· ,O,tn,t n+b···) + I,

where t. E ~r(.) and t.+ 1 = cI».(t.), "ts > n, and 1= {(tb···,tm,O,·",o,···)lm

> 1,ti E ~r{i)}.

596

Proposition 15.5.6.

P

--

Let G = lim{.~r(n), cI»n} be a dimension group, and

= UncI»noo(2Z~(n»).

Then (G,P) is a countable ordered group, and (G,P) has also the Riese interpolation property, i.e., if a, b, C, d E G with a, b < c, d, then there exists e E G such that a, b < e < c, d. Notice that cI»n keeps the order, and (2Z r(n) , 2Z~(n») has the Riesz interpolation property, "In. Thus the conclusions are obvious. Q.E.D.

Proof.

Definition 15.5.7.

Let G

=

lim{2Z r(n),cI»n} be a dimension group, D n =

--

{(n,m)ll < m < r(n)},D = UnD n, and U = {cI»nln}. Then {D,U} is called a diagram of G. Clearly, any (AF)-algebra admits a dimension gruop. Conversely, if G is a dimension group with a diagram (D, U), the we can construct an (AF)-algebra A such that A admits a diagram {D, d, U}. Indeed, it suffices to pick {d( n, i)} such that r(n} d(n + l,i) > Ls~i)d(n,j), 1
--

Proposition 15.5.8. Let G = lim{2Z r(n), cI»n} be a dimension group with a diagram {D, U}. Then there is a bijection between the collection of all order ideals of G and the collection of all ideal subsets of D ( see Definition1.4.3). Moreover, if J is an order ideal of G and E is the ideal subset of D corresponding to J, then J is a dimension group with a diagram {E, UIE}. Let J be an order ideal of G, and E = {(n,k)lcI»noo(e~n)) E J}, where {e~n)ll < k < r(n)} is the canonical basis of 2Zr(n). If (n,k) E E and (n,k) ~ (m,p), by Definition 15.4.2 we have

Proof.

cI»noo(e~n») =

cI»moo(cI»m-l

>

0··· 0

cI»moo(e~m))

cI»n(e~n»))

> o.

Since J is an order ideal, it follows that (m, p) E E. Now let x = (n, k) E D n , and {y E D n +1 lx ~ y} c E. Noticing that J+ + J+ c J+ and cI»noo (e~n») =

L

S~;)cI»n+l,oo (e~n+l»),

i

(n+l,i)EE

we have cI»noo(e~n)) E J+, i.e., (n,k) E E. Therefore, E is an ideal subset of D.

597

Further, let J(E} be the subgroup of G generated by {~noo(e~n»)I(n,k) E E}. Clearly, J(E) C J, and J(E) is a dimension group with a diagram {E,U]E}. Let a E J+. By G+ = Un~noo(2Z~{n») there are non-negative integers AI, ... ,Ar(n) such that a = ~noo Al:e~n»). Since J is an order ideal, it

(L I:

follows that ~noo(e~n») E J+ if AI: > 0, i.e., (n, k) E E if AI: > O. Therefore, a E J(E), and J = J(E). Conversely, let E be an ideal subset of D, and define J = J(E) as above. We claim that J is an order ideal of G. In fact, since E is an ideal subset, it follows that

Vn. Thus , we have

and J = J+ - J+. If a, b E G+ with a < band b E J+, from the expression of J + we can see that a E J +. Thus J is an order ideal of G. Moreover, if (n, k) E D with ~noo(e~n») E J+, then we can write

~noo (e~n») =

L

Ai~moo (e~m»),

i

(m,i)EE

where m > n and Aj E 2Z+, Vj. Thus, there is p with p > m, n such that

L

~np(ein») = ~mp(

A;e;m»)

i

(m,i)EE

E {

~ J.tie~P)

lJ.ti E

2Z} .

(p,ilEE

This means that every descendant of (n, k) in D p belongs to E. Since E is an ideal subset of D, it follows that (n, k) E E. Q.E.D.

--

Proposition 15.5.9. Let G = lim{2Zr (n ) , ~n} be a dimension group with a diagram {D, U}, and J = J(E) be an order ideal of G, where E is an

598

ideal subset of D.

Then G I J is also a dimension group with a diagram

{D\E, UI(D\E)}. Proof.

For any n, let

2Zr (n) = 2ZP( nL-i-2Zq(n} , where 2Zp (n) = [e~n)l(n,k) E E],2Zq(n) = [ein)l(n,k) fI. E], and {e~n)11 < k < r(n)} is the canonical basis of 2Zr (n }. By this decomposition, we have projectons Pn : zr(n) ---+ 2ZP (n ) and Qn = (1 - Pn ) : ~r(n) ---+ ~q(n), Vn. Further, let Wn = Qn+l(~nl~q(n»), Vn. Then the dimension group lim{2ZQ(n) , Wn} admits --+ a diagram {D\E, UI(D\E}}. We have that: TIn = Tln+l

0

Wn

where TIn : 2ZQ(n} ---+ G I J and TIn (tn ) = ~noo (t n ) + J, Vtn E 2ZQ(n ), and Indeed, since E is an ideal subset of D, it follows that TIn (tn )

= ~n+l.ooQn+l ~n(tn)

=

~n+l.oo Wn(tn)

n.

+ ~n+l.ooPn+l ~n(tn} + J

+ J = Tln+l(Wn(t n)),

Vtn E 2ZQ (n) and n. Hence, we can define a map as follows:

TI :

Q(n}, lim{2Z Wn} -..,.-+

----+

GIJ

H t n E 2ZQ(n) with ~noo(tn) E J, then t n = 0 since E is an ideal subset .i.e., " is injective. From G I J = Un( ~noo (2Zr (n )} + J} = Un(~noo(2ZQ(n»)

is also surjective. Moreover, since isomoprphic to lim{~Q(n), Wn}.

TI

--+

TI

and

TI -1

+ J), keep the order,

G

IJ

is order Q.E.D.

Proposition 15.5.10. Let G be a dimension group with a diagram {D, U}, and J = J(E) be an order ideal of G, where E is an ideal subset of D. Then J is prime if and only if E is prime (see Definition 15.4.7.) Proof. By Proposition 15.5.9 and replacing G by G I J, we may assum.e that J = {O} and E = 0. Let the order ideal {a} be prime. For any Xi E D, put E i = {z E Dlxi ---+ Z, i.e., z is a descendant of Xi}, J, = J(Ei), i = 1,2. Suppose that Fi is the ideal subset of D generated by Ei,i = 1,2. Then J i = J(Fi),i = 1,2. By the assumption, we have J 1 n J 2 =I- {o). Thus, F 1 n F 2 =I- 0. Pick y E F 1 n F 2 • By Xl

599

arid y E F l , there is Zl E D such that Xl -4 Zl and y ~ Zl' From y E F 2 , we have also Zl E F2 • Further, by Zl and X2 E F2 , there is zED such that Zl ~ Z and X2 ~ z. Therfore, Xl ~ Z and X2 ~ z, i.e., 0 is a prime subset of D. Conversely, let 0 be prime. Suppose that Ji = J(Ei ) is a non-zero order ideal of G, where E i is an ideal subset of D, i = 1,2. Since E 1 n E 2 =I 0, it follows that J 1 n J 2 =I {O}. Therefore, {O} is a prime order ideal of G. Q.E.D.

Example 1. The CAR (canonical anticommutation relation) algebra i.e., the (UHF)-algebra of type {2 n}. It has a diagram as follows: 2

2

......

2

-----t

-----t

-----t

2

Thus we need to consider the dimension group:

G = lim{mr(n) ~ }, _ ,n where r(n) = 1, ~n = [2], "In. Define a map: 0 t t n+1 , . (0 , ... "n,

0

.)

+I

-----t

n

2t n'

where t« E mr(n) , t n+r = 2rt n, Vro Then we can see that G is order isomorphic to the dyadic rationals {21:n Ik E m, n = 1,2,ooo} = m[I/2] ( relative ordering in JR ).

Example e. Let H be a separable infinite dimensional Hilbert space and K = C(H). From Section 1.3, K has a diagram as follows: 1

1

-----t

-----t

-----t

123

n

n+l

So we have a dimension group G = lim{mr(n), ~n}, with r(n) = 1 and ~n = [1], "In. Clearly, G m ( usual ordering). ,.y

-

Example 9. The dimension group of the GICAR algebra. From Example 3 of Section 1.3, its dimension group G will be the inductive limit of the following system:

i.e.

600

where

°

1

Cb n

1

=

(n+2) x (n+l),'In>O.

1 1

°

Let a, b E IR with 4b > a 2 • Then there exists a positive integer N such that all coefficients of the polynomial (x + l)N (x 2 - ax + b) are non-negative.

Lemma 15.5.11.

Clearly, b >

Proof.

(X

Then for

+ 1)

N(

° o. 2

x - ax +

b) _

-

Write

Nt . i+ 2 • L- (. )' (N _ .),C i + 2 X i=-2 l + 2 . l . N

~

2 we have

(i + 2)~f

- i)!{C1- aC}tl + bC}t2}

(i + l)(i + 2) - a(i + 2)(N - i) + b(N - i)(N - i-I) (1 + a + b)(i - (b + ~)(1 2

+(b - T)(l

+ a + b)-lN)2

+ a + b)-l N 2 - (2a + b)(N - i) + (3i + 2)

2

> (b - T)(l + a + b)-lN 2 - (2a + b)N. Moreover,

C. = o

C N +1

(N+2)! b NI'

C1 =

_ (N+l)! ( )

-

NI

N - a,

{N+l)1 NI

C N +2

(Nb - a ) , _ (N+2)!

-

NI

•

Therefore, if N is large enough, any coefficient of the polynomial (x+ l)N (x 2 ax + b) is non-negative. Q.E.D.

Theorem 15.5.12.

The dimension group G GICAR algebra is order isomorphic to

(P7Z([O, 1]),

P~([O,

= lim{2Zn +1 , Cbnln > o} ---+

of the

1])),

where Pz([O, 1]) is the additive group of all polynomials on [0,1] with integer coefficients , and P~([O, 1]) =

{f E P7Z([O, l])lf(t) > 0, \It E (0, I)} U {Ole

601

Proof. Let u E G. Then there is n(> 0) and an element (ao,"" an) of JZn+1 such that u = 'l»noo((ao,"· , an)). From (ao,' .. ,an), we have unique (bo,' .. ,bn) E JZn+1 such that aoxn + ... + an = bo(x + l)n + ... + bn, \Ix > O. Define a homomorphism 'I» : G 'I»(u)

-+

pz([O, 1]) as follows:

= p(t) = bo + bIt - ... + bntn,

where (bo,' ", bn) E JZn+1 is determined by u as above. First, we must show that 'I» is well-defined. If (a~, ... ,a~+d = 'l»n((ao, ... , an)), then ao, = ao; aIj = aj-l + aj, 1< _ J'< _ n; a'n+ 1 = an' Thus a~xn+l + ... + a~+l = (x + l)(aox n + ... + an), \Ix. Let (b~,· ", b~+l), in JZn+2, satisfy b~(x + l)n+l +... + b~+l = a~xn+l + ... + a~+l' \:Ix. Then we have

\Ix. So , bi = b;,O < j < n, and b~+l = O. Thus, the definition of 'I» is independent of the choice of n. Moreover, if 'l»noo((ao,'" ,an)) = 0, then there is m(> n) such that 'l»nm((ao,"', an)) = 0, where 'l»nm = 'l»m-l 0 ••• 0 'l»n. Since each '1»1: is injective, it follows that ao = ... = an = O. Therefore, 'I» is well-defined. Clearly, 'I» is an isomorphism from G onto P7Z([O, 1]). So it suffices to show that 'I» is also an order isomorphism. Let u E G+ \{O}. Then there is (ao,"', an) E JZ.;+l\{O} such that u = 'I» noo ((ao, ... , an)). Thus bo(x + l)n + ... + bn = aox n + ... + an > 0, \:Ix > O. Therefore, we have

t _ bo(x + 1) n + ... + bn p( ) (x + l))n

> 0,

\It E (0.1).

t-_1_

-,2:+1

Conversely, let p(t)

f(x)

= bo +

= bo(x + l)n +

+ bntn E P~([O, l])\{O}. Then + bn = aox n + ... + an > 0, \Ix> O.

We need to prove that 'l»noo((ao,··· ,an)) E G+ , or to show that there exists m(> n) such that 'l»nm((ao,' .. ,an)) E JZ;+l. Clearly, it is equivalent to prove that there exists a positive integer N such that all coefficients of the polynomial (x + l)N f(x) are non-negative.

602

Since f(x) > 0, Vx > 0, we can write

f(x) = C II(x + Ai) II(x - a;)(x - a;), i

;

where C > 0, Ai > 0, and a; E a:\JR, Vi,j. Now applying Lemma 15.5.11 to each (x - 0.;) (x - a;), we can find a positive integer N such that all coefficients of the polynomial (x + l)N f(x) are non-negative. Q.E.D.

Notes. Proposition 15.5.6 is indeed a characterization of dimension groups. We have the following Effros-Handelman-Shen theorem: if G is a countable ordered group, and G satisfies the Riesz interpolation propperty, then G is a dimension group. Let A be an (AF)-algebra with a diagram {D, d, U}. If D = UnD n; D« = {(n, m) 11 < m < r(n)}; U = {cI»nln}, then the dimension group G = lim{ ~r(n), cI»n} is indeed the Ko-group of A.

--

References. [39], [40], [43], [59}, [103], [133], [160].

15.6. Scaled dimension groups and stablly isomorphic theorem Definition 15.6.1. Let G be a dimension group. A subset r of G+ is called a scale for G, if: 1) G+ is generatred by r, i.e., G+ = r + r + ... ; 2) for any a, bEG+ with a < band b E I', we have also a E r. In this case, we say that (G, G + = P, I'] is a scaled dimension group. For example, if G has an order unit u, then r = [0, u] = {v E GIO < v < u} is a scale for G. In a scale r, we can define a partial addition, i.e., a, b E r is said to be additive, if (a + b) E r. Lemma 15.6.2. Let ai, {3; E ~+, 1 all let ')'11 = at, II; = 0,2 < j < s, then we need to find {,i; I 2 {3h let 111 = {31, IiI = 0, 2 < i < {liil1 < i < r,2 < j < s}(c 7Z+) such that H a1

a

r, then we need to find

a

L

L

Ilk = a1 - {3I,

k=2

li1:

= 01, 2 < i <

r

k=2

and

r

L

e;

l1:j =

2

< j < s.

k=l

Q.E.D.

Repeating this process, we can get the proof.

Proposition 15.6.3. Let G i be a dimension group, I', be a scale for Gi, i = 1,2, and CP be an isomorphism from r 1 onto r 2 , i.e., CP and cp-1 keep the partial addition. Then cp can be uniquely extended to an order isomorphism from G 1 onto G 2 • Proof.

Let tli, bi E

r 1 and a1

+ ... + a, = b1 + ... + ba •

Then by Lemma 15.6.2 we have {cii} C (G 1 )+ such that

,

a

ai =

L

Cik,

1:=1

bi

=

L

Ckj,

1

< i < r, 1 < j < s.

k=l

Since r 1 is a scale, it follows that cii E r, Vi,j, and {Ciklk}, {ckilk} are additive in rbVi,j. Then {CP(Cik)lk},{cp(Ckj)lk} are also additive in r 2 , and

,

a

CP(ai)

=

L k=l

cp(Ci1:),

cp(b j }

=

L

cp(C1:i),

Vi,j.

k=l

Hence , we have

in G 2 • Moreover, since r 1 is a scale for G 1 and G 1 = (Gtl+ - (G 1 )+, cp can be uniquely extended to a homomorphism from G 1 to G 2 • We shall still denote this extension by eJ. From «>(r 1) = r 2 and r 2 is a scale for G 2 , it follows that cp(G1) = G:t, cp(( G 1 )+) = (G 2 )+. Now it suffices to show that cp is injective. Let a, b E (G 1 )+ and CP(a) = cp(b). Write

604

where

~,bj

E rbVi,j. From the preceding paragraph, we have

q.(al) + ... + q.(ar } = 4l(b 1 ) + ... + 4l(b r ) . By Lemma 15.6.2, there is a subset {c:4iI1 that "

q.(ai) = L diAa

< i < r,1 <

j

< s}

C

(G2 )+ such

r

q.(b;} = L d ki ,

k=1

Vi,j.

k=1

Since q.(~), q.(bi ) E r 2 , Vi,j, it follows that c:4j E r 2 , and {c:4klk}, {dkilk} are additive in r 2 , vi.i. Pick Cii E r 1 with q.(cii) = ~j, Vi,j. Since q. is an isomorphism from r 1 onto r 2 , {Ciklk} and {cA:ilk} are also ad,ditive in r 1 , and II

II

q.(L Cik} = L ~(Cik) = q.(ai), k=1 r

=L

k=1

4l(ckj)

= q.(bj ) ,

1

<j <

S.

k=1

Further, we have II

r

k=1

k=1

L Cik = ai, L Ckj = bj, at

< r,

r

4l(L Cki)

Therefore, a =

1
k=1

vi.i.

+ ... a r = bI + ... + b, = b, i.e., 4l

is injective.

Q.E.D.

Propostion 15.6.4. Let A = unA n be an (AF)-algebra with a diagram {D, d, U}, and G = lim{ztr(n), q.n} be its dimension group, where D = UnDn,D n = {(n,m)11 < m < r(n)},Vn, and U = {q.nln}. For each n, let t n = (d(n,l), ..., d(n,r(n)))(E zt~(n)), and [O,t n] = {t~ E ~~(n)lt~ < tn}. Then r = unq.noo ([0, tnD is a scale for G. Moreover, if E is the dimension range of A ( all equivalent classes of Proj(A), see Definition 1.2.2), define

-

where Ak is the rank of Pk in An,k' 1 < k < r(n), and P = PI +.. '+Pr(n) E pnA n ( see the proof of Theorem 1.2.9), then \II is an isomorphism from E onto r. Proof.

It is obvious.

Q.E.D.

Theorem 15.6.5. Let A, B be two (AF)-algebras. Then A and Bare * isomorphic if and only if their scaled dimension groups ( see Proposition 1.6.4) are scaled isomophic. Proof. The result follows immediately from Propositions 15.6.4, 15.6.3 and Theorems 15.2.8, 15.2.9. Q.E.D.

605

Corollary 15.6.6. Dimension groups of * isomorphic (AF)-algebras are order isomorphic. Consequently, the dimension group of an (AF)-algebra is uniquely determined up to order isomorphism. Now we consider the (AF)-algebra K = C(H), where H is a separable infinite dimensional Hilbert space. From Sections 15.3 and 15.5, K = unKn, where K n is n X n matrix algebra, and the embedding matrix
1 --+

---+

n1

n2

1 --+

n3

nk

.

nk+l

Moreover, for any C*-algebra A, there is only one spatial C*-norm ao(') on A ® K n, and o:o-(A ® K n) = A ® K n, \In( see Lemma 3.6.1). Thus, there is only one spatial C*-norm 0:0(') on A ® K, i.e. K is a nuclear C*-algebra.

Lemma 15.6.7. Let A = UnA n be an (AF)-aglebra, and G(A) be its dimension group. Then there exists a subsequence {n1 < n2 < ...} of positive integers such that the scale r of the dimension group corresponding to ao-(A ® K) = Uk(A k ® Knl:) ( see Proposition 1.6.4. ) is G(A)+. Moreover, the dimension group of o:o-(A ® K) is order isomoprhic to G(A). Proof. For any n1 < n2 < "', it is obvious that o:o-(A®K) = Uk(A k ® KnJ:)' Since the embedding matrix from A k ® Knl: into A k+1 ® Knl:+l is
r = u, , and nl = l,nk = 2k-1S1"'Sk_l,Vk i,j

> 2. Moreover, put lk = (1,' .. , 1) E Z"(k) ,Vk. Then tk > lk, Vk. For any a E G(A)+, there exist Nand n such that a. Pick m(> n) with 2m- 1 > N. Then n m = 2m- lS I ' " sm-l > N Sn'" Sm-1' Since ~k(lk} < sk1k+h Vk, it follows that

nmt m > n mlm >
r=G(A)+.

~noo(Nln)

1 0 .•• 0

~n(Nln)'

<
E

r.

Therefore,

Q.E.D.

606

Theorem 15.6.8. Let A = UnAn, B = UnBn be two (AF)-algebras, and G(A), G(B) be their dimension groups respectively. Then G(A) and G(B) are order isomorphic if and only if ao-(A ® K) and ao-(B ® K) are * isomorphic, where K = C(H) and H is a separable infinite dimensional Hilbert space.

Proof. The sufficiency is obvious from Corollary 15.6.6 and G(A) = G(ao-(A~ K)), G(B) = G(ao-(B(®K)). Now let G(A) and G(B) be order isomorphic. Pick {nl < n2 < ...} and {ml < m2 < ...} such that the scales of the dimension groups corresponding to UkAk ® K n"" Uk(Bl: ® K m ", ) are G(A)+, G(B)+ respectively. Thus, the scaled dimension groups of Uk(A k ® Knl:) and Uk(Bl: ® K m ",) are scaled isomorphic obviously. By Theorem 15.6.5, ao-(A ® K) and ao-(B ® K) are * isomorphic.

Q.E.D. References.

[11], [38], [40], [103], [160].

15.7. The tracial state space on an (AF)-algebra Let A = UnAn be an (AF)-algebra with an identity e and a diagram as follows: eE

where An the (k

X

Al

~l

~

A2

~2

~ •••••• ,

= M(t(n») = E£)~~l Mt1R) , ten) = (tin), ... ,t~(l») E ]Nr(n) ,Vn,

and Ml: is

k) matrix algebra, Vk. Then we have the scaled dimension group:

(G, G+ = P, r), where

G = l~{1Zr(n}, (Pn}, P = Un(Pnoo(1Z~(n}),

and

r

=

[0, u] = {v E GIO < v < u]

= Un(pnoo([O, t n }]) , where u = (pnoo(t(n})(Vn) is an order unit of G ( notice that the definition of u is independent of n since e E Al and t(n+l) = (pn(t(n») ,Vn). Let r be a tracial state on A, i.e., r be a state on A with r (ab) = r(ba), Va, b E A. For any v E r we can find some (sr,"', Sr(n») E 2Z~(n}

607

with Sic < tin)(l < k < r(n)) such that v =
we can define

p(v) = T(q). Then p is an additive function from r to [0,1]. From the proof of Proposiiton 1.6.3 , P can be extended to a state on the ordered group (G, P, u). Conversely, let p E Su (G), and

Ai 1 < k < r(n). Then Ai n

= p(
)

n )

> 0, \lk, and

Ai + ... + A~(l) = n

)

p(
= p(u) = 1.

For any a E An, we can uniquely write

al

a =

where ak E

Mt(n),

1< k

< r( n).

+ ... + ar(n) ,

Then define

l:

r(n) Tn(a)

=

L

A~n)tr(ak)'

k=l

where tr(.) is the unique cononical tradal state on Mk(\lk). Clearly, Tn is a tradal state on An, \In. We claim that

Fix nand k E {I,·,·, r(n)}. It suffices to show that

Tn+l(P) = Tn(p) = A~n) /t~n), where P is a minimal projection of

Mt(n).

"

In fact, let
... , sen) ) Hence A(n) = ten) ~ A(n+l) s(n)/ln+l) and ( s(n) llc , r(n+l)1c . k k L-,

,Ie,

;=1

A(n)

Tn(p) =

(:)

r(n+l)

=

tic

L

;=1

,

A(n+l) s(~)

' (n+l) = Tn+l(P). t;

Therefore, we have a tradal state T on UnA n such that TIA n = Tn, \In. Clearly, r(n)

IT(a) I <

L

1c=1

A~n) Itr(ak) I <·llall,

608

Va = al + ... + ar(n) E a tradal state on A.

An, Vn. It

follows that r can be uniquely extended to

From the above discussion, we have the following. Theorem 15.7.1. With the above assumptions and notations, the tradal state space T (A) on A is affinely homeomorphic to the state space Su(G) on G.

Example. The tradal state space of the GICAR algebra. Let A be the GICAR algebra (see Section 15.3, Example 3). By the above discussion, any tradal state on A is determined by {).~n)IO O}. Let It~n) = ).~n) / ci. Then we have an inverse Pascal's triangle: (n) _

Iti

(n+l)

- Iti

+ Iti+l (n+l) > 0 - ,

VO O. H put rn = It~n), then we need to find all sequences {rnln > O} of non-negative numbers with ro = 1 such that (n+l) ItHl -

HI "'(

i

L- -1 )iei+lrn-i+;

> 0,

i=O

VO O. By Theorem 15.5.12, the dimension group of A is

(G, P) = (P~([O, 1]), P~([O, 1]))' and function 1 = u is an order unit. Since G = un>oHn, where H n zt+ztt+··· +~tn, Vn > 0, each state p E Su(G) is determined by rn =

p((I - t ))n), Vn>O.

By Lemma 15.5.3, {r n} satisfies the following:

sup

ai,m E zt,m

n

L

ai ti

> O,and

+ mt n +1 > 0, Vt E (0,1)

i=O

ai,m E ~,m > O,and

<

inf

L" ~ti i=O

mt"+1

> 0, Vt E (0,1)

609

Vn > 0, and ro

= 1. For any r E [0,1] and n > 2, we have

Lemma 15.7.2.

n-1

sup

L

~,m

~ti

E 7Z,m > O,and

+ mtn > 0, Vt E (0,1)

;=0

Proof. First, let n = 2. If (a + bt + mt 2 ) > 0, Vt E (0,1), then (a -m- 1(a + br). Hence, we have

+ br + mr 2 ) > 0,

and r 2 >

I

2 > { a + br a, b, m E ~,m > 0, and } r - sup - m (a + bt + mt 2 ) > 0, Vt E (0,1) .

Now it suffices to show that for any c > 0,

sup{v-v] > r 2 or to find a, b, m E

~

and m >

-m- 1 (a + br) > r 2 Clearly, it must be a > b > O,m > such that

°

-

-

e,

°such that

e, and (a + bt + mt 2 ) > 0,

Vt E (0,1).

° and b < 0. So we want to find a, b, m E

and

a b 2 - -r + r < c m m > 0, Vt E (0,1). Suppose that 4ma > b2 • Then automatically, -

and (a-bt+mt 2 )

~+

a > 0, and (a - bt + mt 2 ) > 0, Vt E (0,1). Now the problem is to find a, b, mEN such that

a b 4ma > b2 , and (- - -r + r 2 ) < c. m m Pick p, q E N with q > p such that p 2 - - r q

+

p2 "2 - r 2 q

< c/2.

Then the problem becomes to find a, b, mEN such that

4ma > b2 , b < 2m,

a

b

p

p2

-m - -m . -q + -q2 < c/2.

610

Now take a, k E IN such that

p2 q

a

a

p2 q

1

- > -, and (- - -) . - < c/2, k

k

q

and let m = kq, b = 2kp. Then we have

4ma > b2 ,

b < 2m,

and

a

b

m

m

p q

p2 a p2 1 = (- - - ) . q2 k q q

[- - _. - + -I

< c/2.

Hence, the conclusion holds for n = 2. For general n > 2, it is obvious that n-l

Lair' n-l

i=1

L

m

ai, m E .?Z, m > 0, and ait'

+ mi" > 0, Vt E (0, I)

i=1

> sup { _

arn-2 + brn- 1

n-2.

- r

I

m

sup

{_a

+ br I m

(at n- 2

a, b, m E .?Z, m > 0, and

+ btn- 1 + mtn) > 0, Vt E

} (0, I)

a,b,m E Yh,m > O,and } (a + bt + mt 2 ) > 0, Vt E (0,1)

Q.E.D.

from the preceding paragraph. That comes to the conclusion.

Lemma 15.7.3. For each r E [0, 1],Pr is an extreme point of Su(G} , where Pr((1 - t}n} = r", Vn > 0.

Proof.

Let P, a E Su (G) and A E (0, 1) be such that Pr = AP + (1 - A) a

Then we have r" where Sn

= p((1 -

t)n), t; a

S2>SUP

{-

= AS n + (1 - A)t n ,

= u((1 + bS I m

t)n), Vn > 0. Since

I(a+bt+mt a, b, m E .?Z, m > 0, and

2}>0,VtE(O,I)

}

,

611

it follows from Lemma 15.7.2 that S2 > s~. Similarly, t 2 > t~. Now from r = AS1 +(I-A)O"l and r 2 = AS2+(I-A)t~.it must be Sl = t 1 = r,s2 = t 2 = r 2 • We assume that Sic = tic = ric, I < k < n - l(n > 3). Then n-1

L~Si Sn > sup

i=O

m

n-1

~,m

E .z,m > 0, and

(2: ~ti + mtn)

> 0, Vt E (0,1)

i=O n-1

La;ri

= sup

i=O

~ -

m

rn

,

and t« > r". But r" = AS n + (1- A)tn, it follows that Sn = t« = r". Therefore, we have P = 0" = p,., and p,. is an extreme point of Su{G). Q.E.D.

Proposition 15.7.4. Let A be the GICAR algebra, and G be its dimension group. Then we have

ExSu(G) = {PrIO < r < I}, where Pr((1 - t)n) Proof.

= r n , Vn > 0, Vr E [0,1).

First, we show that

Su(G) = Co{p,.lo < r <

If'"

in (lR G ) "' . In fact, if there exists apE Su(G)\Co{p,.IO < r < If'" , then by the separation theorem we can find a finite subset F of G = Pz([O,I)) and Ap E lR, v» E F such that

> sup

2: ApP,.(p)

O$,.$l pEF

=

sup

L

App(r).

0$,.$1 pEF

Using rational numbers to approximate each Ap ,we can obtain q E Pz([O, 1]) such that p(q) > sup q(r). 0$,.$1

Pick m E IN, n E .z such that n

p(q) > - > sup q(r). m

0$,.$1

612

Then (n - mq) E G+. By P E Su(G) we get

p(n - mq) > 0, and p(q)

~

n/m,

a contradiction. Therefore, Su(G) = cO{Prlo < r < If". Moreover, r -+ Pr is continuous from [0,1] to Su( G) obviously. Hence, {PrIO < r < I} is a closed subset of Su(G). Now by the Krein-Milmann theorem we have ExSu(G) C {PrIO < r < I}. Further, from Lemma 15.7.3 we obtain that ExSu(G)

= {PrIO < r < I}. Q.E.D.

Remark. The tradal state space T (A) of the GICAR algebra A is affinely homeomorphic to Su(G). So T(A) is a "Bauer simplex» [i.e., its extreme points subset is closed, see [128]) in (A*, O'(A"', A)), and the set of all extreme points of T (A) is homeomorphic to the connected space [0,1].

References. [103], [128].

Chapter 16 Crossed Products

16.1. W*-crossed products In Sections 7.3, 9.5, a discrete crossed product M X a G is defined, where G is a discrete group. Also in Section 8.2, a W·-system (M, lR, u) appears. Now we consider the general case.

(M, G, a) is called a VN-dynamical system ,if M is a VN-algebra on a Hibert space H, G is a locally compact group, a is a homomorphism from G into Aut(M), where Aut(M) is the group of all * automorphisms of M, and t ~ (at (x) '7) is continuous on G, Vx EM, and

Definition 16.1.1.

e,

e,'7EH. In this case, we claim that for each x E M t ~ at(x) is also continuous from G to (M, r(M,M.)). In fact, since Ilat(x) II = Ilxll, Vt, and at(xt = at(x·), it suffices to show that t ~ at(x) is continuous from G to (M, strong top. ) . We may assume that x = u is unitary. Let a net tt --? t in G. Then

II(atl(u) - at(u))eI1 2 = 211e1l 2

-

(atl(u)e, at(u)e)

-(at(u)€,atl(u)e)

~ 0,

VeE H, i.e., at,(u) ~ at(u) strongly.

Definition 16.1.2. Let (M, G, a) be a VN-system . For any x E M, define

(1r(x)f)(s)

=

a,,-l(x)f(s),

Vf

E

L 2(G,H),

where H is the action space of M, i.e., M c B(H);L 2(G,H) = H®L 2(G),L2 (G) = L 2 (G, 11.), and 11. is a left Haar measure on G. We shall write dll. (s) = ds simply. It is easily verified that 1111" (x) II < II x II, and {11", L 2 ( G, H)} is a * representation of M.

614

Proposition 16.1.3. Let (M, G, a) be a VN-system with Me B(H). Then {7r,L2 (G , H )} is a faithful W*-representation of M. Moreover, 7r(M) c M®Z, where Z = {m/lf E LOO(G,J.t)} is the multiplicative algebra on L 2 (G) ( see Definition 5.3.11). Proof. Let 7r(x) = 0 for some x E M. Then for any e E H and compact subset K of G, we have

o = (7r(X)XK ® €, XK e €)

= IK (a.-l (x)

e, €)ds

Since s --+ (a.-l(X)e,e) is continuous on G, and K is arbitrary, it follows that (a.(x)e, €) = 0, Vs E G, e E H. Thus, x = 0, i.e., 7r is faithful. Now let {Xl} be a bounded increasing net of M+ and x = sup x,. Clearly I

7r(XI) /' Y = sup 1r(XI) strongly. We need to prove that y = 7r(x). By the Dini I

theorem, for any e E H and compact subset K of G we have (a,,-l(xl)€, €) /' (a.-l (x) e, e) uniformly for s E K. Then we can see that

(7r(x,)f,f)

--+

(7r(x)f, f),

Vf

E

L 2 (G , H ).

Therefore, y = 7r(x), and {7r,L2 (G , H)} is a faithful W*-representation of M. Fix x E M. Suppose that e~k)(.); G --+ H is continuous and there is a compact subset K of G such that suppe(k)(.)

C

1 < k < n.

K,

For any e > 0, noticing that

II a;1(x) €(k) (t) - a t 1(x)e{k) (t) II < Il(a;l(x) - a,l(x))e(k)(S) II + II(a;l(x) - a;-l(x))(e(k)(s) < II(a;l(x) - a;-l(x))€(k)(S) II + 2I1xll.II€(k)(s) - €(k)(t)ll,

e(k)(t)) II

there is an opern neighborhood U" of s such that Ila~l(x)e(k)(t) - a;l(x)€(k)(t) II < '1,

Vt

E U", 1

< k < n,

where 11(> 0) satisfies '12J.t(K) = c. By the compactness of K, there are SI, ••• ,Sm E K and o, = 1 < i < m, such that U~1 U, :) K. Pick continuous functions gl,' .. ,gm on G such that

u.;

and

m

n

L:gi(t) < 1, Vt E G;

L: gi (t) = 1, Vt E K.

i=1

i=1

615 m

Let a

=L

a~l(x)

e m'i E M®Z.

Since

i=l

lI a€11 2 = f II

~ Yi(t)a~l(x) e(t) 11 2dt I

< IIxll 2 f(~Yi(t))2I1e(t)112dt < II xl1 2 ·lleIl 2 ,

•

Ve E L 2(G, H), it follows that Iiall < IIxll. Further, we have II(a - 1r(x))e(klll

- III ~Yi(t)a;/(x)e(kl(t)-

- IK <

II

• ~ Yi(t)( a~l(x) - a;-l(x)) e(A:) (t) 11 2 dt • 2Jl(K)

1(LYi(t)f/)2dt K

a;-l(x)e(k l(t)11 2dt

= 17

= s,

1 < k < n.

i

Q.E.D.

Therefore, 1r(x) E M®Z. Definition 16.1.4. any t E G, define

Let (M,G,a) be a VN-system with Me B(H). For

(x(t) I) (s) = Clearly, t

L

2(G,

~

1(t -1 s) , VIE L 2 ( G, H) .

.\(t) is the strongly continuous unitary representation of G on

H).

Proposition 16.1.5. Let (M, G, a) be a VN-system with M c B(H). Then {1r,.\, L 2 ( G, H)} is a covariant representation of the system (M, G, a), i.e.,

Proof.

(.\(t)1r(x).\(t) * I)(s)

=

(1r(x).\(t- 1)/)(t-1s)

-

ar1t(x)(.\(t-1)/)(t-1s) = a,,-lt(x)/(s)

-

a.-l(at(x))/(s) = (1r(at(x))/)(s),

Vs, t E G, x EM, 1 E L 2 ( G, H).

Q.E.D.

If we consider the VN-system (1r(M), G,,8), where ,8t(1r(x)) = 1r(at(x)), Vt E G, x E M, then the action ,8 on 1r(M) is unitarily implemented by .\('), i.e., ,8t(1r(x)) = .\(t)1r(x) .\(t)., Vt E G, x E M.

616

Let (M, G, a) be a W*-system ( i.e., M is a W*-algebra , a : G --+ Aut(M) is a homomorphism, and t --+ at(x) is continuous from G to (M, u(M, M*)), Vx EM). From Proposition 1.1.5, we can find a faithful W*-representation of M such that the action a will be unitarily implemented.

Definition 16.1.6. VN algebra

Let (M, G, a) be a VN-system with M c B(H). The { 11" ( x), A( t)

Ix EM, t E G}"

on L2(G,H) is called the crossed product of M by the action a of G, denoted by M X a G. When G is discrete, this definition coincides with Definition 7.3.3. Noticing that

1I"(X)A(t)1I"(Y)A(S) = 1I"(xat(Y))A(tS), (1r(x) A( t)) * = 11"( at l( x*)) A(t- 1), Vx, Y E M, s, t E G, we have M X a G = [1I"(X)A(t) Ix E M, t E Gt'. Later, we shall show that the definition of M x a G is independent of H up to * isomorphism.

Now consider the case of unitary implement.

Proposition 16.1.7. Let (M, G, a) be a VN-system with M C B(H) and s --+ u, be a strongly continuous unitary representation of G on H such that u,Mu; = M and a,(x) = u,xu:, Vs E G, x E M. Define

(W I)(s) = u,/(s),

VI E L 2(G, H).

Then W is a unitary operator on L 2 (G, H ), and

1I"(x) = W*(x Vx E M, S E G, where L2 (G). Consequently, M

S

@

l)W,

A(S) = W*(u,

@

A,)W,

A, is the left regular reperesentation of G on G is spatially * isomorphic to the VN algebra

--+

Xa

{x@ 1,u, @ A,lx EM,s E G}"

Proof.

Clearly, W is unitary, and

(W* /)(s) = u:/(s),

Vs E G, / E L 2 (G, H).

Moreover,

(1I"(x)/)(s) = arl(x)/(S) = u:xu,/(s) = (W*(x ® l)W /)(s) ,

617

and

e

(W·(u.

A.)W f)(t) = u;((ua ® A.)W f)(t)

u;u.(W f)(S-lt) Vs,t E G,x E M,f E L 2(G, H).

= f(S-lt) = (A(s)f)(t), Q.E.D.

Lemma 16.1.8. Let (MI, G, a(l»), (M2 , G, a(2») be two VN-systems. Then (Ml ®M2' G, a) is also a VN-system, where at = ap) ® a~2), Vt E G. Proof. Let M, C B(Hi), i = 1,2. Suppose that t --+ u~i) is a strongly continuous unitary representation of G on Hi, and a~i) (Xi) = U~i) XiU~i)., Vt E G, Xi E M i , i = 1,2. Clearly, t --+ U~l) ® U~2) is a strongly continuous unitary representation of G on HI ® H 2 • Thus (Ml ®M 2 , G, a) is a VN-system, where

at(') = (a~l) ® a~2»)(.) = (U~l) ® U~2») . (uP) ® U~2»)., Vt E G. For general case, by Proposition 16.1.5 we have VN-systems:

(1I"i(Mi),G,a(i»),i = 1,2, where {1I"i' L2(G, Hi)} is a faithful W·-representation of M, (see Proposition 16.1.3. ), and

a~i)(1I"i(Xi)) = 1ri(a~i}(xi))

=

Ai(t)1ri(X)Ai(t)"

Vt E G, Xi E M i , i = 1,2. From the preceding paragraph, (1rdMl ) ®1r2 (M2 ) , G, ti) is a VN-system , where a = a(l) ® a(2). By Theorem 4.3.4, 11"1 ® 11"2 is also a faithful W·-representation of M 1 ®M2, and

1I"1(Ml)®1I"2(M2) = (11"1 ® 1I"2)(M1®M2). Notice that

at (( 11"1 ® 11"2)(a)) (Al(t) ® A2(t)) . (11"1 ® 11"2) (a) . (Al(t) ® A2(t)t (11"1 ® 1I"2)(a~l) ® a~2)(a)) = (1r1 ® 1I"2)(at(a)) , Va E M 1®M2. Now since (1I"1®1I"2)-1 is a-a continuous, t -+ at(a) is continuous from G to (M1®M2 , u), Va E M 1®M2 • Therefore, (M1®M2, G, a) is also a VNsystem. Q.E.D.

Definition 16.1.9. of G on L 2(G), i.e.,

Let t

--+

.At, Pt be the left, right regular representation

(At€)(X) = €(t-lS),

(Pt€)(s) = ~(t)1/2€(st),

618

Vs, t E G, € E L 2(G). Clearly, (B(L 2(G)), G, adp) is a VN-system, where

adpt(a) = Ptap;, Va E B(L 2 (G)), t E G. Now let (M, G, a) be a VN-system with M C B(H). By Lemma 16.1.8, (M"®B(L 2 (G)), G, 0) is also a VN-system, where 0t = at ® adph Vt E G. From Proposition 16.1.3, 1r(M) C M®Z c M®B(L 2 (G)). Moreover, A(s) = l®A a E M®B(L 2 (G)), Vs E G. Thus, we get

M

Xa

G

C

M@B(L 2(G)).

In the following, we shall prove that M system (M®B(L 2(G)), G, 0), i.e.,

Xa

G is the fixed point algebra of the

We shall identity Z with LOO(G). Then 1 @ gEM @ B(L 2(G)), and

Ot(l @ g) = 1 e adpt(g) where gt(')

= g(·t), Vg

Lemma 16.1.10.

= 1 e gt,

E LOO(G), and 1 = 1H •

For any hE L 1 (G) n LOO(G) , we have

f Ot(l

@

h)dt =

f h(t)dt .

1H l8l L 2(G) '

in the a-weak topology.

Proof,

For any

€, 1J

E L2 (G, H), by the Fubini theorem we have

I(Ot(l@h)€,TJ)dt =

=

11 dt

h(st)(€(s),TJ(s))ds

1 t) dt ·1 (e( h(

s ) , 1J ( s )) ds .

Q.E.D.

That comes to the conclusion.

Lemma 16.1.11. Denote all continuous functions on G with a compact support by K(G). Let I,» E K(G), and x E M@B(L 2 (G)) . Then Ot((l @

1

f)x(l

@

g))dt E M®B(L 2 ( G)) in the a-weak topology, and it is a-continuous

for x. Proof.

For any

e, TJ E L

2

( G,

H), notice that

I(Ot( (1 @ f)x(l e g)) €, TJ) I = llxll·II Ot(l e g)€11 ·lIOt(l @ f)TJll

619

and

From Lemma 16.1.10, we have

If (0,((1 e l)x(1 (I 0

11 , (1 ® g)€11 2dt)1/2 . (/11 0, (1 ® 7)TJ 1I 2 dt )1/2

< Ilxll' Hence,

V€,TJ

II XII·11/1I2 ·llgl12 ·1I€1I·IITJII,

E

L2(G,H).

f 0,(1 ® l)x(1 e g))dt E ~B(L2( G)) in the u-weak topology, and

III Of((1 e l)x(1 X

® g))€, TJ)dtl

® g))dtll

< Ilxll·11/112 ·11/112'

Now let g = I, and let {XI} be a bounded increasing net of (~B(L2(G)))+, = sup and € E L 2 (G, H ) with a compact support. Since I

x"

o < (0,((1

®

l)x,(1 ® g)) €, €) / (0,((1 ® l)x(1 e g)) €, €)

Vt E G, by the Dini theorem the convergence is uniform for t E K, where K is any compact subset of G. Since (Of{1 ® g)€)(x) = g{st)€{s) and supp€ is compact, it follows that

I

I

(Of{(1 ® I)XI(1 e g))€, €)dt /

Further, from

(Of{(1 e l)x(1 e g))€, €)dt.

III Ot((1®/)y(l®g)) dt ll < lIyll·11/1I2 ·llgIl2(Vy E ~B(L2{G)))

and {€ E L 2 {G , H ) lsupp € is compact} is dense in L2{G,H), we can see that

1Of({1

®

l)x,{1 ® g))dt

-----+

in the u-weak topology. Then conclusion holds also for any

1Of((1

®

I, g E K( G)

l))x(1 ® g))dt

by polarization.

Lemma 16.1.12. IT X E M and IE K(G), then

I O,(x in the u-weak topology.

®

I)dt = II(t) 1I"(a,(x))dt

Q.E.D.

620

Proof.

For any

€, TJ

E L 2 ( G l H), by the Fubini theorem we have

!(Ot(X®/)€,,,,}dt = !(at(x) ® It€,TJ}dt

- ! dt ! I (st)(at(x) e(s), TJ (s)} ds - ! ds f I (t)(a,.-l (at(x))€(s), TJ(s)}dt - ! ds ! l(t)((1I"(at(x))e)(s)'TJ(s)}dt - ! (I (t) at(x)) e, TJ} dt. 11" (

That comes to the conclusion.

Q.E.D.

Lemma 16.1.13. For any I,g E K(G) and a E M®B(L 2 (G )) , we have that Ot((1 ® l)a(1 ® g))dt E M X a G.

!

Proof.

First, let a = x ® hA., where x E M, h E K(G), s E G. Then

Ot((l e I){x

®

= Ot(x ® k)Ot(1

hA.)(1 e g)) = Ot(x ® IhA.g)

®

A.) = Ot(x ® k)A(S),

where k(·) = 1(·)h(·)g(S-l.). By Lemma 16.1.12, we have

! Ot((1 -

®

I)(x e hA.)(1 ® g))dt

I k(t)1r(at(x))A(S)dt E M

Xa

G.

Since {hA.ls E G,h E K(G)}" = B(L 2(G)), it follows from Lemma 16.1.11 that I Ot((1 ® I)a(l ® g))dt E M

Lemma 16.1.14.

Xa

G, Va E M®B(L 2(G)).

Let h E K(G), and K be a compact subset of G. Let

tPK(S) = IK h(st)dt. Then tPK E K(G), 1 e tPK = IK Ot(l ® h)dt, and u- I W(1 ®

Proof.

For any

e,,,, E L

2

(

tPK) = I h(t)dt . IH~L2(G).

G, H), by the Fubini theorem we have

IK(Ot(1 ® h)e, TJ}dt = IK dt I h(st) (e(s), TJ( s)}ds

- ! (e(s), TJ (s)}ds IK h(st)dt -

Q.E.D.

((I®tPK)€,TJ).

=

I tPK(S )(€(s), ",(s)}ds

621

Thus,

Ix Ot(l e K)dt

H~(l e
= 1

= / h(t)dt .

e

1H~L2(G).

Moreover, since I
IIhll l

and

111 e

a-Hf1(1 ®
=

f h(t)dt ·lH~L2(G}. Q.E.D.

Theorem 16.1.15. we have

M

Let (M, G, a) be a VN-system with M

Xc. G

C

B(H). Then

= {a E M®B(L2(G))IOt(a) = a, Vt E G},

where Ot = at ® adp«, Vt E G.

Proof. Clearly Ot(A(s)) = (at ® adpt)(l ® A,) = 1 ® A, = A(s), Vs, t E G. For any x E M and f E K(G), since the integral in Lemma 16.1.12 exists in the a-weak topology and * automorphisms are a-weakly continuous, it follows that / O,(Ot(x ® f))dt = / f(t)O,(1r(at(x)))dt. On the other hand, we have

/ O,(Ot(x ® f)))dt = / Od(X ® f)dt =

f Ot(x ® f)dt

Thus, by Lemma 16.1.12 and f E K(G) is arbitrary, we get O,,(1r(at(x)))) = 1r(at(x)), Vs, t E G. In particular, O,,(1r(x)) = 1r(x), Vx E M. Therefore, M Xc. G C {a E M®B(L2(G))IOt(a) = a, Vt E G}. Now let a E M®B(L 2(G)) be such that Ot(a) = a, Vt E G. Pick hE K(G)

obviously.

with h > 0 and

f h(t)dt

= 1. For any compact subset K of G, let
/K h(st)dt. Then by Lemma 16.1.13, we have aK =

f Ot((l

®

Xc.

G.

We claime that

aK(l ® g)

-+

a(l ® g),

(1 ® g)aK - + (1 ® g)a

in the a-weak topology, Vg E K(G). In fact, let L = (suppg)-l(supph). Clearly, L is compact, and the function h(·t)g(·) is 0 if t ~ L. Thus, we have

aK(l®g)

=

/Ot(l®
=

i 0,((1 e
®

h))dt(l ® g)

622

Similar to Lemma 16.1.11, we can see that x

~

i

Ot(x)dt is a a-ro continuous

map on M®B(L2(G)). Therefore, from Lemma 16.1.14 an 16.1.10 we have

aK(1 ® g)

a

-----+

=

h

Ot(l ® h)dt(1 e g)

a / Ot(1

=a

e h) . (1 e g)dt

i

Ot(1

=

a(l ® g)

e h) (1 e g)dt

in the u-weak topology. By Lemma 16.1.14 and the Fubini theorem, we can see that

(1 ® g)aK

= / (1 ® g) . Ot(1 e 4>K) . a' Ot(1 e h)dt = / (1

e g) . /K 0f8(1 e h)ds . a· Ot(1 ® h)dt

= IK ds / (1 e g) 6. (s)-1. Ot(1

e h)· a· 0b-1(1 e

h)dt

= / dt IK-l (1 e g) . Ot(l ® h) . a . 0f8(1 e h)ds

= / (1 e g) . Ot(1 ® h) . a' Ot(1 e 4>K-l)dt. Now by the same discussion in the preceding paragraph, we get (1 e g)aK

-----+ /

(1 e g) . Ot(1 ® h)adt = (1 e g)a

in the u-weak topology. For any a' E (M x , G)', since aK E M

Xa

G, it follows that

(1 ® g)a'a(1 ® g) = u-lim(1 ® g)a'aK(1 ® g) K

u-lim(1 ® g)aKa'(1 ® g) K

=

(1 ® g)aa'(1 ® g),

Vg E K(G). Pick a net {gil C K(G) such that l®gi -----+ I H I8l L 2(G) in the u-weak topology. Then a'a = aa', Va' E (M x, G)', and a E M X a G. Q.E.D.

Proposition 16.1.16. Let (M, G, a) be a VN-system with M C B(H) , and a be spatial, i.e., there is a strongly continuous unitary representation s --+ u, of G on H such that a,(x) = uaxu:, Vx E M, s E G. Then we have

(M x , G)' = {x' ® 1, u, ® Palx' EM', s E G}". Proof. Since Ot(a) = a 16.1.15 that

M

Xa

¢::::}

(Ut ® pt)a = a(ut ® Pt), it follows from Theorem

G = (M®B(L 2(G))) n [u, ®

p,ls E G}'.

623

Therefore, we have

Q.E.D. Let (M,G,a),(N,G,P) be two VN-systems with M c B (H), NcB (K) respectively. If T is a * isomophism from M on N with T(at(x)) = Pt(T(X)), 'It E G, x E M, then r is a * isomorphism from M x , G onto N xfJ G, and r(1r a(x)) = 1rfJ(T(X)) , 'Ix E M, where r = T ® id: M®B(L 2(G)) -----+ N®B(L 2(G)), and {1r a,L 2(G, H)}, {1rfJ,L 2(G,K)} are faithful W· - representations of M, N respectively as in Propositon 16.1.3.

Proposition 16.1.17.

Proof.

For any x E M, hE K(G), by Lemma 16.1.12 we have

! at (x) e pthp;dt ! h(t)1r (at (x))dt. ! h(t)r a(at (x))dt ! Pt(T(X)) e pthp;dt = ! h(t)1rfJ(Pt(T(X)))dt. =

Thus

a

0 1r

Since h is arbitrary, it follows that

r(1r a(x))

=

1rfJ(T(X)) ,

'Ix EM.

Clearly, r(A(s)) = A(s), 'Is E G. So r(M X a G) = N xp G. Moreover, r is a * isomorphism from M®B(L 2(G)) onto N®B(L 2(G)) by Theorem 4.3.4. Therefore, r is also a * isomorphism from M X a G onto N xfJ G. Q.E.D.

Definition 16.1.18. (M, G, a) is called a W*-dynamical system, if M is a W·-algebra, a is a homomorphism from G to Aut(M), and t -+ at(x) is continuous from G to (M, u(M, M.)), 'Ix E M. Clearly, t -----+ at(x) is also continuous from G to (M, T(M, M.)), 'Ix E M. Definition 16.1.19. Let (M, G, a) be a W*-system. The fixed point algebra of the W*-system (M®B(L 2(G)),G,0),{a E M®B(L2(G))10t(a) = a,Vt E G}, is called the crossed product of M by the action a of G, and denoted by M x , G, where M®B(L 2(G)) is the W·-tensor product of M and B(L2(G)) ( see Section 4.3 ), and 0t = at ® adpt, 'It E G. Now let {x, H} be a faithful W*-representation of M. Then (1r(M), G, a(1r)) is a VN-system, where

624

Futher, we have VN-system (1r(M)®B(L 2(G)), G, 0(11'»), where 0(11') = 0:(11') ® adp. Clearly, (1r®id) is a * isomorphism from M®B(L 2(G)) onto 1r(M)®B{L 2 (G)), and Therefore, M x, G is * isomorphic to 1r(M) xac..) G.

Notes. In full generality W·-crossed products first appeared in M. Takesaki's paper. The version presented here is taken from A.Van Daele completely.

References.

[176], [189].

16.2. Takesaki's duality theorem In this section, we assume that the locally compact group G is abelian, and denote its dual group by G. Let p" il be the Haar measures on G, G respectively, such that the Fourier transform

[(p) =

f (s,p}f(s)ds,

Vf E L 2(G)

is a unitary operator from L 2 ( G) onto L 2 (G). Moreover, write dp,(t) = dt, dil(p) dp simply. Now for each pEG, define a unitary operator Vp on L 2 (G) as follows:

(Vpf)(s) = (s,p)f(s), Clearly, p

L

----+

Vf E L 2(G).

Vp is a strongly continuous unitary representation of

G on

2(G).

Lemma 16.2.1. Let (M, G, 0:) be a VN-system with M (1 ® Vp)a(1 ® Vp ) · EM x a G, Va E M x, G, where 1 = I H .

C

B(H). Then

Proof. Since (VpA,Vp• f)(t) = (t,p) (Vp• f)(S-lt) = (s,p}f(s-lt) = (s,p)(A.f) (t), Vf E L 2(G), it follows that VpA,V~: = (S,p}A" Vs E G,p E G. Moreover, by ((1 ® Vp)1r(x)e)(s) = (s,p}(1r(x)e)(x) = (s,p}o:,-l(X)e(S)

= o:,-l(X)(S,p)e(s)

Ve E L 2 (C, H ), we have (I®Vp)1r(x) =

= (1r(x)(1 ® Vp)e)(s),

1r(x)(I®Vp),Vx

(1 e Vp)(M x a G)(1 e vpt = M x, G.

E

M,p

E

G. Therefore, Q.E.D.

625

Definition 16.2.2. Let (M, G, a) be a VN-system with M c B(H), and define ap(a) = (1 ® Vp)a(l ® Vp)*, Va E M x , G,p E

a.

a,

Then the VN-system (M X a G, a) is called the dual system of (M, G, a), and a is called the dual action of a.

Proposition 16.2.3. Let (M, G, a), (N, G, (3) be two VN-systems, and a * isomorphism from M onto N such that

Then '1 = 1" ® id (see Proposition 1.1.17) is also a onto M xp G with

p

* isomorphism from M

1"

be

Xa

G

where a, are the dual actions of a, f3 respectively. Consequently, (M X a G) X (; and (N X p G) X p are * isomorphic.

a

Proof.

a

Notice that

;(ap(a)) = ;(1 ® Vp) . '1(a) . '1(1 ® Vp)* =

(1 e Vp)'1(a)(l e Vp)* = pp(r(a)) ,

......

Va E M x, G,p E G. Now the conclusion is obvious from Proposition 16.1.17.

Q.E.D. In the following lemmas, let (M, G, a) be a VN-system with M c B(H), and a be spatial, i.e., there is a strongly continuous unitary representation S ---+ U$ of G on H such that a$(x) = u"xu:, "Ix E M, s E G.

Lemma 16.2.4. Let Ro = (M x , G) x(; a( c B(H) ® L2(G) e L 2( a)). Then Ro is spatially * isomorphic to R}, where

R1 = {x ® 1 ® 1, U"

e A" e 1,1 e Vp ® Aplx E M, S

...... E G,p E G}

"

is a VN algebra on H ® L 2 (G) ® L 2 (a), and S ---+ A", p ~ Ap are the left regular representations of G, a on L 2(G), L 2 (a ) respectively.

Proof. Since a is spatial, it follows from Proposition 16.1.7 that (MxaG) X(; is spatially * isomorphic to

a

{a ® 1,1 ® Vp ® Apia E M

Xa

G,p E a}It.

626

Define a unitary operator W on L2(G, H) as follows:

From Proposition 16.1.7 we have

W(M x a G)W* = {x® 1,u. ® A.lx EM,s E G}" . .....

Since W(l ® Vp) = (1 ® Vp)W, v» E G, it follows that

(W ® l){a ® 1,1 e V, ® ApIa E M x., G,p E G}"(W ® 1)* {x ® 1 e L,u, Therefore, (M x , G)

A. e 1,1 ® V, ® A,lx EM,s E G,p E

G}".

G is spatially * isomorphic to R I •

Q.E.D.

* isomorphic to R 2 , where ... " R 2 = {x e 1 e 1, u. e A, e 1, 1 e V, e V, Ix EM, s E G, pEG} VN algebra on H e L 2(G) e L 2(G). R 1 is spatially

Lemma 16.2.5.

is a

X2

e

Proof.

Let 7* be the inverse Fourier transform from L 2(G) onto L2(G) :

(7*/)(8)

=

f (s,p)/(p)dp,

V/ E L 2 (G).

Then 1 e 1 e 7* is a unitary operator from H ® L 2 (G) L 2 (G) ® L 2 ( G). Now it suffices to show that

7*A,7 = V"

e

L 2 (G) onto H

e

.....

Vp E G.

For any / E K(G), we have

(7* Ap/)(S) = =

f (s, q){A,/)(q)dq f (s, q) /(p-lq)dq (s,p) f (s,q)/(q)dq (s,p){7* /)(s)

= (V,7* /)(s) ,

=

=

Vs E G,p E

G. Q.E.D.

Lemma 16.2.6.

R 2 is spatially * isomorphic to R s ®
... n { x ® 1, u, ® A., 1 ® V, Ix E M, s E G, pEG} is a VN algebra on H e L 2(G). Rs =

627

Proof.

Define a unitary operator U on L2 (G) ® L 2(G) as follows : (Uf)(s,t)

"If E L 2(G)

= f(st,t),

e L 2(G).

It is easily verified that U*(Vp e Vp)U

= Vp e 1,

U*(A.

e l)U = A" e 1,

"Is E G, pEG. Therefore, we have

(1 ® U*)R 2(1 ® U) (1 ® U*){x® 1 e 1,11,11

{x

e

All ® 1,1 e Vp e Vpl . . . }"(1 e U)

e 1 e 1,11,,, e A" e 1, 1 e Vp e 11x E M, s E G,p E G}"

Rg®OJIL~(G)'

Q.E.D. Lemma 16.2.7. R g is spatially * isomophic to M®B(L 2(G)). Proof.

Define a unitary operator W on H ® L 2 (G) as follows: (Wf)(s) = 11,,,f(s),

"If E L 2(G,H).

From Proposition 16.1.7, we have W*(11,.

e A.)W = 1 ® A.,

W*(x

e l)W

= 1r(x),

"Ix E M, s E G. Clearly, W(l e Vp) = (1 e ~)W, Vp E G. 1) {Vplp E a}" is the multiplicative algebra Z on L 2(G). In fact, VI' is' the multiplicative operator associated with the function

f

a.

a,

(.,p) on G, Vp E H h E L 1(G) is such that (s,p}h(s)ds = 0, Vp E then by the uniqueness of Fourier transform we liave h = O. This means that [(·,p)lp E a] is w*-dense in LOO(G). Therefore, we have {Vplp E a}II = Z. 2) {Vp,A.ls E G,p E = B(L 2(G)). This is obvious since Z is a maximal abelian VN algebra on L 2(G). 3) W*(M®Z)W = M®Z. In fact, by Propositon 16.1.3, we have W*(x®l)W = 1r(x) E M®Z, "Ix E M. Moreover, W*(l e Vp)W = 1 e Vp, Vp E Thus, W*(M®Z)W c M®Z. On the other hand, it is easy to see that W(x ® l)W* . (x' ® 1) = (x' ® 1) . W(x ® l)W*, "Ix' EM', x E M. Thus

a}"

a.

W(x

e

l)W* E (M' ® 1)' = M®B(L 2(G)), "Ix E M.

Clearly, W(x ® l)W* E {1 ® Vplp E a}' = B(H)®Z, "Ix E M. By Proposition 1.4.14, we have W(M®l)W* c M®Z. Therefore, W(M®Z)W* c M®Z, and W*(M®Z)W = M®Z.

628

Finally, R g is spatially

* isomorphic to

W*{x ® 1,1 ® Vp,u a ® Aalx EM,s E G,p E G}"W {W*(M"®Z)W, W*{u.

e Aals E G}W}"

{M®Z,1 ® Aals E G}" = M®B(L 2(G)).

Q.E.D. Let adVp(A) = VpaVp*, Va E B(L2(G)),p E 16.2.1, we can see that

G.

By the.proof of Lemma

(adA.)(adVp) = (adv;,)(adAlJ) on B(L2(G)), 'Is E G,p E G. Now let (M, G, a) be a W*-system. Then (id ® adV) is an action of M"®B(L2(G)), and O(id e adV) = (id ® adV)O,

G on

where 0 = a®adp = 0. ® adA. By Definition 16.1.19, ((id®adV)IM x , G) is an action of G on M Xer G.

Definition 16.2.8. Let (M, G, a) be a W*-system, and G be abelian. Then the W*-system (M, G, a) is called the dual system of (M, G, a), and a is called the dual action of a, where M = M Xer G, and ti = (id®adV)IM x, G. Two W*-systems (M, G, 0.) and (N, G, (3) are called isomorphic, if there is a * isomorphism r from M onto N such that r(at(x)) = {3t [r (x)) , 'It E G, x E M.

--

Proposition 16.2.9. Suppose that two W*-systems (M, G, a) and (N, G, (3) ....-.... """" are isomorphic. Then their dual systems (M, G, ti) and (N, G, ti) are also isomorphic. ~..-...

Proof. Let r be a * isomorphism from M onto N with r(at(x)) = (3t(r(x)) , V x E M,t E G. Then r = r ® id is a * isomorphism from M®B(L 2(G)) onto N®B(L 2(G)), and

r(id ® adVp) = (id ® adVp)r,

v»

E

G, t

r8;a) = O~,8)T,

E G, where o(er) = a ® adA,O(,8) = {3 ® ad): Therefore,

isomorphism from (M

Xa

G, G, a) to (N x,8 G, G, P).

r

is an Q.E. D.

Theorem 16.2.10. Let (M, G, a) be a W*-system, and G be abelian. Then _

A.

the double dual system (M, G = G, &) is isomorphic to (M"®B(L 2(G)), G, o = a ® adp).

629

Proof. By Propositions 16.2.9 and 16.1.5 , we may assume that (M, G, a) is a VN-system with M C B(H)' and there is a strongly continuous unitary representation s ----4 u. of G on H such that a. (x) = u.xu;, "Is E G, x E M . Let (M,G,a) be the dual system of (M,G,a). Then M = M x , G C B(H ® L2(G)) , and ap = ad(1 ® Vp), vv E a. By Proposition 16.1.7, let ..-..

.A.

..-..

(W f)(s,p) = ((1 e Vp)f)(p)(s) = (s,p)f(s,p) "If E H

e

L 2 (G) e L 2 ( a). Then

W is a unitary operator, and WRoW· = ~ = {a e 1,1 e Vp e Aplp E C}",

..-

Ro

..-..

......A.

= M = (M x , G) XCi G. Since li. =ad(1 ® 1 ® V.), where (V.g)(p) = (s,p)g(p), Vg E L 2(a),s E G, and W commutes with (1 ® 1 ® V.), Vs E G, so

where

to (Ro, G, a). Clearly, (W e 1) commutes with (1 e 1 ® V.), "Is E G. By Lemma 16.2.4, (Ro, G, &) is isomorphic to (RI, G, &), where &. = ad(1 ® 1 ® V.), "Is E G. By Y·V.Y = A., "Is E G, and Lemma 16.2.5, (Rt, G, a) is isomorphic to (R 2, G, ad(1 e 1 e A)). By Lemma 16.2.6 and U·(1 e A.)U = A" e A", (R 2 , G, ad(1 e 1 e A)) 1S isomorphic to (R s®
(Ro, G, &) is isomorphic

Remark. Let (M, G, a) be a W*-system, and G be alelian. On M"®B( L 2 (G))®B(L2(G)), we have three actions: ail) = a. ® adA. ® id(s E G)j a~2) = lip ® adAp = id ® adVp ® adAp(p E a)i and aiS) = id ® id ® adV.(s E G). Clearly, these actions commute each other; the fixed point algebra of (M®B(L 2(G))®B(L 2 (C)), G, a(I)) is (M x aG)®B(L 2(G))j the fixed point al.......... -. 2(a)),a,a(2)) gebraof((Mx aG)®B(L is (MxaG)x;a = M,and & = a(S)IM. Now let M be au-finite VN algebra on a Hilbert space H, ep and t/J be two faithful normal states on M, {urlt E lR} and {ut It E m} be the modular automorphism groups of M corresponding to ep and t/J reprectively ( see Definition 8.3.1). Then we get VN-systems (M,lR,uV') and (M,m,u¢).

Proposition 16.2.11. VN algebras M are spatially * isomorphic.

XuP

JR and M

X u+

JR on L 2 (lR, H)

Proof. By the unitary cocycle theorem ( Proposition 8.3.3), there is an oneparameter s(M, M.)-continuous family {utlt E JR} of unitary elements of M

630

such that

ut(x)

= Utui(x)u;,

UH.

= Utui(u.) ,

Vs,t E lR,x E M. Define a unitary operator U on L 2(lR,H) as follows

Vf E L 2(JR, H).

(Uf)(s) = u-.f(s),

Clearly, (U· f)(s) = u*-!f(s), Vf E L 2(lR, H). Let 'K1p,1rTjJ be the faithful W·representations of M on L2(lR, H) corresponding to cp, t/J respectively ( see Proposition 16.1.3). Then it is easily verified that

U'KIp{x)U·

= 'KTjJ(x), U>..(t)U· = 'KTjJ(u;) >..(t) ,

Vx E M, t E JR. Thus, we have U(M Similarly, it is easy to see that

U· >..(t)U

XU\"

lR)U· c (M

XU'"

JR).

= 1r1p(Ut) >..(t) , U·1rTjJ(x)U = 'Kip (x) ,

Vx E M, t E JR. Therefore, we obtain that U·(M U(Mx u\" lR) U· = M xU'" JR.

Proposition 16.2.12. The dual systems (M JR, (jTjJ) are isomorphic.

XU'"

XU\"

JR)U c (M

XU'"

JR), and Q.E.D.

lR, JR, alp) and (M

XU'"

JR,

Proof. Let ')'(a) = Uall", Va E MXu\"JR, where U is defined as in Proposition 16.2.11. It suffices to show that

ot(')'(a)) = ')'(&i(a)),

Vt E lR,a EM

XU\"

JR.

Since &i and &t are unitarily implemented by 1 ® Vi( on L 2 (JR , H) = H ® L2(JR)), and (1 ® Vi)U = U(1 ® Vi),Vt E JR, the conclusion is obvious.

Q.E.D. Remark. For au-finite W· -algebra M, by Propositions 16.2.11 and 16.2.12 it is reasonable to write a W·-system (M xuJR, JR,u). Moreover, we can prove that the W· -algebra M Xu JR is semi-finite. References. [127], [165], [176], [189].

16.3. Group algebras and Group C·-algebras This section is a survey on group algebras and group C·-algebras, and we shall not give the proofs for most of conclusions in this section. Indeed, this section is the preliminaries of next section: C·-crossed products.

631

Lacally compact groups Let G be a locally compact group, and JLl be a left invariant Haar measure on G, i.e., JLI be a regular Borel measure on G with JLI(tE) = JLI(E), lit E G and Borel subset E of G. It is well-known that: 1) JLl is uniquely determined up to multiplication by a positive constant; and JLI(U) > 0 for any Borel open subset U of Gj 2) G is compact if and only if JLl (G) < 00; 3) G is discrete if and only if JLr( {e}) > 0, where e is the unit of G. Let JL,.(E) = JLr(E- 1 ) for any Borel subset E of G. Then JL,. is a right invariant Haar measure on G i.e., JL,(Et) = JLr(E), "It E G and Borel subset E of G. Since JLI and JL,. are equivalent, we can write that JLI = 6 . JL,.. The function 6(·) is called the modular function of G, and it is positive and continuous on G, and

6 (e) = 1, 6 ( st) = 6 (s) 6 (t),

6 ( s -1) = 6 ( s)- 1, "Is, t E G.

IT write dJLI(S) = ds simply, then we have

d(ts) = ds, or

d(st) = 6(t)ds, ds- 1 = 6(S)-lds, "It

E G,

fa f(ts)ds = fa f(s)ds, fa f(s-')ds = fa ~~~) ds, 6(t)

L

f(s)ds =

L

f(s)d(st) =

L

f(st- 1)ds,

lit E G, f E K(G), where K(G) is the set of all continuous functions on G with a compact support. G is said to be unimodular, if 6(·) _ 1. In this case, IJ-l = JL,. is an invariant measure on G. For examples, compact groups, abelian groups and discrete groups are unimodular.

Measure algebras and group algebras Let G be a locally compact group, and denote the collection of all bounded Radon measures on G by M(G). It is well-known that M(G) = C~(G)*. In M (G), define the multiplication

Lf(s)d(JL * v)(s) = LLf(st)dJL(s)dv(t) and

* operation

L

f(s)dJL*(s) =

L

f(S-I)dlJ-(s) ,

lifE C~ (G), JL, v E M( G). Then M(G) is a Banach * algebra with an identity De, and 1I1J-*11 = IIIJ-II, VJL E M(G). This Banach * algebra is called the measure algebra of G.

632

L 1(G) = L 1(G,ds) is called the group algebra of G, its multiplication and

* operation are as follows: (I * g)(t)

=

Ll(s)g(s-l t)ds = Ll(ts)g(s-l)ds,

I*(t) = 6(t)-1 l(t-l), V/,g E L 1(G). Clearly with the norm II . lit, L 1 (G) is also a Banach * algebra and 11/*lh = 11/111, VI E L 1(G). With the map : I ----+ I (s)ds, L 1 (G) can be * isometrically embedded into M(G), and becomes a closed * two-sided ideal of M(G). We have the formulas:

(v. f)(t) Vv E M(G),

=

LfV't)d1.'(s), (f. 1.')(t) Lf(ts-') ~~~ , =

IE Ll(G).

In particular,

(06 * I)(t) = l(s-lt),

(I * 06)(t) = 6(s)-I/(ts- l ) ,

Vs E g, I E L 1(G). Moreover, it is well-known that: 1) G is abelian ¢=> Ll(G) is abelian ¢=> M(G) is abelian; 2) L 1 (G) has an identity if and only if G is discrete; 3) Let U be any neighborhood of e in G , and Zu E K( G) with supp Zu C U, Zu

> 0, and

L

Zu (t)dt

=

I

1. Then {zu U} is an approximate identity

for L 1(G), i.e.,

Ilzu * I - 1111 ----+ 0, III * Zu - 1111 ----+ 0, VI E L 1(G); 4) M(G) and L 1(G) are semi-simple, but in general they are not hermitian. Positive linear functionals, the GNS construction, and tations

* represen-

A linear functional p onL 1(G)( or M(G)) is said to be positive, denoted by p > 0, if p(a*a) > O,Va E Ll(G)( or M(G)).{1I",H} is a * representation of L 1 (G) ( or M (G)), if H is a Hilbert space, and 11" is a * homomorphism from the Banach * algebra L 1(G) ( or M(G)) to B(H). We have the following. 1) Let p be a positive linear functional on Ll(G) ( or M(G)). Since L 1(G) admits an approximate identity, it follows from the Cohn factorization theorem (see F.F. Bonsall and J. Duncan, Complete normed algebras, Berlin, Springer, 1973.) that p is bounded and hermitian. Moreover, we have

Ilpll = and

limp(zu

u

* zu)

633

Va,b E L 1(G) ( or M(G)), where 11(') is the function of spectral radius. 2) If 11" is * representation of Ll(G) ( or M(G)), then 1111"11 < 1. 3) IT {1I",H} is a nondegenerate * representation of L 1(G), then it can be uniquely extended to a * representation of M(G). It suffices to define that

or

VI E L 1(G), eE H.

4) A positive linear functional p on L 1 (G) is called a state , if IIpII = 1. For each state p on Ll(G), by the GNS construction there is a cyclic * reperentation {1I"p,H p, p} of L1(G) such that

e

Then p can be extended to a state on M(G). Clearly, each nondegenerate * representation of L 1 (G) is a direct sum of a family of cyclic * representations, and each cyclic * representation of Ll (G) is unitarily equivalent to the * representation generated by a state. 5) Let p be a stete on L 1 (G). Then p is a pure state (an extreme point of the state space on L 1 (G)) if and only if the * representation {11" p, H p } generated by p is toplogically irreducible. 6) For each non-zero a E L 1 (G), there exists a toplogically irreducible * representation 11" of L 1(G) such that 1I"(a) =I O. (7) The left regular representation {..\,L 2(G)} of L 1(G) is faithful, where

1* g, VI E L 1(G),g E L 2(G). Indeed, let ..\(/) = 0 for some I E L 1(G).

..\(/)g

=

Since

any compact neighborhood U of e and ..\(/)zu = 1= 11·111 - lim! u * Zu = o.

Zu

E

L 1(G) n L 2(G) for

I * Zu =

Unitary representations of G and nondegenerate of L 1(G)

0, it follows that

* representations

Let G be a locally compact group. {u., H} is called a unitary representation of G, if u, is a unitary oerator on H for each S E G, U,d = U"Ut, U e = 1, and S ~ (U"e,17) is a continuous function on G, \Ie, 17 E H. In this case, S --+- U" is also continuous from G to (B (H) , 1"( B (H) ,T (H)) ). Moreover, we have the following facts. 1) Let {u.,H} be a unitary representation ofG. If for any 11 E M(G) define

11"(11)

=

L

U"dll(S),

634

i.e., (1I"(v)e,1J) = L(u ae,1J)dv(s),Ve,1J E H, then {1I",H} is a of M(G), and 11"(6,,) = u", Vs E G.

~ 1I"(f) = L f(s)u"ds

Moreover, f

* representation

* representation of

is a nondegenerate

L 1(G). Conversely, let {11", H} be a nondegenerate * representation of L 1 (G). Then it can be uniquely extended to a * representation of M(G), still denoted by {11", H}. Further {u. = 11"(6.), H} is a unitary representation of G. Therefore, there is a bijection between the collection of all unitary representations of G and the collection of all nondegenerate * representations of L 1(G). 2) Let {u., H} be a unitary representation of G, and {11", H} be the nondegenerate * representation of M( G) correspoinding to {u., H}, i.e.,

11"(/) =

VI E L 1(G), v

f I(s)uads,

1r(v) =

f ~"dv(s),

E M( G). Then we hvae.

1I"(M(G))" = 1I"(L 1(G))" = {u"ls E G}". 3) For each s E G with s

#

e, there exists a topologically irreducible unitary representation {u., H} such that U a # 1H. 4) Let {A.,L 2 (G)} be the left regular representation of G, i.e.,

(A" 1)( t) = I (S -1 t) , Vs E G, I E L 2 ( G) . Then

A(f) =

f f(S)A"ds,

Vf E L 1(G)

is exactly the left regular representation of L 1 (G).

Positive linear functionals and continuous positive-definite functions n

A function cp on G is said to be positive-definite ,if

L

AZAkcp(St 1 Sk) >

k,l=l

0, Vs},· . "

Sn

E G, A},' .. , An E (C.

We have the following facts. 1) Let cp be a positive-definite function on G. Then we have

2) Let {u.,H} be a unitary representation of G, and (u·e, e) is a continuous positive-definite function on G.

eE H. Then cp(.) =

635

Conversely, if tp is a continuous positive-definite function on G, then there exists a cyclic unitary representation {u.,H, €} of G such that

tp(.) = (u.,€,€). 3) Let p be a positive linear functional on L1(G), {1r p, H p, €p} be the cyclic

*

representation of L (G) generated by p, and { u~p) , H p} be the unitary representation of G corresponding to {1r p, H p}. Then tp(.) = (u~p)€p, €p) is a continuous positive-definite function on G, and 1

Conversely, if tp is a continuous positive-definite function on G, then p(f) =

f f(s)tp(s)ds(Vf

E

L1(G)) is a positive linear functional on Ll(G).

In particular, there is a bijection between {pip is a state on L 1 ( G)} and {tpltp is continuous and positive-definite on G, and p(e) = I}. 4) Let tph P2 be two continuous positive-definite functions on G. Then tpltp2 is still positive-definite. 5) Let {tp,} be a net of continuous positive-definite functions with p,(e) = 1, VI, and pz be the state on L1(G) corresponding to PI, Vl. Then {p,} converges to a state p in w* -toplogy, i.e., there is a continuous positive-definite function

f

f

tp on G with p(e) = 1 such that j(s)PI(s)ds ~ f(s)p(s)ds, Vf E L1(G), if and only if , p,{s) ~ p(s) uniformly for s E K, where K is any compact subset of G. 6) ( R.Godement's theorem) Let p be a continuous positive-definite function on G, and p E L 2 ( G). Then there exists 1/J E L2 (G) such that p (.) =

(>...1/J,1/J). The enveloping C*-algebra of a Banach

* algebra

Let A be a Banach * algebra, and suppose that A admits a bounded approximate identity {a,} , and Ila*11 = Iiall, Va E A. A positive linear functional p on A is continuous automatically, and Ilpll = sup{p(a*a)la E A, Iiall < I} = liFP{a,) = liFP(a;a,). For any * representation {1r, H} of A , we have also II1rll < 1. Let p be a state on A ( i.e., p > 0 and Ilpll = 1). By the GNS construction, there is a cyclic * representation {1r p, H p, €p} such that p(a) = (1r p(a)ep, ep) , Va E A. Moroever, p is pure if and only if 1r p is topologically irreductible. For any a E A, we define

Iiall e =

sup{II 1r(a)II 11r is a

*

representation of A}.

636

Then we can prove that

Iialle

= sup{ 117r( a) JlI7r is topologically irreducible} = sup{p( a* a) 1/21p is a state on A} = sup{p(a·a)l/2Ip is a pure state on A} = sup{a(a·ap/2Ia is a C·-seminorm on A}

Va E A.

In other wored,

II . lie N

< Iiall,

is the largest C·-seminorm on A . Let =

{a E Allialle = o).

Clearly, N is a closed two-sided ideal of A, and II ·lle can become a C·-norm on A/N. Then completion of (A/N, II . lie) is called the enveloping C· -algebra of A, and denoted by C*(A). Now let A admit a faithful * representation. Then N = {o), II . lIe is the largest C*-norm on A, and C·(A) is the completion of (A, II . lie). Moreover, since Iiallc < lIall, Va E A, {a,l is still an approximate identity for C*(A). If p is a state on A, then by jp(a)1 = 1(7rp(a)ep, ep)1 < l1 7r p(a)II < lIallc, Va E A, and p(a,) -+ 1,p can be uniquely extended to a state on C·(A). Conversely, if p is a state on C·(A) , by p(al) -+ 1 then (piA) is a state on A. Therefore, the state spaces of A and C*(A) are the same. Group C·-algebras and reduced group C·-algebras Definition 16.3.1. Let G be a locally compact group, and II . lie be the 1 largest C·-norm on L (G) ( notice that the left regular representation of Ll (G) is faithful). Then the enveloping C*-algebra of L1(G), i.e., the completion of (L1(G), 11·llc), is called the C*-algebra of the group G, and denoted by C·(G). From the preceding paragraph, {zu} is still an approximate identity for C*(G); and the state space of C*(G) is equal to the state space of L1(G). Proposition 16.3.2. Let G be abelian. Then C*(G) is ..... ..... Cgo(G)' where G is the dual of G.

* isornrophic

to

Proof.

Since C* (G) is abelian, so the spetral space of C* (G) is the pure state space on C* (G). But pure state spaces of C* (G) and L 1(G) are the same. Therefore, the spactral space of C·(G) is 8, the spectral space of

L1(G).

Q.E.D.

Definition 16.3.3. Let G be a locally compact group, and {A, L 2 (G)} be the left regular representation of L1(G). Then Ilfllr = IIA(f)II(Vf E L1(G)) is

637

a C· -norm on L1(G). The completion of (L 1(G), II . II,.) is called the reduced C·-algebra 0/ the group G, and denoted by C;(G). Clearly, 11/11,. < lillie' VI E L 1(G). So the identity map on L 1(G) induces a * homomorhpism from C·(G) onto C;(G). Therefore, C;(G) is * isomorphic to a quotient C·-algebra of C·(G). Moreover, the VN algebra R(G) = {,,\~ Is E G}" on L 2(G) is called the VN algebra 0/ the group G.

Amenable groups Definition 16.3.4. Let G be a locally compact group. G is said to be omenable, if there exists a left invariant mean m on L 00 (G), i.e., m is a state on Loo(G) ( see Loo(G) as a C·-algebra) and m(~/)

= m(/),

where ~/(') = 1(8- 1 . ) , Vs E G, I E Loo(G). If G is amenable, then we can prove that there exists also a right invariant mean and a two-sided invariant mean on Loo(G). Remark. If G is discrete, then G is amenable if and only if R(G) has the property (P) (see the Remark under Lemma 13.4.6 and [153]). Example 1. If G is a compact group, then G is amenable. Indeed, we have an invariant Haar measure J.L on G with J.L(G) = 1. Define

Clearly, m is an invariant mean on Loo(G). Example e. If G is abelian, then G is amenable. In fact, let M be the mean ( state) space on Loo(G). Clearly, M is a compact convex subset of (L oo ( G)" w·- top.). For any s E G, define (T~m)(/) = m(~/),

Vm E

M,I E Loo(G).

Then T~ is an affine continuous map from M to M, Vs E G. Since G is abelian, it follows that T~Tt = TtT~, Vs, t E G. Now by the Markov-Kakutani fixed point theorem, there exists rna E M such that T~ma = ma, Vs E G. Clearly, ma is an invariant mean on L 00 ( G) . Example 9. Let F 2 be the free group of two generators u, v with discrete topology. We say that F 2 is not amenable.

638

In fact, if m is a left invariant mean on lOO(F2 ) , let E z be the set of elements in F 2 beginning with x, Vx E {u, v, u-l, V-I}, then

On the other hand, by the left invariance of m we have 1

= m(G) = m(Eu) + m(uEu-l) =

m(Ev}

+ m(vEv-l)

+ m(Eu-l) m(Ev) + m(Ev-l).

= m(Eu)

=

This is a contradiction. Therefore, F 2 is not amenable. For amenability, there are many classical descriptions. But for our purpose, it suffices to point out the following Go dement ' condition: G is amenable if and only if there is a net {"p,} C L 2 (G) such that

("pI, At'tPI)

----+

1

uniformly for t E K,

where K is any compact subset of G.

Main theorem of this section Definition 16.3.5. Let A be a C·-algebra, and {11", H} be a * representation of A. A state ( or positive linear functional ) p on A is said to be associated with 11" I if there exists E H such that

e

p(a) = (1I"(a)e, e), Va E A. Now let 11"},11'"2 be two * representations of A. We say that contained in 11'"2 ( or 11'"2 weakly contains 1I"1), if ker 11'"2 C ker 11"1'

11'"1

is weakly

Lemma 16.3.6. Let H be a Hilbert space, and p be a state on the C* -algebra B (H). Then p belongs to the o (B (H) '" , B (H)) -closure of Co{ (·E, e) Ie E H,

Ilell = 1}.

Proof. If the conclusion is not true, then by the separation theorem there is a E B(H) such that

Rep(a) > sup{Re(aE, e)le E H, II ell = 1}. Let h = l(a

+ a*), then we have p(h) > sup{(hE, e)le E H, Ilell = 1} = max{AIA E u(h)}.

639

On the other hand, it is obvious that

p(h) < max{,xl,\ E u(h)}, a contradiction. Therefore, the conclusion holds.

Q.E.D.

Proposition 16.3.7. Let A be a C·-algebra, and {1I"b HI}, {11"2' H} be two * representations of A. Then the following statementas are equivalent: 1) 11'"1 is weakly contained in 11"2; 2) Each positive functional on A associated with 11"1 is a w*-limit of sums of positive functionals associated with 11"2; 3) Each state 'on A associated with 11"1 is a w*-limit of states which are sums of positive functionals associated with 11"2' Proof. 1) ==? 3). Let p be a state on A associated with 11"1 • Since ker 11"2 C ker 11"1, p can become a state on AI ker 11"2 ( Proposition 2.4.11). Clearly, we may assume that AI ker 11"2 C B(H2 ) . Since p can be extended to a state on B(H2 ) , then by Lemma 16.3.6 we have the statement 3). 3) ~ 2). It is obvious. 2) ~ 1). For any a E ker 11"2, and E H, by the condition 2) we have (lI"l(a)e, e) = o. Therefore, a E ker 11"}, and ker 11"2 C ker 11"1' Q.E.D.

e

Theorem 16.3.8. Let G be a locally compact group. Then the following statements are equivalent: 1) G is amenable; 2) Any * representation of C*(G) is weakly contained in its left regular representation, where the left regular representation of C* (G) is the unique extension of the left regular representation of L 1 ( G); 3) The left regular representation of C* (G) is faithful; 4) C*(G) = C;(G). Clearly, the statements 2),3) and 4) are equivalent. 1) ==> 2). Let G be amenable. By Godement's condition, there is a net {,p,} C L 2(G) such that (,xt, '!-'I, ,p,) --+ 1 uniformly on any compact subset of G. Since K(G) is dense in L 2(G), we may assume that ,p, E K(G), Vl. Clearly, (,\.,ph ,p,) E K(G), Vl. If p is any positive functional on C*(G), then there exists unique continuous positive-definite function

! f(s)
L 1(G). Clearly, tp(t)(,xt,p",pl) ~ tp(t) uniformly on any compact subset of G, and for each l,tp(·)(,x.,p",p,) E L 2(G) and
is continuous positive-definite. By the Godement 's theorem, we can write

640

where CPI E L 2(G),VI. Since {llcpdI2II} is bounded, it follows that Pt o(A.,A), where A = C*(G) and

PI (I)

=

--+

pin

! I( )(AtJcpl, cpt)ds = (A(/)cpl, CPt), s

VI

E Ll(G) and 1. Therefore, any positive functional on C*(G) is a w*-limit of positive functionals associated with the left regular representation. By Proposition 16.3.7 , we have the statement 2). 2) =:> 1). By Proposition 16.3.7, for any continuous positive-definite function cP on G with cp(e) = 1 there are cP~l),··· ,cp~l E L 2 (G) 'such that

CPI(t) = L{AtCPJ') , CPJO)

--+

cp(t)

i

uniformly on any compact subset of G and CPI(e) = 1, Vi. Since K(G) is dense in L 2 ( G), we may assume that cP~t) E K(G), VI, i. By (A.cp~'), cpJ')) E L 2 ( G), Vi, and the Godement's theorem, we can write

CPI(t) = (At.,p".,p,), where .,p, E L 2(G), VI. Picking amenable.

ip

=1

'It E G,

and by Godement's condition, G is

Q.E.D.

References. [27], [58], [61], [125], [127].

16.4. C· -crossed products Definition 16.4.1. (A, G, a) is called a C* -dynamical system 1 if A is a C*-algebra, G is a locally compact group, a is a homomorphism from G into Aut(A), where Aut(A) is the group of all * isomorphisms of A, and t --+ at( a) is continuous from G to A, Va E A. Definition 16.4.2.

Let (A, G, a) be a C*-system. Define

L1(G A a) = , ,

{

I

f is measurable from G to ( and lG Il/(s) IIAds < 00

By the norm the multiplication

(I * g)(t) =

£I(s)a.(g(s-lt))ds,

A,}.

641

and the

* operation

f· (t)

= 6(t)-lat(/(t- 1)).,

\/I,g E L 1(G,A,a),L 1(G,A,a) becomes a Banach

11/111, VI E L

1(G,

Clearly, L

1(G)

* algebra,

and

Ilf*lIl

=

A, a). ® A is dense in L1(G,A,a); and L1(G,(]J,id) = Ll(G).

Proposition 16.4.3. Let (A,G,a) be a C·-system, {zu} be an approximate identity for L 1(G) as in section 1.3, and {a,l be an approximate identity for A. Then {zu(t)at(a,)I(U,l)} is an approximate identity for L 1(G,A,a).

It suffices to show that

Proof.

Ilzu(·)a.(al) * ga - galll

~ 0, and

Ilga * zu(o)a.(a,) -

galll

~ 0,

\/g E L 1(G), a E A. Notice that

Ilga * zu(·)a.(a,) -

galh

£ £g(s)aa t)a,,-lt)(a,))ds - g(t)all £dtll £g(s)zu{s-lt)aat(az)ds - g(t)all f dtl(g * zu)(t) - g(t)I'llali + ! dtl(g * Zu Ht) 1·llaat(a,) - all dtll

and

3(zu(s-l

! dtl(g * Zu Ht) I· lI aat(a,) - all

< (I(g * zu)(t)I'llat-1(a) . a, - at- 1(a) J1dt + 211all {

I(g * zu)(t)ldt, k~ where K is a compact subset of G. Since {at- 1 (a) It E K} is a compact subset of A, and IJba, - bll --+ uniformly for b E B, where B is any compact subset

k

°

of A, it follows that

Ilga * zu(·)a.(a,) - galll ~ 0, \/g E L1(G),a E A.

Moreover, by

! Zu (s)

ds = 1 we have

IIzu(·)a.(a,)

* ga - galll

- £dtll Lzu(s)g(s-lt)a,,(ala)ds -1 zu(s)g(t)dsll < £dt £zu(s) ·lg(S-l t) - g(t)lds ·lla,a - all +llglll

L

zu(s)lla

3(a)

- alldso

642

Thus, IIzu (.)a.(a,)

* ga -

gall 1

~

0, Vg E Ll(G), a E A.

Q.E.D.

Remark. For any I E Ll(G, A, a) and e > 0, we can find gi E Ll(G), ai E A such that

II L giai - fill < e, ,

where j(t) = at-1(/(t)),Vt E G. Thus [g(·)a.(a)lg E Ll(G),a E A] is also dense in L1(G, A, a). Then we can prove that {zuad(U, I)} is also an approximate identity for L1(G, A, a).

Lemma 16.4.4.

L1(G, A, a) admits a faithful

* representation.

Proof. We may assume that A c B(H) for some Hilbert space. Define a representation {11'", L 2 (G, H)} of L 1 (G, A, a) as follows:

(1I'"(/)e)(t) =

*

L

at-1(/(s))e(s-lt)ds,

VI E L1(G,A, a), € E L 2(G,H). Now let IE L1(G,A,a) be such that 11'"(/) and fI E H, we have

e,

o =

(11'" (/)g ®

€, h ® fI)

=

f f (at-

s))

1 (/(

= O. Then for any g,h E K(G)

e, n)g(s-lt)h(t)dsdt..

Since h E K( G) is arbitrary, it follows that

Notice that

If (at- (/(s)) e, t])g(s-lt)ds - f (ar-l (/(s))e, fI)g(s-lr)dsl < f !((at- ar-l)(/(S))€,fI)I'lg(s-lt)lds + f l(ar-l(/(s))e,fI)I'lg(s-lt) - g(s-lr) Ids < f 11/(s)I!·llell·llflll·lg(s-lt) - g(s-lr) Ids 1

1 -

+211 ell· Ilflll . Ilglloo ·111 - L Ii ® aiIILl(G,A,a) +~

,

f 1((at-

i

1 -

ar-l)(~)e,fI)I'l/i(S)g(s-lt)lds,

643

where Ii E L 1(G),ai E A, Vi. Thus, t tinuous on G. Further, we have

-----+-

! (at-1(I(s))€, l1)g(s-lt)ds

f (at- (/(s)) €, 11)9(S-lt)ds 1

is con-

= 0,

'It E G, €,11 E H,g E K(G). In particular,

f (I (s) €,

11) 9 ( S -1 ) ds

= 0, V9

E K (G),

€, 11

E H.

Therefore, we get

(/(s)€,l1) = O,a.e.,V€,TJ E H. Now let H be the Hilbert space of the universal Then we can see that

F(/(s))

=

* representation

of A.

O,a.e., 'IF E A*.

Since I(G) can be contained in a separable linear subspace of A, we may assume that A is separable. Let {Fn } be a countable w*-dense subset of {F E A*IIIFII < 1}. Then there is a Borel subset E of G with JLl(E) = Osuch that Fn(/(s)) = 0, 'In and s ff- E, i.e., I(s) = 0, a.e, Therefore, L 1(G, A, a) admits a faithful

* representation. Q.E.D.

From Proposition 16.4.3., Lemma 16.4.4, and the general theory in Section 16.3, we have the following facts: each positive functional on L 1(G, A, a) is bounded and hermitian automatically; 1111"11 < 1 if 11" is a * representation of L 1 (G, A, a); there exists the GNS construction for each positive functional on L 1(G,A,a); and there exists the largest C·-norm 11·11 on L 1(G,A,a) with II . II < II . 111'

Definition 16.4.5. Let (A, G, a) be a C·-system. The completion of 1(G,A, (L a), II· III is called the crossed product of A by the action a of G, and denoted by A x , G, where II ·11 is the largest C· -norm on L 1 ( G, A, a). Clearly, an bounded approximate identity of L 1 ( G, A, a) is also an approximate identity for A x a G; the state spaces of L 1(G, A, a) and A x a G are the same.

Example. IT

a

= id,

(I * g)(t) =

then we have

! I(S)g(S-lt)ds,

I·(t) = 6(t)-1 l(t- 1)*,

644

vf, 9 E L 1 ( G, A, id).

Of course, A ® L l ( G) is dense in A Xid G. Thus, A is the completion of A ® L 1 (G) with respect to the norm sup{111I"(')II 111" is a

Xid

G

* representation of A ® L 1(G)}.

By Proposition 3.3.2, we have that

A

Xid

G = max-(A ® C*(G)),

where max-(A ® B) means the projective tensor product, and 11·llmax(= al(·) in Chapter 3 ) is the maximal C*-norm on A ® B.

Definition 16.4.6. {11" , U, H} is called a covariant representation of a C*-system (A, G, a), if {11", H} is a nondegenerate * representation of the C*algebra A, {u, H} is a strongly continuous unitary representation of the group G, and 1I"(a 8 ( x ) ) = U811"(x)u:, Vx E A, s E G. With a covariant representation {1I",u,H} of (A,G,a), we can define a representation {11" Xu, H} of L 1 ( G, A, a) as follows: (11"

X

uHf) =

! 1I"(f(t))utdt,

*

Vf E L 1(G,A,a),

.

i.e.,

Theorem 16.4.7. Let (A,G,a) be a C*-system. Then {1I",u} - - t 11" X u is a bijection between the collection of all covariant representations of (A, G, a) to the collection of all nondegenerate * representations of L 1 ( G, A, a). Proof. First, we prove that {11" x U, H} is nondegenerate. In fact, let € E H be such that (11" x u)(f)€ = 0,Vf E L 1(G, A, a) . In particular, we have

! g(t)(1I"(a)ute,1])dt

= O,Vg E L 1(G),a E A, 11 E H.

e

Then (1I"(a)Ut€, 11) = 0, Vt E G, a E A, 1] E H. Further, 11" (a) = 0, Va E A. But 11" is nondegenerate, so = 0, i.e., (11" X u) is nondegenerate. Now let {p, H} be a nondegenerate * representation of Ll (G, A, a) , and {g,} be an bounded approximate identity for L 1(G,A, a) . For any x E A and f E L 1(G, A, a), put (xfHt) = x·f(t), Vt E G. Clearly, xg,*f = x(g,*f) ~ xf in L1(G, A, a). Then we can define

e

1I"(x) = s-lim P(x9,), I

Vx E A.

645

In particular, 1r(x)p(f)€ = p(xf)€,Vx E A,f E L 1(G,A,a). It is easy to see that {1r, H} is a * representation of A. Further, define u,

= s-limp(Or(g,(r- 1.)), I

Vr E G.

Noticing that (a r (g, (r -1. ))

* f) (t)

/ a r(g,(r- 1s))a

8(f(s-1t))ds

ar((g, * f)(s-1t))

-t

ar(rf)

in L1(G,A,a),

we have

urP(f) € = p(ar(rf)) €, Vf E L 1(G, a, 0), € E H. Since f· * f = ar(rft * ar(rf), it follows that U r is unitary, Vr E G. Moreover, r ~ ar(rf) is continuous from G to L 1(G, A, a), Vf E K( G, A). Thus, {u, H}

is a strongly continuous unitary representation of G. Since

Ur1r(x)u;p(f)€

= Urp(xar-l~-lf))€

p(ar(x)f)€ = 1r(ar(x))p(f)€, Vx E A,r E G,f E L1(G,A,a),e E H, {1r,u,H} is a covariant representation of (A, G, a). Finally, for any f,g,h E L 1(G, A , a ), €, rJ E H, we have

I (1r(f(t))utp(g)€, p(h)rJ)dt I (p(f(t)at(tg))€, p(h)rJ)dt (p(f * g)€,p(h)rJ) = (p(f)p(g)€,p(h)rJ). Therefore, p

= 1r

X

u. Moreover, since p is nondegenerate and

p(f) = / ut1r(at- 1 ( f ( t ) ) d t , 1r

Vf E L 1(G, A , a ), Q.E.D.

is also nondegenerate.

Remark. It is easily verified that (1r

x u)(A x, G)" = {1r(x),u,,[x E A,s E G}".

Definition 16.4.8. Let (A, G, a) be a C"'-system. A map CP : G said to be positive-definite, if n

L i.i=1

cp(s;1 s;)(a 37"1(a;a;)) > 0 •

---+ A"'

is

646

V'n'Sl"" ,Sn E G, and a.,··· ,an E A. ~ : G -----+- A* is said to be continuous positive-definite, if definite, and t -----+- ~(t) (x) is continuous on G, 'Ix E A.

Let (A, G, a) be a C·-system, and

Proposition 16.4.9. positive-definite. -----+- ~(t)(x*at(x)) quently, ~(e) > 0 on A.

1) t

~

~

:G

ep(.)~(.)

also positive-definite. 4) For a covariant representation {1r,u,H} of (A,G,a), and ~(t)(X) = ~

:G

Proof.

-----+-

A· be

---+

is a positive-definite function on G, 'Ix E A. Conse-

2) II e(t) II < 211 ~ (e) 11, 'It E G. 3) H ep(.) is a positive-definite function on G, then

Then

is positive-

:G

-----+-

€ E H,

A* is

let

(1r(x)Ute, e), 'Ix E A, t E G.

A· is continuous positive-definite.

1) For any s.,· .. ,Sn E G, AI, ... ,An E (C, we have

L ~(s;lSj)(x* a.;-l8j (X)) AiAj i,i

L

~(s;lsj)(a3il(aai(..\iXr . a 8 j (..\jx)))

>

O.

i,i

So t -----+- ~(t)(x·at(x)) is positive-definite on G, 'Ix E A. 2) Define [y, x]t = ~(t)(x· at(Y)), 'Ix, yEA. By 1) and Section 1.3, we have ~(e)(z*z),V'z

I[z,z]tl < [z,z]e =

EA.

Then from polarization, we can see that 1~(t)(x·at(y))1 =

<

~(e)(y*y

I[y,x]tl

+ x·x) <

11~(e)II(llxI12

+ Ily112).

Let {a,} be an approximate identity for A. Then 1~(t)(x*)1 = liml~(t)(x·at(a,))1 I

<

1I~(e)Il(lIxI12

+ 1).

Further, 11~(t)ll = sup 1~(t)(x*)1 < 211~(e)II,V't E G. 1I:1:1I~l

3) For

Sb ...

,Sn E G, ab ... an E A, write l

647

where Ai; = p(S;lSj ), J-Li; = .(s;lSjHa.7"l(ai aj)), Vi,j. Since (Aii) and (J-Lij) are two n X n positive matrices, it follows that

L

AijJ-Lij

i,;

L Aij"llci'"'llcj = L(L Aij"llci'"'llc;) > 0, =

iJ,lc 1c

ii

where (J-Lii) = (,i;)* . ('"'Ii;). Thus p(J) is also positive-definite. 4) It is obvious.

Q.E.D.

Theorem 16.4.10. Let (A, G, a) be a C*-system. Then there is a bijection between the collection of all positive linear functionals on Ll(G, A, a) and the collection of all continuous positive-definite maps from G to A *. In detail, let (J) : G ---+ A* be continuous positive-definite, and let

F(/) =

I(J)(t)(/(t))dt,

VI E L 1 ( G, A, a).

Then F is positive on L 1 ( G, A, a) . Conversely, let F be a positive liear functional on L 1 (G , A , a), {p,H, e} be the cyclic * representation of L 1(G,A, a) generated by F, and {11'", U, H} be the covariant representation of (A, G, a) such that p = 11'" x u, Define

(J)(t)(x) = (1I'"(x)Ute, e), 'It Then (J) : G

---+ A *

E G,

x E A.

is continuous positive-definite, and

F(/) =

f (J)(t)(/(t))dt,

VI E L1(G,A,a).

Moreover, we have that 11.(t)11 < 11(J)(e)11 = dence.

IIFll(Vt E G) in

above correspon-

Proof. Let F be a positive linear functional on Ll(G, A, a), and {p, H, {11'", U, H} and (J) be as above. Then

F(/)

=

(p(/)e, €)

f VI

E L1(G,A,a). If

e},

= (11'" X u)(/)e, e)

(11'" (I (t))Ut

e, e} dt = f (J) (t)(I (t )) dt,

{I,} is a bounded approximate identity for L1(G,A,a),

then

II FII =

limF(ll) = lim(p(f,) l l

e, e} =

II e1l 2 •

648

On the other hand, since {11'", H} is a nondegenerate follows that 11~(e)11

= sup{Iq,(e)(x)llx E

A, Ilxll

* representation of A, it

< I}

e, €)llx E A, Ilxll < I} = liF I(11'" (a,) e, €)I = II €11 = IIFII > Ilq,(t)II, 'It E G, = SUp{1 (11'" (x)

2

where {a,} is an approximate identity for A. Now let q, : G ---+ A· be continuous positve-definite. Since L 1 (G) ® A is dense in L 1 (G, A, a ), t ---+ q,(t)(f(t)) is measurable on G,Vf E L 1(G, A , o:). By Proposition 16.4.9, we have Iq,(t)(f(t)) I < 21!q,(e)II·llf(t)II,Vf E L 1(G,A, a ). Thus , we can define a linear functional

F(f) = / q,(t)(f(t))dt,

'If E L 1 (G, A , a ).

For any gl,· .. ,gle E K( G), al,· .. ,ale E A, Sb· .. ,s, E G, notice that

L

ijnm

q,(s;;1 sm)(a;

L

(i,n).(i,m)

where

Sin =

Sn,acn

~ -,J

=

Oll;lllm

~aj))gi(sn)gj(sm)

q,(s;,.ISjm)(all:-l(a;najm)) > tn

0

9,(sn)O:'n(a.),Vi,n. It follows that

f / q,(s-lt)(a;oll-lt(aj))9i(s)9j(t)dsdt >

i.e.,

0

f

~ q,(t)(a;O:t(aj))(g; * gj)(t)dt > 0, -oJ

'191,·' . ,91e E K( G), ai,

... ,ale E A. Therefore, F is positive on L 1 ( G, A, a).

Q.E.D. Corollary 16.4.11. There is a bijection between the state space of Ax a G and e q, : G ---+ A· is continuous positive-definite, } { and 11q,(e}11 = I,Le., q,(e) is a state on A .

I

Let (A, G, a) be a C·-system, and {11'", H} be a Define

* reperesentation

of A.

(1r(x)€)(t) = 1I'"(at-1(x))€(t), { (A(S)€)(t) = €(s-lt), 'Ix E A,s E G, E L 2(G,H). Then {1r,A,L2(G,H)} is a covariant representation of (A, G, a), i.e.,

e

r=

A(S) 11'" ( x) A(S

11" ( 0: 8

(

x)) ,

'Ix E A, S E G

649

( it is similar to Proposition 16.1.5). Further, we have a {1r x A,L2(G,H)} of Ax a G:

(11"

X

A)(f) =

L1I"(f(t))A(t)dt,

* representation

"If E L 1(G,A,a).

The * representation {1r X A, L 2(G, H)} is called the regular representation of Ax a G induced by the * representation {11", H} of A, and denoted by {Ind1r,L2(G,H)}, i.e.,

Definition 16.4.12.

Ind1r(f) or

(Ind1r(f)€)(t) =

=

L

1r(f(S))A(S)ds

L

1r

0

at-1(f(s))€(s-lt)ds,

"If E L1(G,A,a),€ E L2(G,H). Now we make the following discussions.

1) Let {1r, H} be a * representation of A, {€i liE I\} (c H) be a cyclic set of vectors for 1r(A), and {/;Ii E A}{C L 2(G)) be a cyclic set of vectors for A{L1(G)), where {A,L 2(G)} is the left regular representation of L 1 (G). Then {Ii ® €i!i,i} is cyclic for Ind1r(A x., G). In fact, let € E {Ind1r(A x., G)(f; ® €i) li,i}J.. Then for any f E K( G) and x E A ,we have

°

=

= =

Since {f

* fi

(Ind1r(f ® x)(f; ® €i), €)

f f (1r(ate(t))dsdt f at-1(x)€i' €(t))(f * f;)(t)dt. 1(x))f(s)f;(s-lt)€i'

(1r 0

= A(f)!;!f E K(G),j} is total in L 2 (G), it follows that

(1r 0 at-1 (x) €i, €(t))

= 0,

a.e., "Ix E A, i.

For any compact subset E of G with lEI > 0, and e > 0, by the Lusin theorem there is a compact subset F of G with FeE and IE\FI < e such that €(.) : F ~ H is continuous, where IBI = XB(t)dt for any Borel subset B of G. From Proposition 5.1.2 and 0 < IFI < 00, we can write

f

F =

u,«; U N,

where INI = 0, {Kn } is a disjoint sequence of compact subsets with the following property: for any opern subset U of G, if Un K n =1= 0 for some n, then IU n Knt > O. Now it is easily verified that (1r 0 at-1(x)€i, €(t))

= 0, "It

E UnKn, x E A,i.

650

Since {1I"(x)€ilx E A,i} is total in H ,it follows that €(t) = 0, Vt E UnKn. But e(> 0) is arbitrary, so €(t) = 0, e.e., on E for any compact subset E of G, i.e., € = 0 in L 2(G,H). 2) Let cp be a positive linear functional on A, and

II, 12

E K(G). Define

(1)

vx E L1 (G, A, a) . IT {11"tp, Htp, €tp} is the cyclic * representation of A generated by cp, then (Ind1l"tp(x)/l ® €tp,/2 ® €tp)

1(I 1I"tp

11

cp 0

0 at- 1

at-1

(x(s))/t (s- l t )€tp ds , / 2(t )€tp )dt

(X(S))/1(S-lt)/2(t)dsdt =

(2)

~/d~(x),

Vx E L 1(G,A,a).

Now notice the following fact:

{Ind1l"tp(y)1 ® etpll E K(G), y E K(G, A)} is dense in L 2(G,Htp), where v C K(G,A) means that y(.)

G

~

A is

continuous and supp y is compact. Indeed, for any 9 E K(G) and a E A, we have (Ind1l"tp(g ® a)zu ® €tp)(t)

= (g * zu)(t)1I"tp

0

at-1

(a) €tp,

where {zu} is an approximate identity for L1(G) as in Section 16.3. Moreover,

II (g * Zu (.)11" tp

f 111I"tp

0 at- 1

0

a.-l (a) €tp

(a) €tp!l2

- g(.)1I"tp

0 0.-1

(a) etp Ili2 (G,H ., )

·Ig * zu(t) - g(t)1 2dt

f

II al1 2 ·Ilcpll· Ig * zu(g) - g(t)1 2dt < lIa11 2 • llcpl! . (1Igiioo + Ilglloo . sup I ~ (s) 1-1) IIg * Zu - gill IJEU

<

Thus, the closure of {Ind1l"tp(L9, ® ai)zU ® €tplaa E A,gi E K(G),U} i

in L 2(G,Htp) contains the following subset L;

£

=

{Lgi(·)1I"tp i

0 0.-1

(aa) €tplai E A,gi E K(G)}.

~ O.

651

Similar to 1), f, is dense in L 2(G,HV'). Therefore, { Ind1rV' (y)f ® eV'lf E K(G), Y E K(G,A)} is dense in L 2(G,HV'). Now from (1) , (2) and above fact, we get

II Indsr., (x) II

su { Ind1r xy f ® e l f E K(G), v E K(G, A), and} p Ind1r~ v f ® e~ IIInd1r~(y)f ® €~II > 0

=

T

(3) 3) For any

v E K(G, A)

and f E K(G), define

I

O( f) = { cP is a positive functional on A, } y, cP and epff(Y*Y) > 0 . And for any

(4)

* representation 1r of A, let M( ) - { 1r -

I

cP > 0 on A, and 1r~ is weakly } cP contained in 1r, i.e., kern C kern ~ .

(5)

Clearly, there is a subset 1\ of M(1r) such that 1r = Then Indz

=

ED~EAInd1r~ ED

ED~EA1r V'

ED O.

0, and by (3), (4) we have

IIInd1r(x)II = sup{IIInd1r~(x)lllcp E I\}

< su {epff(Y*X* XYf/ I Y E K(G,A),f 2

-

p

epff(y*y)l

2

E K(G),} and cP E O(y, f) n M(1r)

Vx E L 1 (G, A, a). Conversely, for any cP E M(1r), we want to prove that IIInd1r(x) II > IIInd1r~ (x)II,Vx E L 1(G,A,a). This is divided into three steps. (i) Let cp(.) = (1r(')€' e) (some e E H 1r ) . Pick {€I} c H 1r such that € E {€I}, and H 1r = ED,H,EDHo, where H, = [1r(A)6], VI, and 1rIHo = O. Let 1rl = 1rIH"VI. Then 1r = ED,1r1 EDO, and Indrr = EDIInd1rl EDO. Thus, II Inder( x) II > II Inder, (x) II, VI, in particular, IIInd1r(x) II > II Indrr ~ (x) II, Vx E Ll (G, A, a). n

(ii) Let cP = LCPi, where CP.(·)

=

(1r(')€" 6), and

ei E H

1r

,1

< i < n.

We

i=1

claim that

where 1ri = 1r ~i' 1 0, let

",,(i) ( • ) a _ ",,(i) ( ... ) ai -_ tpII Y Y ,fJi - tpII Y x xy ,

where tP~J = (~) 11,1 

O}

< max IIInd1ri(x) 11 2 • I

Further, by (3) we have that IIInd1r~(x)II

< max II Inder, (x) II, Vx E L 1 ( G, A, a). I

(iii) Let tp E M(1r) with Iltpll = 1. From Proposition 16.3.7, there is a net {tp,} of states on A such that each tpl is a sum of positive functionals associated with 1r, and tpl(a) ~ tp(a), Va E A. Clearly, for any a E A,tpl(at(a)) ~ tp(at(a)) uniformly on any compact subset of G. Thus tP~}(g ® a) - t tPII(g ® a),

Vf, 9 E K(G), a E A, where $~} = (7{;;)II' VI. By (1) , IItP~}1I < IIflloo ·Ilfllb VI. lt follows that

tP~}(z) ~ tP II(Z), Vz E L 1 (G, A, a).

Now from (3) and (ii) , we get lIInd7l"~(x)1l < sup IIInd1l"~f(x)11 < 11Ind1r(x)ll, I

Vx E L 1(G, A, a). Therefore, we have that II Indzfz] II = sup{IIInd1r~(x)llltp E M(1r)}

{$//(Y·X.X y )I/2 IY E K(G,A),f E K(G),} - sup tPII(Y"y)1/2 tp E M(1r) n ll(y, f) , _

(6)

Vx E L1(G,A,a). From the proof of Lemma 16.4.4, if {1r u,Hu } is the universal * representation of A, then {Ind1l"u, L2(G, Hu ) } is a faithful * representatio of LI(G,A,a) . Thus, Ilxllr = sup{IIInd1r(x)lll1r is a * representation of A} will be a C*-norm on L 1 ( G, A, a). Let (A, G, a) be a C·-system. The completion of a), 11,11,) is called the reduced crossed product of A by the action a

Definition 16.4.13. (L

1(G,A,

653

= sup{IIInd1r(x)lIl1r is

a * repre-

{rpff(Y*X*Xy)1/2IYEK(G,A),fEK(G),} sup rpff(Y*Y) 1/2 tp E O(y, f) ,

(7)

of G, and denoted by A X a,. G, where IIxllr sentation of A}, Vx E L1(G,A,a). By (6), we have that -

II x II r Vx E L1(G,A,a).

Lemma 16.4.14. Let {1r, H} be a * representation of A, and t E G. Then the covariant representations {1I",..x, L2 (G, H)} and {11" 0 at, ..x, L2(G, H)} of (A, G, a) are unitary equivalent, and the * representations {Inds, L 2(G, H)} and {Ind1roat,L2(G,H)} ofAxaG are unitary equivalent. In detail, if define (Ute)(·) = .6(tp/2e(·t), Ve E L 2(G, H), then we have

o;

7r (

a) o, = 11" 0 at (a) ,

U (t)

x(s) = x(s) U (t) ,

Ut*In&rr(x)Ut = Ind(1r 0 at)(x), Va E A, s, t E G, x E A

Xa

G.

For any e E L 2(G, H), a E A, r, s ; t E G, x E L1(G, A, a), we have

Proof.

(ut 11" (a) Ute)(s)

=

=

(1I"(a)Ute)(st- 1) .6 (t)-1/2 1J" 0 atjl-l(a)(Ute)(st- 1) .6 (t)-1/2

= (1r

and

0

at)

tu; Indz (x)Ut eH r) =

0

ajl~l(a)e(S) = (11"

1r

0

0 a,.-l

that comes to the conclusion.

at(a)e)(s),

.6(t)-1/2(Ind1r(x)Ute)(rt- 1 )

J 0 at,.-1(x(s))(Ute)(s-lrt-1)ds J at) (x(s)) e( )ds (1r

0

s-lr

=

.6 (t)-1/2

(Ind( 1J"

0

at)( x) eH r).

Q.E.D.

Theorem 16.4.15. Let (A, G, a) be a C*-system, and {1r, H} be a * representation of A. Then the following statements are equivalent: 1) ffi1r 0 at is a faithful * representation of Aj

L

tEG

II Ind1r(x) II = Ilxll,., Vx E L 1 ( G, A, a). Consequently, if {1r,H} is a faithful * representation of A, then {Ind1r,L 2(G, H)} can be uniquely extended to a faithful * representation of Ax ar G. 2)

654

Proof. Suppose that IIInd1r(x)1I 1r 0 at(a) = 0, 'It E G, then

(Ind1r(g e a)e)(t) Vg E K(G),

=

=

IIxllr,Vx E L1(G,A,a). If a E A satisfies

! g(05)1r

0 at-1

(a) e(s-lt)do5 = 0,

eE L2(G, H). Thus we have

all r = IIInd1r(g ® a) II = 0, Vg E K(G), and a = 0. So L EB1r at is a faithful * representation of A. tEG Conversely, let L EB1r at be a faithful * representation of A. IIg ®

0

0 By PropositEG tion 16.3.7, any state on A is a w"'-limit of states which are sums of positive functionals associated with {1r 0 atlt E G}. By Lemma 16.4.14, we have

IIInd1r(x) II = llInd(1r 0 at)(x) II,

'It E G, x E L 1( G, A, a).

Then from the above discussion 3) (ii) and (iii), and the formulas (6) , (7), we get IIInd1r(x) II

sup {

tPff(Y· x· Xy~I/21 Y E K(G, A), f E K(G), and } epff(y·y)12 tp E O(y, f) n (U tEG.M(1r 0 at))

sup {

fPlf(Y·X*Xy~I/21 u E K(G, A), f E K(G), tp E O(y, f), } ,.; fl ('" \..I" tp Y Y)1 2 tp = tpl + ... + tpn, tpi E .M (1r 0 at, ) ,vI

fP f / (y* X*Xy )I /2 ! y E K(G,A),j E K(G), and} sup { epff(Y"'y)1/2 tp E O(y, f)

Q.E.D. Example. Let a = id, and 1r be a faithful any 9 E L1(G) and a E A we have

Ind1r(g ® a) on L 2 ( G, H)

= L 2 ( G) ® H. A

X

* representation of A.

= -X(g) ® 1r(a)

Therefore, we get ar G = min-(A e C;(G)),

where II ·lImin(= ao(') in Chapter 3) is the spatial C*-norm.

Then for

655

Proposition 16.4.16. Let (A, G, a) be a C·-system, and B be a C·subalgebra of A with at(B) = B, "It E G. Then we have that B X a r G ~ A

X ar

Proof.

G. It suffices to show that

Fix a x E L1(G, B, a).

Clearly, IlxllAxarG < IlxllBxarG' Conversely, let {1I",H} be a * representation of B. Since each state on B can be extended to a state on A, and 11" EBrpE1\1I"rp, where 1\ is a subset of the state space on B, there is a * representation {p, K} of A such that H C K, p(b)H C H, and p(b) IH = 1I"(b), Vb E B. Then we have "-J

(Indp(x)e)(t)

L

11" 0 at-1

=

fa po

at- 1

(x(s))e(s-lt)ds

(x(s)) e(s-lt)ds

=

[Indz (x) e)(t) ,

"Ie E L 2(G, H) C L 2(G, K). Thus, II

xllAxarG>

IIIndp(x)II > IIInd1l"(x)ll·

Q.E.D. Theorem 16.4.17. Let (A, G, a) be a C·-system, and G be amenable. Then A x a G = A X a,. G. Proof. It suffices to show that IIXII,. > Ilxll, "Ix E Ll(G, A, a). For any state ~ on L1(G, A, a), by the GNS construction there is a cyclic * representation {1I"rp, Hrp, erp} of Ax a G with Ilerpil = 1. By Theorem 16.4.7 , we have unique covariant representation {p, u, Hrp} of (A, G, a) such that 1I"rp = P X u. Since G is amenable, by Godement's condition there is a net {g,} C L 2(G) such that

uniformly on any compact subset of G. We may assume that IIgdl2 = 1, VI. Let ~1(Y) = (Indp(y)6, 6), vv E L1(G, A, a), where 6(05) = g,(s) U.-l erp, \:Is E

656

G,l. Clearly, 6 E L 2(G,H'P) and 11611 = 1,'v'I. Notice that
f (p(y(t))A(t)6, 6)dt f f (p (y(t)) 6(t- 1s), 6(x))dsdt f f g,(t- 1s)g,(s)(u,,(p (y(t)))u:Ut€'P' €'P)dsdt 0 a.,-l

0 a,-l

f (Atg"g,) . (p(y(t))Ut€'P' €'P)dt ~ f (p(y(t))Ut€'P' €'P)dt
=

=

vu E L1(G, A, a). Further,
=

liF(Indp(x*x)6,

e,) < II x ll; ,

'v'X E Ll(G, A, a). Therefore, we get IIxII2 = sup{
< IIxll~,

'v'x E L1(G, A, a).

Q.E.D. Corollary 16.4.18. Let (A, G, a) be a C*-system, G be amenable, and B be a C*-subalgebra of A with at(B) = B, 'v't E G. Then B Xcv G '----t A x , G. Now we discuss the embedding of A and G into M(A XCV G) for a C·-system (A,G,a), where M(A XCV G) is the multiplier algebra of A x , G (see Section 2.1.2). Let (A, G, a) be a C·-system., and {p, H} be a nondegenerate faithful * representation ofAxcvG. Clearly, {p, H} is still nondegenerate and faithful for L1(G,A,a). By Theorem 16.4.7, there is a covariant representation {1r,u,H} of (A, G, a) such that p = 1r X u, Since

p(f) =

"If

L

1r(f(s))u,ds =

L

u,,1r(a,,-I(f(s)))ds,

E Ll(G, A, a) and

p(g ® a) =

fG 1r(g(s)a)u,ds = 1r(a)

Lg(s)u"ds,

Vg E Ll(G) and a E A, it follows that {1r, H} is also faithful and nondegenerate for A.

657

For any a E A and

I

E Ll(G, A,

(Lal)(t) = al(t),

0:), define

(Ral)(t) = l(t)O:t(a) ,

Vt E G.

Clearly, La and R a are bounded on L 1 { G, A, 0:) with norm < Ilall. Further, we have p(Lal) = 1r(a)p{/), and p{Ral) = p{/)1r(a). Hence,

IIp{Lal) II < Iiall . 1I/IIAXaG, and IIRa/llAxaG < lIall·II/IIAxa G ' Then, La and s, can be extended to bounded linear operators on A x , G with IILall, liRa II < lIall, and IILa/llAXaG

=

LaX = p-l(1r(a)p(x)),

Rax = p-l(p(x)1r(a)),

Va E A, x E A X a G. Of course, these expressions are independent of the choice of {p, H}. Moreover, it is easily verififed that

Vx,y E A

G. So that (La, R a) is a double centralizer of Ax a G, Va E A (see Definition 2.12.5). Then we get a * isomorphism a -+ (La, R a) from A into M(A x , G). By Section 2.12, we can write that Xa

a = (0-- or s-) lim(Lall ) = (0-- or s-) lim(RaI,) , Va E A, where {I,} is an approximate identity for L 1 ( G, A, 0:) (then for A G), and 0- or s-topology is in (A x, G)**. Now for any s E G and I E L 1(G, A, 0:), define

(L 61)(t) = 0:6(/(s-lt)),

(RIII)(t) =

~(S)-l l(ts- 1 ) ,

Xa

Vt E G.

Clearly, LII,RII are bounded on L 1(G,A,0:). Further, we have

Hence, L II and R II can be extended to bounded linear operators on A and

Lllx = p-l(U P{X)), Il

Xa

G,

R,x = p-l(p(X)UII) ,

Vs E G, X E A X a G. It is easily verified that (L II , R,) is a double centralizer of Ax a G, and (LII,RlI ) is a unitary element of M(A X a G). Then we get a faithful representation s -+ (L., R IJ ) of G into the group of all unitary elements of M(A x , G). By Section 2.12, we can write that t = (0- or

Vt E G.

s-) lim(Ltll } = (o-- or s-) lim(RtI,) ,

658

We say that s ---+ (L IJ , R IJ ) is continuous with respect to the strict topology in M(A x , G) (see Definition 2.12.11). In fact, it suffices to show that

IIL,f - fll in A

---+

and

0,

IIR,f - fll

---+

0

Xa

Gas s

---+

II L lJ f -

flit <

fa Ilf(s-lt) - f(t) !IAdt + Ll1(a, - IHf(t)) IIAdt,

IIRlJf - fill <

e in G,Vf E K(G,A). But it is obvious since

L

16.(s-I)I· Ilf(ts- l )

-

f(t) IIAdt +

L

16.(s)-1 -

11 ·llf(t) IIAdt,

and IlglIAxaG < IIgllt, Vg E L 1(G, A, a). Finally, regarding A and G as subset of M(A X a G), and regarding {p = ". x u, H} as a faithful * representation of M(A x a G) (see Proposition 2.12.9), we have that

p(a) = 1r(a),

p(s) = u.,

and

sas-1 = a.(a),

Va E A, s E G. In fact, the equalities p(a) = ".(a) and p(s) = u. are obvious. Further, by

p(sas- l) = ulJ1r(a)u; = ".(alJ(a)) = p(alJ(a)) we have sas-1 = a.(a) in M(A X a G).

Summing up the above discussion, we have the following.

Proposition 16.4.19. Let (A, G, a) be a C·-system, and M(A xaG) be the multiplier algebra of Ax a G. Then A and G can be embedded into M(A x a G) such that

sst = s" S

= 1,

sas-1

=a

lJ (

a),

(sfHt) = (LlJfHt) = alJ(f(s-lt)), (fs)(t)

= (RlJf)(t) = 6.(s)-lf(ts- I ) , (afHt) = (LafHt)

=

af(t),

(fa)(t) = (Raf)(t) = f(t)at(a), in the sense of M(A x , G), "Is, t E G, a E A, and f E L1(G, A, a). Moreover, if {"., u, H} is a covariant representation of (A, G, a) such that {". X U, H} is faithful and nondegenerate for A x a G, then we have

p(a) = ".(a),p(s) = where {p,H} is the faithful

U IJ ,

Va E A, s E G,

* extension of {1r x u,H}

on M(A

Xa

G).

659

Notes. Crossed products of C·-algebras with discrete group were introduced by T.Turumaru. Later G.Zeller-Meyer carried out a penetrating analysis. General crossed products were defined by S.Doplicker, D.Kastler and D.W.Robinson, and Theorem 16.4.7 is also due to them. The reduced crossed products were defined by G.Zeller-meyer for discrete groups, and generalized by H.Takai. References.

[29], [127), [166], [186], [202].

16.5. Takai's duality theorem Definition 16.5.1. Two C·-systems (A, G, a) and (B, G, (3) are said to be isomophic, denoted by (A,G,a) (B,G,f3), if there is a * isomorphism () from A onto B such that () 0 at 0 ()-1 = f3t, Vt E G. "'J

Proposition 16.5.2. H (A, G, a) * isomorphic. Proof. Define

Let () be a

~

(B, G, (3), then A x or G and B xp G are

* isomorphism from A onto B with () W(/)(t) = ()(/(t)),VI

E

0

at

= f3t 0 (), Vt E G.

L 1(G,A,a).

Clearly, W is a * isomorphism from Ll(G,Aa) onto L 1(G,B,f3). Moreover, if ". is a * representation of Ll (G, B, (3), then ". 0 W is a * representation of L 1 ( G, A, a) obviously. Conversely, for any * represetation p of L 1 ( G, A, a) , there is a * represntation ". of L 1 (G, B, (3) such that p = ". 0 W. Therefore, we have

IIW(/)II = sup{ll1l" W(/)III'" is a * represntation of L 1(G,B,f3)} = sup{lIp(/)lllp is a * representation of L 1(G,A,a)} = lilli, VI E L 1(G, A, a), and W can be uniquely extended to a * isomorphism from A x or G onto B X P G. 0

Q.E.D. Proposition 16.5.3. Let (A,G,a) and (B,G,f3) be two C·-systems. Then there is a ( tensor product) C·-system (min-(A ® B), G, a ® (3) such that

(a ® (3)t(a ® b)

= at(a) ® f3t(b) ,

Vt E G,a E A,b E B, where min-(A ®B) means the injective tensor product, and 11·lImin(= ao(') in Chapter 3) is the spatial C·-norm on A ® B.

660

Proof.

It suffices to show that n

II L

n

at(tJi) ® Pt(bi ) Ilmin =

II L

.=1

tJi

e bill,

i=1

Vt E G, tJi E A, b. E B, 1 < i < n. Bu that is obvious from Theorem 3.2.5.

Q.E.D. Lemma 16.5.4. Let G be a locally compact group, and t ---+ Pt be the right regular representation of G on L 2 ( G), i.e., (Pt e)(s) = 6. (t) 1/2 (st), VeE L 2 ( G) . Then (C(L 2(G)),G,adp) is a C· -system, where adpt(a) = Ptap;,Vt E G,a E

e

C(L2(G)). Proof.

It suffices to show that

IIpt ap; -

all

=

Ilpt a-

apt I

---+ 0

as

t

---+

e in G,

Va E C(L 2 ( G)). Since the subset of all finite rank operators is dense in C (L 2 ( G) ), we may assume that a is an one- rank operator ® 77, where 77 E L 2(G). For any ~ E L 2(G) with 11~112 < 1, we have

e

e,

Il(pt a - apt)~112 = 11(~,77)Pte - (Pt~,TJ)eIl2

< <

e)ll2 + II (~, TJ) e 1177112 ·llpte - ell2 + lIell2 ·1177 II(~, TJ )(p,e

(~, Pt-1 TJ)

-

Pt-l 77 II

ell2

---+ 0

Q.E.D.

as t ---+ e in G. That comes to the conclusion. Now let (A, G, a) be a C·-system, and

Co(G, A) = min-(Co(G) ® A)

-

{I: G

---+

Allis continuous, and

11/(,)11

E Co(G)}.

By Proposition 16.5.3, (Cgo (G, A), G, ,) is a C*-system, where, = .x ® a, and (.xtg)(s) = g(t-1s), Vg E Co(G) (notice that g is uniformly continuous on G). Lemma 16.5.5.

Let (A, G, a) be a C·-system. Then ( C~ (G, A) x'1 G, G, p)

is also a C"-system, where

o(G,A),,),

(p(t)/)(s) = ptl(s),Vt E G,I E L 1(G,C

661

and (Ptg)(s) = g(st), Vt E G,g E Cgo(G, A). Since Pt'. = ,.Pt on Co(G, A) and p(s)p(t) = p(st), Vs, t E G, Pt is a * isomrophism of Ll(G, Cgo(G, A),,), Vt E G. If {1I",u} is a covariant representation of (Cgo(G,A),G,,), then {1I"0Pt,u} is also a covariant representation of (Co(G,A),G,,),Vt E G. Thus, we have

Proof.

IIp(t)fll sup

{II f

sup{1I

11" 0

pt(f(s))u.

d

sll

I

{1I",u} is a covariant } representation of (Co(G,A),G,,)

f 1I"(f(s))u.dsll 1{1I", u}is as above} = Ilfll,

Vf E L1(G, Cgo(G, A), ,), and p(t) can be uniquely extended to a phism of Cgo(G, A) x,., G, Vt E G. If f = 9 ® h, where 9 E L 1(G), hE K{G, A), then

IIp(t)f - fll

< IIp(t)f -

* automor-

fill

f lIptf(s) - f(s)IIc~{G,A)dS = f Ig(s)lds . sup Ilh(rt) - h(r)IIA =

---+

0

rEG

as t

-+

e in G. Therefore, (Cgo( G, A) x,., G, G, p) is a C· -system.

Q.E.D.

Theorem 16.5.6. Let (A, G, a) be a C·-system. Then the C·-systems ( 2(G)), min-(A ® C(L G, a ® adp) and (Co(G, A) x,., G, G,p) are isomorphic.

Proof. We may assume that A c B(H). Define a faithful 2(G,H)} {1I",L of Cgo(G,A) as follows: Vf E C~(G,A),

(1I"(f)E)(t) = f(t)€(t), It generates a faithful Cgo( G, A) x,., G, i.e.,

* representation

(Ind1r(z)E)(s, t)

(I

11"

=

{Inds: =

* representation

€ E L 2(G,H). 1r X

A, L 2 (G x G, H)} of

(I (1r(z(r, ·))A(r)E)(s, ·)dr)(t)

0,.-1 (z(r, '))€(r- 1s, ·)dr)(t)

f a.-l(z(r,st))E(r-

1s,t)dr,

Vz E K(G,cgo(G,A)),€ E L 2(G x G,H). Define a unitary operator w on

662

L 2 (G x G, H) as follows:

(we)(x,t) = L}.(tp/2e(st,t), { (w*e)(s,t) = L}.(t)-1/2e(st-1,t),'v'e E L 2(G x G,H). Then we have

(w*Ind1r(z)we)(s, t) = (Ind1r(z)we)(st- 1, t) . L}.(t) -1/2

- ! ate-1(z(r,s))(We)(r-1st-1,t)dr ! (z(r, S)) e(r- 1 t)dr, ate-1

L}.

(t)-1/2

s,

'v'z E K(G, Co(G, A)), e E L2(G X G, H). In particular, if z(r, s) = o:,(a)f (r- 1 s) g(s), where a E A,f,g E K(G), then (w*Ind1r(z)we)(s,t)

= =

'v'e E L 2(G

X

g(s)O:t(a)! f(r-1s)e(r-1s,t)dr

! f (r) e(r, t) dr/

9(s) at (a)

G,H). Further, define a faithful

(1) L}.

(r),

* representation {1f,L 2(G,H)}

of A as follows :

Then we have

w*Ind1r(z)w = 1f(a) ® vl'fJ E 1f(A) ® C(L 2(G)), L

.

where z(r, s) = a,,(a)f(r-1s)g(s), i,» E K(G), f'(r) an one-rank operator on L 2(G) :

= f(r)/

L}.

(2)

(r), and vJ'fJ is

Now we claim that

[J(r-1s)g(s)o:,(a)la E A,f,g

E

K(G)]

is dense in Co(G, A) x, G. In fact, define ~ : K(G as follows:

X

G, A)

-t

K(G

X

G, A)

vu E K(G X G,A). Clearly, ~ is a bijection, and keeps the maximal norm. For any y E K(G X G, A) and c > 0, there is a compact subset F of G such that suppy C F

X

F.

663

We may assume that e E F and F K(F), tlt E A such that

=

F-l. Pick 6

= c/IF2 1, and

/i,gi

E

~~ Ily(r, s) - ~ fi(r)gi(s)all(ai) IIA < 6. t

Then we have max 11(~y)(r,s) - z(r,s)IIA ",IIEG

< 6,

where z(r,s) = Lft(r- 1s)9i(s)all (tlt). Clearly, supp ~y and supp z

C

F 2 XF2.

i

Thus

Of course, K(G

II~y - Zllc~{G,A)X,.G

< II ~y -

{ max II (~y)(r, s) l« IIEG

z(r, s) IIAdr <

ZIILl(G,cO"(G,A)•..,)

61F21 =

e.

G, A) is dense in Co(G, A) x.., G. Therefore, [f{r-1s) g{s) QII(a)la E A,f,g E K(G)] is also dense in Co(G,A) x.., G. Then by (2) , we obtain that X

w*Ind1r(C~(G,A) x..,

G)w = (1f x id) min-(A @ C(L 2(G))).

Further, for any 0 E G,z(r,s} = f(r-1s)g(s)alJ(a), where a E A,f,g E K(G), by Lemma 16.5.5,

Then by (1) , we have

(w*Ind1r(p(o)z)we}(s, t) g(so)atq(a)

f f(ru)e(r, t)dr / ~ (r)

(1f(a u(a)) @ PuVl'gp;e)(s,t) , Ve E L2(G x G, H), i.e,

w*Ind1r(p(u)z)w = (1f @ id)(au @ adpu}(a @ vll g). Therefore, the C*-systems (Cgo(G,A)x..,G,G,p) and (min-(A@C(L2(G))),G, a @ adp) are isomorphic. Q.E.D. In the following, let G be abelian, and G be its dual. Let ds, dp be the Haar measures on G, G respectively such that the Fourier transform

j(p)

=

I (s,p)f(s)ds,

VI E L 2(G),

664

is a unitary opertor from L 2(G) onto L 2 (a ) . Let (A,G,a) be a C·-system, and define

ap(/)(t) = (t,p)/(t),

v» E a,t E G,I E L1(G,A,a).

Then ap is a * isomorphism of Ll(G,A,a), and apapl Moreover, since

= appl,Vp,p'

E a.

* represnetation of Ll(G, A, a)} = sup{II1r(/)III1r is as above} = lilli, VI E L1(G, A, a), lip can be uniquely extended to a * automorphism ofAx a G. For

Ilap(/)11 = sup{ll1r

0

a p(/)III1r is a

then I E L1(G, A, a), we have

any

lI ap(/ ) - II] < II lip (I) - Ilh

<

e

II (p, t) - 11 . II I (t) dt II

----+

0

a) a

as p ~ (the unit of in since (p, t) ~ 1 uniformly on any compact subset of Gasp ~ e in G. Therefore, we obtain a new C·-system (A x , G, G, ti). ~

~

Definition 16.5.'1. Let (A, G, a) be a C·-system, and G be abelian. The above C·-system (A x a G, ti) is called the dual system of (A, G, a), and ti is called the dual action of a.

a,

If (A, G, a) '" (B, G, (3), and G is abelian, then we

Proposition 16.5.8. have

(A

Xa

G,

a, a) ~ (B

Xp

G,

a, Pl.

Proof. Let ~ be a * isomorphism from A onto B with ~ 0 at = (3t 0 ~, Vt E G. Then 'If is a * isomorphism from A X a G into B xp G by Proposition 16.5.2. Now it suffices to show that 'If 0 ap 0 'If-I = p , Vp For any pEG and IE L I(G,B,(3), let g = 'If-I (f)(E Ll(G,A,a)). Then

P

'If 0 up 0 'If-1(/)(t)

=

Ea.

'If 0 up(g)(t)

= ~(ap(g)(t)) = (t,p)~(g(t)) = (p, t) I(t) =

That comes to the conclusion.

Theorem 16.5.9. we have that

Pp(f) (t).

Q.E.D.

Let (A, G, a) be a C·-system, and G be abelian. Then

665

1) Let {11", H} be a faithful * representatio of A. By Theorem 16.4.15, = 11" X A, L 2(G, H)} is a faithful * representation ofAx a G. Further, {1t, L2 (G x G, H)} is a faithful * representation of (A x a G) X (; G, where 1r = Indflnd-r] = Ind1r x A, and {Ind1r,A,L 2(G x G,H)} is a covariant representation of (A X a G, G, a). Now let IE K(G x G,A), t E L2(G x G,H), and consider 1t(/)t. For any pEG, t E G, we have

Proof. {Ind1l"

~

~

~

~

~

(1t(/)t)(p,t)

=

((Ind1r

X

A)(/)t)(p,t)

(I Ind1r(/(q, '))Aqtdq)(p,t) (I Indz (f(q, .)) t(q-1p, ·)dq) (t) fa dq Ie dss: (a s. .)) (s) t(q-1p, s-1t) fa dq Ie ds(s, p)1r (f( q, s)) t( «?». s-1t) 0 ap-l

0 Qt- 1

p-

l (/(

0 Qt- 1

Thus , we can write that

1t(/) = '" 'VI E K(G

X

1laxG I. f(q, s)u(s)v(q)dsdq,

G, A), where

(at)(p,t) = 11" 0 Qt-1(a)t(p,t), (u(s)t)(p,t) = (s,p)t(p,s-1t), { (v(q)t)(p,t) = t(q-1p,t), 2

~

~

'Va E A,s,t E G,p,q E G,t E L (G X G,H). 2) Define a unitary operator on L 2 (G X G, H) as follows:

(Jt)(p,t) 'Vt E G,p E

= (t,p)t(p,t), (J+t)(p,t) = (t,p)t(p,t),

G, t E L2(G X

G,H). By 1), we have

(J1r(/)J* t)(p, t)

(t,p)

1lexG f", dqds(s,p)1I"

I laxG dq ds(s, q) (t, q) f....

0

at- 1(f (q, s))(J +t )(q- 1p, s- 1t )

11" 0 at -1

(f (s, s))t (q-1 p, S -1 t) .

Thus, we can write that

Ji(/)J+ =

1laxG i. I(q, s)u'(s)v'(q)dsdq,

666 ......

Vf E K(G x G,A), where

(ae)(p, t) = 1r 0 a,-l (a) e(p, t), (u'(s)e)(p,t) = e(p,s-It), { (v'(q)e)(p,t) = (t,q)e(q-Ip,t), .....

2

......

VaEA,s,tEG,p,qEG,eEL (GxG,H) . ...... 3) Consider the C·-system (A, G, idle Clearly, A X'd G

""-I

min-(A ® C~(G))

=

Cgo(G,A).

Define a homomorphism f3 : G ---+ Aut(L I( G, A, id)) as follows:

f3t(1)(p) = (t,p)O:t(l(p)), Vt E G,p E G,! E LI(G,A,id). If {L, V} is a covariant representation of (A, G, id), then L(a)v p = VpL(a), Va...... E A,p E G. Thus {L 0 at, (t, ·)V.} is also a covariant representation of (A, G, id), Vt E G, and A

A

d II 11 f3,(rr J II = sup {II! L(f3t (f'J ()) P Vp p

{L, V} is a covariant } representation of (A, G, id)

= su p {ll! L 0 at(!(p))(t,p) V p dplll{L, V}is = Ilfll, Vf E LI(G, A, id), t E G Further, f3, can be uniquely extended to a Moreover, notice that

lIf3t(.i) -

* automorphism ofAx'd G, Vt E

G,

G.

ill < IIf3t(1) - illl =

fa II(t,p)at(i(p)) - j(p)lldp

<

fa I(t,p) -

11'lIi(p)lldp +

vi E LI(G,A,id). As t ---+ e in G, since (t,p) subset of

as above}

we can see that the first term

fa II at(j(p)) -

j(p)lldp,

~ 1 uniformly on any compact

fa I(t,p) -

11 . Ilj(p)lldp

---+

0; if

pick f(p) = g(p)a, where g E LI(G) and a E A, then we also have

fa II

at (f(p))

-l(p)lldp - + 0

as t ---+ e in G.

Therefore, (A X'd G, G, (3) is a C·-system. 4) Let {1r,H} be a faithful * representation of A. Then {(1r ® id) A,L2 (G, H )} is a faithful * representation ofAx'dG. Further, {1f,L 2(G

X

X

667

G,H)} will be a faithful * representation of (A Xid G) xf3 G, where 1f = Ind((7r ® id) X ,x). For anyl E K(G x G,A) and EE L 2 (G X G,H), compute i(/) E as follows:

(1f(/) E) (p, t)

(t id) ,x)(/(·, s)),x8Eds){p, t) (t(( id) ,x) t1t- s)) E(" S-l t )ds)(p) {" dqds(t, q)7r q,s)) E( q-1p, s-lt). I laxG (7r ®

X

7r ®

X

0

1 (/(',

0 (It-1 ( / (

Thus, we can write that

1f(/) = 'til E K(G

'tIa

E

A, s, t

X

I laxG t. I(q,s)v'(q)u'(s)dqds,

G, A), where

E G, p, q E

G, ~ E L 2 (G X G, H).

Since v'(q)u'(s) = (s, q)u' (s)v'(q),

it follows from 2) that

1f(/)

I ("

= laxG g(q,s)u'(s)v'(q)dqds = Jif(g)J·,

where g(q,s) = (s,q)/(q,s), 'til E K(G X G,A). 5) Notice that K(G X G, A) is dense in (A Xid G) xI' G and (A SO if define

~(/)(p,t) = (t,p)/(p,t),

'til E K(G

then by 2), 4) ~ can be uniquely extended to a G) xf3 G onto (A X a G) x;; G. 6) It is well-known that the Fourier transform

f

E

L 1( G)

-----t

can be uniquely extended to a

I(s) =

X

Xa

G) x;; G.

G,A),

* isomorphism from

fa (s, p)l(p)dp

E

(A

Xid

cgo (G)

* isomorphism from C·(G)

onto Co(G). Then

668

can be uniquely extended to a * isomorphism \11 from min-(C"(G) ® A) A Xid G onto min-(Co(G) ® A) = Co(G,A). Morevoer, by 3) we have \11

0

f3t 0 \11-1(1 ® a) = \11 ® (it(f e a)

= \11 ( (t, .) j (.)at ( a) ) =

f (.,p)(t,p)f(p)dp

= l(t-l.)

®

®

at(a)

at( a),

E Co(G), a E A. Thus \11 is an isomorphism from (A Xid G, G,(i) onto (Co(G,A),G,')'), where')' = oX ® Q. By Proposition 16.5.2, the Fourier transformation E K(G x G,A) - t I(s,t) = (s,p)f(p,t)dp

VI

fa

t

can be uniquely extended to a Co(G, A) x'1 G. 7) By 5) and 6), the map

*

isomorphism from (A

.....

CIJ: z(s,t) = al(s)g(t)

-----t

Va E A, I E K(G),g E L1(G) and g(t)

Xid

G) Xp G onto

(s,p)al(s)g(p),

= k(t,p)g(p)dp

E

C~(G), can be .....

uniquely extended to a * isomorphism from Co(G, A) x'1G onto (A xaG) x(iG. Further, for any z(s,t) = al(s)g(t), and rEG, since

~(p(r)z)(s,p) = ~(al(·)gr(·))(s,p)

(s,p)(r,p)al(s)g(p) = (r,p)i(z)(s,p)

fir 0

~(z)(s,p),

Va E A,I E K(G),g E L 1(G) and g(.) = k(-,p)g(p)dp, it follows that ~

~

p(r) = aroCIJ,Vr

~

E G. Therefore,

i

0

~

(Co(G,A)x'1G,G,p) '" ((AxaG)xaG,G,a).

Finally, from Theorem 16.5.6, we obtain that

((A x a G) x a G, G, &) '" (min-(A ® C(L 2(G))), G, a ® adp). Q.E.D. Notes. In the study of the structure of Von Neumann algebras of type (III), M. Takesaki obtained a duality theorem for crossed products of Von Neumann algebras ( Theorem 16.2.10). At the same time, he also conjectured about its C"-algebra version. Then H. Takai showed that conjecture is affirmative.

669

References.

[127], [166], [176].

16.6. Some examples of crossed products 1) The relation between C* -and W*-crossed products. Let (M, G, a) be a W*-system. By Definition 16.1.19, the W*-crossed product is

M

Xa

G

= {a E M®B(L2(G))IOt(a) = a, Vt E G},

where Ot = at ® adpt, Vt E G. IT pick a faithful W* -representation {11", H} of M, and let (1I"(x)t)(s) = a.-l(x)t(S), (A(t)t)(S) = t(t-1s)

Vt(·) E L 2(G,H),s,t E G. Then M x., G is * isomorphic to the VN algebra {1I"(M),A(G)}" on H ® L 2(G) = L 2(G,H). Hence, M X a G is the W* -subalgebra of M®B(L2(G)) generated by {(11" ® id)-l1r(x), 1 ® A.. lx E M, s E G}. Now let A be a C*-subalgebra of M, A be ot M, M*)-dense in M, at(A) C A, Vt E G, and (A, G, a) be a C*-system. Further, we assume that G is amenable. IT {11", H} is a faithful W*-representation of M, then by Theorem 16.4.15 {Indsr = 1r X A, L2(G, H)} is a faithful * representation ofAx a G. From the norm on M®B(L 2(G)), the largest C*-norm on L1(G, A, a) is as follows:

11(11" ® id)-l

fa 1r(f(s))A(s)dsll,

Vf E L1(G,A,a).

G)" = {1r(M) , A(G)}". Hence, A x a G is the norm closure of {(1I"®id)-1 1I"(f(S))A(S) dslf E L1(G, A, a)} in M®B(L 2(G)), and the a-closure of A X a G in M®B(L 2(G)) is M X a G. Moreover, (11" ® AHA

Xa

L

2) Let G be an amenable group, M = LOO(G), A = C~(G), and at(f)(s) = f(t-1s), Vf E LOO(G), t, s E G. Then A is a-dense in M, (M, G, a) is a W*system, and (A, G, a) is a C*-system. By the discussion 1), A X a G is a-dense in M X a G. Put H = L 2(G), and

1I"(f)g = fg, Then M x a G is , where

Vf E LOO(G),g E L 2(G).

* isomorphic to the VN algebra {1r(M) , A(G)}" on L 2(G x G) (1r(f)t)(s,t) = f(st)t(s,t), { (A(r)t)(s,t) = t(r-1s,t),

670

Vt E L2{G x G),s,t,r E G. Define a unitary operator W on L 2(G follows:

Vt E L 2 (G

G) as

X

(Wt){s,t) = D.(t)l/2t(st,t), { (W*tHs,t) = D.(t)-1/2t(st- 1,t), X

G). Then it is easy to check that

W*1r(f)W = 1r(f) ® 1, W* A(r)W = x, ® 1 on L2(G) ® L 2(G) = L 2(G X G), Vf E LOO(G), rEG. Since 1r(M) is maximal commutative in B(L 2(G)), it follows that MxaG is * isomorphic to B(L 2(G)). Moreover, if z(r,s) = f(r-1s)g(S){E L1(G,A,a)), where Ls E K(G), then we have

W* (1r

A)(Z)W = v/'fJ ® 1, is one rank operator on L 2(G), i.e., vl'fJ11 = (11, 7)g, V11 E L2(G) X

where vl'fJ ( see the proof of Theorem 16.5.6). Therefore, A

C(L

Xa

G is

* isomorphic

to

2(G)).

3) Every (UHF) algebra can be represented as a C* -erossed product.

(i) Let m(·) be a function from lN to IN. For each n EN, put G n = ~m(n)' Clearly, G n is a finite abelian group, and Gn = G m i.e.,

(j, k) = e2ri; /r,/ m(n),

vi

E

c; k E c.;

Vn. Now consider the discrete abelian group OO G G = EB n= 1 n

( 0 0 ) I n = 1, 2, ... , ={ S = S17 ••• , Sn, ,"', ,'" s; E G i, 1 < i < n.

Then

}

00

8=

II G n =

{oo

= (oo/r,h~lloo/r,

E G/r" Vk}

n~l

is a compact abelian group ( product topology ), and

(00, s) =

II (u/r"

,..,

s/r,),

Va E G, s E G.

/r,~l

(ii) For each n, denote by Cn = C( G 1 X •.. X G n ) the set of all complex functions on G 1 X ••• x G n • Clearly, Cn = C(G 1) ® ... ® C(G n) ( C*-tensor product). If identify c; with c; ® In+b then we have c; C-....+ Cn+l, i.e., f(Sl,"',Sn,Sn+l) = f(Sl,"',Sn),

Vf

E

Cn, Si E Gi, 1 0, since I is uniformly continuous on G, there is a neighborhood W = WI X ••. X Wn X Gn + I X G n + 2 X •.• of (here Oi E Wi C G i , 1 < i < n) such that

°

II (s) - f (t) I < s, Vs, t E G and (s -

t) E W.

H define In(sl,···,sn) = l(st,···,sn,O, ...),Vn,si E G i,1 < 1< n, then In E Cn and Il/n - 11100 < c. (iii) For each n, let H n = 12(Gn). Then Hi; is m(n) dimensional Hiblert space. Put c~n}(Sn) = 66,.. ,0,.. , VS n E c.;

co

where On is the zero element of G n , Vn, and = ®nC~n). Then we have 12(G) = tt; ( see Section 3.8). Now the (UHF) algebra

®!o

A oo

= ao- ®n B(Hn ) = ao- ®n Mm,(n)

is a C·-algebra on H = 12(G). Define

(Uoc)(S) = (O",s)c(s), Vc E 12(G) = H,u E G,s E G. Clearly, 0" --+ U o is a strongly continuous unitary representation of G on H, and U o = ®nuo,.., yO" = (O"n) E G, where (uo,..g)(sn) = (O"m sn)g(sn), Vg E H n = 12(G n), Sn, O"n E G n,Vn. Hence,

(A oo , G, ti) is a C·-system, where

For each

I

E C ( G), define (mf~)(s)

= I(s)€(s),

Vc E H

= 12(G),s E G.

( notice that G can be regraded as a dense subset of G), and let A = {m f II E C(G)}. We claim that A is the fixed point algebra of the system (A oo , G, ti). Clearly, tio(mf) = uomfu; = mj, VO" E G, I E C(G). Conversely, suppose that x E A oo and tio(x) = z, VO" E G. For c > 0, pick y E ®~=lB(Hk) ® ®;>n1; such that lIy - xII < c. Define

z = (m(I)··· m(n))-I

L

Qo(Y).

oEG1x···xG,..

Then lio(z) = z, VO" E G, z E ®~=IB(Hk), and Ilx - zll < c. Since zu; = UoZ, VO" E G I x··· X G n , it follows that z = mf for some I E Cn C C(G). c(> 0) is arbitrary. Hence, x E A.

672

(iv) Let 05

---4

A8 be the left regular represectation of G on 12(G), and define

a 8(x) = A8XA:, Vx E B(12(G)). Then (A, G, a) is a C·-system, and

a 8(m/) = m8f, where (81)(t) = I(t - 05), Vs E G, tEa, I E C( a). We say that the (UHF) algebra A oo is * isomorphic to the crossed product A xQG. Define

(11" (m/) €)(s, t)

=

(a 8-1(m/)€(s))(t)

= I(s

+ t)€(s, t),

(A(r)e)(S,t) = €(s - r,t),

VI

C(a),e E 12(G,H) = 12(G X G),r,05,t E G. Since G is abelian, {11" x A, 1 ( G, H)} is a faithful * representation of Ax G. In particular E

2

Q

((11"

X

A)(Z)e)(S,t) =

(2: arl(z(r, ·))(A(r)€)(s))(t) rEG

= 2:z(r,s+t)€(s-r,t), rEG

Vz E K(G, C(a)), e E 12(G as follows:

X

G). Define a unitary operator W on P(G x G)

(W€)(s,t) = €(s + t,t), { (W· e)(s, t) = €(s - t, t),

V€ E 12(G x G) . Then we have

W·(1I"

X

A)(Z)W =

(2: mgrA

r) @

1,

rEG

where gr(·) = z(r,.) E C(a), Vz E K(G, C(a)). Hence, A X Gis * isomorphic to the C·-algebra on 12 (G) generated by {m" Ar II E C (a), rEG}. On the other hand, @k=lB(Hk ) is generated by {A,, mill E Cn, r E G 1 X ••• x G n} obviously, Vn. Since C(G) = UnCn and G = EBnGn, A oo and A X G are * isomorphic . Q

Q

4) Semi-direct product.

(i) Let H, K be two groups, and p be a homomorphism from K to Aut(H), where Allt(H) is the automorphism group of H. Define

(h,k)(h',k') = (hp(k)(h'),kk'),

673

Vh, h' E H, k, k' E K. By this multiplication, H x K becomes a group. This group is called the semi-direct product of (H, K, p) , and denoted by G = H «,«. Now if H, K are two locally compact groups, and the map (k, h) ---i> p(k)h is continuous from K x H to H, then we have the following facts: (1) G = H x p K is also locally compact with respect to product topology; (2) If dh is the left invariant Haar measure on H, then for any k E K there is a positive constant 8(k) such that dp(k)(h) = 8(k)dh. Moreover, 8(·) is continuous on K,o(eK) = 1 and o(k 1k2 ) = 8(k1)o(k2 ) , Vk1,k2 E K; (3) IT dk is the left invariant Haar measure on K, then d(h, k) = 8(k)-ldhdk is the left invariant Haar measure on G = H x p K; (4) IT 6. H , 6. K , 6. a are the modular functions on H, K, G respectively, then we have 6G(h, k) = 8(k)-1 6. H (h) 6. K (k), Vh E H, k E K. Moreover , 6. H is p-invariant, i.e., 6. H (p(k)( h)) = 6H(h),

Vh E H, k E K.

The proof of these facts can be found in [70]. (ii) Let G = H x p K. If define (akf)(h) = 8(k)-1 f(p(k)-l(h)),

Vk E K,h E H,f E L1(H), then we can obtain a C"'-system (C*(H),K,a). Moreover, let C)(f)(k,·)

= 8(k)-lf(·,k),

Vf

E L 1(G),k E K,' E H.

Then C) is a * isomorphism from L1(G) onto L1(K,L1(H),a). The proof is easy. (iii) Let G = H x p K. Then we have C*(G) ,... C*(H)

X~

K.

In fact, by L1(G) ~ L1(K, L1(H), a) c L1(K, C*(H), a) C C"'(H) X~ K, it suffices to show that every * representation of L1(K, L1(H), a) can be extended to a * representaton of L1(K, C*(H), a). Put B = L1(H),A = C·(H). Let {g,} be a bounded approximate identity for L1(G). Then {It = C)(g,)} is a bounded approximate identity for L1(K,B,a). Since Ll(K,B,a) is dense in L1(K,A,a) ,{It} is also a bounded approximate identity for L1(K, A, a). Now let {p, H} be a nondegenerate representation of L1(K, B, a), and define

674

(notice that f,(·) E B,V· E K),Vb E B,k E K. Then {1r,u,H} is a covariant representastion of (B, K, a), and p = 1r X u. Clearly, {1r, H} can be uniquely extended to a * representation of A, and {1r, U, H} is also a covariant representation of (A, K, a). Hence {p = 1rX U, H} can be extended to a * representation of L 1(K, A, a).

Example.

Let G be a locally compact group, and define p : G

-4

Aut(G)

as follows:

p(s)(t) = sts-t, Then

(allfHt) = .6(S)f(S-ltS),

Vs, t E G.

Vf E L 1(G),s,t E G,

and (C·(G), G, a) is a C·-system. By above discussion, we have

C*(G)

x~ G"'"

C·(G x , G).

5) The periodic action and mapping torus. Let (A, ZZ, a) be a C·-system, and an = id, where n is a fixed positive integer, and a is a * automorphism of A. (i) Let A be the closure of 1~(ZZ, A, a) = {f E ,11(ZZ, A, a) If(k)

in A

X~

= 0, Vk t O(mod n)}

ZZ. Then .It is a C·-subalgebra of A x, ZZ, and

can be uniquely extended to a * isomorphism from sl onto A XidZZ ,.... C(T, A), where f(k) = f(kn), Vk E ZZ, and T is the group of unit circle, i.e., T = {z E

a:llzl = I}. Proof.

Since (fg)(k) =

L

f(m)am(g(k-m)),f·(k) = ak(f(-k)·),Vf,g E

mE7Z

11(ZZ,A,a) and k E ZZ,.It is a C·-subalgebra of zt x ; ZZ, and g homomorphism from 11(ZZ, A, ~'d) to 1~(ZZ, A, a), where v

g

(k) =

{O,g(m),

Then g -4g can be extended to a we have

-4g

is a

*

0

if k t (mod n), if k = mn.

* homomorphism from

A x id ZZ to .It, and

675

Now suppose that A C B(H) for some Hilbert space H. By Theorem 16.4.15, there is a faithful * representation {p,12(~,H)} of Ax a ~ (noticing that ~ is amenable):

(p(f)E)(k)

L

=

a-l:(f(i))E(k - j),

iE~

Vf E 11(~,A,a), EE

12(~,H),

and k E ~. Let

12(~, H) = l~(~, H) EB l~ (~, H)

be an orthogonal decomposition of 12(~, H), where 1~(~,H) = {E E z2(~,H)IE{k) = O,Vk

t O(mod n)}

and l~(.~, H) =

{E E 12(ZZ, H) IE(kn) = 0, Vk E ZZ}.

Clearly, if g E 11(ZZ, A, id)' then l~(~, H) and l~ (~, H) are invariant for p(g). Hence, we get

= IIp(g)ll > IIp(y) 11~(ZZ, H) II· by Theorem 16.4.15, there is a faithful * representation {a,z2{ZZ,H)}

II g I!Ax

aZ

Also A Xid ZZ :

(u(g)E)(k) =

L

of

gU)E(k - j),

iE~

vs E 11(ZZ,A,id), EE 12(~,H), and k E ZZ. Define a unitary operator U from 12(ZZ, H) onto l~ ( ZZ, H) as follows: (UE)(k) = { EU), if k = nj for some j , 0,

otherwise,

VE E 12{~,H). Clearly, we have

Ua{g)U·

=

p(g)ll~{ZZ, H),

Vg E 11(~, A, idle

Thus, Further, we obtain that

and A is

* isomorphic to A

X id

ZZ.

Q.E.D.

(ii) IT define ak{a + A) = al:(a) + A, Vk E ZZ, a E A, A E (C, then we have C·system (A-+a:, ZZ, a). Since ll(~, A, a) is a * two-sided ideal of 11(ZZ, A-+a:, a),

676

and 7Z is amenable, it follows from Proposition 16.4.16 that A x O! ZZ is a closed two-sided ideal of (A-t€) XO! 7Z. Put A E 11(7Z, A-t
# 0, and A(1)

Then A is an invertible element of (A-t€)

Ai (k) = 0i,le,

XO!

= 1.

ZZ, and

(/Ai)(k) = I(k - i),

vi. k E 7Z, I E 11(7Z, A, a). = 0, Vk t

Hence AAi is the closure of {I E 11(ZZ, A, a) I/(k) j(mod n)} in A XO! ZZ, vi E ZZ. We claim that

+ +... +AA

A x O! 7Z = A AA In fact, for any

I

n-l •

E 11 (ZZ, A, a) and j E {O,'" ,n - I}, let if k - ~(mod n), otherwise.

{(k) = {/(k), 0,

1

Then t, E AAi , and I =

10 + ... + In-I'

Hence

11(ZZ, A, a) c A + AA +

... + AA n -

l

.

Notice the following fact. HIE l~(ZZ, A, a), g E 11(7Z, A, a) and g(kn) = 0, Vk E 7Z, then

III + gllAx

> 1I/IIAX 2Z' Indeed, along the notations: p, 12(7Z, H) = l~(ZZ, H) have p(g)I~(7Z, H) c l~ (7Z, H). Hence, o

2Z

o

ffi l~ (ZZ, H) in (i) , we

IIp(1 + g)€11 2 = IIp(I)€11 2 + IIp(g)€11 2 > IIp(l)eIl 2 ,

"Ie E l~(7Z,H). By the proof of (i) we obtain IIp(1 + g)11 > IIp(I)II~(7Z,H)11 = 1I/IIAX A,O < i < n - 1, be such that

III + giIAX Now let G.i E

a 7JJ

=

a

7JJ '

ao + alA + ... + an_lA n- 1 = 0. For each j E {O"",n -I}, pick a sequence {/~i)} of 1~(7Z,A,a) such that I!i) --+ ai in A XO! 7Z. Then n-l

L

I~i) Ai

--+ 0

as k

--+ 00.

;=0

By the fact of the preceding paragraph, we have I~O) --+ 0 in A x O! ZZ. Hence, ao = O. Similarly, from (al + a2A + ... + an_lAn~2)A = 0 we have al = O. Generally, ai = 0, "10 < j < n - 1. So A + AA + ... + AA n - 1 = A+AA+ ... +AA n - l .

677

Finally, for any a E A I~U~, A, a) such that n-l

L

Xl)!

ZZ, pick sequences {lii)lk}(O

11i ) >..i

a as

----t

k

n - 1) of

----t 00.

i=O n-l

Then

L (11;) -

I,U»)>,,;

----t

0 as k, I

----t 00.

From the preceding paragraph, it

;=0

must be

(/Ie(;) - Jlf(i») So for each a

i

----t

k I ~

as

0

"

00

0

< J. < n -

1.

A such that Il;) ~ ai' Therefore, ZZ = A+A>..+ ... +A>..n-l,

E {O, ... ,n - I} there is ai E

= aD + al>" + .. ,+ an_l>..n-l,

and A

Xl)!

(iii) By (i) and Proposition 16.3.2, the Fourier transformation:

I

E 1~(7Z, A,

a) ~ F(z) =

L

I(nk)zk

E C(T, A)

leEZ

can be uniquely extended to a * isomorphism from A onto C(T, A). Denote this * isomorphism by ~. Now on C(T, A) X ••• X C(T, A)(n times) define multiplication, * operation and norm as follows: (~i)O$i$n-1

. (C:;)O$;$n-l

;

(L ~Ic k=o

n-l •

alcC:i_ 1e +

L

~k·

Ie=i+l

aleG n+i _ 1e • z)O$iSn-b

* (Fi )O$iSn-l -- (F.*0' a F*n-l· -Z, a 2F*n-l· -Z,' •• ,an-1F*I . -) Z ,

11(~i)O$i$n-lll =

n-l

II L

~-l(Fi)>..iIlAxa2Z,

i=O

where (~. akc:)(z) = ~(z)ak(c:(z)), (~. akG· z)(z) = z~(z)ak(c:(z)), (a kr. z)(z) = zak(~(z)*), 'IF, G E C(T, A), z E T, k E ZZ. Then C(T, A) X •• , X C(T, A) ( n times) is a C*-algebra, and through the following map W :

(Va; E A,O < i < n -1),A Xl)! 7Z and C(T,A) X ••• X C(T, A) (n times) are * isomorphic. In fact, by (i), (ii) w is a linear isomorphism from A Xl)! ZZ onto C(T, A) X •• , X C(T, A) (n times) obviously. Further, from the definition of norm W is

678

an isometry. For any fo,"', fn-l, 90,' • "

9n-1

E 1~(7Z, A, a) we have

and n-l

(L /;>J)* = f; + (af~_l' ..\.-n)..\.+.+ (an-If;' ..\.-n)..\.n-l, i=O

where (a1:f)(j) = a1:(f(i)),Vf E 11(LZ,A,a),j,k E LZ. If write Fi = ~(9k), then we obtain \If

(nJ~_-Ol f;A; . ~_-Ol

91:..\.1:)

~

=

(t

Ft· o!G;-t +

1:=0

E Ft· atG

= ~(/;),G1:

nH _ t .

z) O~i~n-l

1:=J+l

and n-l

\If((L h..\.i)*) = (F;, aF~_1 . z,"', a n - l F: . z). i=O

Therefore, \If is a ( n times) .

* isomorphism from

A X~ LZ onto C(T, A)

X ••• X

C(T, A)

(iv) Since an = id, we can also consider the C*-system (A,7Zn,a). ll(LZn, A, a) is a Banach * algebra. For any a = (a")O~,,~n-b b = (b")O~,,~n-b we have the following formulas:

n-l

ab

=

(L atat(b,,_t))O~,,~n_l' t=o

lI alll = llaoll + ... + Il an-lll, where the foot index of any integer is understood in the sense of (mod n). Let II ·11 be the largest C*-norm on ll(LZn,A, a). Then the crossed product A x~ LZn is the completion of (ll(LZn,A,a), 11·11). Assume that A c B( H) for some Hilbert space H. Then we have a faithful * representation {p, H n } ofAx~ LZn as follows: n-l

p(a)€ =

(L a-"(a,)e,,-t)O~,,~n-b t=o

679

'Va = (a.)O~.Sn-1 E 11(ztn, A, a), € = (E.)O~.~n-l E Hn , where H n = H E9 ••• E9 H (n times ). Consequently, n-l n-l = sup L II L a-·(a,)€._tI1 2 eEH t=o 8=0 n-I > sup L Ila- 6(a.)EoI1 2 eoEH,lIeoll:O:;I,,=o > Ila- 6(a,,)11 = Ila.lI, Vs E {O,"', n - I}, n,lIell:O:;1

.i.e., Va =

(a6)0~6:O:;n-l E ll(ztn, A, a). Hence, II (ztn, A, a) is C·-equivalent ( see

Definition 2.14.21), i.e., as linear spaces we have A

Moreover, A

Xl:\!

ztn

X l:\! ~n

= 11(~n, A, a)

= A X ••• X A(n

times).

admits a matrix representation as follows. Define

fJ : a = (a.)0~6~n-l

----+

(a-i(~_j))O:O:;i,j:o:;n-l'

*

(a.)o~.:O:;n-1 E A Xl:\! ~n' Then it is easy to see that fJ is a isomorphism from A X OI ztn into Mn(A) = A®Mn = {(aij)O~i,j:o:;n-llaij E A,Vi,j}. Further,

Va = let

U=

o

1

1

o

o

1

0

Then an element (~j)O:O:;iJ:o:;n-l of Mn(A) belongs to fJ(A X OI ~) if and only if U(aij)U· = (a(ai;))' In fact, first it is easy to check that U(a-i(~_i))U· =

(a(a-i(ai_;)))'

Conversely, if U(ai;) U· = (a( ~j)), then

Ul:( aij )U·l: Let bn -

j

= (al:( aij)) ,

Vk > o.

= ao;, 0 < j < n - 1, and bo = bn. We need to show that a-i(bi _ ; ) = Oti;

It is equivalent to prove that

Vi,j.

680

where b1c and aO,1c for any k E 7Z are understood in the sense of ( mod n). Notice that ai(asi) is the (i,j) element of ui (a1cl) tr-, Vi,j. Then by the form of U, we can obtain the conclusion. Now consider the dual sysem (A x a ZZn, ZZn, a) of (A, 7Zn, a). By Definition 16.5.7, we have a.... ( ( a.) O:5.:5n-l ) = (-2ri./n) e a. O:5.:5n-l, V(a.)O:5a:5n-l E A Xa ZZn. The mapping torus of a on A Xa Zn is defined as follows:

M....(A a

X

a

2Z )

=

n

{(R.(t)). ,

(v) A X a ZZ is In fact, define

t ---+ F;(t) is con!inuous from_ [0,1] } O:5,:5n-l to A,Vj, and (F;(I))j = ii(F;(O))i .

* isomorphic to M;;(A X a n-l

L

ZZn).

aj>/ ~ (e- 21fit;jnFi (e- 2rit ))0:5;:5n_1,

i=O

where Fj = Cf) (aj) ,0 < j < n - 1, and Cf) is the * isomorphism from A onto C(T, A) ( see (iii)), Vai E A,O < j < n - 1. Then it is a * isomorphism from A X a ZZ to M;;-(A X a ZZn) obviously. Conversely, if (Fj(t))O:5i:5n-l E M;;(A X a ZZn), then

Fj(l) = e-21fii/n F;(O),

0

1.

So the function e2ritjjnFi(t)(t E [0,1]) picks the same values at the points t = 0 and t = 1, Vj. So for any j E {O," . ,n - I} there is Fi E C(T, A) such that Fi (e -2rit) = e21fiti/n Fi (t), Vt E [0,1]. Let aj E A be such that ~(ai) = Fj, Vj. Then n-l

L

a;)..i ~ (e- 2ritj Fj(e-21fit))0:5i:5n_l

i=O Therefore, A

Xa

= (F;(t))O:5;~n-l'

7Z and M;;-(A x , ZZn) are

* isomorphic.

(vi) A x a ZZ also admits a matrix representation. In fact, define ·< 1 ~ (",-i ri-, D . . • z~(i-j»)o<' '< 1 , (F.,·)o<_,_n_',,_nLL

VF; E C(T, A), 0 < j < n - 1, where IC(.) is a function on ZZ as follows:

lC(k) = {I, ~f k < 0, 0, If k > 0,

Vk E 7Z.

681

Then by (iii) (J is a M n • Moreover, let

* isomorphism from A X a 7Z into Mn(C(T, A)) o U(z) =

1

=

C(T, A) ®

... z .

o

(n X n) 1

0

Vz E T. By the discussion of (iv) we have

(J(A -

Xa

7Z)

{(~i(z)) E

Mn(C(T, A)) IU(z)(Fij(z))U(z)+ = (aFij(z)) , Vz E T} .

Notes. The example 3 appeared in [175]. To understant it, M. Takesaki had to develop the duality theory. The proof about mapping torus (the example 5) presented here follows from B.R. Li and Q. Lin. References.

[11], [70], [127], [175].

Chapter 17

J ones Index Theory

17.1. The coupling constant Suppose that M is a VN algebra on a separable Hilbert space H. By Proposition 1.14.3, M. is separble and M is a-finite. Let (Mh be the closed unit ball of M. From the example under Definition 10.1.1, ((Mh, a(M, M.)), ((Mh, s(M,M.)) and ((Mh,s·(M,M.)) = ((Mh,,.-(M,M.)) are Polish spaces. If rp is any faithful normal state on M, then by Lemma 1.11.2 a proper metric d(., .) for the Polish space ((Mh,s(M,M.)) is d(a, b) = rp((a-b)*(a-b))1/2,Va,b E (Mh. Moreover, if {x n } is a countable dense subset of the Polish space ((Mh,a(M,M.)), then {x n} generates M (i.e., {xn,x~ln}" = M) obviously. Hence, M is countably generated.

Proposition 17.1.1. Let M be a W*-algebra. Then the following statements are equivalent: 1) M admits a faithful W*-representation on some separable Hilbert space; 2) M is a-finite and countably generated; 3) ((Mh,a(M,M.)) is a (compact) Polish space, where (Mh is the closed unit ball of M. From the above discussion, we have that 1) implies 3). Now let ((Mh, a(M, M*)) be a Polish space. Then M is countably generated obviously. Moreover, since ((Mh, a(M, M*)) is metrizable, M. is separable. Hence by Proposition 1.14.3 M is also a-finite. Finally, let M be a-finite and countably generated. Suppose that {x n } is a countable generated subset for M, and {s, H} is a faithful nondepenerate W·representation of M. Since M is a-finite, there is a sequence {en} of H such that PnPm = onmPn, Vn, m, and En Pn = 1H , where Pn is the projection from H onto 1r(M)' en, Vn. Clearly, Pn E 1r(M),Vn, and {€n} is a separating sequence Proof.

683

of vectors for 1r(M). Let p' be the projection from H onto [1r(M)tnln), and A be the * subalgebra over the field of complex rational numbers generated by {1r(x n),1r(xm)*ln,m}. Then A is countable and strongly dense in 1r(M), and rIH = [Atnln] is separable. If c(p') is the central cover of p' in 1r(M)' , then we have

c(p')H = [1r(M)'p' H] => [1r(M)'tnln] = H, and c(P') = lH. Hence, a - t arl is a * isomorphism from 1r(M) onto 1r(M)p'. Consequently, {p'1r, p'H} is also a faithful W·-representation of M, and p'H is separable.

Q.E.D.

In this chapter, every W· -algebra is assumed to admit a faithful W·representation on some separable Hilbert space, and every Hilbert space is separable. For simplification, we shall not repeat these assumptions. Now let M be a finite factor. Then there is unique faithful normal tradal state r on M. Using this r, we define an inner product on M:

(x, y) = ,,-(y. x),

Vx, Y E M.

Denote by L 2(M) the completion of (M, (,)), and by 11·112 the norm on L 2(M), i.e., IIxll2 = r(x*x)l/2, Vx E M. Since r is a trace, the 1I·112-topology is equivalent to ,,-(M,M*) in the closed unit ball (Mh, and ((M)., ,,-(M,M.)) is a Polish space. Thus it follows that L 2(M) is a separable Hilbert space. Clearly, IIxl12 = IIx*1I2' Vx E M. Hence, J : x - t x*{Vx E M) can be uniquely extended to a conjugate linear isometry on L 2(M), still denoted by J. Then we have (Ja,Jb) = (b,a),Va,b E L 2(M). For each x E M, let A(X)Y = xy, Vy E M. Clearly, IIA(X)ylh < !Ixll . lIy112' Vy E M. Thus, A(X) can be uniquely extended to a bounded linear operator on L 2(M), still denoted by A(X). Then we obtain a faithful * representation {A, L2 (M)} of M. We claim that this is also a W· -representation. It suffices to show that for any y, z E M, (A(·)y, z) is a continuous function on ((M)., u(M, M.)). But it is obvious from the normality of the trace ,,- . Hence, A(M) is a VN algebra on L2(M), and is * isomorphic to M. Consider the opposite algebra MOPP of M ( i.e., x 0 y yx, Vx, Y EM). It is also a finite factor, and {p, L 2(M)} is a faithful W*-representation of MOPP, where p(x)y = yx, Vx E MOPP,y EM.

=

Hence, p(MOPP) is a VN algebra on L 2(M), and is

Proposition 17.1.2.

With the above notations, we have

= p(M) = JA(X)J = p(x·), A(M)'

* isomorphis to MOPP.

JA(M)J,

p(M)'

Jp(x)J

= A(X*),

= A(M) = J p(M)J, Vx E M.

684

Clearly, A(X)p(y) = p(Y)A(X), JA(X)J V z, Y EM. Thus, we have

Proof.

A(M) c p(M)',

p(M) c A(M)';

JA(M)J

= p(x*), and

J p(x)J

= A(X*),

= p(M),Jp(M)J = A(M).

Let

B = {a ,

E

L 2(M) Ithere exists A(a) E B(L2(M)) such that } A(a)x= p(x)a,VxE M

and

B = {b E L 2(M) Ithere exists p(b) E B(L2(M)) such that } r p(b)x = A(x)b, Vx E M It is easily verified that A(a) E p(M)', Va E B,; and p(b) E A(M)', Vb E B r •

Conversely, if t E p(M)', then ta E B, and A(ta) = tA(a), Va E B ,. Consequently, t = A(tl), where 1 is the identity of M(c L 2(M)). Similarly, if s E A(M)', then sb E B; and p(sb) = sp(b), Vb E B r • Consequently, s = p(sl). Thus , we have

p(M)' = {A(a)la E B,l,

and

A(M)' = {p(b)lb E B r } .

Now it suffices to show that A(a)p(b) = p(b)A(a), Va E B" b E B«. In fact, let a E B" b E B r • Since A(a)* E p(M)' and p(b)* E A(M)', there is some c E B , and some dE B; such that A(a)* = A(c),p(b)* = p(d). Noticing that

(a, x) = (a, p(l)x) = (p(l)a, x) = (A(a) 1, x) = (1, A(C)X) = (1, p(x)c)

= (p(x*)I, c) = (x*, c) = (Jc,x), Vx E M, we get a = J c. Similarly, we have b = J d. Pick {x n }, {Yn} C M such that X n - - t a,Yn - - t b in L 2(M). Then x~ = JXn - - t Ja = c,Y: = JYn - - t b = d. Further,

(A(a)p(b)x, y) = (p(b)x, A(C)Y) -

(A(x)b, p(y)c) = lim(A(x)Yn, p(y)x~)

-

lim(p(x)x n, A(Y)Y:) = (p(x)a, A(y)d)

-

(A(a)x, p(d)y) = (p(b)A(a)x, y),

Therefore, we have A(a)p(b) = p(b)A(a).

Vx, Y E M.

Q.E.D.

685

Remark. We also have B , = B,. = M. In fact, from the above proof it follows that )'(B,) = p(M)' = )'(M). Now if ),(a) = ),(X) , where a E B" x E M, then x = )'(x)l = ),(a)1 = p(1)a = a. Hence, B , = M. Similarly, B; = M. Moreover, )'(M) and ),(M)' = p(M) are finite factor on the separable Hilbert space L2(M), and they admit a cyc1ic-separating vector l(E M c

L 2(M)). Let M be a finite factor, and {1r,H} be a faithful nondegenerate W*-representation of M. Then there is a projection p' of ()'(M) ® lK}' , and a unitary operator u from H onto p'(L 2{M) ® K) such that u1r{x) = (),{x) ® 1K)u, Vx E M,

Proposition 17.1.3.

i.e., the VN algebras 1r(M) ( on H) and ()'(M) ® 1K)p' are spatially * isomorphic, where K is a countably infinite dimensional Hilbert space. Consequently, u1r(M)u* = p'(),(M) ® lK) and u1r(M)'u· = p'(),(M) ® lK)'p'.

Proof.

~:

),(x)

~

1r(x)(Vx E M) is a * isomorphism from )'(M) onto

1r(M). Now the conclusion follows immediately from Theorem 1.12.4 and the separability of H.

Q.E.D.

Let M be a finite factor, and {1r, H} be a faithful nondegenerate W*-representation of M. Generally, 1r(M)' is not finite. Of course, 1r{ M)' is semi-finite. Then there exist faithful semi-finite normal traces on 1r(M)~ ( uniquely determined up to multiplication by a positive constant, see Proposition 7.1.2 ) . Now we define a natural trace Trk on 1r(M)~ as follows. (i) IT {1r,H} = {), l8l1 K,L2(M) ® K}, where K is a countably infinite dimensional Hilbert space, then ()'(M) ® lx)' = J)'(M}J®B(K} is infinite. Pick an orhtogonal normalized basis {ei} of K. Then for each t' E (),(M)®1 K ) ' we can uniquely write t' = (J),(Xi;}J), where Xi; E M, Vi,j. Define the natrual trace as follows:

Definition 17.1.4.

Tr~2(M)@K(t') =

L

r(xii),

Vt' = (J),(Xij)J) E ()'(M) ® lK)~'

i

where r is the unique faithful normal tracial state on M. Clearly, Tr~2(M)@K is a faithful semi-finite normal trace on (),{M) l8llK )~. If {Ii} is another orthogonal normalized basis of K, and t' = (J)' (Xij) J) is the matrix representation of t' in the decomposition L 2(M) ® K = ffi[L 2(M} l8l ei], then the matrix

L i

representation of t' in the decomposition L2(M) l8l K =

(tij)(J),(Xi;)J)(t i j)*, where tit

= (e;, Ii}' Vi,j.

L

EB[L2(M)

e Ii] is

i

Since (ti ; ) is unitary, the definition of Tr~2{M)~K is independent of the choice {ei}'

686

(ii) For general faithful nondegenerate W*-representation {a, H} of M, by Proposition 17.1.3 there is a projection p' of (A(M) ~ 1K ) ' and a unitary operator u from H onto p' (L 2 (M) ~ K) such that u1r(x)u* = (A(X)

~

1K)p',

Vx EM,

and

u1r(M)'u* = p'(A(M)

~

1K)'P'.

Then we define the natural trace as follows: Tr~(t') = Tr~2(M)~K(ut'U*),

Vt' E 1r(M)~.

Here we must prove that this definition is independent of the choice of u and p' . In fact, if Ul and U2 are two unitary operators from H onto pi (L2(M) ~K) and p~(L2(M) ~K) respectively such that Ui1r(X)U; = (A(X) ~ lK)p~,i = 1,2, then (A(X) ~ 1K)U1U; = Ul1r(X)U; = UIU;p~(A(X) ~ lK) = UIU;(A(X) ~ 1K), Vx E M. Thus UIU; E (A(M)~1K)'. Similarly, U2U~ and ult'u; = ult'ui'U1U; E (A(M)~ 1K)', Vt' E 1r(M)~. If t' E 1r(M)~ is such that Tr~2(M)~K(ult'un < 00, then Ult'U~ E M, where M is the definition ideal of Tr~2(M)~K' By Proposition 6.5.2, we have Ult'u; = ult'ui . U1U; E M, and Tr~2(M)~K(ult'uD

= Tr~2(M)~K( ult'u; . u2 u D

= Tr~2(M)~K( u2u i . U1 t'U;) = Tr~2(M)~K( U2 t'U;). Similarly, ift' E 1r(M)~ is such that Tr~2(M)~K(U2t'U;) < u;) = Trt2(M)~K(U1t'ui). Therefore, we have Tr~2(M)~K(U1t'U;)= Tr~2(M)~K(U2t'U;),

00,

then Tr~2(M)~K(U2t'

Vt' E 1r(M)~.

Moreover, Trk is a faithful normal trace on 1r(M)~ obviously. Now if t'(:;f 0) E 1r(M)~, then by the semi-finiteness of Trt2(M)~K there is 0 :;f a' E (A(M) ~ lK)~ with a' < ut'u* such that Tr~2(M}~K(a') < 00. Clearly a' E p'(A(M}~1K}'p' = u1r(M),u*. So s' = u*a'u E 1r(M}~ and s' < u*(ut'u*}u = t', and Trk(s'} = Tr~2(M}@K(a') < 00. Therefore, Trk is also semi-finite on 1r(M)~.

Definition 17.1.5. Let M be a finite factor, and {1r, H} be a faithful nondegenerate W*-representation of M. Then dimM(H) = Trk(1) is called the coupling constant between 1r(M) and 1r(M)', where Trk('} is the natural trace on 1r(M)~. Since dimM(H) = Tr~2(M)~K(UU"') and H ~ uu*(L 2(M) ~ K), dimM(H) is also the " dimension " of H as aM-module ( see Section 7.1 ).

Proposition 17.1.6. Let M be a finite factor.

687

(i) If {1rb HI} and {1r2, H2} are two faithful nondegenerate W*-representations of M, then

(ii) If {1ri,Hi},i = 1,2"", is a sequence of faithful nondegenerate W*representations of M, then

L dimM(H

dimM(L ffiHi ) =

i);

i

i

(iii) The natural trace Tr~2(M) is exactly the restriction of the unique faithful normal tracial state 1"' of A(M)' = p(M) on A(M)~, and dimM(L 2 (M » = 1; (iv) If {7r, H} is a faithful nondegenerate W*-representation of M, then

1r(M)' is finite

~

Trk is finite on 7r(M)~

<

~ dimM(H)

00.

( In this can, Tr'n can be uniquely extended onto 7r (M)', but it is not necessarily normalized ) .

Proof. (i) First, we notice the following fact. Suppose that N is au-finite and semi-finite factor. If P is an infinite projection of N and Po is a non-zero finite projection of N, then there is an orthogonal sequence {Pn I n > I} of projections of N such that 00

P '" LPn,

Pn '" Po, \In

and

> 1.

n=l

In particular, all infinite projections of N are equivalent (relative to N). In fact, by the Zorn lemma and the u-finiteness of N, there is a maximal orthogonal sequence {Pn I n > I} of projections of N such that Pn

< P,

Pn '" Po,

\In

> 1.

00

Let q

=

P-

L Pn·

Since N is a factor and {Pn} is maximal, it follows that

n=l

q

-< Po· Then it is easily verified that

00

00

n=l

n=l

L: Pn '" LPn + q = p.

Now let Ui be the unitary operator from Hi onto P'i (L 2 (M) ® K) such that dimM(Hi ) = Tr~2(M)~K(UiU;),p~ = UiU; E (A(M) ® l K )' , i = 1,2. Then by the above fact we can see that: dimM(H1) =dimM(H2) if and only if r/l = UIU; '" r/2 = u2ui (relative to (A(M) ® Ix)'). Since {1ri, Hi} '" {(A ® lK )p~, 1t.(L2(M) e K)}, i = 1,2, it follows that dimM(H1 ) = dimM(H2 ) ~

{ 1r

I, H 1 }

::' { 7r 2, H 2 } .

688

(ii) Let

1r

=

L

EB 1ri ' H =

i

L

ffiHi . Clearly, {s, H} is still a faithful nonde-

i

generate W*-representation of M. Let p'(L 2(M) 18l K) such that

U

be a unitary operator from H onto

u1r(x) = (A(X) l8l1K)u,

Vx EM,

and p' = UU· E (A(M) ® l K )' . Then dimM(H) = T r k (l ) = Tr~2(M)~K(P')' Fix i, and let Ui = uqi, p~ = UiU:, where qi is the projection from H onto Hi. Clearly, qi E 1r(M)', and U:Ui = qi. We claim that

Ui1ri(X) = (A(X) 1811 K )ui'

Vx EM,

In fact, Ui1ri(X) = U1r(X)qi = (A(X) ® l K)uqi over, since p~ = UiU: = uqiU* and

and

p~ E

= (A(X)

(A(M) ® l K )' .

® lK)ui' Vx E M. More-

pHA(X) ® l K )p~ = Uqi' U*(A(X) 1811 K )u . qi U* uqi1r(X)qiU* = U1r(x) . qiU* (A(X) 1811 K ) . uqiU· = (A(X) ®

lK)p~,

Vx EM, it follows that p~ E (A(M) ® lK)'.

Now we have that dimM(Hi ) = Tr~2(M)I8lK(pD, Vi. Clearly, p~pi = Oij pi, Vi, j and p~ = p'. Then by the complete additivity

L i

of Tr~2(M)I8lK we have LdimM(Hi ) a

= LTr~2(M)I8lK(pD a

Tr~2(M)I8lK(P')

= dimM{L ffiHi ) . i

e,

e

(iii) Let u(a) = a ® Va E L 2(M), where E K and Ilell = 1. Then U is unitary from L 2(M) onto L 2(M) 18l [e]. If p' is the projection from L 2(M) 18l K onto L 2 (M )®[e), then p' E (A(M)®I K)' and UA(X)U· = (A(x)®I K)p', Vx E M. Let {ei} be an orthogonal normalized basis of K with el = Then p' has a matrix representation (JA(Xij)J) such that Xu = 1, Xij = 0, V(i,j) i- (1,1) Thus, we have

e.

dimM(L

2

(M )) = Tr~2(M)(I) = Tr~2(M)I8lK(P') = 1,

and Tr~2(M) can be uniquely extended to the faithful normal tracial state r' on A(M)'. (iv) By Proposition 7.1.2, we can see that

1r(M)'

is finite

-<===> p' = uu· is finite in (A(M) ® l K )' -<===> Trb(M)I8lK(P') <

00

-<===> dimM(H) = Trk(l) <

00.

Q.E.D.

689

Proposition 17.1.7.

Let M be a semi-finite factor on a Hilbert space H. Then there exist faithful normal semi-finite traces p, p' on M+) M~ respectively such that p(ee) = p'(ee), va =1= E H,

e

where ee, ee are the cyclic projections from H onto M' €, M € respectively. In particular, if a, a' are any faithful normal semi-finite traces on M+, M~ respectively, then there exists a positive constant c such that u( eel = co' (ee)' va =1= € E H. Consequently, if M, M' are finite, and "-,,,-' are the unique faithful normal tradal states on M, M' respectively, then the number CM

= ,,- (ee) /,,-' (ee)

is independent of the choice of €(I- 0). Let cp be a faithful normal state on M) and {1rlp' H rp, €lp} be the faithful cyclic W*-representation of M generated by cpo Clearly, trp is also separating for N = 7rlp(M). Since ((Mh,s(M,M.)) is a Polish space and a proper metric on ((M)"s(M,M.)) is d(a,b) = cp((a - b)·· (a - b))l/2(Va,b E (Mh), the Hilbert space Hrp is separable. Now by Theorem 1.12.4, M is spatially * isomorphic to (N ® l K ) p', where K is a countably infinite dimensional Hilbert space, p' is a projection of (N l8l lK)'. Fix a vector Tlo E K with IITlo II = 1, and let p be the projection from tt; ® K e K onto Hlp e K l8l [Tlo]. Clearly, p E Nl8l1 K®B(K), and Nl8l1 K is spatally * isomorphic to p(Nl8l1 K®B(K))p. Further, M is spatially * isomorphic to p(N ® l K ®B (K ))p . (p' ® l K ) . Since €rp is cyclic-separating for N, by Theorem 8.2.7 there is a conjugate linear isometry j on Hlp with j2 = 1 such that j N j = N'. We may assume that K = 12 • Define

Proof.

Then t can be uniquely extended to a conjugate linear isometry, still denoted by t, on K ® K with t 2 = 1. Clearly, t(lK l8l B(K))t = B(K) l8l l K = (I K ® B(K))'. Let J=j®t, and T=Nl8l1Kl8lB(K). Then J is a conjugate linear isometry on (Hlp ® K ® K) with J2 = 1, T and T' = N'®B(K) l8l lK are semi-finite factors on (Hlp ® K l8l K), and JT J = T'. Let cP be a faithful normal semi-finite trace on T+, and define

cp'(t') = cp(Jt' J),

'It' E

T~.

Then c)' is a faithful normal semi-finite trace on T~. For any a t= ~ E (Hlp l8l K e K), denote by Pt , P; the cyclic projections from ( Hlp ® K ® K) onto

690 T'~,T~

respectively. It is easy to see that JP;J = PJ(O Since T is a factor, it follows that either PJ( -< P, or p( -< PJ(O Suppose that PJ( -< Pt. By Theorem 1.13.2 we have p~( -< P;. Then P, = J p~(J -< J P;J = PJ(. So we obtain that P, ,..., PJ ( and

cp'(P;)

cp(JP;J)

=

=

cp(PJd = cp(p(),

i- ~ E ( HII' ~ K ~ K). Since T is a factor, the central cover of p in T must be 1 . Hence O( t') = .t' p(Vt' E T') is a * isomorphism from T' onto T' p. Let "10

CPt

=

~1(pTp)+,

cP~

= cP' 0

r-,

Then CPt, cP~ are faithful normal semi-finite traces on (pTp)+, (T'p) + respectively. For any 0 i- ~ E p(Hrp(iS)K~K) = (Hrp~K~[11o)), denote by Q(, Q~ the cyclic projections from (HII' ~ K ~ [110)) onto T'p~, pTp~ respectively. Clearly, Q( = p(, Q~ = pP; Thus , from the preceding paragraph we have 0

cp~(Q~) =

"10

i-

~ E

p(H rp ~ K

cp'(P;) = cp(p()

=

cpdQ!),

~ K).

Similarly, there are faithful normal semi-finite traces CP2, cP~ on (pTp . p' lK )+, (p' ~ 1K . T'p . p' ~ lK)+ respectively such that cp~(R~) =

CP2(R(),

"10

i-

~ E

(p' ® lK)p(Hrp

(is)

K

(is)

(is)

K),

where R(, R: are the cyclic projections from (p' (is) lK)p(Hrp ~ K ~ K) onto (P' ® 1K)T'~,pT~ respectively. Therefore, we can find faithful normal semi-finite traces p, p' on M+, M~ respectively such that

Finally, since the faithful normal semi-finite traces on M+ and M~ are uniquely determined up to multiplication by a positive constant , the rest conclusions are obvious. Q.E.D.

Proposition 17.1.8. Let M be a finite factor on a Hilbert space H,M' be also finite, and r, r' be the unique faithful normal tradal states on M, M' respectively. Then we have that: (i) dimMpl(p' H) = r'(p')dimM(H), where p' is any non-zero projection of M'· (ii) Let 0 i- € E H, and ee, ee be the cyclic projections from H onto M'€, Me respectively. Then dimM(H) = CM = r(ee)/r'(e~) l

and

CM

is independent of the choice of €(i- 0);

691

(iii) dimM(H)dimM,(H) = 1; (iv) dim"Mp(pH) = r(p)-ldimM(H), where p is any non-zero projection of

M', (v) H L is a finite dimensional Hilbert space, then dimM( H ® L) = dimL .

dimM(H). Proof. (i) Let u be a unitary operator from H onto uu*(L 2 (M ) ® K) such that ux = (A(X) ® 1K )u, 'Ix E M. Then for any non-zero projection p' of M' we have

dimMp,(p'H) =

Tr~2(M)~K(UP"

(up')·)

= Trk(p')· Since M' is finite, it follows that Trk(') = r'(·)Trk(1). Therefore, we obtain that dimMpi (p' H) = r' (p') Trk (1) = r' (p') dimM ( H) .

(ii) By Proposition 17.1.7, there is a positive constant c-

where

I,."

Tr~2{M)~K(/~) = tr(I,.,) ,

e such that

va =1= TJ E L 2(M) ® K,

I~ are the cyclic projections from

L2(M) ® K onto (A(M) e 1K )'TJ ,

(A (M) ® 1K ) TJ respectively, tr is the unique faithful normal tracial state on A(M) ® l K i.e., tr(A(x) ® 1K) = r(x), 'Ix E M. Now we compute the constant e. Pick" = I ® k with a i= k E K. Then , 1,.,(L2(M) €I k) = (A(M) ® 1K)'(I ® k) = (p(M) ® B(K))(1 ® k) = L2(M) e K, i.e., I,., = 1; and 1~(L2(M) e K) = ().(M) ® l K)(1 ® k) = L 2(M} ® [k]. Thus, we have tr(I,.,) = 1 = Tr~2{M)~K(/~), and c = 1. Let u be a unitary operator from H onto p'(L 2 (M ) ® K), where p' = uu* E (A(M) €I Ix)' such that ux = ().(x) ® lx)u, 'Ix E M. For any a =1= E H, it is

e

easy to see that

*='pfue,

Ufe U

' · = Jue' 1" andUteu

Since lue E A(M) ® l K and ute = lueu, it follows that lue the preceding paragraph, we have

= A(te} €I 1K.

Moreover, Tr~2(M)~K(f~e) = Tr~2(M)~K(UfeU*) = Trk(t

e)=

r'(ee)T riI(1) = r'(ee)dimM(H).

Therefore, we obtain that dimM(H) = eM. (iii) It is immediate from (ii) .

From

692

(iv) From (iii) and (i) we have diIIlpMp(pH) = [dimMlp{pH)]-l

=

-#

(v) Pick 0

[T(p)dimMI(H)]-l

= T{p)-ldimM(H).

eE Hand 0 -# 1 E L. Clearly, (M ® IL)(e ® l)

=

(ee ® PI)(H ® L),

and

(M ® IL)'(e ® 1) = (ee ® IL)(H ® L), where PI is the projection from L onto [l]. Since M ® M'®B(L) are finite, it follows from (ii) that

lL

and (M ® I L ) ' =

_ trM01L{ee ® I L ) dimM(H ® L) - tr~/~B(L)(e~ ® PI)

T( eel di (H) = (dimL)-IT'(ee) = d'mL 1 • ImM .

Q.E.D. Proposition IT.l.9. Let M be a finite factor on a Hilbert space H, and M' be finite too. Then dimM(H)

= CM > I -<===> M admits a separating vector,

and dimM(H)

Consequently, CM

= CM < 1 -<===> M admits a cyclic vector.

= I -<===> M admits a cyclic-separating vector.

The sufficiency is obvious. We claim that either M or M' admits a cyclic vector. In fact, since H is separable, by the Zorn lemma we can find a maximal sequence {ei} of non-zero elements of H such that eiej = ci;ei, and e~ej = ciie~, Vi,i, where ei,e~ are the cyclic projections from H onto M'Ei,Mei respectively, Vi. Put I = 1- ~ei'f' = 1- ~e~. If II' -# 0, pick 0 -# e E II'H, then we have Proof.

i

eeei = 0 and

eee~ =

i

0, Vi. It is impossible since the family {ei} is maximal. Thus, If' = O. Suppose that f' -# O. Since M' is a factor, x ---+ x]' is a * isomorphism from M onto M I'. Hence, I = 0, i.e., ~ e, = 1. We may assume that II ei I < 2- i,Vi, and let

i

e = ~ ei' Then M' e => -=-M-='---:-e~""""e = M' i

Hence

ei =

eiH, Vi.

eis a cyclic vector for M' . Similarly, if I -# 0, then eis cyclic for M.

693

Q.E.D.

Now the necessity is also obvious.

Remark. Let M be a type (In) factor on a m-dimensional Hilbert space H, where n, m < 00. Clearly, njm, and m = np. Then we can write H = H n 18l Hp and M = B(Hn) 1811p, where H m Hp are n-dimensional, p-dimensional Hilbert spaces respectively. Pick 0 =f:. E E Hn,O =f:. '7 E tt; Then (B(Hn) 1811p)( E18l'7) = H nl8l'7, (B(H n) 1811p)'( El8l'7) = (l nl8l B (Hp ) )( El8l'7)) = El8lHp • Thus e~@'7 = p~18l1p and ee@'7 = In 18l P'7' where P~,P'7 are the projections from Hn,Hp onto [E], ['7] respectively. Therefore, we have

where

T, T'

are the canonical tradal states on B(Hn), B(Hp ) respectively.

Notes. Except for the presentation, all the material of this section comes from the original papers by F.J. Murray and J.Von Neumann.

References. [28], [60J, [75], [111], [112J, [113J.

17.2. Index for subfactors Definition 17.2.1. Let M be a finite factor. N is called a subfactor of M, if N is a W·-subalgebra of M with the common identity, and is a factor ( so N must be also finite) . The index of N in M, denoted by [M : N], is dimN(L 2 (M )).

Lemma 17.2.2. Let M be a finite factor on a Hilbert space H, and M' be finite too. Then there is a finite subset {El"", En} of H such that n

H =

E EB[MEiJ, and there exists a common positive integer m

such that for

i=1

any a(M, M.) -continuous positive linear functional lp on M, we can find m

'71,"', '7m E Hand lp(x)

=

E (X11;, 11;), \:Ix

E M. Consequently, the weak

i=1

topology and u(M,M.) -topology in M are equivalent. Pick a I-dimensional Hilbert space L(l < 00) such that dimM,(H 18l L) > 1. Then by Proposition 17.1.9 M' 18l 1L adimits a separating vector ~ = (~I," " ~l) E H 18l L. Thus , ~ is cyclic for (M' 1811 L)' = M 18l B(L). Now it is easy to see that {~l,"" ~l} is a cyclic subset for M. Pick €I = ~1, and let ~: = (l-p~)~i' 2 < i < I, where p~ is the projection from H onto [M€I]. Clearly, Proof.

694

= [M~:l2 1. Thus M ® l K admits a separating vector in H e K. By Proposition 1.13.6, for any tp E (M*)+ there exists 11 = (11I,' . " 11rn,) E H ® K such that rn,

tp(x) = ((x® lK)11,11) = 2:(X11i,11i),'ixE M. ;=1

Q.E.D. Corollary 17.2.3. Let M, be a finite factor on a Hilbert space Hi, and MI be finite too, i = 1, 2. If. is a * isomorphism from M 1 onto M 2 , then we can write • = .3 0 .2 0 .1,

where .1{X) = x ® lK, 'ix E MI, and K is a finite dimensional Hilbert space; .2(') = 'p','i. E M 1 ® l K, and p' is a non-zero projection of (M1 ® lK)'; and is a spatial * isomorphism from (M1 ® l K )p' onto M 2 •

.3

Proof.

It is immediate from Lemma 17.2.2 and the proof of Theorem 1.12.4.

Q.E.D. Proposition 17.2.4. Let M be a finite factor, N be a subfactor of M, and {1T, H} be a faithful nondegenerate W*-representation of M. If dimM(H) < 00 ( i.e., 1T(M)' is finite) , then we have that

[M: N] = dimN(H)/dimM(H). In particular, [M : N] < 00 if and only if for some (then for any) faithful nondegenerate W*-representation {1T, H} of M with dimM{H) < 00 we have dimN{H) < 00. Proof. Let {1Tl, HI}, {1T2' H 2} be two faithful nondegenerate W*-representations of M, and dimM(Hi ) < oo,i = 1,2. By Corollary 17.2.3, there is a finite dimensional Hilbert space K and a non-zero projection p' of (1T2(M) ® l K)' such that {1Tl , HI} 1T 2 ® 1K)p' , p' ('H2 ® K)}. I"oJ

{(

Thus, we have dimN(Htl = dimN(p'(H2 ® K)). If dimN{H2 ) < 00, then by Proposition 17.1.8 we have dimN(H2 ® K) < 00, and dimN(Ht} < dimN(H2 e K) < 00. Hence,

695

Consequently, [M : N] = dim N(L 2(M)) < 00 ~ dimN(H) < 00, where {11'", H} is some (then any) faithful nondegenerate W·-representation of M and dimM{H) < 00. Now let dimM(H) < 00. By Corollary 17.2.3, there is a finite dimensional Hilbert space K and a non-zero projection p' of (A(M) ® l K )' such that

{11'", H}

~ {(A ® l K)p',p'(L 2(M) ® K)}.

By Proposition 17.1.8 we have

dimN(H} = dimN(p'(L 2(M) ® K)) = r'(p')dimN(L 2(M) ® K) = r' (p') dimK dimje (L 2 (M)),

where r' is the unique faithful normal tracial state on (A(M) ® l K ) ' = A(M)'® B(K); and

dimM(H) = dimM(p'(L 2(M) ® K)) = r'(p')dimK. Therefore, [M: N] = dimN(L2(M)) = dimN(H)/dimM(H).

Q.E.D.

Remark. IT N = B(Hn ) ® I p , M = B(Hn ) ® B(Hp ) = B(Hm ), where m = np, then dimL 2(M) = m 2. By the end of Section 17.1 we have

[M: N] = dimN(L 2(M)) =

m2 -2

n

= p2.

Lemma 1"1.2.5. Let M be a finite factor, P be a subfactor of M, and a E M. IT there is a sequence {b n } C P such that bn ~ a in L2(M), then a E P . Consequently, L 2(P} n M = P ( regard L 2(P) and M as linear subspaces of

L 2(M)).

IIbn - all2 ---+ 0 and bn

E P, it follows that xa E L 2(P), \:Ix E P. Thus, p(a)P C L 2(P}, and L 2(P) is invariant under pea). Put t = p(a)IL2(p). Then t is a bounded linear operator on L 2(P), and tx = xa = A(x)a, \:Ix E P.

Proof.

Since

Hence

I

2(P))

a E B,. = {b E L 2(P) there exists p(b) E B(L such that} p(b)x = A(x)b,\:Ix E P Now by the Remark under Proposition 17.1.2, we have a E P.

Proposition 17.2.6. Let M be a finite factor. (i) IT N is a subfactor of M, then we have

[M : M]

= 1,

and

[M: N] > 1;

Q.E.D.

696

(ii) If N is a subfactor of M, M c B(H), and N' is finite, then we have

[M: N] = [N' : M'l <

00;

(iii) If Q is a subfactor of M, P is a subfactor of Q, then we have

[M: Pl = [M : Q] . [Q : P],

and

[M: P] > [M : Q]j

(iv) If Q is a subfactor of M, P is a subfactor of Q, [M : P] [M: P] = [M: Q] , then we have Q = P.

<

00,

and

Proof.

(i) A(M) admits a cyclic-separating vector I(E Me L2(M)). So 1 is also separating for A(N). Now by Proposition 17.1.9 we have

(ii) Since M, N' are finite, N, M' are also finite. Now by Proposition 17.1.6 we have dimM (H) < 00, dimM' (H) < 00, dime (H) < 00 and dimj» (H) < 00. Further, from Propositions 17.1.8 and 17.2.4 we obtain that

[M: N] = dimN(H)/dimM(H)

= dimN,(H)-l/dimM,(H)-l =

dimM,(h)/dimN,(H) = [N' : M'] <

00.

(iii) If [M : Q] < 00, then we can pick a faithful nondegenerate W*representation {11", H} of M such that 1I"(M)' is finite and dimQ (H) < 00 ( i.e., 11" ( Q)' is finite). Hence

[M . Q] . [Q . P] . .

=

dimQ(H) . dimp(H) dimM(H) dimq(H)

=

dimp(H) dimM(H)

= [M . P]

..

Let [M : Q] = 00, and pick a faithful nondegenerate W*-representation {1I",H} of M such that 1I"(M)' is finite. Then 1I"(Q)' is infinite, and 1I"(P)'(~ 1I"(Q)') is also infinite. Hence dimp(H) = 00 and [M : P] = 00. (iv) By (iii) it suffices to show that M = P if [M : P] = 1. Since [M : P] = 1 = dimp(L 2(M)), A(P)' is finite. Let r, r' be the unique faithful normal tradal states on A(P), respectively, and e, e' be the cyclic 2(M) projections from L onto A(P)'I, API respectively, where 1 is the identity of M( C L2(M)). Since 1 is cyclic-separating for A(M), and A(P)' ~ A(M)', it follows that e = l L 2(M )' Now from

t(jl'

= dimp(L2(M)) = r(e)/r'(e'), we have r'(e') = 1 and e' = l L 2(M ) , i.e., A(P)1 = PI = L2(M). Hence, for 1 = [M : P]

a E M there is {bn } C P such that bn = bnl ----+ a in L 17.2.5 we have a E P. Therefore, P = M.

2(M).

any Then by Lemma Q.E.D.

697

Proposition 17.2.7. Let M, be a finite factor, and N, be a subfactor of M i , i = 1,2. Then we have

Proof. Let Tl, T2 be the faithful normal tradal states on M 1 , M 2 respectively. Then TI ® T2 is the faithful normal tradal state on M 1 ®M2 • Hence we have L 2(MI®M2) = L 2(Mt} ® L 2(M2 ) , A(M1®M2 ) = A(Mt}®A(M2 ) , and A(N1®N2 ) = A(NI)®A(N2 ) . By Theorem 6.9.12, >..(N1®N2 ) ' is finite ¢::=> both A(Nt}' and A(N2 ) ' are finite. Now we may assume that A(NI ) ' and >..(N2 ) ' are finite. Let TLT~ be the faithful normal tradal states on >..(M1 }' , A(M2 )' respectively, and Ei = I, be the identity of M i , i = 1,2. Then Ei is cyclic-separating for A(Mi ) ( in L 2(Mi )) , i = 1,2. Clearly, A(Ni)'Ei = L2(Mi ), i = 1,2, and >..(N1®N2)'(EI ® E2) = L 2(M1 ® M 2 ) . On the other hand, we have A(N1®N2HEl ® E2) = (e~ ® e~)(L2(MI) e L 2(M2 )) , where e~ is the cyclic projection from L 2(Mi ) onto A(Ni)Ei' i = 1,2. Therefore, we have

[MI®M2 : N 1®N2 ] = dimNl~N2(L2(Ml®M2))

_

-

(Tl~T2~(l)

(1'-r~T~)(el~e~)

_

-

I . 1 ~ 1'"2(e~)

= dimN 1 (L 2 (M d ) . dimN2 (L 2 (M 2 )) = [M1

:

Nil· [M2 : N 2 ].

Q.E.D. Example. Let G be an ICC group, and H be an ICC subgroup of G. Suppose that s ~ A8 is the left regular representation of G on 12 (G). Then R(G) = M is a type (lid factor on 12(G), and the faithful normal tradal state T on R(G) is as follows:

T(X) = (xce , Ce ),

Vx E R(G),

where ce(s) = Ce , 8 , Vs E G, and e is the unit of G. Clearly, {c8ls E G}, {A8 1s E G} are orthogonal normalized bases of 12 ( G), L 2 (M) respectively. Then U is a unitary operator from 12(G) onto L 2(M), where UC8 = A8 , Vs E G. Thus, we have dimR(G)(12(G)) = dimM(L 2(M)) = 1. Similarly, we have dimR(H) (12(H)) = 1. Pick a subset E of G such that H g' =f:. Hg",Vg',g" E E and g' =f:. s", and {Hgig E G} = {Hgig E E}. Then we have 12(G) = 2: EBI 2 (H g). gEE

698

Now by Proposition 17.1.6 we obtain that

[R{G) : R(H)J =

dimR(H) (L 2 (R(G)))

#{Hgig E E} = [G : H] Notes. Index for subfactors was inlroduced by V.F.R.Jones, and he showed that this definition agrees with the ring-theoretic one. References.[60], [75], [76J.

17.3. The fundamental construction Let M be a finite factor, r be the unique faithful normal tradal state on M, and N be a subfactor of M. For each hEM with 0 < h < 1, define

lPh(y) = r(hy) = r(h 1/ 2yh1/ 2), Vy E N. Clearly, lPh E (N.. )+. If Y E N+ l then lPh(Y) = r(hy) = r(yl/2hyl/2). So we have 0 < lPh < (rIN). By Theorem 1.10.3 there exists to E N with 0 < to < 1 such that r(hy) = lPh(Y) = r{toyt o) = r(t~y), Vy E N. Let E(h) = t~{E N). Then we obtain that r(hy) = r(E(h)y), vu E N. Generally, for any x E M there is E(x) E N such that

vv E N.

r(xy) = r(E(x)y),

We claim that E(x) is uniquely determined by z. In fact, if zEN is such that r(zy) = 0, Vy E N, then we have r(zz*) = 0 and z = 0 since r is faithful. Hence, x ---+ E(x) is a linear map from M onto N, and clearly,

E{z)

=

z, Vz E N,

and

E 2(x) = E(x), Vx E M.

From the above discussion, we also have 0 E{x*) = E(x)*, Vx E M. Moreover, since

r(E(aE(b))y)

=

r(a . E{b)y)

=

< E{h) <

Ilhll

for h E M+. So

r(E{a)E{b)y),

Vy E N, it follows that E{aE{b)) = E{a) . E(b), Va, b E M. Similarly, by

r{E{E{a)b)y) = r(E(a)by) = r(b.yE{a)) = r(E(b)yE(a)) E N, we have E(E(a)b) = E(a)E(b), Va, b E M. Hence

o < E((x -

E(x))*(x - E(x)))

=

= r(E{a)E(b)y),Vy

E(x*x) - E(x)* E(x),

699

and

o < E(xtE(x) < E(x*x) <

IIx*xll = IIx1l2,

Vx EM.

Therefore, E(.) is a projection of norm one from M onto N. If h E M+ is such that E(h) = 0, then r(hy) = 0, Vy E N. In particular, T(h) = O. But T is faithful, so h = 0, i.e., E(.) is faithful. Now suppose that {Xl} C M, X, ----+ 0 in o(M, M*), and IIxlll < 1, Vl. Let {11", H, €} be the faithful cyclic W*-representation of N generated by (rIN). Then we have

(lI"(E(XI) )7r(y) €, 7r(z) €) = r(z* E(Xl)Y) -

r(E(xl)YZ*) = r(x,yz*)

----+

0,

Vy, zEN.

Since 1I1r(E(xl))1I < 1, \ll, it follows that 1r(E(Xl)) ----+ 0 ( weakly) . Further, E(x,) ----+ 0 in o(N, N.). Therefore, E(·) is a(M, M.) - o(N, N.) continuous. By Theorem 4.1.5 we have the following.

Proposition 11.3.1. Let M be a finite factor, and N be a subfactor of M. Then there is a linear map E(·) : M ----+ N such that (i) E(·) is a projection of norm one from M onto N; (ii) E(·) is faithful, i.e., if E(x) = 0 for some x E M+, then X = 0; (iii) E(.) is o(M, M.) - a(N, N.) continuous; (iv) E(·) is completely positive, in particular,

E(M+) = N+, and E(x*) = E(x)*, \Ix E M;

(v) E(aE(b» = E(E(a)b) = E(a)E(b), \la,b E M; (vi) E(x)* E(x) < E(x*x), Vx E M; (vii) r(E(x)y) = r(xy) ,\Ix E M,y E N. In particular, E(·) keeps r{·).

Proposition 17.3.2. Let M be a finite factor, r be the unique faithful normal tracial state on M, and N be a subfactor of M. Using r ; we construct the Hilbert spaces L 2{M) and L 2(N ). Naturally, L2(N} can be regarded as a closed linear subspace of L 2(M) ( i.e., L 2(N) = N, the closure of N in L 2(M». Let P be the projection from L 2(M) onto L 2(N), and E(·) be as in Proposition 17.3.1. Then we have the following: (i) P(x) = E(x), Vx E M, (ii) PA{X}P = A(E(x»P, \Ix E M; (iii) IT x EM, then x E N ~ PA(X} = A(X)P; (iv) A(N)' = {A(M}', P}" = {p(M), P}", and A(N) = {p(M), P}'; (v) JP = PJ, and JPJ = P.

700

Proof.

(i) For any x E M, by Proposition 17.3.1 we have

(x - E(x), y)

=

r(y*x) - r(y*E(x))

=

r(xy*) - r(E(x)y*) = 0,

\/y E N.

Thus, (x - E(x)) 1.. N = L 2(N) = P(L 2(M)), \/x E M. Further, by E(x) E N c L2(N) we get P(x) = E(x), \/x E M. (ii) For any z, y E M, by (i) we have

PA(X)Py = P(xE(y)) = E(xE(y))

= E(x)E(y)

= A(E(x))Py.

Hence, PA(X)P = A(E(x))P, \/x E M. (iii) IT x EN, then for any y EM,

PA(X)Y = E(xy)

=

xE(y)

Hence, PA(X) = A(X)P. Conversely, let PA(X)

x

=

A(x)1 = A(x)Pl = PA(x)1

=

A(X)Py.

= A(X)P. Then we have

= Px

= E(x)

E N,

where 1 is the common identity of M and N. (iv) Clearly, A(N) c p(M),. Further, by (iii) we have

A(N) c {p(M), P}/. Conversely, if t E {p(M),P}', then t E p(M)' = A(M). So t = A(X) for some x E M. Now by A(X)P = PA(X) and (iii) , we get x E N. Thus,

t = A(X) E A(N). (v) For any x E M, it is obvious that

JPx = JE(x) = E(x*) = Px" = PJx. Since M is dense in L 2 (M), it follows that J P = P J.

Definition 17.3.3. 17.3.2. Then, define

Q.E.D.

Keep the assumptions and notations in Proposition

(M, P) = {A(M), P}". It is a VN algebra on L 2 (M).

Proposition 17.3.4. Using the assumptions and notations in Proposition 17.3.2 and Definition 17.3.3, we have the following: (i) (M, P) = J A(N)' J, and (M, P) is a factor; (ii) {LA(Xi)PA(Ydlxi'Yi E M} is a weakly dense * subalgebra of (M,P)j i

701

(iii) x ~ .\(x)P is a (iv)

* isomrophism from

(M, P)

is finite

N onto P(M, P)P;

<=> .\(N)' is finite <=> [M : N] < 00;

(v) If M and N are type (Il.], and [M : N] < 00, then (M, P) is also type

( 111)' Proof.

(i) By Proposition 17.3.2, we have

J.\(N)'J

=

J{.\(M)',P}"J

= {J p(M)J, J PJ}"

=

{.\(M), P}"

=

(M, P).

Moreover, since .\(N)' is a factor, (M, P) is also a factor. (ii) Noticing that

.\(x)P.\(y)P.\(z)a =

xE(yE(za)) xE(y)E(za) = .\(xE(y))P.\(z)a,

\:la, z, y, z E M, we can see that

x = {.\(xo) + L

'\(XdP'\(Yi)lxo,Xi,Yi EM}

i

is a * subalgebra of (M, P). Since .\(M) C X and P E X, it follows that = (M,P). Now let

r

Y =

{L .\(xi)P.\(yd lXi, Yi EM}. i

Clearly, Y is a * two-sided ideal of X. Thus, y<1 is a o -closed two-sided ideal of (M, P). But (M, P) is a facotr, so by Proposition 1.7.1 Y is weakly dense in (M,P). (iii) For any x E N, y E M, we have

P).,(x)Py = E(xE(y)) = xE(y) = ).,(x)Py, i.e., P).,(x)P = ).,(x)P, \:Ix E N. So ).,(N)P C P(M, P)P. Since A(X)P P).,(x), \:Ix E N (see Proposition 17.3.2), x ~ .\(x)P is a * homomorphism from N to P(M, P)P. It is also injective. Indeed, if .\(x)P = 0 for some x E N, then xy = 0, \:Iy E N. So x = O. Now by (ii) , it suffices to show that

P).,(x)P .\(y)P E ).,(N)P,

\:Ix, y E M.

It is equivalent to prove that P).,(x)P E ).,(N)P, \:Ix E M. But it is immediate by P).,(x)P = ).,(E(x))P E ).,(N)P, \:Ix E M.

702

(iv) Since (M, P)

(M, P)

= J "A(N)' J is * isomorphic to "A(N)', it follows that is finite

¢::::::?

"A(N)' is finite

¢::::::?

[M : N]

=

dimN(L 2(M)) <

00.

(v) By (iv), "A(N)' is a finite factor. Since "A(M)' = J"A(M)J is continuous and "A(N') ~ "A(M)', it follows that "A(N)' is type (lId. Now since (M, P) is * isomorphic to "A(N)', (M,P) is also type (III)' Q.E.D.

Proposition 17.3.5. Let M be a finite factor, N be a subfactor of M with [M : N] < 00, and r be the unique faithful normal tracial state on M. Then r has the Markov property of modulus (3 = [M : N], i.e.,

tr("A(x)) = r(x),

(3tr("A(x)P)

=

r(x),

\/x E M,

where tr(·) is the unique faithful normal tracial state on (M, P) ( noticing that (M, P) is also a finite factor by Proposition 17.3.4). In particular, tr(P) =

[M: N]-l. Moreover, we have [(M, P) : M] = [M : NJ. Proof. Since tr 0 "A ( .) is also a faithful normal tracial state on M, it follows that tr("A(x)) = r(x), \/x E M. By Proposition 17.3.4, x ----+ "A(x)P is a * isomorphism from N onto "A(N)P, and P E "A(N)'. So we have

tr("A(x)P) = kr(x), where k is some positive constant. Clearly, k From [M : N] < 00, "A(N)' is finite. Then

(M: N]

\/x E N,

= tr(P).

= dimN(L 2(M)) = tr(e)jtr'(e'),

where tr' is the unique faithful normal tracial state on "A (N)', and e, e' are the cyclic projections from L 2(M) onto "A{N)'1, "A{N)1 respectively. Since "A(N)' ::) "A(M)', it follows that e = 1L 2(M ). Moreover, "A(N)1 = N = L 2(N). Hence, [M : N] = tr'(p)-l. Further, by (M,P) = J"A(N)'J we have [M : N] = tr'(p)-l = tr(JPJ)-1 = tr(p)-l. Then,

tr("A(x)P)

=

tr(P"A(x}P)

=

tr("A(E(x}}P}

= [M : N]-lr(E(x)) = [M : N]-lr(x),

\/x E M.

Finally, by Proposition 17.2.4 we obtain that

[(M, P) : M] = dimM(L 2(M))jdim(M,p}(L2(M)) =

dim).(N)' (L2(M))-1

= dimN(L 2(M)) = [M : N].

Q.E.D.

703

Now let M be a finite factor, N be a subfactor of M, and [M : N] < 00. By Proposition 17.3.5, we obtain a new pair of finite factors ..\(M) and (M,P) = {..\(M),P}" on L2(M). Let MJO) = NjMJl) = N,Ml 1 ) = M;M1 2 ) = ..\(MP») , PI = P, and

M~2) = (M, P) = (M12 ) , PI) = {Mi 2), PI}". Since [MJ2) : MI2)] = [MIl) : MJI)] process. Thus, we have the following:

=

[M : N] <

00,

we can continue this

MJO)

1

M~I) C

MJ2) = (Mi 2 ) , PI)

1..\

Hence, there is an inductive system of finite facotrs:

°

·0

." M(A:+I) P) M (O) -----+ M(l).. 1 -----+. • • -----+ A:+I -_ (M(A:+l) A: ,A:

-----+'..,

where M!~il) is a finite factor on L 2 (M!A:»), PA: is the projection from L 2 (M!A:») onto L2(M!~1), and ~A: is a * isomorphism from M!A:) onto M!A:+l), \lk > 1. We consider the inductive limit of {M!A:) , ~ A: Ik}. By Section 3.7, let

XA:M!A:) = {( aA:) A:?:O IaA: E M!A:) ,\lk > O},

I

A = {(aA:) E XA:Mt) there exists ko such that}, a.+l = ~,,(a.), \Is > k o

and

I

= {( aA:)

E A Ithere exists k o such that }. a.

= 0, \Is >

ko

Then XA:M!A:) is a * algebra in a natural way, A is a I is a * two-sided ideal of A , and

* subalgebra of XA:M!A:) ,

704

Let

M" =

{a E AI I Itherewhere exists (0"" a.+l

,0, a", a"+l"") E = ~6(a.), Vs > k Vk > O. Then we have the following commutative diagram:

a, },

IE IE

c

c

c

c

c

where W" is a * isomorphism from M~") onto M" as follows: for any a" E M~"), let W,,(a,,) = a, then (O,···,O,a",~,,(a,,),···)EO:,

Vk > O. Define e" = W,,+l(P,,),Vk > 1. Since M~~il) = (ep,,(M~")),P,,), it follows that M"+l = (M", e,,), i.e., M"+l is generated by M" and a projection e", Vk > 1. Clearly, M" is a finite factor, and

Definition 17.3.6. The above chain 1 E M« C ... C M"+l = (M", e,,) C ... is called the tower of finite factors induced by a pair {N c M} of finite factors with [M : N) < 00. Theorem 17.3.7. Let M be a finite factor, N be a subfactor of M with [M : N) < 00, and 1 E M o C M 1 C C M k+ 1 = (M",e,,) C be the tower of finite factors induced by {N eM}. Then we have the following: (i) the pair M« C M 1 is * isornorphisc to N eM; (ii)M" is a finite factor, and [M"+l : M,,) = [M : N), Vk > 0; (iii) IT rIc is the unique faithful normal tradal state on M", then rIc has the Markov property of modulus of (3 = [M : N), i.e., r"+lIM" = rIc, and

(3r"+I(xe,,) = r,,(x),

Vx EM",

Vk > 1. In particular, r(e,,) = {3-1, Vk > 1, where T is the tradal state on u"M" such that riM" = rIc, Vk; (iv) M"+l = (M", eb ... , e,,), i.e., M"+l is generated by M" and {eb"', e,,}, Vk > 1; (v) the sequence {e"lk > I} of projections satisfies the following relations:

Proof.

It suffices to prove (v) .

705

Let (k - j) > 2 and j > 1 . Notice that

and fj E MIe-I, fie E M Ie+1. So the relation elefj = fjele is equivalent to that -1 -1 - _ (Ie-I) - _ <.[) Ie 0 <.[) k~l 0 WIe-I (ej) and WIe~l (ele) commute. Define N - <.[)1e-dMIe- 1 ), M M!Ie) ,P = Pie, and M 1 = M Ie~il). Then P is the projection from L 2 (M) onto L 2(N), and M 1 = (A(M),P) = {A(M),P}". Since ~Ie 0 ~Ie-l 0 W;~l(ej) = ~Ie 0 ~1e-l(Pi) E A(N), and WiJl(fle) = P, it follows from Proposition 17.3.2 (iii) that ~Ie 0 ~1e-l(Pj) and P commute. Therefore, we have that flefj = ejele, \;;Ilk - jl > 2. (Ie) (Ie) (Ie+l) For k > 1, let N = MIe_1,M = M le ,~= <.[)Ie(= '\)jM1 = M Ie +1 ,w = <.[)Ie+l (= A); and M 2 = M~~~2). Then we have

where PI is the projection from L 2(M) onto L 2(N),M 1 is a finite factor on L 2 (M ) generated by ~(M) = ,\(M) and PI; P 2 is the projection from L 2(M 1 ) onto L2(~(M)), and M 2 is a finite factor on L 2(Mr) generated by W(MtJ = A(M1 ) and P2 • Denote by E, F the conditional expectation from M onto N, M 1 onto ~(M) respectively, i.e., E = (P1IM), F = (P2IMr). Then the relations: flefle+lfle = /3-1ele, and fk+lfleele+l = /3-lele+l are equivalent to

P2W(PtJP2 =

/3-1 P2 ,

and

W(PdP2W(Pd = /3-1W(P1).

First we show that F(Pr) = /3-11. Indeed, since F(Pr) that

F(P2) =

/3-11

~

(PI -

<=> {(PI -

/3-11)

...L L2(~(M))

= P2(P1), it

in

/3-11), '\(x+)) = 0, \;;Ix E

follows

L 2 (M 1 )

M.

If tr is the canonical tracial state on Ml, then the equality F(Pr) = /3-11 is equivalent to the following:

/3tr('\(x)P1)

= tr('\(x)) = T(X), "Ix E M,

where T is the canonical tradal state on M. This is exactly the Markov property of modulus /3 of T ( see Proposition 17.3.5). Hence, F(P1 ) = /3-11. Now for any x E M1(c L 2(M1 ) ) , we have

P2W(P1)P2X

=

F(P1 • F(x)) = F(Pr)F(x)

= /3-1 F( x) =

/3-1 P2 x.

706

Hence, P2w(PdP2 = p-I P2. By Proposition 17.3.4 and Theorem 1.6.1, the set

{L ~(Xi)PI~(Yi) lXi, Yi E M} I

is dense in L 2(M 1)' Now it suffices to show that W(PtlP2W(PI)(~(X)PI~(Y))= p-IW(PI)(~(X)PI~(Y)),

\lx,yEM. Fix x, y E M. Since

PI ~(X)PI ~(y)z = E(xE(yz)) = E(x)E(yz)

=

~(E(X))PI~(Y)Z,

\lz E M( C L 2(M) ), it follows that PICI>(x)P1~(y) = ~(E(X))Pl CI>(y). Then

w(PdP2W(Pd (~(X)PI ~(y)) -

PIF(PI~(X)PI~(Y))

-

PI ~(E(x))F(PdCI>(y) =

=

PIF(~(E(x))PI ~(y))

p-l PICI>(E(x))~(y).

Noticing that PI~(E(x))z

= E(E(x)z) = E(x)E(z) =

\lz

E

E(xE(z)) = PI ~(X)PIZ,

M(c L 2(M)) , we have

W(P1)P2 w(P1) (~(X)PI ~(y))

= p-I PI CI>(x)P1 CI>(y) = P-1w(PI)( ~(X)PI CI>(y)),

Q.E.D.

\Ix, y E M. Therefore, W(PdP2W(Pd = P-1W(P1).

Remark. Let M be a type (1m) factor, and N be a type (In) subfactor of M. Then it must be follows:

nlm.

From Chapter 15, there is a Bratteli diagram as .

n

p

--+

m'

where m = np. Clearly, L 2 (M ) = M, L 2 (N ) = N, and P = E : L 2 (M ) --+ L 2(N ). By Proposition 17.3.4, we have (M,P) = {A(M),P}" = {LA(Xi)P,\ i

(Yi)lxi,Yi EM}. Further, (M, P) = End~(M) =

{t .. M

--+

M i t is linear, and } t(xy) = (tx)y, \Ix E M, YEN. .

707

In fact, it is easy to see that EndJv(M) = p(N)'. Since P = E is the conditional expectation from M onto N, it follows that Pp(y) = p(y)P, \/y E N. Hence (M, P) =

{L A(X,)PA(y,) IXi, y, E M} i

C

p(N)' = EndJv(M).

Conversely, if t E (M, P)', then

tx = tA(x)l = A(x)t1 = p(tl)x,

\/x E M,

and t = p(t1). Moreover, since P(tl) = tPl = tl, it follows that (t1) E N. Hence, t E p(N), i.e., (M, P)' c p(N). Therefore, we obtain that (M, P) =

p(N)' =

End~(M).

By [(M, P) : M] = [M : N] = m 2 In 2 , (M, P) must be a type (I,) factor, where 1 = m 2 In. Let 1 E Mo C

u, C M 2 =

C M Ic + l = (MIc,elc) C

(Ml,el) C

.

be the tower of finite factors induced by {N eM}. Then M/c is a type (Inpl) factor, \/k > O. Thus, we have a Bratteli diagram as follows: p

--t

n

. . ... .

P

--t

-P -t

..

np

Notes. The technique of towers of algebra and Theorem 17.3.7 are due to V.F .R.Jones. About Proposiiton 17.3.4(ii), M.Pimsner and S.Popa proved that (M, P) = {L A(Xi)PA(Yi)!Xi, Yi E M} indeed. i

References. [60], [7S], [129].

17.4. The values of index for subfactors Define a sequence {PIc(A)lk = 0,l,2,"'} of polynomials as follows:

Po = I,Pl = I,PIc +l(A) = PIc(A) - APIc-l(A), \/k > 1. In particular,

P2(A) = 1 - A, PS(A) = 1 - 2A,

P,(A) = 1 - 3A + A2 , Ps(A) = 1 - 4A + 3A 2 ,

P6(A) = 1 - SA + 6A 2

-

AS,

P7(A) = 1 - 6A + 10A 2

-

4A S.

708

Lemma 17.4.1. Consider an integer k > 0, and set m = [iI. Then (i) The polynomial PIc is of degree m. Its leading coefficient is (-I)m if k = 2m is even and (-I)m(m + 1) if k = 2m + 1 is odd. Consequently, lim P,,(A) = +00. ).--00

(ii) PIc has m distinct roots which are given by

1,2, ... ,m. (iii) Assume k > 1. Let A be a real number with 1 ---1r=-2

< A<

4 cos ( k ) +2

1 2

4 cos (k

1r

+1

)

Then Pl(A) > 0, P2(A) > 0," . ,P,,(A) > 0, and P,,+dA)

< 0.

Proof. Claim (i) is easily checked by induction. For (ii), we consider A > 1/4 and let

We say that P,,(A) = (ILl - JL2)-1(llt+ l - Il;+l) for each k > O. In fact, it holds for k = 0,1. Now assume the conclusion holds for each j < k. Then by A = JLlll2 we have that

P,,+dA) = P,,(A) - AP"-l(A) = (JLl - 1l2)-1[JL~+1 - Il;+l - JLlll2(llt - JL~)] = (JLl - 1l2) -1 (JLt+ 2 - 1l~+2).

Hence, we have P,,(A) = (Ill - 1l2)-1(JLt+1 - JL~+l),VA > 1/4 and k > 0. Consider now a real number 0 with 0 E (O,~) and set A = 1 (> 1/4), so ~ 4 cos 20 that JLl

=

is

2 cos 8 and JL2

=

e:"

2 cos (). Then

P,,(A)

=

sin((k + 1)0) 2" cos" (0) sin 0

which vanishes when 0 = k J; 1 with j = 1" .. , m. Claim (iii) is obvious for k = 1, and we may assume k > 2. For 1 E {2," " k}, the smallest root of P, is 1 1r ,and P,(A) > for A < 2 4cos (-1 - ) +1 1 \ 1 < 1 1T ,we have PI(A) > 0. The ---=--""'1T-' As " < 1T 2 4 cos 2 ( - - ) 4cos 2 ( ) 4cos (-1 - ) 1+1 k+l +1

°

709

two smallest roots of P Ic +1 are

Al =

particular PIc+ 1(A) <

1 2

4cos ( k

1r'

+2

)

o.

Q.E.D.

Now let {eiii > I} be a sequence of non-zero projections on a Hilbert space H such that {Jeiciei

= ei,

if

Ii - jl = 1;

where {J is a constant with {J > 1. Assume that Pi (t) # 0,

cici

0

= eiei,

if

Ii - jl > 2,

<j
where k > 1, t = {J-1, and the polynomials {Pi} are as above. Define inductively operators on H by

Lemma 17.4.2.

For m E {I,··· ,k}, we have:

(i)

(ii) 2

(Omem )

=

Pm(t) Pm -

( ) Omem.

1

t

Furthermore, if m < k - 1 : (iii) Om+l is a projection, and (l-om+1 ) is a linear combination of monomials in {el,··· ,em}; (iv) ejOm+l = Om+lei = 0,1 < j < mj (v) Om+l = 1 - sup{cl,' .. ,em}. Proof. that k

Recall that the integer k is fixed in this discussion; we may assume > 2. Since the lemma is obvious for m = 1, we proceed by induction on

710

m. Thus assume that m > 2 and conclusions (i) to (v) hold for 1,2,'" ,m-1. Put 2 (Pm-2(t) ) a = ( cmom ) = Cm Om-l - P ( ) 0m-ICm-IOm-1 cmom' m-l

t

Since (1 - om-.) is a linear combination of monomials in {CI,"', Cm-2}, it follows that Om-ICm = CmOm-I' Applying (v) for the values (m-l) and (m-2), we have Om < Om-I' Hence a

s: = CmUm

~m-2fijt Um-ICmCm-lCmUm s: s:

= CmUs:m -

~m-2ftt s s: = ~m t t CmUsm· t tUm-lCmUm m-l m-l

m-l

The conslusion (i) follows. Using From (i) we have

* operation, we get the conclusion

(ii).

Hence,

So (iii) follows. For j E {I"", m - I}, we know by induction that CjOm that CjOm+l = Om+lCj = by definition of Om+l' Moreover

°

CmOm+l

Pm - 1 (t ) (

= OmCj = 0,

so

)2

= cmOm - Pm(t) cmOm = 0

by (i). Take * operation, then Om+lem = O. So (iv) holds. Finally, if p is any projection on H with p > cil1 < j < m, then we have p(l- Om+l) = (1- om+.)P = (1- om+.) since (1- Om+l) is a linear combination of monomials in {c""" em}, i.e., (1 - Om+l) < p. By (iv) it is obvious that (1 - Om+l) > ci' 1 < j < m. Therefore, we have (1 - Om+l) = sup{ c.,· . " cm}.

Q.E.D. Lemma 17.4.3. Assume that k > 3. (i) IT 0k-l = Ok, then Ck-l < 1 - Ok-2; (ii) If k > 4 and 0k-l = Ok, then Ok-3cic; = 0, Vi,;" > k - 2 and

Ii - jl > 2.

Proof. (i) Let p = Ck-l (Ok-2 - Ok-I). By Lemma 17.4.2 (iv) and Ok-l = Ok, we have ck-lOk-l = ck-lOk = O. Since 8k - 2 is a linear combination of monomials

711

in {I, Ct," " Ck-S}, it follows that a proj ection. Further,

ck-lhk-2

= hk-2Ck-l' Hence, p = Ck-lhk-2 is

Since Pk-1(t) =1= 0, it follows that p = 0, i.e., ck-lhk-2 = 0, and (ii) Put q = hk-SCk-2Ck. Clearly, q is a projection. Since

it follows that q q

=

= CkCk-2(Ok-S -

qq"

=

hk -

ck{ ck-2( Ok-S -

Ck-l

< 1- hk - 2 •

Then

2) .

Ok-2)2 c k-2}ck

_ Pk _ 4 {)2 (t )2ck{ ( )( )C } ck-2 Ok-sck-S Ok-sck-S 0k-SE:k-2 ck k-S t = j:k-4~tt Ck{Ck-2 0k-S Ck-S Ok-S Ck-2}Ck - p

k-S

=

ll

t tCk- 4 t k-S

ck-2 0k-s ck C ck Ok-s

= 1 - ~k-2ftl) t q. k-S (

0, we get q = 0, i.e., 0k-SCk-2Ck = 0. '-k H } > k, let Vj = ( 3 9 CkCk+l ••• Cj, then

As Pk - 2 (t )

=1=

viCkVi

Hence,

Ok-SCk-2Cj

=

= c;,

and

Vj(Ok-sck-Z)

Vihk-SCk-2CkVj

=

(hk-sck-2)Vj'

= 0.

i-k±2

Finally, if k-2 4 and 2

2 1r

1r

4 cos k _ 1 < (3 < 4 cos k' Then Pj(t) =1= 0, V} < k - 1 , and

Ok-l =

Ok.

712

Proof. By Lemma 17.4.1 (iii) we can see that Pj(t) =1= O,'v'j P k- 2(t ) > 0, Pk-1(t) < O. Then from Lemma 17.4.2 (i) we have

o «Ok-lek-lok-l)2

< k -1, and

= ok-l(ek-lok-l)20k-1 s: = ~k-lffi t Ok-lek-luk-l < O. k-2

Hence, 0 = 0k-lek-lok-l = (ek-lok-l)*(ek-lok-l), and ek-lok-l = 0 = 0k-lek-l' Since 0k-l = 1 - sup{eb' .. ,ek-2}, it follows that ek-l < sup{eb ... ,ek-2}' Therefore, we obtain that

Q.E.D. Theorem 17.4.5. Let {eili > I} be an infinite sequence of non-zero projections on a Hilbert space H such that f3eieiei

=

ei, ifli - jl

=

1;

eiei

=

eiei, ifli - jl > 2,

where f3 is a constant with f3 > 1. Then it must be either f3 = 4 cos 2 ~ for some integer q > 3, or f3 > 4. Suppose that f3 E (0,1) but f3 tf. {4cos 2 ~Iq Proof. find a contradiction. In fact, pick an integer k > 4 such that 4cos 2

11"

= 4,5,"

.}. Then we can

11"

< f3 < 4cos 2 - . k-l k

Then there are oeprators 01,"', Ok on H with 0k-l = Ok by Lemma 17.4.4. Clearly, 01 =I- 02 since el =I- O. If l - e l = 02 = 03 = I-sup{el,e2}, then el > e2' Further, by e2 = e2ele2 = f3-1e2 we get f3 = 1 , a contradiction. Hence,02 =I- 03. Let 1 be the smallest value in {I"", k} such that 01-1 -I 01 = 01+1' Clearly, 3 1, let 1m 01-2e,-2+m' Since 0,-2el-2+m = e'-2+mOI-2' 1m is a projection. Moreover, f3lmlnlm

=

f3 0,-2el-2+mel-2+ne,-2+m

= 01-2 e,-2+m = 1m, if

Imln

= Inlm, if

1m - nJ =

1m - nl > 2.

Replacing {ej} by {Ii} we can obtain that Ik-l

< sup{ II, ... ,lk-3}

1;

713

from Lemmas 17.4.4 and 17.4.3 (i). On the other hand, since (1 0(1+1)-1 = 0Hb it follows from Lemma 17.4.3 (ii) that

<

+ 1) > 4

and

< k-

3. Hence, Ik-1 = O. Further, by f3lmlm+1/m = 1m, m = k - 2, ... , 1, we can see that 0 = II = O,-zcl-1. Then, CI-1 < sup{ Ct, ... ,cl-S}' Further sup{ Ct, ... ,cl-2} = sup{ Ct, ••• , cl-1},

VI

m

and 0'-1

= 0,.

Q.E.D.

This contradicts the definition of I.

Theorem 17.4.6. Let M be a finite factor, and N be a subfactor of M with [M : NJ < 00. Then it must be either

[M : N] = 4 cos 2 or

7r

q

for some integer q

>

3

[M: N] > 4.

Proof.

It is immediate from Theorems 17.3.7 and 17.4.5.

Q.E.D.

Notes. Theorem 17.4.6 is due to V.F.R.Jones, and the proof presented here is taken from H. Wenzl. For any f3 > 4 or (3 = 4 cos 2 ~ (some integer q > 3), the infinite sequence {c; Ii > I} of non-zero projections on a separable Hilbert as in Theorem 17.4.5 does exist. And V.F .R. Jones proved that: R = {I, Ct, CZ, ••• }" is a hyperfinite (lId factor; and [R : R~] = (3, where R~ = {I, cz, cs, .. -}". Now let R be a hyperfinite (lId factor on a separable Hilbert space H. We may assume that H = LZ(R). By Theorem 6.9.12, R(;9R is a hyperfinite (Ill) factor on H (;9 H, and (R ® 1)' = R'(;9B(H) is an infinite factor on H ® H . From Propositions 4.3.6 and 1.8.10, we have L 2(R(;9R) = L 2(R) (;9 L2(R). Therefore, by Proposition 17.1.6 we obtain that

Moreover, the index theory, Markov property, the tower of algebras and etc. can be generatized to the cases: a pair of finite dimensional C"-algebras, and a pair of finite VN algebras with a finite dimensional center. References. [60], [75], [195].

Appendix Weak Topology and Weak

*

Topology

Let X be a Banach space, and X'* be its conjugate space. Denote by u(X, X'*) and a(X* ,X) the weak topology in X and weak * topology in X'* respectively. Clearly, (X, o(X, X*)) and (X"', o(X*, X)) are locally convex Hausdorff linear topological spaces. This appendix is a survey about o(X, X'*) and o(X*, X). All proofs can be found from the references.

Theorem A.I. have that

(Representation of o-continuous linear functionals ) We

(X,o(X,X"'))'* = X'*,

and

(X'" ,o(X'*,X))'* = X.

Theorem A.2. (Asoli-Mazur, Separation theorem) (i) Let K be a (norm-) closed convex subset of X, and there is 1 E X* such that

Xo E

X\K. Then

Rel(xo} > sup{Rel(x)lx E K}. Consequently, a convex subset of X is ( norm - ) closed if and only if it is

o(X, X*)~closed. (ii) Let K'* be a o(X*, Xl-closed convex subset of X*, and 10 Then there is x E X such that

Relo(x) > sup{Rel(x)11

E

K"'}.

Theorem A.3. (Bipolar theorem) (i) Let A be a subset of X. Then A O = {I E X*II/(x)1

<

1, \/x E A}

E

X*\K*.

715

is a circled convex O'(X"', Xl-closed subset of X"', and

is the circled convex u(X, X"')~losure of A.

(ii) Let A'" be a subset of X"'. Then A~ = {x E

XII/(x)1 < 1, VI E A"'}

is a circled convex O'(X, X"')-closed subset of X, and

is the circled convex u(X"', Xl-closure of A"'.

Corollary A.4. (i) Let E be a linear subspace of X. Then

EO = E.L = {I E X"'I/(x) = 0, Vx E E} is a

0'

(X* ,X) -closed linear subspace of X"', and

is the norm- or O'(X,X*)-closure of E. In particular, E is nonm-or O'(X,X*)closed if and only if E = (E1. )1.. (ii) Let F be a linear subspace of X"'. Then

Fa = F.L = {x E XI/(x) = O,VI E F} is a norm - or O'(X, X"')-closed linear subspace of X, and

is the O'(X"', Xl-closure of F. In particular, F is O'(X*, X) -closed if and only if F = (F1.) 1. •

Remark. If E is a (norm- ) closed linear subspace of X , it is well-known that E'" = X* I E.L' and (XI E)* = E1.. Now if M is a O'(X"', Xl-closed linear subspace of X*, then we have

Hence M is the conjugate space of the Banach space M. (M.)* = (XIM1.) *, and M. = XIM.L is a predual of M.

= XIM.L,

i.e., M =

716

Theorem A.5. (Banach -Alaoglu ) Let S· be the closed unit ball of X·, i.e., S· = {f E X·lllfll < I}. Then (S"a(X.,X}) is a compact Hausdorff space. Theorem A.6. (Goldstine) Let S, S" be the closed unit balls of X, X" respectively. Then by the canonical embedding, S is u(X··, X·)-dense in S"", By Theorems A.S and A.6, we can see that the topological space (S, a(X, X·)) is compact if and only if X is reflexive. Remark.

Theorem A.T. (Metrizability) (i) Let S· be the closed unit ball of X·. Then (S*, a( X., X)) is metrizable if and only if X is separable. (ii) Let S be the closed unit ball of X. Then (S, a(X, X·)) is metrizable if and only if X· is separable. Remark. We also have the following fact. (X, u(X, X·)) or (X* ,u(X· ,X)) is metriable if and only if dimX < 00.

Theorem A.S. (Krein-Smulian) Let K* be a convex subset of X·. Then K* is u(X., Xl-closed if and only if K* n AS· is a(X*, Xl-closed , VA > 0, where S· is the closed unit ball of X·. Consequently, a linear subspace F of X" is u(X*, Xl-dosed if and only if F n S· is a(X" Xl-closed. Moreover, a linear functional 8 on X· is a(X"', X) -continuous if and only if (J is a( X*,X)-<:ontinuous on S·. Theorem A.9. (Banach -Alaoglu ) (i) Let U be a a(X, X*)-neighborhood of 0 in X. Then

UO = {f E X·llf(x)1 < 1, Vx E U} is a circled convex u(X*, Xl-compact subset of x ". (ii) Let U· be a a(X., Xl-neighborhood of 0 in X*. Then

U; = {x E Xllf(x)1 < 1, Vf E U·} is a circled convex u(X, X*)-compact subset of X.

Theorem A.IO. (Eberlian -Smulian ) Let A be a subset of X. Then the following statements are equivalent; (i) The a(X, X*)-closure A of A is a(X, X*)-compact; (ii) A is weakly sequentially compact, i.e., any sequence in A admits a subsequence which converges weakly to an element of X;

717

(iii) Every countably infinite subset of A has a weak cluster point in X, i.e., a point such that every weak neighborhood contains an element in the infinite subset ; (iv) A is [norrn-] bounded, and for any pair of sequences {x n } C A and {fm} C X* with Ilfmll < 1, \1m, we have lim lim 1m (X n) n m

=

lim lim fm(X n) m n

whenever both of the limits exist; (v) If {G n } is a decreasing sequence of closed convex subsets of X such that T n c; =I- 0, \In, then T n (nnGn) 1= 0.

Remark. As a corollary, we have the Kakutani theorem: If X is reflexive, then (S,a(X, X*)) and (S+, a(X+, X)) are weakly sequentially compact, where S, S* are the closed unit balls of X, X* respectively.

Theorem A.II. (James) Let A be a (norm-) bounded and a(X, X*)closed subset of X. Then the following statements are equivalent: (i) A is a(X, X*)-compact; (ii) For each IE X+, Ref(·) attains its supremum on A; (iii) For each I E X+, 1/(') I attains its supremum on A. Corollary A.12. (Krein-Smulian) Let A be a a(X, X*)-compact subset of X. Then its closed convex hull GoA is also a(X, X*) -compact. Remark. About norm topology, we have the Mazur theorem: Let A be a (norm - ) compact subset of X. Then GoA is also ( norm - ) compact.

Theorem A.l3. (Mackey) (i) Let J be a locally convex Hausdorff linear topology in X. Then (X, J) * = X* if and only if a(X, X*) C J C norm topology, i.e., the Mackey toplogy in X is the norm topology. (ii) Let J* be a locally convex Hausdorff linear topology in X*. Then (X*, J*)* = X if and only if a(X*, X) C J* C r(X*, X), where r(X+, X) is the Mackey topology in X*, i.e., the uniformly convergent topology on any a(X,X*)-compact subset of X, or {AOIA is any a(X,X*)-{:ompact subset of X} is a neighborhood basis of 0 with respect to r(X*, X). References. [31], [71], [74], [89].

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Natation Index

A+ 75 A H 76 A'" 107 A = 1Tu (A)" 119 ®i=lA i 165 (®i=lAd+ 166 a-®~lAi 165 Aut(M) 327 A(X) 539 (A, G, a) 640 A x , G 643 A X a r G 653 (A x , G, G, a) 664 A = A(H) 436 B(H) 1 BV(A) 228 Bu(M) 375 B 237,414 Bloc 237 (Cl

C(H) 1 c(p) 34 Cr(X) 88 CoF 96 C·(G) 636 C;(G) 637 C(E) 426 D(·) 316 D(A) 126

D = {D,d,U} 588 dimM(H) 686

[E) 7 E(A) 579 ExK96 F(H) 1

Y = Y(H) 438 (G, 0, Il) 331 H 10H2 20 HI ® H 2 22 ® rEA 202 11. 372 1(E) 372 Ind1T 649 K(a) 274 K 1(lR) 371 i; 43, 86 La'P 46 L(M) 114 LH(A) 123 L 1 ( G, A, a) 641

u,

l~{Aa, cl»P,a

a,p E JI and a < ,B} 196 l~{ 'Pa, p,a I a,,B E JI and a < ,B} 200 M'15 M p 17 M'17 p (Mh 35 M(A) 123 MH(A) 123 Mn(A) = A M.19

J

e u;

187

730 M 1®M224 [M: N] 693 min-®bl Ai 171 max-®i=l Ai 171 (M, G, a) 613 M X a G 616 (M X a G, G, a) 625 M(a,E) 375 MU 385 M(X) 539 n(.) 264 IN 337 INOO 408

s(M(A}, A) 128 s(rp) 43 Sl(rp),Sr(rp} 48 s, s" 349 spa 375 sPu(x) 375 T(·) 282

IN~ ..

r(M,M*) 19 U(M) 16 W(X*) 429 'W(H) 430 X* 7 ®i=lXi 163 a-®?:=l Xi 163 x --+ y 590 LZ 335 ao(') 167 al('} 171 a*(·) 164 {3X 127

n"

PtA)' 81

408

Prim(A) 108 Proj(M) 16 v> q 30 p -< q 30 p€ 63 p€ 63 P(X) 539 Q(A) 123 Q 272

JR7

Rarp 46 RH(A) 123 R( G) 337, 637

T(H) 4 top.z ",top. Y 10 top. x:) top. Y 10 Tr'H 685 T = T(A) 608

r(B(H)' T(H)) 9

Irpl47 1(') 163

R>. 403

6. 348

r(G) 399

r(M) 390 r(a) 385 A(·) 163

Rep(A, H) 474 S(A) 79 S(H) 2 S(M) 390 Sn(M) 221 Sa 559 Su(G) 594 S(®i=lA i ) 176

PI '" P2 495 J.L '" v 240 J.L -< v 240 J.L

-< v( C .M.)

540

v(a) 72 7f1 '" 7f2

-<

s(B(H), T(H)) 8 s*(B(H), T(H)) 8

7f1

s(M,M*) 19 s*(M,Ms) 19

7fI..l7f2

7f2

85

116

7fl ~ 7f2

{7fIp'

118 116

Hlp, EIp} 44, 86

731

{1r u,Hu } 119 l1(a) 72 l1(B(H), T(H)) 8 l1(M,M*) 19 {l1i I t E lR} 360

fEe H(t)dv(t) 452 fEe a(t)dv(t) 456 fEe M(t)dv(t) 461 fEe (t)dv(t) 463 fEe 1r(t)dv(t) 533

Subject Index

Abelian algebras 68 Abelian projections 256 Absolute value of cp 47 Action of G on A 640 Adjoint functional of f 48 Affine functions 539 Algebraic tensor products of C·-algebras 165 Banach spaces 163 Hilbert spaces 20 Algebraically irreducible * representations 98 Amenable groups 637 Ampliation isomorphism 61 Analytic subsets 410 Analytic vectors 350 Approximate identity of A 90 Approximately finite-dimensional * algebras (AF)-algebras 570 VN-algebras 324 Arens multiplication 121 Arveson spectrum 375

Baire spaces 493 Banach algebras 71 Banach * algebras 71 Barycenter of J.1- 538 Bauer simplexes 612

Borel cross section 422 Borel maps 414 Borel spaces 414 Bratteli diagrams 588 Bounded operators 1

C·-algebras 71 C*-condition 146 C*-dynamical systems 640 C*-equivalent algebras 140 C*-norms 165 C'" -seminorms 173 Calkin algebra 527 CAR algebra 599 CCR (liminary) algebras 504 Center of M 18 Central cover (or support) 34 Central decomposition 555 Central measure 556 Central valued trace 282 Commutant of M 15 Commutative algebras 68 Completely additive functionals 41 Completely additive * homomorphisms 60 Completely positive linear maps 188 Composition series 507 Concrete C*-algebras 71 Conditional expectations 218

734

Connes spectrum 385 Constant fields of Hilbert spaces 450 operators 475 VN algebras 476 Continuous VN algebras 268 Convex functions 539 Countably decomposable VN algebras 66 Coupling constant 686 Covariant representations 615 Cross norms 163 Crossed products of C* -systems 643 VN systems 616 W*-systems 623

Cyclic projections 63 Cyclic sequences of vectors 66 Cyclic vectors 44, 85

Decomposable operators 456 Decomposable VN algebras 461 Definition ideal of a trace 288 Descendant of x 590 Diagonal algebras 458 Diagonal operators 458 Diagrams of (AF)-algebras 588 dimenesion groups 596 Dimension functions 316 Dimension groups 595 Dimension theory 35 Direct integrals of Hilbert spaces 452 operators 456 * homomorphisms 463 * representations 533 VN algebras 461 Direct sums of C* -algebras 503 VN algebras 37

Discrete VN algebras 268 Disjunction of * representations 116 Double centralizers 124 Double commutation theorem 19 Dual action 664 Dual norm 164 Dual system 664

Effros Borel structure 436 Elementary C* -algebras 501 Enveloping VN algebra 119 Equivalent projections 30 Ergodic decomposition 567 Ergodic states 564 Ergodic type theorem 269 Essential ideal 127 Essential subspace of 1r 91 Extremal decomposition 553

Factors 18 Factorial * representations 117 Factorial states 204 Faithful positive linear functionals 40 Faithful * representations 85 Finite projections 267 Finite VN algebras 267 Fixed point algebra 618 Fundamental sequences of measurable vector fields 450

G-abelian systems 565 GCR (postliminary) algebras 506 GICAR algebra 589 G-invariant states 560 Glimm condition 511 G-N conjecture 148 GNS construction 44, 86 Group C* -algebra 636 Group measure spaces 331

735

Ergodic 332 free 331 measurable 332 non-measurable 332 Group VN algebra 637

Hereditary C·-subalgebras 114 Hereditary cones 114 Hermition Banach * algebras 135 Hermition linear functionals 48, 88 Hilbert-Schmidt norm 2 Hyperfinite VN algebras 324 Hyperstonean spaces 247

ICC groups 337 Ideal subsets 590 Index for subfactors 693 Induction of M' onto M; 17 Inductive limits of C·-algebras 196 states 200 Infinite projections 267 Infinite tensor products of C· -algebras 201 Hilbert spaces 202 states 204 Infinite VN algebras 267 Injective tensor products 171 Inner * automorphisms 363 Integral representations of states 553 Irreducible * representations 102

Jacobson topology 491 Jordan decomposition of

f

Left centralizers 123 Left kernel of rp 86 Left multipliers of A 123 Left regular representation of G 337 Left support of rp 48 Linear C·-norms 146 Locally countable property 238 Localizable measure spaces 242 Lower semicontinuity 497

Mackey topology 9 Mapping torus 680 Markov property 702 Matrix algebras 187 Matrix units 575 Maximal abelian * subalgebras 135 Maximal C·-norm 171 Maximal vectors 257 Measurable fields of Hilbert spaces 448 operators 455 * homomorphisms 463 * representations 533 vectors 448 VN algebras 459 Modular automorphism group 358 Modular operator 350 Modular units 104 Multiplication algebra 253 Multiplicative linear functionals 82 Multiplicity functions 264 Multiplier algebra 123

50

Kadison transitivity theorem 101 Kaplansky density theorem 35 KMS condition 350

n-homogeneous 304 n-positive linear maps 188 Natural trace 685 NGCR (antiliminary) algebras 506 Nondegenerate * representations 91

736

Nondegenerate subspaces 347 Non-hyperfinite type (Ill) factors 342 Non-nuclear C·-algebras 343 Normal elemennts of * algebras 16 Normal linear functionals 40 Normal measures 246 Normal * homomorphisms 60 Normal states 41 Normal universal * representation 221 Nuclear C· -algebras 205 Nuclear spaces 214

Properly infinite projections 268 Properly infinite VN algebras 268 Property (f) 341 Property (P) 521 Property (T) 214 Pure states 81 Purely infinite projections 267 Purely infinite VN algebras 267

Quasi-equivalent * representations 118 Quasi-matrix systems 527 Quotient C·-algebras 92

One parameter groups 350 Operators of finite rank 1 Operators of Hilbert-Schmidt class 2 Operators of trace class 4 Radon-Nikodym theorem 50 Order ideals 595 Ratio sets 399 Order units 594 Real C·-algebras 148 Real C·-equivalent algebras 153 Ordered groups 594 Orthogonal decomposition of f 50, 88 Reduced crossed product of a C·Orthogonal normalized basis of system 652 -, Reduced group C·-algebra 637 H(·) 550 Outer equivalence 388 Regular left ideals 104 Outer multiplier algebra 123 Regular maximal left ideals 104 Regular * representation induced by 11" 649 Representing measure of x 538 Polar decomposition of

737

Semi-finite projections 268 Semi-finite VN algebras 268 Separating sequences of vectors 66 Separating vectors 64 Simplexes 542 Singular functionals 226 Skew-hermitian real Banach * algebras 151 Smooth C* -algebras 528 Sousline criterion 413 Sousline subsets 410 Spatial * isomorphisms 61 Spatial C*-norm 170 Spatial theory 67 Spectral radius of x 72 Spectral subspaces 376 Spectrum of A 493 Spectrum of a 376 Spectrum of x 72 Stable isomorphism theorem 606 Standard Borel spaces 415 !-Standard Borel spaces 416 States 79 States on algebraic tensor products 176 Stonean spaces 242 Stone-Cech compactification 127 Strict topology 128 Strictly positive elements 133 Strong (operator) topology 8 Strong * (operator) topology 8 a-Strong (operator) topology 8 a-Strong * (operator) topology 8 Subfactors 321 Submultiplication 146 * subrepresentations 116 Support of a Positive linear functional 43 * representation 557 Symmetric real Banach * algebras 151 a-finite projections 68 a-finite VN algebras 66

* operation 71

* representations 44,

85

Tensor products of Banach spaces 163 C*-algebras 165 Hilbert spaces 22 VN algebras 24 W* -algebras 223 Tomita-Takesaki theory 354 Topologically irreducible * representations 98 Towers of factors 704 Traces on a VN algebra faithful 288 finite 288 normal 288 semifinite 288 Trace norm 4 Tradal decomposition 568 Tracial states 280 Transitivity theorem 101 Type I C*-algebras 528 Type (I) VN algebras 268 Type (In) VN algebras 304 Type (II) VN algebras 268 Type (lId VN algebras 268 Type (II oo ) algebras 268 Type (III) VN algebras 268 Type (IlIA) factors (0 < x < 1) 392

Uniform (operator) topology 9 Uniformly hyperfinite (UHF) C* -algebras 202 Unital maps 199 Unitarily equivalent * representations 85 Unitarily equivalent states 495 Unitary cocycle theorem 368 Unitary elements 16

738

Unitary involution 350 Universal * representation 119

Von Neumann (VN) algebras 15 VN-dynamical systems 613

W'"-algebras 219 W'"-dynamical systems 623 W'"-representations 221 W'"-tensor products 223 Weak (operator) topology 8 u-Weak (operator) topology 8 Weakly sequential completeness 232

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Real operator algebras

Operator algebras and topology

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Lectures on Operator Algebras

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Introduction to Lie Algebras

Springer Undergraduate Mathematics Series Advisory Board M.A.J. Chaplain University of Dundee K. Erdmann Oxford Univer...