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REAL OPERATOR ALGEBRAS

RE4M |

oNaWfioR ALGEBRAS

This page is intentionally left blank

REAL OPERATOR ALGEBRAS

Chinese Academy of Sciences, P.R.Chinc

World Scientific New Jersey • London • Singapore • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

REAL OPERATOR ALGEBRAS Copyright © 2003 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-380-8

Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface

The theory of operator algebras is generally considered over the field of complex numbers and in the complex Hilbert spaces. So it is a natural and interesting problem: How is in the field of real numbers? Which results are still true in the real case? Which results don't hold in the real case? And which results are needed to change some respects and forms? Up to now, the theory of operator algebras over the field of real numbers seems not to be introduced systematically and sufficiently. The material in [17, 26] seems few and not enough. Moreover, we can also find some respects of the real case in some papers (see References). Similarly to the complex case, a real operator algebra, precisely speaking, is a * algebra consisting of bounded (real) linear operators on a real Hilbert space, i.e., a * subalgebra of B(H), where B(H) is the collection of all bounded (real) linear operators on a real Hilbert space H, and * is the adjoint of operators. 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. Similarly to the complex case, we find that the closures of real operator algebras with respect to usual locally convex linear topologies in B(H) are just two classes: weak closure and uniform closure. Hence, we need mainly to study the weakly closed real operator algebras (real Von Neumann algebras or real W*-algebras) and the uniformly closed real operator algebras (real C*-algebras). The aim of this book is to set up the fundamentals of real operator algebras and to give a systematic discussion for real operator algebras. There are the systematic treatments on complex Banach algebras and complex operator algebras in [1] and [26] respectively. In some sense, this book is a real analogue of [1] and [26] on Banach algebras and operator algebras. Moreover, we shall put other results on real operator algebras into this book

VI

Real Operator Algebras

as more as possible, and give a uniform treatment. Since our treatment is from beginning (real Banach and Hilbert spaces, real Banach algebras, real Banach * algebras, real C*-algebras and W*algebras, and etc.), and some basic facts are given , we can get some results on real operator algebras easily. However, for the aim of systematic discussion, many results in this book seem to be trivial (i.e., to be a simple movement from the complex case). In this case, we shall just give the statements, and not give their proofs. Readers can find the similar proofs from [1] and [26]. Generally, there are two methods for proving the results on real operator algebras: To move and to change the proofs of the complex case into the real case; and first to go to the complexification and then to go back to the real case. Sometimes just one method is available and another method is not available. Sometimes we must use these two methods simultaneously. In this book, we shall describe the differences between the complex case and the real case. Moreover, since A = (B, —) (real C*-algebra, see Chapter 5) and M = (N, —) (real W*-algebra, see Chapter 6), we shall stress the bar "-" operation throughout this book. This book is also an introduction for real operator algebras, written in a self-contained manner. For reading i t , you just have the general knowledge of Banach algebras and operator algebras, e.g., [1] and [26]. Moreover, we shall use the results in [1] and [26] freely, and not give their proofs. This book consists of 10 Chapters. Chapter 1 is the preliminaries. Section 1.1 discusses the complexifications of real Banach spaces and real Hilbert spaces. In particular, we let ||f + ir}\\ — ||£ — i77||(V£, rj), i.e., the bar "-" operation is an isometry. Then we can get Propositions 1.1.4 and 1.1.5, and it is important for this book. Section 1.2 is the spectral decomposition in real Hilbert spaces. For (real) normal operators, we need use spectral pair. And for (real) self-adjoint operators, the spectral decomposition theorem is the same as in the complex case. Chapter 2 contains the complexifications of real Banach algebras, spectrum, divisible real Banach algebras, radical, Arens products, abelian case, and etc. The complexification of a real Banach algebra can be chosen to be a complex Banach algebra, and the bar "-" operation is still isometric. The spectrum of an element must be denned in the complexification, and it is symmetric under the complex conjugation. Proposition 2.4.6 gives a basic fact (cr(x) n l = {0},Vx € R(A)), and we shall use it later. About Arens products, we have Proposition 2.6.4 and etc. Then the regularity of

Preface

vn

real C*-algebras can be obtained easily in Chapter 5. Section 2.7 is the Gelfand theory for abelian real Banach algebras. In particular, we give a systematic discussion for the general case (with or without identity). Chapter 3 is real Banach * algebras. Lemma 3.1.3 gives a basic fact ([f(A)] D AK)- About abelian case, Theorem 3.2.3 is similar to the complex case, but we must put the hermitian condition. Sections 3.3, 3.4 and 3.5 are GNS construction, * representations, and * radical. Section 3.6 discusses symmetric real Banach * algebras. In particular, the right form of Ptak's theory in the real case is given. Chapter 4 is the fundamentals of real Von Neumann algebras. It is a movement of [26, Chapter 1]. Of course, there are many differences between the complex case and the real case, for examples, [P{M)\ = MJI(C M generally), [U(M)] C M( generally, but \U(M)] = M), and etc. Proposition 4.3.3 (M c , = Mt+iM„) seems interesting and useful. Moreover, the important Von Neumann's double commutation theorem and Kaplansky's density theorem and etc. are still true in the real case. Chapter 5 is the fundamentals of real C*-algebras. We use the complexification to define real C*-algebras, and it is equivalent to the definition in [3, 17]. In section 5.3, although n-transitivity (n > 2) is not true in the real case generally, but we can still prove that 1-transitivity holds and a topologically irreducible * representation is also algebraically irreducible for a real C*-algebra. In section 5.5, we point out that any real C*-algebra is regular, and the Arens product in its bidual is the multiplication of operators indeed. The uniqueness of * operation in any real C*-algebra is obtained in section 5.6. Section 5.7 is the structure theorem of finite -dimensional real C*-algebras, and a method of proof in the theory of operator algebras is given . Section 5.8 is the enveloping real C*-algebra of a hermitian real Banach * algebra, and it is a continuity of section 3.6. Chapter 6 is real W-algebras. It is the abstraction of real Von Neumann algebras, and similarly to Chapter 5 we use the complexification to define real W-algebras. There is also an equivalent definition similar to Sakai's theorem . Section 6.3 discusses abelian real W*-algebras and theorem 6.3.6 seems very interesting . Unitaries and partial isometries in real W*-algebras are studied in section 6.4. Gelfand -Naimark conjecture is very famous, and it is a basic problem for the theory of C*-algebras. Its real analogue is studied in Chapter 7. Similarly to complex case, we have an affirmative answer for GelfandNaimark conjecture in the real case. These results seem very interesting, and in particular, Theorem 7.2.4 is remarkable. Moreover, the technique of

VU1

Real Operator Algebras

"sin" and "cos" is introduced here. Of course, compare with the complex case, there are many open questions. Chapter 8 discusses the classification of real W*-algebras. It is a real analogue of Von Neumann - Murray theory. About first classification (dimension theory), we can study it with the complexification. But for second classification, the real case is more complicated. We have semiabelian projections, semi-discrete real V7*-algebras, semi-continuous real W*-algebras and etc. Here, results are very few, and there are many open questions. Chapter 9 is a real analogue of reduction theory. The proofs of many results seem to be simple, and the important point is the systematic creation of concepts and notations. In particular, Theorem 9.4.2 seems very interesting compare with the complex case. Chapter 10 is an introduction for (AF) real C*-algebras. An equivalent analytic definition is proved, and it is more complicated than the complex case since finite -dimensional real C*-algebras have complicated structure. Moreover, more results on (AF) real C*-algebras (e.g., Bratteli diagrams, if-theory and etc.) can be found from References. Moreover, a real C*-algebra (or a real von Neumann algebra) can be regarded as a complex C* -algebra (or complex von Neumann algebra) with an appropriate conjugation (just look at the complexification) or an appropriate * anti-automorphism. In general, we may have two distinct real C* -algebras (or real von Neumann algebras) whose complexifications are the same. An easy example is the real quaternion ring H and M2(R). They are all non-commutative 4-dimensional real C*-algebras (and real von Neumann algebras), but their complexification should be the unique 4-dimensional complex C*-algebra (and complex von Neumannn algebra) M2(C) (see Chapter 10). There is a number of research work of the * antiautomorphisms on complex von Neumann algebras or complex C*-algebras by E. Stormer, P. J. Stacey, T. Giordano and etc. ([14, 56-59], also see some references in H. Schroder's book [60]). In particular, E. Stormer proved the analogue of Connes' result that a type 7/j-factor is injective iff it is hyperfinite, for real von Neumannn algebras ([58]). A complex operator algebra can be expressed as the complexification of some real operator algebra if and only if it has a * anti-automorphism. It is still a (very interesting) open question if every complex operator algebra has a * anti-automorphism, i.e. it is an open question if every complex operator algebra is a complexification of some real one. Though some special cases are known (see Stormer's paper).

Preface

IX

Finally, the author acknowledges gratefully the supports of the NSF of China and his home Institute. And the author is also very grateful to Professor Zhong-Jin Ruan for his recommendation. Bingren Li

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Contents

Preface

v

1.

Real Banach and Hilbert Spaces

1

1.1 Complexification of real Banach and Hilbert spaces 1.2 Spectral decomposition theorem in real Hilbert spaces . . .

1 9

2.

3.

4.

Real Banach Algebras

15

2.1 Definition and complexification 2.2 Divisible real Banach algebras 2.3 The topological group of invertible elements and its principal component 2.4 Radical 2.5 Functional calculus 2.6 Arens products 2.7 Abelian real Banach algebras

15 20

Real Banach * Algebras

37

3.1 3.2 3.3 3.4

Some basic lemmas Abelian real Banach * algebras Positive linear functionals and GNS construction * Representations and topologically irreducible * representations 3.5 * Radical 3.6 Symmetric real Banach * algebras

37 40 42

Fundamentals of Real Von Neumann Algebras

59

xi

21 22 25 26 28

45 48 51

xii

Real Operator Algebras

4.1 4.2 4.3 4.4

5.

6.

Banach spaces of operators on a real Hilbert space Locally convex topologies in B{H) Von Neumann's double commutation theorem Kaplansky's density theorem, tensor product commutation theorem, and comparison of projections 4.5 Positive linear functionals 4.6 ^-Finite real VN algebras

68 72 76

Fundamentals of Real C*-Algebras

77

5.1 Definition and basic properties 5.2 Positive functionals and equivalent definition of real C*algebras 5.3 Pure real states, their left kernels, and irreducible * representations 5.4 Ideals, quotient algebras and extreme points 5.5 The bidual of a real C*-algebra 5.6 The uniqueness of * operation 5.7 Finite-dimensional real C*-algebras 5.8 The enveloping real C*-algebra of a hermitian real Banach * algebra 5.9 * Representations of abelian real C*-algebras

77

Real Vf*-Algebras 6.1 6.2 6.3 6.4

7.

8.

59 61 63

83 88 98 99 104 108 110 120 123

Definition and basic properties 123 Normal linear functionals and singular linear functionals . . 130 Abelian real W*-algebras 132 Unitaries and partial isometries 141

Gelfand-Naimark Conjecture in the Real Case

147

7.1 Real C*-equivalent algebras 7.2 The closed unit ball of a unital real C*-algebra 7.3 Gelfand-Naimark conjecture in the real case

147 152 161

Classification of Real W*-Algebras

167

8.1 8.2 8.3 8.4

167 172 175 176

Classification of real W*-algebras Finite real W*-algebras Properly infinite real W*-algebras Semi-finite real VF*-algebras

Contents

9.

10.

xiii

8.5 Purely infinite (type III) real W-algebras 8.6 Properties on other classes of real VF*-algebras 8.7 Real factors and tensor products

183 184 188

Real Reduction Theory

191

9.1 9.2 9.3 9.4

191 197 200 205

Real Real Real Real

measurable fields of Hilbert spaces measurable fields of operators measurable fields of VN algebras reduction theory

(AF) Real C*-Algebras 10.1 Standard matrix unit 10.2 Technical lemmas 10.3 Definition and basic properties

211 211 218 228

Bibliography

233

Notation Index

237

Index

239

Chapter 1

Real Banach and Hilbert Spaces

1.1

Complexification of real Banach and Hilbert spaces

Let X be a real Banach space. Then Xc = X + iX becomes naturally a complex linear space. First we want to ask the question: is there a norm on Xc which makes Xc a (complex) Banach space and induces the original norm on X. The answer is affirmative, and there are infinitely many ways to do so. (1) Let

\\t + iv\\ = su P {(/(0 2 + f(v)2^ \f e x\ H/II = l},V£,v e X, where X* denotes the continuous dual of X. Clearly, (Xc, || • ||) will satisfy our requirements, and

\\Z + iv\\ =

\\t-ivl

max(||£||, |M|) < U + iv\\ < (Uf

+ IMI 2 ) 1 ,

^,V

e X.

(2) For 1 < p < +oo, let | • \v be the £ p -norm on Xc such that (Xc, \ • \p) is a real Banach space and X,iX are closed (real) linear subspaces of Xc,i.e.,

ie+*7ip = (iKiip + N r ) * ,

I
and le + ii?|oo = max(||e||>||i7||), l

2

Real Operator Algebras

V£, r) € X. Furthermore, let 11^ + iriWv = c~l sup{|e i0 (£ + irj)\p \0 € R}, V£,r7G X, where c p = sup{(|cos0| p + |sin0| p )p|0 6 R}(1 < p < +00), and Coo = 1- Then (Xc, || • || p ) will satisfy our requirements, and U + iv\\v = U-iri\\pt

Ve,^eX.

Indeed, it suffices to prove the triangle's inequality of || • || p , which is equivalent to \e"(ti + ith) + ei9(Z2 + im)\P < | e * ( 6 + itft)| P + |e*(fc + ii»)| p , V0GR,6,6,»7i,%eXLet ei — f l c o s ^ — Vism ^>

' / i = C i s m # + Vi c o s #>

£j = £2 cos 0 — 7/2 sin 0,

T/2

= £2 sin 0 + 772 cos 0.

Then we need to show

ice;+«?i) + (e 2 +»v 2 )ii, < lei+Wi\ P + ie 2 +«? 2 i P But ((Xc, I • \p) is a real Banach space, so the last inequality is obvious. Definition 1.1.1. Let X be a real Banach space, and Xc = X+iX. {Xc, || • ||) is called a complexification of X, if (Xc, || • ||) is a complex Banach space, ||-|| \X is the original norm of X(i.e.,||£+iO|| = ||f||,Vf € X), and||£ + ir7|| = |l£-"/1l,Vf,77€X. In this case, we have max(||£||, ||r7||) < ||f + J??||,V£,?? € X. In fact, lltll < |(lle + iv\\ + lie - iv\\) = lie + ^11- Similarly, ||ry|| < ||£ + ir,\\. Remark. The condition "||f + ir)\\ = \\£ - irj\\, Vf,77 € X " in the Definition 1.1.1 of a complexification is very important for our purpose (e.g., see Propositions 1.1.4, 1.1.5 and etc.). By the above discussion, we have the following. T h e o r e m 1.1.2. For any real Banach space, there is a unique (up to equivalece) complexification of it.

Real Banach and Hilbert Spaces

3

Definition 1.1.3. Let X be a real Banach space, and Xc be a complexification of X. Define bar operation— : Xc —> Xc and — : X* —> X* as follows:

T+*i = Z-iri,

/(&) = /&)>

V£,r,€X,tceXc,feX*c. Clearly, the operations "-" are conjugate linear, isometric, and —2 = id; X = {(ce Xc\(c = &}; and if / e X*c, then J = f ^ f(Q e l , V ( € X. Proposition 1.1.4. Let X be a real Banach space, and Xc be a complexification of X. (i)Iffe X* andj = / , then f\X G X*, and \\f\X\\ = ||/||. (ii) For any g 6 X*, denote gc(£ + in) = g(£) + «#(??), V£, n G X. Then gc G X*,g~c = gc, and \\g\\ = \\gc\\- In particular, if f £ X* and f = f, then

U\x)c = f. (Hi) In the sense of (ii), X* can be isometrically embedded into X*,X* = {/ e X*\f = f}, and X* = X*+iX* is a complexification of X*. Moreover, f + ig = f — ig,Vf,g € X*. Proof (i) Since /(£) e R,V£ e X, it follows that f\X e X*. Clearly, \\f\X\\ < \\f\\. Now for any e > 0, we can find £, n € X such that ||£ + M J | | < 1 ,

aDd\\f\\
+ iri)+e.

By /(£), f(v) € R, we have /(r/) = 0, and / ( £ + in) = /(£)• On the other hand, ||£|| < ||£ + ir}\\ < 1. Thus

\\f\\<m+e<\\f\X\\+e. Hence, H/ll < | | / | X | | , and H/ll = ||/|X||. (ii) It is obvious by (i). (iii) For any / e X*, we can write

/ = | ( / + 7) + *(^(/-7))Now the desired conclusion follows from (i) and (ii) and the discussion following Definition 1.1.3. Q.E.D. Remark.

Let E be a complex Banach space. We consider the following

4

Real Operator Algebras

question: Is there a closed real linear subspace X of E, such that E = X+iX is a complexification of XI Clearly, this is equivalent to the following question: Is there a conjugate linear isometry "-" on E with —2 = id? Moreover, it seems to be an open question in general. Proposition 1.1.5. Let X be a real Banach space, and Xc = X+iX be a complexification of X. Suppose that there is a (complex) Banach space Xct such that Xc = (Xc*)*, and the "-" operation in Xc is a(Xc,Xci,)continuous. Then X is a(Xc, Xct)-closed in Xc, and there is a closed real linear subspace X* of Xclr such that X = (X*)* and Xc* = X*+iX* is a complexification of X*. Proof. Since the "-" operation in Xc is <j(X c ,X c *)-continuous, and X — {€c € Xc\£c = f c } , so X is cr(X c ,X c *)-closed in Xc. For any / s Xc*, we define

/(&) = /&), V£ceXc. By the o(Xc, X c ,)-continuity of the "-" operation in Xc, and duality theory (see [26, Appendix]), we have / € Xc*- Clearly, ||/|| = ||/||. Let X* = {/ € Xc*\f = / } . Then X» is a closed real linear subspace of Xc*, and Xc* = X*+iX* is a complexification of X*. For each £ € (X*)*,£ can be naturally extended to an element of (Xc»)* = Xc. Moreover,

1(f) = W) = 1(7)= €(/), v/€*.. Thus £ = £ € X. Conversely, for each £ € X,

£(7) = f(7) = £(7) = £(/)eR, v / a „ Thus £ € (X,)*, and X = (-X,)* as real linear spaces. Now it suffices to show that

||£|| = su P {|£(/)||/eX.,||/||
||eii = suP{|e(/)| 1/ e X„, ll/H sup{|£(/)||/ £X„ ll/H <1}, V£eX.

Real Banach and Hilbert Spaces

5

For any F G X* C X*c with ||F|| < 1, we can find a net {/;} C Xc* such that H/ill < 1,W, and fi —> F in CT(X*,XC). Since F = F, it follows that ^(/i+7«)—•*,in*(*c.*c). Clearly, f f / , ! / , ) ? ! , , and | | I ( / , + / , ) | | < 1,VZ. Therefore, ||f|| =

Sup{\F(0\\FeX*,\\F\\
= suP{|C(/)| 1/ e X., H/ll < l}, V ( e l Q.E.D. For a complex Banach space E, we can naturally regard E as a real Banach space with original norm, and denote this real Banach space by Er. On the relation between E* and E*, we have the following. P r o p o s i t i o n 1.1.6. we define

Let E be a complex Banach space. For any f G E*, (Rem)

= Ref (f),

Vf G Er.

Then f —> Ref : E* —> E* is a (real linear) bijection and norm preserving, i.e., (E*)r = E*. Proof. Assume R e / = 0 for some / G E*, i.e., Re/(f) = 0,Vf G E. Then R e / ( 0 = 0, R e / (if) = 0 = - I m / ( f ) , and / ( f ) = 0,Vf € E. Hence, / = 0. Clearly, / ( f ) = R e / (f) - i R e / (if),V/ G £*,f G E. For any 3 G £r*, let / ( f ) = 5 (f) " *s(*f),Vf G E. Then / ( i f ) = i/(f), R e / (f) = 5 ( f ) , and l/(OI 2 = 9(02 +9(i02 < 2||5l| 2 ||f|| 2 ,Vf G E. Hence, / G £?*, and 5 = R e / . Clearly, ||Re/|| < ||/||,V/ G E*. Conversely, for f G E let / ( f ) = reid. Then

r=|/(f)|=/(e-ief),

||e-ief|| = ||f||,

and

Re/ (e~ief) = |/(01. Therefore, | | R e / | | > H/ll, and H/ll = ||Re/||,

V/ € E*.

Q.E.D.

Remark. Let E = F*, where F is a complex Banach space. By the preceding Proposition, f —¥ Ref is an isometric isomorphism from Er onto

6

Real Operator Algebras

F*, where Re£(/) = Re£(/),

\/£zEr,f€Fr,

and the action of Er on Fr is just the real part of the action of E on F. Definition 1.1.7. Let E, F be two Banach spaces over same field W(— R or C). Denote the set of all bounded F —linear operators from E to F by B(E,F). Moreover, if E = F, then we will write B(E,E) = B(E) simply. Proposition 1.1.8. Let X,Y be two real Banach spaces, and XC,YC be the complexifications of X,Y respectively, (i) For any a G B(X, Y), we define ac{£ + iff) = a£ + iarj,

V£,n € X.

Then ac 6 B(XC,YC), and there is a positive constant K such that ||a|| < Udell < #||a||, V a e B ( X , y ) . (ii) If we identify a with ac(Va 6 B(X,Y)), then B(X,Y) can be embedded into B(XC, Yc), and B(XC,YC)

=

B(X,Y)+iB(X,Y).

(Hi) Let a7Tib = a-ib,

Va,6 6

B(X,Y).

Then

11*11 = 11*11, iCc = Wc VteB(Xc,Yc),

£ceXc,

and "-" is conjugate linear, and —2 = id. (iv) B(X, Y) = {t e B(XC, Yc)\t = t}, and if t e B(XC, Yc), then t = t*=^Wc

= t£cMc

e l c ^ d c K

(v) If X = Y, then we also have si = si,

Vs,t€B{Xc).

Real Banach and Hilbert Spaces

7

Proof. Regard (Xc, || • ||) as a real Banach space. Then || • || ~ | • |i on Xc, i.e., there is a positive constant K such that |£ + ir]\\ = ||£|| + ||?7|| < K\\Z + iri\\, V ^ T J G X .

Hence, l|ac(e + iv)\\ = K + iav\\ < \\a\\(U\\ + IMI) < K\\a\\ \\£ + ivl Ve,?? € X, and ||a c || < K\\a\\, Va € B(X,Y). For any t € B(XC, Yc), we define 14 = *if + U2Z, where *i£,i 2 £ e Y, V£ € X. Then ti,t 2 € B(X,K), and i = tx + it2 in B(XC,YC) obviously. Moreover, ||i£ c || = \\WC\\ = fell, and ||| c || = ||&||, V£c e Xe. Thus, \\i\\ = \\t\\,VteB(Xc,Yc). Q.E.D. Remark. Generally we don't have ||a|| = ||a c ||, Va € B(X, F ) , so that B(XC,YC) is not a complexification of B(X, Y) in general. Let X be a real Banach space, and Xc be a complexification of X,a 6 B(X). If the spectrum of a is defined by {A € R|a — A is not invertible in B(X)}, then the spectrum of a may be empty. For example, X = R 2 and a = Definition 1.1.9.

2 -3

3 0

Let a S B(X). The spectrum of a is defined by

a{a) = a(ac) = {A € C\a — A is not invertible in B(XC)}. Clearly, a(a) is a non-empty compact subset of C. Proposition 1.1.10.

Let a £ B(X).

Then

a{a) = cr(a), where "-" is the complex conjugation. Proof. Let A € C Clearly, a — A is invertible in B(XC) if and only if a — A = a— A is invertible in B(XC), i.e., A ^ a(a) if and only if A ^ a(a).

8

Real Operator Algebras

Therefore, a(a) = cr(a). Q.E.D. Now we consider the case of Hilbert space. Let H be a real Hilbert space. Then Hc = H+iH becomes a complex Hilbert space if we define (t + iV, Z' + W) = «,€') + (V,v')+i(v,0

~ i«,i/>,

Vf > f'> V: v' € H, where (,) is the inner product in H. Clearly,

lie + ^ll 2 - U - "/ll 2 = U - iv\\2 = lieil2 + IMI2, ve, v e H, and the complex Hilbert space Hc is a complexification of H. Moreover, the equality ||f + ir)\\2 = ||£|| 2 + |M| 2 (V£,77 € H) does not mean that H±iH in Ac.

P r o p o s i t i o n 1.1.11. Let H be a real Hilbert space, and Hc be as above. Then B(HC) = B(H)+iB(H) is a complexification of B(H), and itcVc) = (ZcVc) = (Vc^c), V£cVc S Hc, t € B(HC), t, Vt € B(HC). Proof. For a <E B(H),

t* = ¥,

a*c = (a*)c,

and a,b € B(H),

(ab)c = acbc,

where t* is the adjoint of

£, n e H,

I M e + ^)H 2 = ll< + ia J7 || 2 = K | | 2 + I M I 2 < ||a|| 2 (||£|| 2 + |M| 2 ) = ||a|| 2 ||£ + ir,\\2. Hence, ||a c || < ||a||, and ||a c || = ||a||. By direct calculation, ||a + i6|| = ||a — i6||, Va, b € B(H). By Proposition 1.1.7, B(HC) is a complexification of B(H). Clearly, (fc, rjc) = (T/C, f c ), V£C) r\c G Hc. Now for any t € B(HC), fc, nc e Hc, {t*€c,ric) = (tctVc) =

(ZctiQ

= (tvetc) = <jic,rtc) = (vc,Wc) = ineve). Therefore F = V,\/t eB(HC).

Q.E.D.

Real Banach and Hilbert Spaces

9

Finally, let (K, (,)) be a complex Hilbert, and let

Then (Kr = K, (,) r ) is a real Hilbert space, and the norms in K and Kr are the same. Clearly, B(K) can be isometrically embedded into B(Kr) (i.e., \\a\\K = \\a\\Kr, Va e B(K)), and if a € B(Kr), then o € B(K) if and only if ait; = ia£,V£ e K. Moreover, it is well-known that B(K) = (T(K))*, where T(K) is the space of all tracial class operators on K. If t € T(K), then there exist a normalized orthogonal basis {ei\l £ A} of if and a set {A;|Z € A} of non negative numbers such that (t*t)iet

= Aiej,

VleA

and ||t||i,K- = S j A / . In this case, {ei,iei>\l,I' € A} will be a normalized orthogonal basis of Kr, and (t*t)?ei = A;ej, (t*t)*ieii = \\iei', V/,/' e A. Thus t € T(Kr) and \\t\\itKr = 2]r,A; = 2||i||i,K. By Remark following Proposition 1.1.6, we have the following diagram: B(K) I tTK T{K)

B(K)r -=-» 4- Retrx T(K)r

c C

B{Kr) I tixr T(Kr).

For any a e B(K) and t € T(tf), tr/f r .(ai) = X],(ate;,e() r + = 2Y,i(atei,ei)r

References.

1.2

^2l,(atiev,iei>)r =

2RetiK(at).

[1], [26], [27], [30], [42], [43], [50], [51], [52], [55].

Spectral decomposition theorem in real Hilbert spaces

Definition 1.2.1. Let H be a real Hilbert space, {ei(-), e 2 (-)} is called a spectral pair over C, if for any Borel subset A of C, ei(A) and e 2 (A) e B(H) and satisfy the following: (i) ej(-) is countably additive in strong operator topology of B(H), j = 1,2. (ii) ei(A)* = ei(A),e 2 (A)* = - e 2 ( A ) , V Borel subset A c C.

10

Real Operator

Algebras

(iii) ei(A) = ei(A),e 2 (A) = - e 2 ( A ) , V Borel subset A C C, where "-" is the complex conjugation, (iv) d ( A i n A 2 ) = e 1 (Ai)ei(A 2 ) - e 2 (Ai)e 2 (A 2 ), e 2 (Ai n A 2 ) - c a (Ai)ei(A 2 ) + ci(Ai)ea(A 2 ), V Borel subset A a , A 2 C C. (v) e i ( C ) - 1, el{cj>) - e a (0) = e 2 (C) = 0. Proposition 1.2.2. Let H be a real Hilbert space, and ej(A) G B(H), V Borel subset A ofC, j — 1,2. Then {ei(-),e 2 (-)} is a spectral pair overC, if and only if, in Hc = H+iH, e(-) = (ei(-) + «e2(-)) is a spectral measure ( [52]) over C and e(A) = e(A), V Borel subset A c t Moreover, i/{ei(-),e 2 (-)} is a spectral pair over C, then (1) ||ej(A)|| < 1, V Borel subset A C C, j = 1,2. (2) e i ( A ) = e i ( A ) 2 - e 2 (A) 2 , e 2 (A) = e i ( A ) e 2 ( A ) + e 2 ( A ) e i ( A ) , V Borel subset A C C. (3) For any Borel partition {A fc |l < k < m} ofC, {Afc|l < k < m} C R and ( e f f with ||£|| < 1, we have m

\\Y.\kei(Ak)t\\<

max |A*|,

j = l,2.

Proof. Clearly, (i) of Definition 1.2.1 •«=>• e(-) is countably additive in strong operator topology of B(HC). (ii) of Definition 1.2.1 <^=>- e(A)* = e(A), V Borel subset A C C. (iii) of Definition 1.2.1 <=> e(A) = e(A), V Borel subset A C C. (iv) of Definition 1.2.1 <=>• e(Ai n A 2 ) = e(Ai)e(A 2 ), V Borel subsets A i , A 2 C C. (v) of Definition 1.2.1 <==> e() = 0 and e(C) = 1. (iv) of Definition 1.2.1 = » (2) <=> e(A) 2 = e(A), V Borel subset A C C. Therefore, {ei(-),e 2 (-)} is a spectral pair, if and only if, in Hc e(-) is a spectral measure over C and e(A) = e(A), V Borel subset A c C. Now let {ei(-),e 2 (-)} be a spectral pair over C. Then e(-) is a spectral measure. In particular, e(A)|| < 1, V Borel subset A C C.

Real Banach and Hilbert Spaces

11

From Definition 1.1.1 and Proposition 1.1.11, we have ||e.,(A)|| < ||e(A)|| < 1, Borel subset A C C,

j = 1,2.

Let {Afc|l < k < m} be a Borel partition of C, {Afc|l < k < m} c R, and {, € H with ||£|| < 1. Then II E r = i Afcei(AfeKH2 + || E L i Afce2(Afc)e(|2 = II E L i A fcei (A fc )e + i E L i A fe e 2 (A,)^|| 2 = II Er=x A*e(Afc)£H2 = E L i A!l|e(A fc K|| 2 < maxi< fc < m |Afc|2 E r = i lle(A*;)£ll2 ^ maxi
||^Afcej(AfcK||<

max |Afc|,

t—* fc=l

_

Kk<m _

j = 1,2.

Q.E.D.

T h e o r e m 1.2.3. Let H be a real Hilbert space, a £ B(H) and a be normal (i.e., a*a = aa*). Then there exists a unique spectral pair {ei(-), e2(-)} over C (indeed over cr(a), noticing o~(a) = o-(a) by Proposition 1.1.9) such that a= I

Rez dei(z) — /

J a {a)

Imz de^iz)

Ja(a)

and /

Imz de\{z) = I

J tr (a)

Rez dei(z) = 0.

J a (a)

Proof. Since a is still normal in B(HC), it follows that there is a unique spectral measure e(-) over a(a) such that a= I

zde(z)

Jala) r(a)

in Hc. Moreover, a= I J a (a)

zde(z) = / J a (a)

zde(z)

12

Real Operator Algebras

since a(a) = a(a). But e(T) is still a spectral measure over a(a) and by the uniqueness, we have e(A) = e(A) or e(A) = e(A), V Borel subset A C C. Now let e(-) = ei(-) + ie2(-), where ei(-) and e2(-) € B(H). Then by Proposition 1.2.2, {ei(-),e2(-)} is a spectral pair over a(a). Clearly, a=

I

Rezdei(z) — I

J a {a)

liazde2(z),

J a {a)

and /

Imzdei(z) + /

J a(a)

Rezde2{z) — Q.

J a(a)

By ei( 7 ) = ei(-) and a{a) — a(a), we can see / Imzdei (z) = 0 Jala) Further, I

Rezde2(z) = 0

Q.E.D.

T h e o r e m 1.2.4. Let H be a real Hilbert space, and a* = a € B(H). Then there exists a unique spectral family {e\\\ e cr(a)] C B{H) such that a= I \de\ = / Xdex Jala) JR

Proof. Since a* = a in Hc and a(a) C R, it follows that there is a unique spectral family {CA|A € cr(a)} C B(HC) such that Xde\ Ja(a )

in Hc. Moreover, a = a= I

Xdex,

Jala(a)

and {CA|A € c(a)} is still a spectral family, and by uniqueness we have ex = ex € B(H),

VA e CT(O).

Q.E.D.

Real Banach and Hilbert Spaces

The proof of following theorem is similar to the complex case. Theorem 1.2.5. Let H be a real Hilbert space, and t S B(H). there is a unique polar decomposition

13

Then

t = v\t\, where \t\ = (t*t) 5 , and v is partial isometry in B(H). Notes. proof.

Theorem 1.2.3 is due to R.K. Goodrick, and here we give another

References.

[9], [18], [30], [43], [52].

This page is intentionally left blank

Chapter 2

Real Banach Algebras

2.1

Definition and complexification

Definition 2.1.1. A real algebra A is called a real Banach algebra, if it is a real Banach space and ||a6|| < ||o|| ||6||,

Va,6eA.

Any real Banach algebra A can be isometrically embedded into a real Banach algebra with identity, for example, A = A+M. with ||o + A|| = ||o|| + |A|,

VaeA,AeR.

Let A be a real Banach algebra. Now we want to ask the following question: is there a norm on Ac = A+iA (clearly Ac becomes naturally a complex algebra) such that Ac becomes a (complex) Banach algebra and keeps the original norm on A. The answer is affirmative, and there are infinitely many ways to do so. We may assume that A has an identity 1 and ||1|| = 1. For 1 < p < +oo, let || • ||p be the norm on Ac as in section 1.1. Then (Ac, || • || p ) becomes a complex Banach space keeping the original norm on A, and ||a + ib\\p = ||a — ib\\p,\/a,b G A. Further, let \\a + ib\\'p = \\La+ib\\p

= sup{||(o + ib)(c + id)\\p\c,d <E A, \\c + id\\p < 1},

Va, b 6 A. Then (Ac, \\ • \\') will satisfy our requirements. In fact, since ||1 + i0|| p = ||11| = 1, it follows that ||a + i 6 | | ^ > | | a + *6||p, 15

Va,b € A.

16

Real Operator

Algebras

On the other hand, || • ||p ~ | • |i on Ac (see section 1.1 and see Ac as a real Banach space). Hence there are constants Kp,K'p > 0 such that ^ ( N + I N ) < I | O + ^ I I P < ^ ( I H I + ||6||), Va, b € A. Further, ||a + i&||^=sup{||(ac-6d) + i(ad+&c)|| p | ||c + id|| p < 1} < Kpsup{||ac

- bd\\ + \\ad + bd\\ \ \\c + id\\p < 1}

< K'p(\\a\\ + ||6||)sup{||c|| + ||d|| \\\c + id\\p < 1} < K'pKp\\\a\\

+ ll&ll) < KpK-2\\a

+ ib\\p,

Va,& S A. Therefore, || • ||' ~ || • ||p on Ac, and (.Ac, || * || p ) is a (complex) Banach algebra. By ||c + id\\p = \\c — id||p(Vc, d G .A), we also have ||o + » 6 | | ; = | | o - t 6 | | ; ,

Va,beA.

Moreover, for any a € A \\a\\ = \\a\\p < \\a\\'p = sup{||a(c+id)|| p | ||c + id|| p < 1} = c" 1 sup|| c+id || <j sup0 £R (||accos0 — adsin0|| p + ||acsin# + a d cos0|| p )p < llall su P||c+id|| p
= H2j ajaj\aj

6

U'aj

e

C,Vj, and £V \aj\ < 1}.

Then V is an absolutely convex and absorbent subset of Ac, and V is a semi-group. Let p(-) be the Minkowski functional of V, i.e., p(x) = inf{A > 0\x € XV},

Va; € Ac.

Real Banach

17

Algebras

Then (Ac,p(-)) is a (complex) Banach algebra; V = {x € Ac\p(x) l};p(a) = ||a||,Vo G A;p(a + i6) = p(a - i6),Va, & € A; and

<

max(||a||, H&H) < p(a + ib) < 2max(||a||, ||6||), Va,&G A. In fact, for any x = a + ib £ i4 c (o, & G A), take p > max(||a||, ||6||). Then

M e t / , and 2/4

2

/i

2

^i

Hence, V^ is an absolutely convex and absorbent subset of Ac- Clearly, ^2akak

-^TPjbj

= ^2akPjakbj

6 V

H aklbj € U,ak,Pj G C,Vfc,j and £ f c l«fc| < l,Z)j l#jl ^ 1, «-e.,V is also a semi-group. If p(x) = 0 for some x € Ac, then we can find en —• 0+ and {xn} C V such that X — cnXfi,

V71.

By the definition of V, we can write xn = an + *&n; where an, bn € U,\/n. Therefore, enan —> 0,enbn —> 0, x = 0, and we can see that (A c ,p(-)) is a complex normed algebra. Clearly, x e V if p(x) < 1. Conversely, let x G V. Then we can write x — ^2jCej A > max,, ||a 3 ||. Then ^ G U, Vj, and f = £V « j i f e V. Therefore, p(cc) < A < 1, and V = {x e Ac\p(x) < 1}. Let a G A By ^ € 17 C V , p ( j ^ ) < l,Ve > 0, V n A = U, and by the preceding paragraph, we have p(a) — \\a\\. Clearly, V = V, where V — {a - ib\a,b G A, and a + i t g F } Thus, p(a + ib) = p(a - ib), Va, 6 G A. Finally, let a, b € A. Take/x > max(||o||, ||&||). Then ^ f = i - a + i - i e V, and p ( ^ j f ) < 1. Hence, p(a + ib) < 2max(||a||, ||6||). If A > 0 is such that a + ib G AV, then f + i± G V. By V C 17-Mt/, we have f, f 6 U, and ||a|| < A, ||6|| < A. Therefore, max(||a||, ||6||) < p(a + ib)<2 max(||a||, ||6||), and (Ac,p(-)) is also complete.

18

Real Operator

Algebras

Definition 2.1.2. Let A be a real Banach algebra. (Ac, || • ||) is called a complexification of A, if (Ac, || • ||) is a complex Banach algebra, || • || \A is the original norm of A(i.e.,||a + z0|| = ||a||,Va e A), and \\a + ib\\ — \\a-ib\\,Va,beA. Prom the above discussion, we have the following. Theorem 2.1.3. For any real Banach algebra, there is a unique (in equivalent sense) complexification for it. Remark. For a real algebra A, we can define the operation — : Ac —> Ac as in Definition 1.1.3. Clearly, xy = x y,Vx,y e Ac. Similarly to Definition 1.1.9, we make the following. Definition 2.1.4. Let A be a real Banach algebra, and Ac be a complexification of A. For any a S A, the spectrum of a is defined by a(a) = GAc{a) = {A € C\a — A is not invertible in Ac}, where A = A if A has an identity; A = A+R if A has no identity, and Ac = A+iA. Denote the spectral radius of a by r(o) = max{||A| |A 6 o-(a)}. Similarly to Proposition 1.1.10, we have a{a)

= CT(O),

and r ( a ) = i n f | | a n | | i = l i m | | a n | | » < ||a||, n

n

Va e A([l}). Clearly, we have the following. Lemma 2.1.5. Let B be an algebra with identity over C (or R), a, b e B and ab = ba. Then ab is invertible if and only if a and b are invertible. Proposition 2.1.6. Let Abe a real Banach algebra, a € A, and A , / i e R . Then A + i/j, G
Real Banach

Algebras

19

Proof. By Lemma 2.1.5, Definition 2.1.4, and a(a) = a(a), we have (A + in) 0 a(a) <=$> (A + in) and (A — ifx) $ a(a) «=>• a — (A 4- in) and a — (A — i^) are invertible in Ac <=>• (a — A — ifi)(a — A + in) = (a — A)2 + n2 is invertible in Ac 4=> (a - A)2 + n2 is invertible in A. Q.E.D. Now let B be a complex Banach algebra. Then A — BT is a real Banach algebra by the original norm and multiplication. Proposition 2.1.7. Let B be a complex Banach algebra, and A = Br. Then for any a € A we have aA(a) = 0-3(0) U as (a).

Proof. First we assume that B has an identity. By Lemma 2.1.5 and aA(a) = crA(a), A + in £ o-A(a), where A, n € R <$=>• X±i/j, £ aA(a) <=3~ a— (X+in) and a — (A — in) are invertible in Ac, where a— (X±in) €. Ac = A+iA <==>• (a — A — in)(a — A + in) = (a — A)2 + n2 is invertible in A c •£=> (a - A)2 + fi2 is invertible in A = Br = B -<=> (a — A — i^i) and (a — A + i/Lt) are invertible in B, where (a — A — in) and ( a - A | i / i ) € f l <=>• A ± in 0 <7s(a), i.e., 0^4(0) = CTB(O) U 0^(0),Va e A. Now we assume that B has no identity. By Proposition 2.1.6, A + in & o-A(a), where A, n € R •«=> (a - A)2 + jz2 is invertible in A = A+R. On the other hand, by Lemma2.1.5, A ± i / x ^ 0-3(0,)

4=> (a — A — i/i)(o — A 4- i/i) = (0 — A)2 + n2 is invertible in B + C , where (a — A — in) and (a — A + ijz) € B + C «=> ( a - A ) 2 + /i 2 = (a 2 - 2Aa) + (A2 +n2) is invertible in B + R = A+R. Therefore, we also have ovi(a) = <Js(a) U as(a), Va 6 A Q.E.D. References.

[1], [23], [27], [30], [42].

20

2.2

Real Operator

Algebras

Divisible real Banach algebras

Definition 2.2.1. Let A be a real Banach algebra with identity. A is said to be divisible, if for any non-zero a e A, a is invertible in A. Example 1. F = R or C with the norm of absolute value is a divisible abelian real Banach algebra. Example 2. Let H be the quaternion algebra, i.e., H is a 4-dimensional real linear space with basis {l,i,j,k} and multiplication such that 1 is the identity and i2 = j 2 = k2 = - 1 ,

ij = k =

jk = i — —kj,

ki = j — —ik.

-ji,

Let \\x\\ = (a2 + p 2 +

2 1

+ 52)k>,

Vx — a + (3i + 7 j + 5k, a, (3,7,8 £ R. Clearly, || • || is a norm on H. By xy = (aa' - (3/3' - 77' - 55') + (a/3' + Pa' + -y5' - 5-y')i + ( a 7 ' + ja' + 5P' - P5')j + {a5' + 5a' + Pi where a; = a+Pi+jj+5k,y R, we have

= a'+p'i+j'j+5'k, IM = M

jp')k,

a,nda,P,-y,5,a',P',Y,S'

e

||y||,Vx,2/€H.

Hence, H is a real Banach algebra. Moreover, for x = a + Pi + jj + (5/c(a,/3,7,<J€R) let x* = a — Pi — 7,7 — 5k. Then x*x =• xx* = \\x\\2. Therefore , M is a divisible (non-commutative) real Banach algebra. Theorem 2.2.2. Let Abe a divisible real Banach algebra. Then there exists an algebraic isomorphism T from A onto D such that \\Ta\\ = r(a), Va £ A, where D = R, C or H.

Real Banach

Proof.

Algebras

21

See [1].

Moreover, in the abelian case we have a simple proof as follows. Let A be a divisible abelian real Banach algebra. If there is a £ ^4\R, then a has no real spectrum. Let a + i@ e cr(a), a, /3 e R. Then (3 ^ 0. By Proposition 2.1.6, (a — a ) 2 + /32 is not invertible in A. Since A is divisible, it follows that ( a - a ) 2 + / 3 2 = 0. Let b = (a - a)//3. Then 62 = - 1 , and a = a + /36. If there is another b' £ A such that b'2 = — 1, then 0 = b2-b'2

= (b -b')(b

+b').

Since A is divisible , it follows that b' = b or -b. Moreover, b=(a-a)/0€ A\R. Therefore, A = {A + fj,b\A,\i € R} = C. Q.E.D. References. [1].

2.3

The topological group of invertible elements and its principal component

Let A be a real Banach algebra with identity. Denote the subset of all invertible elements of A by G = G(A). Clearly, G is a topological group in the norm topology of A. Denote the connected component containing the identity of G by G0 = G0(A). Similarly to the complex case, we have the following. Theorem 2.3.1. Let A be a real Banach algebra with identity, G = G(A) and Go — Go (A). Then (i) Go is a clopen normal subgroup of G; any connected component of G has following form: aGo = Go a (some a s G); and G/GQ is discrete in the quotient topology. (ii) GQ = {eai • • • ean\ai 6 A, In particular, Go is also path connected.

\
Vn}.

22

Real Operator

Algebras

Now let A be a real Banach algebra with identity, and Ac be a complexification of A. Clearly, G{A) = G{AC) n A,

Go(A) c G0(AC) n A,

and (a;) - 1 = (a:_1),Va; G G(-Ac). Naturally we want to ask whether G0(A) =

G0(Ac)nA?

Generally, the answer is negative. A typical example is as follows. Let A = R. Then G(R) = R _ 1 = R\{0} has two connected components: {A G R|A > 0} and {A G R|A < 0}; and G 0 (R) = {A G R|A > 0}. But Ac = R-i-iR = C, and G 0 (C) = G(C) = C\{0} is connected. Further, G 0 (C) n R = R\{0} = G(R) D G 0 (R). References. 2.4

[1], [27], [42],

Radical

Definition 2.4.1. Let A be a real Banach algebra. A (real) linear subspace J of A is called a left (right) ideal of A , if J ^ A and A J c J(JA C J ) . A left (right) ideal J of A is said to be regular, if there is u G A. such that a — au G J (a — ua G J),Va G A. This u is called a modular unit for J. Clearly, any left (right) ideal of A is regular if A has an identity. Similarly to the complex case, we can prove that any maximal regular left (right) ideal of A is closed. Definition 2.4.2.

Let A be a real Banach algebra.

R{A) = fl{L\L

maximal regular left ideal of A}

is called the radical of A. If R(A) = {0}, then A is said to be semi-simple. Similarly to the complex case, we have the following. Proposition 2.4.3. Let A be a real Banach algebra. Then (i) R{A) = n{i?|.R maximal regular right ideal of A}, and R(A) is a closed two-sided ideal of A. (ii) R(A) = R(A).

Real Banach

(Hi) A/R is a semi-simple where R = R(A).

23

Algebras

real Banach algebra with quotient

norm,

Theorem 2.4.4. Let A be a real Banach algebra. Then for any a £ A the following statements are equivalent: (i) a € R(A). (ii) 1 + ba has a left inverse in A, V6 € A. (Hi) 1 + ba is invertible in A, V& e A. (iv) 1 + ab has a right inverse in A, V6 € A. (v) 1 + ab is invertible in A,Vb 6 A. Lemma 2.4.5. Let Abe a real Banach algebra, Q^p2=p€A,B and B ^ A. Then oB(b)u{0}

= oA(b),

— pAp

V6 € A.

Proof. We may assume that A has an identity 1. Clearly, p ^ 1, B c = B+iB = pAcp, B has the identity p, and trB(b) = aBe(b),

aA{b) = aAc{b),

V6 € B.

Let A 0 aA(b), and c = (6 - A) _ 1 (e A c ). Then p = pc(b — X)p = pep • (b — Xp) — p(b — X)cp — (b — Xp) • pep, i.e., X & crB(b). Thus, aB(b) C aA(b). Let 0 / A £ crB(b), (b - Xp)'1 = c(e B c ), and d = Ac + p(€ B c ). Then (d - p^X-ib

-p) = (X-H - p)(d

-p)=p,

X~*bd - A _1 6 -d = X-rdb - \-*b - d = 0. Thus, (1 - A 6)(l - d) = (1 - d)(l - A _1 6) = 1, i.e., A £ o^fr). Now it suffices to show that 0 € aA{b). Otherwise, b has an inverse b'1 <EA. Then _1

p = p{bb~l) = pb • fc-1 = bb-1 = 1. It is impossible.

Q.E.D.

24

Real Operator Algebras

Now we notice the following important facts. Propostion 2.4.6 Let A be a real Banach algera. Then (i) a{a) RR = {0},Va 6 R(A). (ii) a(x)u{0} D o-(£)U{0} and (a(x)r\R)L){0} = (<7(x)nR)U{0},V:r 6 A, where x = x + R € A/R and R = R(A). Proof, (i) We may assume that A has an identity. By Theorem 2.4.4, 1 + 6a is invertible in A,Wb e A. In particular, 1 + \ia is invertible in A,\fO ^ \i € R, and a{a) D (R\{0» = cf>. Moreover, 0 € a{a). Therefore, < r ( a ) n R = {0},Va€ R{A). (ii) First we assume that A has an identity. If A + in & cr(x), A, /i € R, then (x — A)2 + n2 is invertible in A (see Proposition 2.1.6). It follows that (x — A)2 +/j? is also invertible in A/R. By Proposition 2.1.6, X + i/j, £ a(x). Thus a(x) D a(x),

Vx e A.

Let A s R and \$. a{x). Then there exists y e A such that (x - X)y = y(x - A) = I , i.e., u = 1- (x - X)y and t; = 1 - y(:r - X) e R. By Theorem 2.4.4, 1 - u and 1 — v are invertible in A. By (x - A)y(l - u ) " 1 = 1 = (1 - v^yix

- A),

x — X is invertible in A, i.e., X # o(x). Thus a{x) n R = cr(i) D R,

Vx € A.

If A has no identity, maybe A/R has an identity. Now by Lemma 2.4.5, we get the conclusions for the general case. Q.E.D. Remark. For any complex Banach algebra B, we have o{b) = {0}, V6 e R(B) and
Real Banach

25

Algebras

Proof. (i) If a £ B, then 1 + xa is invertible in Ac, Vz G Ac. In particular, 1 + 6a is invertible in A,Vb 6 A. By Theorem 2.4.4, a € i?(A). If Ic is a maximal regular left ideal of Ac, and u e Ac is its modular unit, then J c is also a maximal regular left ideal of Ac with modular unit u. Thus R(AC) = R(AC) and i?(A c ) = B+iB. (ii) Let || • ||' be a norm on A such that (A, || • ||') is also a Banach algebra, and (Ac, || • ||') be a complexification of (A, || • ||')- Since A is semi-simple,

it follows from (i) that Ac is also semi-simple. Now by Johnson theorem ([1]), II • II' ~ II • II o n Ac-

In

particular, || • ||' ~ || • || on A.

Q.E.D.

Remark. It is interesting to ask: Whether R(AC) = R(A)+iR{A)? the abelian case, it is affirmative (see Proposition 2.7.6). References.

2.5

In

[1], [27], [30], [42].

Functional calculus

Let A be a real Banach algebra with identity, a G A, and Ac be a complexification of A. Denote -ff(°) — {/I/ analytic in a neighbourhood of

a(a)}.

Then for any / € H(a),

f(") =

^-iJrf(z)(a-zr1dzeAc,

where T is any smooth simple closed curve in an open neighbourhood U of CT(O) (/ analytic in U) and T is enclosing a(a). Naturally we want to ask: when does / ( a ) e A? By a{a) — a (a), we

caa tote T = T e U = U. Ii f (z) = 7lz),^z & U, then

i.e., f(a) € A. = ^-iJrf(z)(a-z)-'dz

= f(a),

26

Real Operator Algebras

Generally, for / G H(a) W)

= ^l!TW){a-z)-'dz=

^^giz^a-z^dz

= ^-iIr9(z)(a-z)-1dz where g(z) = f(z),Vz

= g(a),

G U, and g G H(a). Moreover, f(a) = b + ic,

where b = \{f{a) + g(a)) G A, and c = ^ ( / ( a ) - g(a)) G A. References.

2.6

[1], [27].

Arens products

Let A be a real (or complex) Banach algebra, A* dual of A, and A** bidual of A. For any a e A,f € A*,m G A**, we define (/o)(6) = f(ab),

(af)(b) = f(ba),

(mf)(b) = m(/6),

(/m)(6) = m(6/),V6 G A.

Then it is easy to see that fa,af,mf,

fm G A*, and

||/a|| < ll/H ||a||, ||a/|| < ||a|| ||/||,

l|m/||<|H|||/||,||/m||<||/||||m|lMoreover, if m = a G A (canonical embedding A '-> A**), then mf af,fm = fa,Wf€A*. Definition 2.6.1. as follows:

The first and second Arens products in A** are defined (mn)(f) (m-n)(f)

V/

=

=

m(nf),

=

n(fm),

€A*,m,n€A**. It is easy to see that A** will be a real (or complex) Banach algebra with first or second Arens product, and A is a real (or complex) Banach subalgebra oi A**.

Real Banach

27

Algebras

Definition 2.6.2. A real (or complex) Banach algebra A is said to be regular, if mn = m-n, Vro, n G A**. Proposition 2.6.3. Let A be a real (or complex) Banach algebra. Then the following statements are equivalent: (1) A is regular. (2) For any fixed m 6 i**, the map x—+mx:A**-^

A**

is o(A**, A*)-continuous (clearly, the map x —> xm : A** —> A** is automatically a(A**, A*)-continuous). (3) For any fixed n G A**, the map £ —> z • n : A** —+ A** is a(A*'*, A*)-continuous (clearly, the map x —> n • x : A** —> A** is automatically a(A**, A*)-continuous). Proof. By Definition 2.6.2, (1) ==> (2) is obvious. (2) = • (1). By assumption, for any / G A* and m € A**, (m-)(f) will be a a{A**, A*)-continuous linear functional on A**. Then by [26, Appendix], there is a linear map Tm : A* —> A* such that mn(f) = n(Tmf), Now (T r o /)(a) = a(Tmf) A, i.e.,Tmf = fm. Thus

=

(mo)(/)

VeA**. =

m(af)

{mn)(f) = n{fm) = (m • n ) ( / ) ,

-

(fm){a),Va

6

V/ e A*

or mn = m • n,Vm, n G A**, and A is regular. The proof of (1) <^=> (3) is similar to that of (1) <=> (2).

Q.E.D.

Let A be a real Banach algebra, and Ac ~ A+iA be a complexification of A as Banach algebra. By Proposition 1.1.4, A* = A*+iA* and A** = A**+iA** are the complexifications of A* and A** as Banach spaces respectively. The "-" operations in A* and A** are induced by the "-" operations in Ac and A* respectively (see Definition 1.1.3). Moreover, we also have first and second Arens products in A**. Clearly, the first and second Arens products in A**, are the natural extensions of the first and

28

Real Operator Algebras

second Arens products in A** to A** respectively. Therefore, we have the following. Proposition 2.6.4. Let A be a real Banach algebra, and Ac = A+iA be a complexification of A (see Definition 2.1.2). Then A is regular, if and only if, Ac is regular. Moreover, in this case Arens product in A** is the natural extension of Arens product in A**. References.

2.7

[4], [30].

Abelian real Banach algebras

Let A be an abelian (commutative) real Banach algebra, Ac = A+iA be a complexification of A, and fic be the spectral space of Ac. Definition 2.7.1. fl — il(A) = {p\A\p £ fic} is called the spectral space of A. In other words, fi is the set of all non-zero complex valued multiplicative (real) linear functionals on A. The "-" operation in Ac can be transferred onto il, i.e., we can define p~(a) = p(a),

Va e A,

or p(x) = p(x),

Vz € Ac,

Vp€$l Theorem 2.7.2. Let A be an abelian real Banach algebra, and ft be its spectral space. Then (1) p is continuous on A, and \\p\\ < 1 (notice p# A* generally), Vp € $1 Moreover, p = p <£=> p € A* <^=^ p(A) C R. (2) Q is a locally compact Hausdorff space in the topology of pointwise convergence on A, and il is compact when A has an identity. (3) p —> p(Vp € ft) is a homeomorphism of fi with period 2. (4) o~(a) = {p(a)\p € fi} (when A has an identity), or a(a) — {0} U {p{a)\p € fi} (when A has no identity), andr(a) — sup{|p(a)| \p € fi},Va € A.

Real Banach

Algebras

29

(5) The Gelfand transformation a —> a(-) is a homomorphism from A into C0(fl,—) (as real algebras), where a(p) = p(a), Va G A, p G fi, C0(fJ) = {/|/complex valued continuous function on Cl, and vanishes at oo}, and C 0 ( f i , - ) = { / € Co(fi)|/(p) = 7(p), Vp G n } . Proof.

By the definition of ft, they are obvious.

Q.E.D.

Now we consider the relation between U and the set of all maximal regular ideals of A ( see Definition 2.4.1). First, we assume that A has an identity. If p G ft and p = p, then p € A*, and A = J-i-R, where J = { a € A|p(a) = 0} is a maximal ideal of A. Clearly, Ac = Jc+C, and 7C = J+iJ is also a maximal ideal of Ac. If p G fi and p^p~, then p(-) can take non-real value on A. Since p(l) = 1, it follows that there is v £ A such that p(v) = i. Clearly, p(l + v2) = 0. Then A=

J+[l,v],

where J = {a G A|p(a) = 0}, [l,u] = {a + 0v|a,/3 G R}. Thus A/ J S C, and J is a maximal ideal of A (notice that C contains no any non-trivial ideal). Conversely, let J be a maximal ideal of A. Similarly to the complex case, we can show that J must be closed and A/J is divisible. By Theorem 2.2.2, we have A/ J S R o r C , i.e., A-

J + R o r A = J+[l,v]

where 1 + v2 G J. If A = J-i-R, then Ac = J c 4-C and J c = J+iJ is also a maximal ideal of Ac- Further, there is a unique p — ~p £ fl such that J ={a€ A\p{a) = 0}. UA = J+[l,v], then Ac = ( J c + C ( l + iw))+C

= (J c +C(l - w))+C

30

Real Operator Algebras

Since v(l ± iv) = (=R')(1 ± iv) ± i(l + v2) and (1 + v2) G J, it follows that (J c -i-C(l + iv)) and (J c 4-C(l — iv)) are two maximal ideals of Ac, and their intersection is J c = J+iJ. Thus Jc is not a maximal ideal of Ac, and there is p G fi such that p ^ p and J = {a e A\p(a) = 0} (indeed, p(a + a + (3v) = a + j3i,Va e J,a,(3 € R). Now let a € ft be such that J = {a G A|
= p}

and {J\J maximal ideal of A, and A/J = R} such that J — {a € A\p(a) = 0},A = J+R, and Jc = J+iJ maximal ideal of Ac = Jc+C (2) There is a bisection between

is also a

{{p,p}|pS ft, and p + ~p} and {J\J maximal ideal of A, and A/J = C} such that J— {a € A\p(a) = 0},A = J + [ l , u ] , where [l,v] = {a + /3v\a, (3 € R}, p(v) = i, 1 + v2 G J, and v + J is unique in A/J. Moreover, in this case, Jc = J+iJ is not a maximal ideal of Ac, and Jc = IPC\ Ip, J = Ip <11-p fl A, where Ip - J c + C ( l + iv) - {x G Ac\p(x) = 0} and Ip = lp = J c + C ( l iv) = {x G Ac\p~(x) = 0} are two maximal ideals of Ac. Now suppose that A has no identity, and A = A+R. Notice the following. Lemma 2.7.4.

There is a bijection between {J\J

maximal regular ideal of A}

31

Real Banach Algebras

and {I\I maximal ideal of A, and I ^= A} such that I = J-i-R(l — u), where u € A is a modular unit for J (i.e., a — au £ J,\/a e A). Moreover, for any maximal regular ideal J of A the modular unit u is unique up to the addition of an element of J, and A/J s A/I, J = InA. Proof. Let J be a maximal regular ideal of A, and u, u' be two modular units for J. Then u — uu' and u' — u'u € J. Since uu' = u'u, it follows that (u — u') € J. Let J be a maximal regular ideal of A, u be a modular unit for J, and I = J-i-R(l - u). Clearly, I is an ideal of A, I±A and InA = J. If V is an ideal of A and I C / ' , then I'r\A = Ir\A = J. Indeed, if A n V D J, then I' n A = A and u e V. Since 1 - u e J C / ' , it follows that 1 e / ' and 7' = A. This contradicts V ± A . Thus / ' f l A = / n i = J. Now let a; = a + X € V, where o e i , A e K . Then a + Xu = x - A(l - u) e 7' n A = J, x = (a + AM) + A(l — u) £ I, I' = I, and I is maximal. Let I be a maximal ideal of A, I ^ A, and J = I C\ A. Clearly, J is an ideal of A. Since I ^ A, it follows that there is 1 — u e /\^4, where u £ A. By a — au — a{\ — u) e In A = J,

Va e A,

J is a regular ideal of A with a modular unit u. Similarly to the preceding paragraph, we can show I = J-j-R(l — u). Let J' be an ideal of A, and J' D J. Clearly, u is also a modular unit for J', and I' = J'-i-E(l — u) is an ideal of A. Thus J-f R(l - u) = I = I' = J'-i-R(l - u), J' = J, and J is maximal. Finally, let J be a maximal regular ideal of A,t*bea modular unit for J, and I = J-j-R(l — u). Clearly, a + J —> a + 7(Va € A) is a homomorphism from A/ J to A/I. Since u + I = 1 + I, it follows that this homomorphism is surjective. Moreover, if a + I = I for some a & A, then a € 7 n A = J and a + J = J. Therefore, A / J £* A/J. Q.E.D. If p € fi and p — ~p, then p € A* and there is u e A such that p(u) = 1,

32

Real Operator

Algebras

and A = J+Ru, where J = {a e A\p(a) = 0}. Clearly, J is a maximal regular ideal of A. and u is a modular unit for J. Moreover, Jc = J+iJ is also a maximal regular ideal of Ac = Jc+Cu. Let p £ Q, and p ^~p. p can be naturally extended to A, and p ^ p on A obviously. By Theorem 2.7.3 and Lemma 2.7.4, we have A/ J S A/I = C, where I — {x £ A\p(x) = 0}(^ A) is a maximal ideal of A, J = {a £ A\p(a) = 0} = I n A is a maximal regular ideal of A. Since A/ J = C, it follows that there are u,v € A such that A/J= [u,v\,u is the identity of A/ J, and u + v2 — 0, where u = u + J,u = i; + J, 0 = J. Clearly, u is a modula unit for J. As p ^ 0 on .4, p(u) = 1. Moreover, we may assume that p(v) = i. Then A = J+[u,u], where [u,v] = {au + f3v\a, 0 6 K}. By Theorem 2.7.3 and Lemma 2.7.4, we have 7 = J-i-R(l -u),A = I+[l, v}. Conversely, let J be a maximal regular ideal of A, and u be a modular unit for J. By Lemma 2.7.4, I = J-i-R(l - u){^ A) is a maximal ideal of A, u = u + J is the identity of A/ J, and

A/J^A/I^R

or C.

A = J+Ru,

A = I+R.

If A/J 2 i / i " S R, then

Further, there is a unique p = ~p € Q, such that J = {a £ A\p(a) = 0} and p(u) = 1. If A / J = C, then there is v £ A such that A/ J = [u, v\, u+v2 = 0. Thus A-J+[u,v],

u+

v2£j,

and ^ = (J c +C(u + iu))+Cu = (J c 4-C(u -

iv))+Cu.

Real Banach

Algebras

33

Since u(u±iv) = (u±iv) + (u2 — u)~^i{v-vu) andi>(u±if) = —(v — vu)± i(u+v2)+(?pi)(u±iv), it follows that Jc+C(u+iv) and Jc+C(u-iv) are two maximal regular ideals of Ac, and their intersection is Jc = J+iJ. Thus Jc is not a maximal regular ideal of Ac, and there is p G fi such that p ^ ~p and J = {a € A|p(a) = 0} (indeed p(a + au + [3v) = a + /ft, Va G J, a, (3 G R). Now let a G fi be such that J = {a € ^4|cr(a) = 0}. Since (a — au) G J, Va G A, andCT^ 0 on A, it follows that a(u) = 1. By u + v2 G J, o-(u) = i or - i , i.e., a = p OT~p. From the above discussion, we have the following. Theorem 2.7.5. spectral space.

Let A be an abelian real Banach algebra, and fl be its

(1) There is a bisection between {p G n\p = p} and {J\J maximal regular ideal of A, and A/J = R} such that J — {a e A\p(a) = 0},A = J+Ru,p(u) — 1, where u G A is a modular unit for J, and u + J is unique in A/J, and Jc = J+iJ is also a maximal regular ideal of Ac = Jc-i-Cu. (2) There is a bisection between {{p,p}\pen,

andp^p}

and {J\J

maximal regular ideal of A, and A/J = C}

such that J = {a G A\p(a) = 0}, A = J+[u,v],p(u) = l,p(v) = i, where u G A is a modular unit for J, [u, v] = {au + (3u\a, j3 G R}, u + v2 G J, and u + J,v + J are unique in A/J. Moreover, in this case, Jc = J+iJ is not a maximal regular ideal of Ac, and Jc = IPC\ Ip, J — Ip D I-p D A, where IP = Jc+C(u + iv) = {x G Ac\p{x) = 0},lp = Ip = Jc+C{u - iv) = {x G Ac\~p{x) = 0} are two maximal regular ideals of Ac. Proposition 2.7.6.

Let A be an abelian real Banach algebra. Then

R{AC) = R(A)+iR{A),

and R(A) = {a G A\r(a) = 0}.

34

Real Operator

Algebras

Proof. By Theorem 2.7.5 and Definition 2.4.2,

R{AC) n A = n{ip n A\p e n} = n{ipnipr\A\pefi} = n{ J\ J maximal regular ideal of A} — R(A). Now by Proposition 2.4.7, we have R(AC) =

R(A)+iR(A).

Moreover, since R(AC) = {x € -Ac|r(a;) = 0} (a result in the complex case), it follows that R{A) = R{AC) n A = {a e A\r(a) = 0}.

Q.E.D.

Now let B be an abelian complex Banach algebra. Then A = Br is an abelian real Banach algebra by the original norm and multiplication. Let ft(B), Q(A) be the spectral spaces of B, A respectively. By Definition 2.7.1, we have a natural embedding fi(B) «-> Sl(A). We point out the following. Theorem 2.7.7. A = Br. Then

Let B be an abelian complex Banach algebra, and

fl(A) = n(B)un(B), where "-" operation is in Cl(A) (see Definition 2.7.1), and under the embedding fl(B) <-* Q(A),Q(B) and Cl(B) are two disjointed clopen subsets of Q(A). Proof.

Let p € U(A). For any a,be A, we have p{iafp(b)2

= p{ia • bf = p(-a2b2)

-

-p{a)2p(bf.

Taking b 6 A such that p(b) ^ 0, we can see p(ia)2 - -p(a)2,

Va € A.

Further, p{ia) — ip(a) or -ip(a),Wa G A. If there are oi, 02 6 A such that p(iai) = ip{ai),

p(ia2) =

-ip{a2)

Real Banach

35

Algebras

and p(ai) ^ 0,p(a2) ^ 0, then p(ai)p(a2) = ip(ai) • -ip{a2) - p(iai • ia2) =

-p(a1)p{a2).

This is impossible. Thus p(ia) — ip(a),Va G A, or p(ia) = —ip(a),Va € A, i.e., p is complex linear or conjugate complex linear. Therefore, p e Q,(B) or p 6 fi(S), and Q(A) = Sl{B) U fi(B). Now it suffices to show that Q(B) is an open subset of Cl(A). Let tp e Q(B), and a € A,e > 0 be such that | £. Consider the open neighbourhood U(
& = a or ia}.

Then p(a) ^ 0,Vp € U( |p(ia) - (ia)| = |p(a) + 2\ 2\
[23], [27], [30].

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Chapter 3

Real Banach * Algebras

A is called a real Banach * algebra, if A is a real Banach algebra, and there is a (real) linear operation * in A such that a** = (a*)* = a,

(ab)* = b*a*, Va, b € A.

Define the extension of * : A —> A to * : Ac —> Ac naturally. Then Ac = A+iA is a complex Banach * algebra by Theorem 2.1.3. 3.1

Some basic lemmas

Lemma 3.1.1. Let A be a real Banach * algebra, and B be a maximal abelian * subalgebra of A. Then Bc — B+iB is a maximal abelian * subalgebra of AC,B is closed in A, and CTB(6) U {0} = 0-^(6) U {0},V6 € B. Proof. Clearly, B'c = B'+iB' = B+iB = Bc, where B'c, B' are commutants of Bc, B in Ac, A respectively. Thus Bc is also maximal abelian in Ac. Now from the complex case we can get the other conclusions immediately. Q.E.D. Lemma 3.1.2. (Ford lemma in the real case) Let A be a real Banach * algebra with identity, and h* — h € A. (1) Ifr(l — h)
«=E(i)(*-Dn = i s / r ^ f c - ^ where the series X^n>o''' *s absolutely convergent in norm, F is a smooth 37

38

Real Operator

Algebras

simple closed curve in {z € C| \z — 1| < 1} enclosing cr(h), and T — T. (2) If h > 0 (i.e., h* = h and a(h) C (0,+oo)), then there is unique u & A such that u > 0 and u2 = h. (3) IfRea{h) > 0, then there is unique u* = u € A such that Rea(u) > 0 and u2 = h. Proof. (1) By functional calculus (see Section 2.5) or the convergence of the series, we have u G A and u2 = h. Now by Ford lemma ([12]) in Ac,u* = u and r ( l — u) < 1 obviously. (2) and (3). We can find a positive integer n such that r ( l — £) < 1. Then the conclusions are obvious. Q.E.D. Lemma 3.1.3. Let A be a real Banach * algebra with identity, U(A) = {u e A\u*u = uu* = 1} (the subset of all unitary elements in A), and AK = {k € A\k* = — k} (the subset of all skew-hermitian elements in A). Then \U(A)] D AK, where [U(A)] is the (real) linear span ofU(A). Proof.

Let k e AK and r(k) < 1. Then r ( l - (1 + fc2)) = r(k)2 < 1.

By Lemma 3.1.2, there is h* = h G A such that h2 = 1 + k2, and h indeed is the limit of real polynomials in 1 + fc2. Thus hk = kh. Now let u± = k ± h. Clearly, u± € U(A), andfc= \{u+ + w_) e [U(A)].

Q.E.D.

Remark. If B is a complex Banach * algebra with identity, then the (complex) linear span of U(B) is B , i.e., \U(B)] = B. But this is not the case of real Banach * algebras in general. Lemma 3.1.4. If A is a semi-simple real Banach * algebra, then the * operation on A is automatically continuous in norm. Proof. By Proposition 2.4.7, Ac is also semi-simple. By Johnson's theorem ([1]), the * operation is automatically continuous on Ac, hence on A in norm. Q.E.D.

39

Real * Algebras

Now by Definition 2.4.2, Proposition 2.4.3 and Lemma 3.1.4, We have the following. Corollary 3.1.5. Let A be real Banach * algebra, and R = R(A) be its radical. Then R is a closed * two-sided ideal of A, and the * operation is continuous on A/R. Now let A be a real Banach * algebra. For any / £ A*,m 6 A**, we can define f*(a) = f(a*),

m*(f)=m(f*),

Va 6 A.

It is easy to show the following. Lemma 3.1.6. The * operation on A* and A** are (real) linear and with period 2. Moreover, m* = a* if m = a G A, and (af)* — f*a*,(fa)* = a*f*, (mf)* = f*m*, (fin)* = m*f* (see Section 2.6). In order to make A** into a real Banach * algebra with the first or second Arens product (see Definition 2.6.1) and * operation defined above naturally, we must check that ( m n ) ' = n*m*, or (m • n)* = n* • m*, Vm.ner. Let A be regular (see Definition 2.6.2). Then (mn)*(f)

= (m • n)(/*) = n(rm)

= n*{m*f) = n*m*{f),

V/ € A*. Thus (mn)* = n*m*, Vm,n e A**. Conversely, let (mn)* = n*m* or (ra • n)* = n* -m*, Vra, n e ^4**. Then (mn)*(f)

= (n*m*)(f)

= n*(m*f) = n(f*m) = (m • n)*(f),

or (m • n)*(f) = (n* • m*)(f) = m*(fn*) = m(nf*) = V/ G A*, i.e., mn = m • n,\/m, n G A**, and A is regular. By the above discussion, we have the following.

(mn)*(f),

40

Real Operator Algebras

Proposition 3.1.7. Let A be a real Banach * algebra. Then A** is a real Banach * algebra by the first or second Arens product and the * operation, if and only if, A is regular. Remark. This Proposition is also valid for any complex Banach * algebra B. Of course, we must define f*(a) = f(a*),m*(f) = m(f*),\/a e

B,f

eB*,meB**.

References.

3.2

[1], [12], [23], [27], [30], [36].

Abelian real Banach * algebras

Let A be an abelian real Banach * algebra, and ft be its spectral space. Now in addition to the "-" operation on ft (see Definition 2.7.1), we also have the * operation on ft, i.e., p*{a) = p{a*),

Mae A,

p 6 ft.

Clearly, * operation is also a homeomorphism of ft with period 2. Note that "— = id" on ft is equivalent to the Gelfand transform a(p) € R(Va € A,p e ft); what about "* = id" on ft? Lemma 3.2.1.

Let A be an abelian real Banach * algebra, and r(a)2 =

r(a*a),\/a € A. Then A is skew-hermitian,

i.e., a(k) C iR,Vfe € Ax-

Proof. Suppose k* = — k € A, and 1 + ip, G 0. Then

p(a) =

(\+l+in)mp(k).

Clearly, 0 < 1 + p? = \p(k)\2 < r(k)2 = r{k*k). Also [(A + l ) 2 + p2]m\p(k)\2 = r((A 2 - k2)mk*k)

= \p{a)\2 < r(a)2 = r(a*a) < r{\2 -

k2)mr{k*k).

Taking mth root and letting m —> +oo, we have

(A + l) 2 + ^ < K A 2 - f c 2 ) < A 2 + r(fc)2

Real Banach * Algebras

41

or 2X + 1 +fj,2
VAX).

It is impossible. Therefore, A is skew-hermitian.

Q.E.D.

Definition 3.2.2. Let A be a real Banach * algebra. A is said to be hermitian, if a(h) C R, V7i e AH = {a 6 A\a* = a } , A is said to be skew-hermitian, if a(k) C iR,\/k € AK = {fl£ A\a* = —a}. Theorem 3.2.3. Let A be an abelian real Banach * algebra. Then the following statements are equivalent: (1) A is hermitian and skew-hermitian; (2) A is hermitian and skew-hermitian; (3) Ac is hermitian; (4) * = id on fi, i.e., p* — p,\/p G fi; (5) each maximal regular ideal of A is *-closed; (6) the Gelfand transform Y : A —> Co(fl, —) is a * homomorphism, where / ' ( • ) = 7 0 ) . V / G C 0 ( n , - ) ; (7) A is hermitian and r(a)2 = r(a*a),Va 6 A. Proof. The equivalences of (1), (2) and (3) are obvious. (3) -$=> (4) follows from the complex case ([1]). By Theorem 2.7.5, we have the equivalence of (4) and (5). (4) •«=» (6) is obvious. (3) =*> (7) follows from complex case ([1]). (7) = > (1) follows from Lemma 3.2.1. Q.E.D. Corollary 3.2.4. Let A be an abelian, hermitian and real Banach * algebra. Then

skew-hermitian

R(A) = {a e A\r(a) = p(a) = 0} = p _ 1 (0) = R{AC) n A, and R(AC) -

R{A)+iR{A),

where p(a) = r(a*a)?,Va € A. Proof. By Theorem 3.2.3, r(-) = p(-) on A. Now from Proposition 2.7.6, it is obvious. Q.E.D. References.

[1], [27], [30], [36], [40], [41], [53].

42

3.3

Real Operator

Algebras

Positive linear functionals and G N S construction

Definition 3.3.1. Let A be a real Banach * algebra, and / ( : A — • M) be a (real) linear functional on A. f is said to be hermitian, if /* = / , i.e., f(a*) — f(a),Va € A, or f\AK = 0. / is said to be positive, denoted by / > 0, if f(a*a) > 0,Vo € A. Remark. In the complex case with identity, / > 0 implies that / is hermitian. This is no longer valid in the real case. Proposition 3.3.2. Let A be a real Banach * algebra, / > 0 and hermitian on A. Then fc > 0 and hermitian on Ac,fb(-) > 0 and hermitian on A, and /&(•) e A*, and f(b*a)2 < f(a*a)f(b*b)

(Schwartz inequality),

\fb(a)\ < f(b*b)p(a), where fb(-) = /(&* • b),p(a) = r(a*a)* ,Vo, b € A. Proof. Clearly, fc > 0 and hermitian on Ac. Then other conclusions follow from the complex case ([1]). Q.E.D. Definition 3.3.3. Let A be a real Banach * algebra, {TT, H} is called a * representation of A, if H is a real Hilbert space, and 7r is a * homomorphism from A into B(H). Since {-K, H} can be naturally extended to a * representation {irc, Hc} of Ac, it follows that 7r: A —> B(H) is continuous and ||7r(o)|| 0 and hermitian on A, and Lf = {a e A|/(o*a) = 0} be the left kernel of / . By Schwartz inequality, Lf is a left ideal of A. Define an inner product (,) on A/Lf as follows: (a, b) = f(b*a),

43

Real Banach * Algebras Vo € a = a + Lf,b € b = b + Lf G A/Lf. Then its completion Hf (A/Lf, (,))"" is a real Hilbert space. For any a € A, let

=

TTf(a)b — ab, V6 € A/Lf.

By Proposition 3.3.2, we have ||7r/(a)6||a = f(b*a*ab) =

fb{a*a)

0 and hermitian on A. Then we have a * representation {nf,Hf} of A such that fb(a) = (717 (a)b, 6),

Vo, 6 6 A.

Moreover, if A has an identity 1, then£f = 1 is a cyclic vector for (i.e., iTf(A)£f = Hf) and f(a) = (-Kf(a)l;f,Zf),

{irf,Hf}

Va€A.

If A has no identity, the following class of positive functionals is important. Definition 3.3.5. Let A b e a real Banach * algebra without identity, A = A+R, / > 0 and hermitian on A. f is said to be with positive extension, if there is / > 0 on A such that f\A = / . Clearly, / must be hermitian on A. Denote £{A) = {/|/ > 0, hermitian on A,

and with positive extension }.

Proposition 3.3.6. Let A be a real Banach * algebra, / > 0 and hermitian on A. Then f G £ (A), if and only if, there is a positive constant K such that f{af

< Kf(a*a),

Va € A.

44

Real Operator Algebras

Proof, f e £{A) 4=> we can define / ( l ) such that / ( ( a + A)*(a + A ) ) > 0 ,

VaeA,

A € R.

4=> we can define / ( l ) such that f(a*a) + 2A/(o) + A 2 /(l) > 0,

Va € A,

A€ R

<£=> we can define / ( l ) such that / ( a ) 2 < f(l)f(a*a),

Va 6 A

<^=> there is a constant •&"(= / ( l ) ) > 0 such that / ( o ) 2 < Kf(a*a),

Definition 3.3.7.

Va G A

Q.E.D.

Let A be a real Banach * algebra with identity. Denote

5(^4) = {p\p > 0, hermitian on A, and p(l) = 1}. S(A) is called the real state space of A. p € S(A) is also called a real state on A. P r o p o s i t i o n 3.3.8. Let A be a real Banach * algebra with identity. S(A) is a a (A*, A)-compact convex subset of A*.

Then

Proof. By Proposition 3.3.2, 5(^4) C A*. Clearly, S(A) is convex and a(A*, A)~closed. Now it suffices to show that S(A) is bounded. Since * operation is continuous on AC/R{AC), where R{AC) is the radical of Ac, it follows that there is a constant K > 0 such that P I < K2\\a\\,

Va£Ac/R{Ac).

For any p e S(A),pc e S(AC) and pc\R(Ac) = 0 obviously. Thus, we can define pc € S(AC/R(AC)), i.e., pc(a) = pc(a),Va €ae AC/R(AC). Then |p(a)| 2 = | p c ( S ) | 2 < r s ( a * a ) <||a*a||<*: 2 ||a1| 2 <.K: 2 ||a|| 2 , Va 6 A, where B = AC/R{AC),

i.e., \\p\\ < K.

Q.E.D.

Real Banach * Algebras

45

Definition 3.3.9. Let A b e a real Banach * algebra with identity. The following * representation {7r„ = ©pes^TTp,

Hu =

®pes(A)Hp}

is called the universal * representation of A. References.

3.4

[1], [27], [30], [35], [36], [40], [41], [53].

* Representations and topologically irreducible * representations

Definition 3.4.1. Let A be a real Banach * algebra. A * representation {IT, H} of A is said to be topologically irreducible, if E = {0} or H are the only closed (real) linear subspaces of H which satisfy -K{a)E C E,\/a € A. Remark. In Section 5.3, we will see that {7r,.ff} is topologically irreducible, if and only if {ir{A)')H = R, or n(A)' ^ E, C, or H. Definition 3.4.2. Let A be a real Banach * algebra with identity. V(A) — exS(A) (the set of all extreme points of 5(A)) is called the pure real state space of A. Each p G V(A) is also called a pure real state on A. Proposition 3.4.3. Let A be a real Banach * algebra with identity, p be a real state on A, and {irp,Hp} be the * representation of A induced by p. Then {Trp,Hp} is topologically irreducible, if and only if, p € V{A). Moreover, if {it, H} is a topologically irreducible * representation of A, then there is a p 6 V{A) such that {TT, H} = {irp, Hp} (unitarily equivalent). Proof. The former part is similar to the complex case. Now let {n, H} be topologically irreducible. Clearly, 7r(l) = 1 (the identity operator on H). Take i e H and ||£|| = 1. Then

p(-) = M0£,OeS(A). Define U : H —> Hp such that Uir(a)£ — a = a + L p ,Va € A. Then the conclustion is immediate. Q.E.D. Now we consider the existence of non-zero * representations and non-zero

46

Real Operator Algebras

topologically irreducible * representations. Proposition 3.4.4. Let A be a real Banach * algebra with identity. Then {{ir, H}\{ir,H}

{{7T, H}\{K,

is a non-zero * representation of A} ^ 4>

H} is a non-zero topologically irreducible * representation of

A}^4> <=> S(A) ^ <j>. Proof. If S{A) ^ <j>, then V(A) ^ (j> by Proposition 3.3.8 and KreinMilmann theorem. Thus, there are non-zero * representations of A and non-zero topologically irreducible * representations of A. Conversely, let {n, H} be a non-zero * representation of A. By restricting 7r onto a subspace of H, we may assume that there is £ € H such that n(A)Z = H and ||£|| = 1. Then

p(-) = M-)€.0e5(A), and S(A) ^ .

Q.E.D.

Now let A be a real Banach * algebra without identity, and A = A+R. (1) Let p0(a + A) = A, Vo e A, A € M. Then po is a pure real state on A, and the * representation {TTO,HO,£O} induced by p 0 is an one-dimensional topologically irreducible * representation of A. In fact, the left kernel IQ of po is L 0 = {( a + A)|a e A, A € R, p0((a + A)*(a + A)) = 0} = A, and Ho = A/Lo = R,5ro(a + A) = Al, Vo £ A, A € R. (2) Clearly *S{A) > 1 («=» {po} S 5 ( 2 ) < = • {po} CJP(2)) «=> {/I/ > 0, hermitian on A, and / | A ^ 0} # . (3) *S{A) > 1 «=>• there is a non-zero * representation of A «=» there is a * representation n of A such that TT\A ^ 0,

Real Banach * Algebras

47

where #E means the cardinal number of any set E. In fact, if {n, H} is a non-zero * representation of A, then it can be extended to a * representation of A (defining 7r(l) = 1) . Further, let a e_A, and £ G H such that 7r(a)£ ^ 0 and ||£|| = 1. Then p(-) - <7r(-)£, 0 E S(A) \{p0}. Conversely, let p € S(A)\{po}, and {n, H,£} be the cyclic * representation of A induced by p. We claim that ir\A ^ 0. Otherwise,

p{a) = (ir{a)t,t)=0,

Vo € A

and p = po- This is a contradiction. (4) * S(A) > 1 <£=> there is a non-zero topologically irreducible * representation of A <=>• there is a topologically irreducible * representation n of A such that n\A ^ 0. In fact, if * S{A) > 1, then * V(A) > 1 and there is p € V(A)\{p0}. Let {IT, H, £} be the cyclic * representation of A induced by p. We claim that -K\A ^ 0. Otherwise,

p(o) = ( 7 r ( o ) e , 0 = 0 ,

VaGA

and p = po- This is a contradiction. Clearly, n is topologically irreducible for A. Conversely, if A has a non-zero topologically irreducible * representation, then * S(A) > 1 by (3). By the above discussion, we have the following. Proposition 3.4.5. Let A be a real Banach * algebra without identity, and A = A+R. Then the following statements are equivalent: (i) There is a non-zero * representation of A. (ii) There is a non-zero topologically irreducible * representation of A. (Hi) # S{A) > 1. (iv) £(A)\{0} ? 4,. References.

[1], [27], [30], [35], [36], [40], [41], [53].

48

3.5

Real Operator Algebras

* Radical

Definition 3.5.1.

Let A be a real Banach * algebra. We call

R* = R*(A) = n{ker7r|7r

is a * representation of A}

the *-radical of A. A has no non-zero * representation, iff R* — A; if R* = {0}, then A is said to be * semi-simple. Since any * representation of A is continuous, R* is a closed * two-sided ideal of A. Clearly, A/R* will be * semi-simple, and R*(A) = R*(A) — n{ker7r|7r is a * representation of A}. Moreover, if R — R(A) is the radical of A, then R is also a closed * twosided ideal of A by Theorem 2.4.4. Remark. We don't know that R(AC) = R(A)+iR(A) Section 2.4). But we have R*{Ac) =

(see the end of

R*{A)+iR*(A).

In fact, let a + ib € R*(AC), where a, b € A, and {n, H} be any * representation of A. Then {TTC,HC} is a * representation of Ac, where 7rc = Tr+iir, Hc = H+iH, and wc(a + ib) = ir(a) + in(b) = 0. Hence, R*(A)+iR*(A). n(a) = n(b) = 0, a,b& R*(A), i.e., R*(AC) C Conversely, let o € R*(A), and {a, K} be any * representation of Ac. Let H = Kr (Section 1.1), and n = a\A. Then {-K, H} is a * representation of A. Hence, a(a) = 7r(a) = 0, and a G R*(AC), i.e., R*(A) C R*{AC), and R*{A)+iR*{A) c R*(AC). Proposition 3.5.2.

Let A be a hermitian real Banach * algebra. Then J R(A)Cp-

1

(0)ci?*(A).

Consequently, if A is also * semi-simple, then A is semi-simple. Proof. If 7r is any * representation of A, then ||7r(a)|| < p(a), Va € A. Thus, p-^O) C kervr, and p _ 1 (0) C R*.

Real Banach * Algebras

49

Let a € R. Then a*a & R. Since A is hermitian , it follows from Proposition 2.4.6 that a(a*a) = a(a*a) n R = {0}. Therefore, p(a) = r(a*a)? = 0, i.e., R C p _ 1 (0). Proposition 3.5.3.

Q.E.D.

Let A be a real Banach * algebra.

(i) J S(A) =£ 1, if A has an identity.

-*' (ii) If R* C A, then

R* — n{ker7r|7r topologically irreducible * representation of A} = n{ker7r|7r topologically irreducible * representation of A}.

Proof. (i) It follows from Propositions 3.4.4 and 3.4.5. (ii) By R* (A) = R* (A) we may assume that A has an identity. If a £ R*, then there is a * representation {n,H} of A such that 7r(a) ^ 0. Further, we may also assume that 7r(l) = 1. Take f € H such that 7r(a*)£ ^ 0 and lieil = 1. Then p(-) = <*(.)£,£> e 5 ( 4 ) , and p(aa*) > 0. By Krein-Milmann theorem and Proposition 3.3.8, S(A) is the a(A*, A)-closme of CoV(A), where CoE is the convex hull of a subset E in a, linear space. Thus , there is p S V(A) such that p(aa*) > 0. Let {•7Tp,Hp} be the topologically irreducible * representation of A induced by p. Clearly, a* 0 Lp (the left kernel of p). Further, we claim that aa* & Lp. Otherwise, by Schwartz inequality 0 < p(aa*)2 < p(aa* • aa*) = 0. It is impossible. Now •Kp(a)a* = aa* ^ 0 in Hp, and a $ kevnp. Therefore, (R* c ) n {ker7r|7r topologically irreducible * representation of A} C R*. Q.E.D. T h e o r e m 3.5.4.

Let A be a real Banach * algebra.

50

Real Operator Algebras

(i) If A has an identity, and R* £ A, then R* = n{ker7i>|p G S(A)} = n{ker7r p |p G V{A)}

= n{Lp\P e S(A)} = n{Lp\p e V(A)}. (ii) If A has no identity, and R* C A, then R* = n{ker 7 r / |/ G £(A)} = n{Lf\f

G £(A)}

= n{ker7T/|/ G £ (A), and ker7r/ C L/} n{ker7T/|/ > 0 and hermitian on A}.

Proof, (i) Clearly, R* C k e r ^ C Lp,Vp € S{A). Conversely, if a & R*, then there exists a * representation {n, H} of A such that 7r(a) ^ 0. This shows that there is £ G H such that n(a)£ ± 0. Without loss of generality, we may assume ||f|| = 1 and n(A)£ = H. Then p(-) = (*•(•)£,£) € 5(A), and p(a*a) = ||7r(a)£||2 > 0, i.e., a & Lp. Since CoV{A)° = 5(A), it follows that n{Lp\p G V{A)} = n{Lp\p € 5(A)}. Moreover, Lp D ker7rp D n{ker7rCT|a G V{A)} D R*,Vp G V{A). Therefore, we have the conclusion (i). (ii) By (i), we have R* = R*(A) = n{ker7r> G S(A)} = n { L > G 5(A)}. Let p0(a + A) = A,Va G A, A G R. Then po € 5(A),ker7r 0 = A,L0 = A, where LQ is the left kernel of po and {7P0, HQ} is the * representation of A induced by po- Thus R* = n{Lp\p G 5(A)} n{(LpnA)|pG5(A)\{p0}} = n{L/|/ G

£(A)\{0}}

= n{Lf\fe£(A)}.

Real Banach * Algebras

51

Noticing that every * representation of A is a direct sum of some cyclic * representations of A and a zero * representation of A, we have R* = n{ker7r|{7r, H, £} is a cyclic * representation oi A}. For a cyclic * representation {n,H, £} of A, let /(•) = (7r(-)£, £). Then / 6 E(A), and aekevn

<=>Tr(a)n(b)£ = 0, <=> f{b*a*ab) = 0, <=>• afc € Lf,

V6 € A V6 6 A

V6 € A <=*> a G ker 717,

i.e., ker7r = ker7r/. Moreover, in this case f(a*a) = ||vr(a)^||2 = 0 if a € ker7r, i.e., ker7r = ker717 C Lf. Therefore, we have R* = n{ker7r/|/ € £(A), and ker717 C i / } = n{ker7r/|/e5(^)} = n{ker7T/|/ > 0 and hermitian on

A}. Q.E.D.

Remark. If A has an identity, generally R* C n{p _ 1 (0)|p € 5(^4)} since P\AK = 0,Vp € <5(J4). This is different from the complex case. References.

3.6

[1], [27], [30], [35], [36], [40], [41], [53].

Symmetric real Banach * algebras

Definition 3.6.1. Let A be a real Banach * algebra, a e A is said to be positive, denoted by a > 0, if a* = a and IT (a) C R+. Denote the subset of all positive elements in A by A+, and o > b, if a* — o, b* — b € A and (a - b) e A+. Denote p{a) = r(a*a)?, Va € A. Theorem 3.6.2. Let A be a real Banach * algebra, which is hermitian and skew-hermitian. (1) Ifa€A and a(a) c R, then r(a)2 < r(a*a).

52

Real Operator Algebras

(2) If hi = a*a, h2 = h2 € A, then r(hi h2) < r(hi)r(h2). (3) A+ is a cone, i.e., if a,b> 0, then a + 6 > 0. (4) If h* = hi &A,i = 1,2, thenr(h1+h2) 1 and e > 0, we need only show that 1 — b is invertible, where X(r(a*a) + e)i' In fact , this implies that \(r(a*a) + e)i & cr(a), VA € R with |A| > 1. Since a(a) C R, it follows that r(a)<(r{a*a)

+ e)i,

Ve > 0.

Thus , we have r(a)2 < r(a*a). Now we prove that 1 — 6 is invertible. Clearly, r(b*b) = r(bb*) < 1. Since A is hermitian, it follows that 1 — b*b > 0 and 1 — bb* > 0. By Lemma 3.1.2, there are u > 0, v > 0 such that 1 - b*b = u2, 1 — bb* = v2. Noticing that (1 + 6*)(1 — b) = u(l + u~x{b* — 6)u _ 1 )u and A is skew-hermitian, 1 — 6 has a left inverse. Similarly, (1 - 6)(1 + 6*) = v(l + v-^b* - & > " > implies that 1 — 6 has a right inverse. Hence, 1 — 6 is invertible. (2) Since A is hermitian, we have a{hi h2) U {0} = a(ah2 a*) U {0} C R. Accoding to (1), r(hi h 2 ) < r(h2 /if h2)% - r{h\ h2)1* <---
h22)^

<\\hf\\^-\\h22\\^,

Vn. Let n —> +oo. Then we get r{h\ h2) < r(hi)r(h2). (3) Without loss of generality, we may assume that A has an identity. First we claim that 1 + a + b is invertible. Noticing l + a + 6=(l-£)(l + ^

+ ^ ) ,

Vee(0,l),

Real Banach * Algebras

53

we may assume a > 0. Then 1+

a

+ b = (1 + o)(l - uv)(l + b),

where u= (l + a ) _ 1 a > 0,v = (1 + 6 ) _ 1 6 > 0. Clearly, r ( l - u ) < 1, r(u) < 1, r(u) < 1. By Lemma 3.1.2, we can write u = w2,w > 0. From (2) , we have r(uv) < r(u)r(v) < 1. Then 1 — uv is invertible, and hence 1 + o + b is invertible. Since X + a + b = A(l + A - 1 a + A _1 6) for any A > 0, it follows from the preceding paragraph that A + a + b is invertible, VA > 0. Therefore, a+b> 0. (4)-(7). The proofs are similar to the complex case. Q.E.D. Remark. The inequality "r(a) 2 < r(a*a)" or 'V(a) < p(a)" is called the Ptak's inequality ([40, 41]). If B is a hermitian complex Banach * algebra, then this inquality is valid for each element in B. But in the real case, we must assume A € R (see the proof of (1)). Thus we don't know whether the Ptak's inequality is valid for each element in the real case. Theorem 3.6.3. Let A be a real Banach * algebra. Then the following statements are equivalent: (1) A is hermitian and skew-hermitian; (2) A is hermitian and r(9^-) < p(a), Vo € A; (3) A is hermitian and r(a) = p(a),V normal a G A; (4) A is hermitian and p(-) is sub-additive on A. Proof. (1) => (2) = > (3) and (1) =>• (4) follows easily from Theorem 3.6.2. (4) => (2) is obvious. (3) ==^ (1). Consider a maximal abelian * subalgebra containing a. Now it is in the abelian case, and this is just Theorem 3.2.3. Q.E.D. Definition 3.6.4. a*a > 0, Va € A.

A real Banach * algebra A is said to be symmetric, if

Theorem 3.6.5. Let A be a real Banach * algebra. Then the following statements are equivalent: (1) A is symmetric; (2) A is hermitian and skew-hermitian;

54

Real Operator

Algebras

(3) 1 + a*a is invertible in A, Va G A; (4) Rea(a*a) > 0, Va € A; (5) - A g cr(a*a), VA > 0, o e A Proo/. (1) =>- (2). It is obvious. (2) =$> (1). We may assume that A has an identity. Suppose that there is a € A such that 8 = inf{A|A € 0), we may assume that 5 e (—1, —1/3). Put b = 2a(l +

a*a)-\

Then 1 - 6*6 = (1 - a*a)2(l + a*a)-2 > 0, and a(b*b) C ( - o o , l ] . Write 6 = h + k, where ft* = ft, fc* = -fc. By Theorem 3.6.2, 1 + 66* = 2(ft2 - k2) + (1 - 6*6) > 0, and
5)2€a(b*b).

Thus \A5/(l + S)2\ < 1, i.e., 6|<5| < 1 + S2. Hence, \8\ < 1/3 since 1 + .52 < 2. This contradicts S 6 ( - 1 , - 1 / 3 ) . Therefore, a*a > 0, Va € A. Moreover, A symmetric -<=>• A hermitian and skew-hermitian <J=^ A hermitian and skew-hermitian <£=> A symmetric. Therefore, (1) -4=> (3) is obvious. Finally, (1) =*• (4) =*• (5) =>• (3) are obvious. Q.E.D. Remark. It is natural to ask: whether A is symmetric = > A c is symmetric? In the abelian case, the answer is affirmative (see Theorem 3.2.3 and note the fact that Ac is hermitian •<=*• Ac is symmetric [1]). Theorem 3.6.6. Let A be a real Banach * algebra with identity. the following statements are equivalent: (1) A is symmetric; (2) A is hermitian , and r(u) = 1, Vu € U(A); (3) A is hermitian, and r(u) < 1, Vu € U(A);

Then

Real Banach * Algebras

55

(4) A is hermitian, and there is a constant a > 0 such that r(u) < a, Vu G U{A). Proof. (2) = > (3) =>• (4) are obvious. (4) =4> (3). For u G f/(i4), un is also in [/(A) for each n e N. Then r(u) n = r(u n ) < a, and r(u) < a " , Vn. Therefore, r(u) < l.Vu G I7(i4). (3) => (2). It is a consequence of the fact that u € U(A) implies u~l € U{A). (1) ==>• (2). It is obvious from Theorems 3.6.5 and 3.6.3. (2) =$» (1). By Theorem 3.6.5. it suffices to prove that A is skewhermitian. Let k* = —k € A with r(k) < 1, let B b e a maximal abelian * subalgebra of A containing k, and let Q be the spectral space of B. As in the proof of lemma 3.1.3, we have u = h+k

€ U(A),

where h* = h and h G B. Let p € fi. Since r(u) = 1 = r(u*), we have p(h) + p{k) = p(u) = eia, p(J0-p(A)=p(u*)=e*, where a, /3 € K. Since A is hermitian , p(h) is real. Then 0 < p{hf

< \{\p{h) + p(k)\ + \p(h) - p(k)\)2 < 1.

At the same time p(h)2 - p(k)2 = p(h2 - k2) = p(u*u) = 1. Therefore, p(k)2 = p(h)2 - 1 < 0, and p(k) G iR, Vp G fi. Further, by Lemma 3.1.1 and Theorem 2.7.2, a(k) C iR. Q.E.D. Remark. There are some other results about symmetric real Banach * algebras in Section 5.8. Proposition 3.6.7. Let A be a symmetric real Banach * algebra with identity, f be a hermitian linear functional on A, and / ( l ) = 1. Then the following statements are equivalent: (1) / ( a ) > 0, Va G A+; (2) f > 0 on A, i.e., f G S(A);

56

Real Operator Algebras

(3) f(h2) > 0, Vft* = ft G A; (4)\f(h)\ (6) = • (4). (2) =>• (1). If o > 0, then a + e > 0 for any e > 0. By Lemma 3.1.2, there is u > 0 such that a + e = u2 = u*u. Then / ( a ) + e = f(a + e) = /(u*u) > 0, Ve > 0. Thus ,/(a) > 0. (3) = > (4). Let ft* = /i € A and r(/i) < 1. Then r ( l - (1 ± ft)) < 1. By Lemma 3.1.2, there are u* = u and v* = v such that 1 + ft = u 2 and 1 - ft = v2. Then / ( l ± ft) > 0 by (3), i.e., |/(ft)| < 1. (2) =>• (5). It is obvious from Proposition 3.3.2. Now it suffices to show (4) = > (2). For any ft* = ft G A, we have a(ft)c[a,/3] since A is hermitian, where a = mincr(ft),/? = maxa(ft). Let p = \{a + /3),5 = §(/? - a). Then |A - p| < 5,VA € a{h), i.e., r(h - p) < S. By (4), \f(h)-p\

=

\f(h-p)\
Thus, /(ft) > p — <5 = a = mincr(ft). Since A is symmetric, it follows that minora*a) > 0,Va G A. Therefore, f(a*a) >mina(a*a)

> 0,

Va G A Q.E.D.

Lemma 3.6.8. Let A be a symmetric real Banach * algebra with identity, and ft* = ft G A. Then for any A G o(h), there is p G S(A) such that p(h) = A. In particular, there is p G <S(A) such that \p(h)\=r(h).

Proof. Clearly, A — AH+AK- For ft* = ft G A and A G cr(ft), define a (real) linear functional p on [1, ft] = {a + /3h\a,f3 € K} as follows: p{a + /3ft) =

57

Real Banach * Algebras

a + /3A, V a , / ? e Let

Then p(a) > 0,Va € [1, /i] n A+,p(l) = 1 and p{h) = A. E is a linear subspace of AH, and 1, h £ E;

(E,PE)

PE is a linear functional on E, PE{1) = 1, P£;(a) > 0 , V a 6 £ 0 , 4 + , andp E |[l,/i] = p

and {E,pE) < (F,PF) if E C F and ( P F | - E ) = PE- By Zorn lemma, £ admits a maximal element (E,PE). We claim that .E = AH- In fact, suppose that there is a £ AH\E. Let F = S-j-Ra. Since 1 € .E, - r ( a ) l < a < r(o)l, and A+ is a cone (see Theorem 3.6.2), we can define PF on F such that (pp\E) = PE, and sup{pjs(6)|6e E,

b
< PF{O) < inf {PB(C)|C € E,a<

c}.

For any d + Aa € A+, where d € E, A 6 K, we need to prove pp{d + Ao) = PE(d) + Xpp(a) > 0. When A = 0, it is obvious. If A > 0, then a > -d/X,pF(a) > pE{-d/X), i.e., 0 < pF(d + Xa) = pE(d) + XpF(a). If A < 0, then a < —d/X, PF(O) < PE{—d/X), i.e., 0 < pF(d + Xa) = ps(d) + XpF(a). Therefore, (F,pF) € £ and (E,pE) < (F,pF). This is a contradiction. Further, let p\An — PE and P\AK = 0. Then p € S(A), and p(/i) = A. Q.E.D. Notes. In the complex case, S.Shirali and F.W.M. Ford proved that a Banach * algebra B is symmetric if and only if B is hermitian. Using their method, we can also get Theorem 3.6.5. in the real case. V. Ptak set up an important inequality, and from this he simply proved Shirali-Ford theorem in the complex case. Ptak's theory is considered in the real case by J. Vukman and this section. References.

[1], [27], [30], [35], [36], [40], [41], [47], [53].

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Chapter 4

Fundamentals of Real Von Neumann Algebras 4.1

Banach spaces of operators on a real Hilbert space

Definition 4.1.1. Let H be a real Hilbert space. 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. || • || will be the operator norm on H; The identity operator on H is denoted by Iff, or 1 if no confusion arises; and Hc = H+iH is the Hilbert complexification of H (see Section 1.1). Moreover, F(HC),C(HC) and B(HC) are the corresponding operator spaces on Hc. Then B(HC) =

B(H)+iB{H)

is a complexification oiB(H) in operator norm || • || (see Proposition 1.1.11). Clearly, F(HC) = F(H)+iF(H) as linear spaces. Proposition 4.1.2.

Let H be a real Hilbert space. Then C{H) =

(F(H),\\-\\)-;

C(H) is a closed * two-sided of B(H); Banach space if dim H = +oo; and C{HC) = is a complexification ofC(H)

C(H) is not the dual space of any

C{H)+iC{H)

in || • ||.

Proof. By the prof in the complex case (see [26, Section 1.1]), and Proposition 1.1.11, it suffices to show that if (a + ib) € C{HC), then a, b e C(H), where a, b G B(H). 59

60

Real Operator Algebras

For any bounded sequence {£„} C H, clearly there is a subsequence {£nfc} such that {(a + ib)€nk} is convergent. Then {a£„fc} and {b£nk} must be convergent. Therefore, a,b G C(H). Q.E.D. Definition 4.1.3. Let H be a real Hilbert space. The set of all operators of trace class on H is denoted by T(H), i.e., a G T(H) if a € C(H) and X^n An < +00, where {An} is the set of all positive eigenvalues of (a*a)* (counting the multiplicity). For a G T(H), ||a||i = ^Z n An is called the trace norm of a, and tr(a) = J2t{a£i, &) is called the trace of a, where {&} is any normalized orthogonal basis of H. Similarly to the complex case, tr(-) is well-defined on T(H). Similarly to the complex case (see [26, Section 1.1]), we have the following. Theorem 4.1.4. Let H be a real Hilbert space. (1) T(H) = (F(H), || • Hi)", and T{H) is a * two-sided ideal of B(H); (2) C(H)* = T(H), i.e., for any f G C(H)* there exists a unique a G T(H) such that 11/11 = ||a||i, andf{c) = tr(ac),Vc € C(H); conversely, for any a € T(H), tr(a-) is a continuous linear functional on C(H) with norm ||a||i; (3) T(H)* = B(H), i.e., for any F G T(H)* there exists a unique b € B(H) such that \\F\\ = ll&H, andF(a) = tr(ab),\/a G T(H); conversely, for any b G B(H), on T(H) with norm \\b\\.

tr(-b) is a continuous linear functional

Remark. T{H) is called the predual of B{H), and also denoted by B(H), = T(H). Proposition 4.1.5. Let H be a real Hilbert space. T(H)+iT(H) is a complexification ofT(H) in || • ||i.

Then T(HC) —

Proof By C(H)* = T(H), C(HC)* = T(HC), Propositions 4.1.2 and 1.1.4, the conclusion is obvious. Q.E.D. References.

[26], [30], [50].

Fundamentals of Real Von Neumann Algebras

4.2

61

Locally convex topologies in B(H)

Let H be a real Hilbert space. Similarly to the complex case (see [26 , Section 1.2]), we introduce the following locally convex topologies in B(H) : (1) weak (operator) topology; (2) strong (operator) topology; (3) strong * (operator) topology; (4) cr-weak (operator) topology, and it is also equivalent to a(B(H), T(H))\ (5) a-strong (operator) topology, and it is also equivalent to s(B(H),T(H)); (6) a-strong * (operator) topology, and it is also equivalent to s*(B(H),T(H)); (7) Mackey topology

T(B(H),T{H));

(8) uniform (operator) topology. Similarly to the proofs in the complex case ([26 , Section 1.2]) and by the duality theory (see [26 , Appendix ]), we have the following. Proposition 4.2.1. Let H be a real Hilbert space, and f be a linear functional on B(H). (1) If f is a(B(H),T(H))-continuous, then there exists a unique a e T(H) such that f{b) =

tr{ab),VbeB{H)

and there are {£n}> {%} C H with X)n(ll£nll2 + ll^nll2) < +°o such that

/(&) = $ > £ » , 1*0.

V6ej5(ff).

n

Moreover, if f > 0 (i.e., f(b*b) > 0,V6 6 B(H)), then we can take £ n = ?7„,Vn. (2) If f is weakly continuous or strongly continuous, then there exists a unique v € F(H) such that f(b) = tr(bv),

V6 e B{H)

and there are £i, • • •, £ n and r}\, • • •,rjn € B(H) such that 'n

Moreover, if f > 0, then v > 0 or we can take & — rji, 1 < i < n.

62

Real Operator Algebras

Theorem 4.2.2.

The relations between topologies (l)-(8) top.3)

D

n top.8)

D

top.7)

D

top.2)

D

are as follows: top.l)

n

top.6)

D

where "D " means finer. Moreover, in any bounded ball ofB(H), top.5), top.3) ~ top.6).

top.5)

n , D

topA)

we have top.l) ~ top4), top.2) ~

Remark. We will see top.6) ~ top.7) in any bounded ball of B(H) (see the end of Section 4.3). Corollary 4.2.3. Let f be a linear functional on B(H). Then the following statements are equivalent: (1) f is cr(B(H),T(H))-continuous; (2) f is s(B(H),T(H))-continuous; (3) f is s*(B(H),T(H))-continuous; (4) f is T(B(H),T(H))-continuous; (5) f is weakly continuous on any bounded ball of B(H); (6) f is strongly continuous on any bounded ball of B(H); (7) f is strongly * continuous on any bounded ball of B(H). Proposition 4.2.4. Let K be a convex subset of B{H). (1) The following statements are equivalent: (i) K is a(B(H),T(H))closed; (ii) K is s(B{H),T(H))-closed; (Hi) K is s*(B(H),T(H))-closed; (iv) K is T(B(H),T(H))-closed; (v) KnXS is weakly closed, VA > 0; (vi) K D \S is strongly closed, VA > 0; vii) K C\\S is strongly * closed, where S is the closed unit ball of B(H). (2) K is weakly closed, if and only if, K is strongly closed. Now we consider the problem of topological restriction from B(HC) B(H). Proposition 4.2.5. Let H be a real Hilbert space Then (top. j) in B{HC)\B{H)

-

top. j) in

B{H),

i.e., ai — > 0 with respect to top. j) in

ai — > 0 with respect to top. j) in

B(H)

B(HC),

to

Fundamentals

of Real Von Neumann

Algebras

63

where net {ai} C B(H), and j — 1,2, • • •, 8. Proof. It suffices to show that the conclusion holds for Mackey topology (top. 7)), and it is enough to notice the following fact. If E is a or(r(F c ),B(if c ))-compact subset of T(HC), then Re E = {a € T(H) | there is b e T(H) such that (a + ib) € E}, Im E = {6 € T ( # ) | there is a e TCtf) such that (a + ib) € £ } are a(T(H),B(H))compact subsets oiT(H), and Re E+i liaE — {(a+ib)\a e Re E, b 6 Im £ } is a
4.3

[26], [30], [50].

Von N e u m a n n ' s double c o m m u t a t i o n t h e o r e m

Definition 4 . 3 . 1 . Let if be a real Hilbert space. A * subalgebra M of B(H) is called a real VN (Von Neumann) algebra (on H), if M = At", where M' — {b € B(H)\ba — a6,Va € M} (the commutant of M ) , and M " = (M')' (the double commutant of M). If E is a subset of B(H), and M is the smallest real VN algebra containing E, then M is called the real VN algebra generated by E. Clearly, M = n{N\N real VN algebra on H, and E C N}. Remark. Let M be a * subalgebra oiB(H), and Mc = M+iM. Clearly, M'c = M'4-iM', and M^' = M"+iM". Therefore, M is a real VN algebra on H, if and only if, Mc is a VN algebra on Hc = H+iH. Similarly to the complex case (see [26, Section 1.3]) and by the duality theory (see [26 , Appendix]), we have the following. P r o p o s i t i o n 4.3.2. Let H be a real Hilbert space. (1) If E is a subset of B(H), then (EUE*)' is a real VN algebra, and the real VN algebra generated by E is (EDE*)". In particular, the commutant of any real VN algebra is a real VN algebra. (2) If M is a real VN algebra on H, then M is weakly closed , and M

64

Real Operator

Algebras

is the dual space of real quotient Banach space M* = T(H)/M±,

where

M± = {ae T{H)\tr(ab) = 0,V6 € M } . (3) If {Mi} is a family of real VN algebras on H, then M = Hi Mi is also a real VN algebra, and M' is the real VN algebra generated by U;M/. Proposition 4.3.3. Let M be a real VN algebra on a real Hilbert space H, and Mc = M+iM. Then the "-" operation is weakly continuous or a(Mc, Mc*)-continuous in Mc, M is a(Mc, Mc»)-closed in Mc, and Mct = M*+iM* is a complexification of M*. Proof.

By Proposition 1.1.11, we have (xtcVc) = (r}c,x€c),Vx € Mc,^c,r]c G Hc.

Thus, the "-" operation in Mc is weakly continuous, and M = {x G Mc\x = x} is weakly closed in Mc. Since tr(xt) = tr(xi),Vx G Mc,t G T(HC), it follows that the "-" operation in Mc is also <7(Mc,Mc*)-continuous. From Proposition 1.1.5. Mc* = N+iN is a complexification of N, where Mc* = T(HC)/MC± , and N = {/ G Mc*\f = / } is a(Mc, Mc*) -closed in Mc. Clearly, Mc± — Mj_+iM±. Thus, / = t+Mc± if/ - t+Mc±, where t G T(HC). Since T(HC) = T{H)+iT{H), we have N = {t + By M» = T(H)/M_i,

Mc±\teT(H)}.

it suffice to prove

||* + M C ± || = ||t + M i | | ,

VteT(H)

i.e., inf{||t + b + ic\\i\b, c € M i } = inf{||f + b^b € M x } , Vt G T(H). By Proposition 4.1.5, T(HC) = T(H)+iT{H) is a complexification of T(H) in || • ||i. Thus ||f+6+ic||i > ||t+6||i,V* G T{H),b,ce M x ( c T{H)), and ||i+M c J .|| = ||t + M i | | , V i G T ( i 7 ) . Q.E.D.

Fundamentals

of Real Von Neumann

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65

Proposition 4.3.4. Let M be a real VN algebra on a real Hilbert space H, and P(M) = { p € M\p* = p = p2} (the subset of all projections in M). (1) P{M) is a complete lattice with respect to the inclusion relation, i.e., if {pi\l G A} C P{M), then sup ( p; and inf;p; € P(M). Moreover, sup, e A p; = ( strongly ) - lim F sup; 6 i rPi = the projection from H onto [Ui&\piH] and iniieAPi = ( strongly ) - limF infieF Pi = the projection from H onto

Di^ApiH,

where F is any finite subset of A. (2) Let a G M, and a = vh be the polar decomposition of a. Then v,h G M. In particular, the projection vv* from H onto aH is in M. (3) Let a* = a G M and a=

I Xde\ be the spectral decomposition of a.

Then e\ e M,VA g R. In particular, [P(M)\ = MH, where [P{M)\ is the norm closure of the (real ) linear span of P(M). (4) M — MH+MK, MH = M+ - M+ = [M+] (the (real) linear span of M+), where M+ = {a € M\a > 0} (the subset of all positive operators in M). Proof. (1) It is obvious if we consider the conclusion in Mc (see [26 , Proposition 1.3.5]) . (2) By Theorem 1.2.5, v,h G B{H) D Mc = M. (3) By Theorem 1.2.4, eA G B(H) n Mc = M,VA G E. (4) From spectral decomposition, we can see MH = M+ — M+. Q.E.D. Remark. If N is a (complex) VN algebra, then the (complex) linear span [P(N)] is norm dense in ./V, and the (complex) linear span [N+] — N. Proposition 4.3.5. Let M be a real VN algebra on a real Hilbert space H, and U(M) = {u G M\u*u = uu* = 1} (the subset of all unitary operators in M). (1) Let a G M be normal, i.e., a*a = aa*. Then there exists a spectral

66

Real Operator

Algebras

pair {ei(-),e2(-)} (see Definition 1.2.1) in M such that a=

I

Rezdei(z) — /

Ja(a)

Imzde^z)

Jcr(a)

and /

Imzdei(z) — \

J a (a)

Rezde^{z) — 0.

J a {a)

(2) The (real ) linear span \U{M)) of U{M) (a * subalgebra of M) is norm dense in M, i.e., [U(M)} = M.

Proof. (1) Let a=

I

zde(z)

Jo(a)

be the spectral decomposition of a. Then e(-) € Mc = M+iM. Now the conclusion follows from Theorem 1.2.3. (2) By Lemma 3.1.3, we have [U(M)} D MK. On the other hand, if p € P(M), then p = I ( 2 p - l ) + iG[tf(M)]. Hence, P(M) c [U(M)]. Now by Proposition 4.3.4(3), we have \U{M)} = M.

Q.E.D.

Remark. The result U[U(M)] = M" is important. For any complex VN algebra N, we have N — [U(N)] (the complex linear span of U(N)). But for real VN algebra M, [U{M)\ C M generally. For example, M = i~([0,1]) onH = L2r([0,l}). Definition 4.3.6. Let M be a real VN algebra. Z = Z{M) = M n M' is called the center of M. If Z = R, then M is called a real factor.

Fundamentals

of Real Von Neumann

Algebras

67

Clearly, Zc - Z+iZ is the center of VN algebra Mc = M+iM; and M is a real factor if and only if Mc is a factor.

Proposition 4.3.7. Let M be a real VN algebra on a real Hilbert space H,p e P(M), and Z be the center of M. (1) Mp = pMp and M' = M'p are two real VN algebras on the real Hilbert space pH, and (M p )' = Mp, (2) Mp n M'p = Zp. In particular, if M is a real factor, then Mp and Mp are also real factors. Proof.

(1) By a result of complex case (see [26 , Proposition 1.3.6]) , pMcp = pMp+ipMp,

and M'j> =

M'p+iM'p

are two VN algebras on pHc = pH+ipH, and (pMcp)' = M'j). Now from the Remark following Definition 4.3.1, Mp and M' are two real VN algebras on pH, and (Mp)' = M'p. (2) It is similar to the complex case (see [26 , Proposition 1.3.8]). Q.E.D.

Remark. Since [C/(M)] C M generally, we can't simply "copy" the proof in the complex case to (1) directly.

Similarly to the complex case (see [26, Theorem 1.3.10]) , we have the following.

Theorem 4.3.8. (Von Neumann's double commutation theorem) Let H be a real Hilbert space, and M be a * subalgebra of B{H). Then M is a real VN algebra on H if and only i / l g M and M is weakly closed.

Now we consider the topologies in a real VN algebra. Let M be a real VN algebra on a real Hilbert space H. Since M = (M»)*, where M* = T(H)/M±, we can also introduce the following topologies in M : cr(M, M»), s(M, M*), s*(M, M„) and Mackey topology T(M, M»). Clearly,

68

Real Operator Algebras

we have the following relations: a(M, M.) ~ a(B(H), s(M,Mt)

~

s*(M,M>) ~ T(M,M.)

C

T(H))\M,

s(B{H),T(H))\M, s*(B(H),T(H))\M, T{B(H),T(H))\M.

Similarly to the complex case (see the end of [26 , Section 1.3]) , we still don't know if r(M,Mt) ~ T(B(H),T(H))\M1 Moreover, by Proposition 4.3.3 and similarly to the proof of Proposition 4.2.5, we have
Since T(MC,MC*) ~ s*(Mc,Mc*) in any bounded ball of Mc (see [26 , Section 1.11]), we also have T(M, M„) ~ s*(M, M.) in any bounded ball of M. References.

4.4

[6], [26], [30], [50].

Kaplansky's density theorem, tensor product commutation theorem, and comparison of projections

Theorem 4.4.1. (Kaplansky's density theorem) Let H be a real Hilbert space, N, M be two * subalgebras of B(H), and TV C M. If N is weakly dense in M, then (N)i is T(B(H),T(H))-dense in (M)i, where (iV)i and (M)i are the closed unit balls of N,M respectively. Proof. Clearly Nc = N+iN is weakly dense in Mc = M+iM. Then by Kaplansky's density theorem in complex case (see [26 , Theorem 1.6.1]) , (iVc)i is T(B(Hc),T(Hc)~deme in (M c )i In particular, for any a G M with ||a|| < 1 there exists a net {o; + ibi} C {Nc)i such that (a; + ibi) ->• o

Fundamentals

of Real Von Neumann

Algebras

69

in T(B(HC),T(HC)), where ai,bt € N,Vl. It is easy to see that a; -* a in T(B(Jf),T(fl')). Moreover, ||ai|| < ||aj +i&j|| < 1, VJ. Q.E.D. Corollary 4.4.2. Lei H be a real Hilbert space, and M be a* subalgebra ofB(H). (1) The closures of M for top.l),- • • ,top.7) (see Section 4-2) are the same. (2) IfleM, then M is a real VN algebra <=> M is a(B(H),T{H))closed •<=>• (M)i is weakly closed. Let H\,H2 be two real Hilbert spaces. Similarly to complex case (see [26 , Section 1.4]) , we can construct a real Hilbert space Hi® H2, the tensor product of Hi and H2. And it is easy to see that (H1®H2)C

=

{H1)C®{H2)C,

i.e., Hi <8> H2+iHi ®H2 = (Hi+iHi)

{H2+iH2).

Now let Mj be a real VN algebra on Hj,j = 1,2. The real VN algebra on i?i <8> H2 generated by {ai ®a2\aj G Mj,j = 1,2} is called the tensor product of Mi and M 2 , and denoted by M{®M2, i.e., Mi®M2 = {ai a 2 |oj € M,-,j = 1,2}" Since {Mj+iMj)'

= M'j+iM'j,

j = 1,2,

and (Mi+iMi)®(M 2 +iM 2 ) = (Mi®M 2 )+i(Mi®"M 2 ), it follows from tensor product commutation theorem in the complex case

70

Real Operator

Algebras

(see [26 , Theorem 1.4.11]) that (Mi®M 2 )'+i(M 1 ®M 2 )' = (Mi+iMi)'®(M a -i-iMa)' = =

(M[+iM[)®(M^+iMi)

(M{®M2')+i(M{®M2').

Therefore, we have the following. T h e o r e m 4.4.3. Let Mj be a real VN algebra on a real Hilbert space Hj,j= 1,2. Then ( M i ® M 2 ) ' = MJ®M^' on Hi ® H2. Similarly to the complex case (see [26, Propositions 1.4.12 and 1.4.13]) , we also have the following. Proposition 4.4.4. Let H\,H2 be two real Hilbert spaces. (1) IfMj is a real VN algebra on Hj, and Zj is the center ofMj,j = 1 , 2 , then Z = Z\®Z2 is the center of M\®M2- In particular, the tensor product of two real factors is again a real factor. (2) Let Mj,Nj be two real VN algebras on Hj,j = 1,2. Then ((Mi®M a ) U (JVi®"JV2))" = (Mi U N1)"®(M2 U N2)", (Mi®M2) n (Ni®N2) = (Mi n Ni)®(M2 n N2).

Definition 4.4.5. Let M be a real VN algebra on a real Hilbert space H, and p, q € P(M). p and q are said to be equivalent (with respect to M), denoted by p ~ q, if there exists a v G M such that v*v — p and vv* = q; p is weaker than q, denoted by p ^ q, if there is r E P(M) such that p ~ r and r < q. Moreover, for each p € P(M), there exists a minimal central projection c(p) in M such that p < c(p). In fact , c(p) is the projection from H onto [MpH]. c(p) is called the central cover of p. Similarly to the complex case (see [26, Propositions 1.5.2, 1.5.3 and 1.5.8]), we have the following.

Fundamentals of Real Von Neumann Algebras

71

Proposition 4.4.6. Let M be a real VN algebra. (1) Let {pi}ie\i {li}i€A be two orthogonal families of projections in M, andpi ~ qi,Vl e A. Then p = £ ; e A P * ~ 9 = X ) ( € A ^ (2) Let p,q € P(M), andp ^ q, q^
€ P(M).

Then

qz •< pz, and p(l — z) -< q{l — z). By the proof in the complex case (see [26, Theorem 1.5.6]) and Theorem 4.4.3, we also have the following. Theorem 4.4.8. in M such that

Let {pi\l € A} be an orthogonal family of projections

Pi ~ P i ' , VZ,Z' € A, and J ^ p ; = 1. i

Then M ^

MP®B(K),

where p ~ pi,Vl,K is a real Hilbert space with dim if = # A , and "=" means spatially * isomorphic, i.e., there exists a unitary operator u from H onto pH K such that uMu* =

References.

[6], [26], [30], [50].

MP®B(K).

72

4.5

Real Operator

Algebras

Positive linear functionals

Definition 4.5.1. Let M be a real VN algebra. A linear functional p on M is said to be positive, and denoted by p > 0, if p(a) > 0,Va G M+, and p(&) = 0,V6* = - 6 e Af*. Clearly, if p > 0 on M, then we have p(a*) = p(a),Va G M, and the Schwartz inequality holds: |p(6*a)|2 < p(a*a)p(b*b),

Va,6 G M;

and pc > 0 on M c , where p c is the natural extension of p onto Mc — M+iM. Since p c G M* and ||/9C|| = p c (l) = p(l), it follows from Proposition 1.1.4 that p € M*, and ||pj| = p(l). Moreover, if 0 on M c , then p > 0 on M, where p(o) = Re
Let M be a real VN algebra, and p > 0 on M. p is said supp(oj) = p ( s u p a ( ) i

i

for any bounded increasing net {a;} C M+\ p is said to be a real normal state, Up is normal and ||p|| = p(l) = 1; p is said to be completely additive, if

Ylp^

PC^PI) = i

i

for any orthogonal family {pi} of projections in M. Similarly to the complex case (see [26 , Theorem 1.8.6]) , we have the following. Theorem 4.5.3. Let M be a real VN algebra, and p > 0 on M. Then the following statements are equivalent: (1) p G M*, i.e., p is a(M, M*)-continuous; (2) p is normal; (3) p is completely additive. In particular, if p is normal or completely additive on M, then pc is normal or completely additive on Mc. Now let M be a real VN algebra on a real Hilbert space H, and p be a

73

Fundamentals of Real Von Neumann Algebras

real normal state on M. Then pc is a normal state on Mc. If L = {a 6 M\p(a*a) = 0} is the left kernel of p, then Lc = L+iL is the left kernel of pc. It is easy to see that (Mc/Lc,

(, ) c ) - {MIL, (, ))+i{M/L,

(,))

where (,) and (,) c is the inner products in M/L and Mc/Lc and pc respectively. Then we can see that {^Pc = np+inp, HPc =

induced by p

Hp+iHp},

where {-KP,HP} is the * representation of M induced by p, and is the * representation of Mc induced by pcLet Sn be the real normal state space on M, and {n — ®pes„np, H =

{^pc,HPc}

®PesnHp}-

We claim that {IT, H} is faithful. In fact, if ir(a) — 0 for some a G M, then (n(a*a)lp, lp) - p(a*a) = 0,

Vp G <S„.

Thus, we have tr(a*af) = 0,Vt G T(H)+, and ti(a*ah) = 0,V/i* = /i G T(H). By the definition of trace, it is easy to see that tr(a*afc) = ti(aka*) = 0,

Vfc* = -k G T ( F ) .

Therefore, we must have a* a = 0 and a = 0. Of course, {7r, i / } can be naturally extended to a faithful * representation of Mc. Thus, we have the following. Proposition 4.5.4. Let M be a real VN algebra, and Sn be the real normal state space on M. Then for each p € <Sn, the * representation {np,Hp} of M induced by p (GNS construction) is a — a continuous, itp(M) is a real VN algebra on Hp, and \\np\\ < 1. Moreover, {n = ©pe5n7Tp,i7 = ®PesnHp} is a a — a continuous, isometric, faithful * representation of M. Lemma 4.5.5. Let f G M*. Suppose there exists a G M+ with \\a\\ < 1 and ll/H = / ( a ) . Then / > 0.

74

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Algebras

Proof. By Proposition 1.1.4, ||/ c || = ||/|| = f{a). Now from a result of complex case (see [26, Proposition 2.3.3]) , fc > 0 on Mc. Therefore, / > 0 on M. Another proof is direct. It is as follows. By 0 < a < 1, we have - 1 < 2a - 1 < 1. Since / ( l ) < ||/|| = f(a), it follows that ||/|| 0. In fact, we may assume ||/|| = 1 and 0 < 6 < 1. Then 1 > ||1 - &|| > 1/(1 - 6)1 = |1 - A|, and A > 0. Now for c* = - c G M, we must prove /(c) = 0. Let /(c) = fi(e K), and

11/11 = 1. Then |A + M| = | / ( C + A ) | < | | C + A|| = (||C|| 2 + A 2 ) 5

and A2 + 2A/x + (i2 < A2 + ||c|| 2 ,

VA 6 R.

Therefore, /(c) = n = 0.

Q.E.D.

By Lemma 4.5.5 and similarly to the complex case (see [26, Theorem 1.9.3]), we have the following. Theorem 4.5.6. Let M be a real VN algebra, and
0 and unique v € M such that

where Rvuj(a) — uj(av),Va e M, and s(ui) is the support of w, i.e., 1 - s(u) = sup{p|p G P(M),

and w(p) = 0}.

Remark. The unique expression
0 } .

Fundamentals

of Real Von Neumann

Algebras

75

Theorem 4.5.7. Let M be a real VN algebra, and

Now by Proposition 1.1.4, the uniqueness follows from the complex case (see [26, Theorem 1.9.8]) . Clearly, 0 such that Vc =P+ -P-,

and \\(pc\\ = \\
Let ip± = Re(p±|M). Then tp± e M* + , and \\(p±\\ = Rep±(l) = p±(l) = \\p±\\, and
ip > 0. Then (1) there exist t0 € M with 0 < to < 1 such that ip(a) = | , t/iere eiisis h E M with A > ft > 0 suc/i tftot i>(a) = h + ik > 0 a n d ^ ( a ) = (p(a(h + ik) + (h + ik)a), Va e M. Therefore, ip(a) = h > 0. Q.E.D. References.

[3], [26], [30], [50].

76

4.6

Real Operator

Algebras

a—Finite real V N algebras

Definition 4.6.1. A real VN algebra is said to be a-finite, if any orthogonal family of non-zero projections in M must be countable. Similarly to the complex case (see [26 , Proposition 1.14.2]) , we have the following. Proposition 4.6.2. Let M be a real VN algebra on a real Hilbert space. Then the following statements are equivalent: (1) M is a-finite. (2) M admits a separating sequence {£„} of vectors in H, i.e., if a £ M and if a£n = 0, Vn, then a = 0. (3) M' admits a cyclic sequence {r]n} in H, i.e., [a'r]n\a' € M',n] is dense in H. (4) There is a faithful real normal state on M. Proposition 4.6.3. Let M be a real VN algebra on a real Hilbert space H. (1) M is a-finite, if and only if, Mc = M+iM is a-finite. (2) If M* is separable, then M is a-finite. In particular, if H is separable, then M is a-finite. (3) If M is ablelian and a-finite, then M admits a separating vector. Proof. (1) The sufficiency is obvious. Now let M be cr-finite. Then M admits a separating sequence {£„} C H. Clearly, {£„} is also separating for Mc. Therefore, Mc is cr-finite. (2) Since Mc* = M»+iM», it follows that Mc* is separable. Therefore, Mc (see [26, Proposition 1.14.3]) , and then M, is cr-finite. (3) It is similar to the complex case (see [26, Proposition 1.14.4]). Q.E.D. Proposition 4.6.4. Let M, N be a-finite real VN algebras on real Hilbert spaces H, K respectively. Then M~®N is a-finite on H ®K. Proof. Clearly, (M~(§N)C = MC®NC is cr-finite. Therefore, M®N is crfinite. Q.E.D. References.

[6], [26], [50].

Chapter 5

Fundamentals of Real C*-Algebras

5.1

Definition and basic properties

Definition 5.1.1. Let A be a real Banach * algebra. A is called a real Calgebra, if Ac = A+iA can be normed to become a (complex) C*-algebra, and keeps the original norm on A. Note that, as it is well-known the C*-norm on Ac (if it exists) is unique. Proposition 5.1.2. Let A be a real Banach * algebra. Then A is a real C* -algebra, if and only if, A can be isometrically * isomorphic to a uniformly closed * subalgebra of B(H) on a real Hilbert space H. Proof. Let A be a uniformly closed * subalgebra of B(H) on some real Hilbert space H. Then Ac = .A-M-A is uniformly closed in B(HC), where Hc = H+iH. Indeed, if ||(o„ + ibn) - (a + ib)\\ —> 0 in B(HC), where o n , bn 6 A,Vn, and a,b g B(H), then \\{an + ibn)l;-(a

+ ib)Z\\2=\\(an-a)l;\\2

+

\\(bn-b)Z\\2^0

uniformly for £ € H with ||f|| < 1. Thus, a, b e A and (a + ib) € Ac. Now by Proposition 1.1.11, A is a real C*-algebra. Conversely, let A be a real C*-algebra. We may assume that Ac is a uniformly closed * subalgebra of B(K) for some (complex) Hilbert space K. Let H = Kr = (K, (, )r — Re(,)), where (,) is the inner product in K. Then if is a real Hilbert space, and A is a uniformly closed * subalgebra oiB(H). Q.E.D. Remark. A C*-algebra is isometrically * isomorphic to a concrete C*algebra. However, a concrete real C*-algebra must be defined as a uni77

78

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formly closed * subalgebra of operators on a real Hilbert space. Therefore, by Proposition 5.1.2 our Definition 5.1.1 is natural. We stressed the complexification of real C*-algebras. On other hand , a (complex) C*-algebra B can also be defined as a (complex) Banach * algebra satisfying ||6*6|| = ||&||2,V6 € B. Such a similar and equivalent definition for real C*-algebras will be given in Section 5.2. Let A be a real C*-algebra. From Propositions 5.1.2 and 1.1.11, the C*algebra^4 c = A-i-iAmust be a complexification of A {i.e., ||a+i6|| = \\a—ib\\ on Ac,Va, b e A). Moveover, by Proposition 1.1.4 we have A* = A*+iA*. Now in Ac, we have two operations "*" and "—". They are conjugate linear * isometric bijections on Ac with period 2. "—" is an algebraic isomorphism (i.e., xy = x y,Vz, y £ Ac), and its fixed point set is A, a real algebra. On the other hand , "*" is an anti-algebraic isomorphism (i.e., (xy)* = y*x*,\/x,y € Ac), and its fixed point set is AH (hermitian part of A), not an algebra in general. Conversely, let B be a (complex) C*-algebra, and "-" be a conjugate linear * algebraic isomorphism of B with period 2. We may assume B C B(K) for some (complex) Hilbert space K. Let H = KT. Then D = Br C B(H). By Proposition 5.1.2, D is a real C*-algebra. Clearly, "-"will be a * isomorphism on D, and "-" can be naturally extended to a * isomorphism on Dc- Thus, "-" must be isometric, i.e., \\b\\ = ||6||,V6 € B. Therefore, A is a real C*-algebra, and B — A+iA is a complexification of A, where A = {b € B\b — b}. From the above discussion and Proposition 1.1.4, we have the following. Proposition 5.1.3. Every real C* -algebra A is a fixed point algebra of (B,—),i.e., A = {b G B\b = b}, where B is a (complex) C* -algebra, and "-" is a conjugate linear * algebraic isomorphism of B with period 2. Moreover, B = A+iA and B* = A*+iA* are the complexifications of A, A* respectively. Proposition 5.1.4.

Let A be an abelian real C'-algebra.

Then

A s c0(n, -) = {/€ c0(n)\f(t) = /(*). vt e n}, where fi is the spectral space of A, "-" is as in Definition 2.7.1, and "=" is the G elf and transform (see Theorem 2.7.2), a * isomorphism.

Fundamentals

of Real C* -Algebras

79

Proof. Clearly, A c =* C0{il). By Theorem 2.7.2, a(t) = a(t), Vi e ft, a G A, where a(-) is the Gelfand transform of a. Now let / G Co (ft, —). Then there are a, b G A such that / ( i ) = a(t) + ib(t), Vi € ft. Since o(i) - iW) = (a + ib)(t) = JJfj = f(t) = a(t) + ib(t) = a(i) + ibjtj, Vi G ft, it follows that 6(f) = 0, Vi G ft, b = 0, and /(f) = a(f), Vf € f l Q.E.D. Proposition 5.1.5. Let A be a real C*-algebra. (1) As a real Banach * algebra, A must be hermitian, skew-hermitian and symmetric. Moreover, A is semi-simple, and \\a*a\\ = ||a|| 2 , ||a*|| = ||a||,Vo€ A. (2) If A has an identity, then ||1A|| = 1, and cr(u) C {A G C| |A| = l } , V u € 17(A). (3) If A has no identity, then (A+R) can be normed as a real C* algebra, and the original norm on A remains unchanged. Proof. Since Ac is a C*-algebra, it suffices to show that A is semisimple. By Proposition 2.4.3, we may assume that A has an identity. If a G R{A), then 1 + 6a is invertible in A,V6 G A, from Theorem 2.4.4. Consequently, a*a — A is invertible in A,VA G IR\{0}. Since o-{a*a) C R, we have a(a*a) = {0}. Moreover, ||(a*a) 2 "|| = ||a*a|| 2 ",Vn. Therefore, ||a|| 2 = ||a*a|| = r(a*a) = 0, and A is semi-simple. Q.E.D. Remark. If A has no identity, now from complex case (see [26, Proposition 2.1.2]) we only have ||a + A|| = sup{||ax + Aa;|| \x G Ac, \\x\\ < l},Va G A, A G R. Later (Section 5.2) we can see that \\a + A|| = sup{||o6+ A6|| \b G A, \\b\\ < l},Va G A, A G R. Moreover, a real Banach algebra with identity is said to be unital, if ||1|| = 1. Thus, if A is a real C*-algebra with identity, then A must be unital, and in this case, A ^ [C7(A)] (the real linear span) in general (it is different from the complex case). For example, A = CV([0,1]). Proposition 5.1.6. Let A be a real C* -algebra. (1) Any closed * subalgebra of A is still a real C*-algebra. (2) Let A be unital, a G A be normal (i.e., a*a = aa*), and C*(a) be the real C*-subalgebra of A generated by {a, 1}, i.e., the norm closure

80

Real Operator Algebras

of {p(a, a*) | p(-,-) polynomial with real coefficients } . Then \\a\\ = r(a), and C*(a) = C(a(a),—), where "-" is the complex conjugation (it is wellknown that cr(a) = cr(a), see Definition 2.1.4), and a will be the function z on cr(a) under the * isomorphism ' = " . Consequently, C*(h) = Cr(a(h)) (all real valued continuous functions on o-(h)), V7i* = / i £ i . (3) Let a € A be normal, and CQ (a) be the real C* -subalgebra of A generated by {a}, i.e., the norm closure of {p(a,a*)\p(-,-) polynomial with real coefficients, andp(Q,Q) — 0}. Then r(a) = \\a\\, and C£(a) £ C 0 (a(a)\{0}, - ) .

Proof. (1) is obvious by Proposition 5.1.2. (2) and (3) follow from the complexification. Q.E.D. Remark. If {0} is an isolated point of cr(a), then C^(a) is unital . Moreover, if A is unital and 0 $ a(a), then 1 € Cg (a). Proposition 5.1.7. Let A be a real C*-algebra, and B be a real C*subalgebra of A. If A is unital and 1A € B, then aB(b)=aA(b),

VbeB.

Generally, we have aB(b)U{0}

= oA(b)L>{0},

V6 € B.

Proof. It follows from the complexification and Lemma 2.4.5.

Q.E.D.

Lemma 5.1.8. Let B be an abelian real C*-algebra, and \\ • ||i be an algebraic norm on B (i.e., \\ab\\i < ||o||i||6||i,Vo, b € B). Then || • || < || • ||i on B. Moreover, if B\ = (B, \\ • ||i)~ (the completion of (B, || • ||i)^, then
V6 6 B.

Proof. Let B\ = (B, || • ||i) . Then B\ is an abelian real Banach algebra. Let {B\)c = Bi+iBi be a complexification of B\ (see Theorem 2.1.3).

Fundamentals

of Real C

-Algebras

81

Clearly, (B\)c D Bc — B+iB. Thus , || • ||i can be extended to an algebraic norm on Bc. Since (Bc, \\ • ||) is an abelian C*-algebra, it follows from the complex case (see [26, Proposition 2.1.9]) that || • || < || • ||i on Bc, and then M l < Mil onS. Now let ft be the spectral space of the real C*-algebra (B, \\ • ||), and Qi be the spectral space of B\. Since || • || < || • ||i on B, it follows that t is also continuous on B in norm || • ||i, and t can be uniquely extended to a p G fii,V£ G ft. Conversely, for any p G fii,t = p\B G ft. Now from Theorem 2.7.2, we have aB(b) U {0} =
V6 G B. Q.E.D.

Proposition 5.1.9. C*-norm on A, i.e.,

Let A be a real C*-algebra, and \\ • \\i be another

\\db\\i < IMIiH&Hi, and ||o*a||i = ||a||f,

Va,6 G A.

Then || • || = || • ||i on A. Proof. Let h* = h G A, and B = CQ (h) be the abelian real C*-subalgebra of A generated by {h}. By Lemma 5.1.8, we have rB(h) = rBl(h), where Bi = (B, || • | | i ) - . By Proposition 5.1.6, ||ft|| = rB{h). Then ||A|| = rB(h) = rBl(h)

= Km Wh^Wf" = \\h\\i, n

V7i* = h G A. Further, ||a||i = ||o*o||J = l l o ' o l l ^ H o l l ,

VoGA Q.E.D.

Proposition 5.1.10. Let A be a real C* -algebra, n be a positive integer, and Mn(A) = {(aij)|ojj G A, 1 < i,j < n}. Then there is a unique C*-norm on Mn{A) such that Mn(A) is a real C*-algebra. Moreover, Mn{A)* = Mn{A*) = {(fi^lfij &A*,1< i,j < n}, and

11(^)11 = s u p { | ^ / « ( o « ) | | ( ^ ) 0 i e ^ ) ' , and }.

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Real Operator Algebras

Proof. We may assume that A C B(H) for some real Hilbert space H. Then Mn{A) is a uniformly closed * subalgebra of B(Hn), where Hn = H © • • • © H (n times) is a real Hilbert space. Now by Proposition 5.1.9, there is a unique C*-norm on Mn(A) such that Mn(A) is a real C*-algebra. Clearly, as Banach space norms we have 1

max \\ai

on Mn{A). Then \(fij)((aij))\

=

\^2fij(aij)\

< maxi,j \\aij\\ -Y^WfaWThus, for any fy 6 A*, 1 < i,j < n, we have {fij) € Mn(A)*. if F e Mn{A)*, then /y(.) = F{eiy) e A*, and F = ( / y ) .

Conversely, Q.E.D.

Examples. (1) Let B b e a (complex) C*-algebra. Then A = Br (regard B as a real * algebra with original norm ) is a real C*-algebra. Moreover, O~A{O)

—

O-B(O-)

Uasia),

Va € A.

In fact, we may assume that B C B(K) for some (complex) Hilbert space K. Then A — Br is a uniformly closed * subalgebra of operators on a real Hilbert space H = Kr. Thus, A is a real C*-algebra. The conclusion on spectrum follows from Proposition 2.1.7. (2) H (see Section 2.2 Example (2) is a real C*-algebra with p* = —p, Vp = i,j, k. Indeed,

(a + (3i + 7 j + 8k)

(a -p (3 a 7 5 \S - 7

-7 -S\ —5 7 a —0 P OL )

(Va, (3,7, S e R) is a * isomorphism from HI into 5 ( R 4 ) . Moreover, it is easy to see that

fa -P a P

-7 -8

-S\ 7

-P

7

8

a

\s

-7

P

a )

(t\ V X

W

= (a2 +(32 + 7 2 +8*)(e

+ r,2 + A2 +n2),

Fundamentals

of Real C* -Algebras

83

Va, /3,7,5, £, 77, A, /z € R. Therefore, the above * isomorphism is also isometric, and H is a real C*-algebra. (3) C0(fl,-) = {/ € C 0 (n)|/(f) = 7 W » V t e n > i s a n abelian real C*-algebra, where fi is a locally compact Hausdorff space, and "-" is a homeomorphism of ft with period 2. In fact, it follows from Definition 5.1.1 immediately. Moreover, by Proposition 5.1.4 it is also the general form of abelian real C*-algebras. (4) A = C{X,Y) = {/ e C{X)\f{Y) c R} is an abelian real C*algebra, where X is a compact Hausdorff space, and Y is a closed subset of X. Moreover, A Si C(Sl,-), where fi = (X\Y) U F U (X\Y) (disjointed topological union ) such that y = y,Vy € Y, and "-": x G first or second (X\Y) —> x e second or first (X\Y). References. 5.2

[3], [17], [24], [25], [26], [30].

Positive functionals and equivalent definition of real C—algebras

Definition 5.2.1. Let A be a real C*-algebra. a € A is said to be positive, denoted by a > 0, if a* — a, and a{a) C R+. Denote A+ = {a € A|a > 0}. Proposition 5.2.2. Let A be a real C*-algebra. (1) A+ = {Ac)+ fl A is a closed cone, and A+ (1 (—A+) = {0}. (2) If a £ A+, then there exists a unique a? € A+ such that (a^) 2 = a, and a3 g

CQ(O).

(3) If a e A, then a £ A+, if and only if, there is b € A such that a = b*b. (4) For any h* = h e A, there are unique h+ and /i_ € A+ such that h = h+ — h-,

and h+ • h- = 0.

Moreover, h+ and h- € C^Qi). Consequently, AH = {a £ A\a* = a} = A+ — A+ = [A+] (the (real ) linear span). Proof. From the complexification and [26, Section 2.2], it is obvious. Q.E.D. Remark. For any (complex) C*-algebra B, we have B = [B+] (the (complex) linear span).

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Real Operator Algebras

Definition 5.2.3. Let A be a real C*-algebra. {d/} is called an approximate identity for A, if {di} is an increasing net in A+ such that 0 < di < 1, VI, and {\\adi - a\\ + \\dta - a\\) —> 0, VoeA Clearly, {d{\ is also an approximate identity for Ac. Proposition 5.2.4. identity.

Every real C*-algebra A admits an approximate

Proof. Let {u; = di + ici} be an approximate identity for Ac, where di, ci € A,Vl. Clearly, {d{\ will be an increasing net in A+, and 0 < di < 1,VZ. Moreover, since Ac is a complexification of A, it follows from Definition 1.1.1 that \\adi — a\\ + \\dia — a\\ < \\aui — a\\ + \\uia — a\\ —> 0,

Va e A.

Therefore, {di} is an approximate identity for A.

Q.E.D.

Remark. Let A be a non-unital real C*-algebra, and {di} be an approximate identity for A. Since {di} is also an approximate identity for Ac, it follows from the complex case (see [26, Proposition 2.4.4]) that the C*-norm on (A+R) is as follows: \\a + A|| = limj \\adi + Xdi\\ = limj \\dia + Xdi\\ = sup{||a6 + A6|| \b£A,

\\b\\ < 1 } ,

Va € A, A € R. Definition 5.2.5. Let A be a real C*-algebra. A (real) linear functional / on A is said to be positive, denoted by / > 0, if f\A+ > 0, and f\Ax = 0 (i.e., /(a*) = / ( a ) , Va G A ), where AK = {a e A\a* = -a}. Let / > 0 on A. / i s called a real state on A, if ||/|| = 1. Denote the set of all real states on A by S(A). Proposition 5.2.6.

Let A be a real C*-algebra.

Fundamentals

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-Algebras

85

(1) If f > 0 on A, then we have the Schwartz inequality: \f(b*a)\2 < f(a*a) • f(b*b),

Vo, b G A.

(2) If f > 0 on A, then fc > 0 on Ac. Consequently, f G A*, and 11/11 — ll/dl = u m / ( ^ ) = lim/(^f), where {di} is an approximate identity for A. (3) Let f G A*. Suppose there exists a G A+ such that \\a\\ < 1 and

ll/H = /(a). Then / > 0 on A. (4) (5) r(h). (6) (7)

Let
on Ac. Then p = Re( 0 on A, and \\p\\ = \\ 0,Vp G S(A), then aeA+. Let A be non-unital, and p > 0 on A. Define p(a + A) = p(a) + Xno,

V o e A , A e R , where p,0 > \\p\\. Then p > 0 onA = A+R. (8) Let B be a real C* -subalgebra of A. Then each real state on B can be extended to a real state on A. Proof.

From the complexification and Lemma 4.5.5, all are obvious. Q.E.D.

Proposition 5.2.7. Let Abe a real C*-algebra, and S{A) be its real state space. (1) S(A) is a convex subset of A*. (2) (S(A),a(A*,A)) is compact, if and only if, A is unital. (3) If A is non-unital, then S(Aj

= Co(V(A)u{Q})a

= Q(A),

where V(A) = exS(A) (the subset of all extreme points of S{A)), Q(A) — {/ G A*\f > 0, ll/H < 1}, and "-"" means the a{A\ A)-closure. Proof. (1) By Proposition 5.2.6(2), it is obvious. (2) Clearly, S(A) is cr(A*, A)-compact if A is unital. Conversely, if A is non-unital, then Ac is also non-unital, and 0 G S(AC) (see [26, Proposition 2.5.5]). Thus, there is a net {ipi} C S(AC) such that (pi —> 0 in a(A*, Ac). Then pi = R e ( ^ | A ) —> 0 in o{A*, A), and 0 G S(A)". Therefore, S(A) is not a{A*, A)-compact.

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Real Operator Algebras

(3) Clearly, Q(A) is convex and a(A*, A)-compact, and exQ(A) — V(A) U {0}. Thus, by Krein-Milmann theorem, Q(A) = Co{V(A) U {0}) J . On the other hand, by the proof of (2), (V(A) U {0}) C S ( I f C Q(A). Clearly, S(A)

is convex. Therefore,

S(Aj

= Co(V(A)u{0}f

= Q(A). Q.E.D.

Remark.

By Krein-Milmann theorem, S(A)

= CoV{A)

if A is uni-

tal. Theorem 5.2.8. Let Abe a real C*-algebra, f 6 A* is said to be hermitian, if f(a*) = / ( a ) , V o e A (i.e., f\AK = 0 ) . Let f € A* be hermitian. Then there are unique /+ and / _ > 0 on A such that / = / + - / - , and ll/H = ||/ + || + ||/_||.

Proof. Since / c is also hermitian on Ac, it follows from [26, Theorem 2.3.23] that there are unique 0 on Ac such that fc =
+ \\
Now let f± = Re( 0 on A, and | | / ± | | = ||v?±||. Then, / = / + - / _ , and ll/H = ||/ + || + ||/_||. Let / = p+ - p- be another such decomposition. Then fc = p+c — P-c, and ll/cll = 11/11 = ll/>+ll + IIP-ll = \\p+c\\ + \\p-c\\- By the uniqueness, 0 on A}. Then by Theorem 5.2.8, A*H

= A*+

_ A*+

=

[ A .+j

Fundamentals

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C*-Algebras

87

(the (real) linear span). This is different from the complex case. For any (complex) C*-algebra B, we have B* = [B*+] (the (complex) linear span). For each p € S(A), we have GNS construction {np,Hp,£p}, a cyclic * representation of A induced by p. Moreover, p c G S(AC). Then we also have {TrPc,HPc,£pc} for Ac. Clearly, HPc = Hp+iHp, £Pc = fp, and irPc = Trp+iTTp. Further, by Proposition 5.2.6(5), we have the following. Theorem 5.2.9. tation)

Let A be a real C*-algebra, and (universal * represen-

{*M= S pes(A)

*P>

H

u= 0

Hp}.

peS(A)

Then {nu,Hu} is a faithful * representation of A, and A is isometrically * isomorphic to the uniformly closed * subalgebra TTU(A) of B{HU) on the real Hilbert space Hu. Consequently, A is * semisimple. Theorem 5.2.10. Let A be a real Banach * algebra. Then A is a real C*-algebra, if and only if, A is hermitian, and \\a*a\\ = \\a\\2, Va G A. Proof. The necessity is obvious. Now let A be hermitian, and \\a*a\\ = ||a|| 2 ,Va e A. We may assume that A has an identity. In fact, if A has no identity, then (A+R) is also hermitian obviously. Further, let \\a + A|| = sup{||a& + A6|| \b € A, \\b\\ < 1}, Va € A, A G R. Similarly to the proof in the complex case (see [26, Proposition 2.1.2]), we also have ||(o + A)*(a + A)|| = ||a + A||2, Va € A, A e R. For any k* = —k G A, since * operation is isometric, it follows that (etk)* - e~tk,

1 = ||(e tfe )V fe || = ||e tfc || 2 ,

Vt € R. Thus , a(k) C iR, i.e., A is also skew-hermitian. By Theorem 3.6.5, A is symmetric. By Lemma 3.6.8 and the assumption on A, A admits a faithful * representation {TT,H}, where -K = iru,H = Hu (see Definition 3.3.9). Let

Nil = IkWII, VaeA

88

Real Operator Algebras

Then || • ||i is a C*-norm on A. For any h* = h G A, we have a(n(h)) C a(h) obviously. Thus, \\h\\i = r(n(h)) < r(h) - ||ft||. On the other hand, ||7r(fc)||>K7r(fc)l p ,l p )| = |p(ft)l, Vp G 5(A). By Lemma 3.6.8,

W i = lkWII>r(ft) = |W|. Thus, ||/i||i = \\h\\,Vh* = heA.

Therefore,

||o||i = ||a*a||12 = ||o*o||2 = ||a||,VaG A, and A is a real C*-algebra from Proposition 5.1.2.

Q.E.D.

Corollary 5.2.11. Let A be a real Banach * algebra. Then the following statements are equivalent: (1) A is a real C* -algebra; (2) A can be isometrically * isomorphic to a uniformly closed * subalgebra of B(H) on a real Hilbert space H; (3) A is hermitian, and \\a*a\\ = ||a|| 2 ,Va G A; (4) A is symetric, and \\a*a\\ = ||a|| 2 ,Va € A; (5) 1 + x*x is invertible in A,Vx e A, and \\a*a\\ = ||a|| 2 , Va G A. Remark. In Theorem 5.2.10, the hermitian condition of A is necessary. To see this, consider C as a real algebra, and let ||A|| = |A| and A* = A, VA € C. Then we get a counterexample. Notes. Theorem 5.2.10 is due to L.Ingelstam [21, 39]. Moreover, K. R. Goodearl ([17]), C.H.Chu et al. ([3]) define a real C*-algebra as Corollary 5.2.11(5). References.

5.3

[3], [17], [21], [24], [25], [26], [30], [39].

Pure real states, their left kernels, and irreducible * representations

Definition 5.3.1. Let A be a real C*-algebra, S(A) be its real state space, and V(A) = ex<S(A) (see Proposition 5.2.7). Each p G V(A) is called a pure real state on A.

Fundamentals

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89

Proposition 5.3.2. Let A = C(fl, - ) be a unital abelian real C* -algebra, where Q is a compact Hausdorff space, and "-" is a homeomorphism of tt with period 2. Then its real state space is S{A) — {p\p probability measure on Q, and p,o — = p}, and its pure real state space is

Proof. Let p € S{A). Then pc € S(AC), and there is a probability measure p on Cl such that

PM = f g(t)dn(t),Vg € C(fi). For any / G C(ft, - ) , we have pc(f) = p(f) G R. Thus

Jfdp = Jfdp = Jf(t)dp(t) = Jf(t)d»Ct) = Jf(twct), V/ G C ( f i , - ) . Moreover, f fdp = f f(t)dp(t),\/f

G C(fi). Therefore,

p o — = p, and supp/Li = supp^i. If p G V(A), then supp^i = {t,t} for some t € fi. Since p o — = p, it follows that p = \(5t + <%). Q.E.D. Remark. Generally, a pure real state on an abelian real C*-algebra is not multiplicative. This is different from the complex case. Moreover, in the above, p(f) = \{f{t) + f(t)) (V/ G C(il, -)) is a pure real state, but Pc(9) = \(g(t) + g(i)), is not a pure state on C(H) if £ ^t.

Vg€C(Cl)

On the other hand, we have

Re(
c(n).

90

Real Operator Algebras

Proposition 5.3.3. Let A be a real C*-algebra, and p be a pure real state on A. Denote Ap — {

T;(

on Ac,

V
is convex. Now let {
0 on Ac, \\2, where ip\,ip2 € <S(AC), and A e (0,1). Then/? = Api + (1 — X)P2, where pi = Re( is pure on Ac. Q.E.D. Proposition 5.3.4. Let A be a real C*-algebra, ip be a pure state on Ac, p = Re((p\A)(€ S(A)), and pc = \{

and V({Zn

as n,m —> +00. Since
=

-• +00. Let d = x'x + y'y. Then by y'y e L^ we have <£((z„ - d)*(zn - d)) = (f((zn - x'x)*(zn - x'x)), Notice that (x'x)^ Therefore,

=

-K^X^X^

= nv(x')£v

= (x')v,

Vn.

i.e., (x'x — x') e Lv.

y((zn - d)*{zn - d)) = tp({zn - x')*{zn - x')) —+ 0 as n —»• +00. Similarly, ~ +00. Moreover, we have (zn + Lc) —> (d + Lc) as n —V +00 in Hc, i.e., Ac/Lc

is complete, and Hc — Ac/Lc.

Q.E.D.

Remark. The following is an open question: is p pure on A (notations given in Proposition 5.3.4)? In the abelian case, it is affirmative (see Proposition 5.3.2).

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Real Operator Algebras

By proofs similar to these of Lemma 3.6.8 and Proposition 5.3.3, and by Propositions 5.1.6 and 5.2.6, we have the following. Proposition 5.3.5. Let A be a real C*-algebra (1) Let A be unital, E be a* (real) linear subspace of A, and 1 € E. Let

S{E) = jp:E

p (real) linear, p(l) = 1, p\(En A+) > 0, and p(a*) = p(a),Va

1 &E)

(real state space on E), and V{E) = exS(E) (pure real state space on E). Then each real state or pure real state on E can be extended to a real state or pure real state on A. (2) Let B be a real C* -subalgebra of A. Then each real state or pure real state on B can be extended to a real state or pure real state on A. (3) Let h* = h € A, and 0 ^ A € a(h). Then there is a pure real state p on A such that p(h) = X. (4) Let a* = a € A and p(a) > 0,Vp € V(A). Then a > 0. Definition 5.3.6. Let A be a real C*-algebra, and {w, H} be a * representation of A. {n, H} is said to be topologically irreducible, if E — {0} or H are the only closed linear subspaces of H which satisfy ir(a)E C E, Va € A, {TT, H} is said to be algebraically irreducible, if E = {0} or H are the only linear subspaces of H which satisfy n(a)E C E, Vo 6 A. Clearly, if {n, H} is algebraically irreducible for A, then {-IT, H} is also topologically irreducible for A. Proposition 5.3.7. Let A be a real C* -algebra. (1) Let {TT,H} be a * representation of A. Then {~K,H} is topologically irreducible for A, if and only if, (n(A)')H = R(2) Let p € S{A), and {TTP,HP} be the * representation of A induced by p. Then {np, Hp} is topologically irreducible for A, if and only if, p 6 V{A). And in this case, Hp = A/Lp, where Lp is the left kernel of p. (3) If {n, H} is a topologically irreducible * representation of A, then there is a p E V(A) such that {n,H} = {np,Hp} (unitarily equivalent). Proof. (1) It is obvious from Definition 5.3.6 and Proposition 4.3.4. (2) It is Similar to the complex case (see [26, Proposition 2.7.1]), and by Proposition 5.3.4. (3) Take f e H with ||£|| = 1, and let p(o) = (7r(o)e,0,

VaeA

Fundamentals

of Real

C*-Algebras

93

Further, define U : H -> Hp such that £/7r(a)£ — ap — a + Lp,\/a G A. Then the conclusion is immediate. Q.E.D. Remark. If (TT{A)')H and /x > 0 such that

= K, then for any O ^ a ' e n(A)' there are A > 0

a*a = A, and a a* = p,. Thus, a' has a left inverse and a right inverse, a' is invertible, and ir(A)' is divisible. By Theorem 2.2.2, ir(A)' = R,C or H as real Banach algebras. Later we can see that TT(A)' S R,C or i also as real C*-algebras (see Proposition 5.6.6). As well-known, for a (complex) C*-algebra B, and a * representation {a, K} of B, {a, K} is irreducible for B, if and only if, a(B)' = C. Proposition 5.3.8. Let A be a real C*-algebra, p be a pure real state on A, and L, Lc be the left kernels ofp,pc respectively. If pc — \{ip+Tp), where (p is a pure state on Ac, and Lv, L^ are the left kernels of(p,lp respectively, then we have the following: (i) L is a regular closed left ideal of A; (ii) Lc = Lv D Ljp\ (Hi) L is a maximal left ideal of A. Proof, (i) Let {n, H, f } be the cyclic * representation of A induced by p. From Proposition 5.3.4, H — A/L. Then there is u € A such that u + L = £. Thus a + L = 7r(a)£ = n(a)(u + L) = au + L, i.e., (au — a) S l , V a € A. Therefore, L is regular. (ii) Clearly, a + ib € Lv <=$> a — ib € Ljp, where a,b € A. Thus, if a + ib € Lp D Ljp, then a, b € L, and a + ib e Lc, i.e., L^ n L^ C Lc. Conversely, let a+ib G Lc, where a,b G A. Then a,b€ L, a, b G L^flL^, and a + ib G Lv n Ljp. Therefore, Lc — LVC\ Lip. (iii) We have two norms on H — A/L : ||o + L||i = p(a*a)* and \\a + L|| 2 = dist(a, L),

94

Real Operator Algebras

Va S A.(H, || • ||i) is a real Hilbert space, and (H, || • ||2) is a real Banach space. Clearly, \\a + b\\>p((a + b)*(a + b))? = \\a + b+ L\\i = \\a+ L\\i, Va € A,beL.

Thus ||a + L||i < | | a + L|| 2 ,Va£ A

By Banach theorem , there is a constant A > 0 such that X\\a + L\\2 < \\a + L\\i < ||o + L\\2,

Va € A.

Let £ = {L'\L' left ideal of A, and L c L'}. Since L is regular, it follows from Zorn lemma that C contains a maximal element, i.e., there is a maximal left ideal V of A such that L C L'. Clearly, L' is also regular. Similarly to the complex case (see [26, Theorem 2.7.7]), V must be closed. By L' 7^ A, we can take a € A such that dist(o, L') > 0 Then for any c g i ' , ||o — c + £||i > A||a — c + L||2 = Ainf{||o-c-6||

\beL}

> A i n f { | | a - d | | \d € L'} = Adist(a, L') > 0, i.e., ||(a + L) - (c + L)\\i > Adist(a,L') > 0,Vc e L'. Thus , in (H = A/L, || • ||i) we have dist(a + L, L'/L) > 0, and L'/L is not dense in H. On the other hand, L'/L is invariant for n(A) obviously. Since p is pure , and -K is topologically irreducible (see Poroposition 5.3.7), it follows that L'/L — {0},Z/ = L. Therefore, L is maximal. Q.E.D.

Fundamentals

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95

Theorem 5.3.9. Let Abe a real C*-algebra. (i) There is a bijection between the collection of all pure real states on A and the collection of all regular maximal left ideals of A: for any pure real state p on A, its left kernel Lp is a regular maximal left ideal of A; conversely, if L is a regular maximal left ideal of A, then there exists a unique pure real state p on A such that L = Lp. (ii) If L is a maximal left ideal of A, then L is regular if and only if L is closed. (Hi) If L is a closed left ideal of A, then L = n{Lp\p pure real state on A, and L C Lp} = ri{L'\L' regular maximal left ideal of A, and L c L'}. (iv) Let p be a real state on A. Then p is pure if and only if its left kernel Lp is a maximal left ideal of A. Proof. Similarly to the proofs of the complex case (see [26, Lemmas 2.7.4, 2.7.5 and Theorems 2.7.6, 2.7.7]), we can see (iii) and (ii). (i) From Proposition 5.3.8, Lp is a regular maximal left ideal of A for any pure real state p. Conversely, let L be a regular maximal left ideal of A. Then by (ii) and (iii), there is a pure real state p on A such that L = Lp. Now if p' is also a pure real state on A such that L = Lpi, then on A/L there are two Hilbert norms

II-IU

II-UP'

and a Banach norm ||o + L | | = i n f { | | o + 6|| \beL},

Va € A.

Prom the proof of Proposition 5.3.8(iii), these norms are equivalent each other on A/L. Thus, there exists a constant A > 0 such that p(a*a) > Xp'(a*a),

Va € A.

We may assume that A S (0,1). Then

p" =

(l-\)-l(p-\p')

is also a real state on A, and p = Xp' + (1 — \)p"• Since p is pure, it follows that p = p'. (iv) Suppose that Lp is a maximal left ideal of A. Of course, Lp is closed. From (ii), Lp is also regular. Further, by (i) there is a pure real

96

Real Operator Algebras

state p' such that Lp = Lp> = L. On A/L there are a Hilbert norm || • || p , and a Banach norm || • ||, and || • \\pi ~ || • || (see the proof of Proposition 5.3.8 (iii)). Moreover, on A/L there is a pre-Hilbert norm || • \\p, and || • \\p < || • || on A/L (see the proof of Proposition 5.3.8(iii)). Thus, there is a constant A > 0 such that Xp(a*a) < p'(a*a),

Va € A.

We may assume that A € (0,1). Since p' is pure, it follows that p' = p. Therefore, p is pure. Q.E.D. Remark. All of the above relations of pure real states and their left kernels are similar to the complex case (see [26, Section 2.7]). But for a pure real state p, generally we don't have N(p) ^ Lp + L*p, where N(p) = {a € A\p(a) = 0} is the null space of p, and Lp is the left kernel of p. This is different from the complex case. Let B be a (complex) C* -algebra, and {a, K} be a topological irreducible * representation of B. Then we have the well-known n-transitivity property for any positive integer n: if £i, • • •, £ n ; 771, • • •, r]n € K and {&, • • •, £„} is linearly independent, then there exists b £ B such that
1

Consequently, {a, K} is also algebraically irreducible. Thus we can define irreducible * representations for (complex) C*-algebras. However, the above n—transitivity property (n > 2) is not true generally in the real case. For example, consider following real C*-algebra A on real Hilbert space R 2 : A = {XI + where I = (

1 , and U — I

fj,U\X,p,eR}, 1 . Obviously, A' = A, (A')H =

AH = R, and A is irreducible on R2. Moreover, we can't find A, p, 6 R such that

(AJ + ^ ) ( J ) = ( A / + MC/)(J)^0. So for £1 = I °4i

=

1 , £2 = I

I 1 and 771 = 772 7^ 0, there is no a € A such that

Viii ~ 1>2. This means that 2-transitivity property is not true for

Fundamentals of Real C* -Algebras

97

the above real C*-algebra A. Notice that {XI + pU) > 0 4=> A > 0. Thus S(A) = V(A) = {p}, where p(I) = 1, and p(U) = 0. Further, N(p) = MU,LP = {0}, and N(p)^Lp + L*p. For a (complex) C*-algebra B and a pure state (p on B, in the proof of N(
Proof. By Proposition 5.3.7, we may assume that {n,H} — {TTP,HP}, and H = A/L, where p is a pure real state on A, and L is the left kernel of PLet E be a linear subspace of H such that Tv(a)E c £ , V a € A. Denote L' = {a€A\(a

+

L)€E}.

Then L' is a left ideal of A containing L. Since L is maximal (see Theorem 5.3.9), it follows that L' = L or A, i.e., E = {0} or H. Therefore, {K,H} is also algebraically irreducible for A. Q.E.D. Definition 5.3.11. A * representation for a real C*-algebra is said to be irreducible, if it is topologically irreducible or algebraically irreducible. Notes. The proof of Proposition 5.3.4 follows essentially that of H. Halperin ([19]) in the complex case. Moreover, in the proof of Proposition 5.3.8(iii), we introduced two norms || • ||i and || • H2 on H — A/L, and proved they are equivalent. M.Takesaki ([49, 50]) proved || • ||i = || • ||2 for the complex case.

98

Real Operator

References.

5.4

Algebras

[19], [26], [29], [30], [49], [50].

Ideals, quotient algebras and extreme points

Proposition 5.4.1. Let A be a real C* -algebra, and I be a closed twosided ideal of A. Then I* = I, and A/1 is also a real C* -algebra. Moreover, Ac/Ic = i.e., dist(a,I) = dist(a,Ic),Va ideal of Ac = A+iA).

A/I+iA/I,

6 A, where Ic = I+H (a closed two-sided

Proof. Similarly to the complex case (see [26, Theorem 2.4.8]) , we have I* = I, and ||a*a|| = ||a|| 2 ,

Va = a + I,

aeA

Let h* = h € A/1. We may assume h* = h. Clearly, a(h) U {0} C a{h) U {0} C R (see Lemma 2.4.5). Thus, A/I is hermitian, By Theorem 5.2.10, A/I is a real C*-algebra. Now for any a € A, dist(a, i") > dist(a, Ic) obviously. On the other hand, since Ac is a complexification of A, it follows that \\a-(b+ic)\\>\\a-b\\,

V6,cG/

(see Definition 1.1.1). Thus dist(a, Ic) > dist(a,/), and dist(a, 2") = dist(a, Ie), Va € A.

Q.E.D.

Proposition 5.4.2. Let A be a real C*-algebra, and I be a closed twosided ideal of A. (1) If p is a real state (or a pure real state) on A such that p\I = 0, let p(a) = p(a), Va = a + / 6 A/1, a € A, then p is a real state (or a pure real state) on A/1. Conversely, ifpis a real state (or a pure real state) on A/1, then there is a unique real state (or a pure eal state) p on A such that p\I = 0 and p(a) = p~(a),Va € A.

Fundamentals of Real C* -Algebras

99

(2) Let B be a real C* -subalgebra of A. Then B+I = {b+c\b eB,c€l} is also a real C*-subalgebra of A, and the real C* -algebras (B + I)/I and B/(B C\I) are canonically * isomorphic. (3) Let $ be a * homomorphism form A into another real C* -algebra B. Then $(A) is a real C*-subalgebra of B. In particular, if {ir,H} is a * representation of A, then n(A) is a real C*-algebra on H. These proofs are similar to the complex case (see [26, Section 2.4]). Theorem 5.4.3. Let A be a real C*-algebra, S be its closed unit ball, i.e., S — {a € A\ \\a\\ < 1}, Sc be the closed unit ball of Ac , and xo € S. Then the following statements are equivalent: (i) xo € ex S (the set of all extreme points of S); (ii) x0 € ex Sc; (Hi) (1 - XoX0)A(l - X0XQ) — {0}. If XQ £ ex S, then xo is a partial isometry, i.e., XQXQ and XOXQ are projections. Moreover, S has at least an extreme point if and only if A is unital. Proof. Let Xo € ex S. Similarly to the complex case (see [26, Theorem 2.5.1]) and by Proposition 5.1.4, we can prove that x0 is a partial isometry, and (1 — XoXo)A(l — XOXQ) = {0}. Further, (1 — XOXQ)AC(1 — XOXQ) = {0}. Thus, xo € ex Sc. The other conclusions are easy. Q.E.D. References.

5.5

[26], [30].

The bidual of a real C*—algebra

First we consider the complex case. Let B be a (complex) C*-algebra, and S = S(B) be the state space of B. Then the universal * representation

^

¥>es

of B is faithful, and 7T.(I)(6) = t r ( i TTu(b))

J

100

Real Operator Algebras

(Vb e B,t € B*) is an isometry from B* onto B*, where B — nu(B)" is a VN algebra on B„,B\, = T(HU)/B±,B~± = {s <E T(Hu)\tr(sx) = 0,Vx e B}, B = (B*)*, t e T{HU) and F = (f + B ± ) € B*. Further, (TT*)* is an isometry from B** onto (B*)* = B , and (7r»)*|B = nu. For simplicity, we denote (TT*)* = TT„ : B** -=*• B = 7r„(B)". Transferring operator multiplication of B to B**, we shall prove it is the first and second Arens products in B** (see Definition 2.6.1). Lemma 5.5.1.

For any t € T(HU),X, Y 6 B**, we have {Y)t),

7r.(t)X = 7r.(t7r u (X)),

where F = t + Bx,nJY)t = Ttu{Y)t + B±,t^JX) and YTrt(t),Tvt(t)X € B* (see Section 2.6).

= tnu(X)

Proof. For any b, c € B, (7r.(t)6)(c) = 7r„(t)(bc) = tr(f7r„(6)7ru(c)) = 7r*(t7r„(6))(c).

Thus n*(tTLjb)) = 7r»(t)b, Vb € B. Further, by

yvr* (f)(6) = r(7r.(t)6) = y(7r»(fS&)) = nu(Y)(t^{b))

= tr(7r„(y)i7r„(6))

= ^(7^00(6),

we have F7r*(t) = Similarly,

VbeB,

ir*(nu(Y)t).

&7r,(i)(c) = 7r*(f)(c6) = tr(t7r„(c)7r„(6)) = tr(7ru(b)t7r„(c)) =

n*(nu(b)t)(c),

+ B±€

B»,

Fundamentals

of Real C -Algebras

101

Vc G B, and &7r*(f) = 7r*(7r„(6)t),V6 6 B. Further, ir*(t)X(b) = X(6TT*(2)) =

X{^(^h)i))

= iru(X)(irJfi)t)

=

= ti{tiru(x)nu{b)) V6 6 B. Therefore, irt(t)X =

tr{iru{X)iru(b)t)

=

ir*(tnjX))(b), Q.E.D.

TT,(^(X)).

Proposition 5.5.2. Lei B be a (complex) C*-algebra. Then B is regular, and by Arens product and natural * operation, B** is a W* -algebra such that B is a C*-subalgebra of B**. Moreover, {iru,Hu} is a faithful W*-representation of B**, and Tru(B**) = iru(B)" = B. Proof. Let X,Y G B**, XY and X • Y be first and second Arens products of X, Y respectively (see Definition 2.6.1). Then TTU(XY){7) = XY(lT*(7)) =

X(Y*.(fi)

= X M T T ^ F ) * ) ) = tr(7r u (X)7r u (y)t)

=

(nu(X)nu(Y))(T)

and •KU{X • Y)(T) = {X • y)(7T.(t)) = = Y(7tt{t^jX)))

Y(lT*(fiX)

=

= tr(iru(X)nu(Y)t)

tr(Tru(Y)tnu{X)) =

(TT„(XK(Y))(?)

by Lemma 5.5.1, Vt 6 B*. Therefore iru(XY)

= TTU(X)TTU(Y)

= nu(X • Y),

XY = X • Y, \/X, Y G B**, and B is regular. Now define naturally X* (F) = XjF^),

F* (b) = F~(F),

VX G B*\ F G B*, b G B. We s_ay TT U (X*) = TT U (X)*, VX € B**. Indeed, 7T„ = (7r»)* is cr(S**,B*) - < T ( B , B « ) continuous. For X e B**, take z* G B such that xi ->• X in <7(J3**,B*). Then a;,* ->• X* in cr(B**,S*) obviously.

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Real Operator Algebras

Further, by 7ru(z(*) = TTU(XI)*, VI, and nu(xi)* ->• iru(X)* in a(B, B*), we have TTU(X*) = nu(X)*. Q.E.D. Remark. By TT„(X*) = TT U (X)*, VX 6 B**, we can also see that B** is a * algebra in first Arens product (nu((XY)*) = (TTU(X)TTU(Y))* = nu{Y*X*), and (XY)* = Y*X*, V X , F € B " ) . Then from Proposition 3.1.7, B is regular. Now we consider the real case. Let A be a real C*-algebra. Then Ac — A+iA is a (complex) C*-algebra, and Ac is regular by Proposition 5.5.2. Form Proposition 2.6.4, A is regular. Further, by Proposition 3.1.7 A** is a real Banach * algebra in Arens product and natural * operation such that A is a closed * subalgebra of A**. Since Ac is a complexification of A, it follows from Propositions 1.1.4 and 2.6.4 that A*c* =

A"+iA"

and A** is a closed real * subalgebra of A**. Thus, A** is also a real C*algebra from Definition 5.1.1. Clearly, Arens product in W*-algebra A*c* is a(A**,A*) — a(A**, A*) continuous in each variable. Thus , Arens product in A** is also -a are a(A**,A*) — a(A**, A*) continuous from A** to A**. Moreover, A*c* = A**+iA**. Proposition 5.5.4. Let A be a real C*-algebra, and {K, H} be a * representation of A. Then {IT, H} can be uniquely extended to a* representation {W, H} of A** such that n is cr(A**, A*) - a(B(H),T(H)) continuous and H{A**) = n(A)". Proof. Clearly {nc — Tr+in,Hc = H+iH} is a * representation of Ac = A+iA. Then {TTC, HC} can be uniquely extended to a * representation {nc,Hc} of A** such that Wc is a{A**,A*c) - a(B{Hc),T(Hc)) continuous and 7fc(A**) = nc(Ac)". Moreover, Tfc{A*c*) = 5r c (i4")+i7f c (i4"),

7rc(Ac)" = 7r(A)"+nr(A)".

Fundamentals of Real C-Algebras

Let 7r = irc\A**. Then the conclusion follows immediately. T h e o r e m 5.5.5. space, and

103

Q.E.D.

Let A be a real C*-algebra, S = S{A) be its real state

K

= 0TT P , pes

HU =

Then A** is * isomorphic to A = nu(A)",

@HP}. pes

and n, : A» -11* A*, where

7r*(f)(a) = tr(t 7ru(a)), Va € A, r = t + I i € l » =

T(HU)/~A±.

Proof. By Proposition 5.5.4, {wu, Hu} can be uniquely extended to a * representation {wu, Hu} of A** such that Wu is cr(A**,A*) - a(B(Hu), T(HU)) continuous and nu(A**) = nu(A)" = A. We claim that Wu is still faithful for A** (tru is faithful for A, see Theorem 5.2.9). In fact, if Wu(a) = 0 for some a G A**, then it is easy to see (a*a)(p) = 0, Vp € <S. Further, by Theorem 5.2.8, (a*a)(/)=0,

V/eA*H.

Therefore a*a — 0,a = 0, and Wu is faithful for A**. Since nu : A** -—> A is cr — o- continuous, it follows that there is 7r* : A* —>• A* such that (7r*)* = Wu. Clearly 7r*(/)(a) = f(nu(a)) Va € A, f = t + A±€ T(HU)/AX

= A*.

= tr(t 7ru(a)), Q.E.D.

Remark. Since [A*+] = A*ff (C A* in general), the proof of Theorem 5.5.5 is some different from the complex case. Remark. Now we can regard the multiplication and * operation in A** as transferred from the multiplication and adjoint of operators in A. It is similar to the complex case (see [26, Theorem 2.11.3]) . References.

[3], [25], [26], [30].

104

5.6

Real Operator

Algebras

The uniqueness of * operation

It is well-known that * operation in a (complex) C*-algebra is unique, i.e., if (B, || • ||) is a (complex) Banach algebra, and operations * and # o n B are such that (B,*, \\ • ||) and ( £ , # , || • ||) are two (complex) C*-algebras, then we must have * = # on B (see [42, Theorem 4.8.18]). The same question for the real case is very interesting, i.e., given a real Banach algebra (A, \\ • ||), and operations * and # on A such that (A, *, || • ||) and (A, # , || • ||) are two real C*-algebras, is it true * = # on i ? In this section, we shall give an affirmative answer. Lemma 5.6.1. Let Ai,A2 he two united real C*-algebras, and T be a unital linear isometry from A\ onto A?.. Then we have T*S2 = <Si, and T{Ai)K

=

{A2)K,

where Si = S(AS) is the real state space of Ai, i — 1,2. Proof. Clearly, r(T>2)(i1) =

P2(Ti1)

=

P2(i2)

= i,

\ III^H = HftH = 1, Vp2 G «S2. By Lemma 4.5.5, we have T*p2 € Si, i.e., T*S2 C Si. Furthermore, T*S2 = S\. Leta 2 € A2. By Proposition 5.2.6(6), a 2 G (A2)K if and only if p 2 (a 2 ) = 0, Vp2 e 5 2 . Now let ai € (AI)K- Since T*p2 € 5 i for any p 2 e S2, it follows that p 2 (Ta 1 ) = (r*p 2 )(ai) = 0, and Tai €

{A2)K,

i-e., T(AI)K

C

T(Ai)K

(A2)K-

Vp2e52,

Furthermore,

= {A2)K.

Q.E.D.

Corollary 5.6.2. Let Ai,A2 be two unital real C*-algebras, and T be a unital linear isometry from Ai onto A2. IfT(Ai)}j C (A2)u, then T keeps the * operation, i.e., Ta\ - (Toi)*,

Vai e Ai,

Fundamentals

of Real C* -Algebras

105

and T(AX)H,K

=

(A2)H,K.

Proof. Let hi G {Ai)H, and fci € (Ai)K. {A2)K (Lemma 5.6.1), it follows that T(/i 1 +

Since Th\ G (A2)H and T(fci) e

fc1)*=T/i1-Tfci = (Th1 + Tk1)* = (T(h1 + k1))*,

i.e., T keeps the * operation. Now let h2 € (A2)H• Then there are hi € (-Ai)ff and fci G {A{)K such that T(hi+A; 1 ) = /i2. Now by Lemma 5.6.1 and T(AI)H

C (A 2 )ij,

Tfci = ft2 - T / i i G ( J 4 2 ) K n ( i 4 2 ) f f = { 0 } .

Therefore, ki = 0, and T(A{)H

= (A2)H.

Q.E.D.

Lemma 5.6.3. Let A\,A2 be two unital real C*-algebras, and T be a unital linear isometry from Ai onto A2. IfT is also an algebraic isomorphism, then T{A{)H

Proof. Let a* = ai G {A{)H, (A2)K. For any p2 G <S2, p2{Ta\y=

c

(A2)H.

and Tai = h2 + k2, where h2 e (A2)n,

k2 G

p2((Tai)2) = P2{h22) + p2{kl) + p2(h2k2 + k2h2).

Clearly, h2k2 + k2h2 G (A2)K and p2$A2)K p2{Ta$ = p2{h\) + p2{kl),

= 0. Thus, Vp2 G S2.

(5.1)

106

Real Operator Algebras

On the other hand, T~lk2.

T - % = ai -

By Lemma 5.6.1, T _1 fc 2 € (^i)i<:. Then for any p± G Si, / 9 1 (T-

1

/i|) = / 9 1 ((a 1 -T- 1 fc 2 ) 2 )

= Pl{a\) + piiT^kl)

- px(ax • T~lk2 + T~xk2 •

ai).

Clearly, ax • T~lk2 + T~lk2 • ai € (Ai)jf, and /9i|(Ai)/r = 0. Thus, p1(a?)=p1(T-1^)-/9i(T-1fc2),

Vpie5i.

(5.2)

By T*S2 = 5i (Lemma 5.6.1), we have (T> 2 )(a?) =

P2{Ta\)

p2(h22) + p2(k22),

^

and {T*p2){a\)

(

=2) ( T ^ ) ^ - 1 ^ ) -

= P2{h\) -

(T'toXT-1®

p2{kl),

V/92 G S2. Therefore p2{kl) = 0, Since fc2 g

(A2)H,

Vp2 e 5 2 .

it follows that fc2 = fc2 = 0, i.e., T a i = h2 € (.A2)#. Q.E.D.

T h e o r e m 5.6.4. Let ^4i,A 2 fee two real C*-algebras, and T be a linear isometry from A\ onto A2. If T is also an algebraic isomorphism, then T keeps the * operation, and T{A!)H,K

=

(A2)H,K-

Proof. If A\ or A2 has an identity, then it is easy to see that A\, A2 are both unital and T is unital. By Lemma 5.6.3 and Corollary 5.6.2, the conclusions are obvious. Now let Ai and A2 be non-unital. On -Aj-i-R, define ||aj + A|| = sup{\\aibi+Xbi\\\bi

€ -Aj,||&i|| < 1|},

Fundamentals

of Real C

-Algebras

107

Vci; e At, A 6 R, i = 1,2. Then Ai+R and A 2 +R are unital real C*algebras, and T can be naturally extended to a unital algebraic isomorphism from A i + R onto A2+R. Moreover, it is easy to see that T is still isometric from Ax+R to A 2 +R. Therefore, T keeps the * operation. Q.E.D. Theorem 5.6.5. The * operation in any real C*-algebra is unique, i.e., if (A, || • ||) is a real Banach algebra, and (A, *, || • ||), (A, # , || • ||) are two real C*-algebras, then * = # on A. Proof. Consider T = id : Ax = (A, *, || • ||) — • A2 = (A, # , || • ||). Now by Theorem 5.6.4, the conclusion is obvious. Q.E.D. Proposition 5.6.6. Let A be a unital real C* -algebra. If A is divisible (in particular, AH = R, see the Remark following Proposition 5.3.7), then A is isometrically * isomorphic to the real C*-algebra R, C or H. Proof. Let D be the real C*-algebra R, C or H. By Theorem 2.2.2, there is an algebraic isomorphism T from A onto D such that \\Ta\\ = r(o),

Va € A.

\\Ta\\ < \\a\\,

VoeA

In particular,

Noticing that ||a*|H|a|HHI2=l|a*a||=rVa) = ||Ta* • Ta|| < | | T a * | | - | | T a | | < | | a * | | • ||a||, Va € A, we have ||ra|| = ||a||, Va € A. Now by Theorem 5.6.4, T keeps the * operation, i.e., A = D as real C*-algebras. Q.E.D. References.

[26], [28], [38], [42].

108

5.7

Real Operator

Algebras

Finite-dimensional real C*-algebras

Theorem 5.7.1.

Let A be a finite-dimensional real C*-algebra.

Then

A*Mni(D1)®---®Mnh(Dk), where Di = R, C or H, 1 < i < k. Proof. By Theorem 5.4.3, A has an identity 1 obviously, Denote the center of A by Z. Let Z = C(Cl, —). Since Z is finite-dimensional, we can write (fi, - ) = {tj,Sk,Sk\l

<j
m},

where tj = tj, ~s~k y£ Sk, Vj, A;. Then Z ^ [XJ, /fc°|l ) = 8k,k', fi1](sk') = i8k,k>, fl1\sk') = -i6k,k', Vj,j',k,k',l,imd n

m

j=l

fc=l

Thus we may assume that Z = R (i.e., .A is a real factor), or Z = C. (1) Let Z = C. Then we have Z = {a + /?x|a,/? G R } , where x is a central element of A such that a:* = —x, and x 2 = — 1. Consider the (complex) C*-algebra Ac = A+iA and its elements z\ = -(1+ix),

and z2 = - ( l - H ( - z ) ) .

Clearly, z\ and 22 are two orthogonal non-zero central projections in Ac, and z\ + Z2 = 1. Since the center of Ac is (complex) two-dimensional, it follows that z\ and zi are minimal central projections in Ac. Hence, AcZj is a (complex) finite-dimensional factor, and AcZj = Mnj(C), j = 1,2, i.e., Ac =

Mni(C)®Mn2(C).

Now we claim that as real C*-algebras A =i AcZl

S Acz2.

Then we have A = M n (C), where n = n\ —U2.

Fundamentals of Real C* -Algebras

109

Define a map from Acz\ onto A as follows: ^ , • .,, a — bx • .ax + b

, > a — bx,

Vo, b € A. If (a+ib)zi = 0, then a - bx = 0. So $ is well-defined. If a — bx — 0, then ax + b = (a — bx)x = 0, and (a-j-ib)zi = 0. Hence, $ is injective. Clearly, $ is real linear. Moreover, since ((a+ib)zi)* = (a*+i(-b*))z!, (a+ib)(c+id)zi

(a - bx)* = a* + b*x,

= ((ac — bd)+i(ad + bc))zi,

(a — bx)(c — dx) = (ac — bd) — (ad + bc)x, Vo, 6, c, d e A, $ is also * preserving and multiplicative. Of course, {a — bx\a,b G A} — A. Therefore, Aczx S A. Similarly, define a map: . . .,. a + bx • .b— ax (a+ib)Z2 = — H—

, > a + bx,

Vo, b G A. We can see that ACZ2 = A. (2) Let Z = R, i.e., A is a finite-dimensional real factor. Then we can take an orthogonal family {e^-11 < j
that 2_jei

=

•*•" ^ o w ^ Theorem 4.4.7, we have e» ~ e^, Vi,j. Moreover,

by Theorem 4.4.8,

A^Mn(^)®pAp, where p is a minimal projection in A. Since (pAp)n — Rp and A is a real factor, it follows from Proposition 5.6.6 that pAp = R or HI. Therefore, A S M n (R) or M„(H).

Q.E.D.

Remark. There is another proof of the structure Theorem 5.7.1 of finite-dimensional real C*-algebras in [17], but it involves the Wedderburn theorem. The proof here is elementary, and fits in the method of operator algebras. References.

[17], [26], [28].

110

5.8

Real Operator

Algebras

The enveloping real C*-algebra of a hermitian real Banach * algebra

Lemma 5.8.1. Let B be a real Banach * algebra, \\b*b\\ = \\b\\2, V& G B, and H = {b G B\b* = b}. If there exists a (real) linear subspace W C H such that a(w) c R, Vu; G W, and W = H, then B is hermitian and B is a real C* -algebra. Proof.

If B has no identity, we consider B+R and ||b + A|| = sup{||&c + Ac|| \c G B, \\c\\ < 1},

V6 G, A s M.. Similarly to the complex case (see [26, Proposition 2.1.2]), we have ||(6 + A)*(6 + A)|| = ||b + A||2, V i e B . A e R . And by ||6*6|| = ||6|| 2 ,||6*|| = ||b||, V 6 e B , we have \\b\\ = sup{||6c|| \c G B, \\c\\ < 1},

V6 € B.

Clearly, (W+R) C H+R, and (W+R) = H+R. Thus, we may assume that B has an identity. Since ||6*6|| = ||6|| 2 , V6 e B, if follows that |M! = r(a),

Va € H.

(5.3)

Now for any a G H and t G R, | | e « l < | | c o s ( t a ) | | + ||sin(ta)|| in Bc. Take {an} C W such that a„ —> a. Then cos(ta n ) —> cos(ta) and sin(ta n ) —> sin(to), Vi G E. Since a(tan) C K, Vn,i € R, it follows from (5.3) that || cos(to„)|| = r(cos(ta n )) < 1, || sin(ta n )|| = r(sin(to n )) < 1, Vn,t G R. Therefore, ||e i t a || < 2,Vt € R, and a (a) C R, i.e., B is hermitian. By Theorem 5.2.10, B is a real C*-algebra. Q.E.D. Remark. This lemma is essentially from B. important for this section.

Yood ([54]), and it is

Fundamentals of Real C* -Algebras

111

Definition 5.8.2. Let (A, || • ||) be a real Banach * algebra. || • || c is called a C* -quasi norm on A, if it is a quasi norm on A, and I H I c < NIclHIc,

||a*a|| c = |Mlc,

Va,6eA

Theorem 5.8.3. Let A be a hermitian real Banach * algebra. (1) If || • || c is a C*-quasi norm on A, J = {a e A\ \\a\\c = 0}, then J is a closed two-sided * ideal of A, R* — R*(A) (see Definition 3.5.1, the * radical of A) c J, ||a|| c < p(a) = r(a*a)?, Va e A, and B = (A/ J, \\ • ||~)~ is a real C*-algebra, where || • ||~ is the C*^norm on A/J induced by || • || c , and B = (A/ J, || • | | ~ ) _ is the completion of (A/ J, \\ • ||~). (2) There is a correspondence between the set of all C*-quasi norms on A and the set of all * representations of A, i.e., if \\ • \\c is a C*-quasi norm on A, then there exists a * representation {IT, H} of A such that \\a\\c — I K M I I J Va € A; conversely, if {TT,H} is a * representation of A, then || • ||c = ||7r(-)|| is a C* -quasi norm on A. (3) All C* -quasi norms on A are uniformly bounded in original norm || • ||, i.e., there is a constant K > 0 such that ||a|| c < if ||a||, V a € 4 and any C*-quasi norm || • ||c on A. Proof. (1) Clearly, J is a two-sided * ideal of A, and || • ||~ is a C*-norm on the real * algebra A/ J. Then B = (A/ J, \\ • ||~)~ is a real Banach * algebra, and ||6*6|| = ||6|| 2 , V6 e B. Since A is hermitian, it follows that o-B(a + J)U {0} C crA/J(a + J)U {0} C cr(o) U {0} C R, Va* = a £ A. Clearly, W — {a + J\a* = a € A} is a dense linear subspace of H = {b e B\b* = b}. By Lemma 5.8.1, B is a real C*-algebra. Now we can take a faithful * representation {??, H} of B. Clearly, ||5r(o)|| = ||a||~,

VaeA,

a = a + J.

Let 7r(a) = i?(a),

V a e A,

a = a+ J.

Then {TT, H} is a * representation of A, and ker n = J. Since 7r is continuous (see Definition 3.3.3), it follows that J is a closed two-sided * ideal of A.

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Real Operator

Algebras

Clearly, \Wc = \\S\\Z = \\*(fi)\\ =

\\*(a)\\
Va € A, and R* C ker7r = J by Definition 3.5.1. (2) It has been proved in the proof of (1) indeed. (3)Let R = R(A) be the radical of A. Since the * operation is continuous on A/R (see Lemma 3.1.4), it follows that there is a constant K > 0 such that \\{a + R)*\\
Va € A

If || • ||c is a C*-quasi norm on A, by Proposition 3.5.2 and (1) we have R C R* C J = {a € A\ \\a\\c = 0}. Then there is a C*-quasi norm || • ||~ on A/R induced by || • ||c. Similarly to the proof of (1), A/R is still hermitian. Therefore, from (1) we have IWIc = \\a\\7 < rA/R(a*a)? 1

^ lla*slll/fl ^

#1I«IU/H

^

K

Ml

where a = a + R, a* = a* + R, Va € A.

Q.E.D.

Now by Theorem 5.8.3, we can make the following. Definition 5.8.4. Let A be a hermitian real Banach * algebra. m(a) = sup{||a|| c | || • ||c is a C* — quasi norm on A} = sup{||7r(a)|| \K is a * representation of .4} is called the maximal C* -quasi norm on A. Clearly, R* = R*(A) = m " 1 ^ ) , m(a) < p(a) = r(a*a)i, Va € A, and m(-) is continuous in the original norm || • || on A by Theorem 5.8.3, i.e., there is a constant K > 0 such that m(a) < K\\a\\, Va € A. Proposition 5.8.5.

Let A be a hermitian real Banach * algebra.

Fundamentals

of Real C* -Algebras

113

(1) If A has an identity, then m(a) = sup{f{a*a)?\f =

G S(A)}

sup{f{a*a)i\feV(A)}

= sup{\\irf(a)\\ \f € S(A)} = sup{\\nf(a)\\

\feP(A)},

Va G A,

and | / ( a ) | < m ( a ) , V/ € S(A),

a G A.

(2) If A has no identity, A = A+R, then m{a) = sup{f{a*a)i\f

6 S(A)}

= sup{\\irf(a)\\ \f G S(A)} = sup{||7r/(o)|| | / > 0 and hermitian on A}, Va G A, where £(A) is defined in Definition 3.3.5. Proof. (1) Let {n,H} be a * representation of A, 7r(l) = 1, and a € A. Take f G H and ||f|| = 1. Then p(-) = ( T T ( - ) ^ O G S{A). Moreover, Ma)£\\2

= p(a*a) < sup{/(a*a)|/ e

S(A)}

Thus, by Krein-Milmann theorem we have m{a) <sup{f(a*a)$\f

€S{A)}

= sup{/(a*a)51/ G P(A)};

Va e A.

On the other hand, m(a)a>sup{||7r/(a)||2|/€5(A)} > sup{||7r / (a)e / || 2 i/ G S(A)} = sup{/(a*a)|/ € 5(A)}, and m(a) 2 > s u p { | | 7 r / ( a ) | | 2 | / € P ( i 4 ) } > { l k / ( a ) e / | | 2 | / G P(i4)} = sup{/(a*a)|/ G V(A)},

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Real Operator

Algebras

Vo € A. Therefore, we have the equality for m(-). Moreover, by Schwartz inequality we have \f(a)\ < f{a*a)? < m(a),

Va € A, f e S{A).

(2) It is obvious by (1).

Q.E.D.

Now let A be a hermitian real Banach * algebra, and R* = R*(A). Clearly, A/R* is still a hermitian real Banach * algebra. Let m(-) be the maximal C*-quasi norm on A/R*. Since A/R* is * semi-simple (see Definition 3.5.1), m(-) is a C*-norm on A/R* indeed. Moreover, there is a bijection between the collections of all * representations of A and A/R*. Thus, m(a) = fh(a),

Va € A,

a = a+ R*,

where m(-) is the maximal C*-quasi norm on A, i.e., fh o g(-) = m(-) on A, where q : A —> A/R* is the canonical quotient map. Moreover, by Theorem 5.8.3(1) and R* = m _ 1 (0), B = (A/R*,m(-))is a real C*-algebra. Definition 5.8.6. Let A be a hermitian real Banach * algebra, m(-) be the maximal C*-quasi norm on A, and R* = R* (A) be the * radical of A. Then the real C* -algebra.

B = (A/R*,m(-)r is called the enveloping real C*-algebra of A, where m(-) is the C*-norm on A/R* induced by m(-), and m(-) is indeed the maximal C*-quasi norm on the hermitian real Banach * algebra A/R*. Theorem 5.8.7. Let A be a hermitian real Banach * algebra, and B be the enveloping real C* -algebra of A. (1) If A has an identity, then there is a bijection between S(A) and S(B), i.e., if f € S(A), let f(a) = / ( a ) , Va e a 6 A/R*_, then J can be uniquely extended to a real state on B; conversely, if f G <S(-B), let / ( a ) = 7(H), where a = a + R* € A/R* C B, Va e A, then f € S(A). (2) If A has no identity, / > 0 and hermitian on A, then f e £{A), if and only if, there is a constant K > 0 such that |/(o)| < K m(a),

Va 6 A.

Fundamentals

of Real C*-Algebras

115

Moreover, in this case we have f\R* = 0. (3) If A has no identity, then there is a bijection between £{A) and the collection of all positive linear functionals on B, in a manner similar to ( ! ) •

Proof. (1) It is obvious by Proposition 5.8.5(1). (2) and (3). Let / € £ (A). Then we can assume / > 0 and hermitian on A = A+M- By Schwartz inequality and Proposition 5.8.5, we have | / ( a ) | 2 < Kf(a*a)

K2m(a)2

<

and | / ( a ) | < Km(a), Va € A, where K — / ( l ) . Moreover, let {717, # / , £ / } be the * representation of A induced by / . Then /(o) = (7r / (o)O,e/>=0,

VaeR*,

i.e., f\R* = 0. Now we can define f(a) = f(a),

VaeZeA/R*.

Clearly, / > 0 and hermitian on A/R*, and \f(a)\ = \f(a)\ < Km(a) = Km(a), Va G A/R*. Thus , f can be uniquely extended to a positive linear functional on B. Now let / > 0 on B, and f(a) = /(H),

Va 6 A and o = a + R* G A/R* C B.

By Proposition 5.2.6, there is a constant K > 0 such that f(b)2
VbeB.

Thus, / ( a ) 2 < Kf(a*a), Va € A, and / 6 £(A) by Proposition 3.3.6. Finally, if / > 0 and hermitian on A, and there is a constant K > 0 such that |/(o)| < Km(a),

Va € A,

then f\R* = 0 obviously. So we can define f(a) = f(a),

Vaea

= a + R*

eA/R*.

Clearly, / > 0 and hermitian on A/R*, and |/(a)| = | / ( a ) | < Km(a) — Km(a), Va € A/R*. Therefore, / can be uniquely extended to a positive linear functional on B, and / € £{A) by the discussion of the preceding paragraph. Q.E.D.

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Real Operator Algebras

P r o p o s i t i o n 5.8.8. Let A be a hermitian real Banach * algebra without identity. (1) If f € £ (A), let {TT, H} be the * representation of A induced by f, then there is a £ € H such that 7(AJZ = H,

/ ( a ) = <7r(a)£,0,

n(a)£ = a + L,

Va € A, where L is the left kernel of f. (2) Let {IT, H, £} be a cyclic * representation of A, and /(•) = (vr(-)^, £). Then f G £ (A), and {TT, H, £} = {717, # / , £ / } (unitarily equivalent), where {ftf,Hf,£f} is the cyclic * representation of A induced by f (as (1)). (3) Let f g £ (A). Then f has a minimal positive extension on A = A-j-R, i.e., there exists a minimal p, > 0 such that any extension of f to a linear functional f on A is positive if and only if / ( l ) > /J,. Moreover, this M = ll£l| 2 = IMIJ where {TT,H,£} is the cyclic * representation of A induced by f (as (1)), and ip is the positive linear functional on B corresponding to f (see Theorem 5.8.7(3)). Proof. (1) Let / € £ (A), and {-K, H} be the * representation of A induced by / . Then H = (A/L, {,))-,

and TT(O)(6 + L) = ab + L,

where L is the left kernel of / , and (a + L, b + L) = f(b*a), Va, b e A. By Theorem 5.8.7(2), f\R* = 0. If a e R*, then a*a € R*, and f(a*a) = 0, i.e., a € L; so R* C L. Now we can define f(a + R*) = / ( a ) ,

Va € A,

and / > 0 and hermitian on A/R* obviously. Let {7?, H} be the * representation of A/R* induced by / , i.e., H = {{A/R*)/L,

( , ) ~ ) - , and n{a + R*)(b + R*+ L) = ab +R*+

L,

where L is the left kernel of / , and (a + R* + L, b + R* + L)~ = /((&* + R*)(a + R*)) = f(b*a), Va,be A. By Theorem 5.8.7(3), / can be uniquely extended to a positive linear functional ip on B, where B is the enveloping real C*-algebra of A. Let {ntp,Hv,£
(, )ip)'

Fundamentals

117

of Real C* -Algebras

and M 6 ) ^ = b + L^,

Wb € B,

where Lv is the left kernel of
= (p(b^h),

V61,62 € B.

Considering the maps: (a + L) —• (a + R* + L) —> (a + R* + Lv) and 7r(a) —• 7r(a + i?*) —• 7r„(a + J?*), we can let H = if = if^,, and 7r(o) = 7r(a + i?*) = irv(a + R*), Va € A. Therefore, there is a £ e i7 such that :

4A)I=H,

/(o) = (7r(o)e,0.

7r(a)£ = a + L,

Vae A (2) Let {7r, .ff,£} be a cyclic * representation of A, and /(•) = (7r(-)£,f). Clearly, / € 5(A). Let {7T/,.ff/,£/} is the cyclic * representation of A induced by / as in (1), and Uir(a)£ = a + Lf,

Va e A,

where Lf is the left kernel of / . Then U can be uniquely extended to a unitary operator from H onto Hf. Since Uw(a)ir(b)Z = ab + Lf=

Trf(a)UTr(b)£,

Va, be A, it follows that Un{a)U-1 = 717(a), Va € A. By 7r/(a)£/ = a + L / = U-K{a)TJ-lUt, = 7r/(a)[/£,

*f(a)(Ut-tf)=0, (UZ-Zf,nf(a)t;f)=0,

Vae A

we have U£ — £/. Moreover, M<*)e,0 = / ( a ) = (Uir(a)U-%,if) Therefore, {w,H,£} S {TT,, # , , £ , } .

= (^(atf/.e,),

Va 6 A.

118

Real Operator

Algebras

(3) Let / 6 £{A), and {-K, H, £} be the cyclic * representation of A induced by / as (1). Let / be a positive extension of / on A, and {n, H, 1} be the cyclic * representation of A induced by / . Let U(a + L) = 7r(a)l, Vo € A. Clearly, U can be uniquely extended to a unitary operator from H onto H' = 7r(A)l, and n(a)\H' = Uir(a)U-\ Let r] = UteH'.

Va € A.

Since 7r(o)f7 = UU-xv(a)UZ

= Un(a)$ = 5r(a)T,

and 7?(a)(T - 77) = 0, Va € A, it follows that (I - J])±H' in 5 . Let H" = R(l - 7?). By 5' © #"

= TT(A)1 © R(T - 77) = TT(A)T +

Rl

D5r(i4)T + RT=7r(A)T, we can see H = H' ® H". Therefore, H = H'® H", H' = 7?(A)1, 5 " = R(T - 77), {^|A,H',77}S{7r,^,e}, 7?(o)|H" = 0, V o e A Moreover, /(l) = ||l||2-||77||2 + ||l-77||2 = ||e||2+||l-7?||2. Thus / ( l ) > ||£|| 2 = /i. By /(•) = (7r(-)e,0 and |<7r(a)£,£)| < ||7r(a)£|| • ||£||, we have f(a*a) + 2Xf(a) + A2/^ > 0,

Va e A, A <E R.

Now if / is a linear extension of / on A with / ( l ) > /J, then 7((a + A)*(o + A)) = {f (a*a) + 2A/(a) + A 2 /J) + A2 (7(1) - M) > 0, Va € A, A € R . Hence / is positive on A. Finally, from the proof of (1) above , we have

f(a) = <7r(a)£, 0 = M « + &')&> U = f(a + ^*). Va e A Therefore, ||v>||B = | | ^ | | 2 = ||£|| 2 .

Q.E.D.

Fundamentals

of Real C* -Algebras

119

Definition 5.8.9. Let A be a hermitian real Banach * algebra without identity, p G E{A) (see Definition 3.3.5) is called a real state on A, if the minimal positive extension p of p on A = A-j-R is p(l) = 1. Denote S(A) = {p\p real state on A}. S(A) is called the real state space of A. Clearly, S(A) = S(B) (affine homeomorphism) by Proposition 5.8.8, and it is the same one as in Definition 3.3.7 if A has an identity. Proposition 5.8.10. Let A be a symmetric real Banach * algebra. Then m(-) = p(-) on A and R C p _ 1 (0) = R*. Moreover, if A has no identity, and / > 0 hermitian on A, then f G S(A), if and only if , there is a constant K > 0 such that \f(a)\ < Kp(a),Va € A. Proof. By Theorem 5.8.7(2), and Proposition 3.5.2, it suffices to show that m(-) = p ( - ) on A. By Definition 5.8.4, and the discussion following Definition 3.3.3, m(-) < p(-) on A. On the other hand , p(-) is a C*-quasi norm on A by Theorem 3.6.2(7). Therefore, p(-) < m(-), and p(-) = m(-). Q.E.D. Proposition 5.8.11. Let A be a hermitian real Banach * algebra with identity. Then A is symmetric, if and only if, m(-) = p(-) on A. Proof. If suffices to prove the sufficiency by Proposition 5.8.10. For any a € A, let A = r(a*a), and b — X — a*a. Then for any / € 5(A), we have f(b2) = A2 - 2A/(a*o) +

f(a*(aa*)a).

By Proposition 3.3.2, f(a*(aa*)a) < r(aa*)f(a*a)

= A/(a*a),

and f(b2) < A2 - Xf(a*a) < A2. By Proposition 5.8.5 and p(-) = m(-) on A, we have r(b2) = r(b*b) = p(b)2 < sup{/(6 2 )|/ G S(A)} < A2, i.e., r(b) < A, or r(A — a*a) < A. Since A is hermitian, it follows that a(a*a) > 0, Va G A, and A is symmetric. Q.E.D.

120

Real Operator Algebras

References.

5.9

[26], [36], [40], [41], [53], [54].

* Representations of abelian real C*—algebras

In this section, let A be a unital abelian real C*-algebra, ft be its spectral space (a compact Hausdorff space), and A = C(ft, —) (see Proposition 5.1.4). Proposition 5.9.1. If{n, H} is a cyclic * representation of A, then there is a unique (in the sense of equivalence) regular Borel measure fi on CI such that fio — = (j, and {7r,i/}^{$M,L2(Q,^,-)}, where ($ M (a)/)(i) = a(t)f(t),

Vf G ft,/ G L 2 ( f t , ^ , - ) , a £ A.

Proof. Let £ G H be a cyclic vector for {it, H}. Then {-KC, HC, £} is also a cyclic * representation of Ac, where Ac = A+iA,nc = it+in, Hc = H+iH. By [26, Theorem 5.4.1], there are a regular Borel measure \x on ft and a unitary operator U from Hc onto I/ 2 (ft, ^t) such that (irc(x)£,£) = / x(t)dfi{t), Jn

Vz G Ac,

(Uirc(x)Z)(t) = x(t),

Vx eAc,

te ft,

^ {7r M ,L 2 (ft,/i)}, i.e.,

and U : {KC,HC}

(Unc(x)U-1f)(t)

= x(t)f(t),

a.e.fx,

Vz G Ac, / G L 2 (ft,n), t G ft. Since (7rc(a)f,£) = (n(a)£,£) = (7r(a)f,f), Va G A, it follows that / a(t)dn(t) = / a(t)dfi(t) Jn Jn = [ a(t)d/j,(t) = [ a{t)dfj,(t), Jn Jn Vo G A. Further, / x(t)dfi(t) = J x(t)dfj,(t),

Jn

Jn

Vx G Ac.

Fundamentals

of Real C*-Algebras

121

Thus, \i — \L o - . On the other hand. Uirc(a)£ = Uir(a)£ = o(-) e C(Sl, - ) C L2(Q, /i, - ) , Vo G A, and UH = L2(Ct, /x, - ) . Therefore, {7r,tf}^{
Q.E.D.

Let i J be a real Hilbert space. Similarly to Definition 1.2.1, we can define a spectral pair {ei(-),e2(-)} over (H,fi, —). Theorem 5.9.2. Let {•K, H} be a unital * representation of A. Then there exists a unique spectral pair {ei(-),e2(-)} over (H,Cl,—) such that 7r(o) = / Rea(t)dei(t) — /

Jn

Ima(t)de2(t),

Jn

Va G A, and I Ima{t)dei(t) Jci Moreover, ei(-),e2(-)

= / Rea(t)de2(t) = 0, Jo.

Va € A.

G ir(A)' HIT (A)".

Proof. If {n,H} admits a cyclic vector, then from Proposition 5.9.1 we have {n,H} S { $ M , L 2 ( f t , / / , - ) } . Let (c(A)/)(t) = XA (*)/(*),

V/6l2(fl,/i)

and any Borel subset A of fi. It is easy to see that e(-) is a spectral measure over (L 2 (fl,/z),fi). We claim that e(A) = e(A), V Borel subset A of fi, where e(A) is defined by the decomposition of L 2 (ft,/i) = L2(Cl,fi, —) +iL2(fl, n, - ) . In fact, V Borel subset A of CI, f G L2(fl, /i, - ) , t G Q, (e(A)/)(t) = * * © / ( * ) = XA(t)f(t)

= (e(A)/)(i).

122

Real Operator Algebras

If e(A) = ei(A) + ie 2 (A), where ei(A),e 2 (A) e B(L2(n,

n, - ) ) , then

(i(A)/)(i)=(e1(A)/-tea(A)/)(i) = (c1(A)/)(t)+i(ea(A)/)(t) = (e(A)/)(i) = (e(A)/)(t), V/ e L 2 (ft, £*,-),£ e H. Thus , e(A) = e(A), VA. Moreover, it is easy to see that •KC(X)

= / x(t)de(t),

Ja

Vx € Ac.

Thus, the pair {ei(-)> e 2( - )} satisfies our requirements. For a general representation {TV, H}, since it is a direct sum of a family of cyclic * representations and a zero representation, we can also find the required spectral pair {ei(-),e2(-)}Q.E.D. Remark. By Proposition 5.1.6, Theorem 5.9.2 is a generalization of Theorem 1.2.3. Reference.

[26].

Chapter 6

Real W*—Algebras

6.1

Definition and basic properties

Definition 6.1.1. A real C*-algebra M is called a real W*-algebra, if the (complex) C*-algebra Mc = M+iM is a (complex) W*-algebra. Proposition 6.1.2. Let M be a real C*-algebra. Then M is a real W*algebra, if and only if, M is isometrically * isomorphic to a real VN algebra on some real Hilbert space. Moreover, any real W* -algebra is unital. Proof. If M is a real VN algebra on a real Hilbert space H, then Mc = M+iM is a VN algebra on Hc = H+iH (see section 4.3). Thus, Mc is a W*-algebra, and M is a real W*-algebra. Conversely, let M be a real W*-algebra. Then we may assume that Mc = M+iM is a (complex) VN algebra on some (complex) Hilbert space K. Let H = (Kr, {, )r = Re(,)), and n be the identical * representation of M on the real Hilbert space H. Clearly, {nc = n+in, Hc = H+iH} is a faithful * representation of Mc. Thus, {Trc,Hc} is a W*-representation of Mc, and 7rc(Mc) is a VN algebra on Hc. Since Tt{M)+m(M)

= 7rc(Mc) =

TT C (M C )"

=

n{M)"+in(M)",

we can see that n(M) = n(M)" is a real VN algebra on H, and M = 7r(M). Moreover, if M is a real W-algebra, then the W*-algebra Mc is unital, and the identity of Mc must be in M. Q.E.D. Remark.

From Proposition 6.1.2, we see that Definition 6.1.1 is a nat123

124

Real Operator Algebras

ural abstraction of real VN algebras. Moreover, most results of real VN algebras (see Chapter 4) are true for real W*-algebras. Let M be a real W^-algebra, and Mc = M+iM. Prom Proposition 4.3.3, "-" operation is o~{Mc, Mct) -continuous in M c , and M = {x G Mc\x = x}. Conversely, let TV be a (complex) V7*-algebra, and "-" be a conjugate linear * algebraic isomorphism of N with period 2. By Proposition 5.1.3, the fixed point algebra M = {x G N\x = x) is a real C*-algebra, and N — M+iM is a complexification of M. Thus, by Definition 6.1.1, M is also a real T V algebra. Moreover, since the multiplication in N is a(N, N^-continuous for each variable, it follows that the multiplication in M is also a- and • —> -a are a(M, Mt)-continuous in M; (v) a (N,N.)\M ~ a (M,Mt), s (N,N.)\M ~ s (M,M.), s* (N,N*)\M~s* (M, M»), andr(AT,iV»)|M~T(M,M»). Lemma 6.1.4. Let E be a (complex) Banach space, and V be a (norm) closed (complex) linear subspace ofE* such that (VDS) is a(E*, E)-dense in S, where S is the closed unit ball of E*. If any proper (norm) closed (complex) linear subspace ofV is not cr(E*,E)-dense in E*, then V* = E, i.e., for any F G V*, there is unique x G E such that F = x\v and ||JF|| = ||x||; conversely, for any x G E we have x\v G V* and \\x\\ = \\x\v\\Proof. (1) If E is canonically embedded into E**, and V1

= {X G

125

Real W* -Algebras

E**\X\v = 0}, then E** =

E+VL.

In fact, letx eEnV1. Since V is a(S*,^)-dense in E*, it follows that a; | J; = 0, and x = 0. Now let X € £**\V X . Denote W = { / € n Y ( / ) = 0}. Then W is a proper closed subspace of V as X & V x . Prom the assumption, W is not
i.e., for any x e E we

||x|| = ||x + ^ 1 1 = inf{||x + Z\\ \Z € V x } ; conversely, for any X £ E** there is x 6 E such that X — x € V^-1. In fact, let x G £ and Z € V x . Clearly, \\x + Z\\=Sup{\f(x

+

> s u p { | / ( x + Z)|

Z)\\f£S} \feVnS}

= sup{|/(x)| | / € 7 n S } = W since Z € V1- and since (V D 5) is
126

Real Operator Algebras

L e m m a 6.1.5. Let B be a (complex) C*-algebra, and V be an invariant (complex) linear subspace of B*, i.e., Laf,Rbf G V, where (Laf)(c) = f(ac),(Rbf)(c) = f(cb), \fa,b,ceB,f£V. IfVisa(B*,B)-denseinB*, then (VtlS) is cr(B*,B)-dense in S, where S is the closed unit ball of B*. Proof. For any b G B, regarding Rb as a linear operator from V to V, we have 11*6/11 = sup{|/(a6)| \a G B, \\a\\ < 1} < ||/|| ||b||, hence ||i? 6 || < ||b||, V6 G B. If Rbf = 0, V/ G V, for some b G B, then /(aft) = 0,V/ G V,a G B. Since V is o-(B*, B)-dense in B*, it follows that ab = 0, Va G B, and 6 = 0. Moreover, RaRb = Rab, Va, 6 G B. Thus

116111 = 11^11,

V6GB

is an algebraic norm on B. By [26, Proposition 2.1.9], we have ||6|| < ||6||i = H.R&II, V6* = 6 G B. Moreover, \\Rb\\ = ||6||,

W =

beB.

By ||6||2 = ||6*6|| = ||E 6 . i ) ||<||^.||||fi 6 ||

< \\n m = \\b\\2, we have \\Rb.\\ ||il 6 || = ||6*|| ||6||, V6 G B. But \\Rb*\\ < ||6*|| and ||i? 6 || < ||6||, so \\b\\ = \\Rbl Clearly, ||L 0 /|| < ||a|| ||/||, and Laf ||a|| < 1. Thus,

V&GB. G V n 5, V/ G V n S,a G B and

su P {|/(6)| | / e v n S } < \\b\\ = \\Rb\\ = sup{|/(afc)| | / G V n 5, a G B and ||a|| < 1} = sup{|(L«/)(6)| | / G V fl 5, a G B and ||a|| < 1}

<sup{|/(6)||/evns},

Real W*-Algebras

127

and ||6||=8up{|/(6)||/e^n5},

V&GB.

If there exists 0 ^ / G S\V n S", where F n ^ is the a(B*, £)-closure of V n 5, then by the separation theorem we have b G B such that Re/(6) > sup{Reg(6)|5 € F n 5 } . Since V^ D S is circled, it follows that sup{Re<,(6)|<7 G V n 5 } = sup{| 9 (6)| | f l 6 7 n S } = ||6||; and |/(6)| > Re/(6) > ||6||. This is a contradiction. Therefore, V D S is a(B*, 5)-dense in 5. Q.E.D. Corollary 6.1.6. Let B be a (complex) C*-algebra, V be an invariant (norm) closed (complex) linear subspace of B* and V be a(B*, B)-dense in B*. If any proper (norm) closed (complex) linear subspace of V is not cr(B*,B)-dense in B*, then V* — B, i.e., B is a (complex) W*-algebra, and V is the predual of B. Theorem 6.1.7. Let M be a real C*-algebra. Then M is a real W*algebra, if and only if , there exists a real Banach space M* such that M = (M*)*, and the multiplication in M is o-(M, M*)-continuous for each variable. Proof. The necessity follows from Proposition 6.1.3. Sufficiency. Let Mc» = M*-i-iM«, and define ||/ + ig\\ = sup{|(/ + ig)(x)\ \x G Mc, \\x\\ < 1}, V/,0 6 M*. Since M» «-*• M* <-» M* are isometric, and M* = M*+iM* (see Proposition 5.1.3), it follows from Theorem 1.1.2 that Mc» is a real Banach space, and M* e-> Mc* «->• M* are isometirc. We claim that Mc* is a(M*,M c )-dense in M*. In fact, if a, b G M are such that (/ + ig)(a + ib) = 0, V/,g G M „ then / ( a ) = /(&) = 0, V/ € M», and a = 6 = 0. By assumption, /(-a), /(a-) € M*, V/ e M*,a G M. Thus, (/ + ig)((a + ib)-), (/ + ig)(-{a + ib)) G M c „ V/,g G M*,a, 6 G M, i.e., Mc» is invariant.

128

Real Operator Algebras

Let E be a proper (norm) closed (complex) linear subspace of M c ». Then there exists 0 ^ F € (M„)* such that F\E = 0. Denote x = Re(-F|M„), y = Im(F|M»). Clearly, x,y € M, and F = x+i'y. Thus, x+iy ^ 0, (z + iy)|.E; = 0, and E is not
Let A be a real C*-algebra. Then A** is a real W*-

Proof. It is obvious from Theorems 5.5.3 and 6.1.7.

Q.E.D.

Lemma 6.1.9. Let A be a real C*-algebra, A* be a real Banach space such that A = (A*)*, and A — H+K, where H = {a € A\a* = a},K = {a € A\a* = —a}. Then A is unital, and K is a <j(A,A*)-closed subset of A. Proof. Since S (the closed unit ball of A) is cr(A, A»)-compact and convex, it follows that S admits an extreme point. By Theorem 5.4.3, A is unital. Now it suffices to show that if {ke} c K, \\ki\\ < C (a constant), V7, and h

a(

± 4 0 h = h*, then h = Q. In fact, for sufficiently large n,

||A,+n|| = ||*?-n2||* < ( C 2 + n 2 ) 5 < ( l + n ) < ||ft + n||, where we may assume h ^ 0 and 1 € o(h). Then ||ft + n||
in A. Q.E.D.

Proposition 6.1.10. Let A be a real C*-algebra, At be a real Banach space such that A = (At)*, and A = H+K be as in Lemma 6.1.9. If H is a (A, A*)-closed in A, and the map • —> a • or • —> -a is o(A, A*)-continuous

in A for each a € A, then A is a real W*-algebra.

Real W

129

-Algebras

Proof. By Lemma 6.1.9, K is a(A, A*)-closed in A. If {oj} C A, \\cn\\ < 1,V7, and ai - ^ 0, let a* = /i/ + ki, then ||/i/|| < 1, ||fc/|| < 1,V/. Since the closed imit ball of A is a(A, j4*)-compact, we may assume that hi -—> h, ki -2-» k. Since H and K is
EK(ti'i')

jtft,

-it/l)| =53l(*y^n»7n>|

< ||(*i',-')lll. Vi,J, V normalized orthogonal sequences {£n}, {%} in H. Thus, tij € T(H),Vi,j, i.e., T{Hn) = {{Uj^tij e T ( t f ) , l < i , j < n } . If (%) € M n (A)j_, then (U'j')(eija) = 0,Va e A, and t^- € i4j_, Vi,j. Therefore, Mn(A)» =

T(Hn)/Mn(A)±

S M „ ( r ( f f ) M ± ) = M n (A.). Q.E.D. JVotes. Lemma 6.1.4 is due to J.Dixmier ([7]). Lemma 6.1.5 and Corollary 6.1.6 are due to S.Sakai ([45]). Theorem 6.1.7 is due to B.R.Li ([25]). In the complex case, if AT is a (complex) C*-algebra, and N = (-N*)*, where N* is a (complex) Banach space, the cr-continuity of multiplication in TV for each

130

Real Operator Algebras

variable is satisfied automatically (See Sakai's theorem [46]). Thus, it is a natural question: whether the condition of cr-continuity for each variable can be omitted in Theorem 6.1.7 ([3 ,25])? J.M.Isidro and A.R.Palacios gave an affirmative answer ([22]). Therefore, a real C*-algebra M is a real W*-algebra, if and only if , there exists a real Banach space M» such that M = (M*)*. It is similar to the complex case. References.

6.2

[3], [7], [22], [25], [45], [46].

Normal linear functionals and singular linear functionals

Let M be a real W*-algebra, and M* be its predual (i.e., M = (M»)*). For a linear functional p on M, the concepts of "p > 0", "p real state", "p normal" and "p completely additive" are similar to Definitions 4.5.1 and 4.5.2. By Theorem 4.5.3, if p > 0 on M, then p e M* <==> p is normal •£=> p is completely additive. Denote the set of all normal real states on M by Sn(M). Then by Proposition 4.5.4, we have the following. Proposition 6.2.1. Let M be a real W*-algebra, M = (M*)*, and Sn = Sn(M). Then for each p € Sn, the * representation {np,Hp} of M induced by p (GNS construction) is cr(M, Mt)-o-(B(Hp),T(Hp)) continuous. Moreover, the normal universal * representation {TT = © p e 5 n 7TP,

H = ®pes„ Hp }

is a — a continuous and faithful. Let M be a real V7*-algebra, and M , be its predual. M* M- M* and M <-> M** canonically. Similarly to the complex case (see [26, Proposition 4.2.3]), we have the following. Proposition 6.2.2. Let P(X) = X|M*,VX G M**. Then P is a projection of norm one from M** onto M, and there is a central projection z of M** such that: (1) P is also a * homomorphism from M** onto M, and is a(M**,M*) - cr(M,Mt) continuous. (2) P is a * isomorphism from M**z onto M. If Q(: M -> M**z) is

Real W*-Algebras

131

the inverse of(P\M**z), then Qx = xz,Vx € M, and Q is also a(M,M*) — a(M**,M*) continuous. (3) Let M, 1 = {X e M**\P{X) = X\M„ = 0} = kerP. Then kerP = Mj- = M**(l-z), and M** =

M+M^

(clearly, this decomposition also determines the projection P : M** -> M and (I - P)M** = M^). (4) M* = RZM*, and M* =

M*+R(i-Z)M*,

where F is regarded as a a(M**, M*)-comtunuous functional on M**, and clearly RZF e M*(RZF(X) = F(Xz), VX € M**), VF G M*. Remark. Consider Mc„ = M*+iM*, M** = M**+iM** and

M*+iM*,Mc

PC(XC) = Xc\Mct,

= M+iM,

M* —

VXC 6 M*c*.

Then, we have PC\M** = P,kerP c = k e r P + i k e r P , M^+iM^, kerP c = M j , = M**(l - z), and etc.

Mci

=

Definition 6.2.3. Let M be a real W*-algebra, M* be its predual, and z be the central projection of M** as in Proposition 6.2.2. Then M* = Af.-i-%_,)M*,

M . = fizM*.

Any element of M* (a cr(M, M* )-continuous linear functional on M) is called a normal linear functional on M; and any element of R(^_Z)M* is called a singular linear functional on M. For any F € M*, we have the unique decomposition: F = Fn + F„

Fn = RzFeM*,

F3 =

R(1_z)F,

where Fn, Fs are normal, singular functionals on M respectively. It is easy to see that

11^11 = ||Fn|| + ||Fa||, VPeM*. Similarly to the proof of complex case (see [26, Theorem 4.4.5]), we have the following.

132

Real Operator Algebras

Proposition 6.2.4. Let f € M* and f be hermitian. Then f is normal, if and only if, f is completely additive on M, i.e., f(p) — ^ f(pi), V{p/} orthogonal family of projections in M, and p = ^ pi. Remark. This Proposition is a generalization of Theorem 4.5.3. Moreover, we must require the " / is hermitian" in the above Proposition. This is different from the complex case. Proposition 6.2.5. unique.

Let M be a real W*-algebra. Then its predual M» is

Proof. By [26, Theorem 4.4.5], / G M», if and only if, / G M*, f{M) C E, and / is completely additive on Mc, i.e., f(p) = J^ ; f(pi), V-fj?/} orthogonal family of projections in Mc = M+iM, and p = YliPi- Therefore, M* is unique. Q.E.D. References.

6.3

[26], [50].

Abelian real W*—algebras

Theorem 6.3.1. Let Z be a a-finite abelian real W*-algebra, Z = C(CI,T), where fi is the spectral space (compact Hausdorff space) of Zc = Z+iZ, and T is a homeomorphism of £1 with period 2. Then fi is a hyperstonean space, and there is a normal regular Borel measure v on fi such that u or = v,

supp v = f2, and

C(Q, r) = L°°(Sl, v, r ) = {/ G L°°(Sl, v)\f = f o r } .

Proof. Let Z c B{H), where if is a real Hilbert space. Then by Proposition 4.6.3, Z admits a separating vector fo € H. Clearly, £o is also a separating vector for Zc(c B(HC), where Hc = H+iH). Let / -> m,f be the * isomorphism from C(fi, T) onto Z (it can be naturally extended to a * isomorphism from C(f2) onto Zc), and let v be the regular Borel measure

Real

W*-Algebras

133

on fi such that /

r

fdv = (m/6,,6,),

V/6C((1).

Then by [ 26 , Theorem 5.3.1], we have supp v = il,C(Q) = £°°(n, i/), ft is hyperstonean space, and v is normal on £2. Since iif is a real Hilbert space, it follows that / fdv = (m/f 0 ,£o) = (m/£o,£o) Jn = / /di/ = / fdv — / fdv o T, 7n 7n Jn V/ € C(fi, r ) . Moreover, we have f fdv= Thus, v =

f fdv o r ,

V/ G C(fi).

vor.

FinaUy, from C(fi) = i°°(fi,i/),

c(n) = c(n>T)+iC(n,T), L°°(n,r) = L0o(n,i/,T)+»L0O(n,i',T)> and C(ft, r ) C £°°(ft, i/,r), it follows that C(fi,T) = £°°(ft, i/, T). Q.E.D. Remark. If fi is a compact Hausdorff space, r is a homeomorphism of H with period 2, and v is a regular Borel measure on fi such that VOT = v, then L°°(£l, v,r) is a cr-finite abelian real W*-algebra. In fact, from the cr-flniteness of L°°(il, v) ([26 , Proposition 5.3.2]) and Proposition 4.6.3, this conclusion is obvious. Remark. If £2 is a hyperstonean space, and r is a homeomorphism of fi with period 2, then C(Q, r ) is an abelian real W*-algebra. Moreover, if v is a normal regular Borel measure on fl such that VOT = v and supp v = fi, then C(fi,r) = L°°(ft, I/,T) is also , VZ 7^ I', and r = Uisupp vi is a dense open subset of CI.

134

Real Operator Algebras

Proof. Let {V;} be a maximal family of normal regular Borel measures on Cl such that vi=vio

r, supp vi H supp V{i = (j>, VZ ^ I',

and T = Ujsupp v\,. We need to prove that T is a dense open subset of Cl. Since supp vi is clopen ([26 , Proposition 5.2.6]), VZ, it follows that T is an open subset of Cl, and the closure T of T is also clopen subset of CI. Clearly, T(T)

= T,

T(CI\T)

=

Cl\T.

If (fi\r) ^ <j>, then Xn\r ls a non-zero positive element of C(Cl). Thus there is a normal regular Borel measure \J on Cl such that n'(Cl\T) > 0. Let [i = [i! + /J,' o r. Then [i is a normal regular Borel measure on fi, and n = /x or,/Lt(fi\r) > 0. Moreover, let i/ = Mlrnyrv Then v is normal, I/OT — v, supp v ^ 4>, and supp f PI supp vi = , and T is dense in Cl. Q.E.D. By this lemma and [26 , Theorem 5.3.4], we have the following. Theorem 6.3.3. Let Z be an abelian real W*-algebra, and Z = C(CI,T), where Cl is the spectral space of Zc — Z+iZ, and r is a homeomorphism of Cl with period 2. Then Cl is a hyperstonean space, and there are a dense open subset TofCl (so T is a locally compact Hausdorff space) and a regular Borel measure v onV. such that

r(r) = r, vor = v, suppv = r, and

Z^L°°(T,V,T).

Now we consider a countably generated abelian real W*-algebra Z. Then the abelian W*-algebra Zc = Z+iZ is also countably generated. By [26, Theorem 5.3.7], Zc will be generated by a positive invertible element (a+ib), where a, b € Z. Clearly, a* = a, b* = -b. Let Zc 2 C(Cl), Z * C(Cl, r ) . Then a(t) = a(r(t)) G R,

b(t) = -6(r(f)) e iC,

Vt € Cl, and {a + ib)(t) = a(t) + ib(t) > 0,

135

Real W* -Algebras

(a + ib)(r(t)) = a(t) - ib(t) > 0 ,

Vt € fi.

Thus, a(t) > \b(t)\, Vi € fi, and a is positive and invertible in Z. For any x G Z, there is a net {p;(-, •)} of polynomials in two variables with complex coefficients such that pi(a + ib,a — ib) —> x in Zc. Write pi(a + ib,a — ib) = qt (a,6) + q\ (a,6), where q\ (•,•) and q\ (•,•) are C\\

3

polynomials in two variables with real coefficients, VZ. Then qj '(a, b) —> x in Z. Therefore, Z is generated by {a, 6}. Let ZH be the self-adjoint part of Z. Then it is easy to see that ZJJ is also a countably generated abelian real W*-algebra. If ZJJ+IZH — C(fi) then ZH — Cr{Vt). (all real valued continuous functions on fi). Thus by [26, Theorem 5.3.7], ZH is generated by a positive invertible element c. Further, Z is generated by {c, 6}. From the above discussion, we have the following. Proposition 6.3.4. Let Z be a countably generated abelian real W*algebra, and ZH be the self-adjoint part of Z. Then there are a positive invertible element a and a skew-hermitian element b of Z such that ZH is generated by {a}, and Z is generated by {a, b}. Further, Z is generated by {a + b}. Lemma 6.3.5. Let A be a unital abelian real C*-algebra. Then A does not contain any non-zero minimal projection (a non-zero projection p in A is said to be minimal, if q is a projection in A such that q < p, then q = 0 or p), if and only if, Ac = A+iA does not contain any non-zero minimal projection. Proof Let Ac ^ C(fi), and A £* C{Q, T). If p = a + ib is a projection in Ac, where a,b € A, then a* = a,b* = —b, and

a = a2 - b2 > 0,

b = 2ab.

For any t 6 fi, denote A = a(t) = a(r(t)),iij, = b(t) = -b(r(t)), where A , f i e R . Then A = A2 + fj?,fj, = 2A^. If /x + 0, then A = |,/x = ± | ; If fj, = 0, then A = 0 or 1.

136

Real Operator Algebras

Let iii = {t e n\b(t) = 0} = T-(ni), n2 = {ten\b(t) = -i}, n3 = {te n\b{t) = {} = T(n2). Then fin ^2) fi 3 are disjoint clopen subsets of 17, and fi = fii U J72 U JI3. Clearly, a(£) = 0 or 1 on fi 1 ; and a(f) = | on fi 2 U fi 3 . Thus

p(t) = {

1,

V* G fio,

0,

Vt € fii\fi 0 ,

1, vt e n2, 0,

v* € n3>

where fio = {£ € fi|a(i) = 1} C fii. Then fio = T(QQ) is a clopen subset of fi, andp(-) = xn 0 un 2 (-)Assume that A does not contain any non-zero minimal projection. If p = a + ib is a non-zero minimal projection in Ac, according to above notations, we have fi 2 = 7"(fi3) 7^ <j> (otherwise 6 = 0, and p = a is a nonzero minimal projection in A, a contradiction). If fio 7^ 4>, then 0 7^ po = Xsio 6 A, and po < P,Po ¥" V- This is a contradiction since p is minimal in Ac. Hence, fio — , i-e., a(-)|n! = 0. Let q = Xn2un3- Then q is a non-zero projection in A. Since q is not minimal in A, it follows that there is a non-zero projection qo in A such that qo < q and go ¥" 3- We can write qo = Xn2unj> where <> / 7^ fi 2 C fi 2 , and fi'3 = r(fi 2 ) C fi 3 . Then Po = Xn' is a non-zero projection in Ac, and po < p,po 7^ P- We also get a contradiction since p is minimal in Ac. Therefore, Ac does not contain any non-zero minimal projection. Conversely, assume that Ac does not contain any non-zero minimal projection. Let q = XE be a non-zero minimal projection in A. Then E = T{E) is a non-empty clopen subset of fi. Of course, q is not minimal in Ac. Thus, there is a non-zero projection p = a + ib in Ac such that p < q and p 7^ q, where a, b G A. Using the above notations, p = XfJ0un2 • Clearly, 6 ^ 0 (since q is minimal in A), i.e., fi 2 = r(fi 3 ) 7^ . Since (fi0 Ufi 2 ) C E = T(E), it follows that (fi 0 Ufi 2 Ufi 3 ) C E. Now xn 2 un 3 is a non-zero projection in A, and xn 2 un 3 < , a(-)|fii = 0, and p = xn 2 - Of course, p is

Real W* -Algebras

137

not minimal in Ac. Then, there is a non-zero projection p0 = xn'2 in ^c such that po
,Cl'2 Z&2- Let q0 = Xn'2uW3, where £1'3 = T(Q'2) C JJV Then go is a non-zero projection in A such that qa < q and q0 ^ q. This is a contradiction since q = \E = Xn 2 un 3 is minimal in A. Therefore, A does not contain any non-zero minimal projection. Q.E.D. Now let H be a separable real Hilbert space, Z be an abelian real VN algebra on H, and assume that Z does not contain any non-zero minimal projection. Then Zc = Z+iZ is an abelian VN algebra on a separable Hilbert space Hc = H+iH, and by Lemma 6.3.6, Zc does not contain any non-zero minimal projection. By [26, Theorem 5.3.1] and Theorem 6.3.1, we may assume that Z £* C ( n , T ) = L°°(fi, u, r ) ,

Zc * C(Q) = L°°(S1, v),

where Cl is the spectral space of Zc, T is a homeomorphism of Cl with period 2, and v is a regular Borel measure on fl such that v o r = v, supp u = fi. By Proposition 6.3.4, Z is generated by {a, 6} where 6* = — b € Z, and o is a positive invertible element of Z such that Z # is generated by {a}. We may assume that 0 < a(t) < 1, and \b(t)\ < | , Vt e fi. Then we have a continuous map (a + b)(-) : fj — • • , where D is a rectangle in the complex plan C with vertices: ± | , (1 ± | ) . Clearly, (a + b)(r(t)) = (a + b)(t) = (a - b)(t) = (a + b)*(t),

Vi € Q.

Let p(-) — vo (a + 6) - 1 (-). Then /x is a regular Borel measure on D, and po — — p, where "-" is the complex conjugation. Define $ : L°°(D, p, —) —> L°°(fi,i/,r)by *(/)(*) = / ( ( « + &)(*)),

V/ e L°°(D, /x, - ) , i € fi.

If p (•, •) is a polynomial in two variables with real coefficients, then p (z,~z) e L°°(D, p, - ) , and * ( p ) ( t ) = p ( ( a + 6)(t), ( a + &)(*)) = p ( ( a + &)(*), ( a + &)*(*)),

V*€Jl.

Since Z is generated by {a+6} , it follows that {p((a+6), ( a + 6 ) * ) | p ( - , •) any polynomial in two variables with real coefficients } is a(Z, Z*)-dense in Z. Similarly to the corresponding part of the proof in [26, Theorem 5.3.8], we see that $ is a * isomorphism from L°°(D, p, —) onto L°°(fi, v, r) (as real VT*-algebras).

138

Real Operator Algebras

Since Zc does not contain any non-zero minimal projection, it follows that /J, is non-atomic on • (i.e., n({z}) = 0,Vz € • ) . Now we consider L°°(p,n, - ) = £~([0, l],/x') 0 L°°(D",/i", - ) , where /j,' = fj,\[0,l],L™([0,l],n') is the totality of all real valued functions of L°°([Q,i\,n');\J' = D\[0,1], and p" = /i| n «. Clearly, /*' and /*" are also non-atomic. If fjf £ 0, then L°°([0,1], fi') is an abelian VN algebra on the separable Hilbert space L 2 ([0,1], fi'), and L°°([0,1], //) does not contain any non-zero minimal projection. By [26, Theorem 5.3.8], L°°([0, l],/i') = L°°([0,1]) (as W*-algebras). Therefore, L~([0,1],/*') S< L~([0,1]) (as real W-algebras). On the other hand, clearly L~(D",/*",-) 2 £"(•'",/*"') as real W*-algebras, where D'" = {z e D"\Imz > 0}, y!" = (j."\n>», and / / " is non-atomic. If fj,'" ^ 0, then L°°(D'", jx'") is an abelian VN algebra on the separable Hilbert space L2(D'", /z'"), and L°°(D'", n'") does not contain any non-zero minimal projection. By [26, Theorem 5.3.8], X,°°(D'",//") ^ i°°([0,1]) (as W-algebras). Therefore, L o o ( n " , M " , - ) = i o o ([0,l]) as real W*-aglebras. Since each element in ££°([0,1]) is hermitian, it follows that L£°([0,1]) is not * isomorphic to L°°([0,1]) or L~([0,1]) ©Z,°°([0,1]) as real W*algebras. Denote M = ££°([0,1]) © L°°([0,1]), and let z be the central projection in M such that Mz = Z*£°([0,1]). If there is a * isomorphism $ from M onto L°°([0,1]) as real W-algebras, then Z£°([0,1]) ~ L°°(E), where E is a Borel subset of [0,1] such that <&(z) = \E- It is impossible since each element in Z/£°([0,1]) is hermitian. Therefore, L°°([0,1]) is not * isomorphic to M as real W*-algebras. Prom the above discussion, we have the following. T h e o r e m 6.3.6. If Z is an abelian real VN aglebra on a separable real Hilbert space H, and Z does not contain any non-zero minimal projection, then Z is* isomorphic to L~([0,1]),L°°([0,1]), or L~([0,1]) © ([0,1]) as real W*-algebras. Moreover, the real W*-algebras L~([0,1]),L°°([0,1]) and lv°([0,1]) © L°°([0,1]) are not * isomorphic to one another.

Real W

-Algebras

139

Remark. In the complex case, if N is an abelian VN algebra on a separable Hilbert space, and N does not contain any non-zero minimal projection, then N =* L°°([0,1]) (see [26, Theorem 5.3.8]). Corollary 6.3.7. Let H be a separable real Hilbert space, and Sa be the collection of all abelian real VN algebras on H. For any Z G Sa, denote [Z] = {Y € Sa\Y is * isomorphic to Z}. Then the set {[Z]\Z € Sa} is countable. Proof. If Z € Sa, and p is a non-zero minimal projection in Z, then any non-zero projection in Zp must be p, and (Zp)n = Rp- By the Remark following Proposition 5.3.7, Zp = R or C. Moreover, any two minimal projections in Z must be orthogonal, and the collection of all minimal projections in Z must be countable. Now by Theorem 6.3.6, our conclusion is obvious. Q.E.D. Definition 6.3.8. Let Z be an abelian real VN algebra on a real Hilbert space H. Z is said to be maximal abelian, if there is no abelian real VN algebra on H which contains Z properly. Proposition 6.3.9. Let Z be an abelian real VN algebra on a real Hilbert space H. Then the following statements are equivalent: (1) Z is maximal abelian; (2) Z' = Z; (3) Zc = Z+iZ is maximal abelian on Hc — H+iH. Proof. Clearly, Z C Z' since Z is abelian. Let Z be maximal abelian. For any normal a' € Z', Y = {Z,a',a'*}" will be an abelian real VN algebra on H, and Y D Z. Thus Y must be equal to Z, and a' € Z. Consequently, (Z')H and (Z')K C Z. Moreover, Z' C Z and Z' = Z. Let Z' = Z. Then Z'c = Z'+iZ' = Zc, and Zc is maximal abelian. Finally, let Zc be maximal abelian. Then Z'c = Zc, and Z' = Z.liY is an abelian real VN algebra on H, and Z C Y, then Z C Y C Z' = Z, and Y = Z. Therefore, Z is maximal abelian. Definition 6.3.10.

Q.E.D.

Let fl be a locally compact Hausdorff space, r be a

140

Real Operator Algebras

homeomorphism of 17 with period 2, and v be a regular Borel Measure on 17 such that v o r = v. For any / e L°°(ft, v, r ) , we can define a bounded linear operator fhf on L2(17, Z/,T) as follows: ™>fg = fg,

Vg € L2(Q,

v,r).

Then {rri/|/ € L o c (n,f, r)} is called the real multiplication algebra on L 2 (17,r,,r). T h e o r e m 6.3.11. Let Z be an abelian real VN algebra on a real Hilbert space H. (1) Z is maximal abelian, if and only if, there are a locally compact Hausdorff space 17, a homeomorphism T o/17 with period 2, a regular Borel measure v on £1 with v o T = v and supp v = fi, and a unitary operator u from H onto L2(Q,V,T) such that uZu* is the real multiplication algebra on

L2(Q,I/,T).

(2) Z is maximal abelian and a-finite, if and only if, Z admits a cyclic vector. Moreover, in this case, let Z = C(fi,T),Z C = C(fi), where fi is the spectral space of Zc, (a compact Hausdorff space) and T is a homeomorphism of CI with period 2. Then there are a regular Borel measure v on Cl with v o r = v, supp v = Q and C(fl,r) = L°°{i1,v, T), and a unitary operator u from H onto L2(Q, v, r ) such that umfU~1=fhf,

V/ €

L°°{Q,,v,r),

where f —> ^ / ( V / € C(£1,T)) is the * isomorphism from C(Q,T) onto Z, and fhf is the real multiplication operator on L2(£l,v, r) with symbol (3) If H is separable, then Z is maximal abelian, if and only if , Z admits a cyclic vector. Proof. Since the multiplication algebra L°°(£l,i>) on L2(Q,,v) is maximal abelian ([26, Theorem 5.3.13]), it follows from Proposition 6.3.9 that the real multiplication algebra L°° (17, u, r ) is a maximal abelian real VN algebra on L2(Cl, V,T). Suppose that Z admits a cyclic vector £o 6 H. By Z c Z' and Proposition 4.6.2, we can see that Z is cr-finite, £o is also separating for Z. Let Z c a C ( n ) , Z ^ C ( 1 7 , r ) , and

[ fdv=(m,Z0,t0),

JQ

V/eC(n,r),

Real W* -Algebras

141

where / —»• m/ is the * isomorphism from

C(SI,T)

onto Z. Then by

Thorem 6.3.1, we have v or = u, supp v = fi, and C(fi,r) = L°°(fi,i/,r). Define urn/ft = / ,

V/eC(n,T).

Then u can be extended to a unitary operator, also denoted by u, from H onto L 2 (fi, i', r ) , and um,fU~l — fhf,

V/eC(n,r).

Thus, Z is unitarily equivalent to the real multiplication algebra on L2(Q, V,T), and Z is also maximal abelian. If Z is maximal abelian and cr-finite, then from Proposition 4.6.3, Z admits a separatiing vector £0 £ H. Further, by Z' = Z, £o is also cyclic tor Z. Therefore, conclusion (2) is true. Now (3) is obvious from (2). Finally, Let Z be maximal abelian. Similarly to the proof of complex case ([26, Theorem 5.3.17]) and using (2), we can get the conclusion (1). Q.E.D. References. 6.4

[6], [26], [37], [50].

Unitaries and partial isometries

Let M be a real W*-algebra. Denote the subset of all unitary elements in M by U(M), i.e., U{M) = { u £ M\u*u = uu* = 1}; and the subset of all partial isometries in M by V(M),

i.e.,

V(M) — {v & M\v*v is a projection }. Of course, if v € V{M), then vv* is also a projection. Lemma 6.4.1. Let M be a real W* -algebra, and u G U(M). Then there are a projection p e M and s* = —s 6 M such that u = (1 - 2p)e~s = and {u,p,s}

is commutative.

e"pe~s

142

Real Operator Algebras

Proof. Let W* (u) be the abelian real V7*-subalgebra of M generated by {u}. Form the spectral decomposition of u in (W* (u)+iW*(u)), we can write that u = ei7rbe~s, where b* = b,s* = -s e W*(u). Since u = cos{-Kb)e~s + i sin(7r&)e~s € M, it follows that sin(7rb) = 0. Let Q be the spectral space of W*(u). Then ski7r&(f) = 0, and b(t) € Z,Vi € ft. Let ftn = {t € ft|6(f) = n},Vn € Z. Then ft = U n e zQ n , ftn is a clopen subset of ft, and ftn = ftn,Vn € Z, where "—" is such that W*(u) = C(ft, —) as real C*-algebras. Suppose that p is a projection in W* (u) such that

{

n

0,

t € ftn,

even,

1,

t G n „ , n odd.

Then einb = e™p, e™? = (1 - 2p), and u = (1 - 2p)e~s = e™pe-s.

Q.E.D.

Definition 6.4.2. Let M be a real VT*-algebra. M is said to be continuous, if any projection p in M can be written as P = Pl+P2, where pi, j>2 are two projections in M such that P1P2 = P2P1 = 0, and pi ~ p2 (see Definition 4.4.5). Proposition 6.4.3. Let M be a continuous real W* -algebra. Then U(M) is pathwise connected in the norm topology. Proof. Let u e U(M). By Lemma 6.4.1, u = (1 - 2p)e" s - e i 7 r p e" s ,

Real W* -Algebras

143

where p is a projection in M, s* = —s, and ps = sp. If we can find t* — -te M such that effi = 1 - 2p, then eXnte~Xs

: 1 —» u,

A €[0,1],

i.e., C/(M) is pathwise connected in the norm topology. Since M is continuous, we can write P-P1+P2,

v*v = pi,

VV*-P2,

P\Pi = P2P\ = 0,

where pi,P2 are two projections in M, v 6 V(M). Then vp2 — p2v* = v*p1 = pxv = 0. Let t = v — v*. Then t* = —t, and tp = pt = t = ptp. Since v2 = 0, it follows that r = — it is a self-adjoint unitary element in pMcp, where Mc = M+iM. Thus, a(r) C {±1} in pMcp. Of course, r = 0 on (1 — p).Hc, where ifc = H+iH, and M C B(H). Thus, we can write r = 0 - e o + l - e i - l - e_i, where {eo,ej,e_i} is an orthogonal family of projections in M c , and ei + e_i = p, eo = 1 - p. Finally, _7rt _

„i-nr _ gi-irO-eo . g « r - e i . g —t?r-e_i

- 1 • ((1 - ei) + e ^ e i ) • ((1 - e_i) + e - " e - i ) = (1 - 2ei) • (1 - 2e_i) = 1 - 2p. Q.E.D. Definition 6.4.4. u; if

Let M be a real WA*-algebra, and u, u> € V(M).

P ~ 9,

v «

p' ~ 9'

and (l_p)^(l_g))

(l-p')~(l-5'),

where p = w*w,p' = ww*,q — v*v,q' = vv*. Clearly, v « w, if and only if, p = w*u; is unitarily equivalent to q = v*v in M, and p' = ww* is unitarily equivalent to q' = vv* in M. Moreover, " « " is an equivalent relation in V(M) obviously.

144

Real Operator Algebras

Lemma 6.4.5. Let M be a real W* -algebra, p and q be two projections in M. If\\p — q\\ < 1, then p is unitarily equivalent to q in M. In particular. ifw,v G V(M) and \\v — w\\ < | , then v « w. Proof. Let u' — qp + (1 - g)(l — p), and r = (p - q)2. It is easy to see that u u =uu

= 1 — r,

{u', u'*, r} is commutative, and rq = qr, rp — pq. By ||p — q\\ < 1, the series

(1 - r)"i = £(-!)" M V n>0

^

'

is absolutely convergent in norm. Let u = u'(l — r)~. Then u is a unitary element in M , and up = (1 — r)~u'p = (1 — r)^~qp = (1 — r)~qu' = qu, i.e., q = upu*. Now if v,w G F ( M ) and ||u - iu|| < | , then ||u*u - iw*iw|| < ||u*|| • ||u - iy|| + ||v* -w*\\-

\\w\\ < 1,

and 11TO* — ww*\\ < 1. By the above fact, p = w*w and p' = IMO* are unitarily equivalent to 5 = u*v and q' = vv* respectively, i.e., v zz w. Q.E.D. Lemma 6.4.6. Let M be a real W*-algebra, U(M) be pathwise connected in norm , and i>o,i>i € V(M),VQ W V\. Then there exists a continuous map v(-) : [0,1] —> (V(M), norm) such that v(Q) = vo,

v(l) = vi

andv(t) wt) 0 , Vt e [0,1]. Proof. Let u, w' € U{M) be such that Pi = up0u*,

qi = w'q0u)'*,

where pj = Vj-Vj,qj = VjVj,j = 0,1. Take w G i/(M) such that u;g0 = vxuvoqo,

w(l-q0)

=

i.e., w = viuv^qo + w'(l - qo).

v/(l-q0),

Real

145

W*-Algebras

Now wq0w* — viuv^qo • qoV0u*vl = Vipiv{ = q\, and f wv0(l - po) = 0 = v x u(l - po), \ wvoPo — wqovopo = ViUVoq0v0po = Viup0. Thus, WVQ = v\u,

or WVQU* =

vi.

Since U(M) is pathwise connected in norm, it follows that there are continuous maps io(-) : [0,1] —> (U(M), norm) and u(-) : [0,1] —> (U(M), norm) such that w(0) = l,w(l)

= w,u(0) = l , u ( l ) = u.

Therefore, «(•) = w(-)vou(-)* is a continuous map from [0,1] to (V(M), norm) such that •u(O) = vo, and u(l) = WVQU* =

v\.

Finally, from Lemma 6.4.5 we can see that v(t) « vo, Vi € [0,1]. Q.E.D. Proposition 6.4.7. Lei M i e o reaZ V7*-a/5e6ra, and U(M) be pathwise connected in norm. Then v,w(& V(M)) are in same pathwise connected component of (V(M), norm), if and only if, v « w. Proof. It is obvious from Lemmas 6.4.5 and 6.4.6. Notes.

Propositions 6.4.3 and 6.4.7 are due to K.E.Ekman ([10]).

References.

[10], [33].

Q.E.D.

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Chapter 7

Gelfand—Naimark Conjecture in t h e Real Case

7.1

Real C*—equivalent algebras

Definition 7.1.1. A real Banach * algebra (A, || • ||) is said to be real C*-equivalent, if there is a new norm || • ||' on A such that (A, \\ • ||') is a real C*-algebra, and the new norm || • ||' is equivalent to the original norm || • || on A, i.e., \\ • \\' ~ || • || on A. Lemma 7.1.2. Let E be a compact subset of C, and 0 S E. Then for each A € C, we have max{|A + n\\n G E} > -(max{|^| \» e E} + |A|).

Proof. Since |A + /x| > |/j| — |A|, it follows that max{|A + fi\ \n £ E} > max{|^i| \fi e E} - |A|. In addition, by 0 G E we have 2max{|A + /x| | / J 6 B } > 2|A|.

Therefore, max{|A + /x| |/i 6 £;} > -(max{|/i| |/t 6 £ } + |A|).

Q.E.D.

Lemma 7.1.3. Lei A be a real Banach * algebra. If there is a positive constant K such that Kr(a) > \\a\\, Va* = ± a S A, then the * operation on A is continuous in norm. 147

148

Real Operator

Algebras

Proof. Let {an} C A and a € A be such that ||o„ — a\\ —> 0, where a* = an,Vn; or a* = — an Vn. Then for any e > 0 and sufficiently large n, we have A:(r(a)+£)>ii:r(an)>||an||. Thus, Kr(a) > \\a\\, Va eAHUAK

(7.1)

Now let {a n } C A and a G A be such that ||a„ — a\\ —> 0, where a* = a„, Vn, and a* = —a; or a* = —a„, Vn, and a* = a. Since ( a n + a m ) * = an + am, Vn,m; or (a„ + a m )* = - ( a „ + am), Vn,m, and (a„ + a m ) —>• (a n + a) as ra —> +oo, it follows from (7.1) that \\an + a|| < Kr(an + a) = .ft>((an + a)*) = l f r ( a n - a ) < AT||a n -a||,

Vn,

and \\an + a\\ —> 0. Moreover, 2||o|| < | | a n - a | | + ||a n + a|| — • 0 and a = 0. Thus, A/f = Aff, Ajf = AxIf {an} C A and a £ A are such that a„ —> 0 and a* —• a, then from (a* + a n ) —» a and (a* — a„) —» a, we have a € AH n ^4j<- = .A/j D A X , and a = 0. Therefore, the * operation is a closed linear operator on A, hence continuous in norm. Q.E.D. L e m m a 7.1.4. Let A be a real Banach * algebra. If there is a positive constant K such that K \\a*a\\ > \\a*\\ • \\a\\, Va* = ± a G A, then Kr(a) > ||a||,

Proof. Clearly, K\\o?\\ > \\a\\2,

Va* = ±a e A.

Va* = ±a e A. Moreover,

^2"-1||a2"|| = X 2 " "

1

•••K2K\\a2"\\

> ||a||2",

Vn,a' = ± a £ A. Therefore, Kr(a)>\\a\\,

Va* = ± a € A

Q.E.D.

Gelfand-Naimark

Conjecture in the Real Case

149

Theorem 7.1.5. Let A be a real Banach * algebra. Suppose that A satisfies any one of following conditions: (1) A is symmetric, and there is a positive constant K{> 1) such that Kr{a) > \\a\\, Va* = ± a G A. (2) A is hermitian, and there is a positive constant K(> 1) such that K\\a*a\\ > \\a*\\ • \\a\\, V normal a G A. (3) There is a positive constant K{> 1) such that K^a*a + b*b\\ > \\a*\\ • \\a\\, V normal a,b € A and ab = ba. Then A is real C* -equivalent. Proof. (1) By Lemma 7.1.2 and Theorem 3.6.5, we may assume that A has an identity. Moreover, by Lemma 7.1.3, the * operation on A is continuous in norm. For each real state p on A, from the GNS construction we get a cyclic * representation {np, Hp,£p} of A such that p{a) = (7rp(a)£p, ZP),

Va G A,

where Hp is a real Hilbert space. Moreover, let {-7T = ®peS(A)lTp,

H =

®p£S{A)Hp}

be the universal * representation of A (see Definition 3.3.9), where «S(^4) is the real state space of A. Since the * operation on A is continuous in norm, it follows that ||7r(a)|| = ||7r(a*a)||i=r(7r(a*a))i (7.2)
< ||a*a||i < K'\\a\\,

Va € A, where K' is some positive constant. Let ||o||' = ||7r(a)||, Va e A. We claim that || • ||' is a norm on A. In fact, if ||a||' = ||ir(a)|| = 0 for some a G A, then

|P(«)I = \p(b)\ = \(*P(a)tM\ < K«)ll =«, VpG S(A), where a = b + c, b= ±(a + a*), c= \{a-a*). and Kr{b) > ||6||, we have b = 0, a = c. Similarly, |p(c 2 )| = \{irp(a'a)Zp,£p)\

By Lemma 3.6.8

< ]]*(
Vp G S(A), and c2 = 0. Moreover, ||a|| = ||c|| < Kr(c) = Kr{c2)^

= 0.

150

Real Operator Algebras

Therefore, a = 0, and || • ||' is a C*-norm on A. Finally, by (7.2) it suffices to show that || • || is continuous with respect to || • ||' on A. Let {an} cA be such that ||o„|j' -» 0. Since | | < | | ' = ||a n ||', Vn, it follows that ||&n||' -> 0 and ||c„||' -> 0, where 6* = bn,c*n = —cn, and «n = bn + cn, Vn. By Lemma 3.6.8, we have if ||7r(h„)|| > Kr(bn) > \\bn\\, and K\\n(cn)\\=K\\n(cl)\\h>Kr(cl)l =

Kr(cn)>K\\cn\\,

Vn. Therefore, ||b„|| -» 0, ||c n || —> 0, and ||a n || —> 0, i.e., || • || is continuous with respect to || • ||' on A. (2) By (1) and Lemma 7.1.4, it suffices to show that A is skew-hermitian. By Lemma 7.1.3, the * operation on A is continuous in norm. For a* = -a € A, let f(ta) = e*° - l,Vt € R. Then f(ta) is normal, and f(ta)* = f(-ta), V* e R. Moreover, ivT2r(2 - e t o - e -* a ) = > K\\f(ta)*f(ta)\\

K2r(f(ta)*f(ta))

> ||/(to)*|| • ||/(ta)|| > r 2 ( / ( t a ) ) ,

Vt e R. Let a = max{|ReA| |A € a(a)}, and /? e R be such that (a + i/3) (or ( - a + i(3)) G cr(a). Then for t > 0 (or t < 0) we have 2if 2 (l + e a t ) > K2r(2 - eta - e~ta) >

r2(f{ta))

> |1 - e t ( a + i ^ | 2 = 1 + e2at - 2eat cos (It (or 2iC 2 (l + e-at) > 1 + e _ 2 Q t - 2e~at cos(3t ) . Clearly, it is impossible if a > 0. Therefore, a = 0, cr(a) C iR, and A is skew-hermitian. (3) By (2) it suffices to show that A is hermitian. For a* — a € A, let b = cos a — 1, and c = sin a. Then 6, c are hermitian, and 6c = cb. Thus, || cosa - 1||2 < K\\(cosa - l ) 2 + sin2 a\\ =

2K\\cosa-l\\,

i.e., || cosa — 1|| < 2K, Va* — a e A. Consequently, | cos At - 1| < 2K,

Vt e R,

A s a(a),

a* = a € A.

Therefore, a(a) C R, Va* = a £ A, i.e., A is hermitian.

Q.E.D.

Gelfand-Naimark

151

Conjecture in the Real Case

Remark. Comparing with the result in the complex case (see [26, Theorem 2.14.23]), we require an additional condition " A hermitian" in Theorem 7.1.5(2). It is necessary. Otherwise, there is a counterexample (see the Remark following Corollary 5.2.11). Lemma 7.1.6.

Let A be a real Banach * algebra, W = {a6A\a*

=

a,a(a)cR}

and V = {b e A\b* = -b,a(b) C iR}. If there is a positive constant K(> 1) such that Kr(a) > \\a\\, Va* = ± a € A, then the subsets W and V are closed. Proof. By Lemma 7.1.2, we may assume that A has an identity. Again from Lemma 7.1.3, the * operation on A is continuous in norm. Thus, W C AH, V_C AK. Let a € W. Then a € AH• Take {a„} C W such that \\an — a\\ ->• 0. Clearly, ||c i t e ||<||cos(ta)|H-||8iii(to)|| in Ac = A+iA,Vt e R,cos(ta n ),sin(fa n ) 6 An,Vn,t cos(to„) —> cos(ta),

£ R, and

sin(ta n ) —> sin(ta),

Vt € R. Since a ( a n ) c R,Vn, it follows that ||cos(to n )|| < Kr(cos(tan))

< K,

|| sin(ta„)|| < Kr(sm(tan))

< K,

Vn, t € R. Therefore, || e ita|| <

2R^

Vf

g

Rj

CT(O) C R, and W = W is closed. Let b e V. Then b* = -b. Take {6 n } C V such that ||6„ - b|| -)• 0. Clearly, /etbn

(

+ e-tb„

2

\

j

/ e t6„ _ e -tfc„ \ € j 4 f f

'

(

2

)GA«>

152

Real Operator Algebras

Vn,t e R. Since a(bn) C iR,Vn, it follows that <

Kr

( « " " + « - " • ) < if,

2

Vn, i e R. Therefore, ptbn

\\etb\\ < ||

_L. (,—tbn

7

ptbn

i| + ||5

£> — tb„

^

1| < 2K,

Vi G R,CT(6)C ZR, and V = F is closed. Remark.

Q.E.D.

Lemma 7.1.6 is a generalization of Lemma 5.8.1.

Theorem 7.1.7. Let A be a real Banach * algebra, and W, V be the same as in Lemma 7.1.6. Suppose that A satisfies any one of following conditions: (1) W, V are dense in AH, AK respectively, and there is a positive constant K(> 1) such that Kr{a) > \\a\\, Va* = ± a € A; (2) W is dense in AH, and there is a positive constant K{> 1) such that K\\a*a\\ > \\a*\\ • ||a||,V normal a € A. Then A is real C* -equivalent. Proof. It is obvious from Lemma 7.1.6 and Theorem 7.1.5.

Q.E.D.

Notes. Lemma 7.1.2 is due to B.R.Li ([24]). Theorem 7.1.5 is due to T.W.Palmer ([39]). The corresponding result of Theorem 7.1.5 in the complex case is due to R.Arens. The corresponding result of Lemma 7.1.6 in the complex case is due to B.Yood ([54]). References.

7.2

[24], [26], [39], [54].

The closed unit ball of a united real C*-algebra

Lemma 7.2.1. Let K be a complex Hilbert space, M be a weakly closed * real operator algebra on the real Hilbert space H = Kr, and 1 = 1# G M. Then Co{u G M\u unitary } is weakly dense in S = {a G M|||a|| < 1}. Proof. Since S is weakly compact, so by the Krein-Milmann theorem it

Gelfand-Naimark

153

Conjecture in the Real Case

suffices to show that xo e Co{u € M\u unitary } for any XQ € exS (the subset of all extreme points of S), where "—w" means the weak closure. Let H = (K = Kr, (, ) r = Re(,)). Then H is a real Hilbert space, and M is a real VN algebra on H. Let xo € exS. By Theorem 5.4.3, xo is a partial isometry of M, and (1-XQXO)M(1-XQXQ) — {0}. For any projection p in M, denote the projection from H onto MpH by c(p). Clearly, c(p) is the minimal central projection in M containing p. Since M ( l - x*0x0)H±M(1

- x0x*Q)H

in H, it follows that c(l—XQX0)-C(1 —XOXQ) = 0. Let z = 1—c(l—XQXQ). Then > z, XQXQ >1 — Z. Of course, we may assume that XQ is not unitary. Replaciing {x0, M, H} by {x0z, Mz, zH} and {XQ(1-Z), M(1-Z), (1-Z)H} respectively, we may assume that XQXQ

XQXQ = 1, and XQXQ = p < 1.

Now it suffices to show that under the assumption of X^XQ = 1 and X0XQ = p < 1, for any £i, • • •, £ n , 771, • • •, rjn € H, there is a unitary element u 6 M such that \{(xQ - u)ii,r]i)\ < I, By xg(l — P)HLXQ(\

\
— p)H, Vs ^ t, we can write that 00

£r = tfoeX>zS(i-p)/7, fc=0

and let % be the projection from H onto :Eo(l — p)H, k = 0,1, • • •, and q be the projection from H onto Ho- Clearly, qk,q 6 M, Vfc. Pick a positive integer m such that

11 Y\ 9fc&n < oC1 + *?<»? I M D - 1 .

1 <«<«,

/c>m

and let u = x0 on H0 0 (1 - p)tf 8 • • • 0 z™(l - p ) # , u = x*0(m+1) ^ + 1 ( 1 - p ) t f , and u = 1 on £ f c > m + 1 ®x%(l-p)H, i.e., m

u = x0(q + Yl9k) + x*0(m+1)qm+i+ k=Q

^2 k>m+l

qk

'

on

154

Real Operator Algebras

Clearly, u 6 M. We claim that u is unitary. In fact, by = p we have

XQXQ

— 1 and

XQXQ

m

m

u*u = {q + ^2 qk) + (q + ] T qk)x*0(m+2) qm+i k=o

k=o

m

m

+(Q + ^2Qk)xo J2 qk+qm+ix^+2(q + ^2qk) fc=0

fc>m+l +1

+qm+ipqm+i + qm+ix™ ,

+

V^

2^

9fe:E

*(m+l)

o

,

k=0 qk +

X

m

X

*:>ro+l \~~*

9m+i +

y,

^^(g + ^gj)

fc>m+l

j=0

qk

fc>m+l fc>m+l

= 1 + 11 + III + IV + V + VI + VII + By x*0(m+2)qm+1H By

= x*0(m+2)x™+1(l-p)H

m

VIII.

= x*Q(l-p)H

(7.1) = {0}, II = 0.

m

fc=0 m

fc>m+l

fe=0

= (q + ^2qk)^2®xk0(l-p)H

k>m+l

= {0},

fc=0 fc>m

m = o. Since q 

fc=m+2

and JV = 0. By g m + 1 i J = ^ + 1 ( 1 - p)H CpH,V By qm+ix™+1

£

= qm+1.

©*§(!" P)#

fc>m+l fc>m+l

= 5m+l

X) fc>2m+2

VI = 0.

©3!0(1-P)fl'={0},

Gelfand-Naimark

Conjecture in the Real Case

155

By x0H0 = H0, and m k>m+l

m+l

j=0

k>m+l

j=l

VII = 0. By qkx*0(m+1)xZ+1(l-p)H=

X

X

k>m+l

fc>m+l

VIII = {0}. Therefore, u*u = 1. On the other hand, uu* = x0{q + X9k)zo + Xo (m+1) 3 m+ ia;^ +1 + ^ k=0

Ik-

fc>m+l

m

Clearly, xo{q + ^ J ^ x * , *s a projection. By fe=0

x

o{q + ^2 qk)x*0H = x0(g + X 9fc)# fc=0 m+ + ll m

fe=0 fc=0

fc=l m+l

we have x0{q + X ^ O ^ o = ? + X fc=0 fe=l

piJ, gm+i < p, it follows that XQ x$m+1)qn+ixZ+1H

9fe-

Since X H

°

=

PH'Q™+iH

'qm+ix™+1 is a projection. Further,

= x S ( m + 1 ) 9 m + 1 ^ + 1 ( ^ o © X ©x£(l - p)H) fc=0

= x* (m+1) g m +i(H 0 © X

*o(l-P)*0

fc>m+l

= x* ( m + 1 ) x- + 1 (1 - p)ff = (1 - p)ff =

qoH.

m+l

Therefore, uu* = (q 4- ^

4%) + 9o + 5 3 9fc = 1-

fc=l fc>m+l

c

156

Real Operator Algebras

Finally, \{(x0 - u)Zi,i]i)\ =

\(^2(x0-u)qk^,Vi)\ fc>m

< ||a:o-u|| -II Yl

9fc

^H ' H^ll < -1'

k>m

l
Q.E.D.

Proposition 7.2.2. Let A be a unital real C* -algebra, and XQ be a normal element in A with \\XQ\\ < 1. Then XQ belongs to the closure of Co < cos b • ea

a,b e A,a* = —a,b* — b, and {a, b, XQ} commutative

Proof. We may assume that A is abelian. Then by a result of the complex case (see [26, Theorem 2.14.2]), xo belong to the closure of Co{eih\h* = he

Ac},

i.e., there are A$n) > O ^ A f > = l,a< n) * = -a< n) ,6< B) * = bf

G

A,Vn,j,

3

such that

W^X^expia^+ib^-xoW-^O as n -> 00. Since A is abelian, it follows that I^A^cos^-e^-zol 3

Q.E.D.

a s n —» 0 0 .

Corollary 7.2.3. Let K be a complex Hilbert space, M be a weakly closed * real operator algebra on H = Kr, and 1 = \H G M. Then the subset Co{cosb-ea\a,b

G M,a* = -a,b* = -b}

is weakly dense in S = {a G M\ \\a\\ < 1}. Proof. Clearly, any unitary element is normal. Now the conclusion comes directly from Lemma 7.2.1 and Proposition 7.2.2. Q.E.D.

Gelfand-Naimark

Conjecture in the Real Case

157

Theorem 7.2.4. Let A be a unital real C*-algebra, and S = {a G A\ \\a\\ < 1} be the closed unit ball of A. Then the subset Co{cosb • ea\a, b G A, a* — -a, b* — b} is dense in S. Proof. Let {7r, K} be the universal * representation of the (complex) C*algebra Ac — A+iA, and M — -K{A) . We claim that M n iM = {0} in B(K). In fact, if x G M D iM, then there are two nets {aa}, {bp} such that n(aa) —> x,

Tr{ibp) —> x.

Thus, n(aa — ibp) -> 0 weakly. In particular, for any c, d G A and any state p on Ac we have (n(aa - ibp)-K(c)t,p,-K{d*)£,p) = p(d(aa - ibp)c) —> 0. Since A* = A*+iA*, and A* is the linear span of its state space, we have f{d(aa - ib0)c) —> 0,

V/6i*,

i.e., f(daac)

—> 0, and f(dbpc) —>• 0,

Vc, d G A, f € A*. Furthermore, (7r(a Q )7r(c)^,7r(d*)^) = p(daac) —> 0, (n(bp)n(c)t;p, n(d*)£p) = p(dbpc) —> 0, i.e., {xTr{c)t;p,n(d*)Zp)=0 V state p on A, and c , d e i . By the construction of {n, K}, we have x = 0, and M D i M = {0}. By Kaplansky's density Theorem 4.4.1, (TT(A))I is r-dense in (M)i, where (7r(A))i and (M)! are the closed unit balls of n(A) and M respectively. Since 7r is faithful and isometric, it follows that (7r(A))i = TT(S). Thus, 7r(5) is r-dense in (M)\. Moreover, Co{7r(cos6 • e°)|a, 6 G A, a* = —a, 6* = 6} is weakly dense in (M)i. In fact, by Corollary 7.2.3 it suffices to show that its weak closure

158

Real Operator Algebras

contains cosy • e x , Vx,y € M with a;* = — x and y* = y. Let x,y £ M be such that x* = —x and y* — y. Prom the preceding paragraph, there are nets {aa}, {bp} c A such that n(aa) - ^ x,

n(bp) - ^ y,

and ||a a || < ||x||, H&^IJ < \\y\\, Va,/3. Since * operation is r-continuous, we may assume that a*a = —aa, bp = bp, Va, /3. Clearly, for any polynomials P(-) and Q(-) we have *{P(aa))

-^

P{x),

7r(Q(bp)) - A Q(y).

Therefore, ^(e 0 ") —> ex,

n(cosbp) —> cosy

and 7r(cosbp • ea") —> cosy • ex weakly. Now if there exists x0 € 5\Co{cos6 • ea\a, 6 € A, a* = -a, b* = b}, then we can find / € A* such that f{x0) > sup{/(cos&-e a )|a,&€ A,a* = -a,b* = b}. On the other hand, from the preceding paragraph there is a net {xi} C Co{cosb • ea\a, b € A, a* = —a, b* = &} such that ir(xi) —> 7r(#o) weakly. In particular, for any state p on Ac we have p{xi - x0) = {(n(xi) - 7r(x0))£p,fp) —> °Thus, f(xi) -¥ / ( X Q ) . This is a contradiction. Therefore, Co{cos6-e a |a, b € A, a* = - a , b* — b} is dense in S. Q.E.D. Lemma 7.2.5. ball. Then

Let A be a unital real C*-algebra, and S be its closed unit

- S c Co{cos6-e a |a,6 6 A,a* — -a,b* — b}.

Proof. Let x e S, and x — h + k, where h* = h and k* — —k.

Gelfand-Naimark

Conjecture in the Real Case

159

Suppose that B is the abelian real C*-subalgebra of A generated by {h, 1}. Then B S C(fi, - ) , and h->h{t)=~h(Fj

= h(t),

-l
ViGfi.

Clearly, arccos/i(-) € C(fi, —), where arccosA G [0, ir], if A G [—1,1]. Hence, there is b G B with b* = b such that b(t) = arccosh(t), Vt G ft, i.e., h = cos b. Suppose that C is the abelian real C*-subalgebra of A generated by {k, 1}. Then C Si C(ft', - ) , and k -> fc(f) = -k(t)

= -k(t),

\k{t)\ < 1,

Vi G ft'.

Clearly,-i arcsin(iA;(-)) G C ( f t ' , - ) , where arcsinA G [ - | , f ] , i f A G [—1,1]Hence, there is a G C with a* = —a such that a(t) = —iarcsin(ifc(£)),Vi G ft', i.e., fc = i ( e a - e _ °). Therefore, 2;

1 , „ „ „ /cosfc ea cos7r-e_a\ = c o s & + - ( e ' 1 - es - ' 1 ) = 2 ^ + T + J. i

The desired conclusion follows. Proposition 7.2.6. x G A we have

Q.E.D.

Let A be a unital real C*-algebra.

Then for any

x = 2_\ A? cos fy • eaj, where aj, 6^ € A,

=<»/'• i

3

and a*j — —aj,b*- = bj, Vj

Proof. By Lemma 7.2.5, we can define ||x||i=inf{£|A.,-| !•••},

VxGA

3

Clearly, || • ||i is a norm and || • || < || • ||i on A. Again by Lemma 7.2.5, we have ||x/2||x|| ||i < 1,

Vx G A.

Thus,

||*|| < INK < 2||x||,

VxGA

160

Real Operator Algebras

If ||x|| — 1, then by Theorem 7.2.4 there exists a sequence {xn} C Co{cos6ea\a,b G A,a* = —a, and b* = b} such that \\xn — x\\ —> 0. Moreover, \\xn — z||i —> 0. Since \\xn\\i < l,Vn, it follows that ||x||i < 1. Therefore, INIi < ||z||, and ||x|| = HarHi.Var 6 A. Q.E.D. Proposition 7.2.7. unit ball. Then

Let A be a unital real C* -algebra, and S be its closed

{h e A\h* = h, \\h\\ < 1} U {k e A\k* = -k, \\k\\ < 1} C Co{cos6- ea\a,b € A,a* — -a,b* = b}. and Int(S)

C Co{cos6• ea\a,b€A,a*

= -a,b* = b} c S,

where Int(S) = {x € A\ \\x\\ < 1}. Proof. The former conclusion is contained in the proof of Lemma 7.2.5 indeed. Now let x € Int(S). By Proposition 7.2.6 we can write x — 2~] -\j cosbj • ea>, 3

where \j > 0,aj = -a,j,bj — bj,Vj, and J ^ A j < 1. Then

!-EAi x = V") -\j cos bj • ea> H

j-

i-E A i cos 0 H

j-

cos TT.

3

i.e., x € Co{cos b • ea\a, b € A, a* = -a, b* = b}.

Q.E.D.

Proposition 7.2.8. Let A be a unital real C*-algebra, E be a real normed space, and $ be a bounded (real) linear mapping from A to E. Then

I

$|| = sup< ||$(cos6-e a )

a,b € A, and a* = —a, b* = b

Gelfand-Naimark

161

Conjecture in the Real Case

Proof. From Proposition 7.2.7 we have the right side < ||$|| =

sup ||$(a;)|| xelnt(S) < the right side.

Q.E.D.

Notes. If B is a unital (complex) C*-algebra, then Co{elb\b* = b € B} is dense in the closed unit ball of B (J.G.Glimm and R.V.Kadison [16], B. Russo and H.A.Dye [44], T.W.Palmer [39]). For this result, L.A.Harris gave a simple proof using Mobius transformation ([20]), and L.A.Gardner gave a short and elementary proof ([13]). But in the real case, the methods of Harris and Gardner are not applicable, and we must go back to the methods in [16, 39, 44]. Theorem 7.2.4 is due to B.R.Li ([24, 25]). References.

7.3

[13], [16], [20], [24], [25], [26], [39], [44].

Gelfand-Naimark conjecture in the real case

Theorem 7.3.1. Let A be a real Banach * algebra with identity. If there is a positive constant K (> 1) such that \\ea\\
||cos6||
Va, b € A with a* = —a, b* = b, then A is real C*-equivalent. Moreover, if K = 1 in above, then A itself is a real C* -algebra. Proof. For any a, b S A with a* = —a, b* = b, we have |e A t | < ||e t a || < K,

| cos(M*)l < II cos(t6)|| < K,

V t e R , A e o-(a),fi e a(b). Thus, A G iR,fi e R,a(a) C iR,a(b) C E, i.e., A is hermitian and skew-hermitian. We claim that inf{||a;2|| \x* = ±x <E A, and ||z|| = 1} = e > 0. In fact, let x* = ±x e A and ||x|| = 1. Denote ||x 2 || = rj. Then 0 < rj < 1, and ||z 2 n || < \\x2\\n < r)n,

\\x2n+1\\ < \\x2n\\ < nn,

Vn.

162

Real Operator Algebras

Let 5 = rj3. For any n > 1, || x 2nj| < j3n <

<J2n>

jja;2»*+l|J < S3n

<

S2n+1.

Thus, ||x n || <Sn, V n > 2 . If a* = —a and llall = 1, then K > \\eta\\ > \\ta\\ - 1 -

^tn\\an\\ln\ n>2 jnin

> _ t _ l _ X - * 4 _ >_ t _ c « j 4^ n! n>2 5

i.e., AT + e* > i, V* > 0. Let t = K + 2. Then ||a 2 || > {{K + 2)~Hn2)z. If 6* = 6 and ||6|| = 1, then K>

||cos(f - t 6 ) | | = ||sin(t6)||

= I E(- 1 ) n kTnI" - lM ~ 5Z ll*&ll2"+1/(2n + 1)! n>0

> t - e**,

*•

''

n>l

Vi > 0.

Let i = if + 2. Then ||6 2 || > ( ( # + 2)- 1 Zn2) 3 . Therefore, inf{||x 2 || \x* = ±x€A,

\\x\\ = 1} = £ > 0.

Further, ||z 2 || >£||a;|| 2 ,

Vx* =

±xeA,

and r(x) > e\\x\\, Vz* = ±x e A. By Theorem 7.1.5, .A is real C*-equivalent. Now let K — 1. From the preceding paragraph, there is a new norm || • ||' on A such that || • ||' ~ || • || on A and (A, \\ • ||') is a real C*-algebra. Consider the identity map J : (A, || • ||') —• (A, || • ||). By Proposition 7.2.8, ||J|| < 1, i.e., \\x\\ < \\x\\', Vx € A. If there is x0 € A such that ||a;o|| < H^oll', then r(x*0x0) =

\\x*0x0\\'=\\x*0\\'-\\xo\\'

> \\xo\\ • Ikoll > Ikoacoll > r{x*ox0). This is a contradiction. Therefore, || • ||' = || • || on A, and A itself is a real C*-algebra. Q.E.D.

Gelfand-Naimark

Conjecture in the Real Case

163

Theorem 7.3.2. Let Abe a hermitian real Banach * algebra with identity. If for any normal element x € A we have ^x*x = ||a;*|| • ||x||, then A is a real C* -algebra. Proof From the assumption, it is easy to see that ||x|| = r(x), Vx* = ±x € A. Since A is hermitian, it follows that ||cos6|| = r ( c o s 6 ) < l ,

W =

beA.

On the other hand, by Lemma 7.1.3 the * operation on A is continuous in norm. Thus, for any a* = —a € A we have l = r(l) = ||l|| = ||(e-r-e°ll = l | c - 0 | | - | | e l Of course, A is also skew-hermitian (see the proof of Theorem 7.1.5(2) ). Thus, ||e ± a || > | e ± A | > 1,

VA G or(o),

and ||e a || = l,Va* = - a . Therefore, A is a real C*-algebra by Theorem 7.3.1.

Q.E.D.

Remark. It is still an open question: Does Theorem 7.3.2 hold in the absence of identity? It is well-known that this is true in the complex case ([ll])T h e o r e m 7.3.3. x € A we have

Let A be a hermitian real Banach * algebra. If for any

\\x*x\\ = \\x*\\.\\x\\, then A is a real C* -algebra. Proof. By Theorem 7.1.5(2), A is real C*-equivalent, i.e., there are a new norm || • ||' on A and two positive constants Ki, K2 such that K^xW' <\\x\\
VxeA,

and (J4, || • ||') is a real C*-algebra. Moreover, by the assumption it is easy

164

Real Operator Algebras

to see that ||A|| = r(h) = \\h\\',

Vh*=he

A.

We may assume that A has no identity (otherwise, the conclusion is obvious from Theorem 7.3.2). Let ||x + A|| = sup{||xy + Xy\\ \y G A, and \\y\\ < 1}, and ||x + A||' = sup{||xy + \y\\'\y G A, and ||y|| < 1}, V x e A , A e R . By the Remark following Proposition 5.2.4, (A+R, || • ||') is a unital real C*-algebra. Clearly, Millar + A||' < ||x + A|| < K2\\x + A||', Vi € A,A 6 R. Thus, (A+R, || • ||) is a hermitian real Banach * algebra with identity. Moreover, let {di} be an approximate identity for A. By ||d(|| = \\di\\' < 1, V7, we can see that ||a;|| = sup{||x2/|| \y G A, and ||y|| < 1} Vx€ A. Fix x G A and A e R . For any e > 0, we can find y G A with \\y\\ < 1 such that ||s + A|| > \\xy + Xy\\> ||i +A|| - e. Since (x + \)diy —>• (x + X)y and di(x + X)y —> (x + X)y, it follows that ||i + A|| > ||(x + A)d,|| > ||(x + A)djy|| > \\x + A|| - 2e, and ||x + A|| > \\di(x + A)|| > ||dj(» + A)y|| > ||x + A|| - 2e, for / sufficiently large. Therefore, we have ||x + A|| = lim \\di{x + A)|| = lim ||(x + A)dj||,

Gelfand-Naimark

Conjecture in the Real Case

165

V i e A , A e R . Furthermore, \\{x + X)*|| • ||z + A|| = Urn; ||d,(z + A)*|| • ||(i + X)di\\ = ]imi\\di(x + X)*(x + X)di\\ < lim, \\(x + X)*(x + X)di\\ = \\{x + X)*(x + A)|| < ||(a:-|-A)-||-||(a: + A)||, and \\(x + X)*(x + A)|| = ||(z + A)*|| • ||z + A||, Va; e A, X 6 R. Now by Theorem 7.3.2, (A+R), and then A, is a real C*-algebra. Q.E.D. Proposition 7.3.4. Let A be a real Banach * algebra. Then A is a real C* -algebra, if and only if, ||o'||-||o|| < ||o*o + 6*6||,

Va,6e A.

Proof. The necessity is obvious. Now let ||a*|| • ||a|| < ||a*a + 6*6||,Va,6 6 A. By Theorem 7.1.5(3), A is real C*-equivalent. In particular, A is hermitian. Moreover, by ||o*||-||a|| < ||o*o|| < ||o*||-||o||, we have ||a*a|| = ||a*|| • ||a||, Va 6 A. By Theorem 7.3.3, A is a real C*algebra. Q.E.D. Notes. Theorems 7.3.1, 7.3.2 and 7.3.3 are due to B.R.Li ([24, 25]). In particular, Theorem 7.3.3 gives an affirmative answer for Gelfand-Naimark conjecture in the real case. The corresponding result of Theorem 7.3.1 in the complex case is due to B.W.Glickfeld [15], Moreover, on the characterizations of (complex) C*-algebras, there are many other directions, for example, the submultiplication of linear C*norms, geometrical and duality characterizations, locally C*-equivalent algebras, and etc. (see [8]). Naturally, could these results be moved into the real case? There are many open questions. References.

[8], [11], [15], [24], [25], [26].

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Chapter 8

Classification of Real W*-Algebras

8.1

Classification of real W*-algebras

Definition 8.1.1. Let M be a real V7*-algebra. A projection p in M is said to be finite, if q = p wherever q is a projection in M with q < p and q ~ p (see Definition 4.4.5). p is said to be infinite, if it is not finite, i.e., there is a projection q in M such that q < p, q ~ p and q ^ p. p is said to be purely infinite, if p does not contain any non-zero finite projection, i.e., if q is a projection in M such that q < p and q^O, then
168

Real Operator Algebras

z3 = 0 (see Proposition 8.1.2) . M is said to be properly infinite, if M does not contain any non-zero finite central projection, i.e., z\ = 0 (see Proposition 8.1.2). Moreover, a projection p in M is said to be semi-finite, or properly infinite, if the real W*-algebra Mp(= pMp) is semi-finite, or properly infinite. T h e o r e m 8.1.4. decomposition:

Let M be a real W*-algebra.

Then there is a unique

M = Mi® M 2 © M 3 , where Mi — Mzi is finite, M3 = Mz% is purely infinite, Mi = Mz-i is semi-finite and properly infinite, and z\ + Z2 + 23 = 1. Proof.

It is similar to the complex case (see [26, Theorem 6.1.6]). Q.E.D.

Definition 8.1.5. Let M be a real W*-algebra. A projection p in M is said to be abelian (commutative), iipMp is abelian (commutative); p is said to be semi-abelian, if pMuP is commutative, where MH is the self-adjoint part of M. Clearly, an abelian projection must be semi-abelian. Remark. In the complex case, a semi-abelian projection must be abelian. Thus, these two concepts coincide in the complex case. But in the real case, they are different. For examplex, in the quaterion algebra H, the projection 1 is semi-abelian but not abelian. Proposition 8.1.6. Let M be a real W* -algebra, p and q be two projections in M, and p be abelian. (1) Ifp~q, then q is also abelian. (2) pMp — Zp, Where Z is the center of M. Proof. It is similar to the complex case (see [26, Proposition 5.3.19]). Q.E.D. Proposition 8.1.7. Let M be a real M*-algebra, and p be a projection in M. (1) If p is semi-abelian, then p is finite.

Classification

of Real W* -Algebras

169

(2) Ifp is semi-abelian, and q < p, where q is a projection in M, then q is semi-abelian. (3) Ifpis semi-abelian, and q < p, where q is a projection in M, then q = c(q)p, where c(q) is the central cover of q in M. (4) The following statements are equivalent: (i) p is semi-abelian; (ii) ((pMp)H =)PMHP - ZHp(= Z(MP)H), where Z = Z(M) is the center of M, and ZJJ is the self-adjoint part of Z; (Hi) for any non-zero projection q in M and q < p, we can't write q = Qi + q? for any two projections gi, (iii). Let q be a non-zero projection in M such that q < p,q = <7i + q2, where qi, qv are two projections in M with q\qi = 0 and q\ ~ qi. Then there is v € M such that q\ = v*v and qi = vv*. Since MH is commutative, it follows that (v + v*)qi = qi(v + v*) and v = vqi = (v + v*)qi = Qi{v + v*) = qiv* — v*. Moreover, q\ = v*v — vv* = qi. It is impossible.

170

Real Operator Algebras

(iii) => (iv). Let q be a non-zero non-central projection in M. Theorem 4.4.7, there is a central projection z in M such that qz±(l-

q)z,

By

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

If qz = (1 — g)(l — z) = 0, the 9 = 1 — z is central. This is a constradiction. Thus, we may assume that qz ^ 0 or (1 — g)(l — z) ^ 0. Let ri = qz or (1 — q)(l — z) such that ri ^ 0. Then there is r2 such that 7^2 = 0 and r\ ~ r2. Now by the assumption of (iii), we get a contradiction for r = ri + r2 • (iv) = » (i), In this case, any projection in M is central. By Proposition 4.3.4(3), MH C Z, and MR is commutative. (ii) ==$> (i). It is obvious. (iv) =$• (ii). From the proof of "(iv) = • (i)", we have MH C Z. Clearly, Z f f C M f f . Therefore, MH = ZHQ.E.D. Definition 8.1.8. Let M be a real W-algebra. M is said to be discrete, if for any non-zero central projection z in M, there is a non-zero abelian projection p in M such that p < z. M is said to be semi-discrete, if for any non-zero central projection z in M, there is a non-zero semi-abelion projection p in M such that p < z. M is said to be semi-continuous, if M does not contain any non-zero abelian projection. M is said to be continuous, if M does not contain any non-zero semiabelian projection. Similarly to the complex case, M is said to be type (I), if it is discrete; M is said to be type (III), if it is purely infinite; M is said to be type (II), if it is semi-finite and continuous. T h e o r e m 8.1.9. decomposition:

Let M be a real W*-algebra. Then there is a unique

M -Mi®

M1<2 © M 2 © M 3 ,

where Mi = Mzt,i = 1,2,3 are type (I), (II), (III) real W*-algebras respectively, Mi,2 = Mz\ti is semi-discrete and semi-continuous, and z\ + %\,2 + z2 + z3 = 1. Moreover, Mi = Mi © Mi ] 2 is semi-discrete, M2 — Mi,2 © Mi is semi-finite and semi-continuous, {M2 © M3) is semicontinuous, and (M2 © M3) is continuous.

Classification

of Real W

-Algebras

171

Proof. Similarly to the complex case (see the proof of [26, Theorem 6.1.8]), we have M - Mi © M 2 © M 3 , where M3 = Mz3 is type (III) and z3 is the maximal purely infinite central projection in M; M\ — Mz\ is type (I) and z\ = sup{ z 6 M \z central projection in M such that Mz is type (I)}; M 2 = Mz 2 is semi-finite and semi-continuous and £2 = 1 — z\ — z3. Now (M2 © M3) does not contain any non-zero abelian projection, and (M2 © M3) is semi-continuous. Let p be an abelian projection in M. Since Mc(p) is type (I), it follows that p < c(p) < z\. Thus % + z3 = sup{z € M\z central projection in M such that Mz is semi-continuous}. Let zi = sup{z G M\z central projection in M such that Mz is semi-discrete}. Clearly, z\ < zi. We claim that Mzi = M% is semi-discrete. In fact, suppose that p is a non-zero central projection in M with p < zi. Then there is a central projection z in M such that pz ^ 0 and Mz is semi-discrete. Clearly, pz is central for Mz and pz < z. Thus, there is a non-zero semiabelian projection q in Mz such that q < pz. q is also semi-abelian for Mzi and q < p obviously. Thus, Mzi is semi-discrete. Let p be a non-zero semi-abelian projection in M. We say that Mc(p) is semi-discrete, i.e., c(p) < zi. In fact , suppose that z is a non-zero central projection in M such that z < c(p). Then zp ^ 0, and zpMjjzp = (pMup)z is commutative. 0 ^ zp is semi-abelian, and zp < z. Thus, Mc(p) is semidiscrete. Therefore, M ( l — zi) is continuous, and zi = sup{z € M\z central projection in M such that Mz continuous }. Let Mi,2 = M(zi—zi). Then Mi,2 is semi-discrete and semi-continuous, and Mi = Mzi = Mi © Mi i 2 . By Proposition 8.1.7(1), ziz 3 = 0, zi < zi + Z2, (zi - zi) < zjj. Let Z2 = z 2 - (zi - zi), and M 2 = MZ2. Then M2 is semi-finite and continuous, i.e., Mi is type (II). The uniqueness of such a decomposition is obvious. Q.E.D. Remark. Mi ] 2 is the "pathological" part of M. Because of its existence (for example, H, ££°(r, i/)®H, and etc.) the second classification of real W*-algebras is more complicated than the complex case. References.

[6], [26], [31], [33], [50].

172

8.2

Real Operator Algebras

Finite real PT'-algebras

Definition 8.2.1. Let M be a real W-algebra. A (real) linear functional
0, if (p(a*a) > 0, and (a*a) =
U(M).

Moreover, if

Similarly to [26, Theorem 6.3.8], we have the following. Theorem 8.2.2.

Let M be a finite real W*-algebra. Then *K(a) = 1,

ya* = ae

M,

where K(a) = Co{u*au\u £ U(M)} D Z, "-" means the norm closure, and Z = Z(M) is the center of M. Now we get a map T : MH -> ZH, i-e., {T{a)} = K(a), Va 6 MH- Clearly, T is (real) linear, and T{M+) C Z+, Mz £ZH,a€

T{z) - z,

T{a) = T(u*au),

MH, U € U(M).

Similarly to [26, Theorem 6.3.10], the following is obvious. Theorem 8.2.3. Let M be a real W*-algebra. Then M is finite, if and only if, there exists a faithful family of normal tracial real states on M.

Classification

of Real

W-Algebras

173

Theorem 8.2.4. Let M be a real W* -algebra. Then M is finite, if and only if, the (complex) W*-algebra Mc = M+iM is finite. Proof. The sufficiency is obvious by Definition 8.1.1. Now let M be finite. From Theorem 8.2.3, there is a faithful family T of normal tracial real states on M. We shall show that Tc = {yc\
0, andb* = -b. If (pc(a+ib) = 6 T, then a = 0. Moreover, by ib = a + ib > 0 and a(&) = a(b) C iM, we have 6 = 0. Therefore, Tc is faithful on Mc, and M c is finite. Q.E.D. Now we discuss the (real ) linear map T : MH —> ZR, where M is finite, and {T(a)} = K(a), Va € MH- Since Mc is also finite and U(M) C U(MC), it follows that T is the restriction of Tc on MH, where Tc is the central valued trace from Mc to Zc (the center of Mc, see [26 , Definition 6.3.13]), i.e., {Tc(x)} = Co{u*xu\u € U(MC)} D Zc,

Lemma 8.2.5.

Vx € M c .

Let M be a finite real W* -algebra. Then Tc(x) = Tc{x),

Vx e M c ,

where Tc : Mc —> Z c is t/ie central valued trace, and x = a — ib if x = a + i6, a,b £ M. Consequently, TC(M) c Z,

T C (M K ) C ZK-

Proof. Let x € Mc. Then there is a sequence {xn} C Co{u*xu|u € U(MC)} such that x n -> Tc{x) € Z c . Since ||y|| = \\y\\, Wy € M c , it follows that xn -» T c (x) e Z c . By u*xu = u*xu,

Vu e £/(Mc) = U(MC),

we have x n G Co{u*xu|u € [/(Mc)}, Vn, and Tc(x) € Co{u*xu|u € U(MC)} DZC =

{Tc(x)}.

174

Real Operator

Algebras

Thus, Tc{x) = Tc(x), Vx G Mc. Consequently, TC(M) C Z. Since Tc{x*) = Tc(x)*, Vx G Mc, we have T C (M K ) C ZK. Q.E.D. Definition 8.2.6. Let M be a finite real W*-algebra, and Tc be the central valued trace from Mc to Zc, where Z = Z(M) is the center of M. Then T = TC\M is called the central valued trace on M, i.e., {T(a)} = Co{u*au|u € f/(M c )} D Z,

Va € M.

Proposition 8.2.7. Let M be a finite real W*-algebra, and T : M —• Z be the central valued trace. Then (1) T is (real) linear, T{M+) C Z+, T{MH) C ZH, T{MK) C ZK, T{a*) = T{a)*, Va e M, and T(a) = Co{u*au\u € J7(M)} n Z,

Va € M H ;

^ T is a projection of norm one from M onto Z. Consequently, T(za) = zT(a),Vz G Z, a€M, and T(a)*T(a) < T(a*a), Va G M; (S)T(ab) = T(ba), Va,beM; (4) T(a*a) = 0 if and only if a — 0; (5) {

We still don't know whether {T(a)} = Co{u*au\u G U(M)} n Z,

Va G M.

Proposition 8.2.8. Let M be a finite real W*-algebra, andp,q projections in M. Then p ~ a in M, if and only if, p ~ q in Mc.

be two

Proof. The necessity is obvious. Now let p ~ q in Mc. By Theorem 4.4.7,

Classification

of Real

W*-Algebras

175

there is a central projection z in M such that pz •< qz, and q(l — z) < p(l — z) in M. Let pz ~ qi < qz in M. Clearly, qz ~ pz ~ 91 in M c . Since Mc is finite (Theorem 8.2.4), it follows that gi = qz, i.e., pz ~ qz in M. Similarly, p(l - z) ~ q(l - z) in M . Therefore, p ~ 5 in M. Q.E.D. References.

8.3

[6], [26], [31], [33], [50].

Properly infinite real W*-algebras

Similarly to [26, Theorem 6.4.4], we have the following. Theorem 8.3.1. Let M be a real W* -algebra. Then the following statements are equivalent: (1) M is properly infinite. (2) There is an orthogonal sequence {pn} of projections in M such that y ^ P n = 1, andpn ~ 1, Vn. n

(3) There is a projection p in M such that p ~ (1 — p) ~ 1. Theorem 8.3.2. Let M be a real W* -algebra. Then M is properly infinite, if and only if, Mc = M+iM is properly infinite. Proof. Suppose that M is properly infinite. Then by Theorem 8.3.1, there is a projection p in M such that p ~ (1 —p) ~ 1 in M. Clearly, p ~ (1 — p) ~ 1 in Mc. By [26, Theorem 6.4.4], Mc is properly infinite. Conversely, let Mc be properly infinite. If z is a finite central projection in M, then Mz is a finite real V7*-algbra. By Theorem 8.2.4, (Mz)c = Mcz is finite. Thus, z is also a finite central projection in Mc , and z must be zero , i.e., M is also properly infinite. Q.E.D. References.

[6], [26], [31], [33], [50].

176

8.4

Real Operator Algebras

Semi-finite real W*-algebras

Definition 8.4.1. Let M be a real W*-algebra.
• [0,+oo] is called a trace on M+, if ip(a + b) = 0 (where we define 0-+oo = 0 ) , and (a) < +00, then 02 € A/", a = a 2 • a? € A4+ = A4 n M+. Conversely, let a = ^2 x*yj € A4+ = M n M + , i where Xj, J/J- € A/", V7. Since x*y + y*x = -{(x + y)*(x + y)-(x-

y)*{x - y)},

Wx, y 6 M, it follows that a = | ( a + a*) = J ^ 5 ^ i W +

vfr)

3

= 2 S^^'

+

ViYfai + w) - (xi ~ »i)*fo - Wi)>

By Xj,yj € A/", Vj, we have
= M+ = {a € M+|
Moreover, from (8.1) we can see that MH = [M+] = M+ -

M+,

where A4jy = MC\ MH, and [A4+] is the (real) linear span of M+.

(8.1)

Classification

of Real W* -Algebras

177

Let x e M, and x = uh be the polar decomposition of x. Then h = u*x G M+ n A-t = Af+, ft2

G

A/,

and x = uk* • h?. Thus , M = {ab\a, b G A/"}. Now f can be extended to a (real) linear functional on M (still denoted by <£>), if we define ip(a — b) =
M+)+MK-

It is

—> C,

by (pc(a + ib) = • [0, +oo] as follows: +oo,

if

(a + ib) e

ip(a + ib) =
Mc+\Mc+

(a + ib) G Mc+-

We claim that V is a trace on Mc+, ip\M+ = if, and ip(x) = ip{x), Vz G M c +, where z = a — ib if x = a + ib G Mc, and a, 6 G M. In fact , clearly we have ip(x + y) = ip(x) + tp(y),

xp{\x) = \ip(x),

Va;, y G M c + , A > 0. Moreover, if a, b G M are such that (a + i6)*(a + i&) G Mc+, then (a*a+6*o)G At+,

( a * 6 - b*a) G Af*,

and ^((a + ib)*(a + it)) =
178

Real Operator Algebras

Thus, a,b<EAf, (aa* + 66*) € M+, (6a* - ab*) 6 MK,

and

ip((a + ib)(a + ib)*) = ip{aa*) + ((a + ib)(a + ib)*) = ip((a + ib)*{a + ib)). Therefore, (a + ib)* (a + ib) e Mc+ <=>• (a + ib)(a -M&)* € A4C+ <^=> a, 6 € A/", and ^ is a trace on M t +. Clearly, V'l-M-f = ¥>• Moreover, it is easy to see that x €E Mc+ if x € M c + . Thus, a; € .M c + •<=>• 3; € M.c+- By the definition of ip, we have ^>(x") = ip(x), Va: € M c + . Now we study the definition ideal of ip. Clearly, Mc+ = {x € M+\tp(x) < +oo}. By [26, Proposition 6.5.2], the definition ideal of rp is [.Mc+] (the complex linear span of M.c+). Moreover, from the above discussion we have {x € Mc\ip(x*x) < +00} = Af+iAf = Afc, and [Mc+] = M^+iM2 = M? = M+iM

= Mc.

Therefore, the definition ideal of ip is M.cBy [26], ip can be extended to a (complex) linear functional, still denoted by ip, on Mc- We point out that ip\M = \M+ =
Classification

of Real

179

W*-Algebras

By 6 G M, we have 03 = 04. Therefore tl)(b) = iip(a3 + ib3) - iii>(ai + 164) = i' : Mc+ -¥ [0, +00] is a trace such that ip'\M =
• x G M'c. So we can write

M'c =

M'+iM',

where M' is a (real) linear subspace of M. Clearly, M'+ = M'c D M+ = {a G M+|^(o) = V » < +00} = M+. Let A/"c' - {x G Mc|V>'(a:*a:) < +00}. If (a + i6) G N'c, where a, 6 G M, then (a + *6)*(o + ib) = {a*a + b*b) + i(a*b - b*a) eM'c

=

M'+iM'.

Thus, (a*a + b*b) G M! D M+ = A4'+ = A4+, (a*a + 6*6) < +00, and a,b € Af. Conversely, if a, b G A/", then by xj/\M+ =
N'c = M+iN = Afc, M'c = M'? = AP+iN2 = M+iM = Mc, and M! = M. Now \p' can be extended to a (complex) linear functional on M'c = Mc, which is still denoted by ip'. If (a + ib) G M'c+ = Mc+, where a, b G M, then a G M+, b G MKSince (a — ib) G A^c+ = Mc+, and 93(a) + # ' ( 6 ) = V'(a 4- i6) = V>'(a - i6) =
180

Real Operator Algebras

it follows that yj'(a + ib) =
{

+00, if (a + ib) e

MC+\M'C+,

i.e., ip' = ip. From the above discussion, we have the following. Proposition 8.4.2. Let M be a real W*-algebra, and ip be a trace on M+. Then there is a unique trace xp on Mc+ such that 4>\M+ =

(x) = tp{x),

Vz G Mc+,

where Mc = M+iM. Moreover, let J\f = {a G M\ip(a*a) < +00}, A/",/, = {x G Mc\ip{x*x) < +00}, M. = A/"2 (the definition ideal of
= M+-

M,p = M+iM =

M+,M+ MC,NTI>

= {a G M+\
= N+iM = Mc,

and +00, if

(a + ib) G

(a + ib)

MC+\MC+,

ip(a + ib) = eMc+,

where a,b G M. Still denote the natural extensions of

$\M

by

-
and tp(ab) =
Definition 8.4.3.

ip is called the trace (on Mc+) determined by the trace

Proposition 8.4.4. Let M be a real W*-algebra,

Classification

of Real

W*-Algebras

181

(1)
(b + ic) > 0, it follows that a > b > 0 and c* = —c. If b = 0, then ic = b + ic > 0 and c = 0. It is impossible since b + ic ^ 0. Thus, b G M+ is such that b ^ 0, b < a, and ip(b) < +00, i.e.,
b. Hence, we have 0,

2a' > (a' + ib').

Thus, (a' +ib') G Mc+, il>(a' +ib') = Z0- By the above fact, a; £ M+, VZ > Z0.

182

Real Operator

Algebras

Then l0. Moreover, a £ M+, (a + ib) G MC+\MC+, and ip(a + ib) = +00 = sup; ip(ai + ib{). If i>(ai + ibi) < +00, VI, then (a; + ib{) G Mc+, a/ G M+, and ip(ai + ibi) —
1

if 0 we have 6 = 0. Therefore, ip is also faithful. Q.E.D. Theorem 8.4.5. Let M be a real W*-algebra, and Mc = M+iM. the following statements are equivalent: (1) M is semi-finite; (2) there exists a faithful semi-finite normal trace on M+; (3) Mc is semi-finite.

Then

Proof. (1) <£=» (2) It is similar to [26, Theorem 6.5.8]. (2) => (3) It is immediate from Proposition 8.4.4 and [26, Theorem 6.5.8]. Now let Mc be semi-finite. Then there exists a faithful semi-finite normal trace ip' on Mc+. Let 1>{x) = \{tf{x)

+

tf{x)h

Vx G Mc+.

Clearly, ip is a faithful normal trace on M c + , and tp{x) = i{>(x), Vx G M c + . We claim that ip is also semi-finite. In fact, if x is a non-zero element of Mc+, then there is 0 ^ y G M+ such that y < x and tp'(y) < +00. Of

Classification of Real W* -Algebras

183

course, we have 0 ^ y € Mc+. Then there is 0 ^ ~z € M c + such that z < y, and ^'(2) < +oo. Clearly, we have 0 < z < y < x, z ^ 0, and ip'(z) < ip'(y) < +oo. Hence, ip(z) = \{ip'{z) + ip'(z)} < +oo, i.e., ip is semi-finite. Moreover, let ip = ip\M+. Then
8.5

[6], [26], [31], [33], [50].

Purely infinite (type I I I ) real W*— algebras

Proposition 8.5.1. Let M be a real W*'-algebra. Then M is purely infinite, if and only if, there is no any non-zero semi-finite normal trace on M+. The proof runs similarly to [26, Proposition 6.6.2]. Theorem 8.5.2. Let M be a real W*-algebra, and Mc = M+iM. M is purely infinite, if and only if, Mc is purely infinite.

Then

Proof. Let Mc be purely infinite. If ip is a non-zero semi-finite normal trace on M+, then by Proposition 8.4.4, V is a semi-finite normal trace on Mc+, where tp is the trace on Mc+ determined by tp. Of course, ip is non-zero. This is impossible since Mc is purely infinite. Therefore, there is no any non-zero semi-finite normal trace on M + , and M is also purely infinite by Proposition 8.5.1. Conversely, let M be purely infinite, If ip' is a non-zero semi-finite normal trace on M c +, let tf(*) = \W(x)

+ iP'(x)),

Vx e Mc+,

then V is a non-zero semi-finite normal trace on M c + , and ip(x) =

Real Operator Algebras

184

ip(x), *ix € Mc+. Let ip = ip\M+. Then

8.6

[6], [26], [31], [33], [50].

Properties on other classes of real W*— algebras

Proposition 8.6.1. Let M be a real VN algebra. (1) The following statements are equivalent: a) M is discrete; b) M' is discrete; c) there exists a real VN algebra N such that M is * isomorphic to N, and N' is abelian; d) there exists an abelian projection p in M such that c(p) = 1; e) for any non-zero projection p in M, there is a non-zero abelian projection q in M such that q are also discrete. (3) Let M = X); ©Mj. Then M is discrete, if and only if, each Mi is discrete. (4) Let M be discrete. Then M can be uniquely written as follows:

M=^2 ©M„, neE

where E is a set of different cardinal numbers, and Mn is n-homogeneous discrete, i.e., there exists an orthogonal family {pi\l £ A} of equivalent abelian projections in Mn such that JZ; 6A Pi = ln (the identity of Mn ) and # A = n (notice that if {qr\r G A'} is another orthogonal family with the same property, then # A ' = # A = n, i.e., n is well-defined), Vn € E. Moreover, Mn Si where Hn is an n-dimensional

B(Hn)®Nn,

real Hilbert space, and Nn is an abelian real

Classification of Real W* -Algebras

VN algebra, Vn e E. (5) If M is discrete, then the complex VN algebra Mc — M+iM discrete.

185

is

Proof. The proofs of (1) - (4) are similar to [26, section 6.7]. (5) Let M be discrete. By (1), there is an abelian projection p in M such that c(p) = 1. Clearly, p is also an abelian projection in Mc, and the central cover of p in Mc is also 1. Therefore, Mc is discrete ([26, Theorem 6.7.1]). Q.E.D. Remark. If Mc is discrete, then M may not be discrete. This is the case for M = H, the quaterion algebra. Proposition 8.6.2. Let M be a real VN algebra. (1) If M is semi-discrete, then there is a semi-abelian projection p in M such that c(p) = 1. In particular, M is semi-finite. (2) If M is semi-discrete, andp' is a projection in M', then Mpi is also semi-discrete. (3) Let M = ^2t ®Mi. Then M is semi-discrete, if and only if, each Mi is semi-discrete. (4) Let M be semi-discrete. Then M can be uniquely written as follows: ®Mn,

M=YJ n€E

where E is a set of different cardinal numbers, and Mn is n-homogeneous semi-discrete, i.e., there exists an orthogonal family {pi\l € A} of equivalent semi-abelian projections in Mn such that YlieAPi = 1» (^e identity of Mn) and # A = n (notice that if {qr\r € A'} is another orthogonal family with the same property, then # A = # A ' = n, i.e., n well-defined ) , Vn € E. Moreover, Mn =

B{Hn)®Nn,

where Hn is an n-dimensional real Hilbert space, and Nn is a real VN algebra such that its self-adjoint part is commutative, Vn € E. (5) If MH is commutative, then M is semi-discrete. In particular, if p is a semi-abelian projection in M, then Mp is semi-discrete.

186

Real Operator Algebras

Proof. The proofs of (1) - (4) are similar to [26, section 6.7]. And (5) is obvious. Q.E.D. Proposition 8.6.3. Let M be a real VN algebra. (1) M is semi-continuous, if and only if , there is no any non-zero central projection z in M such that Mz is discrete. (2) If M = Y^i ®Mi, then M is semi-continuous, if and only if , each Mi is semi-continuous. (3) Let M be semi-continuous, andp' be a projection in M'. Then Mp> is aslo semi-continuous. (4) M is semi-continuous, if and only if, M' is semi-continuous. Proof. (1) - (3) are obvious. (4) is also obvious by Proposition 8.6.1(1). Q.E.D. Proposition 8.6.4. Let M be a real VN algebra. (1) M is continuous, if and only if, there is no any non-zero central projection z in M such that Mz is semi-discrete. In particular, if M is purely infinite, then M is continuous. (2) Let M = Yli ©Mj. Then M is continuous, if and only if, each Mi is continuous. (3) Let M be continuous, andp' be a projection in M'. Then Mp> is also continuous. (4) M is continuous, if and only if, each non-zero projection p in M can be written as p = p\ + pi for some projections p\,p2 in M satisfying P\Pi = 0 and pi ~ p2-

(5) If M is continuous and semi-finite, then there exists a decreasing sequence {pn} of finite projections in M such that c{p\) = 1 and (Pn ~ Pn+l) ~ Pn+l,

Vn.

(6) If M is continuous, then the complex VN algebra Mc = M+iM also continuous.

is

Proof. (1) - (3) are obvious. (4) Let M be continuous, and p be a non-zero projection in M. Then p is not semi-abelian. By Proposition 8.1.7(4), there is a non-zero projection q in pMp such that q can be written as q — qi + 92, where giffe = 0 and

Classification

of Real W* -Algebras

187

P\iP2i it follows that c(pi) = c(p). Now by Proposition 8.1.7(3), Pi = c(pi)p = c(p)p = p. Therefore, p = 0, and M is continuous. (5) Since M is semi-finite, we can find a finite projection p in M such that c(p) = 1 . Then by (4) we can get the required {pn}(6) By Theorem 8.5.2, we may assume that M is also semi-finite. Then by (5) there is a decreasing sequence {pn} of finite projections in M such that c(pi) = 1, and (pn — pn+\) ~ p r a + i , Vn. Of course, these properties of the sequence {pn} still hold in Mc. Now by [26, Theorem 6.8.4] Mc is also continuous. Q.E.D. Remark. By Proposition 8.6.4(4), we can see that the definitions of 6.4.2 and 8.1.8 are the same for a continuous real W-algebra. Proposition 8.6.5. Let M be a real VN algebra, and Mc = M+iM. (1) If Mc is discrete, then M and M' are semi-discrete. In particular, if M is discrete, then M' is semi-discrete. (2) If Mc is continuous, then M and M' are semi-continuous. In particular, if M is continuous, then M' is semi-continuous. (3) If there exists a decreasing sequence {pn} of finite projections in M such that c(pi) = 1, (pn — pn+i) ~ Pn+\i Vn, then M is semi-continuous (and semi-finite, of course). Proof. (1) First we claim that M contains a non-zero semi-abelain projection. Otherwise, M is continuous. By Proposition 8.6.4(6), Mc is also continuous. This contradicts our assumption. Clearly, Mcz is discrete for any non-zero central projection z in M. By the preceding paragraph, Mz contains a non-zero semi-abelian projection. Hence , M is semi-discrete. Moreover, (Mc)' = (M') c is also discrete since Mc is discrete. Therefore, M ' is semi-discrete.

188

Real Operator Algebras

(2) If there is a non-zero central projection z in M such that Mz is discrete, then by Proposition 8.6.1(5), (Mz)c = Mcz is discrete. This contradicts our assumption. Thus, M is semi-continuous. Moreover, since Mc is continuous, if follows that (M c )' = (M') c is continuous. Therefore, M' is semi-continous. (3) By the assumption and similarly to the proof of [26, Theorem 6.8.4], M P l does not contain any non-zero semi-abelian projection , i.e., M P l is continuous. By (2), M'pi is semi-continuous. But c(pi) = 1, so M' is also semi-continuous. Moreover, by Proposition 8.6.3(4), M is semi-continuous. Q.E.D. Remark. Here, we have many open questions. For example, if M is semi-discrete, semi-continuous, or continuous, are so Mp, M', M c ? and etc. References.

8.7

[6], [26], [31], [33], [48], [50].

Real factors and tensor products

Definition 8.7.1. Let M be a real W*-algebra. M is called a real factor, if its center Z = Z(M) is trivial, i.e., Z^R. We have the following real factors. (1) Discrete real factors. By Proposition 8.6.1, it must be n-homogeneous for some n, and it must be the form of B(Hn), where Hn is an n-dimensional real Hilbert space. Moreover, B(Hn) is finite or (properly) infinite, if and only if, n is finite or infinite. (2) Semi-discrete and Semi-continuous real factors. Let M be a semi-discrete and semi-continuous real factor. By Proposition 8.6.2, it must be n-homogeneous for some n, and M 3

B{Hn)®N,

where Hn is an n-dimensional real Hilbert space, and N is a non-

commutative real factor with commutative self-adjoint part NH- By Propo-

Classification of Real W-Algebras

189

sition 8.1.7, NH = ZH, where Z = Z(N) is the center of N. But N is a real factor, so NJJ = ZH = Z = R. Now it is easy to see that N is divisible (see Remark following Proposition 5.3.7). Since N is not commutative, N = H (the quaterion algebra). Therefore, M £ B(ff„)®H. Moreover, M is finite or (properly) infinite, if and only if , n is finite or infinite. (3) Continuous and semi-finite real factors. (4) Purely infinite real factors. Using the standard method of group measure spaces (see [26, Chapter 7]), we can find the examples of the real factors of (3) and (4). Proposition 8.7.2. Let Mi,M2 be two real W*-algebras. (1) If Mi is rii-homogeneous discrete, i = 1,2, then M{®M2 is ( n i ^ ) homogeneous discrete. (2) If Mi and Mi are semi-finite, and Mi or Mi is continuous, then M{®Mi is semi-finite and semi-continuous. (3) If Mi®Mi is discrete, then both of Mi and Mi are semi-discrete and semi-finite. (4) If M{®Mi is continuous, then Mi or Mi is semi-continuous. Proof. (1) It is similar to [26, Proposition 6.9.7]. (2) Let Mi be continuous. By Proposition 8.6.4(5), there exists a decreasing sequence {pn} of finite projections in Mi such that c(pi) = 1, and (pn — Pn+i) ~ Pn+i, Vn. Take a finite projection q in M 2 such that c(q) = 1. Using {pn <8> q\n} and by Proposition 8.6.5(3), M{(§Mi is semicontinuous. (3) It is immediate from (2). (4) It is immediate from (1). Q.E.D. Remark.

There are also many open questions. Moreover, further study

190

Real Operator Algebras

of semi-discrete and semi-continuous real W*-algebras is necessary. References.

[6], [26], [31], [33], [50].

Chapter 9

Real Reduction Theory

9.1

Real measurable fields of Hilbert spaces

Definition 9.1.1. (E, B, —) is called a Borel bar space, if (E, B) is a Borel space {i.e., E is a set , and B is a a-Bool algebra of subsets of E), and "-" : E -> E is a Borel isomorphism with period 2 (i.e. t = t, Vt G E, and I e B , VB6B). Definition 9.1.2. H(-) is called a real field of Hilbert spaces on a Borel bar space (E, B, —), if H(i) is a real Hilbert space, and H(t) = H(t), Vt € E. Here the condition "H(t) — H(t),Wt e En is essential for latter development. Let (E,B,—) and H(-) be as above . £(•) is called a real field of vectors (with respect to (E,B,-,H{-)), if f(t) € H(t)c H(t)+iH(t), and £(£) = £(£), where £(£) is defined by the decomposition of H(t)c = H(t)+iH{t), Vt € E. Definition 9.1.3. A real field H(-) of Hilbert spaces on a Borel bar space (E, B, —) is said to be measurable, if there is a sequence {£n(-)}n of real fields of vectors (with respect to (E,B, —, H(-))) such that the subset {£n(t)}n is total in H(t)c (i.e., the (complex) linear span of {£n(t)}n is dense in H(t)c),

Vt e E, and *—+/n,m(0 = «»(*). *m(0>t is a (complex valued ) measurable function on (E,B), \/n,m, where (,) t is the inner product of H(t)c induced by the inner product (, }t in H(t), Vt G E. Clearly, fn,m(t) = fn,m(t), Vn,m and t e E. 191

192

Real Operator Algebras

Definition 9.1.4. Let (E, B, -),H(-) and {fn(-)}n be as above, and £(•) be a real field of vectors. £(•) is said to be measurable, if

is measurable on (E,B),Vn.

Denote.

6 = {£(•)!£(•) is a real measurable field of vectors }. Clearly, © is a real linear space; £„(•) G G, Vn; and if £(•) G 0 and fn{t) = (£(*Un(*)>t> then fn(t) = ~fjj), Vt G E,n. Remark. By [26, Definition 12.1.1], H(-)c with {£n(')}n *s a l s o a measurable field of (complex) Hilbert spaces on (E, B), and the set of all measurable fields of vectors (in H(-)c, see [26, Definition 12.1.1]) is ffl

©c=U(0

G H(t)e,

Vi G £ ;

£ ->• (£(£), f n (£)) t is measurable on (E,B),

Vn

If £(•) G 6 C , then £(•) e 6 if and only if £{t) = £(t), Vf € £ . Now let £(•) G 0 C , and

»?(•) = f(*(-)+tt)), c(-) = ^ ( - ) - I O ) Since m),Ut))t = K»(*U(*)>* ^jCn(*),e(t)r, V* € £ , n . and "-" is a Borel isomorphism, it follows that £(7) S 6 C , and n(-),C(-) G G. Therefore, G c = G+iG.

Proposition 9.1.5. Let (E,B,—), (1) For any n = oo, 0,1,2, • • •,

H(-),{^n(-)}n

and Q be as above.

En = {t G £ | dimi7(i) = n } 6 6 , ond l? n = En. (2) From {£n(-)}n we can construct an orthogonal normalized basis {en(-)}n of the field H(-), i.e., en{-) G G, Vn, and for any t G E, {en(t)}n is an orthogonal normalized basis of H(t)c if dimH{t) c = +oo; or {e n (f)|l < n < k} is an orthogonal normalized basis of H(t)c if dim H(t)c = k < +oo anden(t) = 0, Vn > k. Indeed, {en(-)}n can be obtained through the method of Schmidt orthogonalization of {£n(-)}n, and £,i{t) G [ei(<),-•• ,e„(i)],

Real Reduction

Theory

193

where [e\(t),--- ,en(t)] is the (complex) linear subspace of H(t)c spanned by {ei(t), • • •, en{t)}, VI (£(£),e n (t)) t is measurable on (E,B), Vn. Proof. Clearly, t -+ \\tj{t)\\t is measurable on (E,B),Vj. Let F 0 (o) = { t e £ & ( * ) = 0,Vj} = { t e E|ff(t) = {<>}}, F ^ = {* e S|ei(0_# O } , ^ = {£ e Eft® ? 0,ti(t) = 0,1 < t < j},Vj > 2. By fc(i) = fc(t),Vt G £ , j , we have Ff»=Ff6B1i=0,l,2,

^°)|,--

and {Fj '\j = 0,1,2, • • •} is a Borel partition of E. Let

ei(t) =

0, if fc(*)/ll&(*)llt,if

t e F r(°) j , t6F}G),Vj>l.

Clearly, || e i (t)|| t = l.Vt G F\F 0 ( 0 ) ,e a (i) = ei(t),Vt € F , and fc(i) e [ei(t)], where [ei(i)] is the (complex) linear subspace of H(t)c spanned by {ei(t)},Vi € E . Moreover,

~

X F (o)(t)

t -»• (ei(i),o(t))t = E ]j^)jj7^(')'^W)t is measurable on (E,B),Vj.

Thus, ea(-) 6 6 . If £(•) € ©, then ~ XF(°>(0

t -> (ew.exW)* = E i f ^ ^ ^ * is also measurable on (E, B). Now for n(> 1) suppose that we have (ei(-), • • • ,e n (-)} C 0 with the following properties: (i) if t € E and dimi7(i) > n, then (ei(t),ej(t))t = 5ij,l • <£(£)> e,(0) t is measurable on (F,B), VI < i < n.

194

Real Operator Algebras

Let pn(t) be the orthogonal projection from H (t)c onto [e\ (t), •••, en(t)], Vf € E, and £(•) e 9 . Since p n (f) = p„(i) and n

P»(0£(*) = E fc(*),ei(*)Wi),Vt € £, »=i

it follows that p„(-)e(") and (l(-) -p„(-))£G) = £(•) - P n ( - ) £ » 0 € 9 , where 1(f) is the identity operator in H(t)c,\/t € E. Clearly, (l(t) -p n (*))&(*) = 0,Vte£,l
F0(n) = {te ^|(i(t) -Pn(*))6(0 = o.Vi} = {te^jdimiZ'(t)
^

\

te/i

J

W _ Pn(«), fn+i(t) 7^ 0, and

l(t)-p„(t), m

1

= 0^i
Similarly to the above, FJ n ) = FJn) e S,Vj > 0, and { F J n ) | j = 0,1, • • • , } is a Borel partition of E. Let ,

e„+l(0 - ^ (l(t)- Pn (t))^ + ,(t) I \\(i(t)-Pn(t

if t € F 0 ( n ) , .,. , C , »

Then {ei (•),••• ,e„+i(-)} c 9 also satisfies the above properties (i)—(iv) (replace n by n+1 , and notice that t -> ||(l(t) —P»(*))C»+*(*)llt is measurable on (E, B), Vi). Repeating this process, we get a sequence {e„(-)}„ C 9 . Since {£n(t)}n is total in H(t)c,Vt € E, it follows that {e n (-)} n is an orthogonal normalized basis of H(-). If £(•) is a real field of vectors such that t -> (f (f), e„(i)) t is measurable on (E,B),Vn, then

i

is also a measurable function on (E,B),Wn, and £(•) G 9 . Finally, since i ->• ||ej(t)|| t is measurable on (E,B),Wi, it follows that

«..«.- { . e / S |d t a H(.)-. } -{.6*| - | ; ° 0 ; ^„ £ "' a " d }e B Vn = oo,0, !,•••.

Q.E.D

Real Reduction

Theory

195

Remark. The above proof is indeed similar to the proof of [26, Proposition 12.1.2]. Of course, one needs to notice the bar "-" operation . Proposition 9.1.6. Let (E,B,—),H(-),{^n(-)}n andQ be as above. (1) {Cn(-)}n C 0 is said to be total, if{Cn(t)}n is total in H(t)c, Vt € E. If {Cn(m)}n is total, and £(•) is a real field of vectors, then £(•) € 0 , if and only if, t —*• (£(t),Cn(t))t is measurable on (E,B), Vn. This means that 0 can be defined by any total sequence in 0 . (2) #£(•) € 0 , then t -> ||£(t)|| t is measurable on (E,B). (3) #£(•)> ^ O € ©> tfien t ->• (£(t),r)(t))t is measurable on (E,B). Proof. It is similar to [26, Proposition 12.1.4].

Q.E.D.

Example. The constant real measurable field of Hilbert spaces. Let (E,B,—) be a Borel bar space, and Ho be a fixed separable real Hilbert space. Let H(t) = Ho, f n (t) = £n, Vt € E, where {£n}n is a total subset of H0. Then H{-) is real measurable with {£n(-)}n- Since the (real) linear span of {£n}n is dense in H0, it follows that t(t) € Ho+iHo, m

e=U(0

= £(*), Vt € E,

t -> (€(t), 0 is measurable on (E, B), V£ e H0

where (,) is the inner product of Ho+iHo- Thus, 0 is independent of the choice of the total subset {£™}(c Ho). Clearly, {e„(-)}n is an orthogonal normalized basis of H(-), where e n (t) = e n ,Vt € E,n, and {e n }„ is an orthogonal normalized basis of HoDefinition 9.1.7. Let H(-) be a real measurable field of Hilbert spaces on a Borel bar space (E,B,—),v be a measure on (E,B) and v o — = v. Denote

/e J(E,-)

H{t)du{t) = {{(.) e 01 / ||£(t)||>(t) < +ool, I

\JE

)

and JE

Ve(-)^(0 e /(® v H{t)dv{t). Moreover, / ( | _ } if (t)di/(t) is called the direct integral of H(•) on (£, 6, —) with respect to IA

196

Real Operator Algebras

Proposition 9.1.8. With the notations in Definition JtE -) H{t)dv (t), (,)) is a real Hilbert space, and H+iH = J

H(t)cdu(t)

= U(-) G 9 C \j

9.1.7, (H

\\i{t)ftdv{t)

< +oo | ,

i.e.

/

H{t)cdv{t)

JE

is the complexification of J(E

_.H{i)dv{t).

Proof. By [26, Proposition 12.1.8], fE H{t)cdv{t) is a (complex) Hilbert space. If 77(-), £(•) G H, then by n(t) = tffi, £(i) = W), H(t) = H(t), Vi G E, and v o — = v, we have <»?(•),*(•)> = / fa(*U(*)>t<M*) = / <^(t),C(t))r^(*) JE

JE

JE

JE

Thus, (if, (,)) is a real inner product space. If £(•) G / £ H{t)cdv{i), then by £(") € JE H(t)cdv(t) and the Remark following Definition 9.1.4 we can write

where „(.),C(0 6 H, and ||£(-)|| 2 = \\v{-)\\2 + IICOII2- Therefore, /® H(t)cdv{t) = ff-i-iff, and ff is a real Hilbert space. Q.E.D. Example. Let ff(-) = ffo be the constant real measurable field on (E, B, - ) , and v o - = i/. Then L2(E,B,-,v)®H0,

tf = /

H{t)dv(t) =

H{t)cdv{t)

= L2{E, B, v) ® (ffo+iffo) -

and /

ff+i#.

Real Reduction

197

Theory

In particular, if Ho = K, then H0+iHQ = C, / Rdv(t) = J(E,-)

L2(E,B,-,v)

and L2(E,B,-,v)+iL2(E,B,-,v)

= L2(E,B,v)

= I

Cdv{t),

JE

where L2(E,B,-,u) References.

9.2

= {/ G L 2 (£,B,i/)|/(i) = M ,

«-e- * } •

[6], [26], [34], [50].

Real measurable fields of operators

Definition 9.2.1. Let (E,B,—) be a Borel bar space, H(-) and K(-) be two real measurable fields of Hilbert spaces. A field o(-) of operators from H(•) to K(-) is said to be real measurable, if a(t) is a bounded (complex) linear operator from H(t)c to K(t)c (i.e., a(t) G B(H(t)c,K(t)c) ), Vt € £ , and a(-) : 0(tf(-)) -»• ©(if(•)) («-ev for any £(•) € 6(H(-)), we have (aO(-) G 0(K(-)), where (aOW = a(t)e(i),Vi G £ ) . Proposition 9.2.2. A field a(-) o/ operators from H(-) to K(-) is real measurable , if and only if , a(-) : H(-)c -» K(-)c is measurable (i.e., a(-) : ec(H(-)c) ->• QC(K(-)C), see [26, Definition 12.2.1]), and a(t) = a(i) (Notice that B{H(t)c,K(t)c) = B(H(t),K(t))+iB(H(t),K(t)). _By this decomposition, we have a "-" operation in B(H(i)c,K(t)c), and a(t) is in this sense), Vt € E. Moreover, if a(-) : H(t)c ->• K(-)c is measurable (see [26, Definition 12.2.1]), then a(-) can be uniquely written as a(-) = &(•) + ic(-), where b(-) and c(-) are real measurable from H(-) to K(-). Proof. If a(-) : H(-) ->• K(-) is real measurable, then (a£)(-) G 6(K"(-)), Ve(-) G 6 ( # (•))• In particular,

a(i)tit) = (<*)(?) = RXO = °WU)

198

Real Operator Algebras

Vt G £,£(•) G &(H(-)). Since {$n(t)}„ Definition 9.1.3), it follows that a(t) = o(t),

is total in H(t)c

= H(t)c

(see

Vi G E.

Moreover, by 0 c (if(-) c ) = G(if (-))-Me(ff (•)), we have a(-) : 6 c (ff(.)cj —• e c ( ^ ( - ) c ) , i.e., a(-) : H(-)c -¥ K{-)c is measurable. Conversely, let a(-) : H{-)c -»• fsT(-)c be measurable, and a(t) = a(t), Vi G £ . For any f (•) G 6 ( i f (•)), we have (a£)(-) G e c (ff(-)c), and « ) ( * ) = <*(*) e(t) = a(t)g(t) = (ae)(t),

ViG£,

i.e., (af)(0 G 0(iir(-)). Therefore, a(-) : if (•) -» ff(-) is real measurable. Now let a(-) : if (-)c ->• iT(-) c be measurable. It is easy to see that a(7) : if (-)c -»• i
and c(-) = I ( o ( - ) - ^ ) )

are real measurable.

Q.E.D.

Definition 9.2.3. Let (E,B, —), H(-) and K(-) be as above, and a(-) : if (•) -» K{-) be real measurable . If v is a measure on (E, B) with i/o— = v, and if t —>• ||a(t)||t is essentially bounded on {E,B) with respect to v, (By [26, Proposition 12.2.2], t ->• ||o(i)||t is measurable on (E,B)), then (a£)(.) = a(-)£(0(^£(0 G if) is a bounded (real) linear operator from H to iif, where if = /

•As,-)

H(t)du(t)

and AT = /,® _, K(t)dv{t)

are real Hilbert

spaces. The above operator a is denoted by /•©

a—j

a{i)dv it),

and is called the direct integral of a(-) on ( £ , B, - ) with respect to v. A bounded linear operator a from if to if is said to be decomposable, if a has the above form. In particular, for any / G L°°(E,B,—,i'), the operator

Real Reduction

Theory

199

is decomposable, where (m/f)(£) = f{t)£(t),Vt 6 £,£(•) 6 if. And m / is called a reo/ diagonal operator with symbol f. Let Z =

{mf\feL°°(E,B,-,v)},

and Z is called the real diagonal operator algebra in H = / , £ _. Clearly,

H(t)dv(t).

Zc = Z-HZ is the diagonal operator algebra in Hc = H+iH = fE H{t)cdv{t) Definition 12.2.7]).

(see [26,

Proposition 9.2.4. Let H{-) be a real measurable field of Hilbert spaces on a Borel bar space (E,B,—), v be a a-finite measure on (E,B) with v o — = v, and H = f(E , H{i)dv{t). Then the real diagonal operator algebra Z is an abelian real VN algebra in H, Z' is a-finite, and f —Y m,f is a faithful W*-representation of L°°(E, B, —, v) in H. Proof. It is obvious from [26, Proposition 12.2.9] and Proposition 4.6.3. Q.E.D. Proposition 9.2.5. Let Hi(-),H2(-) be two real measurable fields of Hilbert spaces on a Borel bar space (E, B, —), and v be a measure on (E, B) with v o — = v. Denote Hi = f,E _>Hi{t)dv{t),i = 1,2. If v is a-finite, then a bounded (real) linear operator a : Hi —> H2 is decomposable, if and only if, (i)

amj

(2)

— m) a,

where m\ is the real diagonal operator with symbol f in Hi, i = 1,2, V/ € L°°(E,B, -,1/). Proof. Let a € B(Hi,H2) be decomposable. Then a as an element of B((Hi)c, (H2)c) is still decomposable. Now by [26, Theorem 12.2.10], we have amf] = mfa, V/ € L°°(E,B, - , v). Conversely, let am^ = mf]a, V/ € L°°(E,B,-,v). Then we have am(p = mf]a, V/ € L°°{E,B,v). By [26, Theorem 12.2.10], a e B((Hi)c, (H2)c) is decomposable. Thus, there is a measurable field

200

Real Operator Algebras

a(-) of operators from (Hi(-))c to (i?2(0)c

sucn

that

/•e a= I a(t)di/(t). JE

Let {en(-)}n be an orthogonal normalized basis of -Hi(-). Then f{-)en{-) € HuWf e L2(E,B,-,v),n. Since a € B{H1,H2), it follows that a ( - ) / ( - K ( - ) € ff2, and a(.)/(-)e»(-) 6 e(H a (-)),V/ € L2(E,B,-,v),n. In particular a(?)f(t)en(t)

= a{t)f{t)en{t) = a(t) f(t) en{t) =

V« e £ , / 6 L2{E,B,-,v),n. total in H\{t), it follows that

a(t)f(i)en(t),

Since {f{t)en{t)\f

a(t) = o(t),

9.3

is

Vi 6 £ .

Therefore, o is also decomposable from Hi to #2References.

e L2(E,B,-,v),n}

Q- E. D.

[6], [26], [34], [50].

Real measurable fields of V N algebras

Definition 9.3.1. Let (E, B, —) be a Boreal bar space, and H(-) be a real measurable field of Hilbert spaces on (E,B,—). A field M(-) of (complex) VN algebras on (E, B, —) is said to be real measurable (in H(-)), if M{t) is a (complex) VN algebra in (complex) Hilbert space H(t)c — H(t)+iH(t), Vi G E, and there is a sequence {a n (-)}„ of real measurable fields of operators in H(-) such that the VN algebra M(t) is generated by the sequence {an(t)}n of operators, Vi S E. Proposition 9.3.2. Let(E,B,—) andH(-) be as above. Then a field M'(•) of (complex) VN algebras in H(-) is real measurable , if and only if, M(-) is measurable in H(-)c (see [26, Definition 12.3.1]) and M(i) = M(t), Vi € E. Proof. Let M(-) be real measurable in H(-), and {an(-)}n be as in Definition 9.3.1. Since an(-) is also measurable in H(-)c(see Proposition 9.2.2), Vn, it follows that M(-) is measurable in H(-)c (see [26, Definition 12.3.1]). Moreover, by H(t) = H(t), and an(i) = an(t), Vn, we have

Real Reduction

201

Theory

M(f) = {a^Yi = M m = {<*»(*)}„ = M(t) = {x\x G M(t)}, Vf G £7. Conversely, let M(-) be measurable in .ff(-)c, and M(i) = M(t), Vi G .E. By [26, Definition 12.3.1], there is a sequence {a n (-)}„ of measurable fields of operators such that M(t) is generated by {an(t)}n, Vi G E. By Proposition 9.2.2, we can write On(-) = M 0 + «Vi(-)i where b n (-)> c n( - ) M{t) and

a r e rea

l measurable , Vn. Since bn(t) + icn(t) = an{t) G

bn(t) - icn{t) = bn(t) + icn{t) = an{t) € M{t) = M(t), it follows that bn(t), cn(t) e M(t), Vt G E, n. Further, M(t) is generated by {bn{t),Cn(t)}n, Vf G E. By Definition 9.3.1, M(-) is real measurable. Q.E.D. Definition 9.3.3. Let M(-) be a real measurable field of VN algebras on (E, B, —) in H(-), v be a cr-finite measure on (E, B) and vo — = v. Denote

M= I

r

M{t)dv(t) ,_)

= \a=

© I

I a{t)dv{t) G £ ( # ) |a(t) G M(t), a.e. v \ ,

where if = C s _. H(t)du(t), and M is called the direct integral of M(-) on (E,B,—) with respect to v. Similarly to [26, Proposition 12.3.5], M is a cr-finite real VN algebra in the real Hilbert space H. M is also called the real VN algebra defined by the real measurable field M(-) of VN algebras. A real VN algebra M in H is said to be decomposable, if M has the above form . Proposition 9.3.4. Let (E, B, - ) , H(-), v and H be as in Definition 9.3.3. (1) Let M(-) be a real measurable field of VN algebras in H(-), M = J,E _, M{i)dv{t), {an(-)}n be as in Definition 9.3.1, and ||a n (t)||t < 1, Vf G E,n. Then M is indeed generated by Z and {an}n, where Z is the real diagonal operator algebra in H (see Definition 9.2.3), and a ™ = /(*,-) o.„{t)dv(t), Vn.

202

J,E

Real Operator

Algebras

Moreover, let a € B(H). Then a e M, if and only if, a = sa(t)di/(t) is decomposable, and a(t) = a(t),a(t) e M(t), a.e.v.

(2) In the (complex) Hilbert space H+iH = Hc = f® H(t)cdv{t), (complex) VN algebra M+iM = J

the

M{t)dv(t),

JE r®

where the VN algebra /

M{i)dv{t) is defined as in [26, Definition 12.3.6].

JE

(3) A real VN algebra M in H is decomposable, if and only if, M is generated by Z (the real diagonal operator algebra in H) and a sequence {an = f,E _. a(t)df(t)}n of decomposable operators in H. In this case,

M= I

M{t)dv(t),

where M(t) is the (complex) VN algebra generated by {an(t) in H(t)c, a.e. v. Moreover, M(-) is unique (a.e. v), and Z (4) If H is separable, then a real VN algebra M in H is if and only if, Z C M c Z' (and this is also equivalent to obviously).

G B(H(t)c}n cMcZ'. decomposable, Z C M D M'

Proof. (1) and (2). M(-) is also measurable in H(-)c. By [26, section 12. 3] , the (complex) VN algebra f® M(t)dv{t) in J® H(t)cdv{t) is generated by the diagonal operator algebra Zc and the sequence {JE an(t)dv(t)}n of decomposable operators. Moreover, ZCC

I

M(t)dv(t)

C Z'c

JE

and / JE

M(t)du(t)

= {a = /

a(t)du(t) € B(Hc)\a(t)

€

M{i),a.e.v},

JE

where Hc = / ® H{i)cdv{t) = H+iH (Proposition 9.1.8). For any a = f® a{t)di/(t) € / ® M(t)du(t), by Proposition 9.2.2 we can write a(.) = &(•) + «:(•),

203

Real Reduction Theory

where &(•) = |(a(-) + a(-)),c(-) = ^(o(-) - a(-)) &re r e & l measurable. Since Af(i) = M(t), it follows that b(t),c(t) € M(t), a.e. i/. Clearly, ||fc(*)llt,||c(*)ll* < NOIIt.Vt € £ . Thus, / ( ®,_j fc(t)^(*) and /(£,-) c(t)dp(t) G M , and Mc=

I

M{t)dv{t) = M+iM,

JE

where M = / g ^ M ^ ^ ^ ) - By Z c = Z-i-iZ, and /•©

re

an-

an(t)dv(t) JE

-

I

an{t)dv(t) J(E,-)

under the embedding B(H) <-+ B(Hc),Vn, M is generated by Z and { a n } n . (3) It suffices to show the sufficiency. Let M be generated by Z and {an = J,® »a n (i)di/(i)} n . Suppose that M(t) is the VN algebra in i?(i)c generated by {a„(t)} n ,Vt € E. Clearly, M(-) is real measurable. Let a = /(® _. a{t)dv{t) G JB(#), o(t) G M(t), a.e.v, and a' G M'. Since M' c Z', it follows from Proposition 9.2.5 that a' = /

o'(t)di/(t)

is decomposable. Noticing that a'a n = ana',Vn, we have a'(i) G M(t)', a.e.v, a'(t)a(t) = a(£)a'(£),a.e.i/, and o'a = aa'. Thus, -CD

MD

/ HE,-) J(E,-)

M{t)dv{t).

Clearly, Z and {a„}„ C /(® _} M(t)dv{t)

by (1).

Therefore, M

c

/ J , ) M(i)^(t) and M = Jg ^ Af (i)<M*)Moreover, it is easy to see that M(-) is unique, a. e. v. (4) is immediate from [26, Proposition 12.3.10].

Q.E.D.

Remark. Let (E, B, —), i/(-)> v and i7 be as in Definition 9.3.3. By [26, Proposition 12.2.6], t —>• Cli, and t —>• B{H(t)c) are measurable fields of VN algebras in H(-)c on (E,B). C l r = Clt = Cl t >

B(H(t)c)

= B(H(t)c)

=

Clearly, B{H{t)c),

204

Real Operator

Algebras

Vt e E. By Proposition 9.3.2, t ->• C l t and t -»• B(H(t)c) are real measurable fields of VN algebras in H(-) on (£, S, —). Then we have real VN algebras / Cltdu(t), J(E,-)

and / •/(£,-)

B{H{t)c)dv(t)

in H. By the Remark following [26, Proposition 12.3.7], Z+iZ = /

Cltdu(t),

and Z'+iZ' = /

JE

B(H(t)c)du(t),

JE

where Z is the real diagonal operator algebra in H. Thus , by Proposition 9.3.4 we have Z = f Cltdu(t), J(E,-)

and Z' = f J(E,-)

B{H{i)c)dv{t).

Proposition 9.3.5. Let (E, B, —), H(-), v and H be as in Definition 9.3.3. Let M(-),M n (-) (n = 1,2, •••) be real measurable fields of VN algebras in #(•), andM=$®E_) M{t)dv{t), Mn = / ( ® _ } Mn(t)dv{t), Vn. Then (1) M(-)',(UnMn(-))" and n„M n (-) are also real measurable; (2) M'=

f M{t)'dv(t), J(E,-)

(UnMn)"= f

J(E-)

(UnMn(t))"dv(t),

and r®

n„M n = /

J(E,-)

nnMn(t)du(t);

(3) M (~l M' = Z (the real diagonal operator algebra in H), if and only if, M(t) is a (complex) factor, a.e. v. Proof. (1) By Proposition 9.3.2 and [26, Proposition 12.3.8], M(•)' is measurable. Clearly, M(f)' = M{t)' = M(t)', Vt € E. By Proposition 9.3.2, M(-)' is real measurable. Similarly, (U n M n (-))" and n n M„(-) are real measurable .

Real Reduction Theory

205

(2) By Proposition 9.3.4 and [26, Proposition 12.3.8], we have Af'+iAf'= ( M + t i l f ) ' = /

M{t)'dv{t)

JE

= /

M(t)'du(t)+i

f

M{t)'dv{t).

Thus, M'= /

M(t)'du(t).

Similarly, re

(u n M„)"= /

/•©

(unMn(t))"dv(t),

nnM„ = /

n„Mn(i)dj/(t).

(3) Clearly, M n M ' = Z <=> Mcr\M'c = Zc, where Mc = M+iM, M'c = M'+iM', and Z c = Z+iZ (the diagonal operator algebra in Hc ). Now by [26, Proposition 12.3.9], M n M' = Z, if and only if , M(i) is a factor, a.e. v. Q.E.D. References.

9.4

[6], [26], [34], [50].

Real reduction theory

Let Z be an abelian real VN algebra in a real Hilbert space H. As a real C*-algebra, we have

z - c(n, -) = {/€ c(n)|/(t) = TOO.vt e n>, and Zc = Z+iZ s C(H) = C(n, - ) + t C ( f i , - ) , where fi is the spectral space of the (complex) C*-algebra Zc (a compact Hausdorff space), and bar "-" is a homeomorphism of fi such that —2 = id. Let g —>• mg be the * isomorphism from C(Q) onto Zc. For a fixed vector 77 € H, we have a unique regular Borel measure 1/ on fi such that

(771^,77)= f g(t)dv(t),

V s eC(fl),

206

Real Operator

Algebras

where (,) is the inner product in Hc — H+iH. We claim that v o — = v. In fact, (mfr],r)) g R , V / e C(fi, - ) . Thus, / „ f{t)du{t) = / „ f{t)dv{t) = fnf(t)dv(t)

= fn =

f{t)dv{t) fnf(t)dv(t),

V/ e C(Cl, - ) . By C(fi) = C(n, - ) + i C ( f i , - ) , we have / g{t)du{t) = [ g(t)dv(i),

V 5 € C(fi).

Therefore, ^ o — = v. Now similarly to [26, Theorem 12.4.1], we have the following. Theorem 9.4.1. Let Z be an abelian real VN algebra in a real Hilbert space H, and Z' be a-finite. Then there is a regular Borel measure v on fi such that vo-

= v, suppv = fi, L°°(n, v) = C(il),

L°°{n, v, - ) = C(fi, - ) ,

where fi is the spectral space of Zc = Z+iZ (as abelian C*-algebra), i.e. Zc = C(fi), "-" is a homeomorphism of fl with period 2 such that Z = C(fi, —), and there is a real measurable field H(-) of Hilbert spaces on (CI, —) such that H(t) ^ 0, o.e. v, H ^ /(® _ } H{t)dv{t), and [ J'(n,-) {SI.-}

Clte&/(t),

Z'Si

f J{£1.-) ./(n,-)

B{H{t)c)dv{t),

mil i.e., Z in H is unitarily * isomorphic to the real diagonal operator algebra in / H{t)dv(t). in J(n.-) '(n.-) Lemma 9.4.2 Let Z be an abelian real VN algebra in a separable real Hilbert space. Then Z is generated by a single operator. Proof. By [26, Theorem 5.3.7], Zc is generated by a self-adjoint operator a + ib, where a,b e Z. Clearly, Zc (and then Z) is generated by {a,b}. Since a* = a, b* — —b, and (a + b)* = a — b, it follows that Z is generated by a + b. Q. E. D. Suppose that A is an abelian real C*-algebra with identity 1, and A is generated by { l , a } for some a G A. It is well-known that a(a) = cr(a), and A = C(a(a),—), where a(a) is the spectrum of a as an element in Ac = A+iA, and "-" is the complex conjugation. Moreover, if v is a

Real Reduction

207

Theory

regular Borel measure on a (a) such that v o — — v, then v can be trivially extended to a regular Borel measure v on C such that v o — = v. By Propositions 5.1.6 and 6.3.4, similarly to [26, Theorem 12.4.2], we have the following. Theorem 9.4.3. Let H be a separable real Hilbert space, M, Z be two real VN algebras in H, Z be abelian, and Z C M C Z' (i.e., Z C MnM'). Then there are a finite Borel measure v on C such that vo— = v, where "-" is the complex conjugation, a real measurable field H(-) of Hilbert spaces on (C, —), and a real measurable field M(•) of VN algebras with respect to H{-), such that H*

I

H(t)dv(t),

Ac,-) and Z^

f

Cltcb/(t),

•Ac,-) Moreover, if Z = MDM',

M=

f

M{t)dv{t).

Ac,-) then M(t) is a (complex) factor, a.e. v.

Remark. In the complex case, (C, —) is replaced by R (see [26, Theorem 12.4.2]). Compare with these results, it seems very interesting. Theorem 9.4.4. Let (fi, —) be a a-compact bar space, Vj = Vj o — be a regular Borel measure on Cl, and Hj(-) be a real measurable field of Hilbert spaces on (CI, —),j = 1, 2. If there is a unitary operator u:Hl=

f

Hi(t)dvi(t)

J(Q,-)

—+ if2 -

/

H2{t)dv2(t)

"A",-)

such that um\'u* = m l , where vrvj is the real diagonal operator with symbol f in Hj, V/ € Co(Q, —), j = 1,2, then v\ ~ v2, and there exists a real measurable field v(-) of unitary operators from H$-) onto /•e H2(-) such that u = wvi where v = / v(t)dv\(t) : H\ —> H2 = J(n,-) r®

I

•Aa-)

H2(t)dv\(t){v(i)

: Hi(t)c

—>• H2{t)c unitary, a.e.v$, and w is the

208

Real Operator Algebras

canonical isomorphism from H^ onto Hi, i.e.

K)(.) = p(-)-*e(-),ve(-)€H2,

Proof. Since C0(tl,-)+iC0(Sl,-) = C0{Cl), Hj+iHj = /® Hj{t)cdvj{t), j = 1,2 (Proposition 9.1.8), it follows from [26, Theorem 12.4.3] that v\ ~ vi. Let v = w*u. Then v is a unitary operator from Hi onto H2, and vm\ 'v* = rhf, where rhf is the real diagonal operator in Hi with symbol / , V/ G C0(Sl,-). Since C 0 ( f i , - ) is u;*-dense in L°°(fi,-,i/i), it follows that umy 'u* = m/, V / G L°°(£l, — ,vi). By Proposition 9.2.5, v is decomposable. Q. E. D. Remark. By this theorem, we can see that the measure v in Theorem 9.4.3 is unique in the equivalent sense. But we still don't know that the Borel bar space (C, —) in Theorem 9.4.3 is unique (in the sense of Borel bar isomorphism), and in the complex case this is affirmative (see [26, Theorem 12.4.5]). Now we discuss the relations between decomposable real VN algebras and its components. Since the first classification of real VN algebras is simple (see Theorems 8.2.4, 8.3.2, 8.4.5 and 8.5.2), by [26, Theorem 12.5.10] we have the following. T h e o r e m 9.4.5. Let (E,B,—) be a standard Borel bar space, v a afinite measure on (E, B) such that v o — = v, H(-) a real measurable field of Hilbert spaces on (E,B,—). Let H = J,E _-.H(t)du(t), and let M — J,E _, M{t)dv{t) be a decomposable real VN algebra in H. (1) M is finite, semi-finite, properly infinite or purely infinite, if and only if, M(t) is finite, semi-finite, properly infinite, or purely infinite, a.e.v. (2) There are unique central projections z

j

=

/•e / Zj(t)di/(t), J(E,-)

j = 1,2,3

in M such that z\ © z-i © Zz = 1, z\{t) © z-z(t) © z^{t) = h, a.e.v, and (i) Mi = Mz\ is finite, and M{t)z\{t) is also finite, a.e.v;

Real Reduction

Theory

209

(ii) M2 = MZ2 is semi-finite and properly infinite, and M(t)z2{t) is also semi-finite and properly infinite, a.e.v; (Hi) M3 = Mzj, is purely infinite, and M(i)z3(t) is also purely infinite, a.e.v. The second classification of real VN algebras is very complicated (see section 8.6). We just have following results. Proposition 9.4.6. Let M = /(® _, M(t)dv(t). If M is discrete, or continuous, then M(t) is discrete, or continuous, a.e. v. Proof. It is immediate from Proposition 8.6.1, 8.6.4 and [26, Theorem 12.5.10]. Q.E.D. Remark. Here we have many open questions (see [33]), and further study is necessary. References.

[6], [26], [34], [50].

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Chapter 10

(AF) Real C*-Algebras

10.1

Standard matrix unit

By Proposition 5.1.3, every real C*-algebra A is a fixed point algebra of (B, —), i.e., A = {b € B\b — b}, where B is a complex C*-algebra, and "-" is a cojugate linear * algebraic isomorphism of B with period 2. Now let B be a finite-dimensional complex C*-algebra. We shall study all "-" (as above) operations on B. Proposition 10.1.1. Let B = Mn(C), and let — : B —> B be a conjugate linear * algebraic isomorphism with period 2. Then there exists a unique (up to a multiple of e%e, 0 6 R) unitary matrix u € B such that x = uxcu*,

x € B

{

n is odd,

and u,

if

±u, if n is even, where x = (ttij) if x — (ay) £ M n (C),u T is the matrix transfer of u. In this case, let A = {x € B\x = x} (a real C*-algebra). Then c

[ Mn(R), ifuT = u,

AS* < I M m (H), ifuT = —u and n = 2m. Conversely, if A = M„(R) or Mm(E), M2m(C). 211

then Ac = A+iA = Mn(C)

or

212

Real Operator Algebras

Proof. Let {ey 11 < i, j < n} be the canonical basis of matrix unit of B, and let fy — ^ij-, 1 < *>3 < n- Then {fij} is also a matrix unit of B. Thus there exists a unique (up to a multiple of eie, 0 s R) unitary matrix u € B such that / i j = ueijU*,

Vi,j,

and

V i = ( a y ) e M n (C). Since a; = f = u(ux c u*) c u* = (uu c ) • x • (uuc)* V x e M„(C), it follows that uu c = A7n, for some A G C with |A| = 1. Thus, u* = Xuc and uT = (u*)c = (Xuc)c = Xu. Moreover, 1 = | detu| 2 = det(ttu c ) = det(AJ n ) = An. If n is odd, then A = 1 and uT — u; If n is even, then A = ± 1 and u T = ±u. Since B is an n 2 -dimensional (complex) factor, A must be an n 2 dimensional real factor. By section 8.7, A = M n (K) or M m (H), and when A ^ M m (H) we have n = 2m. Let ui = 7 n . Clearly, Ax = {x € £ | z = xc} = M n (R)

213

(AF) Real C* -Algebras

1 Then

Let n = 2, and let v = I {x G .B = M 2 (C)|x = i>xcu*}

={*(' i) + "C

' H G OlA'M'7'i£R}

-0+T(-I

Generally, let n = 2m, and let

(m times).

u2 \

(V)

0

For x G B = M2m(C), write x = {xij)i(A) = Aj(j = 1 or 2). Clearly, we can find a unitary matrix w G B such that $(x) = wxw*,

Vx G B.

In particular, if x G A, then $(x) G Aj, and IUUX C U*UJ* = IUXW* =

$(x)

= Uj$(x)cU^ =

or f *

A

f*

f

*p

ik

iu UjWux = x w UjWU

UjWcXcW*cUj

214

Real Operator Algebras

VIE € A, then V x e B . Hence wTu^wu = wc*UjWU = A7n for some A e C with |A| = 1. Let v — y,w, where fj? = A. Then vTu*-vu — In, o r u =

{VTU*AV)*

= vCTUjVc,

and

{

u,

if j = 1,

-u,if

j = 2.

Therefore

(

M n (R),

if

M r o (H), if

u T = u, u T = - u and n = 2m.

Conversely, if A = M n (E) or MTO(H), then A is an n 2 -dimensional or 4m 2 -dimensional real factor, and Ac = A-i-iA is an n 2 -dimensional or 4ra 2 -dimensional complex factor. Therefore, Ac s M„(C) or M 2m (C). Q.E.D. Proposition 10.1.2. Let B = B x © £ 2 , S i = £ 2 = M„(C) (complex C*algebras), and let Zj be the central projection in B such that Bj = BZJ , j = 1,2. If "-" is a conjugate linear * isomorphism of B with period 2 such that ~2\ = Z2, then there exists a unique (up to a mutiple of el6,6 e fit) unitary matrix u € M n (C) such that {x\,X2) = (uTx%uc,

ux\u*),

MXJ 6 Bj = M n (C), j — 1,2, and as real C* -algebras, A = {(ziz 2 ) £ B\{xux2)

= (11,12)} = M„(C).

Conversely, let A be the real C*-algebra M n (C). Then B\ = B2 = M n (C) fas complex C*-algebras), and ~z\ = Zi, where Ac — A+iA = B\ © B2, and Zj is the central projection in Ac such that Bj = AcZj, j = 1,2. Proof. — : Bi —> B2 and — : B2 —> B\ can induce two conjugate Unear * isomorphisms of M n (C). By the proof of Proposition 10.1.1, there are two

(AF) Real

C*-Algebras

215

unique (up to a multiple of eie,6 e R) unitary matrixes u,v € M n (C) such that - : (z,0) —> {0,uxcu*) and —>(vxcv*,0),

-:(Q,x)

Var 6 M n (C). Since - 2 is identity on 5 , it follows that uvc = XIn for some A G C with |A| = 1. Then we can take v = uT, and = (u T x£u c ,

{x\,x2) VXJ € Bj = Mn(C),j

ux\u*),

= 1,2. Moreover,

A = {(xi,x2) = {(x!,x2)

€ B\{xi,x2)

=

(xi,x2)}

€ B|x 2 = u x f u * , ^ £ Bi} = M n (C)

as real C*-algebras. Now let A be the real C*-algebra M„(C), and A c = A-MA Clearly, dims Z{A) = dime Z ( A ) = 2, where Z(A), Z(A=) are the centers of A, Ac respectively, and dim* A = dime Ac = 2n2. Hence, we have AC =

B1®B2,

where Bj = AcZj is a complex factor, j = 1,2, and dime Ac = dime Bi + dimeB 2 = 2n2. Let a = In and b = iln 6 A Clearly, | ( a + z6) and 5(a — ib) are two orthogonal central projections in Ac. Now we can think that z

i = 2^a

+

ib

^

•2:2=2(0_ib)-

Therefore, z"i = z 2 , d i m c B i = dim c B2, and Bi = B2 = M n (C) (as complex C*-algebras). Q.E.D. Remark.

Let B b e a finite-dimensional complex C*-algebra. Then B = Bie---eBfe,

where £?Zj = Bj = M„j(C), 1 < j < k, and {zi, • • •, z*,} is the orthogonal family of minimal central projections in B with V • Zj = 1. If "-" is a conjugate linear * isomorphism of B with period 2, then {zi,-• • ,Zfc} = {^1, • • • ,Zk}- Moreover, if z s = zt and s ^ t, then z t = z s , and ns = nt.

216

Real Operator Algebras

By Propositions 10.1.1 and 10.1.2, we can see the structure of the "-" operation, and A = {x e B\x = x} S

Y,®Mmi(Di), i

where Di = R, C or H,VZ. This gives another proof of Theorem 5.7.1. For M n (C), we have a canonical basis of matrix unit {ey|l < i,j < n}, i.e., the matrix e^ is 1 at (i,j)-place and is 0 at other places, 1 < i, j < n. (1) If u = In and x = uxcu* = xc,Vx e M„(C), then Cij

(2) If n = 2, u = ( e n = e22,

=

Cij,

VI, J.

J , and x = uxcu*, Vx € M 2 (C), then e 12 — - e 2 i ,

e 2J = - e 1 2 ,

e 22 = e n .

(3) If n = 2m,

(m times),

u =

7

\ c

and x — ux u*,Vx € M n (C), then

{

e 2 (i-l)+2,2(j-l)+2;

if ft = Z = 1,

e2(i-i)+i,2(j-i)+i7

if k = 1 = 2,

-e2(i-i)+j,2(j-i)+fc' if fc / ^

VI < i, j < m, 1 < ft, Z < 2. Notice that for any i', j ' € {1, • • •, 2m} we have unique i, j € {1, • • •, m} and ft, Z e {1,2} such that i' = 2{i - 1) + ft, j ' = 2(j - 1) + I. „C0i|1 < i,j For M n (C) © M n (C), we have a canonical matrix unit {e^n, ft = 1,2}, i.e., e ^ = ( eij -,0),

e (£2' ) =_ (0,ey),

Vi,j.

<

(AF) Real C* -Algebras

217

Let u = In, and {x\,X2) = ( a ^ x f ) , Vxi,X2 € Mn(£). e,y.'

-

eK'

&•••' = e •'

Vi

Then

i

Lemma 10.1.3. Let Aj be a real C*-algebra, and Bj = Aj+iAj be the complexification of Aj,j = 1,2. If $ is a * isomorphism from A\ onto Ai ($ can be extended to a * isomorphism from B\ onto B2, still denoted by <&), then $(x) = $(x),

Vx € Bi.

Proof. Since $(a) = $(a), Va € A\, it follows that $(x) = $(x),Vx € Bi. Q.E.D. Definition 10.1.4. Let A be a finite-dimensional real C*-algebra, and Ac = A+iA be the complexification of A. Then Ac

= Bi ® • • • ®

Bk,

where Aczi = Bi = Mni (C) is an n;-dimensional complex factor, 1 < Z < k, X);=i n ? = dimit A, and {zi, • • •, Zk} is the orthogonal family of minimal central projections in Ac with Ylizi — 1- {% |1 — l ' J — n'> 1 — ' — ^} is called a fcaszs 0/ matrix unit for A c if e,-•' € B;, e;„. = e\}, e;,- • e i i = SwSji-ef),, Vi,j,l,i',j',V, and E i ^ P = zi,Vl. A basis of matrix unit {e\J \i, j , 1} for Ac is said to be standard, if under the "-" operation (induced by the decomposition Ac = A+iA) { e j } has the following changes. (1) {real transposition) li~zi = zi, and {x e Bi\x = x} = M„,(R), then e)/=e\/,

l
(2) (quaternionic transposition ) If z; = z;, nj = 2m, and {x G B;|x =

218

Real Operator Algebras

x} S Mm(M), then f e(l) e e5

(0

2(i-l)+ S ,2(j-l)+t

if s - t - 1

2(i-l)+2,20-l)+2'

* — t — J-,

e^

if s — / — 2

e

u

2(»-i)+i,2y-i)+i' (')

_

L

e

(t)

T

^

* ~~ l — z >

-f

2(i-l)+t,2(j-l)+s'

VI 

then

-.

By Lemma 10.1.3 and the preceding paragraph, we can find a basis of standard matrix unit for B. Definition 10.1.5. i,j
Let A be a real or complex C"*-algebra, {ej-|1 <

1 < I < k} is called a matrix unit in A, if ej_(') J O _ SmS'-.e®

= e^', and

Vi i / i ' i ' / '

Let A be a real C*-algebra, and i? = .A-i-iA A matrix unit {ej-|1 < i,j
10.2

[2], [14], [17], [26].

Technical lemmas

Lemma 10.2.1. Let A be a complex or real C*-algebra, and let p,q be two projections in A with \\p — q\\ < 1. Then there exists v € A such that p = v*v,q = vv*, and \\v — p\\ < 2\\p— q\\.

(AF) Real C

-Algebras

219

Proof. Let C*(pqp,p) be the abelian C*-subalgebra (with identity p ) of A generated by {pqp,p}. Since ||p — pqp\\ < \\p— q\\ < 1, it follows that pqp is invertible in C* {pqp, p) and there is x € C* {pqp, p) with x > 0 such that pqp • x2 = x2 • pqp = p. Similarly, there is y € C* {qpq, q) with y > 0 such that qpq-y2 = y2-qpq — Let v = qx. Then u*t; = xqx = x • pqp • x = p, and vv* = qx2q = y2 • qpq • px2 • q = V2 • q{pqP • x2)q = y2 • qpq = q. Denote o = {pqp)* in C*{pqp,p). Then, we have 0
0 < a 2 < a,

0

and

lb-(p«p)* II < lb-p«p|l ^ lb-ellMoreover, by a = a; -1 in C*{pqp,p) we have v{pqp)* — q. Therefore,

l b - p | | < lb(p- {pqp) *)II + lb(P9p)' -pll
220

Real Operator Algebras

Proof. (1) Let Log be the principal value of the logarithm in C\(—oo,0]. Since a{u) c {z G C| \z\ = 1} and - 1 £ a(u), it follows from section 2.5 that Log u & A. Moreover, (Log u)* = Log u* = Log u _ 1 = —Log u. Let v = exp(|Log u). Then v € A is unitary, and v 2 = u. By \z? — 1| < |z — 1|, V z S C with |z| = 1, we have ||u — 1|| = max{|z2 — 1| \z G a(u)} < max{|z - 1| \z G a(u)} = \\u — 1||. (2) Similarly to (1), let v = exp(|Log u). Then v G Ac is unitary, v2 = u, and \\v — 1|| < ||u — 1||. Moreover, v = exp(|Log u) = exp(|Log u*) = v*. Q.E.D. Lemma 10.2.3. (real transposition) Let A be a real C*-algebra, and let {eij\l < i,j < n} and {fij\l < i,j G n} be two matrix units in A. If ||ey — fij\\ < e < 1, Vi,j, then there is u G A such that u*u = e, uu* = / , ueijU* —

fij,Vi,j,

and \\u — e\\ < 3ne, where e = £ ? = 1 e«> / = E"=i / « • Proof. By Lemma 10.2.1, there is v G A such that v*v = e n ,

=

TO*

/H,

and ||v - e n | | < 2||en - / n | | .

Let n

u = ^2filveli

(G A).

Then it is easy to see that u*u = e, uu* — / , and ue^u* =

fy,

Vi, j .

221

(AF) Real C* -Algebras

Moreover, ||u -e\\<^2

||/iiuen - eneuW i

< £ ( l l / i i ( » - e n ) e u | | + ||(/ii - e u ) e u | | ) i

<3ne. Q.E.D. Lemma 10.2.4. (quaternionic transposition) Let A be a real C*-algebra, A — A+iA, and let {ejj|l < i,j < 2m} and {fij\l < i,j < 2m} be two standard matrix units in A and quaternionic transposition (see Definitions 10.1.4 and 10.1.5). If ||e»j - / y | | < e < j ^ , Vi, j , then there is u € A such that u*u = e,uu* = f,ueijU* = fa, Vi,j, and \\u — e\\ < 18me, 2m

2m

where e = ^

en, f = ^

i=\

/ w (clearly, e,f € A).

i=\

Proof. By Lemma 10.2.1, there is w £ A such that w*w = eu, ww* = fn,

and | | « ; - e i i | | < 2 | | e n - / n | | .

Let 2m

v = ^2fiiweu. Then it is easy to see that v*v = e, vv* — f, and veijV* = fij, Vi, j . Since the permutations of {e^} and {fij} iinduced by the "-" operation are the same, it follows that ve^*

= fij,

\/i,j.

Let x — v*v. Then xeijX* = v* • ve^v* • v — v* fijV = eij,

Vi, j ,

222

Real Operator Algebras

and xx* = x*x = e, x = x*, \\x - e|| < \\v*{v - e)|| + \\(v* - e)e|| < 2\\v - e\\. Moreover, 2m

\\v -e\\<^2

Wfnweu - eneuW

< ^2(\\fn(w

~ en)eii|| + ll(/a -

en)eu\

< 2m- 3e = 6me < \. Then \\x — e\\ < 1. Since x is a unitary element in { e ^ } ' n e.Ae, it follows from Lemma 10.2.2(2) that there is a unitary element y in {e^}''HeAe such that V = V*, V2 = x, and ||y - e|| < ||z - e||. By y*y — y2 = x = v*v, we have vy* = vy*e = v • y*y • y* = v • v*v • y* = f • vy* = vy*. Let u = vy*. Then u = u& A, and ue^u* = vy*eijyv* — ve^v* =

fa,

Vi,j,

since y € { e ^ } ' D eAe and y*y = e. Moreover, u*u = yv*vy* — e, uu* = vy*yv* = / , and ll«-c||<||(t;-c)tf*|| + ||c(y'-c)|| < \\v - e\\ + \\x - e\\ < 3\\v - e\\ < 18me. Q.E.D. Lemma 10.2.5. (complex transposition) Let A be a real C*-algebra, A = A+iA, and let { e ^ | l < i,j < n,l = 1,2} and {f$\l < i,j < n,l = 1,2} be two matrix units in A such that

$=e%\7$

=

and \\e\J — f>j\\ < e < 1, Vi,j,l.

f%\Vi,j,l
u*u = e, uu* = /, uef)u* = ff), Mi,j,l

223

(AF) Real C* -Algebras

and \\u — e\\ < 6ne, n

where £ ? = 1 e% = eu £ / £ > = / » , ' = 1,2, and e = ex + e 2 , / = h + h _

j=i

(clearly, e~i = ek,fi = /&, 1 , and ||V - e ^ l l < 2\\e$ - /£>!!. Then v*v = e%>, vv*=f£\

and||i;-e(1a1)||<2||ei21)-/1(;)|

Let

i=\

i=l

and U = U\ + U 2 .

Then u\ = u 2 , u 2 — u\, u = u € A, u*u = e, uu* = / , uejyu* = / v , Vi, j,Z. Similarly to the proof of Lemma 10.2.3, we have ||iti - ei || < 3ns,

||u 2 - e 2 || < 3ne.

Therefore, ||u - e|| < 6ne.

Q.E.D.

Lemma 10.2.6. Let A be a unital real C*-algebra, B and C be two unital real C*-subalgebras of A, and diniR.B < +oo. Let B = B+iB,C = C+iC, .(') < i,j < m, 1 < I < k} be a basis of standard matrix unit in and let {e\?|1 B. If for e e (0,1) there is a matrix unit

{/JV|1

C such that ^ 2 i ; / ^ = 1 and

Heg-yflK^,

Vi.j.I,

1 < I < k} m

224

Real Operator Algebras

where N = n\ + • • • + nk, then there exists a unitary element u £ A such that uBu* C C and \\u - 111 < e.

Proof. Denote TJ = 333Ni • Then Hei?-/S ) ||<»7,

Vi,j,i.

(10.1)

Since {e\-} is standard (Definition 10.1.4), we have a map 7 : {ej-} —> {e\j} such that 7(eg>) = $ ,

ViJ,l.

Then we can define the map 7 : {/!• } —• {/»j }> J-e-j ) )

7(/S ) = 4?^7(eg ) = eg,

Vi,j,/. Then ll^T-7(eg))|| + Il7(eg} -f$j>)\\

< \\$-$\\

(10-2)

= liyg ) -eg ) || + | | e g - / g ? | | < 2 7 7 , Vi,j,l. Clearly, 2v < 1. By Lemma 10.2.1, there is ai G C such that a

*iai = / u .

a

W = 7(/n)

and lh-/i?ll<2||7J?-7(/1(i))ll<477,

(10.3)

VZ. Let k

ni

1=1 i = l

Then a is a unitary element in C, a^To*=7(4-)),

Vi,j,/

(10.4)

225

(AF) Real C* -Algebras

and

\\a-l\\<^h(ti%7$-hff$\\

< E{ii7(^)(a, -tbWh + buff) -T!?)7S?II} (10-5) <6NTJ

by (10.2), (10.3). Let V = {/-^i, j,l}'nC (a C*-subalgebra of C). By ^ T ^

= a ^) a *(4) 7 ( / g) ) )

we have o.o-/g>.r-o*=a7(4;V - a / ^ , a - i{ti>j') - hj since 7 2 = id, where 7(e^-) = ej,.,, Vi,j,J, i.e., oa € P . Moreover, by (10.5) ||aa - 1|| < 2||a - 1|| < 12JVij.

(10.6)

Define Jo: = axa*,

Vz € C.

Then J is also a conjugate linear * isomorphism of C with period 2. For any i £ 5 , J x . / g ) = J ( x . j 4 i ) ) = J(a;.a/gya*)

= ^-7(/g ) )) = ^(7(/ig))^) <

£j(Jf$)-x)

=

f$)-Jx,Vi,j,l.

Hence, JZ> = P . Since aa is a unitary element in V, J(aa) — aa (clearly), and „ (6>

| | a o - l | | < 12Ar?7< 1, it follows from Lemma 10.2.2(1) that there is a unitary element w € V such that Jw = w, w2 = aa, and ||u; — 1|| < ||oa— 1||.

226

Real Operator Algebras

Let b = w*. Then ba — w* • aa • a* = w* • w2 • a* = wa* = a* • awa* = a* • Jw = a*w = (ba)* and

ll&o-lH < | | « ; - l | | + | | o - l | | (10.7) < | | o a - l | | + | | o - l | | < 18AT77 by (10.5), (10.6). Clearly, 18iV>7 < 1. Since 6a is unitary in C, it follows from Lemma 10.2.2(2) that there is a unitary element v eC such that v = v*,

v2 = ba,

and \\v - 1\\ < \\ba - 1\\ < 18Nrj. Let gf) = v*ff)v(e

(10.8)

C), Vi, j , I. Then by (4) and b € V we have

= v* • b{aff)a*)b* • v = v*b-y(fV)b*v =

v*bf$b*v

- „* f ( 0 u - o ( , , ) if 7 ( e j ) = e\ilyi,j, I- This implies that the matrix unit {g\)} in C is also standard. Moreover,

ll5S)-eg)||<2||l,-l|| + ||4;)-eg)||

(10.9)

< 37,Nr) = rj, Vi,j,l by (10.1), (10.8). Let £ = the (complex) linear span of {g\j\i,j, and ni

ni

i=l

i=l

I},

227

(AF) Real C* -Algebras

Then E is afinite-dimensionalC*-subalgebra of C,6 = S, and

{gfjlhjj}

is a basis of standard matrix unit for £. (i) For some I, if { e ^ | l < i,j < n;} is real transposition, then {g\^ |1 < i, 3 < n{\ is also real transposition, i.e., e\- = e ^ - , % = g\j , Vi,j. Clearly, 77 < 1. By Lemma 10.2.3, there is ui e A such that

and ||«J-ej||<3ni^. (ii) For some I, if {ej- |1 < i,j

(10.10)

< m} is quaternionic transposition,

then m is even and {g\, |1 < i,j < m} is also quaternionic transposition. Clearly, rj < ^-. By Lemma 10.2.4, there is m 6 A such that uu = eu uu = gi, me]jUt = $ / , Vi,j, and ||ui-ei||<9n;5?.

(10.11)

(iii) For some I ^ m, if e^- = e | j * \ l < i , j < n; = n m , then c^- — 9iT\ 1 < *)j < ™i = ^m (complex transposition). Clearly, rj < 1. By Lemma 10.2.5, there is mm € A such that u

imU*lm = et + e m ,

Wime^u,*™ = g$,

uimuim = gi+ gm, l
= l,m

and \\uim - (e; + e m )|| < 67^77. Now let

U=^2ui+ ei real

"^2 Ul+ ] P Uim. e.\ quat.

ej + e m complex

Then u is unitary, « = u € A, and

u e gV=4\

v;,j,z.

(10.12)

228

Real Operator Algebras

Hence,

uBu* =

£cC.

Moreover, by (10.10), (10.11), (10.12), ll«-l|| <5Zl|ui-ej|| + ^ | | u j - e j | | + real

quat.

$ 3 \\uim - (ei + em)\\ complex

<9JV77 = e. Q.E.D. Lemma 10.2.7. For any n > 0 and positive integer N, there exists S4 — Si(N, 77) > 0 with the following property: Let B be a C*-algebra on a Hilbert space H, and {e^|l < i,j < m, 1 < I < k} be a matrix unit on H with n\ + • • • + nic = N. If there is a subset {bli;j\l < i,j < m,l < I < k} of B such that \\b\j — e{-|| < 64, Vi,j,l, then we have a matrix unit {flj\i,j,l} in B such that \\e\j ~ fij\\ < V, Moreover, if^2nelH

— 1#, then ljj

Vi,j,Z.

G B and {/y} can be chosen as

Hi,i fk = inProof. It is just the lemma 15.1.8. of [26]. Notes.

Lemma 10.2.6 is due to T.Giordano ([14]).

References.

10.3

Q.E.D.

[2], [14], [17], [26].

Definition and basic properties

Definition 10.3.1. A real C*-algebra A is (AF) (approximately finite dimensional) if there exists an increasing (with respect to inclusion) sequence (An)n>i of finite dimensional * subalgebras of A such that their union is dense in A, i.e., A = \JnAn.

(AF) Real C*-Algebras

229

Remark. Let A = \JnAn be an (AF) real C*-algebra , and let ipn be the embedding of An into An+i, Vn. Similarly to the complex case, we have

A^]im{An,(pn} (see [26, section 3.7]).

Proposition 10.3.2. Let A = \JnAn be an (AF) real C* -algebra, where {An}n>i is an increasing sequence of finite dimensional * subalgebras of A. Then (i) Ac — A-U.A = Un(An+iAn) is a complex (AF) C*-algebra; (ii) A has an identity 1 if and only if there exists no such that l n = 1, Vn > no, where l n is the identity of An,"in. Proof, (i) is obvious, (ii) is similar to the complex case ([26, Proposition 15.1.2]). Q.E.D. Proposition 10.3.3. Let A be a real C* -algebra without identity. A is (AF) if and only if A+M. is (AF). Proof. It is similar to the proof of [26, Lemma 15.1.10].

Then

Q.E.D.

Theorem 10.3.4. Let A be a real C*-algebra, and A = Ac = A+iA. Then the following statements are equivalent: (i) A is (AF); (ii) A is separable, and for any ai, • • •, an 6 A and e > 0, there exist a finite dimensional * subalgebra B of A and &i, • • •, bn € B such that hj-bj\\<£,

l<j
(Hi) A is separable, and for any x\, • • •, xn € A and e > 0, there exist a finite dimensional * subalgebra B of A and j/i, • • • ,yn e B such that B = B, and \\XJ — yj\\ < e, 1 < j < n. Moreover, if A is an (AF) real C* -algebra, and B is a finite dimensional * subalgebra of A, then we can find an increasing sequence (An)n>i of finite dimensional * subalgebras of A such that A\ = B, and A = \JnAn.

230

Real Operator Algebras

Proof. The equivalence of (ii) and (iii) is obvious. Moreover, (i) =>• (ii) is also obvious. Now it suffices to show that A is (AF) under the assumptions of (ii) and (iii). By Proposition 10.3.3, we may assume that A is unital. Let { i n } n > ! be a countable dense subset of Int(S c ) = {x € A\ \\x\\ < 1}, and x\ = 0. Clearly, we can take a finite dimensional * subalgebra A\ of A and y\ e A\ = Ai+iAi such that \\y{ — xi\\ < 2 - 1 . Indeed, we can let A\ = R, and y\ = 0 € A\ C A\. Moreover, if B is a given finite dimensional * subalgebra of A, then we can let A± = B. Suppose that we have finite dimensional * subalgebras: 1 € Ai C •••

CAn{cA)

and {yj\l < j < s} C A3 = As+iA such that \\ysj-xj\\<2-s,

l<j<s,

VI < s < n. Let {e\j\l < i,j < ni, 1 0, there exist a finite dimensional * subalgebra B of A and {b\j,ym\l < i,j < ni,l 0 will be determined in the following. Let N = m + • • • + nk, and S < 6 0 will be determined in the following. Then by lemma 10.2.7, there is a matrix unit {/^-|1
K-fij\\
Vi,3,l

= Zi,ifL = l-Let S < 2-(" + 2 ) , and 2r] • 666AT2 < 2 ~ ( n + 2 ) .

By Lemma 10.2.6, there is a unitary element u € A such that uAnu* CB, and \\u- 1|| < 333iV2r/.

(AF) Real C -Algebras

231

Let An+i = u*Bu. Then An C Ai+i = A i + i , dimAi+i < oo, and {uymu*\l < m < n + l } c An+i- Moreover, \\uymu* - xm\\ < 2\\ym\\ • ||u - 1|| + \\ym - xm\\ < 666N2(1 + S) + S < 2"( n + 1 ), 1 < m < n + 1. Therefore, we can find an increasing sequence {Ai} of finite dimensional * subalgebras of A such that An = An, Vn, and A — \JnAn, i-e., A is (AF). Q.E.D. Proposition 10.3.5. Let A = UnAn be an (AF) real C*-algebra, where {An} is an increasing sequence of finite dimensional * subalgebras of A, and B be a finite dimensional * subalgebra of A. Then for any e > 0, there exist a unitary element u € A and a positive integer n such that uBu* C An, and \\u — 1|| < e, where A = A if A has an identity, or A = A+M. if A has no identity. Proof. Suppose that A has an identity. Let A = A+iA, B = B+iB, and {e'j-11 < i,j < ni,l < I < k} be a, basis of standard matrix unit for B. We may assume that Yli i eu = 1- By Lemma 10.2.7, there exist a positive integer n and a matrix unit {f\j} in An — An+iAn such that Y^i,i fu = 1 and

where N — n\-\ f-rife. Moreover, by Lemma 10.2.5, there exists a unitary element u € A such that uBu* C An, and ||u — 1|| < e. Clearly, uBu* C AnNow let A be without identity. Consider A+R = UniAn+R),

B+R, and e > 0.

By the preceding paragraph, there exist a unitary element u e A+R and a positive integer n such that u(B+R)u* c An+R,

and ||u - 1|| < e.

232

Real Operator Algebras

Clearly, uBu* c An.

Q.E.D.

Proposition 10.3.6. Let A be an (AF) real C* -algebra, and p be a projection in A. Then pAp is also a real (AF) C*-algebra. Proof. Since the Lemmas 15.1.3 to 15.1.6 of [26] are still true in the real case, it follows from the proof of [26, Proposition 15.1.12] and Theorem 10.3.4 that pAp is also an (AF) real C*-algebra. Q.E.D. Notes. Proposition 10.3.5 is due to T.Giordano ([14]). Moreover, there are more results on (AF) real C*-algebras (e.g., Bratteli diagrams, if-theory, and etc.), and one can see [14]. References.

[2], [14], [17], [26].

Bibliography

[I] Bonsall, F. F and Duncan, J. (1973). Complete normed algebras, SpringerVerlag, Berlin-Heidelberg -New York. [2] Bratteli, O. (1972). Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc, 171, pp. 195-234. [3] Chu, C.H., Dang, T., Russo, B. and Ventura, B. (1993). Surjective isometries of real C*-algebras, J.London Math. Soc, 47, 2, pp. 97-118. [4] Civin, P. and Yood, B. (1961). The second conjugate space of a Banach algebra as an algebra, Pacific J. Math., 11, pp. 847-870. [5] Dixmier, J. (1977). C*-Algebras, North-Holland. [6] Dixmier, J. (1981). Von Neumann Algebras, North-Holland. [7] Dixmier, J. (1948). Sur un thereme de Banach, Duke Math. J., 15, pp. 10571071. [8] Doran, R. S. and Belfi, V. A. (1986). Characterzations of C* -algebras, The Gelfand - Naimark theorems, Marcel Dekker Inc., New York-Basel. [9] Dunford, N. and Schwartz, J. T. (1958). Linear operators, Vol. I, General theory, Interscience, New York. [10] Ekman, K. E. (1976). Unitaries and partial isometries in a real W - a l g e b r a , Proc. Amer. Math. Soc, 54, pp. 138-140. [II] Elliott, G. A. (1970). A weakening of the axioms for a C*-algebra, Math. Ann., 189, pp. 257-260. [12] Ford, J. W. M. (1967). A square root lemma for Banach * algebras, J. London Math. Soc, 42, pp. 521-522. [13] Gardner, L. A. (1984). An elementary proof of the Russo-Dye theorem, Proc. Amer. Math. Soc, 90, pp. 171. [14] Giordano, T. (1988). A classification of approximately finite real C*-algebras, J. reine angew. Math., 385, pp. 161-194. [15] Glickfeld, B. W. (1966). A metric characterization of C(X) and its generalization to C*-algebras, Illinois J. Math., 10, pp. 547-556. [16] Glimm, J. and Kadison, R. V. (1960). Unitary operators in C*-algebras, Pacific J. Math., 10, pp. 547-556. [17] K. R. Goodearl, K. R. (1982). Notes on real and complex C*-algebras, Shiva Publishing Limited. 233

234

Real Operator Algebras

[18] Goodrick, R. K. (1972). The spectral theorem for real Hilbert space, Acta Sci. Math, (szegel), 33, pp. 123-127. [19] Halperin, H. (1967). Finite sums of irreducible functionals on C*-algebras, Proc. Amer. Math. Soc, 18, pp. 352-359. [20] L. A. Harris, L. A. (1972). Banach algebras with involution and Mobius transformations, J. Functional Anal, 11, pp. 1-16. [21] Ingelstam, L. (1964). Real Banach algebras, Ark. Math., 5, pp. 239-270. [22] Isidro, J. M. and Palacios, A. R. (1996). On the definition of real W - a l g e b r a s , Proc. Amer. Math. Soc, 124, pp. 3407-3410. [23] Kulkarni, S. H. and Limaye, B. V. (1992). Real function algebras, Marcel Dekker Inc., New York-Basel -Hong Kong. [24] Li, B. R. (1975). Real C*-algebras (in Chinese), Acta Math. Sinica, 18, pp. 216-218. [25] Li, B. R. (1979). Real operator algebras, Scientia Sinica, 22, pp. 733-746. [26] Li, B. R. (1992). Introduction to operator algebras, World Scientific, Singapore -New Jersey -London -Hong Kong. [27] Li, B. R. (1992). Banach algebras (in Chinese), Academic Press (Beijing). [28] Li, B. R. (1995). Finite-dimensional real C*-algebras, Adv. Math. (Beijing), 24, pp. 466-471. [29] Li, B. R. (1995). Irreducible * representations of real C*-algebras, Acta Math. Sinica, New Ser., 11, pp. 381-388. [30] Li, B. R. (1996). Real operator algebras, Lectures in Dept. of Applied Math., National Sun Yat-sen Univ., Kaohsiung, Taiwan. [31] Li, B. R. (1996). Classification of real Von Neumann algebras (I), " Functional Analysis in China" edited by B. R. Li et al., Kluwer Academic Publishers, pp. 322-332. [32] Li, B. R. (1996). Real operator algebras, RIMS, Kyoto Univ., 936, pp. 58-69. [33] Li, B. R. (1997). Classification of real Von Neumann algebras (II), Southeast Asian Bull. Math., 21, pp. 421-429. [34] Li, B. R. (1998). Real reduction theory, Sci. in China (Ser. A), 41, pp. 574581. [35] Li, B. R. and Li, M. L. (1998). Banach * algebras (in Chinese), Acta Math. Sinica, 41, pp. 553-562. [36] Li, B. R. and Tam, P. K. (2000). Real Banach * algebras, Acta Math. Sinica, English Ser., 16, pp. 469-486. [37] Li, M. L. and Li, B. R. (1998). Abelian real W - a l g e b r a s , Acta Math. Sinica, New Ser., 14, pp. 85-90. [38] Li, Z. Y. (2000). The uniqueness of * operation in real C*-algebras (in Chinese), Acta Math. Sinica, 4 3 , pp. 611-614. [39] Palmer, T. W. (1970). Real C*-algebras, Pacific J. Math., 35, pp. 195-204. [40] Ptak, V. (1970). On the spectral radius in Banach algebras with involution, Bull. London Math. Soc, 2, pp. 327-334. [41] Ptak, V. (1972). Banach algebras with involution, Manuscripta Math., 6, pp. 245-290. [42] Rickart, C. E. (1960). General theory of Banach algebras, Princeton, New

Jersey, D. Van Nostrand Co., New York.

Bibliography

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Riesz, F. and Sz-nagy, B. (1952). Functional analysis, Budapest. Russo, B. and Dye, H. A. (1966). A note on unitary operators in C*-algebras, Duke Math. J., 33, pp. 413-416. Sakai, S. (1956). On the cr-weak topology of W - a l g e b r a s , Proc. Japan Acad., 32, pp. 329-332. Sakai, S. (1959). A characterization of W*-algebras, Pacific J. Math., 6, pp. 763-773. Shirali, S. and Ford, J. W. M. (1970). Symmetry in complex involutory Banach algebras (II), Duke Math. J., 37, pp. 275-280. Stormer, E. (1967). On anti-automorphisms of Von Neumann algebras, Pacific J. Math., 2 1 , pp. 349-370. Takesaki, M. (1958). On the conjugate space of an operator algebra, Tohoku Math. J., 10, pp. 194-203. Takesaki, M. (1979). Theory of operator algebras I, Springer-Verlag, New York. Taylor, A. E. (1943). Analysis in complex Banach spaces, Bull. Amer. Math. Soc, 49, pp. 652-669. Taylor, A. E. (1958). Introduction to functional analysis, John Wiley & Sons, Inc., New York, London. Vukman, J. (1981). Real symmetric Banach * algebras, Glasnik Math., 16, pp. 91-103. Yood, B. (1970). On axioms for B*-algebras, Bull. Amer. Math. Soc, 76, pp. 80-82. Zaanen, A. C. (1953). Linear analysis, North Holland, Amsterdam. Giordano, T. (1983). Antiautomorphismes involutifs des facteurs de von Neumann injectifs I, J. Operator Theory, 10, pp. 251-257. Stacey, P. J. (1987). Real structure in direct limits of finite dimensional Calgebras, J. London Math. Soc.(2), 35, pp. 339-352. Stormer, E. (1980). Real structure in the hyperfinite factor, Duke Math. J., 47, pp. 145-153. Stormer, E. (1986). Conjugacy of involutive antiautomorphisms of von Neumann algebras, J. Functional Analysis, 66, pp. 54-66. Schroder, H. (1993). K-theory for real C*-algebras and applications, Pitman Research Notes in Mathematics Series 290, Longman Scientific and Technical.

This page is intentionally left blank

Notation Index

M±, 64 M„(yl), 81 M p , 67 M,+, 74 M , H , 74 P(M), 65 fl(^), 22 i?*(A), 48 T(H), 60

( £ , £ , - ) , 191 (M)i, 68 A*+, 86 A*H, 86 A + I 51 AH, 41 A K , 38 Ac, 15 B(£), 6 S(B,F),6 B(H), 8 B r , 19 C ( H ) , 59 C(X,F),83 C*(a), 79 C0*(o), 80 C„(Q,-), 29 £r, 5 F ( # ) , 59 G ( A 21 Go(i4), 21 H(-), 191 ffc,8 7nt(5), 160 Kr,9 Lf, 42 M;, 67 M(-), 200 M„, 64 M + , 65 M/,, 65 M K , 65

r(*o, 9

U{A), 38 £/(M), 65 V(M), 141 X',1 X.,4 Xc, 1 Z(M), 66 Q(A), 28 e , 192

/

H{t)du{t),

J(E,-)

/

a{t)dv(i),:

/

M{t)du{t),

J(E,-)

\im{An,
+,2 C, 6 F, 20 H, 20 R, 2 237

229

238

Real Operator Algebras

Z, 142 M, 176 N, 176 CoE, 85 , 69 j>, 69 L£nyln, 228 F, 8 a(B(H),T(H)), ff(M,M.), 67 o-(a), 7

61

/c, 3 fa, 26 fm, 26 m(-), 112 m • n, 26 mf, 199 mf, 26 mn, 26 p(-), 41 p ^ g , 70 p ~ q, 70

T(B(H),T(H)),61

r(-), 18

T(M,M,),

s{B{H),T(H)),

67

4,15,18 {$ M ,L 2 (fi, / i , - ) } , 120

61

s(Af,M„), 67 S*(B(/f),T(/f)),61

{ T T , / / } , 42

s*(M,M»), 67

{7r p ,if p }, 45 {ei(-),e a (-)},9 {e A }, 12 # £ , 47 o(-), 197 o*,8 oc, 6 a / , 26 c(p), 70 Coo, 2 Op, 2 ezE, 45

tr,9 w « to, 143 £ ( 4 ) , 43 V(A), 45 S W , 85 S(A), 44 <Sn(M), 73 I-loo, 2 | • |P, 1 M , 74 II "Hi. 9

Index

Completely additive functional, 72 Complex transposition, 218 Complexification of real C*-algebra, 77 real W*-algebra, 123 real Banach * algebra, 37 real Banach algebra, 18 real Banach space, 2 real Hilbert space, 8 real VN algebra, 63 Constant real measurable field of Hilbert spaces, 195 Continuous dual, 1 Continuous real (VN) W-algebra, 142, 170 Cyclic * representation, 47 Cyclic vector, 43

C*-quasi norm, 111 a-Finite real (VN) W*-algebra, 76 a-strong * (operator) topology, 61 u-strong (operator) topology, 61 cr-weak (operator) topology, 61 (AF) real C*-algebra, 228 * Operation, 37 * Radical, 48 * Representation, 42 * Semi-simple algebra, 48 Abelian projection, 168 Abelian real Banach algebra, 28 Absolute value of tp, 74 Adjoint operator, 8 Algebraically irreducible * representation, 92 Approximate identity, 84 Approximately finite-dimensional real C*-algebra, 228 Arens product, 26

Decomposable operator, 198 Decomposable real VN algebra, 201 Definition ideal of a trace, 176 Diagonal real operator, 199 Diagonal real operator algebra, 199 Discrete real (VN) W*-algebra, 170 Divisible real Banach algebra , 20 Double commutant, 63 Dual, 1

Bar operation, 3 Bidual, 26 Borel bar space, 191 Center, 66 Central cover, 70 Central valued trace, 173 Commutant, 63 Commutative algebra, 28 Comparison theorem, 71

Enveloping real C*-algebra, 114 Extreme point, 45 Faithful * representation, 73 239

240

Real Operator Algebras

Faithful normal real state, 76 Finite dimensional real C*-algebra, 108 Finite projection, 167 Finite real (VN) W*-algebra, 167 First Arens product, 26 Functional calculus, 25 Gelfand transformation, 29 Gelfand-Naimark conjecture, 147 GNS construction, 43 Hermitian element, 38 Hermitian functional, 42 Hermitian real Banach * algebra, 41 Infinite projection, 167 Infinite real (VN) W*-algebra, 167 Irreducible * representation, 92 Kaplansky's density theorem, 68 Left kernel, 42 Left(right) ideal, 22 Mackey topology, 61 Maximal left (right) ideal, 22 Maximal regular left (right) ideal, 22 Minimal projection, 135 Minkowski functional, 16 Modular unit, 22 n-Homogeneous discrete (semi-discrete) real (VN) W-algebra, 184, 185 n-Transitivity, 96 Non-zero complex valued multiplicative (real) functional, 28 Normal functional, 72 Normal operator, 11 Normal real state, 72 Orthogonal (or Jordan) decomposition, 75 Orthogonal normalized basis for H(-), 192

Partial isometry, 13 Polar decomposition of ip, 74 Polar decomposition of operator, 13 Positive element, 51 Positive extension of / , 43 Positive functional, 42 Predual, 4 Principal component, 21 Projection, 65 Properly infinite projection, 168 Properly infinite real (VN) VK*-algebra, 168 Ptak's inequality, 53 Pure real state, 45 Purely infinite projection, 167 Purely infinite real (VN)W*-algebra, 167 Quaternion algebra, 20 Quaternionic transposition, 217 Quotient real C*-algebra, 98 Radical, 22 Radon-Nikodym theorem, 75 Real C* -algebra, 77 Real C* -equivalent algebra, 147 Real W-algebra, 123 Real Banach * algebra, 37 Real Banach algebra, 15 Real Banach space, 1 Real factor, 66 Real Hilbert space, 8 Real measurable field of Hilbert spaces, 191 operators, 197 vectors, 192 VN algebras, 200 Real reduction theory, 205 Real state, 44 Real transposition, 217 Real VN (Von Neumann) algebra, 63 Regular Banach algebra, 27 Regular left (right) ideal, 22 Schwartz inequality, 42 Second Arens product, 26

241

Index Self-adjoint operator, 12 Semi-abelian projection, 168 Semi-continuous real (VN) W-algebra, 170 Semi-discrete real (VN) W*-algebra, 170 Semi-finite projection, 168 Semi-finite real (VN) W*-algebra, 167 Semi-simple real Banach algebra, 22 Separating vector, 76 Singular functional, 131 Skew-hermitian element, 38 Skew-hermitian real Banach * algebra, 41 Spectral decomposition theorem , 11 Spectral family, 12 Spectral measure, 10 Spectral pair, 9 Spectral radius, 18 Spectral space, 28 Spectrum, 7 Standard matrix unit, 217 Strong (operator) topology, 61 Strong * (operator) topology, 61 Symmetric real Banach * algebra, 53

Tensor product of real Hilbert spaces, 69 real VN algebras, 69 Topological group of invertible elements, 21 Topological irreducible * representation, 45

Trace, 176 Tracial class operator, 9 Tracial norm , 9 Two sided (*) ideal, 48 Type (I) real (VN) W-algebra, 170 Type (II) real (VN) W*-algebra, 170 Type (III) real (VN) W*-algebra, 170 Uniform (operator) topology , 61 Unitary element, 38 Universal * representation, 45 Von Neumann (VN) algebra, 63 Von Neumann double commutation theorem, 67 Weak (operator) topology, 61

REAL OPERATOR ALGEBRAS The theory of operator algebras is generally considered over the field of complex numbers and in the complex Hilbert spaces. So it is a natural and interesting problem: How is the theory in the field of real numbers? Up to now, the theory of operator algebras over the field of real numbers has seemed not to be introduced systematically a n d sufficiently.

The aim of this book is to set up the fundamentals of real operator algebras and to give a systematic discussion for real operator algebras. Since the treatment is from the beginning (real Banach and Hilbert spaces, real Banach algebras, real Banach * algebras, real C*algebras and W*-algebras,

etc.), and some basic facts are given, one can get some results

on real operator algebras easily.

The book is also an introduction to real operator algebras, written in a self-contained manner. The reader needs just a general knowledge of Banach algebras and operator algebras.

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