Introduction to Operator Algebras

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


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
" 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


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 < i < k, where U(Oja},' ",ak,e) = {b E B(H)ltr((b*b + bb*)ai) < e,1 < i < k}. In fact, if there are ab'" ,ak E T(H)+ and e > 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 < i < m} of 0 such that If(b)1 < 1, Vb E U. Let {€I,"', en} be a normalized orthogonal basis of the linear span of {17b' .. ,17m}' Then for enough small e > 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 < i < n. Then f(b) = tr(ab) = L:( ab6, 6) = L:(bei, 77,). I

The last statement on indeed.

i=1

f > 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
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 < i

< n. But a(1 - Po)

= 0, so

k=1

n

ae. - L.p'.kaek = 0,

1

< n,

k=1

i.e., (ae., .. " aen) E jiH. Now by the definition of can find b E M such that

p', for the above e > 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

}

0,1

where aile

X

n-unitary matrix (Uij) such that

< i < n. Further we can write

= UikJ.L~/2, VI < i, J', k < n. Similarly, we can write 1

< i,i < n.

Then

(u,u) =

L(eP),eY»)(e}2),e;2») i,j L(2: Q;ilefJil) . (2: Q;jlefJiz) > 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® Pl > P2,P '" P2· From PI < P '" P2, we also have a projection P3 E M such that P3 < P2,PI '" P3, i.e., P > 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 or P -< q.

Proof. Let c(p),c(q) be the projections from H onto [MpH], [MqH] respectively. Clearly, c(p), c(q) E M n M ' and c(p) > 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

-
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) = '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 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 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· 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 .,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 < P such that

lP(a) = cp(toat o), Clearly, p = s(cp)

Va E M p •

> 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 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
n2' Thus

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

(3)

Vk > rno, n > n2'

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

and {e:,

~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®,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)
,.)-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 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 .

°


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 ';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

.+,. 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 + -[,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
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

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 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 < i < m. Then (~ + hI) < (~+ hl')' By Proposition 2.2.9, we have (~+ hl')~1 < (~+ hz)-I. Since ~(~ + hl,)-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
'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
0, we can pick yEA with

llylll
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


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
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+ 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

< i < n.

Let

i=1

ai = a(Pi), then 1 < If(ai)1 < f(a;ai), Vf E Vi n Oe, 1 < i < n. In particular, n

n

p(I: a;ai) > 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}.

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(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
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

-
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 )® 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


(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
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+ 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
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)


IiI
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
"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 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
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
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
n

and let u

1 2

qkEil1 < -(1

+ ln+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
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 be a bounded real linear mapping from A to a real normed linear space B. Then lIiPll = sup{II(cosb. ea)llla,b E A,a* = -a,b* = b}. Proof.

From Proposition 2.15.22 we have the right side < 1111

=

sup 11(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 < i < n, it follows that

"'Y*(u*) = sup{lu*(u)11 u E ®i=lXi,"'Y(u) < I}

> sup{ju"' (®?=l Xi) I I Xi E Xi, IIXil1 < 1,1 < i < n} supj] ®~=1 Xi(u"')11 Xi E

-

X;\ Ilxill
..(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


..(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 < i < n. For any u E ®i=lAi, define 2 _

ao ( u ) - sup

I

{®i=::ICPi(V*U*uv) CPi E s; 1 < i < n, } n * n n * . ®i=lCPi{V v) v E ®i=lA, and ®i=l CPi(V v) > 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 < i < n, it follows that

II l8l in=l 11"cp; (U)11 2 =

sup

{®~lCPi(V*U*uv) n i,A ( .) } n ( . ) v E ®i=1 andn ®i=l CPi V V > 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


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 < i

2 < i < n.

< n, such

that

1/;(1 1 ® ®?=2ai) - $(1 1 ® ®?=2ai) = D > 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

a(x*x)

= sup{a(x·xu)lu E ®i=lAi,a(u)

> sup{a(u*x·xu)I11. E ®i=lA"a(11.)
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 < i < n - 1, and Xn E Sn' Further, ®?=l CPi is continuous with respect to a(.), VCPi E S, (the state space of Ai), 1 < i < n. By Lemma 3.5.8, we get a(.) > 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


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 (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 ({a E

Al II all
ai('), it suffices to show that a;(·) = a o(') on ®i=IA;. Clearly, there is a * homomorphism 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((u))II u E ®i=lAi,at{u) < I} sup{I*(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 < i < n. Then by the condition 3),

(;r(a)(Ci)' (Ci))

= 1im(1f(Lx~*aijX~))C, c) > 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 (B) C 7f(A)', H = [7f(A)vK], Ilvll = 1111 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((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

•


(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 = ((dnE, E) < !I!I·IIEI1 , and Ilvdl < 11~111/2 ,VI. If I' > l, then (dl' - dz)2 < d - d, and " 2 II(VI' - v')E1I < (((d ,,) - (d,))E, E) ~ 0, 'IE E K. 2

1/2

Thus there is v : K ----+ H with Ilvll < 1111 such that Vt --+ v ( strongly) . Since (vt a® E, TJ) = ( (d, a) E, 71) --+ ( (a) E, 71 )

----

e

VTJ E--.-. K, VI, it follows that v*a ® E= (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,

(a) = v*7r(a)v, Va E A.

2

Consequently, 1111 < llvl1 . Hence Ilvll = 1111

(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)


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 < n. Then 1817=1 €J>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

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 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 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 ,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, WaF (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+ 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

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 < p such that F(q) = O. By Definition 1.8.9, Fn(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
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
0 there exists E = E(e)(> 0) such that l 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)


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) 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 < 2n,X~(t) = 1}.

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 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
') = 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

-
-

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
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

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 < P E LI(O, f.J,), v = o- f.J" and E = {t E 0 I p(t) > 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.

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 cp(fd. Consequently,

I

It and M is finite, it follows by Proposition (p- II). Thus, we get eI < P and el f- p. By the maximum

p. Since el

f'J

6.3.2 that (p- ed rv of {el} and {I,}, we have cp(e) < cp(f) for any projections e and and e < p - eb f < p - II' Let = inf {

Jio

I Ji > 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 < p - e1 , f < P - f 1

Clearly, Jio < 1. We say that 0 < cp(p - el) < Jio. In fact, if cp(p - ed > 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 < p - el, f < P - fl. In particular, cp(p - ell < JiCP(p - fd < cp(p - edcp(p - 11) and cp(p - It) > 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 < P - el, h < P - II f'J

f'J

278

such that cp(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)


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*


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

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



,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 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) = 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,

- 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,

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 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, 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
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,

< i < m.

1

Moreover, N is called a subfactor of M, if N is a factor, N c M, and N contains the identity of M.

Lemma 7.2.5. For any al,"',am E M and e > 0, there is a type (12 ,, ) subfactor N of M (n sufficiently large) and b},· .. , bm E N such that

II ai -

bi 112
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 < i < m and

lI a;- bi ll 2
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 < i < m; 4) M is countably generated, and for any at, ... ,am E M and e > 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




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

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
'(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 < i < 3.

342

Since 0 = cp(u) = b(e) and llbll = properties of F and gl, g2, gs that 1

IIul12 = cp(u*u)l/2 = 1,

it follows from the

(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


: 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)
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(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 =
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

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


[(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 ( 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

" 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
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
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.

428

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

429

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,

430

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)
- /( -x) = /(x),

Vu,v E E.

On the other hand, since x E E, it follows and

Ml;)(/) < - /( -x) = /(x).

432

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
-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).

433

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 ( /

) ]]


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.

435

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*).

436

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' = 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®(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 =


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 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)
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(.) = 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 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 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+ 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 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 < i < n}. Clearly, B(n) is separable, and H(r) is invariant under B(n), \lr > 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 " 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

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
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].

528

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" 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 < i < n. In other words, we can find a closed convex subset K of X with J.L(K) > 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


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) -


{hj}i=l' then there is a partition {I.,···, In} of {I"", m} such that h; = h~, 1 < i < n. Thus, for any j and 1 E I, we have

L

lEI,.

hat) = 0, Further, Sf E Vi if

! fh~dJ1"

cr.~ >

0, Vi E

a.e.J1, on n\Int(V;).

t.; where Ct~ =

! h~dJ1"

Vf E E"'. Then

!

I
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) -


O,Si

E G and

LAi

•

= 1 such that

I(€, [L Ai 7r'{J(a8 ; { a d ) , 7r11'{a2)]€) I
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
.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 < i f. j < n.

°

Proof. For n = 1, it is obvious. Now assume that the conclusion holds for n. For (n + 1) and e > 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
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
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 < 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
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
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 < i < nk, 1 < k < m} such that »> 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 < m;

(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
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 2: s~i)d(n,j), ;=1

and

1

< i < r(n + 1),

r(n+l} ""'" si; (n) L-

> 0,

1
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 < r(nl:)'

Now we may assume that

(n, i)

-+

(n + 1,1),

VI

< i < r(n) and n.

Thus, there is a minimal pojection Pn of An, Vn, such that P« > 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 < r(n + 1), 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(

°< i < N -

Ci+2 =

o. So we may assume that a > 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 < i < r,l < j < s, and al + ... + a r = {31 + ... +{3/J. Then there is a subset {,i;11 < i < r,l < j < s} of ~+ such that /J r ai = L,ik,{3; = Llk;, 1 < r,l < j < s. k=1

k=1

Proof. If {31 > all let ')'11 = at, II; = 0,2 < j < s, then we need to find {,i; I 2 < i < r,l < j < s}(c ~+) such that /J

o; =

L A:=1

r

lik,

2

< i < r,

L A:=2

Ikl = {31 - aI,

603

and

r

=

{3i

L

Iki'

< j < s.

2

k=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
N Sn'" Sm-1' Since ~k(lk} < sk1k+h Vk, it follows that

nmt m > n mlm > 0 - ,

VO < i < n, n > 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 < i < n, and n > 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


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

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

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 =


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


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 < i < n. From this inequality and (i) , we shall get IIInd1r~(x)11 < IIInd1r(x)II,Vx E Ll(G,A,a).

652

In fact, for any x E Ll(G,A,a),y E K(G,A),f E K(G) and tPII(Y·Y) > 0, let

",,(i) ( • ) a _ ",,(i) ( ... ) ai -_ tpII Y Y ,fJi - tpII Y x xy ,

where tP~J = (~) 11,1 < i < n. Since tP~J is positive on Ll(G, A, a), it follows that Pi = 0 if ai = O. Then by (3), = al fit

+ +

+ + f3n an -< max {liil a; i

with o:I

>

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 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 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-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.

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')
-=-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 < i < I]. In this way, we can find {6,···, en} c H 2: EB[M6].

we have (l-pi)H n

such that H =

i=1

Similarly, there is a m-dimensional Hilbert space K such that dimM (H ® K) > 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
[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.


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) - _ 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 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 < i < }-2, let Ui = (3 2 and Ui(Ok-SCj) = (Ok-SCj)Ui. Hence, Ok-scic;

Lemma 17.4.4.

ei, then U;ck-2 Ui = Ci, UiOk-SE:k-2c;Ui = 0. Q.E.D. Ck-2 •..

=

Assume that k > 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