Geometry of State Spaces of Operator Algebras (Mathematics: Theory & Applications)

Mathematics: Theory & Applications Series Editor Nolan Wallach

Erik M. Alfsen Frederic W. Shultz

Geometry of State Spaces of Operator Algebras

Birkhauser Boston • Basel· Berlin

Erik M. Alfsen University of Oslo Mathematical Institute Oslo. N-0316 Norway

Frederic W. Shultz Wellesley College Department of Mathematics Wellesley. MA 02481 U.S.A.

Library of Congress Cataloging-in-Publication Data Alfsen. Erik M. (Erik Magnus). 1930Geometry of state spaces of operator algebras I Enk M. Alfsen and Fred Shultz. p.cm. Includes bibliographical references and index. ISBN 0-8176-4319-2 (alk. paper) - ISBN 3-7643-4319-2 (alk. paper) I. Quantum logic. 2. Axiomatic set theory. 3. Lattice theory. 4. Jordan algebras 5. Functional analysis. 1. Shultz. Frederic W .. 1945- II. Title. QA10.35 .A44 2002 530 12'01'5113-dc21

2002034257 CIP

AMS Subject Classifications. Primary: 03GI2. 06C15. 17C65. 17C90. 32M15. 46A40. 46A55. 46B20. 46B28. 46L05. 46LIO, 46L30. 46L52. 46L53, 46L60, 46N50. 46N55. 47A60, 47L07, 47L50, 47L90, 52A05, 52A07, 8IPIO, 81P15, 8IRI5,8IT05; Secondary: 46B04.46C05, 46L57, 46L65, 47CIO,47CI5,47L30

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Preface

In this book we give a complete geometric description of state spaces of operator algebras, Jordan as well as associative. That is, we give axiomatic characterizations of those convex sets that are state spaces of C* -algebras and von Neumann algebras, together with such characterizations for the normed Jordan algebras called JB-algebras and JBW-algebras. These nonassociative algebras generalize C*-algebras and von Neumann algebras respectively, and the characterization of their state spaces is not only of interest in itself, but is also an important intermediate step towards the characterization of the state spaces of the associative algebras. This book gives a complete and updated presentation of the characterization theorems of [10]' [ll] and [71]. Our previous book State spaces of operator algebras: basic theory, onentatwns and C*-products, referenced as [AS] in the sequel, gives an account of the necessary prerequisites on C*-algebras and von Neumann algebras, as well as a discussion of the key notion of orientations of state spaces. For the convenience of the reader, we have summarized these prerequisites in an appendix which contains all relevant definitions and results (listed as (AI), (A2), ... ), with reference back to [AS] for proofs, so that this book is self-contained. Characterizing the state spaces of operator algebras among all convex sets is equivalent to characterizing the algebras (or their self-adjoint parts) among all ordered linear spaces. The problem was raised in this form in the early 1950s, but remained open for several decades. A brief survey of the history of the subject is given in the preface to [AS], and at the end of each chapter of the present book there are historical notes with references to the original papers. Our axioms for state spaces are primarily of geometric character, but many of them also can be interpreted in terms of physics. As in our previous book, we have included a series of remarks discussing such interpretations. The book is divided into three parts. Part I (Chapters 1 through 6) concerns Jordan algebras and their state spaces. Part II (Chapters 7 and 8) develops spectral theory for affine functions on convex sets. Part III (Chapters 9, 10 and 11) gives the axiomatic characterization of operator algebra state spaces, and explains how the algebras can be reconstructed from their state spaces. We imagine that the backgrounds and interests of potential readers vary quite a bit, and we have tried to make the book adaptable for different groups of readers. Parts I and II are independent of each other, except for a few motivating examples. Part III makes use of the results and concepts from Parts I and II, but those interested in a short route to the state space

vi

PREFACE

characterization theorems may skim Parts I and II, and then proceed to Part III, referring back to the first two parts when necessary. We now will give a brief description of the content of each part. Part I serves as an introduction for novices to the theory of JB-algebras and JBW-algebras, and of their state spaces. No previous knowledge of Jordan algebras is assumed. The reader familiar with C*-algebras and von Neumann algebras may want to keep in mind that the self-adjoint part of a C*-algebra is a JB-algebra, and the self-adjoint part of a von Neumann algebra is a JBW-algebra. Many results in the Jordan context are natural (if non-trivial) generalizations of known theorems on C*-algebras or von Neumann algebras, although there are sometimes significant differences in results and/or in proofs. To stress the similarities, we often have given a reference to the analogous result for C*-algebras or von Neumann algebras in [AS]. We have given full proofs of most results, but some of the more complex proofs we have referred to [67]. In quantum mechanics, observables usually are taken to be self-adjoint operators on a Hilbert space. Jordan algebras were originally introduced as a mathematical model of quantum mechanics (cf. [75]), motivated by the observation that the square of an observable has a physical interpretation, which the product of two observables will not have (unless they are simultaneously measurable). Since the Jordan product a 0 b = ~(ab + ba) can be expressed in terms of squares by a 0 b = i((a + b)2 - (a - b)2), the set of observables should be closed under Jordan multiplication, but not necessarily under associative multiplication. A JB-algebra is a real Jordan algebra which is also a Banach algebra under a norm with properties similar to those of C*-algebras. Not every JB-algebra A admits a representation as a Jordan algebra of self-adjoint operators on a complex Hilbert space, but there is a Jordan ideal J (described in terms of the exceptional Jordan algebra H3 (0) of Hermitian matrices with entries in the octonians) which cannot be so represented, but for which A/ J has a faithful representation on a Hilbert space. A lB-algebra which is monotone complete and has a separating set of normal states is called a lBW-algebra. Chapters 1 through 4 contain standard algebraic background for lBalgebras and lBW-algebras, with a few results that are more specialized for later use. Chapter 5 investigates properties of lB-algebra state spaces, and lBW-algebra normal state spaces. Chapter 6 contains the theory of dynamical correspondences, which was first presented in [12], and has not appeared before in book form. A dynamical correspondence on a lB-algebra is a map which assigns to each element an order derivation, which is a generator of a one-parameter group of affine automorphisms of the state space (and also of the normal state space in the lBWalgebra case). The definition of a dynamical correspondence is motivated by quantum mechanics. Elements of a lB-algebra are thought of as representing observables. But in quantum mechanics physical variables play

PREFACE

VII

a dual role, as observables and as generators of one-parameter groups, and a dynamical correspondence gives the elements of a JB-algebra such a double identity. The main theorem on such correspondences says that a JB-algebra is the self-adjoint part of a C*-algebra (and a JBW-algebra is the self-adjoint part of a von Neumann algebra) iff it admits a dynamical correspondence, which will then provide the Lie part of the associative product. We also discuss the connection of dynamical correspondences with the concept of orientation used by Connes in his characterization of O'-finite von Neumann algebras via the natural cone of TomitaTakesaki theory [36]' and with the notion of orientation of state spaces introduced by the authors and discussed in detail in [AS], where it is shown that orientations of the state space are in 1-1 correspondence with C*products. Part II starts out with convex sets K which are at first very general, but later will be specialized to state spaces of algebras (Jordan or associative). In the case of the state space of a C*-algebra (or JB-algebra), the selfadjoint elements act as affine functions on the state space. A key feature of such algebras is that these elements admit a functional calculus, which, on the algebraic side, reflects a key property of the subalgebra generated by a single element, and, on the physical side, represents the application of a function to the outcome of an experiment. Thus a natural starting point in finding properties that characterize such state spaces is to give conditions that lead to a spectral theory and functional calculus for a space A of affine functions on K. This is the main focus of Part II. This spectral theory is based on axioms involving concepts which we will now briefly describe. A key concept is that of an (abstract) compression P : A -+ A which generalizes a (concrete) compression P : a f---> pap by a projection p in a C*-algebra (or a JB-algebra). Such maps are often thought of as representing the filters in the measurement process of quantum mechanics. Geometrically they correspond, in a 1-1 fashion, to a class of faces of K, called pro]ectwe faces since they are determined by projections in the motivating examples. (In [7] and [9] compressions were called P-projectwns). Two positive affine functions b, c E A are said to be orthogonal if they live on a pair of complementary projective faces; then a = b - c is said to be an orthogonal decomposdion of a. The concept of an orthogonal decomposition plays an important role in the establishment of the functional calculus. Part III contains the main result which characterizes state spaces of C*-algebras among all compact convex sets. A preliminary result gives a similar characterization of state spaces of JB-algebras, which is of interest in itself. The first step in the construction of an algebra from a compact convex set K is the spectral functional calculus for the space A = A(K) of continuous affine functions on K. This step makes use of two assumptions. The first is the existence of orthogonal decompositions in A(K). The other is the existence of "many" projective faces; more specifically, that

pi

PREFACE

viii

the projective faces include all norm exposed faces, that is, all those faces which are zero-sets of positive bounded affine functions. The functional calculus provides a candidate for a Jordan product defined in terms of squares, as explained above. This product need not be distributive, but this is achieved by the crucial Hilbert ball axwm, by which the face generated by each pair of extreme points of K is a norm exposed Hilbert ball (i.e., is affinely isomorphic to the closed unit ball of an arbitrary finite or infinite dimensional Hilbert space). Together with a technical assumption (which may be regarded as a non-commutative generalization of the decomposition of a probability measure in a discrete and a continuous part), these assumptions make A the self-adjoint part of a JB-algebra whose state space is K. There is also an alternative axiom which serves the same purpose. That axiom is less geometrical, but has has a bearing on physical applications in that it involves three pure state properties, of which the crucial one describes the symmetry of transition probabilities in quantum mechanics. Thus, our characterization of JB-algebra state spaces is as follows.

Theorem A compact convex set K zs affinely homeomorphic to the state space of a fE-algebra (with the w*-topology) zjJ K satzsjies the conditions: E A(K) admits a decomposztion a = b ~ c wzth b, c E A(K)+ and b .1 c. (ii) Every norm exposed face of K is proJective. (iii) The (J"-convex hull of the extreme poznts of K is a split face of K and K has the Hilbert ball property.

(i) Every a

Here the condition (m) can be replaced by the alternatzve condztwn (iii ') K has the pure state propertzes. The state spaces of C*-algebras are characterized by adding more assumptions to those for state spaces of JB-algebras. The Hilbert ball property is strengthened to the 3-ball property, by which the Hilbert balls in question are of dimension 0, 1 or 3. If a JB-algebra A with state space K has the 3-ball property, then each extreme point of K generates a split face of K that is affinely isomorphic to the normal state space of B (H) for a suitable complex Hilbert space H. Each such affine isomorphism induces an irreducible representation of A as a Jordan algebra of self-adjoint operators on H, in analogy with the standard GNS-construction. However, different affine isomorphisms can give unitarily inequivalent representations, so in general, for each pure state there are two associated irreducible representations, up to unitary equivalence. It turns out that the 3-ball property is not sufficient to insure that K is a C*-algebra state space. At this point we need to impose our our final

PREFACE

ix

condition. This is the axiom of or'lentabzlzty, which we will now briefly explain. The key problem stems from the fact that a C*-algebra (unlike a JBalgebra) is not completely determined by its state space (as can be seen from the simple fact that a C*-algebra and its opposite algebra have the same state space). Thus we cannot recover a C*-algebra from the mere knowledge of the compact topology and the affine geometry of its state space. Some more structure is needed, and the clue to this additional structure already can be found by looking at the simplest non-commutative C*-algebra, the algebra M 2 (C) of 2 x 2-matrices. Its state space is a three dimensional Euclidean ball, which has two possible orientations, and those two determine the two possible C*-products on Ith(C): the standard one, and its opposite product. In the general case, we need the concept of a global Or'lentatzon of a compact convex set K with the 3-ball property. Loosely speaking, a global orientation of K is a "continuous choice of orientation" of all those faces of K that are 3-balls. Now K is said to be or'lentable if it admits a global orientation, and it is shown that such a global orientation of a compact convex set, with the other properties stated above, will determine a choice of the irreducible "GNS-type" representations such that the direct sum of these representations maps A(K) Jordan-isomorphically onto the self-adjoint part of a C*-algebra with state space K. Thus our main result is the following.

Theorem A compact convex set K is affinely homeomorphzc to the state space of a C*-algebra (wzth the w*-topology) zfJ K satzsfies the following propertzes: E A(K) admzts a decomposztzon a = b - c wzth b, c E A(K)+ and b ~ c. (ii) Every norm exposed face of K zs pro)ectzve. (iii) The (J-convex hull of the extreme poznts of K zs a splzt face of K, and K has the 3-ball property. (iv) K zs orientable.

(i) Every a

If K satisfies these conditions, then A(K) admits a unique Jordan product making it a JB-algebra with the given state space Furthermore, associative products on A(K) + zA(K) that make it a C*-algebra with state space K will be in 1-1 correspondence with orientations of K. Part III also contains characterizations of the normal state spaces of JBW-algebras and von Neumann algebras among all convex sets. One key property of these normal state spaces is that they are spectral, which means that their bounded affine functions admit unique orthogonal decompositions, and that their norm exposed faces are projective. In this context, there may be no extreme points, so the remaining axioms for JB-algebra and C*-algebra state spaces are not relevant here. Instead, the normal state spaces of JBW-algebras are shown to be those spectral convex sets

x

PREFACE

that are elliptic (in that certain distinguished plane subsets are affinely isomorphic to circular disks). The normal state spaces of von Neumann algebras are shown to be those spectral convex sets that are elliptic and enjoy a generalized version of the 3-ball property. (There is no orientability axiom needed in this case.) In this part of the book, it is also explained how the state spaces of C*-algebras and the normal state spaces of von Neumann algebras can be characterized among the corresponding state spaces for JB-algebras and JBW-algebras by assuming the existence of a dynamical correspondence, which will give the Lie part of the associative product (as proved in Part I). This result does not involve a construction from local geometric data like the theorems above, but is of interest because of its relationship to physics. A brief description of the content of each chapter is given in the opening paragraph of that chapter. Acknowledgments We would like to thank Dick Kadison for suggesting that we write this book, and for his constant encouragement. We are grateful for the support for our writing of this book provided by grants from the Norwegian Research Council and from Wellesley College. We thank Ann Kostant of Birkhauser for her support over the years. Finally, we thank our wives Ellen and Mary for their support and patience, especially during our many trips to work on this book and its predecessor. The second author would also like to thank his children Chris, Kim, and Ryan for their patience during the years he has been working on these books.

Erik M. Alfsen Frederic W. Shultz October 2002

Contents

Prefu~

. . . . . . . . . . . . . . . . . .

v

PART I: Jordan Algebras and Thetr State Spaces

1

Chapter 1: JB-algebras .

3

Introduction . . . . . Quotients of JB-algebras Projections and compressions in JB-algebras Operator commutativity . . . . . Order derivations on JB-algebras Appendix: Proof of Proposition 1.3 Notes . . . . . . . . . . . . . Chapter 2: JBW-algebras Introduction . . . . . Orthogonality and range projections Spectral resolutions . . . . . . . The lattices of projections and compressions a-weakly closed hereditary subalgebras Center of a JBW-algebra . . . The bidual of a JB-algebra . . The predual of a JBW-algebra JW-algebras . . . . . . . . Order automorphisms of a JB-algebra Abstract characterization of compressions Notes . . . . . . . . . . . . . . . Chapter 3: Structure of JBW-algebras Equivalence of projections . . . . . . Comparison theory . . . . . . Global structure of type I JBW-algebras Classification of type I JBW-factors Atomic JBW-algebras Notes . . . . . . . . . . . . .

3 16 22 26 30

32 34 37 37

42 46 48 54 55 58 62

70 73 75

76 79 79 83 85

87 96

101

CONTENTS

xii

Chapter 4: Representations of JB-algebras Representations of JBW-factors as JW-algebras Representations of JB-algebras as Jordan operator algebras Reversibility . . . . . . . . . . . . The universal C*-algebra of a JB-algebra Cartesian triples Notes . . . . .

Chapter 5: State Spaces of Jordan Algebras Carrier projections . . . . . . . . The lattice of projective faces . . . . Dual maps of Jordan homomorphisms Traces on JB-algebras . . . . . . . Norm closed faces of JBW normal state spaces The Hilbert ball property . . . . . . . . The a -convex hull of extreme points . . . . Trace class elements of atomic JBW-algebras Symmetry and ellipticity Notes . . . . . . . .

Chapter 6: Dynamical Correspondences Connes orientations . . . . . . . . . . Dynamical correspondences in JB-algebras Comparing notions of orientation Notes . . . . . . . . . . . .

PART II:

Convex~ty

and Spectral Theory

Chapter 7: General Compressions Projections in cones . . . . . . . Definition of general compressions . Projective units and projective faces Geometry of projective faces Notes . . . . . . . . .

Chapter 8: Spectral Theory The lattice of compressions . The lattice of compressions when A = V* Spaces in spectral duality Spectral convex sets . . . . . . Appendix. Proof of Theorem 8.87 Notes . . . . . . . . . . . .

103 103 108 113 120 127 137 139 139 143 144 147 157 168 172 178 183 188 191 191 196 205 206 209 211 211 222 227 241 249

251 251 260 271

291 303 311

CONTENTS

Xlll

PART III: State Space Charactellzatwns . . . . . . . . .

313

Chapter 9: Characterization of Jordan Algebra State Spaces . . . . . . . . . . . . . . . . . . . . .

315

The pure state properties . The Hilbert ball property Atomicity . . . . . . . The type I factor case . . Characterization of state spaces of JB-algebras Characterization of normal state spaces of JBW algebras Notes .....................

315 319 323 325 339 343 354

Chapter 10: Characterization of Normal State Spaces of von Neumann Algebras . . . . . . . . . . . . . .

357

The normal state space of B (H) . . . . . . . 3-frames, Cartesian triples, and blown up 3-balls The generalized 3-ball property Notes .......... Chapter 11: Characterization of C*-algebra State Spaces States and representations of JB-algebras. . . . . . . . Reversibility of JB-algebras of complex type . . . . . . The state space of the universal C*-algebra of a JB-algebra Orientability . . . . . . . . . . . . . . . . . . . . Characterization of C*-algebra state spaces among JB-algebra state spaces. . . . . . . . . . . . . . . . . . . . Characterization of C* -algebra state spaces among convex sets Notes . . . . . . . . . . . . . . . . . . . . . . . . .

357 358 365 373 375 376 382 393 402 406 414 417

Appendix

419

Bibliography

451

Index . . . .

459

PART I

Jordan Algebras and Their State Spaces

1

JB-algebras

This chapter starts with the definition and basic properties of JBalgebras. We show that quotients of JB-algebras are again JB-algebras, and that Jordan isomorphisms between JB-algebras are isometries. Next we investigate properties of compressions (maps a f--> {pbp} for p a projection); these will play a key role in our axiomatic investigations in Part II. This is followed by a discussion of operator commutativity (the Jordan analog of the associative notion of commutativity). The chapter ends with basic facts about order derivations on JB-algebras. (In Chapter 6, we will discuss order derivations in more detail.)

Introduction The motivating example for a JB-algebra is a JC-algebra, i.e., a norm closed real linear space of self-adjoint operators on a Hilbert space, closed under the symmetrized product (1.1 )

a

0

b

=

1

2(ab + ba).

This product is bilinear and commutative, but not associative. A weakened form of the associative law holds for this product, and that leads to the following abstraction. 1.1. Definition. A Jordan algebra over R is a vector space A over R equipped with a commutative bilinear product 0 that satisfies the identity

(1.2)

(a 2

0

b)

0

a

=

a2

0

(b

0

a)

for all a, b EA.

If A is any associative algebra over R, then A becomes a Jordan algebra when equipped with the product (1.1), as does any subspace closed under o. If a Jordan algebra is isomorphic to such a subspace, it is said to be specwl; if it cannot be so imbedded it is exceptwnal. If A is a C *algebra, then A = Asa is the primary example of a Jordan algebra that motivates our work. In spite of the non-associative nature of the Jordan product, calculations in a special Jordan algebra are straightforward because of the associative algebra in which the Jordan algebra lives. There is then a procedure for verifying that the corresponding results hold in general Jordan algebras. This depends on a remarkable result called MacDonald's theorem that says

4

1

JB-ALGEBRAS

roughly that most Jordan identities valid for special Jordan algebras are valid for all Jordan algebras. (We will discuss Macdonald's theorem in more detail when we first make use of it.) In any Jordan algebra we define powers recursively: Xl = X and xn = x n - 1 0 x. A p'NOrl we are in the difficult position of not knowing that the associative law holds for powers of a single element, i.e., that a Jordan algebra is power associative, cf. (A 45). We now set out to remedy this situation. Note that the following special case of power associativity follows at once from commutativity and the Jordan identity (1.2) by taking a = b = x:

(1.3) where for the moment we have indicated multiplication by juxtaposition. We are going to show that this implies power associativity for commutative algebras over R, and thus that Jordan algebras are power associative. We start by finding an identity that is linear in each variable and is equivalent to (1.3). (This is a standard process, called "linearization" of the identity.)

1.2. Lemma. Let A be an algebra over R wdh a product that is commutative and bllmear, but not necessa'Nly associative. Then (1.3) lS equivalent to (1.4)

where the sums are over distinct

l,

j, k, l with 1 ::::; l,], k, l ::::; 4.

Proof. Assume (1.3) holds. Replacing x by A1X1 + A2X2 + A3X3 (for real scalars A1,A2,A3,A4) gives

+ A4X4

where the sums are over all l, ], k, l with 1 ::::; l,], k, l ::::; 4. Each side in (1.5) is a quartic polynomial in the variables A1, A2, A3, A4 with coefficients in A. Let ¢ : A ---> R be a linear functional. Then applying ¢ to both sides of (1.5) leads to two polynomials with real coefficients that are equal for all values of the variables A1, A2, A3, A4. It follows that the coefficients of corresponding terms must be equal. Thus in particular, equating coefficients of the A1A2A3A4 term on each side gives

where the sum is now over all distinct i, j, k, l. Since linear functionals separate elements of A, (1.4) follows.

INTRODUCTION

Conversely, if (1.4) holds, substituting (1.3). D

x

5

=

Xl

=

X2

=

X3

=

X4

gives

1.3. Proposition. Let A be a (not necessa7'lly assocwtive) commutative algebra over R. Then A zs power assocwtzve zft zt satisfies the identity X(xx2) = x2x2 or the ImeaT"ized version (1.4). Proof. The proof relies on Lemma l.2 and a lengthy induction argument. To avoid interrupting the flow of ideas, we have relegated this proof to the appendix of this chapter. D

1.4. Corollary. Every fordan algebra over R is power associative. Proof. As remarked above, (l.3) is an immediate consequence of the Jordan identity. The corollary then follows from Proposition l.3. (A proof can be given that avoids Proposition l.3 (cf. [67, Lemma 2.4.5]), but we need Proposition 1.3 for other purposes anyway.) D For complex Banach *-algebras, by the Gelfand-Neumark theorem (A 64) the norm axiom Ila*all = IIal1 2 characterizes norm closed *-algebras of operators on Hilbert space. It is natural to single out a class of normed Jordan algebras that act like Jordan algebras of self-adjoint operators on Hilbert space.

1.5. Definition. A fE-algebra is a Jordan algebra A over R with identity element 1 equipped with a complete norm satisfying the following requirements for a, b E A: (l.6) (l. 7) (l.8)

Iia 0 bll :::; Ilallllbll, IIa 21 = Il a1 2, IIa 21 :::; IIa 2+ b211·

JB-algebras can also be defined without requiring a unit (multiplicative identity). A unit can always be adjoined, but this turns out to be non-trivial to show, cf. [67, Thm. 3.3.9].

1.6. Example. The self-adjoint part of a C*-algebra (A 57) is a JBalgebra with respect to the Jordan product a 0 b = ~(ab + ba). (We will make the convention that all C* -algebras mentioned have an identity.) The norm properties (1.6) and (1.7) follow from the definition of a C*-algebra. By (A 58) and (A 52), a C*-algebra satisfies the property (1.8). Any norm closed Jordan subalgebra of B(H)sa is easily seen to be a JB-algebra. (The defining norm properties follow from the corresponding properties for B(H)sa.)

1

6

JB-ALGEBRAS

1. 7. Definition. A JC-algebra is a JB-algebra A that is isomorphic to a norm closed Jordan sub algebra of B (H),a' If A is a norm closed Jordan subalgebra of B(H)sa, then we may also refer to A as a concrete iC-algebra, or as a iC-algebra actmg on H. We will make the convention that our JC-algebras have an identity. We will see in Proposition 1.35 that an isomorphism of JB-algebras is necessarily an isometry, so in Definition 1.7 we could replace "isomorphic" by "isometrically isomorphic" . These are the motivating examples for JB-algebras, but JB-algebras are slightly more general. Even in finite dimensions, not every JB-algebra is a JC-algebra. There is essentially just one exception: the algebra H3(O) of Hermitian 3 x 3 matrices with entries in the octonions (Cayley numbers). More precisely, a finite dimensional JB-algebra is a JC-algebra iff it does not contain H3(O) as a direct summand. In infinite dimensions, each JBalgebra A admits an exceptional ideal 1 (closely related to H3 (0)) such that Allis a JC-algebra. We will discuss these results in more detail in Chapters 3 and 4. We will see below (Lemma 1.10) that every JB-algebra not only is a Banach space, but also is an ordered linear space, with the positive cone being the set of squares. With this ordering every JB-algebra becomes an order unit space, whose definition we now review.

1.8. Definition. An order unzt space is an ordered normed linear space A over R with a closed positive cone and an element e, satisfying

Iiall

=

infp

>

0

I -Ae::::; a::::; Ae}.

The element e is called the distmguished order unit. Order unit spaces were investigated by Kadison [76], who called them function systems, since they are precisely the spaces that can be imbedded as subspaces of CR(X) for a compact Hausdorff space X. For more background, see (A 12) and the succeeding results in the Appendix. Remark. Jordan algebras have been proposed as a model for quantum mechanics. One reason for this is that, typically, observables are taken to be self-adjoint operators on a Hilbert space, and the space of such operators is closed under the Jordan product but not the associative product. One is then led to abstract axioms for a Jordan algebra. However, the fundamental Jordan axiom (1.2) has no obvious physical interpretation. It has therefore been proposed by von Neumann [98] and by Iochum and Loupias [69, 70] to replace (1.2) by power associativity. This leads to the notion of an order unit algebra (called a Banach-power-assocwtwe algebra in [69, 70]), which we now define. We will see that JB-algebras are precisely the commutative order unit algebras, cf. Lemma 1.10 and Theorem 2.49.

INTRODUCTION

7

1.9. Definition. A normed algebra is a (not necessarily associative) algebra A with a norm such that Ilabll : : : Ilallllbll for all a, bE A. An order unit space A (A 14) which is also a power associative complete normed algebra (for the order unit norm) and satisfies (i) the distinguished order unit 1 is a multiplicative identity, (ii) a 2 E A+ for each a E A, will be called an order umt algebra. For more background on order unit algebras, see (A 46) and the succeeding results in the Appendix.

1.10. Lemma. Every iE-algebra A ~s a commutative order umt algebra with the gwen norm and pos~twe cone A2 = {a 2 I a E A}.

Proof. A has an identity, and is power associative (Corollary 1.4). Then the norm properties (1.6), (1.7), (1.8) imply that A is a commutative order unit algebra with the given norm and with a positive cone consisting of the squares in A (A 52). D Hereafter we will assume each JB-algebra A is given the order for which the positive cone is A 2 . In Chapter 2 we will prove the converse of Lemma 1.10: that every commutative order unit algebra is a JB-algebra.

1.11. Theorem. If A ~s a iE-algebra, then A ~s a norm complete order umt space wzth the ~dent~ty 1 as d~stzngmshed orde'r umt and with order umt norm coznc~dzng wdh the gwen one. Furthermore, for each aE A (1.9)

Conversely, if A ~s a complete order umt space eqmpped w~th a iordan product for which the d~stzngmshed order umt ~s the identzty and satisfies {l.9}, then A ~s a iE-algebra wzth the order umt norm. Proof. Assume that A is a JB-algebra. By Lemma 1.10, A is a commutative order unit algebra, with the given norm, with a positive cone consisting of the squares, and with the distinguished order unit being the identity of A. By (A 51) the implication (1.9) is valid for commutative order unit algebras, and thus for A. Conversely, let A be a Jordan algebra over R, so that A is a commutative, power associative algebra with identity (Corollary 1.4). If A is also a complete order unit space for which the identity is the distinguished order unit and satisfies (1.9), then A is an order unit algebra (A 51). The norm in a commutative order unit algebra satisfies the properties (1.6), (1.7), (1.8) (A 52), so A is a JB-algebra. D

1

8

JB-ALGEBRAS

Combining Theorem 1.11 and the definition of the order unit norm (A 14), we have the following in any JB-algebra: (1.10)

Iiall = inf{A

sa S

> 0 I -AI

AI},

and the related inequalities (cf. (A 13)): (1.11)

-llalll

sa s

Il a ll1.

Note that by (1.11) a JB-algebra is positively generated, i.e., A =

A+ - A+, since a = (a + Ilalll) -llall1. For each element a in a JB-algebra A we let C(a,l) denote the norm closed subalgebra generated by a and 1. By Corollary 1.4 the Jordan subalgebra generated by a single element and 1 is associative, and by (1.6) multiplication is jointly continuous, so C(a,l) is also associative.

1.12. Proposition. If A ~s a fE-algebra and B is a norm closed associative subalgebra contaming 1, m particular ~f B = C( a, 1) for a E A, then B is ~sometncally (0 rder- and algebra-) ~somorph~c to CR(X) for some compact Hausdorff space X. Proof. This follows from the fact that B is then an associative JBalgebra, that a JB-algebra is a commutative order unit algebra (Lemma 1.10), and that a commutative associative order unit algebra is isomorphic to some CR(X) (A 48). D We will frequently make use of Proposition 1.12, referring to its consequences as following from spectral theory. We give one application, proving that the norm condition (1.8) is not redundant. Let A be the algebra of all continuous functions on the closed unit ball in C, holomorphic in the interior, and real on the real axis, with the supremum norm. Then it is easily verified that A is a commutative Banach algebra satisfying (1.6) and (1. 7). If A satisfied (1.8), then A would be an associative JB-algebra, so by spectral theory 1 + a 2 would be invertible for all a E A. On the other hand, if a E A is the identity function on the unit disk, then 1 + a 2 is the function z f---7 1 + z2, which takes the value 0 at ~, and thus is not invertible. Therefore (1.8) cannot hold for A.

In any Jordan algebra, we define the triple product {abc} by (1.12)

{abc} = (a

0

b)

0

c + (b

0

c)

0

a - (a

0

c)

0

b.

Note that {abc} is linear in each factor, and {abc} = {cba}. In a special Jordan algebra, {abc} = ~(abc + cba) (where the right side involves the ordinary associative product).

INTRODUCTION

9

The special case of the triple product {abc} with a = c will play an important role in the sequel. We will define Ua : A ----+ A by

(1.13)

Uab = {aba} = 2a

0

(a

0

b) - a 2 0 b.

The maps Ua are generally much more convenient to work with than the Jordan multiplication b ----+ a 0 b. For a special Jordan algebra (e.g., a JC-algebra), we have Ua(b) = aba. The following two identities are valid in any Jordan algebra, and show that the Jordan triple product {aba} acts very much like the associative product aba, which is a great aid in algebraic calculations:

(1.14)

{{aba}c{aba}}

(1.15)

{bab}2

= =

{a{b{aca}b}a}, {b{ab 2a}b}.

Note that (1.15) can be rephrased in terms of the maps Ua as (1.16) For special Jordan alge br as {aba} = aba, so (1.14) and (1.15) follow immediately. It now follows from Macdonald's Theorem (discussed below) that these identities hold for all Jordan algebras. 1.13. Theorem (Macdonald). Any polynomwl Jordan 1dentzty mvolving three or fewer vanables (and poss1bly involving the 1dentity 1) which is of degree at most 1 m one of the varwbles and wh1ch holds for all specwl Jordan algebras will hold for all Jordan algebras. Proof. The interested reader can find a discussion with a complete proof in [72, p. 41] or [67, Thm. 2.4.13]. D The following result can be derived from MacDonald's Theorem, and is sometimes more convenient to apply. Note that it implies that calculations in any Jordan algebra involving just two elements can be carried out by assuming that the Jordan algebra is a sub algebra of an associative algebra with respect to the product a 0 b = ~ (ab + ba). 1.14. Theorem (Shirshov-Cohn). Any Jordan algebra generated by two elements (and 1) 1S specwl. Proof. See [67, Thm. 2.4.14]. D Our next goal is to prove the crucial property that the maps Ua are positive, i.e., Ua (A +) c A +. First we need to develop some properties of inverses.

1

10

JB-ALGEBRAS

1.15. Definition. An element a of a JB-algebra A is mvertzble if it is invertible in the Banach algebra C(a,I). Its inverse in this subalgebra is then denoted a-I and is called the inverse of a. The next few results relate this notion to other concepts of invertibility.

1.16. Lemma. Let A be an assocwtzve algebra, and conszder the associated Jordan product a 0 b = ~(ab + ba). Then for a, b m A, ab = ba = 1 zfj (1.17)

a

0

b

=

1

and

a2

0

b

= a.

Proof. If ab = ba = 1, then (1.17) follows easily. Conversely, suppose a and b satisfy (1.17). Then

(1.18)

ab

+ ba = 2

and

(1.19) Multiplying (1.18) on the left by a and then on the right by a gives

which implies

(1.20) Substituting (1.20) into (1.19) gives

(1.21) Multiplying the first of the equalities of (1.21) on the left by b and the second on the right by b gives ba 2 b = ba and ab = ba 2 b. Hence by (1.18), ab = ba = 1. 0

If elements a, b in a Jordan algebra satisfy (1.17) we will say that a and b are Jordan invertzble, and that b is the Jordan mverse of a. We will now show that in the context of JB-algebras this is the same as invertibility as defined in Definition 1.15. (It will follow that the Jordan inverse of an element is unique.)

INTRODUCTION

11

1.17. Proposition. For elements a, b In a iB-algebra A, the following are equwalent:

(i) b ~s the Inverse of a in the Banach algebra C(a, 1), (ii) b is the Jordan Inverse of a In A. Proof. (i) =} (ii). This follows from the associativity of the Jordan product for elements of C(a, 1). (ii) =} (i). Let b be the Jordan inverse of a in A, and let Bo be the Jordan subalgebra of A generated by a, b, and 1. By the Shirshov-Cohn Theorem (Theorem 1.14) Bo is special, so we may assume that Bo is a Jordan subalgebra of an associative algebra C with x 0 y = ~(xy + yx) for x, y in Bo. By Lemma 1.16, ab = ba = 1. Thus the associative subalgebra of C generated by a and b is commutative, and so the induced Jordan product on this subalgebra coincides with the associative product. By norm continuity of the Jordan product on A, the norm closure B of Bo in A is also associative. Thus both Jordan and associative notions of inverse coincide in B, and by Proposition 1.12 there is an isomorphism 1jJ from B onto CR(X) for a compact Hausdorff space X. Since a is invertible in B, then 1jJ(a) will be invertible in CR(X), and so will have compact range Y not containing zero. By the Stone-Weierstrass theorem we can uniformly approximate the function ,.\ f---> 1/,.\ on Y by polynomials. It follows that the inverse of 1jJ(a) in CR(X) is a norm limit of polynomials in 1jJ(a) and 1, and thus that the inverse of a in B (namely b) is a norm limit of polynomials in a and 1. Thus b must lie in C (a, 1). D

1.18. Definition. If a is an element of a JB-algebra A, the spectrum of a, denoted sp(a), is the set of all ,.\ E R such that "\1-a is not invertible. 1.19. Corollary. For each element a In a JB-algebra A, sp(a) is compact and non-empty, and there ~s a umque ~somorph~sm from CR(sp(a)) onto C(a,l) taking the ~dent~ty functwn on sp(a) to a. Th~s isomorph~sm is also an ~sometric order ~somorphism. Proof. We will proceed by constructing the inverse of the desired isomorphism. Let b f---> b be an isometric isomorphism from C(a,l) onto CR(X) for a compact Hausdorff space X (cf. Proposition 1.12). An isomorphism preserves invertibility, so sp(a) = sp(a). By (A 55), sp(a) is the range of so is a continuous map of X onto sp(a); in particular sp(a) is compact and non-empty. If a(x) = a(y) then ;rex) = ;r(y) for n = 1,2, .... By density of polynomials in a and 1 in C(a,l) and the fact that b f---> b is an isometric isomorphism, polynomials in and 1 are dense in CR(sp(a)), and so we conclude that x = y. Thus is a homeomorphism from X onto sp(a). The composition b f---> b 0 a-I is an isomorphism from C(a,l) onto CR(sp(a)), which sends a to the identity map on sp(a). Call this map q,.

a,

a

a

a

12

1.

JB-ALCEBRAS

If III : C( a, 1) ---? C R (sp( a)) is any isomorphism, then III takes 1 to 1 and squares to squares, so must be a unital order isomorphism. By (1.10) III is an isometry. Since polynomials in a are dense in C(a,I), III is uniquely determined by its value on a, and so if III takes a to the identity map on sp(a), then III = ~. Finally, ~-l is the desired unique isomorphism from CR(sp(a)) onto C(a,l) taking the identity map on sp(a) to a. D 1.20. Definition. Let a be an element of a JB-algebra A. The contmuous functional calculus is the unique isomorphism f f-+ f (a) from CR(sp(a)) onto C(a,l) taking the identity function on sp(a) to a. 1.21. Proposition. Let a be an element of a iE-algebra A. The contmuous functwnal calculus lS an lsometnc order lsomorphlsm from CR(sp(a)) onto C(a, 1), and f(a) has ds usual meamng when f lS a polynomwl. Furthermore (1.22) (1.23)

sp(J(a)) (J

0

=

f(sp(a))

for f E CR(sp(a)), and

for 9 E CR(sp(a)) and f E CR(sp(g(a))).

g)(a) = f(g(a))

If f E CR(sp(a)) and f(O) = 0, then f(a) lS m the (non-umtal) norm closed subalgebra C(a) generated by a.

Proof. f f-+ f(a) is an isometric order isomorphism by Corollary 1.19. If ld denotes the identity map on sp(a), and p is the polynomial p()..) = Li CXi)..i, then p(a) = (LCXi(ld)i) (a) t

=

Lcxia\ t

where (ld) i denotes pointwise multiplication of functions, so p( a) has its usual meaning. A unital isomorphism of algebras preserves invertibility and thus spectrum. Therefore sp(J) = sp(J(a)). By (A 55), sp(J) equals f(sp(a)) and so (1.22) follows. The equality (1.23) is clear for polynomials f, and then follows for arbitrary f E CR(sp(g)) by continuity of the functional calculus and the Stone-Weierstrass theorem, since f f-+ f(a) is isometric. If f E CR(sp(a)) and f(O) = 0, then f can be uniformly approximated on sp(a) by polynomials Pn without constant term. Now Pn(a) E C(a) for n = 1,2, ... ; hence also f(a) E C(a). D 1.22. Corollary. IJ a is an element oj a iE-algebra A, then a ~ 0 iff sp(a) C [0, =), and Iiall = sup{I)..1 I ).. E sp(a)}.

13

INTRODUCTION

Proof. Since the functional calculus is an order isomorphism, then iff ~d ~ 0, where ~d is the identity function on sp( a). But ~d is a square iff sp(a) c [0,00), which proves the first statement of the corollary. The second follows in similar fashion from the fact that the functional calculus is isometric. 0

a>

°

Note that if A is a JB-algebra, and B is a Jordan subalgebra containing the identity of A, and B is also a JB-algebra itself for the inherited norm, then a priori we have two notions of spectrum and order in B: those that come from its status as a JB-algebra, and those inherited from A. However, for a in B, we have C(a, 1) c B, so the spectrum of a is the same whether calculated in A or in B. Then by Corollary 1.22, a ~ is equivalent for a viewed as a member of A or of B, so the order on B as a JB-algebra is that inherited from A.

°

1.23. Lemma. Let A be a iB-algebra and a E A. Then a ~s an invertible element iff Ua has a bounded mverse, and in thzs case the mverse map is Ua-I.

Proof. If a is invertible, let c E C(a, 1) be the inverse of a. Then by the associativity of C(a, 1) we have {ac 2 a} = 1. Now by (1.16) we have

and thus Ua is invertible. To calculate its inverse, we first observe that since a-I is the inverse of a in the associative subalgebra C(a, 1), then {aa-Ia} = a. Thus by (1.16)

Now multiplying by the inverse of Ua on both sides shows Ua-I is the inverse of Ua . If a is not invertible, then by spectral theory there are elements {b n } in C(a,l) which have norm 1 and which satisfy IlUab n II --+ 0. If Ua had a bounded inverse, applying it to the sequence {Uab n } would show {b n } --+ 0, a contradiction to II bn II = 1. 0 The assumption that the inverse of Ua is bounded is unnecessary; a purely algebraic proof that a is invertible iff Ua is invertible can be found in [72, p. 52].

1.24. Lemma. If a and bare inveTtzble elements of a iB-algebra, then {aba} is invertible wdh inverse {a-Ib-Ia- I }.

Proof. By Lemma 1.23, if a and b are invertible, then the maps Ua and U{aba} = UaUbUa . By

Ub have bounded inverses, and thus by (1.16) so does

JB-ALGEBRAS

14

Lemma 1.23 it follows that {aba} is invertible. For any invertible element x, calculating in C(x,l) we find Ux-IX = x-I, so applying Lemma 1.23 and (1.16) gives

{aba}-I = U{aba}-l{aba} = U{aba}-I{aba} = (UaUbUa)-I{aba} = Ua-1Ub-1Ua-1Uab = {a- I b- 1 a- I }, which completes the proof. 0

1.25. Theorem. Let A be a iE-algebra. For each a E A the map Ua zs positive, and has norm Ila11 2 .

Proof. Let A-I denote the set of invertible elements of A, and let Ao = A-I n A + be the positive invertible elements. By spectral theory, if b 2': 0, then b is invertible iff b 2': Al for some A > o. It follows that Ao is convex and open. By (1.11), a

2': 0

<==?

1III aill - a I :::: Iiall,

so A+ is closed. Therefore Ao = A-I n A+ is both open and closed in A-I. Note also that Ao is dense in A +, since if a 2': 0 and E > 0, then a+ dE Ao. We first show that Ua is positive for invertible a. Note that (a- I )2 is in Ao by spectral theory. By Lemma 1. 24, Ua(A -1) C A-I, so in particular Ua(Ao) c A-I. We next show that 1 E Ua(Ao). To verify this, note that 1 = Ua((a- 1 )2) (calculate in C(a, 1)). Since Ua(Ao) is convex, it is a connected subset of A-I. Since Ao is closed and open in A-I and meets Ua(Ao) (e.g., 1 E Ua(Ao) n Ao), then Ua(Ao) C Ao . By continuity of Ua and density of Ao in A+, we conclude that Ua(A+) C A+, and so Ua is positive. Now let a E A be arbitrary, and let b E Ao. By spectral theory there is an invertible c such that c 2 = b. Then by the identity (1.15)

and so by the first part of this proof

Thus Ua(Ao) C A+, and by continuity Ua is positive. Finally, for any positive map T on A, it follows from (A 15) that IITII = IITIII· Applying this to Ua we conclude IWal1 = IWal11 = IIa211 =

Ila112.

0

INTRODUCTION

15

1.26. Lemma. If a, bare posdwe elements of a lB-algebm A, then (1.24) and (1.25) aTe equwalent and zmply (1.26): {aba} =0,

(1.24) (1.25)

{bab} =0, a

(1.26) Pmof. (l.24)

¢?

0

b = O.

(l.25) Note that by spectral theory

(1.27) If (l.24) holds, then applying Ua to the inequality at the right side of (l.27) gives

(1.28)

0:::; {ab 2a} :::; Ilbll{aba} = o.

Then by the identity (l.15), {bab}2 = {b{ab 2a}b} = 0, and (l.25) follows. Exchanging the roles of a and b shows that (l.25) implies (l.24). Finally, we show that (l.24) and (l.25) together imply (l.26). The identity

(l.29) is readily verified in any special Jordan algebra and thus holds for all Jordan algebras by MacDonald's theorem (Theorem l.13). If {aba} = 0 and {bab} = 0, then by (l.28) we conclude that {ab 2a} = 0, and similarly {ba 2b} = O. Now a 0 b = 0 follows from the identity (l.29). D

1.27. Definition. Let A be a JB-algebra. Positive elements a, bE A are orthogonal if {aba} = {bab} = O. We will write a ..1 b when a, bare orthogonal. 1.28. Proposition. Let A be a lB-algebm. Each a E A can be expressed umquely as a dzfference of posdwe orthogonal ele;n,ents:

Both a+ and a- will be m the norm closed subalgebm C(a,l) genemted by a and 1.

Pmof. The existence of such a decomposition follows at once from spectral theory. To prove uniqueness, we first show that orthogonal positive elements generate an associative sub algebra. Let b, c E A be orthogonal positive elements and let C(b, c, 1) denote the norm closed Jordan

JB-ALGEBRAS

16

subalgebra generated by b, c, and 1. We are going to show this algebra is associative by showing that

bn

(1.30)

0

c m = 0 for all n, m :2: 1.

We first establish the implication

(1.31)

0::; x,y E A

and

x -.l y

=}

xn -.l y for all n:2: 1.

Let 0 ::; x, yEA We may assume without loss of generality that Ilxll ::; 1 and Ilyll ::; 1. Then by spectral theory xn ::; x for all positive integers n, and so 0 ::; {yxny} ::; {yxy} = O. By Lemma 1.26, y and xn are orthogonal. Thus the implication (1.31) follows. Let m and n be positive integers. Applying (1.31) with x = b, and y = c, we conclude that bn -.l c. Applying (1.31) again with y = bn , x = c, and with m in place of n, gives cm -.l bn for m, n :2: 1. By Lemma 1.26, this implies (1.30), which completes the proof that C(b, c, 1) is associative. Now let a = b - c be any decomposition of a into orthogonal positive parts, and let a = a+ - a- be the orthogonal decomposition of a within C(a,l) (which exists by spectral theory). By Proposition 1.12 we have C(b, c, 1) "'! CR(Y) for some compact Hausdorff space Y. Note that a+, a- E C(a, 1) C C(b, c, 1). By the uniqueness of orthogonal decompositions in CR(Y) we must have b = a+ and c = a-. 0 We will call the decomposition a orthogonal decomposdwn of a.

= a+ - a- in Proposition 1.28 the

Quotients of JB-algebras An zdeal of a 10rdan algebra is a linear subspace closed under multiplication by elements from the algebra. We are going to prove that the quotient of a lB-algebra by a norm closed ideal is again alB-algebra. We will imitate the standard proof of the analogous fact for C*-algebras, suitably "lordanized". We begin by constructing an approximate identity.

1.29. Definition. Let 1 be a norm closed ideal in alB-algebra A. An increasing net {v",} in 1 is an mcreasing approxzmate zdentdy for 1 if o ::; Va ::; 1 for all a and lim Ilv", 0 b - bll = 0 for all b in 1. 1.30. Lemma. If a, b are elements of a lE-algebra, then

(1.32) If a

~

(1.33)

0, then

Iia 0 bl1 2 .s

lIallll{bab}lI.

QUOTIENTS OF JB-ALGEBRAS

17

Proo]. We start with the identity

(which follows from Macdonald's theorem (Theorem 1.13)). Since the norm in A is the order unit norm, {ba 2 b} :s; II{ba 2b}111, and combining this with the last identity gives

II{ab 2a}112 = II{ab 2a}211 = II{a{b{ba 2 b}b}a}11 :s; II{ba 2b}11 II{ab 2a}ll· If {ab 2 a}

-=1=

0, then dividing by II{ab 2a}11 gives

II{ab 2a}11 :s; II{ba 2b}ll,

°

and if {ab 2 a} = then this inequality is trivial. The opposite inequality follows by symmetry, so (1.32) holds. To prove (1.33), using the identity (1.29) and (1.32) (and the JB-axioms (1.6), (1.7)) we have

+ {ab 2 a} + {ba 2 b}11 :s; ~lla 0 {bab}11 + ~11{ba2b}11 :s; ~llallll{bab}11 + ~11{ba2b}ll·

II(aob)112 = II(aob)211 = il12ao {bab} (1.34)

Since a ~ 0, then by spectral theory a 2 :s; Iialla, so

II{ba 2b}11 :s; Ilallll{bab}ll· Combining this with (1.34) gives (1.33). 0 1.31. Lemma. Let a, b be mvertible elements of alE-algebra A. If b, then b- 1 :s; a-I.

°:s; a :s;

Proo]. Denote by b- 1 / 2 the positive square root of b- 1 , which exists by spectral theory. Applying Ub - l / 2 to a :s; b gives {b- 1 / 2 ab- 1 / 2 } :s; 1. By spectral theory and Lemma 1.24,

1.32. Lemma. Every norm closed zdeal 1 in a lE-algebra A contains an increasmg approximate identity. Proo]. Let U = {a E l+ I Iiall < I}. We are going to show that U is directed upwards. Define f : [0, 1) -+ [0,00) and g : [0,00) -+ [0, 1) by f(t)

= t(l - t)-l,

g(t) = t(l

+ t)-l.

18

1.

JB-ALGEBRAS

Note that the compositions fog and gof are the identity on their respective domains. By the functional calculus, since f(O) = and g(O) = 0, we may view f as a map from U into J+ and 9 as a map from J+ into U (d. Proposition 1.21). By (1.23) these maps are inverses. Since f(t) = -1 + (1 - t)-l and g(t) = 1 - (1 + t)-l, Lemma 1.31 implies that f and 9 preserve order, and thus are (nonlinear) order isomorphisms. It follows that U is directed upwards since J+ is. Now we show that U is an approximate identity. We will show that limuEu Iia - a 0 ull = for a E J. We may assume that lIall <::: 1. For each n = 1,2, ... define hn(t) = 1 - lin for It I ::::: lin, h(O) = 0, and define h to be linear on [0, lin] and on [-l/n,O]. Then IIhn(a)1I <::: 1, and hn(a) E C(a) (Proposition 1.21), so hn(a) is in U. Since t 2 (1 - hn(t)) <::: lin for t E [-1, 1], by spectral theory

°

°

lIa 2

-

{ahn(a)a}1I = lIa 2

0

(1 - hn(a))11 <::: lin.

Now if u E U and u::::: hn(a) then

{ahn(a)a} <::: {aua} <::: a 2 , so

Thus

limuEU

II a2 -

{aua} II

lIa - a 0

=

0. Finally, by Lemma 1.30,

ull 2= <:::

lIa

(1 -

0

u)1I2

111- ullll{a(l -

= 111 -

ulllla 2 -

u)a}1I

{uau} II

--)

0,

which completes the proof. 0

AI J

If J is a norm closed ideal in a JB-algebra A, then the quotient space is a Jordan algebra and a Banach space with the usual quotient norm

lIa + JII = (We will show in Lemma 1.34 that

inf

bEJ

AI J

lIa + bll· is a JB-algebra.)

1.33. Lemma. Let J be a norm closed ideal in a iE-algebra A. {va} is an increasing approxzmate ident~ty for 1 then

(i) lima 11{(1 - v a )b(1 - va)}11

=

0 for all bEl,

If

QUOTIENTS OF JB-ALGEBRAS (ii) Iia + JII a EA.

=

lima Iia -

Va 0

all

=

19

lima 11{(1 - v a )a(1 - va)}11 for all

Proof. (i) We first establish

=

lim 11{(I- v aJx 2 (1- va)}11

(1.35)

a

0

for x E 1.

By (1.32), and the fact that (l-va)2 :::; I-va (which follows from spectral theory), we have lim 11{(1 - va )x 2(1- va)}11 = lim II{x(l - va )2x}11 a

a

:::; lim II{x(l- va)x}11 a

=

lim 112x

0

a

(x

(1 - va)) - x 2

0

0

(1 - va)11

=0

where we have used the definition (1.13) of the triple product in terms of Jordan multiplication and the definition of an approximate identity. We next observe that

o :::; bEl

(1.36)

=}

b1 / 2 E 1,

and bEl

(1.37)

=}

b+ Eland b- E 1.

These follow from the fact that we can uniformly approximate the functions A I--Y v'>- and A I--Y IAI by polynomials with constant term zero on compact subsets of R (which is a consequence of the Stone-Weierstrass theorem.) Now combining (1.35) and (1.36) gives (i) for 0 :::; b E 1. Applying (1.37) gives (i) for all bEl. (ii) For arbitrary b in 1 and a in A, Va 0 b --> b so lim sup Iia -

Va 0

all = lim sup Iia -

a

a + b-

Va 0

Va 0

a

=

lim sup 11(1 - Va:)

0

(a + b)11

a

:::; Iia

+ bll·

Thus (1.38)

lim sup Iia a

Va 0

all:::; inf Iia bEJ

+ bll

=

Iia

On the other hand (1.39)

Iia

+ 111

:::; lima inf Iia -

Va 0

all·

+ 111.

bll

20

1.

JB-ALGEBRAS

Combining (1.38) and (1.39) gives the first equality in (ii). To prove the second inequality, first note that by linearity of the triple product in each factor, for arbitrary a E A,

so

Iia + .III ::; liminf 11{(1 ex

(1.40)

v a )a(l -

va)}ll·

For the opposite inequality, by (i) and Lemma 1.26, for each b in have lim sup

11{(1 -

va )a(l- va)}11 = lim sup

ex

.I

we

11{(1- va)(a + b)(l- va)}11

ex

::; lim sup

11(1- va)211 Iia + bll

Q

::; Iia + bll· Thus (1.41)

lim sup a

11{(1 -

v a )a(l -

va)}11 ::;

inf

bE]

Iia + bll = Iia + .III·

Combining (1.40) and (1.41) gives the second equality in (ii). 0 1.34. Lemma. If A zs a .IE-algebra and .I zs a norm closed zdeal, then AI.1 lS a JE-algebra wlth the quotzent norm.

Proof. We first show (1.42)

II(a+J)o(b+J)II::;

Ila+Jllllb+JII·

We have

Iia + JII lib + .III =

inf

xE]

Iia + xii yE] inf lib + yll

::>: inf inf

xEJyE]

::>: inf

zEJ

II(a + :r)(b + y)11

II(ab + zll = Ilab + JII

which implies (1.42). Let {v o,} be an increasing approximate unit for J. \i\!e next show (1.43)

QUOTIENTS OF JB-ALGEBRAS

21

We have from Lemma 1.33 and (1.15)

Iia + .f112 = lim 11{(1 - va )a(l - va)}112 a

= lim a

II( {(I - v cx )a(l - va)} )211

= lim 11{(1 -

Va)

a

{a(l - va )2a} (1 - va)}11

<::::

lim 11{(1 - va) {a(111 - vaI1 21)a} (1 - va)}11

<::::

lim 11{(1 - v a )a 2 (1 - va)}11

a

a

= IIa 2 + .fll· The opposite inequality follows at once from (1.42), so we have established (1.43). Finally, we will prove (1.44) Applying Lemma 1.33 gives

which establishes (1.44). 0

1.35. Proposition. Let A and B be lE-algebras and 7r : A -+ B a homomorphism such that 7r(1) = 1. Then 7r(A) 2S a .fE-algebra, 117r11 <:::: 1, and if 7r is 1-1, then 7r 2S an 2sometry from A mto B. Proof. (1.45)

Assume first that 7r is injective. We first show for a E A

a2 0

-¢==?

7r(a) 2

o.

One direction is clear since 7r takes squares to squares. Suppose now that 7r(a) 2 o. Let a = x-y be the orthogonal decomposition of a into positive and negative parts (see Proposition 1.28). Then 7r(a) = 7r(x) - 7r(Y) is an orthogonal decomposition of 7r(a). Since 7r(a) 20, then by the uniqueness of the orthogonal decomposition 7r(Y) = O. Since 7r is injective, then y = 0, so a 2 O. This establishes (1.45). It follows that for each positive scalar .\, -.\1 <:::: 7r(a) S;'\1

-¢==?

-.\1 S; a S; .\1,

22

JB-ALGEBRAS

117f( a) I = Iia II. Thus 7f is an isometry. It follows in this case that 7f(A) is complete and thus is closed in B, so in particular 7f(A) is a JB-subalgebra of B. In the general case, we factor 7f through All. By Lemma 1.34, All is a JB-algebra. By definition the quotient map A ----> All has norm at most 1, and so by the first paragraph the composition has norm at most 1, and the image is a JB-algebra. 0

SO

The requirement in Proposition 1.35 that 7f is unital (i.e., 7f(1) = 1) is not necessary. Let p = 7f(1). Then p2 = P and we will see in Proposition 1.43 that {pBp} is a JB-algebra with identity p. Thus we can apply Proposition 1.35 to 7f : A ----> {pBp}. Projections and compressions in JB-algebras 1.36. Definition. Let A be a JB-algebra. A pmJectwn is an element p of A satisfying p2 = p. A compresswn on A is a map Up for a projection p. We will now develop some properties of compressions. Let p be a projection in a JB-algebra A. By the definition (1.13) of the triple product, using p2 = p: (1.46)

Upa

= {pap} = 2p 0 (p 0 a) - p 0 a.

1.37. Lemma. Let a be a posztwe element and let p be a proJectwn zn a lB-algebm A. Then {pap} = 0 zJJ p 0 a = O. Proof. If {pap} = 0, then Lemma 1.26 implies po a if po a = 0, then (1.46) implies {pap} = O. 0

= O. Conversely,

If p is a projection in a JB-algebra A, then we denote the projection 1 - p by p'. We record for future use: (1.47)

p

0

a

= ~ (a + {pap} - {p' ap/} ),

which follows directly from (1.46). 1.38. Proposition. Let p be a pmJectzon zn a lB-algebm A. Then and

IlUp I ::; 1, ( 1.48)

If a

E

(1.49)

A is posztwe, then

PROJECTIONS AND COMPRESSIONS IN JB-ALGEBRAS Proof. We can assume pic 0. Since Ilpll Thus by Theorem 1.25, IIUpl1 = IIpl12 = 1. The identity

23

= IIp211 = Ilp112, then Ilpll = 1.

(1.50) follows from MacDonald's theorem, as does

{(I - b){bab}(l - b)} = {(b - b2 )a(b - b2 )}. Substituting p for b gives (1.48). Now let a E A. If U1_pa = a, then applying Up to both sides gives Upa = 0. Conversely, if Upa = 0, then by Lemma 1.37, po a = 0. Letting pi = 1 - p, we have pi 0 a = a. Combining this and (1.46) gives

°: ;

Up,a

=

2p' o (pi

0

a) - pi

0

a

=

2p'

0

a - pi

0

a

=

a,

which completes the proof of (1.49). 0 If a, b are elements of a lB-algebra A, we write [a, b] for the associated order interval, i.e., [a, b] = {x E A I a ::; x ::; b}. If a is any positive element of A, face( a) denotes the face of A + generated by a, i.e., face(a) 1.39. satisfies

(1.51)

= {y

Lemma.

face(p)

E

A I

°: ; y ::; Aa for some A:::- O}.

Let A be a lB-algebra.

= im +Up,

and

Every projection pEA

face(p) n [0,1]

=

[O,p].

°:;

Proof. If a E face(p) then a ::; AP for some A E R. Then Up,a = 0, so by (1.49) a = Upa E im +Up . Thus face(p) C im +Up . Now suppose a E im +Up. Then a = Upa ::; Up(llaI11) = Iiall p, so a E face(p). Thus face(p) = im +Up . To prove the second statement of (1.51), let a E face(p) n [0,1]. Then a E im Up, so a = Upa -:; Up1 = p. Thus face(p) n [0,1] C [O,pJ. The

opposite inclusion is evident. 0 1.40. Proposition. The extreme pomts of the posztwe part [0,1] of the unit ball of a lB-algebra A are the projectwns. Proof. Let a be an extreme point of [0,1]. By spectral theory, a - a 2 and a 2 are in [0, 1]. Since

24

JB-ALGEBRAS

it follows that a = a 2 . Conversely, let p be a projection in A, and suppose (1.52)

with

p=Ab+(l-A)c

b,cE[O,l]

°

and < A < 1. Then band c are in face(p) n [0,1] = [O,p] and so b ~ P and c ~ p (Lemma 1.39). Therefore by (1.52) b = c = p, and thus p is an extreme point of [0, 1]. 0

We will say a positive linear functional on a JB-algebra A with value 1 at 1 is a state, and will refer to the set of all states as the state space of A. This is consistent with the notion of state for order unit spaces discussed in [AS], cf. (A 17). Each positive linear functional is (norm) continuous, with norm equal to its value at 1 (cf. (1.11)). We denote the space of all continuous linear functionals on A by A *. We observe for future reference that for each state CJ on a JB-algebra A, the map (a, b) f---> CJ(a 0 b) is a symmetric positive semi-definite bilinear form, and thus we have the Cauchy-Schwarz inequality (1. 53) and (taking b = 1) (1.54) We now explore some properties of compressions related to states. If T : A ----> A is a continuous linear operator, then T* : A * ----> A * denotes the adjoint map, defined by (T*CJ)(a) = CJ(Ta) for a E A and CJ E A *.

p'

1.41. Proposition. Let p be a pm)ectwn m a lB-algebm A, let p, and let CJ be a state on A. Then 11U; I ~ 1, and

= 1-

(1.55)

CJ(p) = 1

¢=}

(1.56)

U;CJ =

¢=}

Ul~'CJ

(1.57)

IIU;CJII =

°

U;CJ = CJ,

1

¢=}

Proof. By Proposition 1.38, IIUpl1 We next establish the implication

(1.58)

CJ(p')

=

°

=;.

= CJ, U;CJ = CJ. ~

1, which implies that

U;CJ

=

11U;11

~

1.

CJ.

If CJ(p') = 0, by the Cauchy-Schwarz inequality (1.53), CJ(p' 0 b) = 0 for all bE A. Since p' = 1-p, CJ(pob) = CJ(b). Applying this fact several

PROJECTIONS AND COMPRESSIONS IN JB-ALGEBRAS

25

times (with po a in place of b and then with a in place of b), and using the formula (1.46) for Up, we find

IJ(Upa) = IJ(2p

0

(p

0

a)) - IJ(p

0

a) = IJ(2p

0

a) - IJ(a) = IJ(a),

which proves the implication (1.58). We now prove (1.55). If IJ(p) = 1, then IJ(p') = 0, so U;IJ = IJ follows from (1.58). Conversely, if U;IJ = IJ, then applying both sides to 1 gives IJ(p) = 1. To establish (1.56), note that U;IJ = 0 implies IJ(p) = 0, and so by (1.58) (with p' in place of p) we conclude that U;,IJ = IJ. Conversely, if U;,IJ = IJ, then by (1.48) we find U;IJ = U;U;,IJ = (Up,Up)*IJ = O. To prove (1.57), note that as observed above the norm of the positive functional U;IJ is its value at 1. Thus IJU;1J11 = 1 implies (U;IJ)(l) = 1, and so IJ(p) = 1. Now apply (1.55) to get Ul~1J = IJ. The converse implication is trivial. 0 Property (1.57) has a physical interpretation ("neutrality" of the measurement process) which will be discussed in Part II of this book.

1.42. Definition. Let A be a JB-algebra. A lB-subalgebm of A is a norm closed linear subspace, closed under the Jordan product of A. 1.43. Proposition. If p zs a proJectwn zn a lB-algebm A, then {pAp} = Up(A) is a lB-subalgebm wzth zdentzty p.

Proof. By the Jordan identity (1.15), {pAp} is closed under squaring, and since

then {pAp} follows that that {pAp} Proposition

is closed under Jordan multiplication. Since U; = Up it = {a E A I Upa = a}. Now continuity of Up implies is norm closed. For a E {pAp}, Upa = a and U1_pa = 0 by 1.38, and so by (1.47) po a = a. Thus p is an identity for

{pAp}

{pAp}. 0 1.44. Definition. If p is a projection in a JB-algebra A, then Ap denotes the JB-subalgebra {pAp}. 1.45. Lemma. If p zs a projection zn a lB-algebra A, then a jor a E Ap and bE A p"

0

b= 0

Proof. Since A = A+ - A+ (Proposition 1.28), by the positivity of Up and Up' it follows that Ap and A p' are positively generated. Thus

JB-ALCEBRAS

26

to establish the lemma we can assume 0 :::; a and 0 < b. We will show {aba} = 0; by Lemma 1.26 this will imply a 0 b = O. Since 0 :::; b :::; Ilb111, applying Up' gives 0 :::; b :::; Ilbllp'. Then by positivity of the map Ua ,

0:::; {aba} :::; Ilbll{ap'a}. Since p is the identity of the Jordan algebra A p , then p' 0 a = O. From the definition of the map U a in (1.13), {apia} = 0, which implies {aba} = O. D

Operator commutativity We want to describe the Jordan analog of elements commuting in an associative algebra.

1.46. Definition. Let A be a JB-algebra. Ta denotes Jordan multiplication by an element a. Elements a and b are said to operatoT commute if TaTb = TbTa' It is easily verified that in a special Jordan algebra, elements that commute in the surrounding associative algebra will operator commute in the Jordan algebra.

1.47. Proposition. Lei A be a iE-algebra, let a E A, and let p be a pTOJectwn m A. Then the followmg aTe equwalent:

(i) a and p operatoT commute, (ii) Tpa = Upa, (iii) (Up + Up,)a = a, (iv) p and a are contamed m an assocwtwe subalgebra. Proof. (i) =} (ii) Assume a and p operator commute, so that Ta commutes with Tp. By the definition (1.13), Up = 2T; - T p, so Ta commutes with Up. Thus

(ii)

=}

(iii) If Tpa

=

Upa, then by the expression (1.47) for Tp

This rearranged gives (iii). (iii) =} (i) The defining Jordan identity a 2 0 (b 0 a) = (a 2 0 b) 0 a can be restated in the form [Ta2, Tal = 0, where the brackets denote the commutator. This is equivalent to the following identity, called the linearized

27

OPERATOR COMMUTATIVITY Jordan axzom:

To derive this, in the identity [Ta2, Tal = 0 replace a by b + Al C + A2 d and equate the coefficients of AIA2 in the resulting polynomials. For more details, see the analogous argument in the proof of Lemma 1.2. If we take b = C = P in the linearized Jordan axiom, this becomes (1.60) Note that by (1.60), (1.61) i.e., if pod = d, then p and d operator commute. Assume (iii). Let r = Upa and s = Up,a. Then a = r + s, with rEAp and s E A p'. Since p is the identity for Ap and pi for A p' (Proposition 1.43), then po r = r and pi 0 S = s. By (1.61), p operator commutes with r, and pi operator commutes with s. Since p = I-pi, then p also operator commutes with s, so p operator commutes with a = r + s, proving (i). (iii) ¢o} (iv) By Proposition 1.43, p is the identity for the JB-subalgebra Ap. Hence p and Upa generate an associative subalgebra of Ap. Similarly pi and Up,a generate an associative subalgebra of A p'. For x E Ap and Y E A p' we have x 0 Y = 0 (Lemma 1.45). It follows that p, Upa, pi, and Up,a generate an associative subalgebra B. By (iii), a and p are contained in the associative subalgebra B. Conversely, if p and a are contained in an associative subalgebra, adjoining 1 and taking the norm closure gives a subalgebra isomorphic to CR(X) (cf. Proposition 1.12). Then (iii) follows at once. 0

1.48. Lemma. Let a be an element of a JE-algebra A and p a projection m A. Then (1.62)

a ::::: 0 operator commutes wdh p

¢o}

Upa :::; a.

Proof. If a ::::: 0 and p is a projection which operator commutes with a, then Upa + Up,a = a, so Upa :::; a. Conversely, if a ::::: 0 and Upa :::; a, then a - Upa is a positive element such that Up(a - Upa) = o. By (1.49), Up,(a - Upa) = (a - Upa). Thus Upa + Up,a = a and so p and a operator commute by Proposition 1.47. 0

We now observe that operator commutativity and ordinary commutativity coincide in the context of Jordan operator algebras. Recall that a JC-algebra is a norm closed Jordan sub algebra of B (H)sa.

28

1

JB-ALCEBRAS

1.49. Proposition. If a, b are elements of a .fC-algebra A, then a, b commute zff a, b operator commute.

Proof. Let A c 13 (H)bd be a J C-algebra. If a and b commute, it is straightforward to verify from the definition of the Jordan product that ao(box) = bo(aox) for all X, i.e., that a and b operator commute. Conversely, suppose that a and b operator commute. Let Ta denote Jordan multiplication by a on 13 (H)"a. Then one easily verifies the following identity relating commutators of operators and elements:

Thus if a and b operator commute, then [a, b] commutes with every element in A, and thus also with every element of the C*-subalgebra A of 13(H) generated by A. Thus [a, b] is central in A. Let 7r be an irreducible representation of A. Then 7r([a, b]) = [7r(a),7r(b)] is central in 7r(A), and thus is in the commutant of 7r(A). By irreducibility, the commutant equals C1 (A SO), so 7r([a, b]) is a multiple of the identity. But it is well known that the identity is not equal to the commutator of bounded operators ([SO, 3.2.9]), so we must have 7r([a, b]) = O. Since this is true for every irreducible representation of A, and irreducible representations of A separate elements of A (A S2), then [a, b] = 0, and so a and b commute. (An alternative conclusion to this proof can also be based on the Kleinecke-Shirokov theorem [63, p. 12S], which says that a commutator of two elements of a Banach algebra that commutes with one of the elements is quasi-nilpotent, i.e., its spectral radius is zero. Since z[a, b] is self-adjoint, if it is quasi-nilpotent it must be zero.) 0 1.50. Corollary. If A zs a C*-algebra, then the .fE-algebra Asa zs assocwtzve zff A zs abelwn.

Proof. If A is abelian, then the Jordan product coincides with the ordinary product and thus is associative. If Aba is associative, then by associativity and commutativity of the Jordan product a

0

(b

0

c) = (b

0

c)

0

a = b 0 (c 0 a) = b 0 (a

0

c)

for all a, b, c E Asa. Thus TaTb = TbTa for all a, b E A"" i.e., all elements of Aba operator commute. By Proposition l.49 all elements of Aba commute. It follows that A is abelian. 0 1.51. Definition. The center of a JB-algebra A consists of all elements that operator commute with every element of A. An clement Z E A is central if it is in the center. We will usually denote the center of A by Z(A) or just by Z.

OPERATOR COMMUTATIVITY

1.52. Proposition. The center Z of a fE-algebra A fE-subalgebra contammg the identdy 1.

Z2

29 lS

an assocwtwe

Proof. To see that Z is closed under the Jordan product, let be in Z and x E A. Then

Zl

and

so TZIOZ2 = TZI T Z2 · Since TZI and TZ2 commute with every T x , then so does their product TZI OZ2. Hence Zl 0 Z2 operator commutes with every element of A, and thus is in the center. Finally, Z is associative because for Zl, Z2, and Z3 in Z:

By continuity of Jordan multiplication, Z is norm closed. 0 For Z in the center of a JB-algebra A and a E A, we will write za or az instead of z 0 a. (As motivation, note that in a Jordan operator algebra context, by Proposition 1.49, z and a will commute so the Jordan product z 0 a will coincide with the usual associative product za.) We then have for a and b in A and a central element z, (1.63)

z(a

0

b)

= (za) 0 b = a 0 (zb),

and (1.64)

If a is an arbitrary element of A, and (1.63)

Zl, Z2

are central elements, then by

We will omit the parentheses and just write Zlz2a. If c is a central projection, then from the definition of the triple product together with (1.63) (1.65) so a

f-?

(1.66)

ca is a positive map. By (1.63) we also have

c( a

0

b)

= (ca)

0

(cb).

Thus Uc : a f-? ca is a Jordan homomorphism. Recall that the center of a C*-algebra A consists of those elements in A that commute with all elements in A.

30

JB-ALCEBRAS

1.53. Corollary. If A lS a C*-algebra, then the center of the fEalgebra Asa conslsts of the self-adJomt elements of the center of A. Proof. Since the self-adjoint elements span A, an element z = z* is in the center of A iff z commutes with all self-adjoint elements of A. By Proposition 1.49, this occurs iff z operator commutes with all elements in the JB-algebra A sa , i.e., iff z is in the center of the JB-algebra Asa. 0

Order derivations on JB-algebras We will now discuss an order-theoretic concept of derivation which was introduced by Connes [36]. We defined it in the general context of ordered Banach spaces in [AS, Chpt. 1], but for the reader's convenience we will give the definition here for JB-algebras. We begin with some motivating remarks. Derivations occur in many different contexts. What is common for the various notions of a derivation 5 is the fact that 5 generates a oneparameter group of maps e t8 which preserve the structure under study. In many algebraic contexts where the relevant structure is a bilinear product, this is equivalent to the Leibniz rule, which is often taken as the definition of a derivation. vVhen the relevant structure is order-theoretic, the Leibniz rule is not meaningful, and so we are led to the following definition.

1.54. Definition. A bounded linear operator 5 on a JB-algebra A is called an order derwatwn if e t8 (A+) C A+ for all t E R, or which is equivalent, if {et8hER is a one-parameter group of order automorphisms. 1.55. Proposition. A bounded lmear operator' 5 on a fE-algebra A an order derwatwn lff the followmg lmpllcatwn holds for all a E A + and P E (A*)+ :

lS

(1.67)

p(a) = 0

=}

p(5a) = O.

Proof. This is an immediate consequence of (A 67). 0

If A is the self-adjoint part of a C*-algebra A, then for each dE A we define a linear operator 5d on A by (1. 68) for x E A. Note that 5{/(x) = (5 d (x))*. If A is the self-adjoint part of a von Neumann algebra M, then by (A 183) the order derivations are exactly the maps 5{/ for dE M. Recall that we denote the Jordan multiplication determined by an element b of a JB-algebra A by T b •

ORDER DERIVATIONS ON JB-ALGEBRAS

1.56. Lemma. If A order derzvatzon.

1,S

31

a fE-algebra, then for each b E A, Tb IS an

Proof. Suppose p E (A*)+, a E A+, and p(a) = O. By the CauchySchwarz inequality for JB-algebras (1.53),

(1.69) By spectral theory, a 2 ~

By Proposition 1.55,

n

Iialla.

Now it follows from (1.69) that

is an order derivation. D

1.57. Definition. An order derivation 5 on a JB-algebra A is selfadjoint if 5 = Ta for some a E A, and it is skew-adJoznt (or just skew) if 5(1) = O. We will see in the next chapter that the skew order derivations on a JB-algebra are exactly the fordan derzvatzons, i.e., the bounded linear operators 5 which satisfy the Leibniz rule (1. 70)

5(a

0

b)

=

(5a)

0

b+a

0

(5b).

Note that not all order derivations on a JB-algebra satisfy the Leibniz rule. In fact, (1.70) implies 5(1) = 0, and this is not true for any self-adjoint order derivation Ta with a i o.

1.58. Proposition. Let A be a fE-algebra WIth state space K and let 5 be an order derzvatzon on A. FOT" every t E R, let at = e t8 and let a; be the dual map defined on A * by (a; p) = p( at (a)) for pEA * and a E A. Then the followzng are eqmvalent:

(i) 5 IS skew, (ii) at(1) = 1 for all t, (iii) a;(K) c K for all t. Proof. (i) => (ii) Use the exponential series for e t8 . (ii) {o} (iii) This follows immediately from the definition of states and of an order derivation. (ii)=>(i) This follows by differentiating the equality exp(t5)1 = 1 at t = O. D

Note that (iii) of Proposition 1.58 says that an order derivation 6 of a JB-algebra A with state space K is skew iff the members of the dual one-parameter group t f--+ exp(t6*) are affine automorphisms of K.

32

JB-ALCEBRAS

1.59. Proposition. The set D(A) of order derwations of a iEalgebra A ~s a norm closed real lmear space closed under the L~e product

[15 1,152 ] = 15 115 2 ~b2b1' Proof. By (A 68), the set of order derivations is a real linear space, which is closed under the Lie product. It is norm closed by Proposition 1.55. D

1.60. Proposition. Every order derzvatwn 15 on a iE-algebra A can be decomposed unzquely as the sum of a sclf-adJomt and a skew derzvatwn, namely 15 = Ta + 15' where a = 15(1) and 15' = 15 ~ Ta. Proof. By the definition above, 15'(1) = 15(1) ~ Ta(l) = a ~ a 01 = 0, so 15 = Ta + 15' is a decomposition of the desired type. If 15 = + 15" is another such decomposition, then evaluation at 1 gives 15(1) = b 01 = b, so b = a. Therefore the decomposition is unique. D

n

We will investigate skew order derivations further in the next chapter on JBW-algebras. Later, when we study conditions that guarantee that a JB-algebra is the self-adjoint part of a C*-algebra, a key role will be played by skew order derivations.

Appendix: Proof of Proposition 1.3 Proposition 1.3. Let A be a (not necessarzly assocwtzve) commutatwe algebra over R. Then A lS power associatzve ~ff ~t sat~sfies the ~dentdy x(xx 2 ) = x 2x 2 or the Imearzzed verswn (1.4). Proof. Assume A satisfies x(xx 2 ) = X 2X 2 . To prove A is power associative, we follow the proof in [3]. Recall that powers of x are defined recursively: Xl = x and x"+l = xnx. We must prove that (1. 71) We will prove this by induction on n = J + k. By the definition of powers and (1.3) this holds for n :::; 4. Suppose that it holds for J + k < n where n ?: 5. Substituting Xl = xC>, X2 = x(3, X3 = x', X4 = x 8 into (1.4) where a + /3 +, + 15 = nand 1 < a, /3",15 gives 6(xC>:rn~"

(1. 72) Substituting a (1. 73)

+ x(3xn~/,J + x'xn~, + x8xn~8) = 8(x,,+(3x,+8 + x"+'x/3+ 8 + x

= /3 = , = 1,

b

=

n

~

L

,+8 x (3+,).

3 into (1.72) and simplifying gives

APPENDIX PROOF OF PROPOSITION 1.3

For n

=

33

5 this reduces to

(1.74) Thus (1.71) holds for J + k :S 5 and so we will assume hereafter that n :2': 6. Substituting Q = 2, (3 = I = 1, J = n - 4 in (1.72) gives

(1. 75) Substituting (1.73) into (1.75) gives

(1.76) Next substitute

Q

= (3 = 2, 1= 1, J = n - 5 in (1.72); this gives

(1. 77) Substituting

Q

= 3, (3 = I = 1, J = n - 5 in (1.72) gives

(1.78) Combining (1.77) and (1.78) gives

(1. 79) Now substituting (1.73) and (1.76) into (1.79) gives

(1.80) Now we complete the proof that (1.71) holds for J

+k = n

by establishing

(1.81) by a separate induction on k for this fixed n. Note that by (l.80), the identity (1.81) holds for k = 1,2, n - 2, n - 1. Suppose that it holds for all k :S p + 1 with p :S n - 3. Then substituting Q = p, (3 = I = 1, J = n - p - 2 in (1.72) we get

(1.82) By our latest induction hypothesis we conclude that the right side of (l.82) is equal to 3x n , and thus that xp+2xn-p-2 = xn. By induction, we conclude that (l.81) holds for all k < n. This also completes our original induction and thus finishes the proof. 0

34

.IB-ALGEBRAS

Notes The finite dimensional "formally real" Jordan algebras (which are the same as finite dimensional JB-algebras, d. [67, Cor. 3.1.7 and Cor. 3.3.8]), were introduced and classified by Jordan, von Neumann, and Wigner [75]. A Jordan algebra over R is formally real if

Concrete Jordan algebras of self-adjoint operators on a Hilbert space (JC-algebras and JW-algebras) were studied by Effros and St0rmer in [48], by St0rmer in [124, 125, 126]' and by Topping in [128, 129]. The definition of a JB-algebra was first given by Alfsen, Shultz, and Stormer [8], in a paper which gave a Jordan analog of the Gelfand-Naimark theorem (A 57), cf. Theorem 4.19. The norm axioms that define a JB-algebra are a natural combination of the axioms for a C*-algebra (specialized to self-adjoint elements) together with the additional condition IIa 2 11 ::; IIa 2 + b2 11. This last condition is similar to one used by Segal [114] in a paper on axioms for quantum mechanics, and also appears in Arens' [18] abstract characterization of CR(X). This can be thought of as a way of saying that elements should behave like self-adjoint operators (or like real numbers). In finite dimensions the same role is played by the "formally real" condition of [75]. However, this is not strong enough in infinite dimensions, as illustrated by the sub algebra of the disk algebra consisting of functions that are real-valued on the real axis. (See the discussion following Proposition 1.12). We have defined our JB-algebras to include an identity. If there is no identity, it is always possible to adjoin an identity, and to extend the product and norm to get a JB-algebra, as was shown in [25] and in [120]. Most of the results in this chapter can be found in [8]. We have generally followed the presentation there, except for perhaps a more ordertheoretic flavor. We have made use of the notion of an order unit algebra (Definition 1.9), which was studied in [AS], and which was introduced by lochum and Loupias under the name "Banach power-associative algebra" [69, 70]. The proof that Jordan algebras are power associative relies on Proposition 1.3 (proved in the appendix to this chapter), which is due to Albert [3]. \iVe have followed the proofs in [67] for the construction of an approximate identity (Lemmas 1.30, 1.31, 1.32). The fact that in a JCalgebra, operator commutativity is the same as ordinary commutativity (Proposition 1.49) can be found in [66]. Order derivations were first defined by Connes [36]. The characterization of order derivations in Proposition 1.55 was first proved by Connes in the context of self-dual cones, then generalized to a wider class of ordered Banach spaces by Evans and Hanche-Olsen [50]. Similarly, the result that order derivations form a Lie algebra (Proposition 1.59) is in [36] and [50]. A

NOTES

35

more complete discussion of order derivations is given in [AS, Chpts. 1 and 6]. The notion of self-adjoint and skew order derivations was introduced by Alfsen and Shultz in [12], and is further discussed in Chapter 6. There are important connections of JB-algebras with infinite dimensional holomorphy. For example, there is a 1-1 correspondence between symmetric tube domains and JB-algebras. For a brief discussion, see [67, pp. 92-93]. A more complete discussion can be found in the survey article of Kaup [82]. JB-algebras are in 1-1 correspondence with JB*-algebras. A JB*algebra is a complex Jordan algebra with involution and complete norm such that Ilx 0 yll ~ Ilxllllyll, Ilx*11 = Ilxll, and II{xx*x}11 = Ilx11 3 . The self-adjoint part of a JB*-algebra is a JB-algebra, cf. [67, Prop. 3.8.2], and the complexification of a JB-algebra is a JB*-algebra, a result due to J. D. M. Wright [137]. The authoritative work on JB-algebras is the book of Hanche-Olsen and St0rmer [67]. Another good source of information is Upmeier's survey [130]. For general Jordan algebras, some sources are [33]' [72], and [113].

2

JBW-algebras

JBW-algebras are the Jordan analogs of von Neumann algebras (also known as W*-algebras), and include the self-adjoint part of von Neumann algebras as a special case. In this chapter we will develop basic facts about JBW-algebras. We begin with the definition and the relevant topologies. Then we introduce an abstract notion of range projection, and a spectral theorem for JBW-algebras, derived from the spectral theorem for monotone complete CR(X) (A 39). We then study the lattice of projections of a JBW-algebra, and establish a 1-1 correspondence of projections and O"-weakly closed hereditary subalgebras, and a correspondence of O"-weakly closed ideals with central projections. We show that the bidual of a JBalgebra is a JBW-algebra. Then we use the bidual to prove that JB-algebras are the same as commutative order unit algebras, to show that a unital order isomorphism of a JB-algebra is a Jordan isomorphism, and to show that skew order derivations are in fact Jordan derivations. We prove that every JBW-algebra has a unique predual consisting of the normal linear functionalso Then we develop some basic facts about lW-algebras (O"-weakly closed sub algebras of B(H)sa), and we finish this chapter with an order-theoretic characterization of the maps a f---> {pap} for projections p.

Introduction 2.1. Definition. An ordered Banach space A is monotone complete if each increasing net {b,,} which is bounded above has a least upper bound b in A. (We write b" / b for such a net.) A bounded linear functional 0" on a monotone complete space A is normal if whenever b" / b, then

O"(b,,) --7O"(b). 2.2. Definition. A lEW-algebra is a JB-algebra that is monotone complete and admits a separating set of normal states. In [AS], normal positive linear functionals on a von Neumann algebra were defined first (A 90), and then general normal linear functionals were defined to be linear combinations of normal positive linear functionals. We will see later (cf. Proposition 2.52) that every normal linear functional on a JBW-algebra is the difference of positive normal linear functionals, so the definitions of normal linear functionals on von Neumann algebras and on lBW-algebras are consistent. We will call the set of normal states on a lBW-algebra M the normal state space, and denote it by K. We will let V denote the linear span of Kin M*.

38

2

JBW-ALGEBRAS

The definition above is motivated by Kadison's intrinsic characterization of von Neumann algebras as those C*-algebras whose self-adjoint parts are monotone complete and admit a separating set of normal states (A 95). Thus the self-adjoint part of a von Neumann algebra with the usual Jordan product is an example of a JBW-algebra.

2.3. Definition. Let AI be a JBW-algebra, K its normal state space, and V the linear span of K. We will call the topology on M defined by the duality of M and V the a-weak topology. Thus the a-weak topology is defined by the set of seminorms a f---+ Ip( a) I for p E K. The a -strong topology on !vI is defined by the set of semi norms a f---+ p(a 2 )1/2 for p E K. Thus aD' ----) a a-strongly if p((ao: - a)2) ----) 0 for all p E K. For the special case of the self-adjoint parts of von Neumann algebras, the a-weak and a-strong topologies as defined above coincide with the same terms in the von Neumann algebra context, cf. (A 93). What we have called the a-weak and a-strong topologies on a JBW-algebra are referred to as the weak and strong topologies respectively in [67]. Recall that for an element a of a JB-algebra A, Ta denotes Jordan multiplication and Ua is the map Ua : b f---+ {aba} (cf. (1.13)). If A* is the dual space of A, and T : A ----) A is a bounded linear map, then T* : A* ----) A* is defined by (T*a)(b) = a(Tb) for b E A and a E A*. In the context of JBW-algebras, we will now show that U~ and T; map V into V.

2.4. Proposition. Jordan multzplzcatwn on a JEW-algebra !vI zs astrongly and a-weakly contmuous m each varzable separately, and zs ]omtly a-strongly contZn1WUS on bounded subsets. For each a E AI, Ua zs astrongly and a-weakly contmuous, and T; and U~ map V mto V, and map normal ImeaT functwnals to normal lmear functwnals. Proof. We first prove a-weak continuity of Ua and Ta. Note that if a is invertible, then by Lemma 1.23 and Theorem 1.25 the map Ua is an order isomorphism. Thus if a is invertible, then for each normal state a 011 M, b f---+ a(Uab) is positive and normal, so U~a is a multiple of a normal state. Thus U~ maps V into V when a is invertible. For arbitrary a in !vI, choose A > Iia II. Then by spectral theory Al - a and Al +a are invertible, and from the linearity of the triple product (1.12) in each factor we have

where I is the identity map on A. It follows that every a, and so Ua is a-weakly continuous. The identity relating the maps Ta and Ua ) (2.1)

U~

maps V into V for

INTRODUCTION

39

also follows from the linearity of the triple product in each factor. This implies that each map Ta is a-weakly continuous, and that T; maps V into V. The arguments just given, with trivial changes, also show that U; and T; take normal linear functionals to normal linear functionals. To prove a-strong continuity of Uu , suppose that bn --> 0 a-strongly and fix a in M. Then by definition bn 2 --> 0 a-weakly, so by the first part of this proof {ab a 2a} --> 0 a-weakly. Then using the identity (1.15) and a 2 :::; IIa 2 111 we have

which proves that {abna F --> 0 a-weakly, and so {abaa} --> 0 a-strongly. Thus Ua is (J-strongly continuous, and by (2.1) so is Ta. Finally, we prove multiplication is jointly a-strongly continuous on bounded subsets. Note that from spectral theory, a 4 :::; IIa 2 11a 2 (cf. (1.27)). Thus if {aa} is a bounded net converging a -strongly to zero, and (J is a normal state, then

Therefore the squaring map is (J-strongly continuous at zero on bounded sets. From a 0 b = ~((a + b)2 - a 2 - b2) it follows that multiplication is jointly a-strongly continuous on bounded sets at zero. Finally, if {an} and {ba} are bounded nets with aa --> a and ba --> b a-strongly, then it follows from joint continuity of multiplication on bounded sets at zero and separate continuity (proven above), together with the equality

that an

0

ba

-->

a

0

b a-strongly. 0

2.5. Proposition. Let M be a lEW-algebra. Then

(i) a-strong convergence zmplies a-weak convergence. (ii) A bounded monotone net converges a-strongly. (iii) A monotone net of proJectzons converges a-strongly to a projection. Proof. (i) Let {aa} be a net in A1 which converges a -strongly to a. By the Cauchy-Schwarz inequality (1.53), for each normal state a,

and so aa

-->

a a-weakly.

40

2

JBW-ALGEBRAS

(ii) If {act} is an increasing net with least upper bound a, then for normal states cy by (1.27) we have

Thus act ---> a cy-strongly. (iii) By part (ii), a monotone net converges cy-strongly. Now cy-strong joint continuity of multiplication on bounded subsets (Proposition 2.4) implies that the cy-strong limit of projections will be a projection. D It also follows from joint cy-strong continuity of multiplication on bounded subsets and Proposition 2.5 that for a monotone net of elements {a o,} and any element b in a JBW-algebra we have (2.2)

Uanb

--->

Uab (cy-strongly),

(2.3)

Ua"b

--->

Uab (cy-strongly).

2.6. Corollary. Every cy-weakly contmuous lmear functwnal cy on a lEW-algebra M ~s normal. Proof. If aa / a, then by Proposition 2.5, act cy( aa) ---> CY( a). Thus CY is normal. D

--->

a cy-weakly, so

Later (in Proposition 2.52) we will prove that normal linear functionals are differences of positive normal linear functionals, from which the converse of Corollary 2.6 will follow (Corollary 2.56). 2.7. Proposition. Let B be a cy-weakly closed lordan subalgebra of a lEW-algebra M. Then B has an ~dent~ty and ~s a lEW-algebra. Proof. If {ba} is an ascending net in B that is bounded above, let b E NI be its supremum in 1M. By Proposition 2.5, ba ---> b cy-weakly, so

b E B. Thus b is the supremum of {ba} in B, which proves that B is monotone complete. Since norm convergence implies cy-weak convergence, B is norm closed as well as cy-weakly closed. We next show B has an identity. Let B be the linear span of Band 1. Then B is a norm closed subalgebra of lvI, and thus is a JB-algebra containing B as a norm closed ideal. By Lemma 1.32, B contains an increasing approximate identity {va}. Let v be the least upper bound of this increasing net in M (and then also in B). Then Va 0 a ---> a in norm, while Va 0 a ---> V 0 a cy-weakly. Thus V 0 a = a, i.e., v is an identity for B. Finally, the normal states on M restrict to multiples of normal states on B, and so B has a separating set of normal states. D

INTRODUCTION

41

2.8. Definition. If M is a JBW-algebra, a lEW-subalgebra is a fJweakly closed Jordan subalgebra N of M. (Note that by Proposition 2.7 N will itself be a JBW-algebra.) Recall that if p is a projection, then U; = Up, and Up Up' = Up,Up = 0 (Proposition 1.38).

2.9. Proposition. If p zs a pro}ectwn m a lEW-algebra M, then Mp = UpM zs a lEW-subalgebra of M wdh zdentdy p. Proof. By Proposition 1.43, Mp is a JB-subalgebra of M with identity p. It follows that Mp = {a E M I Upa = a}. Now fJ-weak continuity of Up implies that Mp is fJ-weakly closed. D

2.10. Mp

Corollary.

If p zs a pro)ectwn m a lEW-algebra M, then

+ M p' = im (Up + Up') zs a lEW-subalgebra of M.

Proof. Since MpoMp' = {O} (Lemma 1.45), then Mp+Mp' is a Jordan subalgebra of M. Thus Mp+Mp' =imUp+imUp' ={aEMI (Up+Up,)a=a}, so this subspace is fJ-weakly closed as claimed. 0

If a is an element in a JBW-algebra lvI, we will denote by W (a, 1) the fJ-weakly closed subalgebra generated by a and 1. 2.11. Proposition. If B is an assocwtwe lordan subalgebra of a lEW-algebra M, then the fJ-weak closure of B zs an assocwtwe lEWalgebra which zs zsomorphzc to a monotone complete CR(X). In partIcular, this holds for W (a, 1) . Proof. By separate fJ-weak continuity of multiplication, the fJ-weak closure of B is a fJ-weakly closed subalgebra of M, and thus by Proposition 2.7 is a JBW-algebra. A simple argument using separate continuity of Jordan multiplication shows that the fJ-weak closure of B is associative. By Proposition 1.12, an associative JBW-algebra is isomorphic to CR(X) for some compact Hausdorff space X. Since a JBW-algebra is monotone complete, so is CR(X). D

2.12. Corollary. If a pro}ectwn p m a lEW-algebra M operator commutes with an element a, then p operator commutes wzth every element in W(a,l). Proof. By Proposition 1.47, p operator commutes with an element x iff x is in Mp + M p" If p operator commutes with a, then by Corollary

42

2

JBW-ALGEBRAS

2.10, Mp + M p' is a O'-weakly closed Jordan sub algebra of M containing a and 1. It therefore contains W (a, 1). Thus for every x E W (a, 1), x is in Mp + !vIp" and so p operator commutes with x. 0 Orthogonality and range projections

If E is a subset of a set X, then XE will denote the characteristic function of E, i.e., XE(X) = 1 for x E X and XE(X) = 0 for x E X \ E. If p is a projection in a JB-algebra A, recall from Lemma 1.39:

We will use this frequently hereafter without further reference. Recall also from Proposition 1.28 that each a E A has a unique orthogonal decomposition a = a+ - a- where a+, a- are in A+.

2.13. Proposition. For every element a of a lEW-algebra M there zs a unzque pro)ectzon p zn W(a, 1) n face(a+) (O'-weak closure) such that Upa 2: a and Upa 2: O. Furthermore, p zs the smallest pro)ectzon zn M such that Upa+ = a+. Proof. By Proposition 2.11, we can identify W(a,l) with a monotone complete CR(X). Let E = {x E X I a(x) > O}. By (A 38), E is closed and open, a 2: 0 on E, a::; 0 on X \ E, and XEa+ = a+. Let p = XE; then p E CR(X) and p2 = p, so p is a projection in CR(X) such that Upa+ = a+. Since a 2: 0 on E, then pap 2: 0, so Upa 2: O. Since a ::; 0 on X \ E, then pap = XEa 2: a, so Upa 2: a. Furthermore, by (A 38) there is an increasing sequence {an} in W (a, 1) n face ( a+) with supremum p (in W(a,l)). Let b be the supremum of {an} in M. Then an --+ b O'-weakly by Proposition 2.5, and so b is in W(a,l). Thus b = p and an --+ p O'-weakly, which proves that p E W(a, 1) n face(a+). To prove uniqueness, let q be any projection in W(a, l)nface(a+) such that Uqa 2: a and Uqa 2: O. Let q = XF for Fe X, and set G = X \ F. Then

and

so a

2: 0 on F

and

a::; 0 on

G.

ORTHOGONALITY AND RANGE PROJECTIONS

43

By (A 38)(i), E c F, and so p::; q. On the other hand, Upa+ = a+, so a+ E face(p). Thus q E face(a+) C face(p) C im Up.

Then q = Upq ::; Up1 = p, so q ::; p. Therefore p = q, proving uniqueness. Finally, let q be any projection in M such that Uqa+ = a+. Then a+ E im+Uq = face(q), so

P E face(a+) and as above p < q follows. Up a+ = a+. 0

C

face(q)

C

im Uq

Thus p is the least projection such that

The conditions Upa 2 a and Upa 2 0 above will appear again in our axiomatic treatment of spectral theory in Part II.

2.14. Definition. Let a be a positive element of a JB\N-algebra. The smallest projection p such that Upa = a is called the range pTO]ectwn of a and is denoted r(a). For later reference, we collect some elementary facts about r( a).

2.15. Proposition. Let a be a posltwe element of a lBW-algebra !v! with normal state space K. Then the range pro]ectwn r( a) satlsfies the following propertzes. (2.4)

r(a) E W(a, 1) n face(a).

(2.5)

a::; jjajj r(a).

(2.6)

If (J E K, then

(2.7)

a::; b

(2.8)

(J(a) =}

r(a)::;p=p2

=

0

{==}

(J(r(a))

= o.

r(a)::; r(b). =}

Upa=a.

PTOOf. Property (2.4) follows from the definition of r(a) and Proposition 2.13. Since a ::; jjajj1, applying Ur(a) to both sides gives (2.5). Property (2.6) follows from (2.4) and (2.5). Next, suppose a ::; b. By (2.5), a::; b::; jjbjjr(b), so a E face(r(b)) = im+Ur(b). Thus Ur(b)a = a, so r(a) ::; r(b), which establishes (2.7). Finally, if p is a projection such that r(a) ::; p, then a E face(r(a)) C face(p), so Upa = a. 0

Let M be a JBW-algebra, and let W be a (J-weakly closed sub algebra of a JBW-algebra M containing the identity of M. If 0 ::; a E W, then W(a,1) C W, so by (2.4) r(a) E W. Thus r(a) will also be the least projection p in W such that Upa = a, i.e., the notions of range projection in Wand in M coincide.

44

2

JBW-ALGEBRAS

Recall that positive elements a, b are orthogonal if {aba} = 0, and this is equivalent to {bab} = 0 by Lemma 1.26. We now characterize this notion in terms of range projections.

2.16. Proposition. Let a, b be pos7,twe elements of a .IE W-algebra M, Then a ~ b zJj r(a) ~ r(b). Proof. If r(a) ~ r(b), then by definition Ur(a)r(b) = 0, so using (2.5) gives

Thus Ur(a)b = 0, which is equivalent to Ubr(a) = 0 (Lemma 1.26). Hence

so Uba = 0, i.e., a ~ b. If a ~ b, then Uab = 0 implies that Ua annihilates face(b), and therefore Uar(b) = 0 by (2.4). Thus a ~ b implies a ~ r(b), and the same implication applied to the pair r(b), a shows r(b) ~ r(a). D Now observe that if a and b are positive elements of a JBW-algebra such that a ~ b, then by Proposition 2,16, r(a) ~ r(b), so r(a) ~ b, i.e., Ur(a)b = O. This implies Ur(a),b = b (where r(a)' = 1-r(a)), cf. equation (1.49). We record this observation for later reference:

(2.9)

0

:s;

a, b

and

a ~ b

=}

Ur(a)b

= 0 and

Ur(a),b

=

b.

2.17. Corollary. The normal states on a .lEW-algebra !vI determme the order and the norm on !vI, z. e., for each a E AI,

(2.10) (2.11 )

a ~ 0

<¢==}

u(a) ~ 0 for all u E K, and

Iiall = sup{ lu(a)1 I u E K}.

Proof. Let a E AI and suppose that u(a) ~ 0 for all normal states u. Let a = a+ - a- be the orthogonal decomposition of a (Proposition 1.28). Suppose that a- is not zero. Since by the definition of a JBW-algebra the normal states separate points, there is a normal state w such that w(r(a-)) "# o. By (2.6) w(a-) "# O. Let T = U:(a-)w. Then T(l) = (U:(a-)w)(l) = w(r(a-)) "# 0, so T is a non-zero positive u-wcakly continuous linear functional on lvI. By Corollary 2.6, T is a multiple of a normal state. However, Ur(a-)(a+) = 0 (cf. (2.9)), so

ORTHOGONALITY AND RANGE PROJECTIONS

45

This contradicts our assumption that all normal states were positive on a, and thus we conclude that a- = 0, and therefore a ~ o. This proves (2.10). We will now prove (2.11). Since the norm is the order unit norm (cf. (1.10)), and the normal states determine the order, then liT(a)1 :::; 1 for all

iT E

K

¢==}

-1:::; a :::; 1

¢==}

Ilall:::; 1,

which completes the proof. 0 Note that by Corollary 2.17, M and V are in separating order and norm duality (A 21), and the positive cone M+ is iT-weakly closed. Thus for each projection p in M, face(p) = im+Up = imUp n M+ is iT-weakly closed. If 0 :::; a, then r(a) E face(a) (cf. (2.4)) and a E face(r(a)), so face(r(a))

(2.12)

=

face(a)

(iT-weak closure).

2.18. Proposition. Let p, q be pro]ectzons m a lEW-algebra Ai. The Jollowmg are equwalent.

(i) p.l q, (ii) (iii) (iv) (v)

poq = 0, p:::; q' (where q' p+q:::;I, UpUq = o.

= 1 - q),

Proof. (i) =} (ii) This is part of Lemma 1.26. (ii) =} (iii) If po q = 0, then (p + q)2 = P + q, so P + q is a projection. Then p + q :::; 1 so p :::; 1 - q. (iii) .;=;. (iv) This is clear. (iii) =} (v) If p :::; 1- q, then q:::; 1- p so Upq :::; Up(l - p) = o. Then for all a ~ 0,

Since every element of M is a diflerence of positive elements, UpUq = 0 follows. (v) =} (i) If UpUq = 0, then Upq = 0, which by definition means p.l q. 0 We will frequently make use of the equivalence of (i) and (iii) above Without reference.

46

2

JBW-ALGEBRAS

Spectral resolutions

2.19. Proposition. If M 2S a lEW-algebra, a 2S any element of M, and p 2S a proJectwn m M, then (2) and (n) are equwalent:

(i) Upa::::> a and Upa ::::> 0, (ii) Upa::::> 0, Up,a S; 0, and p operator commutes wdh a. For each a m Iv!, r(a+) 2S the smallest pTOJectwn p sat2sfymg (2) and (n) Proof. (i) =? (ii) Assume p satisfies (i). Then Upa - a ::::> O. Since Up(Upa - a) = 0, we can use Proposition l.38 (with Upa - a in place of a

in (l.49)) to conclude that

Hence -Up,a = Upa - a. From this we conclude that a = Upa + Up,a, so p operator commutes with a (Proposition l.47). Also Up,a = a - Upa S; 0, so (ii) holds. (ii) =? (i) Suppose (ii) holds. Then Upa = a - Up,a ::::> a, so (i) holds. Finally, by Proposition 2.13, p = r(a+) satisfies (i). Let q be any projection satisfying (i) and (ii). Then Uqa is positive and is fixed by Uq, so r(Uqa) S; q and similarly r( -Uq,a) S; q'. Thus Uqa and -Uq,a are orthogonal (Proposition 2.16). Since a operator commutes with q, then a = Uqa - (-Uq,a) (Proposition l.47), so this equation must coincide with the unique orthogonal decomposition of a (Proposition l.28). Thus a+ = Uq(a), so r(a+) S; q. D We are now ready to state the spectral theorem. For each finite increasing sequence I = Po, AI, .. . ,An} with Ao < -ilall and An > Iiall, define

Ihll

=

maxPi - Ai-d· ,

2.20. Theorem. (Spectral Theorem) Let Iv! be a lEW-algebra and let a E Iv!. Then there 2S a umque fam2ly {e>.} >'ER of pTOJectwns such that

(i) each e>. operator commutes wdh a, (ii) Ue).a S; Ae>., and U~" a ::::> Ae~, (iii) e>. = 0 for A < -ilall, and e>. = 1 for (iv) e>. S; ep for A < IL, (v) e = e>. (greatest lower bound m Iv!).

1\

IL>>'

A>

Iiall,

"

The family {e>.} is given bye>. = 1 - r((a - A1)+), and is contained m W(a,l). For each fimte increasing sequence I = {Ao, AI, ... , An} with

SPECTRAL RESOLUTIONS

>'0 < -ilall

and An >

Iiall,

47

define n

(2.13)

s,

=

2:= >.,(e,\, -

e'\,_l)·

;=1

Then

(2.14)

lim Ihll-->O

lis, - all = O.

Proof. We will apply the spectral theorem for monotone complete GR(X) (A 39) to W(a,l). Note that by Proposition 2.5, increasing nets in M converge a-weakly to their least upper bounds, so these least upper bounds are the same whether interpreted in W (a, 1) or in M, and thus (v) is unambiguous. Furthermore, by the remarks following (2.4), the range projection of (a->.l)+ calculated in W(a, 1) and in M coincide, so r((a>.1)+) coincides with r((a - A1)+) as defined in (A 39). Thus by (A 39), the family {e,\} defined in the theorem above satisfies (ii) through (v), and is the unique family in W(a,l) with this property. Since each e,\ is in the associative subalgebra W (a, 1), each e,\ operator commutes with a (Proposition 1.47), which proves (i). Finally, we prove uniqueness. Suppose that {e,\} is any family of projections in M satisfying (i) through (v). If A < /1-, then e,\ ::; e J1 , so e,\ E face(eJ1); therefore Ue"e,\ = e,\. Thus ell and e,\ operator commute by (1.62). Then a and all of {e,\} mutually operator commute. From the definition of operator commutativity, it follows that a together with {e>.} generate an associative subalgebra, whose a-weak closure is then isomorphic to a monotone complete GR(X) (Proposition 2.11). Then by the uniqueness statement in (A 39) we must have e,\ = 1 - r((a - A)+) which completes the proof of uniqueness in !vI. D

We will see in Part II (cf. Theorem 8.64) that a purely order-theoretic proof of the spectral theorem is possible, which relies solely on the following property established in Proposition 2.19: For each a EM, there zs a least pro]ectwn p such that Upa 2:: a and Upa 2:: o.

2.21. Definition. The family {e,\} in Theorem 2.20 is called the spectral resolution for a, and will be denoted by {eU when there is need to indicate the associated element a. We will call the e,\ 's the spectral projections of a.

48

2

JBW-ALGEBRAS

If we define the Stieltjes integral of a with respect to {e).} as the norm limit of the approximating Riemann sums s-y in (2.13), then we have shown

a

=

J

>-de)...

We note for later reference (2.15)

The lattices of projections and compressions We let P(M) (or just P) denote the set of projections of a JBWalgebra M, with the order inherited from M.

2.22. Lemma. If P, q are pTOJectwns m a lBW-algebm lvI, then r(p + q) zs the least upper bound of P and q m the set P of pTOJectwns of

M. Prooj. Let r = r(p + q). Then p + q E im+Ur = face(r), so P and q are in face(r) n [0, 1] = [0, r] (using (1.51)), and thus p <::: rand q <::: r. On the other hand, if e is any projection such that e 2': P and e 2': q, then p and q and then P + q are in face (e) = im + Ue , and so by the minimality of the range projection, r <::: e. Thus r is the least upper bound of P and q in P. 0

Since p f-7 pi = 1 - P is an order reversing bijection, it follows from Lemma 2.22 that P is a lattice. For projections p, q E P, the lattice supremum is denoted by P V q and the lattice infimum by p 1\ q.

2.23. Corollary. If PI, ... ,p" are orthogonal pTOJectwns m a lBWalgebm lvI, then PI V ... V Pn = PI + ... + Pn· PTOOj. By Lemma 2.22, PI V P2 = r(pl + P2) = PI + P2. For z > 2, PI <::: p~ and P2 <::: p~, so PI + P2 = PI V P2 <::: p~. Thus PI + P2, P3, ... , p"

are orthogonal. Now the proof can be completed by induction. 0 For the reader's convenience we recall the definition of a complete orthomodular lattice.

THE LATTICES OF PROJECTIONS AND COMPRESSIONS

49

2.24. Definition. A lattice L is complete if every subset has a least upper bound. L is orthomodular if there is a map p f---7 pi that satisfies

(i)

pI!

= p,

(ii) p::; q implies pi ~ ql, (iii) p V pi = 1 and p A pi = 0, (iv) If p::; q, then q = p V (q Api). The map p

f---7

pi is called the orthocomplementatwn.

Note that (i) and (ii) imply that p f---7 pi is an order reversing bijection, so transforms greatest lower bounds to least upper bounds and vice versa. Thus if (i) and (ii) hold, then

(p V q) I = pi A ql

and

(p A q) I = pi V ql.

2.25. Proposition. The set P of pmJectwns of a lEW-algebra zs a complete orthomodular lattzce, wzth orthocomplementatwn p f---7 pi = 1 - p. If {Pc>} zs an zncreaszng net of proJectwns, then zt converges O"-strongly to its least upper bound zn the lattzce P.

Proof. We will verify the properties (i) through (iv) of Definition 2.24. Properties (i) and (ii) are obvious. (iii) Since p 1.. pI, then p V p' = p + pi = 1 (Corollary 2.23). Then p A pi = (p' V p)' = 11 = 0, which proves (iii). (iv) Suppose p::; q. Then p::; (q')' implies p 1.. ql, so (2.16)

q Ap' = (ql V p)1 = (q'

+ p)' = 1- (q' + p) = q -

p.

Since q A pi ::; pi, then q A pi and p are orthogonal, so (2.16) gives

p V (q A pi) = P + (q A pI) = P + q - p = q. Finally, the set of finite suprema of any set of projections forms an increasing net of projections (indexed by themselves). Any increasing net of projections converges O"-strongly to a projection p (Proposition 2.5), which is necessarily the least upper bound of this net in !vI and thus also in P. Thus P is a complete lattice. D The following was established during the proof above and will be used frequently: (2.17)

p ::; q

=}

q A pi = q _ p.

(For motivation, note that if this is applied to the lattice of subsets of a set, it becomes the statement that if E c F, then F \ E = F n E I , where EI is the set complement of E.)

2

50

JBW-ALGEBRAS

If p and q are elements of an orthomodular lattice, we say p and q are orthogonal if p :::; q' (or equivalently, if q :::; p'). Note that for the lattice of projections in a JBW-algebra, this is equivalent to our previously defined notion of orthogonality for positive elements of a JBW-algebra, cf. Proposition 2.18. We next investigate properties of the maps Up (continuing the study begun in Chapter 1). We will generally be interested in the duality of M and the space V = lin K. Recall that M and V are in separating order and norm duality (Corollary 2.17). By Proposition 2.4 each map maps V into V. In what follows we will view U; as acting on V (rather than M*) and thus interpret im U; to mean U; (V). (In other words, we interpret as the adjoint map with respect to the duality of l'v1 and V.) Note that Up ;::: implies U; ;::: 0, and IIUpl1 :::; 1 implies IIU;II :::; 1. We observe that since M = A1+ - A1+ and V = V+ - V+, and since the maps Up and U; are positive, then

U;

U;

(2.18)

°

. Up = nn . + Up - 1m . + Up 1m

. + U*P . an d 'lnl U*p - 'ln1 + U*p - llll

2.26. Proposition. If p) q aTe pro)ectwns m a lEW-algebra l'vI, then the followmg are equwalent:

(i) p:::; q, (ii) im+Up c im+Uq) (iii) im Up C im Uq, (iv) UqUp = Up) (v) im+U*p C im+U*q' (vi) imU; C imU;, (vii) UpUq = Up. Proof. (i)

=}

(ii) If p :::; q, then face(p)

C

face(q), so

im+Up = face(p) C face(q) = im+Uq. (ii) =} (iii) This follows from (2.18). (iii) =} (iv) This follows from U'i = Uq . (iv) =} (v) If UqUp = Up, then U; = Ur:U,;. Let er E im+U;. Since IIU; II :::; 1 and 11U; II :::; 1, then Ilerll = IIU;erll = IIU;U;erll :::; 11U,;erll :::; Ilerll, so IIU;erll = Ilerll. By "neutrality" of compressions (1.57), this implies U*er = er Thus im+U*p C im+U*q' q' (v) =} (vi) This follows from (2.18). (vi) =} (vii) If im U; C im U;, then U;U; = U; and so UpUq = Up. (vii) =} (i) If UpUq = Up, then Upq' = O. Thus by (1.49), Up,q' = qt. Then q' = Up,q' :'S: Up-l = p', which implies p:'S: q. 0

THE LATTICES OF PROJECTIONS AND COlVIPRESSIONS

51

2.27. Corollary. A pmJectwn P operatoT commutes wzth an element a of a lEW-algebra zfJ p operatoT commutes wzth eveTY spectral pmJectwn of a. PTOOj. Recall that the spectral projections of a are contained in W(a,l) (d. Theorem 2.20). Now let p be a projection which operator commutes with a. Then p operator commutes with everything in W(a,l) by Corollary 2.12, and so p operator commutes with every e.\. Conversely, suppose p is an arbitrary projection that operator commutes with every spectral projection of a. By continuity of Jordan multiplication, p will operator commute with norm limits of linear combinations of the spectral projections, and thus with a by the spectral theorem. 0

2.28. Proposition. If p and q aTe pmJectwns in a lEW-algebra M, then p and q operatoT commute zfJ UpUq = UqUp, and then UpUq = Up/\q and p 1\ q = P 0 q. Pmoj. By the definition (1.13), Up = 2T; - T p, where Tpa = po a for a E !vI. Therefore, if p and q operator commute, then Up and Uq commute. Conversely, if Up and Uq commute, then

By (1.62) P and q operator commute. Assuming p and q operator commute, we next show UpUq = Up/\q. Let a E M+, and b = UpUqa = UqUpa. Then Upb = band Uqb = b, so by the definition of the range projection (Definition 2.14), r(b) <::: p and r(b) <::: q. Therefore r(b) <::: p 1\ q. Thus bE face(r(b)) C face(p 1\ q) = im +Up/\q, so Up/\qb = b. Then substituting b = UpUqa gives

Since p 1\ q <::: p and p 1\ q <::: q, applying Proposition 2.26 ((i) this equation gives

=?

(vii)) to

for all a E M+. Since such elements span lvI, then UpUq = Up/\q follows. Finally, suppose p and q operator commute. By Proposition 1.47 Upq = Tpq, so

This completes the proof. 0

52

2.

JBW-ALGEBRAS

We note for future reference the following implications for projections p and q: (2.19)

p<5,q

(2.20)

pJ..q

=> =>

p operator commutes with q. p operator commutes with q.

To verify (2.19), observe that p <5, q implies p E face(q), so Uqp = p. Then q operator commutes with p by (1.62). If p 1.. q, then p <5, q', so p and q' = 1 - q operator commute by (2.19). Thus p and q operator commute.

2.29. Definition. Elements p and q of an orthomodular lattice are compatzble if there exist orthogonal elements r, 5, and t such that p = r V 5 and q = 5 V t. Note that projections p and q in a JBW-algebra M are compatible as elements of the lattice of projections of NI iff there are projections l' and 5 (necessarily orthogonal) such that p = l' + 5 and q = 5 + t (since l' + 5 = l' V 5 and 5 + t = 5 V t, d. Corollary 2.23). Thus the next result shows that projections are compatible iff they operator commute.

2.30. Proposition. Let p, q be pro)ectwns zn a lEW-algebra lvI. Then p and q operator commute zfJ there are mutually orthogonal pro)ectwn5 r, 5, and t such that (2.21)

p

= l' +

5

and

These pro)ectwn5 are umque, wzth r

q

= 5 + t.

= p A q',

5

= P A q, and t = q A p'.

Proof. Assume (2.21) holds. Since 1', 5, and t are orthogonal, they operator commute with each other (d. (2.20)), and thus p = r + 5 and q = 5 + t operator commute. Furthermore,

= (1' + 5) 0 (5 + t) = po q = P A q, = P - 5 = P - P 0 q = p 0 (1 - q) = P A q', 5

l'

and similarly t = q A pi, so orthogonal projections satisfying (2.21) are uniquely determined. Conversely, if p and q operator commute, let l' = P - P 0 q, 5 = P 0 q, and t = q - p 0 q. Then l' + 5 = p, and 5 + t = q. By Proposition 2.28, l' = po (1- q) = pA q', 5 = pA q, and t = q Api, so all three are projections. Finally, r + 5 +t = p+q Api :S p+p' = 1, so r, 5, t are mutually orthogonal, and this completes the proof. 0 The following formula for the range projection will be very useful later.

THE LATTICES OF PROJECTIONS AND COMPRESSIONS

53

2.31. Proposition. Let !vi be a lEW-algebra, p any pm]ectwn in 0 m M. Then

M, and a 2:: (2.22)

r(Upa)

=

(r(a) V pi) Ap = P - (p A r(a)').

Proof. Below we will repeatedly use (2.23)

e ::; j

=}

j - e

=j

A e'

for projections e and j, which is just (2.17) restated. Note that the rightmost equality in (2.22) follows from (2.23) by taking j = p and e=pAr(a)'. By (2.19) pi operator commutes with pi V r(a), and thus so does p. Therefore by Proposition 2.28, Up'vr(a) and Up commute. Since p' V r(a) 2:: r(a), then Up'Vr(a)a = a (d. (2.8)), so

By the definition of the range projection (Definition 2.14), this implies

r(Upa) ::; pi

V

r(a).

Again by the definition, r(Upa) ::; p, so (2.24) We now prove the reverse inequality. We will make repeated use of the fact that for a projection e and b 2:: 0,

(cf. (1.49)). Let w = r(Upa). Then w ::; p, so w, and then Wi, operator commutes with p. By definition, Uw(Upa) = Upa, so Uw,(Upa) = O. Therefore Uw'/\pa = Uw,Upa = 0, which implies U(w'/\p),a = a. Thus Uwvp,a = a, which implies w V pi 2:: r(a). Now w V p' 2:: r(a) V pi follows. Since w ::; p = (pi)', then w is orthogonal to pi, so we have w

+ p' =

w V pi 2:: r (a) V p'.

This implies w 2:: (r(a) V pi) - pi

= (r(a)

V

pi)

A

p,

where the last equality follows from (2.23) with e = pi and j = r( a) V p'. This is the opposite inequality to (2.24), and thus completes the proof. 0

54

2

JBW-ALGEBRAS

a-weakly closed hereditary subalgebras A Jordan subalgebra B of a JB-algebra A is heredItary if (a E A and b E B with 0 S; a S; b) implies a E B. Note that this just says B+ is a face of A +. Since im + Up = face(p) , then im Up for a projection p is an example of a hereditary subalgebra . 2.32. Proposition. Let A1 be a J BW -algebra. Then there are 11 order preserving correspondences of pm]ectwns of !vI, a-weakly closed faces of M+) and a-weakly closed heredItary subalgebras of !vI, gwen by p

+-->

face(p)

+-->

imUp = {pMp}.

Proof. Since face(p)n[O, 1] = [O,p] (cf. Lemma 1.39), then p +--> face(p) is a 1-1 order preserving correspondence of projections and the faces they generate. We will show these faces are precisely the a-weakly closed faces of M+. Since face(p) = im+Up = Up n M+, then face(p) is a-weakly closed. Conversely, let H be any a-weakly closed face of M+. Let Ho = H n [0, 1]. We have a E H

=}

r(a) E H o ,

since r(a) E face(a) C H (cf. (2.4)). If a, bE H o, then r(a + b) E Ho and as; r(a) S; r(a+b) (and similarly b S; r(a+b)) by (2.5) and (2.7). Thus Ho is directed upwards. Since !vI is monotone complete, the net consisting of all elements of Ho (indexed by themselves) must converge a-strongly to an element p, which is then the largest member of Ho. Since p S; r(p) E HOI then p is a projection. Thus Ho = [O,p] and so H = face(p). Thus we have shown that p +--> face(p) is a 1-1 correspondence of projections and a-weakly closed faces of M+. Next we show that p +--> im Up is a 1-1 correspondence of projections and a-weakly closed hereditary subalgebras. Since im+Up n [0,1] = face(p) n [0,1] = [O,p], the correspondence p +--> im Up is bijective. By the remarks preceding this proposition, im Up is a a-weakly closed hereditary subalgebra. Thus what remains is to show that every a-weakly closed hereditary subalgebra has the form iIll Up for some projection p. Let B be a a-weakly closed hereditary sub algebra of M. By Proposition 2.7, B is a JB-algebra, so B = B+ -B+ (cf. (1.11)). By the definition of the term hereditary, B+ = B n M+ is a a-weakly closed face of !vI + . By the preceding paragraphs, B+ = im+Up for some projection p. Thus B = B+ - B+ = im+Up - im+Up = im U1" which completes the proof. 0 In Chapter 5 we will prove a corresponding result for the normal state space K, by showing that every norm closed face of K is of the form im+U; n K = {a E K I a(p) = I} for a projection p.

55

CENTER OF A JBW-ALGEBRA

Center of a JBW-algebra Recall that the center of a JB-algebra A is an associative JB-subalgebra consisting of all elements that operator commute with every element of A (Proposition l.52).

82

2.33. Definition. An element s of alB -algebra A which satisfies = 1 is called a symmetry.

2.34. Proposition. If s lS a symmetry m a lB-algebra A, then the map Us lS a lordan automorphzsm of A, and U; lS the ldentdy map on

A. Proof. For b E A,

where we have used the identity (l.15). Thus Us preserves squares, and therefore preserves the Jordan product (d. (l.59)), so Us is a Jordan homomorphism. By the identity (l.16) the map is the identity map, since U; = UsU 1Us = U{sls} = l. Thus Us is its own inverse, and therefore is a Jordan isomorphism. 0

U;

Note that if s is a symmetry, then p = ~ (s + 1) is a projection such that s = 2p - 1 = p - p' (where p' = 1 - p). Similarly, if p is a projection then s = 2p - 1 is a symmetry. There is the following useful relationship between Us and Up: (2.25) (Here I is the identity map.) To verify this, note that linearity of the triple product in each factor gives each of the following two equations:

+ {p'ap'} - {pap'} - {p'ap}. = {pap} + {p'ap'} + {pap'} + {p'ap}.

Usa = {(p - p')a(p - p')} = {pap} a = U1a = Up+p,a = {(p+p')a(p+p')}

Adding these two equations gives (2.25). Note also that s = 2p - 1 operator commutes with an element a iff p does, which in turn occurs iff (Up + Up,)a = a (Proposition l.47). By (2.25) this is equivalent to Usa = a. Thus if s is a symmetry, then (2.26)

s operator commutes with a iff Usa

=

a.

lIenee if an element a is central, then Usa = a for all symmetries s. The converse is also true, leading to the following convenient characterization of the center in terms of symmetries.

56

2

JBW-ALGEBRAS

2.35. Lemma. An element a of a lEW-algebra IvI Usa = a for all 5ymmetne5 5 mAl.

l5

central lff

Proof. By Corollary 2.27, an element a is central iff it operator commutes with every projection in M. Since the symmetries are exactly the elements 2p - 1 for projections p, this in turn is equivalent to a operator commuting with all symmetries. By (2.26) this is equivalent to Usa = a for every symmetry 5 of IvI. D 2.36. Proposition. The center Z of a lEW-algebra IvI cwtwe lE W-subalgebra.

lS

an aS50-

Proof. By Proposition 1.52, Z is an associative JB-subalgebra. Since for each symmetry 5, the map Us is cr-weakly continuous, then by Lemma 2.35, the center of M is cr-weakly closed, and thus is an associative JBWsubalgebra of M. D 2.37. Lemma. Let p be a pmJectwn m a lEW-algebra !'vI. Then there lS a smallest central pmJectwn c(p) ~ p.

Proof. Let X = {central projections e I c ~ p}. Note that if el, C2 are in X, then their product coincides with their greatest lower bound among projections (Proposition 2.28), so is again in X. Thus X is directed downwards. Let e(p) be the greatest lower bound of X. Then by Proposition 2.5, c(p) is a cr-weak limit of elements of X and is a projection. It is central since the center of A1 is cr-weakly closed. D 2.38. Definition. Let p be a projection in a JBW-algebra M. The smallest central projection e(p) ~ p is called the central cover of p. 2.39. Proposition. A cr-weakly closed subspace 1 of a lEW-algebra !'vI l5 a lordan ldeallff It has the form 1 = cAl for a (necessanly umque) central projection c.

Proof. Let 1 be a cr-weakly closed ideal of lvI. Then 1 is a cr-weakly closed subalgebra of A1, so by Proposition 2.7 has an identity c. For any element a in A1, co a is in 1 so co (c 0 a) = (e 0 a). It follows that Uca = 2e 0 (e 0 a) - co a = co a = Tea, so c is central by Proposition 1.47. Then eM eland 1 = el c cM, so 1 = elvI. D We say a JBW-algebra A is the (algebraic) dlrect sum of subalgebras Ml and M2 (and write M = Ml ffi M 2 ) if M is the direct sum of Ml and M2 as linear spaces, and if b1 0 b2 = 0 for all b1 E lvh and b2 E M 2.

CENTER OF A JBW-ALGEBRA

57

2.40. Proposition. Let M be a lEW-algebra. If M = Ml EB M 2 , then Ml and .M2 are a-weakly closed lordan zdeals of the form cIvI and c' M for a central pro)ectwn c, where c' = 1 - c. Proof. It is straightforward to check that Ml and Ivh are Jordan ideals of M. Since Ml = {x E M I x 0 M2 = O}, then Ml (and similarly M 2 ) is a-weakly closed. Finally, let Cl and C2 be central projections such that Ml = clIvI and Ivh = c2M. Then Cl is an identity for IvIr and C2 for M 2 , so Cl + C2 is an identity for M. Thus c) + C2 = 1. D

2.41. Proposition. Let AI be a lEW-algebra and c a central projection. Then Iv! = cM EB c' !vI, and Te = Ue ZS a homomorphzsm from M onto cM wzth kernel c' AI. Proof. If c is a central projection, then by Lemma 1.45, cAl + c'M = Me + Me' is an algebraic direct sum, equal to Ivi since m = cm + c'm for each m E M. By (1.66), m f-+ cm is a homomorphism. Since cm = 0 iff e'm = m, then the kernel of m f-+ cm is c'IvI. D

2.42. Definition. If {Ma} is a collection of JBW -algebras, then we define EBa Ma to consist of the clements (b a ) of the Cartesian product such that SUPa Ilball < 00. We define a product by coordinatewise multiplication, and use the supremum norm. It is straightforward to verify that EDa Ma is a JB-algebra, and is monotone complete with a separating set of normal states, i.e., is a JBW-algebra. Let {b a } be a net in a JBW-algebra. We define partwl sums to be sums of the form bF = LaEF ba , where F is a finite set of indices. We order such finite subsets F by inclusion, and define Lo: bex to be the a-strong limit of the partial sums (if this limit exists).

2.43. Lemma. If {b ex } zs a net of orthogonal posztwe elements in a lEW-algebra !vI, and 4 SUPa Ilbexil < 00, then Lo:bo: ex:zsts and I La ha II = sup a I ba II· Proof. The partial sums form an increasing net, so it suffices to prove it is bounded above. For each finite set F of indices, by (2.5)

L

(2.27)

ba

aEF

Since the elements so

(2.28)

L aEF

{bn}nEF

Ilboll r(b a )

::;

L

IlbnIlr(ba).

aEF

are orthogonal, so are their range projections,

::;

(m:x Ilball)

L aEF

r(b a

) ::;

(m:x Ilball) 1,

58

2.

JBW-ALGEBRAS

(where the last inequality follows from Corollary 2.23). It follows that the partial sums are bounded above, so converge u-strongly to their least upper bound. Furthermore, (2.27) and (2.28) imply that II La ball:::; sup", Ilba:ll, and the opposite inequality is clear, so the proof is complete. 0 In particular, for any orthogonal collection {p",} of projections, the sum La: Po< converges, and if each Pa is central, then the sum will also be central.

2.44. Proposition. If {c a } lS a maxlmal collectwn of orthogonal central pro)ectwns m a lEW-algebra M, then La Ca = I, and M lS lsomorphlc to EBa: CaM vw a f-7 (caa). (We wnte 11,1 = EBa c",M.)

Proof. By maximality of the collection {c a }, La c'" = 1. Therefore the map a f-7 (c",a) is clearly injective. To see that this map is surjective, suppose ba: E c",IM for each 0, with sup", II ba II < 00. For each 0, let ba = ba: + -b a - be the orthogonal decomposition (Proposition 1.28). By spectral theory, Ilball = max(llb a +11, lib", -II), so sup", lib", +11 < 00 and sup", lib", -II < 00. Thus Lo: b", + and La ba - converge u-strongly (Lemma 2.43). Hence La ba = La (b", + - ba -) converges u -strongly to an element b such that Cab = ba for each 0. 0

The bidual of a JB-algebra For C*-algebras, the fact that the bidual comes with a natural product making it a von Neumann algebra (A 101) is a very useful tool, and we are now going to establish the analogous result for JB-algebras. We are going to prove the related result that the bidual of a commutative order unit algebra is an order unit algebra, and this will allow us to prove both that the bidual of a JB-algebra is a JBW-algebra, and that every commutative order unit algebra is in fact a JB-algebra. We need first to review the notion of a base norm space, which is used in the next proof and which will play an important role in the rest of this book.

2.45. Definition. An ordered normed vector space V with a generating cone V+ is said to be a base norm space if V+ has a base K located on a hyperplane H (0 tt H) such that the closed unit ball of V is co( K U - K). The convex set K is called the dlstmgUlshed base of V. The dual space of every order unit space A is a base norm distinguished base the state space of A, and we can recover A the space A(K) of w*-continuous affine functions on K, cf. (A 20). Thus the dual space of a JB-algebra A is an order unit

space, with from K as (A 19) and space, with

THE BIDUAL OF A JB-ALGEBRA

59

the state space of A as distinguished base. The dual of a base norm space with distinguished base K is an order unit space (A 19), and this dual space can be identified with Ab(K), cf. (A 19) and (A 11). We will see later (Corollary 2.60) that the predual of a JBW-algebra is a base norm space, with the normal state space as distinguished base.

2.46. Theorem. Let A be a commutatwe order umt algebra. Then A ** zs an order umt algebra for a umque product that extends the origznal product on A and whzch zs separately w*-contznuous. Each state on A extends umquely to a w*-contznuous state (necessarzly normal) on A**. If K is the state space of A, then A ** can be zdentzjied as an order umt space with the space Ab(K) of bounded affine functzons on K. Thus A ** is monotone complete wzth a separatzng set of nor-mal states. Proof. For each state a on A define a symmetric bilinear form on A by (a I b) = a(ab). Since squares in A are positive, this form is positive semi-definite, so the Cauchy-Schwarz inequality holds. Define N = {a E A I (a I a) = O}. By the Cauchy-Schwarz inequality,

N = {a

E

A I (a I b) = 0 for all b E A}.

Therefore N is a subspace of A. It follows that (a + Nib + N) = (a I b) is well defined, symmetric, and positive semi-definite, so AI N is a preHilbert space. Let HeJ denote its completion as a Hilbert space, and let ¢eJ : A ----> AI N c HeJ be the quotient map. Note that

where the last inequality follows from the definition of an order unit algebra (Definition 1.9). Thus

Note that for a, b E A we have

(2.29) We identify H(J" with H;* and let ¢~* be the bidual map from A** into H;* = H(J"' Since II¢eJll : : ; 1, then

(2.30)

II¢~*II

::::;

1.

For a, b in A** define a function a * b on the state space K of A by

(2.31 )

(a

* b, a)

= (¢~*(a)

I ¢~*(b)).

60

2

JBW-ALGEBRAS

Note that cjJ~* is continuous as a map from the w*-topology on A** to the weak topology on H". It follows that (a, b) f---' (a * b, 0") is separately w*-continuous. Note also that for a E A, viewing A as a subspace of A**, we have ¢~*(a) = ¢,,(a). Thus if a, bE A, then by (2.31) and (2.29), a * b coincides as a functional on K with the given product ab in A. Since A is an order unit space, then A * is a base norm space with distinguished base K, and A** is an order unit space which can be identified with Ab(K). (See the remarks preceding this theorem.) It follows that A** is monotone complete with a separating set of normal states, since Ab(K) evidently has these properties. Each (J E K is an element of A *, so acts by evaluation as a w* -continuous functional on A **. By density of A in A **, this is the unique extension of (J to a w* -continuous functional on A**. If a a in A** = Ab(K), then a is the pointwise limit of a on K, and thus aCt ---) a in the w* -topology. Thus every w* -continuous linear functional is normal. Next let a,b E A**, and let {aa} and {bi3} be nets in A converging in the w*-topology to a and b respectively. Then for (J E K, Q

/

Q

Thus a * b is a pointwise limit of affine functions be affine. Furthermore, for (J E K,

I (a * b,(J) I = I (¢~*(a) I cjJ~*(b)) I -:: :

Oll

K, and so must itself

11¢~*(a)llllcjJ~*(b)11

-:: : Ilallllbll·

Thus a * b is a bounded affine function on K, with (2.32)

sup I(a * b, (J)I

-:: : Ilallllbll·

"EK

Therefore a * b can be identified with an element of A **. The map (a, b) f---' a * b is our candidate for the extension of the given product on A to a product on A**. Separate w*-continuity of this product and w*-density of A in A ** imply that this product is commutative and bilinear. Since A is power associative, in particular the identity x(xx 2 ) = x 2 X 2 is valid for all x E A. Therefore the equivalent linearized version (1.4) holds in A. By w*-continuity of this product on A** and w*-density of A in A**, the identity (1.4) holds in A**, and then by Proposition 1.3, A** is power associative. To show that A ** is an order unit algebra, by (A 51) there only remains to prove the implication

Let a E A** and assume -1::; a::; 1. Then by (2.32), Ila*all ::; l. On the other hand, by (2.31), a * a:::>: O. By the definition of the order unit norm,

THE BIDUAL OF A JB-ALGEBRA

61

we conclude that 0 :s: a * a :s: 1. This completes the proof that A ** is a commutative order unit algebra. Finally, by density of A in A **, the product on A ** is uniquely determined by the product on A and by the requirement of separate w*continuity. D The proof above can easily be modified to show that the bidual of a JB-algebra is a JBV/-algebra, but this will also follow when we show that order unit algebras and JB-algebras are the same. We remark that the clever proof above is due to Hanche-Olsen [64] (stated there for JBalgebras). Note that although the proof begins by using states to construct a Hilbert space, as in the GNS-construction, the remainder of the proof is fundamentally different. In particular, no representation of A as operators acting on a Hilbert space is involved. Indeed, we will see in Chapter 4 that there are JB-algebras for which no non-zero representation on a Hilbert space exists. 2.47. Lemma. Let A be a commutatwe order umt algebra. Then the set of fimte lmear combmatwns of orthogonal projections IS norm dense m A**.

Proof. By Theorem 2.46, the hypotheses of (A 53) are met, so this is an immediate consequence. D The proof of the following lemma is elementary (after all, one has very little to work with) but fairly lengthy, so we will just give a reference for the proof. In the Jordan context, this lemma states that orthogonal projections operator commute, a result that we already know (cf. (2.20)). 2.48. Lemma. Let A be a commutatwe power assocwtwe algebra over R. If p and q are zdempotents m A such that PrJ = 0, then

p(qx) = q(px)

for all x E A.

Proof. [113, Lemma 5.2] D Recall that every JB-algebra is a commutative order unit algebra (Lemma 1.10). Now we can prove the converse result, which was first proved by Iochum and Loupias [69,70]. 2.49. Theorem. If A is a commutatwe order umt algebra, then A is alE-algebra.

Proof. By (A 52), the norm conditions that define a JB-algebra are satisfied by commutative order unit algebras. Thus we only need to prove

62

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JBW-ALGEBRAS

the defining Jordan identity

(a 2

(2.33)

0

b)

0

a = a2

0

(b 0 a).

Suppose first that a is a finite linear combination of orthogonal projections: = Li A,Pi, and bE A. Then

a

(a 2o b)oa= 2)A;Pi Ob )O (LAjPJ) = 2

~A7Aj(PiOb)op]. t.]

]

Similarly

a2

0

(boa) = (LA7Pi)

0

(bo LA]PJ) = LA;AjPi

t

]

0

(bop]).

',]

Now (2.33) follows from commutativity and Lemma 2.48. For general a, we imbed A in the order unit algebra A ** and use the fact that in A ** every element can be approximated in norm by linear combinations of orthogonal projections (Lemma 2.47). Then (2.33) follows from norm continuity of multiplication on order unit algebras. D

2.50. Corollary. Let A be a lE-algebra. Then ltS second dual A ** zs a lEW-algebra Jor a umque product that extends the orlgmal product m A and whlch is separately w*-contmuous. Each state on A extends umquely

to a w*-contmuous state (necessanly normal) on A **. IJ K lS the state space oj A, then A** can be ldentlfied as an order umt space wzth Ab(K). Proof. This follows from the analogous result for commutative order unit algebras (Theorem 2.46) and the fact that JB-algebras are the same as commutative order unit algebras (Theorem 2.49 and Lemma 1.10). D We will often view A as a space of functionals on A * and thus identify A with a subspace of A **. Passing to the bidual is a very useful technique for proving certain results about JB-algebras, as we will now see.

The predual of a JBW-algebra We will say a Banach space !vI is a Banach dual space if there is a Banach space !vI* such that M is isometrically isomorphic to (IVI*)*. The space M* is then called a predual of M. Note that elements of any predual can be viewed as functionals on M, and thus M* can be isometrically imbedded in M*.

THE PREDUAL OF A JBW-ALGEBRA

63

2.51. Lemma. Let !vI be a .fEW-algebra, and N the subspace of M* consistmg of the normal lmear functwnals. Let .f be the anmfdator of N in M**. Then N zs norm closed, and there zs a central pTOJectwn c m M** such that .f = (1- c)M**, and N = U;(M*). Proof. By Corollary 2.50 each state on M has a unique extension to a normal state on AI**, which is then l1-weakly continuous (by the definition of the l1-weak topology). Furthermore, since !vI is an order unit space, then M* is a base norm space (A 19), so every element of !vI* is a linear combination of states (A 26). It follows that every element of !vi* acts as a l1-weakly continuous linear functional on M**. Thus .f is l1-weakly closed in M**, as well as w*-closed. If a E M, then by Proposition 2.4, T; maps N into N. Since the multiplication on M** is separately w* -continuous (Corollary 2.50), one readily verifies that T;* is multiplication by a on M**. It follows that the annihilator J of N in M** is invariant under multiplication by elements of AI. By w*-density of M in M**, J is invariant under multiplication by elements of M**, and thus is a l1-weakly closed ideal of /VI**. By Proposition 2.39, .f has the form (1 - c)/VI** = U1_c/VI** for a central projection c. Since an increasing net bounded above is eventually bounded, a simple argument with the triangle inequality shows that a norm limit of normal functionals is normal. (See the proof of [AS, Lemma 2.85] for a similar argument.) Thus N is a norm closed subspace of M*. By the bipolar theorem, N is the annihilator of J. Working in the duality of M* and !vI * * , N is the annihilator in /VI* of im U 1- c , and thus equals the kernel of Ut_c on !vI*. Since U 1- c = I - Uc, and Uc is idempotent, then the kernel of Ut-c equals the image of U;, which completes the proof. 0

2.52. Proposition. Let !vI be a .fEW-algebra. Every normal lmear functwnal on /vI zs the dzjJerence of posdwe normallmear functwnals, and the normal state space of !vI zs a splzt face of the state space of M. Proof. Since M* is a base norm space (A 19), and thus is positively generated (A 26), then with the notation of Lemma 2.51, N = U;(M*) is positively generated. Hence every normal linear functional on M is the difference of positive normal linear functionals. Since U; and Ut-c are positive idempotent maps on /VI* with sum equal to the identity map, it is straightforward to verify that the normal state space of !vI is a direct convex summand of the state space of !vI, i.e., a split face. 0

If B is a subspace of a lB-algebra A, we say B is monotone closed in E A implies a E B.

A if {baJ c B and be>: / a

2.53. Lemma. Let A be a .fE-algebra. Then the smallest monotone closed subspace of A** containing A is all of A**. Proof. [67, Thm. 4.4.10] 0

64

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JBW-ALGEBRAS

2.54. Lemma. Let M be a lEW-algebra canonically 'Imbedded m lv!**, and c E lv!** the central pro)ectwn of Lemma 2.51. Then a f-+ ca 'IS an 'Isomorph'lsm from lv! onto cM**.

Proof. By Proposition 2.41, Uc : M** ---> cM** is a Jordan homomorphism with kernel (1 - c)M**. By the choice of c, the kernel is the annihilator of the normal functionals on M, and so contains no element of M. Thus this map restricted to M is a Jordan isomorphism onto the image cM. We are going to show that cM = cAI**. First we show that clv! is monotone closed in clv!**. Suppose that {a o,} C M and can / b in clvI**. Since Uc is an isomorphism on A!, then {an} is an increasing net. Let a be the supremum of {an} in lv!. Then O"(a n ) ---> O"(a) for all normal states 0" on A!. If 0" is any state on AI, then by Lemma 2.51, U;O" is normal on M, so (U;O")(a n ) ---> (U;O")(a). Thus O"(ca n ) ---> O"(ca) for all states 0" on lvI, and so can ---> ca weak*. On the other hand, by Corollary 2.50 the w*-continuous extension of each state on M is a normal functional on M**, so 0"( can) ---> O"(b) for each state 0" on M. We conclude that b = ca E cAl, which completes the proof that cM is monotone closed in cM**. Now let Ma = {b E M** I Ucb E Uc(M)}. If {b n } is an increasing net in Ma with least upper bound b E lvI**, then by O"-weak continuity and positivity of the map Uc , the net {Ucb n } increases and converges O"-weakly to Ucb. Thus Ucb n / Ucb. Since clv! is monotone closed in clVI**, then Ucb E UclVI , proving that A!a is monotone closed in AI**. By Lemma 2.53, Ma must be all of M**, and thus Uc(M**) = Uc(M). 0 2.55. Theorem. AlE-algebra lv! 'IS a lEW-algebra 'Iff 'It 'IS a Eanach dual space. In that case the predual M* 'IS umque and cons'lsts of the normal lmear functwnals on M.

Proof. Assume M admits a predual lVI*. Then the (closed) unit ball [-1,1] of A! is w*-compact, and x ---> ~(x + 1) is a homeomorphism of [-1,1] onto [0,1]' so the positive part of the unit ball is w*-compact and thus w*-closed. It follows from the Krein-Smulian theorem (A 34) that the positive cone lvI+ is w*-closed. By the bipolar theorem, the positive elements of M* determine the order on M, and in particular separate points of M. If we let V = (M*)+ - (M*)+, then the identity map on the unit ball of lVI is continuous from the w*-topology to the O"(lv!, V) topology, and so by w*-compactness of the unit ball is a homeomorphism. Thus the w* and 0"( lvI, V) topologies coincide on the unit ball of 1'0. Now suppose that {a,,} is a bounded increasing net in M. Then for each positive 0" in M* the net {O"(a n )} must converge. Therefore the net {an} is O"(M, V) Cauchy. Since the unit ball is O"(M, V) compact and therefore complete, this net has a O"(M, V) limit a. Since the O"(M, V) and w*-topologies coincide on the unit ball, an ---> a weak*. As shown above,

THE PREDUAL OF A JBW-ALGEBRA

65

the positive functionals in !'vl* determine the order on M, so au / a. Thus we have shown that !'vI is monotone complete and that each functional in M* is normal. We showed above that the positive functionals in M* separate points, and therefore ]\,1 admits a separating set of normal states, and so by definition is a JBW-algebra. Conversely, suppose M is a JBW-algebra, and let N be the space of normal functionals. Let 1 be the annihilator of N in M**, and c the central projection such that 1 = U1 - e(!'vl**) (Lemma 2.51). By Lemma 2.54, Ue is an isomorphism (and thus an isometry: d. Proposition l.35) from M onto Ue(M**). From the fact that Ue is idempotent and IlUell 'S 1, one readily verifies that Ue(M**) is isometrically isomorphic to the dual space of U; (!'vl*) = N. Thus Ue is an isometry of ]\,{ onto the Banach dual of N. By Lemma 2.51, N is a closed subspace of !'vl* and thus is a Banach space. Finally, let V be any Banach space that is a predual of M. By the first part of this proof, each element of V is a normal functional on M. Since V is complete, then V is a norm closed subset of the space N of normal functionals on !'vI. Since V separates elements of M, by the Hahn-Banach theorem we must have V = N. D 2.56. Corollary. The normal, w*-contmuous, and a-weakly continuous lmear functwnals on a lEW-algebra coinczde. The a-weak and w*-topologzes comczde, and thus the umt ball of !vI zs a-weakly compact. Pmoj. By Theorem 2.55, the normal linear functionals are the elements of the unique predual that define the w*-topology, and thus coincide with the w*-continuous linear functionals. By Proposition 2.5, bounded monotone nets converge a-weakly, so a-weakly continuous linear functionals are normal. By the definition of the a-weak topology, every positive normal linear functional is a-weakly continuous. By Proposition 2.52 every normal linear functional is the difference of positive normal functionals. Thus every normal functional is a-weakly continuous. Finally, the a-weak topology is defined as convergence on positive normal linear functionals, which is then the same as convergence with respect to all normal linear functionals. The latter topology is precisely the w*-topology. D

2.57. Corollary. If A zs a lE-algebra, then A zs a-weakly dense m the lE W-algebra A ** . Pmoj. By the bipolar theorem, A is w*-dense in A**. By Corollary 2.56 it follows that A is a-weakly dense in A **. D

2.58. Proposition. If p is a normal functwnal on a lEW-algebra T of positive normal linear functionals

At, then there is a umque pazr a,

66

2

JBW-ALGEBRAS

such that

(2.34)

P= a - T

Furthermore, there

lS

(2.35 )

= Iiall + IITII·

Ilpll

and

a pro]ectwn q m lYI such that U;a

=

a

and

U;T

= o.

Proof. Let p be a normal linear functional on M with Ilpll = 1. Recall that M is the dual of the space of normal linear functionals (Theorem 2.55). Let bE M be an element of the w*-compact unit ball MI on which p achieves its maximum 1. The set of such elements b is a w*-compact face of the unit ball of M, and so by the Krein-Milman theorem, we can choose b to be an extreme point of !'vII. Since a f---> 2a - 1 is an affine homeomorphism of the positive unit ball [0,1] of M onto Ml = [-1,1], it follows from Proposition 1.40 that the extreme points of the unit ball of M have the form 2q - 1 = q - q' for projections q (where q' = 1 - q). Now p can also be viewed as an element of M*, and lYI* is a base norm space (A 19). It follows from the definition of a base norm space that there exist positive functionals a and T such that p = a-T and 1 = Ilpll = Iiall + IITII, cf. (A 26). We are going to show that a and T are normal, and that this decomposition is unique. Now 1

=

p(b)

=

(a - T)(q - q')

::; a(q)

+ T(q')

=

a(q)

::; IlaI!

+ T(q') + IITII =

- a(q') - T(q)

1.

It follows that a(q) = Iiall and T(q') = IITII, and so by Proposition 1.41, U;a = a and U;T = o. Then U;p = a. Since Uq is a-weakly continuous (Proposition 2.4), and monotone nets converge a-weakly (Proposition 2.5), then a = p is normal, and so is T = a - p. This completes the proof that the decomposition (2.34) exists. The construction of q depended only on p and not on the choice of a and T, so the equation p = a implies uniqueness of the decomposition. D

U;

U;

Recall that a von Neumann algebra M also has a unique predual (A 95). The following describes the relationship of the predual of M and the predual of the lBW-algebra M sa. 2.59. Proposition. Let M be a von Neumann algebra wdh predual M*, and let M = M sa , wzth predual M*. Then IvI* + zM* = M*, and AI* has the norm mherzted from M *. Proof. By definition, a positive normal linear functional on M is also normal on M, and thus is in M •. The predual of M consists of the normal

THE PREDUAL OF A .J8W-ALGEBRA

67

linear functionals (A 95), and every normal linear functional on M is a linear combination of positive normal linear functionals (cf. (A 90) and the remarks after Definition 2.2). Thus M* = M* + lA1*. Now let 0' E M*. We need to show that the norm of 0' as a linear functional on M coincides with its norm as a complex linear functional on M. Write 110'11 M and 110'11 M for these respective norms. If 0' ;:: 0, then IIO'IIM = 0'(1) = 110'11 M' d. (A 16) and (A 71). For arbitrary 0' E M*, write 0' = 0'+ - 0'-, where 0'+,0'- EM;, and IIO'IIM = IIO'+IIM + IIO'-IIM (cf. Prop. 2.58). Then

Therefore 110'11 M =

IIO'IIM'

0

Recall in Chapter 1 we showed that a lB-algebra A is an order unit space, and thus its state space is the distinguished base of the base norm space A*.

2.60. Corollary. The predual of a lEW-algebra M lS a base norm space (V, K) where the normal state space K lS the dlstmguished base. Thus M can be ldentlfied as an order umt space wlth Ab (K). Proof. We need to prove (M*h = co(K u -K). If p E (M*)l, choose a decomposition p = 0' - T as in equation (2.34). Then

The second statement of the corollary follows from the fact that the dual of a base norm space (V, K) is Ab(K) (A 11). 0 Recall from (A 24) that positive linear functionals 0', T in a base norm space are orthogonal if 110' - Til = 110'11 + IITII. We will therefore refer to (2.34) as the orthogonal decomposltwn of p.

2.61. Corollary. If A lS a lE-algebra, then each state on A extends to a unique normal state on A **, and the extenswn map lS a bZJectwn from the state space of A onto the normal state space of A ** . Proof. By Corollary 2.50, every state on A extends to a normal linear functional on A**. Every normal state on A** is w*-continuous by Corollary 2.56, so the normal extension is unique. Evidently two distinct states cannot have the same normal extension, so the extension process is one-toone, and is surjective since every normal state on A ** is the extension of its restriction to A. 0

68

2.

JBW-ALGEBRAS

2.62. Proposition. Let NI be a lEW-algebra w~th normal state space K, let p be a pro]ectwn m M, and NIp = Up (A1) . The rest~ctwn map ~s an order preservmg zsometry from UT:(AI*) onto (NIp)*, and gwes an affine zsomorphzsm from Fp = {a E K I O'(p) = I} onto the normal state space of NIp. The mverse of the restrzctwn map zs the map 0' f-7 0' 0 Up.

It is readily verified that 1/J is an isometry from (Mp)* onto U;(M*), whose inverse is the restriction map. If 0' is a normal state on Alp, then 1/J(O')(p) = O'(p) = 1, so 1/J(0') E Fp. If w E F p, then U;w = w (d. (1.55)), so 1/J(wIN1p) = w. Thus 1/J is an affine isomorphism of the normal state space of Alp onto Fp. 0

2.63. Definition. Let NIl, NI2 be JBW-algebras. A positive map from MI to !v12 is normal if whenever a", / a, then 7r(ao:) /7r(a).

7r

Note that any order isomorphism between JB\V-algebras is normal, and then also any Jordan isomorphism.

2.64. Proposition. Let M I , A12 be lEW-algebras. A posztwe map T from Nh to NI2 zs nOT'Tnalzjj zt zs a-weakly contmuous. Proof. Since T is a positive map between order unit spaces, it is norm continuous (A 15). Observe that T is normal iff the dual map T* : !vI'; ---+ NIi sends normal states to normal functionals. In fact, if a", / a in !vh, then Ta", / Ta iff p(Ta",) / p(Ta) for all normal states p on Nh, i.e, iff T* p is normal for all normal states p. Since the a-weak topology is by definition the same as pointwise convergence on normal states, it follows by a similar argument that T is a-weakly continuous iff T* takes normal states to normal functionals. Thus normality for T is equivalent to a-weak continuity. 0

2.65. Theorem. Let A be a lE-algebra and 7r : A ---+ !vI a homomorphzsm mto a lEW-algebra NI. Then there zs a umque normal homomorphzsm IT : A ** ---+ lv1 extendmg 7r. Proof. Note that 7r** : A** ---+ NI** is by definition w*-continuous, so is normal. By w* -density of A in A **, 7r** is a homomorphism. Recall from Lemma 2.54 that there is a central projection c in NI** such that Uc is an isomorphism from M onto Uc(M**). Let 7ra be the inverse of this isomorphism, i.e., 7fa(Uc a) = a. Finally, define IT to be the composition 7F = 7ra 0 U c 0 7r**. Since the isomorphism 7ra is necessarily normal, then 7F is normal, and it extends 7r. Finally, uniqueness follows from the a-weak density of A in A **. 0 7r**

THE PREDUAL OF A JBW-ALGEBRA

2.66. Proposition. Let M and N be lEW-algebras and 7f : M a normal homomorphlsm. Then 7f(M) lS a lEW-subalgebra of N.

69 --->

N

Proof. The kernel of 7f is a a-weakly closed ideal of M, so by Proposition 2.39 has the form U1-cN! for some central projection c in N!. Then 7f is an isomorphism from UcN! onto 7f(M). By Proposition 2.9, UcM is a JBW-algebra, so the same is true of the isomorphic algebra 7f(NI). Finally, the unit ball of M is a-weakly compact (Corollary 2.56). Since IIUcl1 ::; 1 and Uc is idempotent, then Uc maps the unit ball of M onto that of Uc l'.;f , and the latter is mapped onto the unit ball of 7f( l'.;f) by the isomorphism 7f. Thus by a-weak continuity of 7f, the unit ball of 7f(M) is a-weakly compact, and then w*-compact in N. By the Krein-Smulian theorem, 7f(l'.1) is w*-closed, and then a-weakly closed. Thus it is a JBWsubalgebra of N. D 2.67. Proposition. Let A be a lE-algebra and M a lEW-algebra, and 7f : A ---> M a homomorphzsm. Then 7i'(A**) = 7f(A) (a-weak closure.)

Proof. By normality of 7i' and Proposition 2.66, 7i'(A**) is a JBWsubalgebra of lvI, and thus contains 7f(A). The opposite inclusion follows from the fact that A is a-weakly dense in A** (Corollary 2.57), and so by a-weak continuity 7i'(A**) C 7f(A). D 2.68. Proposition. Let M be a lEW-algebra. Then the a-weakly continuous lmear functwnals and the a-strongly continuous lmear functionals comczde. Thus the a-weakly closed and a-strongly closed convex subsets comczde.

Proof. By Proposition 2.5, a-weakly continuous linear functionals are a-strongly continuous. If T is a-strongly continuous, since monotone nets converge a-strongly (Proposition 2.5), then T is normal, and therefore 0'weakly continuous (Corollary 2.56). D We can now derive the very useful Jordan analog of the Kaplansky density theorem for von Neumann algebras.

2.69. Proposition. (Kaplansky density theorem for JB-algebras) If

M is a JBW-algebra and A is a JB-subalgebra that is a-strongly dense in M, then the unit ball of A is a-strongly dense in the unit ball of M. In particular, the unit ball of A is a-strongly dense in the unit ball of A **.

Proof. By Theorem 2.65, there is a normal extension 7f : A** ---> M of the inclusion map of A in M. By Proposition 2.66, 7f(A**) is a-weakly closed in M. Since A is a-strongly dense in M, then by Proposition 2.5, A is a-weakly dense in M. Thus 7f(A**) ~ 7f(A) = A is a-weakly dense in

70

2.

JBW-ALGEBRAS

M, and so 7r(A**) = M. The kernel of 7r will be a a-weakly closed ideal in A **, and so is of the form (1 - c) A ** for a central projection c in A ** . Then 7r will be an isomorphism from cA ** onto M. Thus 7r takes the unit ball of cA** onto the unit ball of M. It follows that 7r takes the unit ball of A** onto the unit ball of M. The unit ball of A is a-weakly dense in the unit ball of A** by the bipolar theorem. Thus by a-weak continuity of 7r, the unit ball of A = 7r(A) is a-weakly dense in the unit ball of M. By Proposition 2.68, since the unit ball of A is convex, its a-weak and a-strong closures coincide, so it is a-strongly dense in the unit ball of M. Finally, since A is convex and a-weakly dense in A **, it is also astrongly dense. Now the second statement in the proposition follows from the first. D

JW-algebras Recall that a JC-algebra is a norm closed Jordan subalgebra of B (H)sa (Definition 1.7). B (H)sa is monotone complete (A 76) and has a separating set of normal states (e.g., the vector states wE, : a f--4 (a~ I ~)), so is a JBW-algebra. In [AS, Chpt. 2] the a-weak topology was defined in terms of specific linear functionals on B (H) (A 84), which were shown to be precisely the normal linear functionals (A 92), so the term a-weak in [AS, Chpt. 2] coincides with our usage here for B (H)sa viewed as a JBWalgebra.

2.70. Definition. A lW-algebm is a JB-algebra that is isomorphic to a a-weakly closed Jordan sub algebra of B(H)sa. A concrete lW-algebra is a a-weakly closed Jordan subalgebra of B(H)sa. 2.71. Example. Of course, B(H)sa is one example of a JW-algebra. Any concrete von Neumann algebra is by definition weakly closed (A 89), and thus also a-weakly closed (A 88), and so the self-adjoint part of a von Neumann algebra is also a JW-algebra. 2.72. Lemma. Every lW-algebra 2S a lEW-algebra. Furthermore, 2f M 2S a concrete lW-algebm, then the a-weak topology on M coincides wzth that mherzted from B (H). Proof. Let B = B (H)sa. As was just discussed, B is a JBW-algebra, and thus so is any a-weakly closed Jordan subalgebra, by Proposition 2.7. Let No be the annihilator of !vI in B* (the predual of B). By the bipolar theorem, since M is a-weakly = a(B, B*) closed, then it is equal to the annihilator of No. Each a E IvI induces a map Ii : B*/No ---7 C by Ii(p + No) = p(a). By (A 32) a f--4 Ii is an isometric isomorphism from

JW-ALGEBRAS

71

M onto (B*/No )*. It follows that the elements of the predual of Mare precisely the restrictions of the elements of the pred ual of B, and thus the two cr-weak topologies coincide on M. 0 2.73. Lemma. Let A and B be Banach spaces and Jr : A -+ B a linear isometry from A mto B. Then Jr** zs an isometry from A ** onto a w*-closed subspace of B** . Proof. We first show that Jr* maps the closed unit ball of B* onto the closed unit ball of A*. For cr E (B*)l we have IIJr*crll = sup{(Jr*cr)(a) I a E Ad =

sup{cr(Jr(a)) I a E Ad ::; IlcrIIIIJr(a)11 ::; 1,

so Jr* maps the unit ball of B* into the unit ball of A **. Let T be in the unit ball of A*. Then Jr(a) f--+ T(a) is a linear functional of norm at most 1 on Jr(A). By the Hahn-Banach theorem we can extend this to a linear functional cr on B of norm at most 1, and then Jr*cr = T, which proves that 7T* maps the unit ball of B* onto the unit ball of A *. We next show that 7T** is an isometry from A ** into B**. Let a E A ** ; then IIJr**(a)11

=

SUp{cr(7T**(a)) I cr E (B*)d

=

SUp{7T*(cr)(a) I cr E (B*)d

= sup{T(a) IT

E

(A*h}

= Ilall·

Finally, note that Jr** is w* -continuous and the unit ball of A ** is w*compact. Since 7T** is an isometry, the unit ball of 7T**(A**) is the image of the unit ball of A**, and so is w*-compact. By the Krein-Smulian theorem, 7T**(A**) is w*-closed in B**. 0

2.74. Proposition. If A and Bare lB-algebms and Jr : A -+ B zs a unital lordan zsomorphzsm from A mto B, then 7T** wzll be a lordan isomorphism from A ** onto a cr-weakly closed subalgebm of B**. Proof. Recall that the w*-and cr-weak topologies on a JBW-algebra coincide (Corollary 2.56). Now w*-density of A in A**, w*-continuity of 71"**, and separate w* = cr-weak continuity of multiplication imply that 71"** is a Jordan homomorphism. Since unital Jordan isomorphisms are isometries (Proposition 1.35), then Jr is an isometry. Hence by Lemma 2.73, 71"** is an isometry onto a w*-closed subspace of B**. Thus Jr** is a Jordan isomorphism from A ** onto a cr-weakly closed subalgebra of B**. 0

72

2.

JBW-ALGEBRAS

Let B be a Banach space, and as usual identify B with a subspace of B**. If A is a closed subspace of B, then we may also identify A with a subspace of B**. If 'if : A -+ B is the inclusion map, then by Lemma 2.73, 'if** is an isometry from A ** into B**, and we will identify A ** with 'if**(A) c B**. 2.75. Lemma. Let B be a Banach space and A a closed subspace. Let A be the w*-closure of A m B**. Then A** = A, and

AnB =A. Pmoj. Let 'if : A -+ B be the inclusion map. Since 'if** is w*continuous and A is w*-dense in A**, it follows that 'if**(A) is w*-dense in 'if**(A**). We identify the former with A and the latter with A** according to the remarks preceding this lemma. Then A ** is w* -closed in B**, so A = A ** follows. Now let V be the annihilator of A in B*, and let VO be the annihilator of V in B**. Then by the bipolar theorem (A 31) for the duality of B* and B**, A = VO. On the other hand, since norm closed subspaces of a Banach space are weakly closed, applying the bipolar theorem to the duality of B and B*, we conclude that VO n B = A. Combining these two equalities gives the result in the lemma. D 2.76. Lemma. If A lS a C*-algebra wlth state space K, then (Asa)** and (A ** )sa can be ldentlfied as lB W-algebras.

Pmoj. By (A 102) and Corollary 2.50, both (A ** lsa and (Asa)** are order isomorphic to Ab(K). The Jordan product on each is determined by the Jordan product on Asa and by separate w*-continuity of this product. The w*-topologies are determined by pointwise convergence on K, so coincide, and therefore the Jordan products do as well. D 2.77. Proposition. If A lS a lC-algebra, then A** zs a lW-algebra.

Pmoj. Let 'if be the inclusion map from A into B (H)sa. Then by Proposition 2.74, 'if** is a Jordan isomorphism from A** onto a a-weakly closed Jordan subalgebra of (B(H)"a)**. By Lemma 2.76 the latter can be identified with the self-adjoint part of the enveloping von Neumann algebra B(H)** (A 101) of B(H). Thus A** is a JW-algebra. D 2.78. Corollary. If A zs a lC-algebra and also a lBW-algebra, then A lS a lW-algebra.

Prooj. Since A is a JC-algebra, A ** is a JW-algebra (Proposition 2.77). Since A is a JBW-algebra, there is an isomorphism from A onto a direct summand of A** (Lemma 2.54). A direct summand of a JW-algebra is a JW-algebra. D

ORDER AUTOl\lORPHISI\IS OF A .JB-ALCEBRA

73

Order automorphisms of a JB-algebra

2.79. Definition. A umtal oT'der automOT'phlSm of a JB-algebra is a bijective linear map ¢ such that ¢(1) = 1, and such that both ¢ and ¢-l preserve order. 2.80. Theorem. A umtal oT'der automOT'phlSm of a lE-algebra A lS a lOT'dan algebra autorrWT'phlSm Proof. Let ¢ be an order automorphism of A. Recall (from Theorem 1.11) that A has the order unit norm, i.e.,

Iiall = infp E

R

I -AI::; a::;

AI}.

Thus ¢ is an isometry. Then the dual map ¢ * : A * --) A * is also an order isomorphism and an isometry, as is the bidual map ¢** : A ** --) A **. It suffices to prove that ¢** is a Jordan algebra isomorphism, and thus it is enough to prove the statement in the theorem when the algebra A is a JBW-algebra. Since the projections in A are precisely the extreme points of the order interval [0,1] (Proposition 1.40), it follows that ¢ maps projections to projections. Since projections p, q are orthogonal iff p+q ::; 1 (Proposition 2.18), then ¢ preserves orthogonality of projections. If ~i AiPi is any linear combination of orthogonal projections, then

¢

((~>,p,n ~1(~Aip,) ~ ~Ai1(P')

~ (~A'1(p,)r ~ H~A'P'))' By the spectral theorem (Theorem 2.20) every element of A is a norm limit of linear combinations of orthogonal projections, so by norm continuity of ¢ We have shown ¢(a 2 ) = (¢(a))2 for all a E A. Since a 0 b = ~(a + b)2 a2 - b2 ), it follows that ¢ preserves the Jordan product. 0 We are now able to relate skew order derivations and Jordan derivations. (See Definition 1.57 and equation (l. 70).)

2.81. Lemma. Let A be a IB-algebm wzth state space K and let <5 be an ordeT' deT'wation on A. FoT' every t E R let 0t = e ts . The followmg are equivalent:

(i)

<5

is skew.

74

2

JBW-ALGEBRAS

(ii) at lS a Jordan autornorphlsrn for all t. (iii) J lS a Jordan denvatwn. Proof. (i) =} (ii) For each t E R, at(1) = 1 (Proposition 1.58), so each at is a unital order isomorphism. Therefore each at is also a Jordan automorphism (Theorem 2.80). (ii) =} (iii) Since at = e tli is a Jordan automorphism, then 5(a

0

b) = lim C1(etli(a

0

b) - a

0

(etlib - b)

0

b)

t~O

= lim C1(etlia) t~O

=a0

+ lim C1(etlia -

a)

0

b

t~O

(5b)

+ (5a)

0

b,

so Leibniz' rule (1.70) holds and thus 5 is a Jordan derivation. (iii) =} (i) By Leibniz' rule (1.70), 5(1) = 5(1 0 1) = 25(1). Hence 5(1) = O. 0 In general a Jordan automorphism need not fix the center. However, if it lies on the one-parameter group generated by a skew order derivation, then it must.

2.82. Proposition. If 5 lS a skew order denvatwn on a JB-algebm A and Z E Z(A), then

(i) e tli z = z for all t (ii) 5z

=

E

R, and

O.

Proof. By Corollary 2.50 the bidual A** is a JBW-algebra. Since 5 is skew, by Lemma 2.81, e tli is a Jordan automorphism. By O'-weak continuity of multiplication 111 each variable separately, the bidual map (e tli )** is also a Jordan automorphism. Furthermore, again by O'-weak continuity of multiplication, the center of A will be contained in the center of A**, so it suffices to prove the lemma for the special case where A is a JBW-algebra. Recall that the center of a JBW-algebra is an associative JBW-algebra (Proposition 2.36). Then by the spectral theorem (Theorem 2.20) applied to the center, it is enough to prove the lemma for the case where z is a central projection. Since e tli is a Jordan automorphism, then e tli z is also a central projection for t E R. Thus e tli z - z is the difference of two central projections. Since the center is associative, it is isomorphic to CR(X) for a compact Hausdorff space X, so the difference of two central projections has norm either zero or 1. Thus Ile tii z - zll is either zero or 1. Since Ile tii z - zll is a continuous function of t which is zero when t = 0, it must be zero for all t. This proves (i). Nowalso oz = limt~Orl(etliz - z) = 0, which proves (ii). 0

ABSTRACT CHARACTERIZATION OF COMPRESSIONS

75

Abstract characterization of compressions We can now give an order-theoretic characterization of the maps Up on lBW-algebras. Let M be a lBW-algebra. We will say that a positive projection (idempotent linear map) P: M ---+ M is normalned if either it is the zero map or else has norm 1. Two positive projections P, Q on an ordered linear space are said to be complementary (and Q is said to be a complement of P and vice versa) if ker+Q

= im+ P,

ker+ P

= im+Q,

cf. (A 4). Also we will say that a positive projection P : M ---+ M is complemented if there exists a positive projection Q : NI ---+ M such that P, Q are complementary, and we will say that a CT-weakly continuous positive projection P : !vI ---+ NI is bzcomplemented if there exists a CT-weakly continuous positive projection Q : AI ---+ !vI such that P, Q are complementary and the dual projections p', Q' : NI. ---+ lvI. are also complementary. Note that if p is a projection in a lBW-algebra M, then by Proposition 1.38 and Proposition 1.41, Up and U1 - p are bicomplementary normalized positive projections. We now show that these properties characterize compressions on lBW-algebras.

2.83. Theorem. Let M be a lEW-algebra and let P : M ---+ M be aCT-weakly contmuous normalned posztwe projection. There exzsts a projection p E NI such that P = Up zfJ P zs bzcomplemented; in thzs case p is unique: p = PI, and the complement pi of P zs also umque: pi = U1 - p . Proof. As observed above, Up and Up' are bicomplementary. Now let P, pi be any bicomplementary normalized positive projections on M. We

will first establish the following properties: (2.36) (2.37) (2.38)

PI PI

+

p ll

= 1,

is a projection,

im +(P)

= face(Pl).

To verify these, first note that PI ::; 1 (since IIPII ::; 1), and P(1 PI) = 0, so P I (1 - PI) = 1 - PI, which implies (2.36). To verify (2.37), by Proposition 1.40 it suffices to show PI is an extreme point of [0, 1]. Suppose that (2.39)

PI

where 0 < A < 1 and a, b pia = Plb = 0 and thus a

=

E

Aa + (1 - A)b, [0,1].

Applying pi to both sides shows

= Pa ::; PI and b = Pb ::; Pl. By (2.39)

76

2

JBW-ALGEBRAS

it follows that a = b = PI, so PI is an extreme point of [0,1]' which verifies (2.37). Finally, suppose that a E face(Pl). Then pia = 0 so Pa = a, i.e., a E im +(P). Now we prove the opposite inclusion. If a is an arbitrary element in im +(P), then applying P to 0 ::; a ::; Ilalll gives 0::; a = Pa ::; IlallPl, so a E face(Pl), which completes the proof of (2.38). Now let p = PI. We will complete the proof of the theorem by showing P = Up and pi = U 1 - p' By (2.38) and (1.51), im +P = im +Up. Since im P and im Up are positively generated, then im P = im Up. By (2.36) we also have P l 1 = 1 - PI = 1 - p = U1 _ p1, so applying (2.38) with pi in place of P we conclude im pi = im U1 - p ' Taking annihilators in M*, we have ker P' * = ker U{_p'

(2.40)

By hypothesis P* and p l * are complementary, and so are U; and Ui_ p by Proposition 1.41. Thus (2.40) implies im + P*

= ker + p l * = ker + U1*-p = im + U*p'

From this im P* = im U; follows. Taking annihilators gives ker P = ker Up. Thus the idempotent maps P and Up have the same image and kernel, and so must be equal. A similar argument shows U1 - p = pi, and this completes the proof. 0 We remark that the same theorem holds for a JB-algebra A with the dual maps P*, p l * living on A *. The proof is the same except for the context being the duality of A and A * instead of !vI and AI*.

Notes Normed infinite dimensional Jordan algebras first appeared in the work of von Neumann [98]. Motivated by axiomatic quantum mechanics, he studied what are essentially JBW-algebras, though defined in quite different language. On physical grounds, he assumed power associativity instead of the Jordan identity (1.2). He proved the key fact that orthogonal projections operator commute (Lemma 2.48), and then showed that power associativity implies the Jordan identity. In modern language, he showed that the type I JBW-factors are not essentially different than those found in finite dimensions in [75]. Infinite dimensional Jordan algebras also appear briefly in Segal's paper [114] on axiomatic quantum mechanics. Topping [128] generalized much of the structure theory for von Neumann algebras to JW-algebras. St0rmer studied JW-algebras and the von Neumann algebras they generate in [125, 126]. Janssen [73, 74] studied a class of infinite dimensional Jordan algebras with a trace.

NOTES

77

Monotone closed JB-algebras with a separating set of normal states, Le., JBW-algebras (Definition 2.2), were used as a key tool in the proof of the Gelfand-N aimark type theorem in [S]. Most of the basic working tools for JBW-algebras described in Chapter 2 appear in [S]. The key technical result that the monotone closed subspace generated by a JB-algebra A in A** is all of A** (Lemma 2.53) we have just referred to [67]' where the proof follows the lines of Pedersen's proof of the analogous result for C*-algebras. JBW-algebras were first formally defined by Shultz [116]' who showed that a JB-algebra is a Banach dual space iff it is monotone complete with a separating set of normal states (Theorem 2.55), and that a concrete JC-algebra admits a faithful weakly-closed representation (i.e., is a JWalgebra) iff it is a JBW-algebra (Corollary 2.7S). This equivalence generalizes the abstract characterizations of von Neumann algebras due to Kadison [78] and to Sakai [110], cf. (A 95). Edwards in [42, 43] studied quadratic ideals of JB-algebras, i.e., those subspaces J closed under the map a f--+ {b 1 ab 2 } for all b1 , b2 in the algebra. He showed that norm closed quadratic ideals are the same as the hereditary JB-subalgebras, and established the correspondence of O"-weakly closed hereditary subalgebras of JBW-algebras with projections in the algebra (Proposition 2.32). The result that the bidual of a JB-algebra is also a JB-algebra (Corollary 2.50) is in [116]' but we have used the much simpler proof due to Hanche-Olsen [64]. Hanche-Olsen's result is for JB-algebras, but we have first used essentially the same proof to show that the bidual of a commutative order unit algebra is an order unit algebra (Theorem 2.46). This is a key step in showing that commutative order unit algebras are the same as JB-algebras (Theorem 2.49), a result due to Iochum and Loupias [69, 70]' and then Corollary 2.50 follows immediately. The fact that projection lattices are orthomodular goes back to Loomis [93] for von Neumann algebras, in his lattice-theoretic axiomatization of dimension theory, and is proven for JW-algebras in [12S] and for JBWalgebras, cf. Proposition 2.25, in [S]. The standard reference for orthomodular lattices is the book of Kalmbach [81]. Orthomodular lattices play an important role in the "quantum logic" approach to the foundations of quantum mechanics. This is discussed in more detail in the notes to Chapter 3, and in the remarks after Corollary S.lS. The proof that unital order automorphisms of JB-algebras are Jordan automorphisms (Theorem 2.S0) is based on the proof of Kadison's similar result for C* -algebras [77].

3

Structure of JBW-algebras

This chapter will begin by describing the notion of equivalence of projections in JBW-algebras. (This generalizes unitary equivalence rather than Murray-von Neumann equivalence of projections in von Neumann algebras.) We will define the notion of type I and type In JBW-algebras, and describe the classification of type In JBW-factors. (For the sake of brevity, most of these results will be stated without proof, with references given to [67].) Finally, we will investigate atomic JBW-algebras, in which every projection dominates a minimal projection. Vie will show that every JBW-algebra admits a decomposition into atomic and non-atomic parts, and that the atomic part is the direct sum of type I factors.

Equivalence of projections Recall that projections p and q in a von Neumann algebra are (Murray-von Neumann) equivalent if there is an element v such that v*v = p and vv* = q, cf. (A 161). Since Jordan multiplication is commutative, it is not obvious how to generalize this notion to JBW-algebras. However, Topping [128] showed that a satisfactory theory of equivalence can be built by defining projections p and q to be equivalent if there is a finite sequence Sl, S2,"" Sn of symmetries such that 81S2 .. · S"PSn'" 82S1 = q. Note that for von Neumann algebras this implies unitary equivalence, and indeed is the same as unitary equivalence (A 179). 3.1. Definition. Projections p and q in a JBW-algebra AI are equivalent if there is a finite sequence Sl, S2, ... , Sn of symmetries such that {Sl {S2 ... {snPsn} ... S2}SJ} = q. In this case we write p rv q. We will often make use of the following stronger property. Projections p and q in a JBW-algebra lvi are S if {sps} = q. (This relation is symmetric, Slllce = 1, but is not necessarily transitive.) We will say p and q are exchangeable if there exists a symmetry that exchanges them. 3.2.

Definition.

e.xchanged by the symmetry

U;

The following class of algebras will playa key role in our analysis of the structure of JBW-algebras.

3.

80

STRUCTURE OF JBW-ALGEBRAS

3.3. Definition. Let B be a real associative algebra with identity and with involution *. Then Mn(B) denotes the *-algebra of n x n matrices with entries in B, equipped with the involution (aiJ) f-t (a;,). We let Hn (B) denote the set of Hermitian (a7J = aJ ,) matrices in Mn (B) with the Jordan product a 0 b = ~(ab + ba) for a, bE Hn(B). Let B be as above, and let {eiJ} be the standard matrix: units in lvin(B). Note that for each I and j, SiJ = eij + eJi + (1 - en - e JJ ) is a symmetry that exchanges e,i and e JJ . Thus 1 can be written as the sum of n projections pairwise exchangeable by symmetries. The lordan coordmatzzatIOn theorem (Theorem 3.27) says roughly that the converse is true: given a Jordan algebra whose identity is the sum of n exchangeable projections (4::::; n < co), then the Jordan algebra is isomorphic to Hn(B) for some associative *-algebra B. For JBW-algebras, we will see that this implies that the algebra is in fact a JW-algebra. Thus it will be very useful to be able to show that the identity can be written as a sum of at least four exchangeable projections.

3.4. Definition. Let M be a JBW-algebra. A partwl symmetry is an element of lvi whose square is a projection. We define exchange of projections by partial symmetries in the same way as for symmetries (Definition 3.2). If e is a projection and s2 = e, then we say s is an e-symmetry. Note that if 8 2 = e where e is a projection, then by spectral theory has spectrum in {O, I}, so 8 3 = 8. Thus 80 e = so 8 2 = 8 3 = s, and therefore s is a symmetry in the JBW-algebra Me = {elvi e}. 8

3.5. Lemma. Let e be a projectIOn m a lEW-algebra lvi, and let p, q be projectIOns m lvie . Then there IS a symmetry exchangmg p and q 7.ff there IS an e-symmetry exchangmg p and q. Proof. Let e be any projection such that lvie contains p and q, and let s be any e-symmetry exchanging p and q. (Note that by the remarks preceding this lemma, 8 ElvIe.) Let t = 8 + 1 - e. Since e is the identity of ]vIe, then (1- e) ox = 0 for all x E ~Me. It follows that t 2 = 1, and that to x = 8 0 x for all :r E Me. Thus by the definition (1.13) of Ut , Utp

= 2t 0 (t 0 p) -

e

0

p

=

2s

0 (8 0

p) - p.

Then

UsP

= 28 0

By hypothesis UsP P and q.

(8 0

p) -

82 0

P

=

28 0 (S 0

= q, so Utp = q. Thus

p) - p

= Utp.

t is a symmetry that exchanges

EQUIVALENCE OF PROJECTIONS

81

Now let t be any symmetry that exchanges p and q, and let f = pVq. Define 8 = {ftf}. Recall that Ut is an automorphism of M (Proposition 2.34) and so Ut takes f = p V q to (Utp) V (Utq) = q V P = f. By the Jordan identity (1.15) 82

=

{ftf}2

=

{f{tf 2t}f}

=

{f{tft}f}

=

{fff}

=

f,

so 8 is an f-symmetry. Since p and q are in face(j) = im+Uj identity (1.16)

,

by the

Thus 8 is a (p V q)-symmetry that exchanges p and q. By assumption !vIe contains p and q. By the previous paragraph, there is a (p V q)-symmetry 8 that exchanges p and q. By the first paragraph of this proof, applied to the pair (]vIe, Mpvq) in place of (M, Me), we can extend the (p V q )-symmetry 8 to an e-symmetry that exchanges p and q.D

In the following, we make use of the general triple product {abc} defined previously (cf. (1.12)):

(3.1)

{xay}

= (x 0 a) 0 y + (a 0 y) 0 x - (x 0 y) 0 a.

Note that this product is symmetric in x and y, i.e., {xay} = {yax}. By linearity of the triple product in each factor,

{(x

+ y)a(x + y)} = Uxa + Uya + 2{xay},

so

(3.2) Recall that if p is a projection in a lB-algebra and t = P - pi, where pi = 1 - p, then t is a symmetry, and each symmetry arises in this way from a unique projection. Note also that the equivalence of (iii), (iv), (vi) in Lemma 3.6 below generalizes the similar equivalence for von Neumann algebras in (A 188).

3.6. Lemma. Let A be a lB-algebm, and let p be a pro)ectwn m A. Let pi = 1 - p, t = P - p', a E A. Then the followmg are equwalent:

(i) Upa = Up,a = 0, (ii) po a = pi 0 a = ~a, (iii) t 0 a = 0, (iv) Uta

= -a, (v) a E {pAp'}.

82

3

= q - q' wheTe q

If a

STRUCTURE OF JBW-ALGEBRAS IS

a proJectIOn, the above aTe equwalent to

(vi) Utq = q'. PTOOj. (i)

=}

(ii) By equation (1.47)

so (i) implies that poa = ~a. Then p' oa = (l-p) oa = ~a, so (i) (ii) =} (iii) Trivial, since t = P - p'. (iii) =} (iv) By the definition of Ut , Uta

=}

(ii).

= 2t 0 (t 0 a) - t 2 0 a = 2t 0 (t 0 a) - a,

so to a = 0 implies (iv). (iv) =} (i) By equation (2.25), Ut = 2Up + 2Up' - I. If (iv) holds, then

so Upa

(i)

+ Up,a =}

= o. Applying Up and Up' to this equation gives (i). (v) By (3.2),

(3.3) Then (i) implies a = 2{pap'} E {pAp'}. (v) =} (i) If a E {pAp'}, then by (3.3), for some b E A, we have a = ~ (I - Up - Up' )b. Applying Up and Up' to both sides gives (i). (iv) =} (vi) Let q be a projection and a = q - q'. If (iv) holds, then Ut (q - q') = q' - q. Adding to this the equation Ut 1 = 1 in the form Ut(q + q') = q + q' gives Ut (2q) = 2q', which implies (vi). (vi) =} (iv) If Utq = q', since U'(: = I, then Utq' = q. Now Ut(q-q') = q' - q follows. 0 3.7. Definition. Orthogonal projections p, q in a lB-algebra are stTongly connected if there exists a (p + q )-symmetry s in {plvl q}. 3.8. Lemma. OTthogonal pTOJectIOns p, q In a lBW-algebm Nl are stTOngly connected iff they aTe exchangeable by a symmetry.

PTOOj. We first show that

(3.4) If x

E

{pM q}, then by (3.2) there exists b E M such that

COl\lPARISON THEORY

83

Then since UpUp+ q = Up and UqUp+ q = Uq (cf. Proposition 2.26),

This proves the left side of (3.4) is contained in the right, and the opposite inclusion is clear, so we have established (3.4). Thus p and q are strongly connected in /lv1 iff they are strongly connected in M p +q . By Lemma 3.5, p and q are exchangeable by a symmetry in M iff they are exchangeable in M p + q ' Thus by working in M p + q , without loss of generality we may suppose that p + q = 1, so that q = pl. Suppose that s is a symmetry, and let t = P - pl. Applying the equivalence of (iii) and (v) in Lemma 3.6 we have

s E {pMp/} ~ to

(3.5)

S

=

o.

By the equivalence of (iii) and (vi) in Lemma 3.6 (with s in place of t and p in place of q)

sot = 0 ~ UsP = pl.

(3.6)

Combining (3.5) and (3.6), we conclude that p and pi are strongly connected iff they are exchanged by a symmetry, which completes the proof of the lemma. D

Comparison theory

In this section we state the basic results on equivalence of projections, just giving a reference for most proofs. Recall that the central cover c(p) of a projection p is the least central projection that dominates p, cf. Definition 2.38. 3.9. Lemma. Equwalent proJectzons zn a lEW-algebra .M have the same central cover.

Proof.

By [67, 5.1.4], if p is a projection in !v1 with central cover

c(p), then c(p) is the least upper bound in the lattice of projections of all projections equivalent to p. The lemma follows. D One virtue of Murray~von Neumann equivalence of projections is countable additivity (A 163). As remarked before, in the von Neumann algebra context the Jordan notion of equivalence is the same as unitary equivalence, so it is easy to see that countable additivity fails. However, We have the following substitute.

84

3

STRUCTURE OF .JBW-ALGEBRAS

3.10. Proposition. Let {Pal and {qa} be orthogonal collectwns of pro]ectwns wzth La (Pc> + qo:) ::; l. If Pc> and qa are exchanged by a symmetry for each a! then P = La p" and q = La qc> are exchanged by a symmetry. Proof. [67, Lemma 5.2.9]. D 3.11. Corollary. If AI IS a lEW-algebra m whlch 1 lS the sum of mfimtely many exchangeable pro]ectwns, then for any n < 00, 1 lS the sum of n exchangeable pro]ectwns. Proof. We may divide the set of exchangeable projections into 17 equinumerous subsets, and form the sum of each. The resulting 17 projections can be exchanged by symmetries by Proposition 3.10. D The following is an analog of the comparison theorem (A 164) for projections in a von Neumann algebra, and directly generalizes (A 180). 3.12. Theorem. (Jordan comparison theorem) Let P and q be projections in a JBW-algebra AI. Then there is a central projection c such that cp is exchangeable by a symmetry with a subprojection of cq, and c'q is exchangeable by a symmetry with a subprojection of c'p. Proof. [67, Thm. 5.2.13]. D The following result relates the centers of IvI and Alp. Its proof relies on the Jordan comparison theorem. 3.13. Proposition. Let p be a pro]ectwn m a lEW-algebm AI. and let Z be the center of M. Then the center of Aip lS Zp. Furthermore,

(i) The map z

f--7 zp (or equwalently, the map Up) lS an lsomorplnsm from c(p)Z onto the center Zp of Mp. (ii) If e lS a central pro]ectwn m M p , then e = c(e)p.

Proof. The fact that the center of Alp is Zp can be found in [67, Prop. 5.2.17]. The statement (ii) is part of the proof of [67, Prop. 5.217]. Finally, to show the map z f--7 zp is I-Ion c(p)Z, we show that its kernel is trivial. Let w be any central projection in c(p)Z such that wp = 0, and let w' = 1 - w. Then w' p = p, so w' ::;:. c(p). Since w E c(p) Z, then w :::; c(p) :::; w', which implies w = O. D

GLOBAL STRUCTURE OF TYPE I .JEW-ALGEBRAS

85

Global structure of type I JBW-algebras JBW-algebras can be classified into types I, II, III (cf. [67, Thm. 5.1.5]) in a manner similar to that for von Neumann algebras. However, we will just be interested in a decomposition into type I and non-type I summands (soon to be defined), and the further decomposition of type I into type In summands for n = 0,1,2, ... ,00. We review the situation for type I von Neumann algebras, cf. (A 168). The type I factors are the algebras B (H) where H is a finite or infinite dimensional Hilbert space (d. [80, Thm. 6.6.1]). The dimension of H is the same as the cardinality of any set of minimal projections with sum the identity. More generally one defines a von Neumann algebra to be of type In if the identity is the sum of n equivalent abelian projections. Every type I von Neumann algebra is the direct sum of algebras of type In for various cardinal numbers n. We will see that the situation is quite similar for JBW-algebras, except that there is a greater variety of type I factors (especially of types 12 and 13).

3.14. Definition. A projection p in a JBW-algebra M is abelwn if the algebra Mp = {pM p} is associative. Recall that Ta denotes Jordan multiplication by a. For elements a, b, c of a Jordan algebra,

so a Jordan algebra is associative iff all elements operator commute (cf. Definition l.46). Hence a projection p in a JBW-algebra M is abelian iff all elements of Mp operator commute. Thus p is abelian iff every element of Mp is central in !vip. If M is a concrete JW-algebra, then it follows from Proposition l.49 that p is abelian iff Mp consists of mutually commuting elements. Thus this generalizes the notion of abelian projections in von Neumann algebras, cf. (A 165).

3.15. Definition. A JBW-algebra is type I if it contains an abelian projection with central cover l. 3.16. Lemma. Let M be a lEW-algebra. Then there zs a unzque decompositzon M = Ml EEl M2 where Ml zs of type I and M2 has no summand of type 1. Proof. [67, Thm. 5.1.5]. 0 As discussed after Definition 3.3, one of our goals will be to describe circumstances when the identity can be written as the sum of (ideally at least four) equivalent projections. In lBW-algebras without type I direct sUmmands we have the following result.

86

3

STRUCTURE OF JBW-ALGEBRAS

3.17. Proposition. Let M be a lEW-algebra wdh no type I d~rect summand. Then there are four pro]ectwns w~th sum 1 such that each pa~r is exchangeable by a symmetry.

Proof. [67, Prop. 5.2.15]. D The following result makes specific one way in which type I lBWalgebras have "many" abelian projections.

3.18. Lemma. If q ~s any pro]ectwn m a type I lEW-algebra .M, then there ~s an abelzan pro]ectwn p w~th p -s: q and c(p) = c(q).

Proof. [67, Lemma 5.3.2]. D Recall that equivalent projections have the same central cover (Lemma 3.9). The following strong converse is true for abelian projections.

3.19. Lemma. If p and q are abelzan pro]ectwns m a lEW-algebra wdh the same central cover, then there ~s a symmetry exchangmg p and q.

Proof. [67,5.3.2]. D 3.20. Corollary. Equwalent abelian pro]ectwns m a lEW-algebra are exchangeable by a symmetry.

Proof. By Lemma 3.9, equivalent projections have the same central cover. By Lemma 3.19, ahelian projections with the same central cover are exchangeable. D 3.21. Definition. Let A1 be a lBW-algebra and n a cardinal number. Then M is of type In if the identity of M is the sum of n equivalent abelian projections. (If n is an infinite cardinal, such a sum converges (J-strongly, cf. Lemma 2.43.) M is of type rXJ if M = €Be> Me; where each Me> is of type Inu for nc; an infinite cardinal.

If M is of type In, and {Po,} is a collection of equivalent abelian projections with sum 1, then the central covers c(Pex) coincide, and must dominate every Pc;. Therefore this common central cover must equal l. Conversely, if {Pex} is a collection of abelian projections with sum 1 and with c(Pe;) = 1 for all a, then these projections are equivalent (Lemma 3.19). Thus in the definition above we could replace the requirement that the projections be equivalent by the requirement that their central covers are all equal to 1. (This is the form of the definition of type In in [67, 5.3.3].) Note that in the definition ahove of a lBW-algehra of type Lxo, we are not assured that the identity is a sum of equivalent abelian projections. Finally, observe that M is of type h iff the identity 1 is an abelian projection, i.e., iff M is associative.

CLASSIFICATION OF TYPE I JBW-FACTORS

87

3.22. Lemma. Let M be a lEW-algebra of type In wzth n < 00. Then n lS umquely determmed by !'vI, and any set of orthogonal non-zero projectwns wlth the same central cover has cardmalzty at most n. Proof. [67, Lemma 5.3.4]. 0

3.23. Theorem. Let M be a lEW-algebra of type 1. Then there is a unique decomposltwn

where each Mn

lS

edher {O} or of type In.

Proof. [67, Thm. 5.3.5]. 0

3.24. Proposition. If M lS a lEW-algebra of type I cx)) then the identity can be expressed as the sum of four exchangeable pro]ectwns. Proof. Write !'vI = EB", !'vI", where M", is of type Ina for n", an infinite cardinal. Fix oc, and write the identity of AI", as a sum of an infinite collection of equivalent abelian projections. These projections are exchangeable (Corollary 3.20), so by Corollary 3.11 we can write the identity of M", as a sum of four projections e"" f"" g"" h", exchanged by symmetries. Repeat this construction for each oc, and let e = 2:", e"" f = 2:", f"" g = 2:", g"" and h = 2:", h",. Then e, f, g, h are projections pairwise exchangeable by symmetries (Proposition 3.10), and 1 = e + f + g + h, which completes the proof. 0

Classification of type I JBW-factors

3.25. Definition. A JBW-algebra is a lEW-factor if its center consists of scalar multiples of the identity. Since the center of a JBW-algebra is a JBW-subalgebra (Proposition 2.36), by the spectral theorem it is generated by its central projections. Thus a JBW-algebra M is a factor iff the only central projections are 0 and 1. Our main purpose in this section is to develop the classification of type I JBW-factors. We will see that for n > 4 the classification of JBW-factors of type In is analogous to that for von---Neumann algebras (except that the Hilbert space can be real, complex, or quaternionic), but that there are new possibilities for types hand 1 2 . The key to the classification of type I factors is the Jordan coordinatization theorem, which describes circumstances under which a Jordan algebra is isomorphic to one of the form Hn(B) (see Definition 3.3). For n = 3, we have to allow algebras B which are not quite associative.

3.

88

STRUCTURE OF JBW-ALGEBRAS

3.26. Definition. An algebra B is alternatwe if it satisfies the identities a 2 b = a(ab)

ba 2

and

= (ba)a.

3.27. Theorem. (Jordan coordmatlzatzon theorem) Let M be a Jordan algebra over R wdh ldentdy 1. Assume that 1 lS the sum of n ldempo tents PI,"" Pn (3 ::; n < (0) wdh each palr Pi) Pj exchangeable by symmetries. Then there lS a * -algebra B over R and a Jordan lsomorphlsm of iliI onto Hn (B) carrymg Pi to the matTlX umt eii for each L Any such algebra B lS alternatwe) and wlll be assocwtive If n 2': 4. Proof. [67, Thm. 2.8.9 and Thm. 2.7.6]. (The statement of the Jordan coordinatization theorem in [67, Thm. 2.8.9] assumes that the projections are "strongly connected", which by Lemma 3.8 is the same as the projections being exchanged by a symmetry.) 0

We will explain the idea of the proof of Theorem 3.27. Suppose that Hn(B) with n = 3. The key is the observation for 3 x 3 matrices that the associative product is to the following extent determined by the Jordan product:

M

~

ab

(3.7)

o o

~) o

= 2 (

o o

~

a*

o

o o b

We can identify the algebra B as a vector space with the matrices of the form

(~

a

o o

i.e., with the subspace {ellMe22}. Then we can use (3.7) to recover the associative multiplication from the Jordan product. However, for this purpose we need to be able to recover ae13+a*e31 from ae12+a*e21 and b*e23+be32 from be12 + b*e21. We can use the symmetries S23 = ell + e23 + e32 and S13 = e22 + e13 + e31, since

and

CLASSIFICATION OF TYPE I JBW-FACTORS

Thus for x, y E B

=

Similarly, letting 512

89

{ellM e22} the equation (3.7) becomes

=

el2

+ e21 + e33,

then we can recover the involution

since

* on

B

= {ellN! e22} by

Now for a Jordan algebra satisfying the hypotheses of the Jordan coordinatization theorem, one imitates this process, replacing the {eid by the given projections {pd, and the {Si)} by the given symmetries that exchange the projections. Finally, the Jordan axiom (1.2) is used to show that B is associative (if n 2:: 4), and alternative if n = 3. Note that this construction depends on the choice of the projections and the symmetries exchanging them, and so the algebra B may not be uniquely determined by M. 3.28. Definition. A non-zero projection p in a JBW-algebra M is minimal or an atom if q ::;: p for a non-zero projection q implies q = p. 3.29. Lemma. A proJectwn P m a lEW-algebra M zs mmimal iff

Mp =Rp. Proof. Every projection in Mp is dominated by p, so if p is an atom, then the only non-zero projection in the JBW-algebra Mp is p. Then by the spectral theorem (Theorem 2.20), Mp = Rp. Conversely, since face(p) = im+Up = M p , every projection under p is in M p , so if Mp = Rp then there are no projections between 0 and p, i.e., p is an atom. 0

3.30. Lemma. A non-zero proJectwn m a lBW-factor zs abelzan zff it is minzmal. Proof. If p is minimal, then Mp = Rp is associative, so by definition p is abelian. If p is abelian, and q is a projection with q ::;: p, then q is central in Mp (see the remarks following Definition 3.14). Then by Proposition 3.13(ii), q = c(q)p, and c(q) must be 0 or 1 since M is a factor. Thus p is minimal. 0

3.31. Definition. H will denote the quaternions, and 0 will denote the octonions (or Cayley numbers) with their usual involutions. We note that H is four dimensional over R, and 0 is eight dimensional. The selfadjoint elements are just real multiples of the identity, and the skew adjoint elements are spanned by the elements b such that b2 = -1. H is associative but 0 is only alternative. See [67, proof of Thm. 2.2.6].

90

3.

STRUCTURE OF JBW-ALGEBRAS

3.32. Theorem. Let M be a .lBW-factor of type In for 3 :S- n < Then 1M ~ Hn(B) where B 28 *-2somoTphl.C to R, C, H 01' O. In paTt2culaT, !vI 28 fimte d2men8zonal. If n ~ 4, then B 28 *-280moTph2c to R, C, orH. 00.

Proof. Let PI, ... , Pn be abelian projections with sum 1 with each pair exchangeable by symmetries. Choose a *-algebra B over R as in Theorem 3.27. (Note that if n = 3, then B might not be associative but rather only alternative.) Let {ei]} be the canonical matrix unit;; of Mn(B). Since el1 = PI is abelian, and Af is a factor, then el1 is minimal (Lemma 3.30). Thus by Lemma 3.29, {ellMell} = Rell, and {ellMell} = {ellHn(B)ell} = B"aell, so (3.8)

We are going to show that B is an (alternative) division algebra over R, and is quadratic (i.e., every element b E B satisfies a quadratic equatioll over R). Then we will make use of the result of Albert [4] that such an algebra must be isomorphic to R, C, H, or 0, cf. [67, Thm. 2.2.6]. Let b E B. Since Boa = Rl, we can write ~(b + b*) = Al for A E R. Then b - Al = b - ~(b + b*) = ~(b - b*) is skew-adjoint, so its square is selfadjoint. Therefore (b-Al)2 = 0'1 for some O'ER. Expanding we conclude that b satisfies a quadratic equation of the form

b2 + {3b

+ ,I = 0

for some scalars (3, f. Next we show that B is a division algebra, i.e., that each element of B is invertible. Let b be a non-zero element of B. If above , -# 0, then b(b + (31) = -,I implies that b is invertible. If, = 0, then b2 = -{3b, so by the alternative law (Definition 3.26)

b(bb*) = b2b* = -{3bb* and

(b*b)b

= b*b 2 = b*( -(3b) = -{3b*b.

If either of the scalars b* b or bb* is non-zero, then we call conclude that b = -{31, where (3 -# 0, and so again we have shown b is invertible. However, both b* band bb* cannot be zero, since if they were, then the fact that be12 + b*e2I E Hn(B) = AI, and

would force b = O. This completes the proof that B is a division algebra.

CLASSIFICATION OF TYPE I JBW-FACTORS

91

We conclude that B equals one of R, C, H, or 0 as algebras over R. Finally, we check that the involution on B is the standard one. Note that B is spanned as a vector space over R by 1 and the elements) such that j2 = -1, so the involution is determined by its action on such elements. We are going to show )* = -). By (3.8) there is a nOll-zero real nUll1ber 8 such that )*) = 8l. By the alternative law we have ()*))) = )*()2), so 8j = -)*. Then )* = -8), so

which implies that 5 = ±l. If 5 = -1, then J* = ), so ) E Bsa = R1, which contradicts J 2 = -1. Thus 8 = 1, so we have shown ) * = -). This completes the proof that there is a unique involution on B, and so the involution is the standard one. D Recall that finite dimensional formally real Jordan algebras are the same as finite dimensional JB-algebras. (See the Notes to Chapter l.) Each of the algebras listed in Theorem 3.32 is formally real [58, Prop. 2.92], and thus is a JBW-algebra. It is straightforward to check that the center is R1 in each case, so each is a JBW-factor of type 1. The analogous classification of finite dimensional formally real algebras is due to Jordan, von Neumann, and Wigner [66]. The methods described so far leave out JB\V-factors of type 12 , which we will now discuss. 3.33. Definition. Let N be a real Hilbert space of dimension at least

2, and let R1 denote a one dimensional real Hilbert space with unit vector l. Let M = NEB R1 (vector space direct sum). Give f-.1 the product 0 defined by (a

+ oj) 0

(b

+ ,81) =,8a + O'b + ((a I b) + O',8)l.

Give M the norm Iia + Alii = IIal12 + IAI (where space norm). Then !vI is called a spm factor.

I . 112

denotes the Hilbert

We will see in Proposition 3.37 that a spin factor is a JBW-factor, and so we will immediately apply the terminology of JB-algebras (e.g., we will refer to an element s of a spin factor as a symmetry if s2 = 1.) 3.34. Lemma. Let !vI be a spm factor, and wnte !vI = N EB R1 as in Defimtwn 3.33. Then N conszsts of all multzples of symmetnes m !vI other than ±l.

Proof. From the definition of the product on N, each unit vector in N is a symmetry, and thus each element of N is a multiple of a symmetry.

92

3.

STRUCTURE OF JEW-ALGEBRAS

Conversely, suppose 8 = n +),1 is a symmetry with n E N. Then 8 2 = 1 = (n + ),1)2 = (n I n)1 + 2),n + ),2l. Then we must have 2),n = O. Either n = 0 (which forces ),2 = 1 and thus 8 = ),1 = ±1), or else), = 0, which implies 8 = n E N. 0

3.35. Definition. Let M be a JB-algebra. Elements a, b in ]vI are Jordan orthogonal if a 0 b = O. Note that in a concrete JW-algebra, elements are Jordan orthogonal iff they anti-commute. In a spin factor M = N EEl Rl, each unit vector in N is a symmetry, and such symmetries are Jordan orthogonal iff they are orthogonal in the Hilbert space N. Thus an orthonormal basis for N is the same as a maximal set of Jordan orthogonal symmetries in N.

3.36. Example. H 2 (R) = M 2 (R)sa and H 2 (C) = Nh(Ck, are spin factors of dimension 3 and 4 respectively. To see this, in each case let N be the trace zero elements, and for x, yEN define

(XIY)=T(XOY), where T is the normalized trace (i.e., T(I) = 1), and 0 is the usual Jordan product of matrices. Note that the set of Pauli spin matrices

(3.9) are symmetries in M 2 (C)sa that are both Jordan orthogonal and orthogonal unit vectors with respect to the tracial inner product. They span the trace zero matrices, so that it is easy to see the Jordan product on .1I12(R)sa and on M2(C)sa coincides with that for a spin factor. For the quaternions we can append to the list of Pauli spin matrices two more orthogonal symmetries Je12 - Je2l and ke12 - ke2l and then verify in similar fashion that with the usual Jordan product H2(H) is a spin factor. Similarly we can show that H 2 (O) is a spin factor. The fact that the usual norm on these algebras matches that defined for spin factors follows easily from the spectral theorem.

3.37. Proposition. Every 8pm factar zs a JEW-factar af type and every JEW-factar af type h zs a spm factar.

h,

Proaf, Let M = N EEl Rl be a spin factor as defined above. We will first show M is a JB-algebra. Note that all of the axioms for the Jordan product and the norm in a JB-algebra refer just to two elements.

CLASSIFICATION OF TYPE I JBW-FACTORS

93

Furthermore, since by definition NoN c Rl, if a and b are arbitrary elements of M, then Rl + Ra + Rb is a subalgebra, which we can write in the form H EB Rl where H is a 2-dimensional subspace of N. It then suffices to show that H EB Rl is a JB-algebra. Note that H EB Rl is itself a 3-dimensional spin factor. However, we saw above that M2(R)sa is a 3-dimensional spin factor, and it is clear that two spin factors of the same dimension are isomorphic. Since .M2(R)sa is a JB-algebra, then so is tl}e isomorphic algebra H EB Rl, and this completes the proof that every spin factor is a JB-algebra, except for proving norm completeness. The norm on !vI is equivalent to the norm on the Hilbert space direct sum N EB Rl. In fact

IIAI EB nl12 = (IAI 2+ IInl12 2)1/2

<:::

IAI + IInl12 = IIAI + nil,

and by the Cauchy-Schwarz inequality for R 2 ,

Thus

(3.10) so M is complete and reflexive. Thus AI is a JB-algebra and a Banach dual space, so is a JBW-algebra. To see that !vI is a factor, it suffices to show that the only central symmetries arc ±1 (so that the only central projections are 0 and 1.) Let t be any symmetry other than ±1. Then from Lemma 3.34, t must be in N. Let 5 be a unit vector (and thus a symmetry) in N orthogonal to t. Then 5 is Jordan orthogonal to t, so Ust = 280 (80 t) - 52 0 t = -to Thus by Lemma 2.35, t is not central. Finally, we show that !vI is of type 1 2 . Let 8 be any symmetry in M. Then 82 = 1 implies that the spectrum of 5 is contained in {O, I}. Since every element of M has the form al + (35 where 5 is a symmetry, it follows that each element of !vI has a spectrum containing at most two numbers. This implies that any set of orthogonal non-zero projections in M has cardinality at most 2. Thus M contains minimal (= abelian) projections, and so 11,'1 is a type I factor. By Theorem 3.23, !vI is of type In for some n. Since M does not contain three orthogonal projections, then n <::: 2. Since 1 is not abelian (indeed, we showed above that there are non-central elements in M), then M is not of type 1 1 , so then M is of type 1 2 . Conversely, let M be any JBW-factor of type 1 2 . By Lemma 3.22, any orthogonal set of non-zero projections has cardinality at most 2. Thus every projection other than 0 or 1 is a minimal projection, and so every spectral resolution (cf. Definition 2.21) contains at most one projection other than 0 and 1. It follows that the spectrum of every element has cardinality at most 2.

94

3.

STRUCTURE OF JBW-ALGEBRAS

Now let N be the linear span of the symmetries in M other than ±1. We are going to prove that

NoN C R1.

(3.11 )

Let s, t be symmetries in N. We will show sot E R1, which will prove (3.11). Let p be the projection such that s = p - p'. By equation (1.47) we have sot

= (p - pi) 0 t = (Up - Up' )t.

Since every projection other than 0 and 1 is minimal in !vi, then the range of Up is Rp and that of Up' is Rp', so sot E Rp + Rp'

(3.12)

= R1 + Rs.

On the other hand, the same argument applies to t in place of s, so (3.13)

sot

= to s

E R1

+ Rt.

If sot E R1, we have proven our claim. Otherwise, by (3.13), tos = Ad+ A2t for scalars AI, A2 with A2 =f. O. Then by (3.12), All + A2t E R1 +Rs, so t lies in R1+Rs, say t = a1+!3s for scalars a,!3. Then t 2 = 1 implies that a or 13 must be zero. Since by assumption t is not a multiple of 1, then t must be a multiple of s, and so sot E R1, which completes the proof of (3.11). We next show that M = N ffi R1. If x is any element of !vi, with spectral decomposition x = ap + !3p', then x

=

~(a

+ 13)1 + ~(a -

!3)(p - pi),

so x is a linear combination of 1 and a symmetry other than ± 1. Thus !vi = N + R1. This sum is direct, since if 1 = L: A,Si for a finite number of symmetries s 1, ... ,Sn other than ± 1, and scalars AI,"" An, then multiplying by any Sj shows that Sj is a multiple of 1, a contradiction. Since NoN C R1, then every element of N is a multiple of a symmetry. We define a symmetric bilinear form on N by x

0

y

= (x I y) 1 for x,

y in N.

Since every element of N is a multiple of a symmetry, it follows that this is positive definite, and thus equips N with an inner product. Furthermore, for s a symmetry in N and A a scalar we have

so the inner product norm coincides with the given JBW-norm on N.

CLASSIFICATION OF TYPE I JBW-FACTORS

95

Define a linear functional T on Ai by T('\l +x) = '\. Then it is easy to check that T is positive on squares and T(l) = 1, so T is a state. Its kernel is N, so N is norm closed (and thus complete) for the JBW-norm. Since the inner product norm coincides with the JB\;V-norm, it follows that N is a Hilbert space. If N is zero dimensional, then !vI = R1 is not of type 12 , If N is one dimensional, say N = Rs for a symmetry s, then one verifies that s is central so !vI is not a factor. Thus N is of dimension at least 2. From the fact that each member of N is a multiple of a symmetry, it follows at once that the Jordan product in !vI coincides with that defined for spin factors. If s is a symmetry not ±1, one easily verifies from spectral theory that

so that the JBW-norm coincides with the spin factor norm. This completes the proof that !vI is a spin factor. D We record the following fact that was established in the proof above.

3.38. Proposition. Every spm factor zs a refiexwe Banach space, and thus every state on a spm factor zs a normal state. Proof. We showed in the second paragraph of the proof of Proposition 3.37 that spin factors are reflexive. It follows that their dual and predual spaces coincide, so every state is normal (Theorem 2.55). D We have now described the type In factors for 1 :s; n < 00. Not surprisingly, the type IOCJ factors are the infinite dimensional versions of Hn(B) for B = R, C, H, i.e., the self-adjoint bounded linear operators on real, complex, or quaternionic Hilbert space. Here is a summary of type I JBW-factors.

3.39. Theorem. The type I lBW-factors can be dwzded mto the following classes (up to zsomorphzsm):

(i) the (ii) the (iii) the (iv) the (v) the the

symmetric bounded operators on a real Hzlbert space, self-adJomt bounded operators on a complex Hzlbert space, self-adJomt bounded operators on a quaternwmc Hzlbert space, spm factors, exceptwnal algebra H3(O) of 3 x 3 Hermztzan matnces over Cayley numbers.

Moreover, these classes are mutually dzsJoint, wzth the exceptwns that the matrix algebras M 2(R),a, M 2(C)sa, M2(H) are all spm factors, and (z), (ii) , (iii) coinczde for Hzlbert spaces of dzmension 1. Proof. Each type I factor is of type In for some n with 1 :s; n :s; 00 by Theorem 3.23. The factors of type 12 are the spin factors by Proposition

96

3

STRUCTURE OF JBW-ALGEBRAS

3.37, and the factors of type In for 3 :s; n < 00 are described in Theorem 3.32. In a JBW-factor of type h, the identity 1 is abelian, so every element is central. For a factor, this is possible only if the factor is R1. Finally, the fact that JBW-factors of type 100 are the bounded self-adjoint operators on a real, complex, or quaternionic Hilbert space can be found in [67, Thm. 7.5.11]. Finally, the disjointness of the isomorphism classes (i) - (v) (with the exceptions noted) follows by considering orthogonal minimal projections p and q and Mp+q. In the five cases this subalgebra will be isomorphic to M 2 (R)sa, M 2 (C)sa, M 2 (H)sa, M, and M2(O). 0

Atomic JBW-algebras Recall that a minimal non-zero element of a lattice is called an atom. In the context of JBW-algebras, the relevant lattice will be the lattice of projections, so an atom will then be a minimal non-zero projection.

3.40. Definition. A JBW-algebra M is atomzc if every non-zero projection dominates an atom. Note that by a simple Zorn's lemma argument, in an atomic JBWalgebra every non-zero projection is the least upper bound of an orthogonal collection of atoms.

3.41. Lemma. If q zs an atom m a JEW-algebra M and p zs any pT"OJectwn, then Upq zs a multzple of an atom. PT"Oof. Since q is minimal, then Uq(M) any element a in M, by the identity (1.16)

= Rq (Lemma 3.29). Thus for

(3.14) Taking a = 1, we see that {pqp}2 is a multiple of {pqp} , so {pqp} is a multiple of a projection w. By (3.14), Uw(M) = Rw, so by Lemma 3.29 w is an atom or zero. Thus Upq = {pqp} is a multiple of an atom. 0

3.42. Lemma. Let IvI be a JEW-algebra, and 'P the lattzce of projectwns m IvI. The least upper bound m 'P of the atoms m M is a central proJectwn z such that zlvI zs atomzc and (1 - z) IvI contams no atoms. Thz8 decomposztwn of IvI mto atomic and non-atormc parts lS umque. Proof. By Proposition 2.25, the lattice of projections in M is complete, so the leaiOt upper bound z of the atoms exists. Let s be any symmetry in M. Then Us is a Jordan isomorphism of M by Proposition 2.34. Therefore

ATOl\IIC JBW-ALGEBRAS

97

Us induces an isomorphism of the lattice of projections, and in particular gives a bijection of the set of atoms onto itself. Thus Usz

=

Us(V{q I q is an atom})

=V

{Usq I q is an atom}

= V{atoms in M} = z. By Lemma 2.35, z is central. To show zAI is atomic, let p be a non-zero projection in zA1. If Upq = 0 for every atom q in /1,1, then p would be orthogonal to every atom in M. By Proposition 2.18, p' would dominate every atom and therefore dominate z, so p would be orthogonal to z. Since p is not zero and p -:=; z, this is impossible. Thus there is an atom q such that Upq is non-zero, and by Lemma 3.41 Upq is a multiple of an atom. Then IjUpqll-IUpq is an atom under p, which completes the proof that zAI is atomic. By construction, z dominates all atoms, so z'M contains no atoms. Finally, if lvI = lvh EEl /1,12 with Ah atomic and Nh containing no atoms, then MI = cAJ and AI2 = c'lvI for a central projection c (Proposition 2.40). Since lvII = c/l,I is atomic, then c must be the least upper bound of atoms by the remark following Definition 3.40. For every atom q, c'q is a projection under q, and so c' q must equal either zero or q. Since c' A1 contains no atoms, then c' q = 0, and so q = cq -:=; c. Therefore every atom is under c, and thus c is the least upper bound of the atoms in NI, i.e., c = z. This establishes the uniqueness of the decomposition of A1 into atomic and non-atomic parts. 0 3.43. Lemma. If p Z8 an atom zn a lEW-algebra lvI, then the central Cover c(p) zs a mzmmal pro]ectzon zn the center of AI. Proof. Let w -:=; c(p) be a central projection. Then wp is a projection under p. Since p is an atom, wp = p or wp = O. In the first case, by the definition of the central cover, w :::. c(p), and then w = c(p). In the second case, (c(p) - w)p = p, so c(p) - w :::. c(p). Thus w = O. Therefore c(p) is minimal among central projections of /1,1. 0

3.44. atom.

Proposition.

A lEW-factor NI zs type I 1.fj zt contazns an

Proof. If Ai is type I, then by definition it contains an abelian projection, and an abelian projection in a factor is an atom (Lemma 3.30). Conversely, suppose that M contains an atom p. Then p is abelian, so its central cover must be 1 (since a factor contains no central projections other than 0 or 1). Thus M is of type I. 0

98

3

STRUCTURE OF .lBW-ALGEBRAS

3.45. Proposition. A lEW-algebra !vI sum of type I factors.

~s atom~c ~ff

zt

~s

the d7rect

Pmoj. Suppose first that IvI is atomic. Let {c",} he a maximal collection of orthogonal minimal elements of the center of !vI. We claim I:", c'" = l. If not, then hy atomicity of M there is an atom p ::; 1- I:", C",. Then c(p) ::; 1 - I:", C a , so c(p) is a minimal clement of the center of M (Lemma 3.43) orthogonal to all C"" This contradicts the mdximality of the collection {c"'}' so we have shown I:", c'" = 1. Let !vI", = c",M. Then IvI is the direct sum of the lBvV-suhalgehras AI",. Any central projection in c",M is central ill IvI, so hy minimality of c'" in the center of M, we conclude that AIa is a factor. Note that hy atomicity of AI, each c'" dominates an atom, say p". Thus hy Proposition 3.44, Ma is of type I. Conversely, suppose that !vI is the direct sum of type I factors. By Lemma 3.18 every projection in a type I lBW-algehra dominates an ahelian projection, and hy Lemma 3.30, every ahelian projection in a factor is an atom. Thus every type I factor is atomic. It follows that every direct sum of type I factors is atomic. 0 We are going to develop the theory of dimension for projections in atomic JBW-algehras. The next two results are useful technical tools for that purpose.

3.46. Lemma. Let p, q be pT'O]ectwns m a lEW-algebra. Then there eX7sts a symmetry s E IvI such that

Us ( {pqp }) = {qpq}.

Pmoj. (Sketch) Let a = p + q - l. Let s he a symmetry in W(a,l) such that so a = lal. Then by a lordanized version of the argument in the proof of [AS, Lemma 6.46]' one shows that s has the desired property. See [67, Lemma 5.2.1] for details. 0 3.47. Lemma. Let p, q be pT'O]ectwns m a lEW-algebra. eX7sts a symmetry s E M such that

Us (p V q - p)

=

There

q - (p /\ q).

Pmoj. By Lemma 3.46 there is a symmetry s such that Us exchanges {p' qp'} and {qp' q}. Since Us is a Jordan isomorphism, it will exchange the associated range projections: (3.15)

Us(r({p'qp'})

=

r({qp'q}).

ATOMIC JBW-ALGEBRAS

99

From Proposition 2.31 and equation (2.17) (3.16)

l' (

{pi qp'}

=

(p V q) i\ pi

=

(p V q) - p

and (3.17)

1'( {qp' q})

= (q' V pi) i\ q = q i\ (q i\ p)' = q - (q i\ p).

The lemma follows from (3.15), (3.16), and (3.17). D The following result will not be needed in the sequel, but seems of independent interest. Let L be a lattice and a, bEL. An ordered pair (a, b) is a modular pmr (written (a, b)M) iffor all x in L with ai\b ::; x::; b there holds x = (x Va) i\ b. A lattice is sermmodular if the relation M is symmetric, i.e., if (a, b)M implies (b, a)M. Topping [128] showed that the projection lattice of a von Neumann algebra is semimodular. 3.48. Corollary. The semzmodular.

latt~ce

of pro]ectzons of a lEW-algebra M

~s

Proof. By Proposition 2.25, the lattice of projections in At is orthomodular. By [96, Cor. 36.14 and Thm. 29.8] in order to prove this lattice is semimodular it suffices to show that whenever p and q satisfy p V q = 1 and p i\ q = 0, then there is a lattice isomorphism that takes p to q' and q to p'. If s is the symmetry in Lemma 3.47, then Us induces a lattice isomorphism that takes (p V q) - p = 1 - P = pi to q - (p i\ q) = q. Then Us also takes p = 1 - pi to q' = 1 - q. Furthermore, Usq' = U;p = p, so Usq = p'. Thus Us is the desired lattice isomorphism. D

We will now discuss the notion of dimension in the lattice of projections of a lBW-algebra. We will say a projection in an atomic lBW-algebra M is finite if it is the least upper bound of a finite set of atoms. The minimum number of atoms whose least upper bound is p is called the dzmenszon of p, and is denoted dim(p). (For general background on dimension in atomic lattices, see (A 41) and the succeeding results in the Appendix.) 3.49. Definition. If p, q are projections in a lBvV-algebra with p ::; q, we say q covers p if q - p is an atom. (This is equivalent to saying that p ::; q and that there is no projection strictly between p and q.) 3.50. Lemma. (The coverzng property) Let p and u be pro]ectzons in a lEW-algebra M wzth u an atom. Then e~ther u ::; p or else p V u Covers p. Proof. By Lemma 3.47, there is a symmetry s such that Us exchanges (p V u) - p with u - (u i\ p). Since u is an atom, then u - (u i\ p) is either

100

3

STRUCTURE OF JBW-ALGEBRAS

equal to u or is zero. Since Us takes atoms to atoms, then (p V u) - p is either an atom or zero. In the former case, (p V u) covers p. In the latter case, p = p V u so u ::; p. 0

3.51. Proposition. Let p be a jinzte pm]ectzon m an atomzc lEWalgebra M. Then p can be expressed as a jinzte sum of atoms, and the cardmalzty of any set of atoms wzth sum p zs dzm(p). Furthermore, every pm]ectzon q < p zs jinzte wzth dzm( q) < dzm(p). Proof. The result stated in the proposition follows from the fact that the projection lattice in a JBW-algebra has the covering property (Lemma 3.50) and from the general lattice-theoretic properties of dimension (A 44). (If q ::; p, (A 44) only states that dim(q) ::; dim(p). However, if q ::; p and dim(q) = dim(p), then q = p, or else p - q could be expressed as a finite sum of atoms, leading to p = q + (p - q) being expressed as the sum of more than dim(p) atoms.) 0

3.52. Definition. If M is a JBW-algebra, then M f denotes the linear span of the atoms in .M. 3.53. Lemma. Let M be a lEW-algebra. Then each element x E ]l.i! can be expressed as a lmear combmatzon of orthogonal atoms, x = Li AiPi. Proof. Note that all atoms by definition are in the atomic part of ]l.i, so the same is true of their linear span. Thus without loss of generality we may assume !vI is atomic. Let x be a linear combination of atoms ul, ... , Uk. Let p =- Ul V ... V Uk. Note that x E !vip. vVe will prove the lemma by induction on dim(p). If dim(p) = 1, then p is an atom and !vIp = Rp, so the result is clear. Suppose that the lemma holds for dim(p) ::; n - 1. To carry out the induction, suppose that dim(p) = n and that x E Mp. If x is a scalar multiple of p, by Proposition 3.51, p (and then x) is a finite sum of orthogonal atoms, so the result holds. Otherwise, there is some A E R such that x - AP l 0 and x - AP "i o. By Theorem 2.20 (i) and (ii) applied to !vIp, there is a projection q in ]l.lp such that

(3.18) and

If q = p, since x E Mp then x - Ap = Up(x - Ap) = Uq(x - Ap) 2: 0, contrary to our choice of A. Similarly p-q =I p. Thus by Proposition 3.51, q and p - q are projections with dimension strictly smaller than dim(p).

NOTES

101

By our induction hypothesis, we can express Uqx as a linear combination of orthogonal atoms in JvIq and Up_qx as a linear combination of orthogonal atoms in !vIp_q. Each atom in JvIq is orthogonal to each atom in !I/[p_q, so by (3.18) we are done. 0

Notes Structure theory for JEvV-algebras is similar to that for von Neumann algebras, but differs in some important ways. In each case, it depends on comparison of projections. However, there is no good analog of Murray-von Neumann equivalence in JEW-algebras, so the definition of equivalent projections relies on exchanging projections by products of symmetries (Definition 3.1). In the context of von Neumann algebras, this is the same as unitary equivalence (A 179). Topping [128] made a thorough study of this notion of equivalence in the context of JW-algebras, and was able to carry over to JW-algebras many of the standard results on comparison of projections and structure of von Neumann algebras. Some of these results were then carried over to the context of JEW-algebras (especially JEW-factors) in [8], and were treated more systematically in [67]. The description of the center of JvIp in Proposition 3.13 appears in [44]. For arbitrary Jordan algebras over R there is the classical coordinatization theorem (Theorem 3.27), cf. [72], which implies that JEW-algebras in which there are n exchangeable projections with sum 1 can be represented as n x n hermitian matrices with entries in a *-algebra B. In the type I factor case, this algebra can be taken to be the reals, complexes, quaternions, or oct onions (if n = 3), cf. Theorems 3.32 and 3.39. To show this, we have relied on the result of Albert [4] that these are precisely the alternative division algebras over R such that each element satisfies a quadratic equation. Alternatively, we could have relied on the classification of finite dimensional formally real Jordan algebras in [75]. The infinite dimensional type I factor classification can be found explicitly in [67]' and appeared first in [125]. Spin factors were first studied by Topping [128, 129]. The result that the spin factors are exactly the JEW-factors of type 12 (Proposition 3.37) is due to St0rmer [125]. Concretely represented atomic JW-algebras were studied in [128]' where the JW-algebra version of the results in the last section of this chapter is proven. The semimodularity of the projection lattice in a concretely represented JW-algebra is in [128]' and is proved for JEW-algebras (Corollary 3.48) in [10]. The covering property (Definition 3.49), and the definition of dimension for atomic lattices with the covering property, is a lattice-theoretic abstraction of dimension for subspaces of a finite dimensional vector space, and is due to Maclaren [95]. Just as Jordan algebras originated in investigations of the foundations of quantum mechanics, there is also a lattice-theoretic approach to these

102

3

STRUCTURE OF JBW-ALGEBRAS

foundations, usually called the "quantum logic" approach, which originated with the paper [29] of Birkhoff and von Neumann. Here the elements of the lattice are thought of as propositions, and one hopes to impose physically meaningful axioms that will lead to the lattice being isomorphic to the lattice of projections of B (H), or to more general lattices such as the lattice of projections of a von Neumann algebra or JBW-algebra. If M is a type In JBW-factor with 3 ::; n < 00, the lattice of projections of IvI will form a projective geometry. For n = 3, this will be a projective plane, with the one dimensional projections being the points of the projective plane, and the two dimensional projections being the lines. If n ~ 4, by the fundamental theorem of projective geometry there will be a division ring D and a finite dimensional vector space V over D such that the lattice of projections of AI will be isomorphic to the lattice of subspaces of V. If n = 3, such a division ring D will exist iff the projective geometry is Desarguesian, cf. [131, Thm. 2.16]. The projection lattice of H3(O) gives an example of a non-Desarguesian projective plane. A projective geometry is the same as an irreducible modular complemented lattice of finite rank. Von Neumann's continuous geometries [99] are one generalization of this to infinite rank, and the lattice of projections of a type II 1 von Neumann factor will be a continuous geometry. However, in general, projection lattices of von Neumann algebras or JBW-algebas (even of B(H) for H infinite dimensional) are not modular. Orthomodularity (and semimodularity) are weakened forms of modularity that are satisfied by these projection lattices, as we have seen. For a more detailed discussion of the quantum logic approach, see the books of Beltrametti and Cassinelli [28], Varadarajan [131]' and Piron [103]. A more complete discussion of the role of the octonions in a variety of fields of mathematics, and of H3(O) in particular, can be found in the survey article of Baez [24]. Some physical applications of the octonions and the associated exceptional Jordan algebra H3(O) are discussed in the papers of Corrigan and Hollowood [38]' and Sierra [119].

4

Representations of JB-algebras

In this chapter we will discuss representations of JB- and JB\V-algebras as Jordan algebras of self-adjoint operators on a Hilbert space. We will see that there is one crucial difference compared to the situation for C*algebras: not every JB-algebra admits such a concrete representation. The "Gelfand-Naimark" type theorem (Theorem 4.19) states that there is a certain exceptional ideal, and modulo that ideal every JB-algebra admits a concrete representation, i.e., is a ,lC-algebra. The first section will make use of the structure theory developed in the last chapter to determine which ,lBW-factors admit concrete representations (i.e., are JVV -algebras). The next section is devoted to the Gelfand-N aimark type theorem for JB-algebras mentioned above. We then examine the relationship between a .IC-algebra A and the real *-algebra it generates. \Ve will see that for many ,lC-algebras, no matter what the representation, the algebra is the self-adjoint part of the real *-algebra it generates. (We will call this "universal reversibility".) In the penultimate section, we study the relationship between a ,lC-algebra and the C*-algebra it generates. Here the situation depends very much on the particular representation: every non-associative ,lC-algebra admits representations where it is not the self-adjoint part of the C* -algebra it generates. However, there is a particular representation which is quite useful, in which the generated C*-algebra is called the universal C*-algebra. This has the property that Jordan homomorphisms of the JC-algebra into any C*-algebra extend to *-homomorphisms on the universal C*-algebra. The final section introduces Cartesian triples in JB\V-algebras. Such triples played a central role in our investigations of orientations for von Neumann algebras, cf. (A 197), and will also playa role in our characterization of normal state spaces of von Neumann algebras in a later chapter.

Representations of JBW-factors as JW-algebras In this section we will show that every JB\V-factor except H3(O) is a JW-algebra, and that H3 (0) cannot be so represented.

4.1. Theorem. Every spm factor

7S

a lW-algebm.

Proof. We first construct representations for finite dimensional spin factors from tensor products of 2 x 2 matrix algebras. Fix n 2: 1. Let 0"1, 0"2, 0"3 be Jordan orthogonal symmetries in M2 = jl.I2 (C) , e.g., the

104

4

REPRESENTATIONS OF JB-ALGEBRAS

Pauli spin matrices (cf. (3.9)). M2 0 ... 0 M2 ~ Nhn by

For 1 ::; J ::; n define elements Sj in

and

(where there are J -1 factors of 0"1 and n - J factors of 1). Note that for any Xl, ... ,Xn,Yl, ... ,Yn in lVh,

as long as Xi and Yi commute for all except possibly one index l. (Evaluate the left-hand side making use of the fact that xJYj = YjXj = x) 0 Yj when Xj and YJ commute.) It follows that {51, ... , S2n} are Jordan orthogonal symmetries (d. Definition 3.35). Let S be the real linear span of 1 and SI, ... , S2n. Let M = N E9 R1 be a (2n + 1) -dimensional spin factor, and let WI, ... , w2n be an orthonormal basis of N. Define 8 : N! ---) S by 8(1) = 1, 8(Wi) = Si for 1 ::; l ::; 2n. Then 8 is a Jordan isomorphism from N! onto S C (lV!2n )sa ~ B (H)sa, where H is a 2n -dimensional complex Hilbert space, which proves that !v! is a JC-algebra. For n 2> 2, the (2n)-dimensional spin factor is a subalgebra of the (2n+1)-dimensional spin factor, so we have shown that all finite dimensional spin factors are JC-algebras. Let S be the (2n + 1) -dimensional spin factor constructed above, and A C M 2 " the C*-algebra generated by S. If B is any C*-algebra and IT : S ---) Bsa a unital Jordan homomorphism, we are going to show that IT extends to a *-homomorphism 1/J : A ---) B. (This says that A is the "universal C*-algebra" of S, a concept to be investigated in more detail later in this chapter, cf. Definition 4.37.) Consider for each J = 1,2, ... , n the elements of A, t J = lS2J-lS2j = 10···0100"1010···01, uJ

=

tl t 2··· tj-lS2J-l

vJ

=

t l t2··· t J- l S2)

= =

10···0100"2010···01, 10···0100"3010···01.

Since 1,0"1,0"2,0"3 span M2 = Nh(C), it follows that 1, t J , u)' v) span C10 ... 0 M2 0 ... 0 C1, so A = 1'>!2". Furthermore, A i::; the linear span (over C) of 1 and all finite products of the elements 81,82, ... ,8 n . Since these elements are Jordan orthogonal (i.e., anticommuting) symmetries, each such product either equals 1 or can be written in the form 5i,8i2 . . . Sik where ZI < Z2 < ... < lk and {zl, ... ,zd is any subset of {1, ... ,2n}. Together with the identity element 1 there are 2 2n such products. Since

REPRESENTATIONS OF JJ3W-FACTORS AS JW-ALGEBRAS

105

M2n

is 2 2 " -dimensional over C, it follows that these products together with 1 form a basis for M 2 n. Now we define 1/J : A -> B by

where the sum is over all k-tuples ~I < l2 < ... < lk for 1 <:::: k <:::: 11. Since S -> Bsa is a unital Jordan homomorphism, it follows that 7r(sd, ... , Jr(S2n) are Jordan orthogonal symmetries in B, and thus anticommute. It is then straightforward to verify that 1/) is a *-homomorphism that extends Jr:

Jr.

Now we are ready for the infinite dimensional case. If NI is an infinite dimensional spin factor, let {S,,}aEI be a maximal set of Jordan orthogonal symmetries in NI \ {l, ~1}. For each finite subset F of the index set I, let NIp be the spin factor spanned by 1 and {sa I a E F}, and Ap the associated C*-algebra constructed above. Then the collection {A1F} is directed upwards by inclusion. Let lvIo be the union of the subalgebras MF for all finite subsets F, and let Ao be the algebraic inductive limit of the associated C*-algebras. (Note that any inclusion FI C F2 of finite subsets of I extends to a Jordan isomorphism of AIFI into A1F2 , and then to a *-homomorphism of AFI into AF2.) Then Ao is a normed *-algebra whose completion A is a C*-algebra. (The defining norm properties for a C*-algebra hold in A o , and thus also in A.) The imbedding of each NIF is isometric (Proposition 1.35), and thus extends to an isometric isomorphism from the union NIo into A. Since NIo is dense in N!, this isomorphism extends by continuity to an isometric isomorphism from M into A. Thus by the Gelfand-Naimark theorem for C*-algebras (A 64), M is a .IC-algebra. Since M is by hypothesis also a .IBvV-algebra, then AI is a .IW-algebra (Corollary 2.78). 0 4.2. Lemma. Let B be a real assocwtwe *-algebra for which A =

Hn(B) is eqUlpped wzth a norm makmg It a lB-algebra. If n ?: 2 and a E Mn(B), then a*a

<::::

0

=}

a

= o.

Proof. We first show for a E Mn(B), (4.1)

a*a = 0

=}

aa* =

o.

If a*a = 0, then the spectrum of a*a in N!n(B) is {O}, so by (A 54) the spectrum of aa* in the associative algebra NI,,(B) is {O}. We will show the spectrum of aa* in the JB-algebra A is {O}, which by Corollary 1.22 will imply aa* = o. Let A E R with A =I o. Then aa* ~ Al is invertible in Mn(B), and by uniqueness of inverses, its inverse is self-adjoint. By Lemma 1.16, aa* ~ Al has a Jordan inverse in A. Since this is true for all

106

4

REPRESENTATIONS OF JB-ALGEBRAS

non-zero A, then the spectrum of aa* in A is {O}, which completes the proof of (4.1). Now let a E Mn(B) and suppose that a*a ::::: o. Let {e,j} be the standard system of matrix units in Mn(B), and write a = Li,j bijei) with bij E B for 2,] = 1, ... , n. For each bE B, each 2 and any] i= 2, we have

(4.2) The inequality a*a ::::: 0 implies

02: eiia*aeii =

2:= bkibkze ii · k

By (4.2) we conclude that bkibkieii = 0 for each 2, k. It follows that (bkieii)*(bkie'i) = 0, so by (4.1) bkie" = O. By the uniqueness of the representation of elements of Mn(B) with respect to the system of matrix units {ei)}' we conclude that each bki is zero, so a = O. D

4.3. Lemma. If M 2S a lBW-algebra such that M zs zsomorphzc to Hn(B) for a real assocwtive *-algebra B wdh n 2: 2, then l'vl zs a lW-algebm. Proof. We identify M with Hn(B). We first show (4.3) The proof we will give for (4.3) is similar to the standard proof of the positivity of x*x for elements x in a C*-algebra, cf. (A 56), but differs in some essential ways since we are not working in a complex algebra. Let x*x = a - b be the orthogonal decomposition of x*x in C(x*x,l) (cf. Proposition l.28), so that a, b are two positive elements of C(x*x, 1) such that a 0 b = O. We are going to show b = 0, so that x*x = a 2: 0 will follow. By the definition of {bab} (cf. equation (l.13)) and the associativity of C(x*x, 1), {bab}

=

2b

0

(b

0

a) - (b

0

b)

0

a

=

-(b

0

b)

0

a

=

-b 0 (b

0

a)

= O.

Since the Jordan triple product {bab} in lvln(B) coincides with the associative product bab, we conclude that bab = O. Calculating in the associative *-algebra Mn(B), we now find that (4.4)

(xb)*(xb)

=

bx*xb

=

b(a - b)b

By Lemma 4.2, xb = 0, so by (4.4) b3 which completes the proof of (4.3).

=

_b 3

:::::

o.

= O. By spectral theory,

b

= 0,

REPRESENTATIONS OF JBW-FACTORS AS JW-ALGEBRAS

107

By spectral theory, every element of M+ has the form a 2 for some a E M. Thus by (4.3) we have M+ = {x*x I x E Mn(B)}. Hence Mn(B) is a *-algebra whose self-adjoint part is a complete order unit space with the positive cone {x*x I x E Mn(B)}. By (A 65), there is a representation 1f of the *-algebra lvln(B) on a complex Hilbert space H such that 7r is an isometry on M = H,,(B). Thus M is a JC-algebra, and therefore by Corollary 2.78 is a JW-algebra. D

4.4. Lemma. If AI zs a lEW-algebra m whlch the zdentdy lS the sum of n pro)ectwns (4 -:; n -:; (0) exchangeable pazrwzse by symmetnes, then M lS a lW-algebra. Proof. We can assume 4 -:; n < 00, for if n = 00, then the identity of M can be written as the sum of four exchangeable projections (Corollary 3.11). Then by Theorem 3.27, M is isomorphic to the self-adjoint part of Mn(B) where B is an associative *-algebra over R. Now the result follows from Lemma 4.3. D Recall that 0 stands for the octonions or Cayley numbers.

4.5. algebra.

Theorem.

Every lEW-factor other than H3(O)

lS

a lW-

Proof. Let M be a JBW-factor. If M is not of type I, then by Proposition 3.17, there are four exchangeable projections with sum 1. By Lemma 4.4, M is a JW-algebra. If AI is of type I, then by Theorem 3.23 M is of type In where 1 -:; n -:; 00. By the definition of type In, if n < 00, then 1 is the sum of n exchangeable projections, and if n = 00 then 1 is the sum of infinitely many exchangeable projections. If 4 -:; n -:; 00, then by Lemma 4.4 M is a JW-algebra. If n = 1, then M = R1 (Theorem 3.39), and so M is a JW-algebra. If n = 2, then by Proposition 3.37, M is a spin factor, and by Theorem 4.1 is then a JW-algebra. If n = 3, then by Theorem 3.32, M ~ H3(B) where B is the reals, complexes, quaternions, or octonions. In the first three cases, B is a real associative *-algebra, and so by Lemma 4.3, M is a JW-algebra D Recall from Chapter 1 (cf. remarks following Definition 1.1) that a Jordan algebra A is specwl if there is an associative algebra A such that A is isomorphic to a Jordan subalgcbra of A with the product a 0 b = ~ (ab + ba). It is exceptwnal if it is not special.

4.6. Theorem. H3(O) is exceptional. Proof. This is a classic result due to Albert [2]. The idea of the proof is as follows. If M were special, say MeA where A is an associative

108

4

REPRESENTATIONS OF JB-ALGEBRAS

algebra, one can arrange that A is a *-algebra with M c As", and use the three exchangeable projections in M to construct a 3 x 3 system of matrix units for A. Then there is an associative *-algebra B and an isomorphism from A onto Mn(B) taking the matrix units in A onto the matrix units of Mn(B) and carrying M = H3(O) onto Hn(B). Now (3.7) gives an isomorphism from 0 onto B, so 0 must be associative. Since the octonions are not associative, this is a contradiction. See [67, Cor. 2.8.5] for the details. 0

Representations of JB-algebras as Jordan operator algebras By Theorem 4.6 some 1B-algebras admit no faithful representation on a Hilbert space, so the concept of a 1B-algebra is strictly more general than that of a 1C-algebra. But we will see in Theorem 4.19 that an arbitrary 1B-algebra has an ideal (referred to as the exceptional zdeal) such that the quotient is a 1C-algebra. For this purpose we will show (in Lemma 4.14) that every 1B-algebra admits a separating set of (abstract) factor representations as defined below. (We will reserve the term concrete representation for a 10rdan homomorphism of a 1B-algebra into B(H)sa.) 4.7. Definition. Let A be a 1B-algebra. A factor representatwn of A is a unital homomorphism 71 from A into a 1BW-algebra such that the a-weak closure of 7I(A) is a 1B\V-factor. It is said to be a JW-factor representatwn if the a-weak closure of 7I(A) is a 1W-factor. By Theorem 4.5 a factor representation 71 of a 1B-algebra A is a 1\"1factor representation unless the a-weak closure of 1f(A) (and then 7I(A) itself) is isomorphic to H3(O). Thus a factor representation 71 of A is a 1W-factor representation except when 7I(A) is isomorphic to H3(O). 4.8. Lemma. If M zs a JEW-algebra of type In, and 71 zs a factor representation, then N = 7I(M) (a-weak closure) zs a type In JEW-factor. Proof. Let Pl,' .. ,Pn be abelian exchangeable projections in !VI with sum 1. Then by density of 7I(M) in N, 7I(pd, ... , 7I(Pn) are also abelian projections in N with sum 1. If Pi and PJ are exchanged by a symmetry s, then 7I(s) is a symmetry that exchanges 7I(Pi) and 7I(PJ). Thus N is of type In. 0 4.9. Lemma. If a zs a normal state on a JEW-algebra M, then there zs a smallest centr'al proJectwn c wzth value 1 on a. Proof. By taking complements, it is equivalent to show that there is a greatest central projection with value 0 on a. We show the set of all

REPRESENTATION AS A JORDAN OPERATOR ALGEBRA

such projections is directed upwards. Assume a( cd C1C2 :::: Cl, then a(clc2) = 0, hence

109

= a( C2) = O. Since

Since a is normal, it has value 0 on the least upper bound of sllch projections, which is itself a projection by Proposition 2.5(iii) and is central by Proposition 2.36. 0

4.10. Definition. Let a be a normal state on a JBW-algebra 1vl. The least central projection with value 1 on a is called the central carrzer of a, and is denoted c( a). 4.11. Definition. Let A be a JB-algebra, and a a state on A. A --> c(a)A** is defined to be the composition of the natural injection of A into A** followed by the homomorphism Uc(u) mapping A** onto c(a)A** . Jr" :

4.12. Definition. A state on a JB-algebra is pure if it is not a convex combination of two distinct states. Pure states on JB-algebras generalize pure states on C* -algebras (A 60), and are a special case of pure states on general order unit spaces (A 17).

4.13. Definition. A state a on a JB-algebra A is a factor state if its central carrier c( a) is a minimal projection in the center of A **. Note that a is a factor state iff c( a)A ** is a JBW-factor, and thus iff is a factor representation. We will study the relationship of states and representations more thoroughly later in this book, after we have discussed the basic properties of Jordan state spaces. For now, we just remark that Jr u as defined above is an abstract analog of the GNS-representation associated with a state on a C*-algebra (cf. (A 63)). Jr"

4.14. Lemma. If a lS a pure state on a iE-algebra A, then a lS a factor state, so Jr u lS a factor representatzon. These factor representatzons Jru separate the elements of A. Proof. Let a be a pure state on A. Since A is a-weakly dense in A **, then Jru(A) is a-weakly dense in c(a)A**. Suppose 0 < w < c(a) is a central projection in c( a)A **. Define .\ = a( w) and note that

4

110

REPRESENTATIONS OF JB-ALCEBRAS

By the definition of the central carrier, neither A nor 1 - A equals 1, so 0 < A < 1. Note that IIU,:all = (U,:a)(l) = a(Uw 1) = a(w) = A. Similarly, IlUi-wall = a(l - w) = 1 - A. Now define

and

Then al and map on A**,

a2

are distinct states, and since Uw

+ U1 - w

is the identity

This contradicts the assumption that a is pure. Thus no such central projection w exists. Then we've shown that c(a) is minimal among central projections in A**, and hence that c(a)A** is a factor. Finally, let c = c(a). Then a(c) = 1, so (l.55) implies that U;a = a. If 7f(J"(a) = ca = 0, then a(a)

= (U;a)(a) = a(ca) = O.

Thus 7f(J"(a) = 0 implies a(a) = O. By the Krein-Milman theorem the pure states separate points of A, so the associated factor representations 7f(J" separate points of A. 0

4.15. Definition. Let A bc a JB-algebra, and 1 a norm closed Jordan ideal. Then 1 is purely exceptwnal if every factor representation of 1 has image isomorphic to H3(O). 4.16. Lemma. Let A be a lE-algebra, and 1 a norm closed 10rdan zdeal. Then every homomorphzsm 7f from 1 mto a lEW-algebra M extends to a homomorphzsm from A into AI. Proof. If 1 E 1, then 1 = A, and there is nothing to prove, so we may assume 1 rt. 1. Let B = 1 + R1 c A. Then B is a JB-algebra. We also denote by 7f : B --> M the unique extension of 7f to a unital homomorphism from B into M. By Theorem 2.65, there is a unique normal homomorphism 7r from B** into lvi extending 7f : B --> lvI. By Lcmma 2.75 we can identify B** with the a-weak closure 13 of B in A **, and 1** with the a-weak closure J of 1 in A **. Thus 7r restricts to a normal homomorphism from J into lvI, extending 7f : 1 --> M. Since 1 is an ideal in A, then J is a a-weakly closed ideal in A **, so by Proposition 2.39 there is a central projection c in A ** such that J = cA **. Then 7r 0 U c (restricted to A) is the desired extension. 0

REPRESENTATION AS A JORDAN OPERATOR ALGEBRA

4.17. Lemma. If 1 AjJ ~s a lC-algebra.

~s

a norm closed

~deal m

111

a lC-algebra A, then

Proof. Let J be the (J-weak closure of 1 in A**. By Lemma 2.75, = 1, so the map a + 1 a + J is an isomorphism from All into A** 11. We will be done if we show the latter is a JC-algebra. By Lemma 2.77, A** is a JW-algebra, and by Proposition 2.39 there is a central projection c such that J = cA**. Thus U 1- c is an isomorphism of A ** I J onto (1 - c)A **. A direct summand of a JW-algebra is again a JW-algebra, so A ** I J is a JW-algebra, which completes the proof. 0

Jn A

f-)

4.18. Definition. If {7rc>} is a family of concrete representations of a JB-algebra A with 7ra : A -> B(Ho)"" then we define the direct sum EBa 7ra by

The following is a "Gelfand-Naimark" type theorem for JB-algebras.

4.19. Theorem. Let A be a lE-algebra. Then there is a umque 1 such that AI 1 ~8 a lC-algebra and 1 ~8 purely exceptwnal.

~deal

Proof. Let J be the set of all ideals 1 that are kernels of JW-factor representations of A, and let 1 be the intersection of all ideals in J. For each K E J, let 7rK be a JW-factor representation of A with kernel K. Let 7r be the direct sum of the representations 7r K for K E J. Then the kernel of 7r is 1, and by construction 7r(A) ~ Allis a JC-algebra. To prove that 1 is purely exceptional, let 7r be a factor representation of 1 and let lvI = 7r(1). By Lemma 4.16, 7r extends to a homomorphism (also denoted 7r) from A into M. If M is a JW-factor, then 7r is a JWfactor representation of A, so its kernel contains 1. Then 7r( 1) = {O} which contradicts 7r being a factor representation of 1. Thus lvI is not a JW-factor, so by Theorem 4.5, lvI ~ H3(O). Thus 1 is purely exceptional. To prove uniqueness, suppose 11 is another norm closed Jordan ideal of A with the same properties as 1. Then 11 is annihilated by every factor representation of A onto a JW-factor, so by the definition of 1, 11 C 1. By hypothesis, All) is a lC-algebra. Let 7r be a factor representation of All1 and let M = 7r(Alld. By Lemma 4.17, 7r(Alld is a JC-algebra, and so cannot be dense in the finite dimensional factor H3(O). Thus by Theorem 4.5, M is a JW-factor. Each such representation pulls back to a JW-factor representation of A, which must then kill 1. Thus each factor representation of All) kills l1J) cAlli' By Lemma 4.14 the factor representations separate points of AI J), so J 11) = {O}, i.e., J C J). 0

112

4.

REPRESENTATIONS OF JB-ALGEBRAS

4.20. Corollary. Let A be a lE-algebra (respectively lEW-algebra). If A admzts no factor representatzon onto H 3 (O), then A zs a lC-algebra (respectzvely, lW-algebra).

Pmof. Assume A admits no factor representation onto H 3 (O), and let 1 be the purely exceptional ideal of Theorem 4.19. If 1 were not {O}, then by definition J would admit a factor representation onto H3(O). By Lemma 4.16 this would extend to a factor representation of A onte H 3 (O), contrary to our assumption. Thus J = {O}. By Theorem 4.19, AI J is a JC-algebra, and thus so is A. Finally, if A is a JBW-algebra as well as a JC-algebra, then by Corollary 2.78 it is a JW-algebra. D For JBW-algebras, we will see that the relevant ideal is cr-weakly closed.

4.21. Lemma. Let M be a lEW-algebra of type h. Then M admzts a unzque decomposztzon lvI = lvIR EEl lvIe EEl MH EEl lI10 where all factor representatzons of M R , Me, M H , Mo are onto H3(R), H 3(C), H 3(H), H3(O) respectzvely.

Proof. [67, Thm. G.4.1].

D

4.22. Lemma. Every lEW-algebra M of type 12 is a lW-algebra.

Proof. By Lemma 4.8, for every factor representation 71' of lIf, 71'(M) is a type I2 factor. Every I2 factor is a JW-algebra by Theorem 4.5. By Lemma 4.14, lvI admits a separating set of factor representations, each of which can then be taken to be a concrete representation. The direct sum of this separating set of factor representations then gives a representation of M as a lC-algebra. By Corollary 2.78, 1\1 is a lW-algebra. D 4.23. Theorem. Let M be a lEW-algebr·a. Then lvI admzts a unzque decomposztzon M = MspEElMexe wzth Msp a lW-algebra and lvIexe a purely exceptzonal lEW-algebra of type 13.

Pmof. By Lemma 3.16 and Theorem 3.23 we can write M = lvh EEl M2 EEl M3 EEl ... EEl lvIoo EEl N where Mn is either {O} or of type In for 1 ::; n ::; 00 and N has no type I direct summand. By Lemma 4.4, lvIn for 4 ::; n < 00 is a lW-algebra. By definition, l'vfoo is the direct sum of lBW-algebras whose identities are the sum of infinitely many exchangeable projections. By Lemma 4.4 each such direct summand is a lW-algebra, and thus so is Moo. By Proposition 3.17, the identity in N is the sum of four exchangeable projections, so by Lemma 4.4, N is also a JW-algebra. lvh is associative, and so is isomorphic to CR(X). The latter is the self-adjoint part of Cc(X), which is a monotone complete C*-algebra with a separating set of normal states. By (A 95), Ce(X) is a von Neumann algebra, so

REVERSIBILITY

113

Ml = CR(X) is a JW-algebra. lvh is a JW-algebra by Lemma 4.22. The remaining case is the 13 summand M 3 , which is a sum of JW-algebras and purely exceptional summands by Lemma 4.2l. The direct sum of all the direct summands that are JW-algebras is again a JW-algebra, which completes the proof. 0 Reversibility We next discuss the relationship between a JC-algebra and the real *-algebra it generates. Recall that one motivating example for Jordan algebras is the self-adjoint part of an associative *-algebra. One might expect that a 1 C -algebra A would equal the self-adjoint part of the real *-algebra it generates. This is true iff A is reverslble as defined below.

4.24. Definition. A Jordan subalgebra A of an associative *-algebra is reverslble if for all n > 0

Note that the symmetric product in (4.5) is just twice the ordinary Jordan product if n = 2, so (4.5) holds for n = 2. The Jordan triple product in a special Jordan algebra satisfies

(4.6) so (4.5) always holds for n = 3. However, (4.5) can fail for n = 4 (cf. the remark after Proposition 4.31). In principle, reversibility for a JC-algebra depends upon the representation, and in fact we will see that for certain spin factors reversibility does indeed vary with the representation. On the other hand, we will show that the self-adjoint parts of C* -algebras are reversible in every concrete representation. If A c 13(H)sa is a concrete JC-algebra, we will let Ro(A) denote the real subalgebra generated by A in 13 (H). (Note that Ro(A) is closed under adjoints, so coincides with the *-algebra generated by A.) We denote by R(A) the norm closure of Ro(A), and R(A) the a-weak closure. Clearly A is reversible iff A = Ro(A)sa. Furthermore:

4.25. Lemma. If A lS a reverslble lC-subalgebm of 13(H),a, then A = R(A)sa. If m adddwn A lS a-weakly closed, then A = (R(A))sa (a-weak closure). Proof. Let x E R(Ak,. Then there is a sequence {xn} in Ro(A) converging in norm to x. For Yn = ~(xn + x;,) we have Yn E Ro(A)sa = A and Yn --+ ~(x + x*) = x. Since by hypothesis A is norm closed, it follows that x E A. The same argument applies if the norm topology is replaced by the a-weak topology. 0

4

114

REPRESENTATIONS OF JB-ALGEBRAS

We will now see that the existence of sufficiently many exchangeable projections with sum 1 is sufficient to guarantee reversibility. It is easy to verify that Hn (B) is a reversible Jordan subalgebra of Mn (B), and this will be the key to proving reversibility for a wide class of JW-algebras. We will use exchangeable projections with sum 1 to construct a system of matrix units and thus to show that Ro(M) = Mn(B) and that M = Hn(B). We recall the definition and the following well known result relating matrix units and matrix algebras.

4.26. Definition. Let R be an associative algebra over the reals with involution *. A set {eij I 1 <::: 2,) <::: n} of elements of R is a self-ad)omt system of 71, x n matru umts if (i) e:j = eJi for all 2, ), (ii) ell + ... + e nn = 1, (ii) eiJekm = 6Jkeim for all otherwise. )

2, ),

k, m (where 6Jk is 1 if )

=

k and 0

4.27. Lemma. Let R be an assocwtwe * -algebra contammg a selfad)omt system of n x 71, matT·lX umts {e ,J }. Let B be the relatwe commutant of {eij}. Then each element x of R admlts a umque representatwn (4.7)

x

=

L bijeiJ i, j

wdh bij E B, and R

lS

* -2somorphlc

to Aln (B).

Proof. [67, Lemma 2.8.2]. See also (A 169). D

4.28. Lemma. Let M be a lW-algebra actmg on a Hdbert space H. Assume that the ldentdy 2S a sum of n pro)ectwns PI, P2, . .. ,Pn (wdh 2 <::: n < (0) such that for each l, Pi and PI are exchanged by a symmetry s,. (We mclude the case 2 = 1, where we can take SI = 1.) Define eij = SiPlSj for 1 <::: l,) <::: n. Then {eij} lS a self-ad)omt system of 71, x n matrix umts for the assocwtwe *-algebra generated by AI m B(H). Proof. For all

2,),

Note that eii = SiPlSi Thus for ) # k,

k we have (eij)*

=

=

e Jl and

Pi by our hypothesis, so eJJekk

which completes the proof. D

= 0 for ) #

k.

REVERSIBILITY

115

We will now establish the main result of this section. Under the hypotheses of Theorem 4.29, we will see that there is a close relationship between JW-algebras M acting on a Hilbert space H and the real algebra Ro(M) generated by M. In particular, M = Ro(M)sa, i.e., M is reversible. Furthermore, we will see in Corollary 4.38 that the norm closed real algebra R(M) generated by IvI is determined up to *-isomorphism by M. A hint of why this might be expected is contained in the proof of Theorem 4.29. It is shown that Ro(M) is isomorphic to Mn(B) for an associative *-algebra B, via an isomorphism carrying M onto the set Hn(B) of hermitian matrices with entries in B. We can identify B with any of the subspaces {bei) + b*e]i I bE B} of Hn(B) for L # J, and then can recover the associative multiplication on B from the Jordan product of elements of these subspaces, d. (3.7), and also (4.8) below. If this sounds familiar to the reader, it should: the proof of the next result contains a proof of the Jordan coordinatization theorem (Theorem 3.27) for the case of special Jordan algebras.

4.29. Theorem. Let l'lI be a JW-algebra actmg on a HIlbert space H. Assume that the IdentIty IS a sum of n pazTwIse exchangeable pTOJectwns (WIth 3 ::; n ::; 00). Then M 1.S reversIble.

Proof. By Corollay 3.11, we can assume n < 00. Set R = Ro(M). Thus R is the real subalgebra generated by M in B(H), and by the remark preceding Lemma 4.25, R coincides with the *-algebra generated by M. Let {eij} be the set of n x n matrix units described in Lemma 4.28. Let B be the relative commutant of {ei]} in R; then by Lemma 4.27, R = Mn(B) and (4.7) holds. We will identify Mn(B) with the sums of the form (4.7). Under this identification, M is a Jordan subalgebra of Hn(B) that generates Iv1n(B) as a real associative algebra. We are going to show that AI consists exactly of the hermitian matrices in Mn(B), i.e., M = Hn(B), which as remarked after Lemma 4.25, will show that M is reversible. For 1 ::; I, J ::; n with 7 # J define Bij

=

{b E B I bei]

+ b*eji

EM}.

We are going to show that these subspaces all coincide, and that in fact they all equal B, i.e., Bi] = B for all I # J. Note that with the notation of Lemma 4.28,

Now note that for distinct

I,

J, k, if b1 E Bij and b2 E Bjk, then

4.

116

REPRESENTATIONS OF JB-ALGEBRAS

This equation is the key to recovering the associative multiplication from the Jordan product. (Note that taking n = 3, Z = 1, j = 3, k = 2, and setting a = bl , b = b2 gives the equation (3.7) which was the key to the Jordan coordinatization theorem.) By (4.8) (4.9) Taking b2

= 1 in (4.8) shows that Bi) C Bik

for

Z,],

k distinct.

Interchanging the roles of the indices ], k shows that

Bi) = Bik

(4.10)

for

Z,],

k distinct.

Furthermore, if b E B ij , then

so (4.11 ) On the other hand, by the definition of B'j we have B;) = B)i, so (4.11) implies that Bij = B Ji . Thus for all z # 1, z #] we have (4.12) We also have B12 = B1J for all ] # 1 by (4.10), and combining this with (4.12) shows Bij = B12 for all z #]. Furthermore, by (4.9) and (4.11), B12 is a *-subalgebra of B. (Here we've used n ~ 3.) We next prove that Bl2 = B. Let M,,(B 12 ) denote the subalgebra of R = M,,(B) generated by the set of matrix units {eiJ} and by B 12 . We will show this contains M, so that A1,,(B l2 ) must coincide with the *-algebra R = M,,(B) generated by M. Let x E M, and write x = I: .. biJeiJ· as in '.J (4.7). Then x = x* implies that btJ = bJi for each Z,]. If z # ], then

so bij E BiJ = B 12 . For each z we have bti = bii and

biieii = eiixeii E M, and for any j

#

z,

REVERSIBILITY

117

Thus bii E Bij = B 12 . Therefore all coefficients {b i]} of x are in B 12 , which completes the proof that M is contained in Mn(B12)' Thus M n (B 12 ) = Mn(B) = R, from which B12 = B follows as claimed. Finally, we show M = Hn(B). Let x E Hn(B). Write x = Li,] bi]eij with each b'j in B. Note that since x = x*, then b]i = bij for all z, j. Then, by the definition of B 'j and the fact that B = B ij , for z =I J, each term bijeij + b]ie]i = bi]eij + bije]i is in M. Also for each z since bii E B = Bi) and bii = biz, then choosing any J =I z,

Then biieii = e·ii(bii(e,i + ejj))eii EM. Thus each x E Hn(B) is a sum of terms in M, so is in !v!. Hence Al = H,,(B), and so M is reversible. D

M

4.30. Corollary. Let M c B (H)sa be a concrete iW-algebra. Then reverszble zff the 12 direct summand of !v! IS reversIble.

IS

Proof. According to Lemma 3.16 and Theorem 3.23, we can write

where Mn is of type In for 1 ::; z ::; 00 and !v!o has no type I direct summand. We first observe that M is reversible iff each of these direct summands is reversible. Indeed, if !v!o, !vh, ... , Moo are reversible, let Ql, ... ,an E M. Let Ci (0::; I ::; (0) be the central projections in !v! such that ci!vl = !vI;. Then cia) E !v!, for each I and J, so by reversibility of M i , bi = ci(ala2'" an + anan-l ... ad E !vI;. Now Li bi converges a-strongly to ala2'" an + anan-l'" aI, so the latter is also in M. Thus M is reversible. The converse is clear. We now show that each summand except !v!2 is reversible, from which we can conclude that !v! is reversible iff !vI2 is reversible. In all of the summands except All and !vh, we can write the identity as the sum of n ~ 3 projections pairwise exchanged by symmetries, so by Theorem 4.29 each summand other than Ml and M2 is reversible. By definition, the identity of Ml is abelian so as remarked after Definition 3.14, all elements of Ml commute. Thus the Jordan product coincides with the associative product in M l , and so All is reversible, and we are done. D 4.31. Proposition. Every spm factor !vI of dimenswn versIble m every faIthful representatwn as a ie-algebra.

<

4

IS

re-

Proof. Write M = Rl EEl N as in Definition 3.33. Then N is of dimension 2 or 3, and each element of N is a multiple of a symmetry (Lemma 3.34). Let S be a maximal Jordan orthogonal set of symmetries in N, i.e., an orthonormal basis for N. Then S contains at most three symmetries,

118

4.

REPRESENTATIONS OF JB-ALGEBRAS

and every element of M is a linear combination of the symmetries in S and the identity 1. Assume !vI is faithfully represented as a Jordan algebra of self-adjoint operators on a Hilbert space. We need to show that if al, ... ,an are elements of M, then al a2 ... an + an ... a2al is in M. By linearity it suffices to prove this for al ... an E S. Since Jordan orthogonal symmetries anticommute, we can rearrange the order of the factors (possibly multiplying the result by -1 in the process). Using the fact that the square of each symmetry is the identity, we can then reduce to the case where al ... an are distinct (and then n ::::: 3.) Thus

either coincides with the Jordan product or the Jordan triple product (see equation (4.6)), so this product is again in M. Thus M is reversible. 0

Remark. The five dimensional spin factor is reversible in no representation. In fact, the implication (4.5) fails in every representation. The six dimensional spin factor admits both reversible and non-reversible representations. Finally, spin factors of dimension 7 or more are again reversible in no representation. We will not need this information; a proof of these remarks can be found in [67, Thm. 6.2.5]. We also note that the adjective "faithful" in Proposition 4.31 is redundant. It is straightforward to verify that spin factors are simple, i.e., have no proper closed ideals, and therefore every representation is faithful. 4.32. Definition. A JC-algebra A is umversally reversible if 7r(A) is reversible for every faithful representation 7r : A - ? B(H)sa. By Proposition 4.31, every spin factor of dimension at most 4 is universally reversible. We next show that the same is true of the self-adjoint part of C*-algebras. To prove this, we will work with the biduals of both a C*-algebra A and its self-adjoint part Asa.

4.33. Lemma. Let A be a lC-algebra. If A ** is reverszble m every representation as a concrete lW-algebra, then A is umversally reverszble. Proof. Let A be a concretely represented JC-algebra, and let B be the C*-algebra generated by A. Let B be the enveloping von Neumann algebra of B, which can be identified with B** (A 101). We therefore view A as a JB-subalgebra of Bsa. By Lemma 2.75, we can identify A with A** C (B ,a)** = (B ** )sa = 13"" and we have An B sa = A. (The identification of (B sa)** with (B **)5a follows from Lemma 2.76). Then for aI, ... ,an E A, by the reversibility of the concrete l'N'-algebra A** = A C 13, we have

REVERSIBILITY

119

which completes the proof that A is reversible in every concrete representation as a JC-algebra. D

4.34. Proposition. The selJ-adJoznt part oj every C*-algebra zs unzversally reverszble.

Proof. Let A be the self-adjoint part of a C* -algebra, faithfully represented as a concrete JC-algebra acting on a Hilbert space H. (Note that we do not assume that A + zA will be a C*-algebra in this representation!) Assume first that A is isomorphic to the self-adjoint part of a von Neumann algebra M of type 1 2 , Then M is *-isomorphic to the 2 x 2 matrices with entries in the center Z of M (cf. (A 170) or [67, Lemma 7.4.5]). Define Sl, S2, S3 to be the Pauli spin matrices (cf. (3.9)), and So = 1. Then Sl, S2, S3 are Jordan orthogonal symmetries, and every element a E A = lvh(Z)sa admits a unique representation a = I:i ZiSi, where ZO, ... , Z3 are in the center Z of A. Indeed, since Z = Z + zZ (Corollary 1.53) we can write each element a of A in the form

where Wi, W2, W3, W4 are in Zsa = Z. Since all elements of A are linear combinations of elements of the form zs for Z E Z and S E {SQ,Sl,S2,S3}, to prove reversibility it suffices to show

+ (zns,,,)··· (Z2 S i )(Zl S ,J Zn(Si, Si 2 ... Sin + Sin'" Si2Si,)

(ZlSi,)(Z2SiJ··· (ZnSin)

=

Zl Z 2'"

2

is in A for all choices of Zi in Z and all choices of Zl,'" Zn in {O, 1,2, 3}. Here (Si' Si2 ... S'n + Sin" . Si2Si,) is in A by the universal reversibility of the spin factor given by the real linear span of {so, Sl, S2, S3} (Proposition 4.31), and universal reversibility of A follows. Next, suppose that A is isomorphic to the self-adjoint part of a von Neumann algebra, and that A is represented as a concrete JW-algebra. Write A = A2 EEl B where A2 is a JW-algebra of type 12 and B has no type 12 direct summand. Note that there is a central projection c such that A2 = cA, so A2 is also the self-adjoint part of a von Neumann algebra, which will also be of type 12 (cf. (A 167)). By Corollary 4.30, B is reversible, and by the first part of this proof A2 is reversible, so A is a reversible JW-algebra. Finally, let A be isomorphic to the self-adjoint part of a C*-algebra A. Then A** is isomorphic to the self-adjoint part of the von Neumann algebra A** (Lemma 2.76), and thus is reversible in every faithful representation as a JW-algebra by the preceding paragraph. Now the proposition follows from Lemma 4.33. D

120

4

REPRESENTATIONS OF JB-ALGEBRAS

The universal C*-algebra of a JB-algebra We will see that the norm closed real *-algebra generated by a universally reversible lC-algebra in a faithful representation is independent of the representation up to *-isomorphism. However, the C*-algebras generated by faithful representations of a lC-algebra will generally depend upon the representation unless the algebra is abelian. A very useful tool in investigating these generated C* -algebras is the universal object described in Proposition 4.36.

4.35. Lemma. Let A be a JB-algebra. Then there zs a Hzlbert space H of dzmenszon large enough that for every Jordan homomorphzsm ¢ : A --+ Asa where A zs a C*-algebra, the C*-subalgebra A¢ generated by ¢(A) can be *-zsomorphzcally imbedded m 3(H). Proof. The cardinality of ¢(A) is at most equal to the cardinality of

A. Thus there is an upper bound, independent of the choice of ¢, on the cardinality of the *-algebra generated by ¢ in A. Then there is a uniform upper bound on the cardinality of the set of all Cauchy sequences in this *-algebra, and hence on its closure A¢. Hence there is a uniform bound on the cardinality of the state space of A¢, and of the dimension of the Hilbert space for each GNS-representation associated with a state on A¢ (A 63). This gives a bound m on the dimension of the universal representation of A¢ (A 83), independent of cp. Now let H be an m-dimensional Hilbert space. 0

4.36. Proposition. Let A be a JB-algebra. Then there zs a pazr (A, 'if) conszstmg of a C*-algebra A and a Jordan homomorphzsm 'if from A mto Asa wzth the followmg unzversal propertzes:

(i) The C*-algebra generated by 'if(A) zs A. (ii) For every pazr (3, 'if I) conszstmg of a C*-algebra 3 and a Jordan homomorphzsm 'ifl from A mto 3 sa ) there 1.S a *-homomorphzsm 'if2 : A --+ 3 such that 'ifl = 'if2 0 'if. Thzs pazr (A,

'if)

zs unzque up to zsomorphzsm.

A

Proof. Let H be as in Lemma 4.35, let {'ifu}<>EI be the set of all Jordan homomorphisms from A into 3(H)sa (organized into a family with index

THE UNIVERSAL C*-ALCEBRA OF A JB-ALCEBRA

set I), and consider the direct sum 7f

=

121

EBaEI 7f". Thus 7f is a Jordan

homomorphism from A into B (H)sa where H = EBaEI Ha with He> = H for all a E I. Let A be the C*-algebra generated by 7f(A) in B(H). Then (i) is satisfied. We now show that (ii) is satisfied. Let (B,7fl) be as in (ii). By replacing the C*-algebra B by the subalgebra generated by 7fl (A) in B, we may assume without loss of generality that B is generated by 7fl(A). Then by the key property of H, we may assume B c B (H), so that 7fl is a *-isomorphism from A into B (H),a. Thus 7fl = 7ff3 for some {3 E I. Let 1/Jf3 be the projection onto the (3-coordinate in the direct sum EBaEI B(H,,) (i.e., 1/Jf3(EBaEI xc» = x(3). Then for each a E A,

Now let 7f2 : A --) B be the *-homomorphism obtained by restricting 1/Jf3 to the C*-subalgebra A of B(H). Then 7fl = 7f2 0 7f, which proves (ii). Finally, if (A', 7f') is a second pair with the same property, then applying (ii) gives a *-homomorphism ¢ from A onto A' carrying 7f to 7f', and a *-homomorphism ¢' from A' onto A carrying 7f' to 7f. The composition ¢/ 0 ¢ agrees with the identity map on the image of A in A, and so must equal the identity map on A by (i). Similarly ¢ 0 ¢/ is the identity map on A'. Thus ¢ is a *-isomorphism, and carries 7f to 7f', which completes the proof of uniqueness. 0 Note that the map 7f2 guaranteed in (ii) is unique since any two such factorizations agree on the image of A, which by (i) generates A.

4.37. Definition. Let A be a JB-algebra. The pair (A,7f) in Proposition 4.36 is called the unwersal C*-algebra for A and we will denote it by (C~(A), 7f). Remark. If A is a JC-algebra, then 7f is faithful, so A can be identified with a Jordan subalgebra of C~(A). See [67, §7.1] for an alternative construction of (C~ (A), 7f). 4.38. Corollary. Let A be a umversally revers~ble iC-algebra. Then the norm closed real *-algebra R(A) generated by A m a fa~thful representation is determzned by A up to *-~somorph~sm. Proof. Let (7f, A) be the universal C*-algebra for A, and let R be the norm closed real *-subalgebra of A generated by 7f(A). Suppose A c B (H) is faithfully represented as a concrete lC-algebra. Let 1/J be the *-homomorphism from A into B(H) such that y6(7f(a)) = a for all a E A. Then y6 maps R onto a dense subalgebra of R(A). We next verify that y6 is isometric on R. For b E R, by universal reversibility of A and Lemma

122

4.

REPRESENTATIONS OF JB-ALGEBRAS

4.25 we have b*b E R(1T(A))ba = 1T(A). Since 1jJ restricted to 1T(A) is a Jordan homomorphism and is 1-1, it is isometric on 1T(A) (Proposition 1.35), and thus we have

111jJ(b)112 = 111jJ(b)*1jJ(b)11 = II
= R(A) follows.

0

4.39. Corollary. A iC-algebra A zs unzversally reverszble zff zts zmage zn ds unzversal C*-algebra zs reverszble.

Proof. Suppose A C B(H)ba is a concrete representation, (1T, C~(A)) the universal C*-algebra for A, and that 1T(A) is reversible in C~(A). By the universal property of C~(A), there is a *-homomorphism 1jJ: C~(A) --. B (H) such that 1jJ( 1T( a)) = a for all a E A. If a1, a2, ... ,an are in A, then

Now by reversibilty of 1T(A), a1a2'" an +a,,'" a2a1 E 1jJ(1T(A)) = A. Thus reversibility of A in C,: (A) implies universal reversibility; the converse is trivial. 0

If A is a C*-algebra, then we denote by A op the opposite C*-algebra. A op as a set coincides with A, and is equipped with the same involution and norm. If a f---> aO P denotes the identity map viewed as a map from A to AOP, then the product on AOP is given by aOPbo p = (ba)OP. Thus a f---> a OP is a *-anti-isomorphism from A onto A °P. 4.40. Proposition. Let A be a iE-algebra. Then there zs a unzque *-antz-automorphzsm of C,:(A) of perzod two that fixes all poznts zn the zmage of A zn C~ (A). (We wzll refer to as the canonzcal *-antzautomorphzsm assoczated wzth C~ (A) .)

Proof. Let A = C,:(A), and let, : A --. A 01' be the identity map, i.e., ,(a) = a°l'. Let 1T : A --. A be the canonical map from A into its universal C*-algebra. By the universal property defining A, the Jordan homomorphism a f---> ,(1T(a)) lifts to a *-homomorphism : A --. A by = ,~1 o1jJ.

THE UNIVERSAL C*-ALGEBRA OF A JB-ALGEBRA

123

Then 1> is a *-anti-homomorphism of A into A that fixes 1T(A). Thus 0 1> is a *-homomorphism of A into itself that fixes 1T(A). Since 1T(A) generates A as a C*-algebra, it follows that 1> 0 1> is the identity map on A. In particular, 1> is a *-anti-automorphism of period two of A. If 1>' is another *-anti-automorphism of A fixing 1T(A), then as above 1>-1 0 1>' must be the identity map, so 1> = 1>'. D

4.41. Lemma. Let B be a C*-algebm, and let A be a umtal lBsubalgebm of Bsa that genemtes B as a C*-algebm. Let 1> be a *-antzautomorphzsm of order 2 of B such that 1>(a) = a for all a E A. Let R(A) denote the norm closed real subalgebm of B genemted by A. Then the followmg relations hold:

(i) R(A) = {x E B j 1>(x) = x*}. (ii) R(A) n zR(A) = {O}. (iii) B = R(A) + zR(A). (iv) For x, y m R(A) we have 1>(x

+ zy) = x* + zY*.

If B zs a von Neumann algebm, and A zs a lBW-subalgebm such that the von Neumann subalgebm of B genemted by A zs all of B, then (z) through (iv) hold for the a-weak closure R(A) m place of R(A). Proof. (i) Note that 1> restricted to Boa is a Jordan automorphism, so 1> is an isometry on Bsa by Proposition 1.35. Then for arbitrary b E B, using the property jjx*xjj = jjxjj2 for elements x of a C*-algebra, we have

jj1>(b)jj2

= jj1>(b)1>(b)*jj = jj1>(b*b)jj = jjb*bjj = jjbjj2,

so 1> is an isometry of B onto itself. Since each a E A is fixed by 1>, and since x f--+ 1>(x)* is a norm continuous (real-linear) *-automorphism of H, it follows that (4.13)

R(A)

c {x

E

B

j

1>(x) = x*}.

To prove the opposite inclusion, let bE B with 1>(b) = b*. Note that since A generates B as a C*-algebra, then B is the norm closure of R(A)+zR(A). Thus there is a sequence {x" + zYn} converging in norm to b with Xn and Yn in R(A) for each n. By assumption 1>(b) = b*, so by (4.13)

Since Xn + ZYn -> band Xn - ZYn -> b, adding gives Xn -> b and so bE R(A). This completes the proof of (i). (ii) For x E R(A), by (4.13), (ix) = z(x) = ix* = -(zx)*. If also ix E R(A), then again applying (4.13), we have (ix) = (zx)*, so zx is not in R(A) unless x = 0, which proves (ii).

124

4

REPRESENTATIONS OF JB-ALGEBRAS

(iii) Each b E B can be written as b = bl bl

=

~(b

+ (b*))

and

b2

=

+ ~b2,

where

-~(b - (b*)).

Since has order 2, one readily verifies that (b j ) = b; for J = 1, 2. Thus by (i) both bl and b2 are in R(A), proving (iii). (iv) This follows immediately from (4.13). Finally, assume that B is a von Neumann algebra, and that A is a JBW-subalgebra that generates B as a von Neumann algebra. Then restricted to Bsa is an order isomorphism, so is O"-weakly continuous there. Since commutes with adjoints and is complex linear, by O"-weak continuity of the adjoint map it follows that is O"-weakly continuous on all of B. Now the proof of (i) - (iv) given above goes through with norm convergence replaced by O"-weak convergence throughout. 0

4.42. Corollary. Let A be a umversally revers~ble iC-algebra, zdentzfied w~th zts canomcal zmage m C~ (A). If zs the *-antz-automorphzsm of perwd 2 assocwted wdh C~(A), then A zs the set C~(A);a of self-adJmnt fixed pomts of .

Proof. By assumption, A is fixed by . By Lemma 4.41, any selfadjoint fixed point of is in R(A)sa, and by Lemma 4.25, R(A)sa = A. D

4.43. Definition. Let M be a JBW-algebra. A pair (M,7r) consisting of a von Neumann algebra M and normal Jordan homomorphism 7r : M -> Msa is a umversal von Neumann algebra of M if the following properties hold: (i) 7r(M) generates M as a von Neumann algebra. (ii) If N is a von Neumann algebra and 7r1 : M -> Nsa is a normal Jordan homomorphism, then there is a normal *-homomorphism 7r2 from Minto N such that 7r2 0 7r = 7r1. We will now show that each JBW-algebra M has a unique universal von Neumann algebra, and we will generally denote this universal von Neumann algebra by (W~(M), 7r).

4.44. Lemma. Let A be a iE-algebra, and (C~(A), 7r) zts umversal C*-algebra. Then (C~(A)**, 7r**) zs a umversal von Neumann algebra of A** . Proof. Let ¢ be a normal Jordan homomorphism from A:'* into Msa for a von Neumann algebra M. Let 1; be the restriction of 1; to A. Let 1j; : C~(A) -> M be the unique *-homomorphism that extends 1;, and

THE UNIVERSAL C*-ALGEBRA OF A JB-ALGEBRA

125

'ljJ : (C~(A))** --> M the unique extension to a normal *-homomorphism (A 103). Let 1f = 7f**. 7r

A**

y

~

1 ¢/ A

C~(A)**

M

1jJ~ 7r

1 C~(A)

Since 'ljJ 0 7f = cj;, it follows by u-weak continuity that 'ljJ o1f cj;. This provides the desired factorization of ¢, so the pair (C~ (A) **,1f) has the necessary universal property. Since 7f(A) generates C~(A) as a C*-algebra, then the von Neumann subalgebra of C~(A)** generated by 1f(A**) = 7f**(A**) contains C~(A), and so by u-weak density equals C~(A)**. 0

4.45. Proposition. Let M be a lEW-algebra. Then there exzsts a unique unwersal von Neumann algebra (W:(M),7f), and a umque *-antzisomorphzsm of W: (M) of perwd 2 fixmg each element of 7f(M). If M is a lW-algebra and its zmage m W: (Nl) zs reverszble, then the set of self-adJomt fixed pomts of zs 7f(M). Proof. It follows easily from the requirements (i), (ii) in Definition 4.43 that if M has a universal von Neumann alge br a (M, 7f), then (M, 7f) is unique up to isomorphism. From Lemma 4.44, AI** admits a universal von Neumann algebra. It follows easily that every direct summand of M** has the same property. In fact, let e be a central projection in M** and let (M, 7f) be the universal von Neumann algebra for NI**. If N is a von Neumann algebra and 7fl : cM** --> Nsa is a normal Jordan homomorphism, then 7fl 0 Ue : AI** --> Nsa is a normal Jordan homomorphism, so there is a normal *-homomorphism 'ljJ : M --> N such thatlj;o7f = 7fl oUe' Let 7fo be the restriction of 7f to eM**. Then 1/J 0 7fo = 7fl. Since e is central in M**, then 7f(e) is central in the JBW-algebra 7f(M**), so commutes with all elements of 7f(NI**) (Proposition 1.49). Since the von Neumann algebra generated by 7f(M**) is M, then 7f(e) is a central projection in M. Define M 0 = 7f( c) M. Then (M 0, 7fo) satisfies the defining properties of the universal von Neumann algebra for cM**. By Lemma 2.54, M is isomorphic to a direct summand of AI**, so M possesses a universal von Neumann algebra. Finally, the existence of with the specified properties follows from Lemma 4.25 by essentially the same argument as in the proof of Proposition 4.40 and Corollary 4.42. 0

126

4

REPRESENTATIONS OF JB-ALGEBRAS

For the case of a universally reversible lC-algebra, we can describe in more concrete terms. The next several results lead to that description. C~ (A)

4.46. Lemma. Let A be a umversally reverszble lC-algebra. Let B be a C*-algebra , 7fo a fmthful lordan homomorphzsm from A mto BSd such that the C*-subalgebra of B generated by 7fo(A) zs all of B. If there exzsts a *-antz-zsomorphzsm B of perwd 2 whose se: of selfadJoint fixed pomts zs 7fo(A), then (B,7To) zs the umversal C*-algebra of

A. Proof. By the universal property of (C1:(A),7T), there is a *homomorphism 'ljJ : C~ (A) ---> B such that 7To = 'ljJ 0 7T. Let

0


0

7T = 'ljJ

0

7T = 7To.

Since by assumption
C~(A)

lrr~ ~ A

71"0.

B

,


B

By the uniqueness of 'ljJ, we must have

Since

=
(4.14) Let 1 be the kernel of 'ljJ. By (4.14),
CARTESIAN TRIPLES

127

4.4 7. Proposition. Let A be a umversally reverszble lC-subalgebra of 13(H)sa. Define 71"0 : A ---+ 13(H) EB 13(H)OP by 71"o(a) = a EB aO P. Let 13 be the C*-subalgebm generated by 71"o(A). Then the pazr (13,71"0) zs the universal C*-algebra for A. The assocwted * -antz-automorphzsm of period 2 of 13 zs the map ( a EB bOP) = b EB aO P.

Proof. Let denote the map a EB bOp ---+ b EB aO P on 13 (H) EB 13 (H)OP. Note that ;p is a *-anti-isomorphism of 13 (H) EB 13 (H)OP of period 2 whose set of self-adjoint fixed points is {x EB Xo p I x E 13 (H)sa}. Since ;p is isometric and fixes 71"o(A), it leaves invariant the C*-algebra 13 generated by 71"o(A). By Lemma 4.41, the set of self-adjoint fixed points of ;p on 13 is R(71"o(A))sa where R(71"o(A)) is the norm closed real *-subalgebra of 13 generated by 71"0 (A). Since A is by assumption universally reversible, by Lemma 4.25, R(71"o(A))sa = 71"o(A), so the set of self-adjoint fixed points of ;p on 13 is 71"o(A). Now the desired result follows from Lemma 4.46. D Now we will describe the universal C* and von Neumann algebras for those Jordan algebras that are actually self-adjoint parts of C* or von Neumann algebras. If A is the self-adjoint part of a C*-algebra A, by Proposition 4.34, A is universally reversible. Therefore by Proposition 4.47, C~(A) is the C*-subalgebra of A EB AOP generated by {a EB aO P I a E A}. If A has no abelian part (i.e., has no one dimensional representations) then the universal algebra is actually equal to A EB A oP, and there is an analogous result for von Neumann algebras. We will now state these results more formally, but will not need them in the sequel.

4.48. Proposition. If A zs a umtal C*-algebm wzth no one dzmenswnal representatwns, then C~ (Asa) = A EB A 01' wzth r-espect to the imbeddmg a ---+ a EB aO P.

Proof. [67,7.4.15]. D 4.49. Lemma. Let M be a von Neumann algebra wzth no abehan dir-ect summand. Then the umveT'sal von Neumann algebra of Msa zs M EB Mop with respect to the zmbeddmg a ---+ a EB a0 1' .

Proof. [67, Thm. 7.4.7]. D Cartesian triples We now introduce the notion of a Cartesian triple in a JBW-algebra, which generalizes the same notion in a von Neumann algebra, first introduced and studied in [AS, Chpt. 6]. In any von Neumann algebra containing a set of 2 x 2 matrix units, the Pauli spin matrices (cf. (3.9)) provide

128

4.

REPRESENTATIONS OF JB-ALGEBRAS

an example of a Cartesian triple. Our purpose here is to prove that the existence of a Cartesian triple of symmetries in a JBW-algebra is sufficient to guarantee that a JBW-algebra is isomorphic to the self-adjoint part of a von Neumann algebra containing 2 x 2 matrix units. Recall that if e is a projection in a JB-algebra, an e-symmetry is an element s satisfying s2 = e. By spectral theory applied to the Jordan subalgebra C(s,l), e 0 s = s, so s E !vIe. Then there are orthogonal projections p, q such that s = p ~ q, in fact p = ~(s + e) ane. q = e ~ p.

4.50. Definition. Let M be a JBW-algebra, and e a projection in !vI. A Carteswn triple of e-symmetnes is an ordered triple (1', s, t) of mutually Jordan-orthogonal e-symmetries such that (4.15 ) When there is no need to specify the projection e, we will say that (1', s, t) is a Carteswn tnple of partial symmetnes. When e = 1, we will say that (1', s, t) is a Carteswn tnple of symmetrzes.

Remarks. Suppose (1', s, t) is a Cartesian triple of e-symmetries, and that e = l. Then multiplying (4.15) on the left by Ur gives Ur = UsUt . Since each of Ur , U" Ut is its own inverse, then taking inverses gives Ur = UtUs . Thus Us and Ut commute. Similar arguments show that any pair of Ur , Us, Ut commute. It follows that (4.15) holds with any order for Un U" U t . For arbitrary e, the same conclusion follows by applying the previous conclusion to Ur , Us, Ut acting on the JB-algebra Ale. We note also that there are triples of Jordan-orthogonal symmetries for which (4.15) fails. For example, if 1', s, t, ware Jordan-orthogonal symmetries in a spin factor, then (4.15) fails since UrUsUtw = ~W, cf. Lemma 3.6. 4.51. Lemma. Let !vI be a spm factor. Then !vI contams a non-zeT'O Carteswn tnple of symmetnes zfJ !vI zs four dzmenswnal.

Pmof. Write M = R1 EB N as in Definition 3.33. Here N contains all symmetries other than ±l. If dim !vI = 4, let 1', s, t be an orthonormal basis of N. Then T, s, t are Jordan-orthogonal symmetries. Representing Al as a JW-algebra (cf. Theorem 4.1), 1', s, t anticommutc, so one easily verifies that UrUsUt is the identity on AI. Now suppose that IvI is of unknown dimension, and admits a Cartesian triple 1', s, t of symmetries. Recall that N is equipped with an inner product such that a 0 b = (a I b)1 for all a, bEN. Suppose the dimension of M were greater than 4. Then N would have dimension greater than 3, so there would be a non-zero element w Jordan orthogonal to 1', S, t. Then

CARTESIAN TRIPLES

129

by the definition of a Cartesian triple and Lemma 3.6,

so w = 0, a contradiction. Thus we conclude M is at most four dimensional. On the other hand, r, s, t will be an orthonormal subset of N, so N is at least three dimensional and thus M is at least four dimensional. Thus M has dimension 4. 0 4.52. Lemma. Let M be the exceptwnal lEW-factor H3(O). Then M contams no non-zero Cartesian tnple of partzal symmetnes.

Proof. Let e be a non-zero projection in !vI and (r, s, t) a Cartesian triple of e-symmetries. We first show that e 1= 1 and that Me is four dimensional. Note that the center of !vIe is Re by Proposition 3.13, so Me is a JBW-factor. Write r = p - q where p and q are orthogonal projections with sum e. By Lemma 3.6 applied in the JBW-algebra !vIe, Us exchanges p and q, so p and q are equivalent projections. We claim both are abelian projections in Me. Suppose one were not abelian, say p. Since Us is a Jordan automorphism of !vIe, then q would also be non-abelian, and thus neither p nor q would be minimal (Lemma 3.30). This would imply that there are at least four mutually orthogonal non-zero projections in M, which by Lemma 3.22 would contradict M being a factor of type 1 3 . Thus e is the sum of the equivalent abelian projections p and q, and so by definition Me is a factor of type 1 2. This implies that e 1= 1, else M = Me would be both an 12 factor and an 13 factor, contradicting Lemma 3.22. Thus Me is a spin factor containing a Cartesian triple of symmetries. By Lemma 4.51, !vIe is four dimensional. On the other hand, let {ei)} be the standard system of 3 x 3 matrix units for H3 (0). Then {e33H3 (0 )e33} = Re33, so e33 is a minimal projection (Lemma 3.29). Since p, q, 1 - e are orthogonal and non-zero, by Lemma 3.22, 1- e must be minimal. lvIinimal projections in a JBW-factor are exchangeable (Lemma 3.30 and Lemma 3.19), so there is a symmetry w exchanging l-e and e33. Then w also exchanges e and l-e33 = ell+e22. Thus Uw will be an isomorphism from !vIe onto !vIeJ1+e22 c:=: H2(O). It follows that the latter must be four dimensional. However, since 0 is of dimension 8 over R, this is impossible. Thus we conclude that M contains no non-zero Cartesian triple of partial symmetries. 0 4.53. Lemma. If a lEW-algebm !vI contams a Carteswn tnple of symmetries, then M zs a lW-algebm, and zs reverszble in every faithful representation as a concrete lW-algebm.

Proof. Let (r, s, t) be a Cartesian triple in M. We first show that M is a JW-algebra. Suppose there exists a factor representation 1f from M onto

130

4

REPRESENTATIONS OF JB-ALGEBRAS

Mo = H3(O). Then (7f(1'), 7f(s), 7f(t)) is a Cartesian triple of symmetries in Mo. This is impossible by Lemma 4.52. Thus M admits no factor representation onto the exceptional factor H3(O). By Corollary 4.20, M is a JW-algebra. Assume M is faithfully represented as a JW-subalgebra of B (H)sa. By Corollary 4.30 it suffices to show that the 12 summand of M is reversible. Let e be the central projection such that eN! is the 12 summand of M. We may assume e =I- 0 (for otherwise there is nothing to prove). Since multiplying by a central projection is a Jordan homomorphism, (e1', es, ct) is a Cartesian triple in el\;[. Thus by replacing M with eM, it suffices to prove that a JW-algebra M of type 12 containing a Cartesian triple of symmetries is reversible. Let So = 1 and let (Sl, S2, S3) = (1', S, t) be the given Cartesian triple of symmetries in N!. Let x E M and define (4.16) (4.17)

Zo = (((x 0 sd 0 sd 0 S2) 0 S2), Zi = (x - zo) 0 s, for I = 1,2,3.

We are going to show that zo, Zl, Z2, Z3 are central and that

2:= ZiSi·

x =

(4.18)

Let 7f be a factor representation of M and define Mo = 7f(M) (()-weak closure). Since M is of type 12, by Lemma 4.8, Mo is a type 12 factor, i.e., a spin factor. Let Wi = 7f(Si) for I = 0,1,2,3. Then (W1,W2,W3) will be a Cartesian triple of symmetries in 11,10 . By Lemma 4.51, Mo is 4-dimensional, so {WO,W1,W2,W3} will be a basis for N!o. Thus we can write 3

(4.19)

7f(X)

2:= Aiwi

=

i=O

for suitable scalars Ao, ... , A3' One easily verifies that (4.20) and (4.21)

Ail = (7f(x) - Ao1)

0

Wi

for

I

= 1,2,3.

Combining (4.16), (4.17), (4.20), (4.21) gives (4.22)

7f(Zi)

=

Ail

for i = 0, 1,2,3.

CARTESIAN TRIPLES

131

Thus for each ~, each a E M, and each factor representation H, the element H(Zi) operator commutes with H(a). Since factor representations separate points (Lemma 4.14), it follows that Zo, ... , Z3 are central in M. Furthermore, from (4.19) and (4.22) it follows that

for all factor representations H, from which (4.18) follows. Finally, if a1,' .. , an are in M, then we can write 3

ak =

L

ZjkSJ

j=o

where each Zjk is central in M. Then a1a2'" an + an" . a2a1 is a sum of terms of the form Zi, Zi2 ... Z'n (Si' Si 2 ... S'" + Si" ... Si2 Si,) where each ZiJ is central and each Si j is in {Sl' S2, S3}. Since the symmetries Si, s·J anticommute for z I.../.. J , each sum (s·~1 s·72 ... s· + s· ... s·72 s·1.1 ) can be rewritten in a similar form with n ~ 3. Such sums coincide either with a Jordan product of elements of !vi or a Jordan triple product, and thus are in M. Reversibility of M follows. 0 1,n

1,n

We finish this section with some technical results on Cartesian triples that we will make frequent use of in later chapters. 4.54. Lemma. Let M C B(H)sa be a JW-algebra, and let Ro(M) be the real algebra generated by .M. If (r, s, t) ~s a Carteswn tnple of symmetnes m M, then J = rst ~s a central skew-adJomt element of Ro(M) satisfying J2 = -1.

Proof. By definition, r, s, t are Jordan orthogonal symmetries, and thus anticommute. This implies that J = rst is skew-adjoint and satisfies p = -1. Furthermore, by assumption the symmetries satisfy UrUsUt = 1 on M. Since conjugation by a symmetry is an algebraic automorphism of B(H), then UrUsUt = 1 also holds on the real algebra Ro(M) generated by M. Thus rstxtsr

This implies that rstx

=

x

for all x E Ro(M).

= xrst, and so

J

= rst is central in Ro(A1).

0

4.55. Lemma. Let M C B(H)sa be a reversible lW-algebra. If the real algebra Ro(M) generated by M contams a skew-adjoint element j which is central m Ro(M) and satisfies J2 = -1, then Ro(M) = M EEl jM

4

132

REPRESENTATIONS OF JB-ALGEBRAS

wdh the mherded product, mvolutwn, and norm, and wzth J as comple1; umt, becomes a von Neumann algebra wdh self-adJomt part M. Thus IvI zs zsomorphzc to the self-adJomt part of a von Neumann algebra. Proof. We first show that

Ro(M) = M EB JM

(4.23)

(direct sum as subspaces). If x E Ro(M), then

x

(4.24)

= ~(x

+ x*)

- ~J(J(x - x*)).

Since Ro(Jv1) is closed under the adjoint operation and IvI is reversible, then x + x* E Ro(M)sa = JlvI by reversibility. Furthermore, since J is central in Ro(M) and skew-adjoint, then J(x - x*) E Ro(M)sa = M. Thus Ro(M) = M + JM. On the other hand, M n JM = {O}, since each element of IvI is self-adjoint and each element of JIvI is skew-adjoint. This completes the proof of (4.23). Since M is norm closed, it follows from (4.24) that Ro(M) = MEBJM is also norm closed. Since J is central in Ro(A), with J* = -1 and J* = - J, then we can view Ro(M) as a complex *-algebra with imaginary unit scalar J, and with the involution inherited from B(H). The norm inherited from B(H) is a norm on Ro(IvI) viewed as a complex linear space, since for

x E Ro(M), 11(0:1

+ ,BJ)xI1 2 =

+ I'J)x)*(o:l + I'J)xll = 11(0:2 + 1'2)x*xll = (0: 2 + 1'2)llxI1 2. 11((0:1

Thus Ro(M) = M EB JM becomes a C*-algebra, with (Ro(M))sa = M. Since Jvf is monotone complete with a separating set of normal states, then so is the C*-algebra Ro (JvI) , so the latter is a von Neumann algebra (A 95). 0

Remark. In the context of the last theorem, it is not necessarily the case that AI + zJlvI is a von Neumann algebra for the product inherited from B(H). For example, suppose that M = Ivh(C)sa, and that A is the universal C*-algebra associated with the lB-algebra JlvI (Definition 4.37). Identify Jvf with its canonical image in Asa. Then A is the direct sum of two copies of M 2 (C) (Proposition 4.48), and by definition the C*subalgebra of A generated by IvI is all of A. By finite dimensionality, A will be the complex *-algebra generated by M. If Ai + iM were closed under the multiplication inherited from A, then the complex *-algebra generated by M would be M + iM, so then M + iM = A. Since M + lM

133

CARTESIAN TRIPLES

is 8-dimensional (as a real vector space), and A is 16-dimensional, this is impossible. Similarly, if Ro(M) were closed under multiplication by z, then Ro(M) would be the complex *-algebra generated by M, so M + j M = Ro (M) = A. As before, by counting dimensions this is shown to be impossible.

x*y on !vI +zM making M +zM We will now discuss products (x, y) into a von Neumann algebra with involution a + zb a - zb. We will say such a product is Jordan compatzble if the induced Jordan product matches the given one on M,i.e., aob=~(a*b+b*a) for a,bEM. f-)

f-)

4.56. Theorem. If a lEW-algebra M contams a Carteswn trzple of symmetrzes (r, s, t), then M zs zsomorphzc to the self-adJomt part of a von Neumann algebra. In partzcular, M = !v! + zN! can be equzpped wzth a umque von Neumann algebra product compatzble wzth the gwen lordan product and satzsfymg zrst = 1. For thzs product, the complex lmear span of 1,r,s,t zs a *-subalgebra *-zsomorphzc to M 2 (C), and thus M ~ M 2 (C)®B for a von Neumann algebra B. Proof. By Lemma 4.53, we may assume M is faithfully represented as a reversible JW-algebra, say M C B(H)sa. Let Ro(M) be the real *-algebra generated by M in B(H). Now by Lemma 4.54 the element j = -r st is central in Ro (!vI) and satisfies J * = - J, J 2 = -1. By Lemma 4.55, Ro(A1) = !v! EB JM becomes a von Neumann algebra for the complex unit J and the inherited norm, multiplication, and involution. Now note that the identity map on !v! extends to a complex linear bijection from M + JM to !v! + zM (with the former viewed as a complex space with complex unit J) taking a + Jb to a + zb for a, bE M. Thus we can use this to carry the product and norm from !v! + JM to !v! + zM, giving Al +z!v! an associative Jordan compatible product for which Al +zM becomes a von Neumann algebra. (In fact, this map is an isometry for the norms on M + J M and Al + z!v! inherited from B (H), but we will not need this result.) This product on M + z!v! will not necessarily coincide with the associative product defined in B(H), though the Jordan products will agree. (See the remark preceding this lemma.) Since 1 = Jr st in N! EB J M, then for the corresponding product on !v! + zAl, we have zrst = 1. Now let M be any von Neumann algebra containing a Cartesian triple of symmetries r, s, t such that zrst = 1. Then the product of allY two of the three symmetries is a multiple of the third, so the complex linear span of 1, r, 5, t is a four dimensional complex *-subalgebra Co of M, which is easily seen to be isomorphic to M2(C). (In fact, the multiplication on Co is determined by the fact that r, 5, t are Jordan orthogonal, and thus are anticommuting symmetries. As we have observed before, the Pauli spin matrices form a Cartesian triple, and thus are also anticommuting symmetries. It is then straightforward to check that the map taking 1 to

4

134

REPRESENTATIONS OF JB-ALGEBRAS

1 and r, s, t to the three Pauli spin matrices (in the order listed in (3.9)) extends to a *-isomorphism from Co onto l\12(C).) Thus M contains a system of self-adjoint 2 x 2 matrix units {eij}, so it follows that M ~ M 2 (C) (>9 B ~ M 2 (B), for B equal to the relative commutant of the set of matrix units {ei]} in M (Lemma 4.27). If M is represented as a weakly closed *-subalgebra of B (H), then the same will be true of B, so B is a von Neumann algebra. Now let * be a second Jordan compatible product on M wch that zr * s * t = 1. We will show * coincides with the given product. We first sketch the idea. The Jordan product on M so. extends uniquely to a complex linear Jordan product on M. As seen above, we can represent elements of M as 2 x 2 matrices. Then the associative product is determined by the Jordan product in the following sense: (4.25) (Note that a similar calculation (3.7) was behind the Jordan coordinatization theorem (Theorem 3.27). In that case, lacking M = Msa+zMsa, the Jordan product on the self-adjoint part of the algebra did not determine the Jordan product on the whole algebra, and so we needed 3 x 3 matrix units instead of 2 x 2.) Now we give the details of the proof. For each associative product on M, we can create 2 x 2 matrix units from linear combinations of 1, r, s, t, as shown above, and so can represent M as the algebra of 2 x 2 matrices over an algebra B, where B is the relative commutant of r, s, t in M. Since r, s, t are symmetries, any element x in M element commutes with each of r, s, t iff UrX = UsX = UtX = x. Since these are Jordan expressions, the relative commutants for the two products coincide. Thus we can represent M with each product as 2 x 2 matrices over a fixed subset B. The two products on M will then coincide iff the two products coincide when restricted to B. Since the two products must induce the same Jordan product on M, the equality of the two products then follows from (4.25). 0 By the bicommutant theorem, an irreducible von Neumann algebra B(H) must be equal to B(H). For JW-algebras, this is not necessarily the case, since for example l\1,,(R) acts irreducibly on c n . However, if we add the assumption that NI is isomorphic to the self-adjoint part of B (H) the analogous conclusion holds, as we will show below (cf. Proposition 4.59). The remaining results will also be useful in our characterization of normal state spaces of von Neumann algebras.

M

C

4.57. Lemma. Let M be a reversible lW-subalgebra of B(H)sa. Suppose that there exists a self-adjoint system {eij} of n x n matrix umts

CARTESIAN TRIPLES

135

in B (H) (wzth 1 < n < 00) such that ei) + eli E M for all z, J. Suppose further that there zs a partwl symmetr·y t such that ell - e22, e12 + e21, t form a Carteswn trzple of partwl symmetrzes zn M. Then there is an element J zn the center of the real algebra Ro(Jv1) generated by M such that J* = -J and J2 = -1. Proof. Note that ei) = eii (ei] + e);) E Ro (lIn for all z i= J, so {ei)} is a self-adjoint system of matrix units for Ro(M). By Lemma 4.27 we can identify Ro(M) with Mn(B) where B is the relative commutant of these matrix units in Ro(M). By assumption Ai is reversible, so M = Ro(M)sa = Hn(B). Define r = ell - e22 and s = e12 + e21. Then by assumption there is an (ell + e22)-symmetry t such that T, s, t form a Cartesian triple of (ell + e22)-symmetries. Define Jo = Tst, and

By Lemma 4.54 applied to M 12 , Jo satisfies J5 = -(ell + e22) and Jo = -jo, and Jo is central in Ro(M12). Since Jo is in Mn(B) and Jo = JO(ell + e22) = (ell + e22)Jo, then Jo admits a unique representation 2

(4.26)

Jo =

L

bijei)

i.j=l with each bi ) in B. We will show that there is a central element z in B such that Jo = z(ell +e22), and that J = L:~l zeii will have the properties specified in the lemma. Since Jo is central in R o (M 12 ), it commutes with ell, so

By the uniqueness of the representation (4.26), b21 = b12 = O. Since jo commutes with e12 + e21, a similar calculation shows bll = b22 . Thus there is an element z E B such that

Since jo = -jo and j5 = -(ell +e22), then the element z satisfies z* = -z and z2 = -1. We are going to show z is central in B. Let b E B, and let x = be12+b*e21. Then x E Ro(M)sa = M, and (ell +e22)x(ell +e22) = x, so x E (ell + e22)M(ell + e22). In particular, Jo must commute with x. Then

136

4.

REPRESENTATIONS OF JB-ALGEBRAS

which gives

This implies zb = bz, so z is central in B as claimed. Finally, define J = zeii· Then J is clearly central in Mn(B) Ro(M), and satisfies J2 = -1, J* = -J. 0

L

4.58. Lemma. Let M be a lEW-algebra zsomorphzc to B (H)sa. If e is a projectIOn m M and rand S are lordan orthogonal e-symmetrzes, then there exzsts an e-symmetry t m M such that (r, s, t) is a Carteszan triple. Proof. It suffices to prove the result for M = B(H)sa. Define t = zrs. Since rand s anticommute, one finds that t is an e-symmetry that is Jordan orthogonal to rand s. Furthermore, rst = -ze and then tsr = (rst)* = (-ze)* = ze. Therefore for x E eB(H)e,

so (r, s, t) is a Cartesian triple of e-symmetries in M. (See also (A 192)). 0 We say a concrete JW-algebra Me B(H)sa is zrreduczble if there are no proper closed subspaces of H invariant under lVI.

4.59. Proposition. If M zs a lEW-algebra zsomorphzc to B(K)sa for a complex Hilbert space K, and lVI zs represented as an zrreduczble concrete lW-algebra m B(H)sa, then M = B(H)sa. Proof. The case where K is one dimensional is obvious, and so we assume that dim K > 1. We first show that there is an element J in the center of the real algebra Ro(M) generated by lVI such that J* = -J and J2 = -1. We are going to apply Lemma 4.57, so we show that M ~ B(K)sa satisfies the hypotheses of that lemma. (Note that there is a subtlety here: associative computations are taking place in B (H) and not B (K).) We can write 1 = Pl + ... + Pn where Pl,P2, ... ,Pn are projections exchangeable by symmetries in Ai and 2 <::::: n < 00. For z = 1,2, ... , n let Si be a symmetry in !vI that exchanges Pl and Pi' Define eiJ

=

SiPlSJ E B(H)

for 1

<:::::

i,J

<:::::

n.

By Lemma 4.28, {eij} is a system of matrix units for Ro(M) C B(H), and eiJ + eJi = SiPlSj + SJPlSi E lVI. Since M ~ B(K)sa, and r = ell - e22 and s = e12 + e2l are Jordan orthogonal (ell + e22)-symmetries, by Lemma 4.58 there is an (ell + e22)symmetry t in M such that (r, s, t) is a Cartesian triple. The existence of j as described above now follows from Lemma 4.57.

NOTES

137

Since M is irreducible, the von Neumann algebra it generates is irreducible, and so has commutant C1 in B(H) (A 80). Thus by the bicommutant theorem (A 88), the von Neumann algebra generated by M is all of B (H), so J is also central in B (H). Therefore J must be a scalar multiple of the identity. The only possibilities are J = ±d. It follows that Ro(M) contains d. Thus Ro(M) is a complex *-subalgebra of B(H). Hence Ro(M) is the von Neumann algebra generated by M, which as remarked above must be all of B (H). The self-adjoint part of a C* -algebra is reversible in every concrete representation (Proposition 4.34), so M is reversible. Hence M = Ro(M),a = B(H),a by Lemma 4.25. D

Notes The result that spin factors are JW-algebras (Theorem 4.1) is due to Topping [129]' but the proof we have given is taken from that in [67]. The fact that H3(O) is exceptional (Theorem 4.6) is a classic result of Albert [2]. The "Gelfand-Naimark" type theorem for JB-algebras (Theorem 4.19) appeared in [8]. We have taken the decomposition of JBW-algebras of type I3 (Lemma 4.21) from [67]; this result is due to Stacey [121]. The decomposition of an arbitrary JBW-algebra into JW and exceptional pieces (Theorem 4.23) appeared first in [116]' but we have followed the proof in [67]. Reversibility in JC-algebras and JW-algebras was investigated by Stormer [124, 125, 126]' and by Hanche-Olsen [66]. Those concrete JCalgebras that are the self-adjoint part of the C*-algebra they generate are characterized in [124]. The key result that a concrete JW-algebra is reversible iff its I2 summand is reversible (Corollary 4.30) is in [125] and in [67, Thm. 5.3.10]. Our proof of the reversibility of a JBW-algebra whose identity is the sum of at least three exchangeable projections (Theorem 4.29) is essentially a proof of the Jordan coordinatization theorem (Theorem 3.27) for the case of special Jordan algebras, and is based on the proof in [67, Thm. 2.8.3]. The concept of the universal C* -algebra for a JB-algebra (Definition 4.37) first appeared in [11], but the construction of the universal C* -algebra in Proposition 4.36 is new. The universal C*-algebra was studied further in [66], where the results in Corollary 4.39, Corollary 4.42, and Proposition 4.48 can be found. The corresponding notion of a universal von Neumann algebra (Definition 4.43) comes from [67]' as does the description of the universal von Neumann algebra for the self-adjoint part of a von Neumann algebra (Lemma 4.49). Cartesian triples were introduced in the characterization of normal state spaces of von Neumann algebras by Iochum and Shultz [71], although not with that name, and the results of the final section of this chapter mostly appear in [71]. Cartesian triples played an important role in the

138

4

REPRESENTATIONS OF JB-ALGEBRAS

treatment of orientations of von Neumann algebras and their normal state spaces, which was first announced in [13], and presented with complete proofs in [AS, Chpt. 7].

5

State Spaces of Jordan Algebras

In this chapter we will discuss properties of the normal state space of JBW-algebras. Since every JB-algebra state space is also the normal state space of a JBW-algebra (Corollary 2.61), these properties also apply to JB-algebra state spaces. We will first define the notion of the carrier projection of a set of normal states, and the concept of orthogonality of states. Then we will characterize those maps that are the dual maps to Jordan homomorphisms. Next we will discuss traces on JB-algebras. These will playa key role in the proof of the main result of this chapter: the fact that there is a 1-1 correspondence between projections in a JRW-algebra and the norm closed faces of the normal state space. For most types of JBW-algebras we reduce this result to the corresponding result for norm closed faces of normal state spaces of von Neumann algebras. For JBW-algebras of type 12 or 13 , we need a different argument based on traces. We then study some properties of the state space that concern the pure states (i.e., the extreme points of the state space). One of these is the Hilbert ball property, which says that faces generated by pairs of pure states are affinely isomorphic to Hilbert balls. This is closely related to the physically interesting property that is called "symmetry of transition probabilities". These results also playa role in our proof of the fact that every normal state on an atomic JB\V-algebra is a countable convex sum of orthogonal extreme points. This is then used to show that atomic JBWalgebras admit a notion of trace class elements that allows us to generalize the well known description of normal states on B (H) in terms of the trace and "density matrices." We finish by describing two properties related to faces and affine automorphisms of the normal state space: "symmetry" and "ellipticity" . Many of the aforementioned properties will reappear in later chapters in our abstract characterizations of those convex sets that are affinely isomorphic to Jordan state spaces or C* -algebras. Carrier projections 5.1. Lemma. Let M be a lEW-algebra, and F a non-empty set of posztive normal functwnals on NI. Then there zs a smallest proJectwn P such that lJ"(p') = 0 (equivalently, lJ"(p) = jjlJ"jlJ for alllJ" m F. Proof.

Let

s = {p E M

j p = p2,

IJ"

(pi) = 0 for all

IJ"

E F}.

140

5

STATE SPACES OF JORDAN ALGEBRAS

Note that for projections p, q and 0" E F,

O"(p') = O"(q') = 0

+ q') = 0 + q')) = 0

<=>

O"(p'

=}

O"(r(p'

=}

O"(p' V q') = 0

=}

O"((p A q)') = O.

(5.1)

(by equation (2.6))

(by Lemma 2.22)

Thus if p and q are in 5, by (5.1) pAq is in 5, so 5 is directed downwards. By Proposition 2.5, {q E 5} converges O"-weakly to a projection p. Then p E 5 and evidently p is the minimal element of 5. 0 The definition below generalizes the notion of the carrier projection of a normal state on a von Neumann algebra (A 106).

5.2. Definition. Let M be a JBW-algebra, and F a non-empty set of positive normal functionals on AI. The smallest projection p such that O"(p) = 110"11 (equivalently, O"(p') = 0) for all 0" in F is called the support pro]ectzon or carner pro]ectzon of F, and is denoted carrier(F). For a single positive normal functional 0" we will refer to the carrier projection of the set {O"} as the carrier projection of 0". If F is any set of normal positive functionals on a JBW-algebra lvI, then we have (5.2)

v

{carrier(w) I w E F}

=

carrier(F).

(The projection on the left takes the value 1 on every element of F, so dominates carrier(F). On the other hand, carrier(F) has the value Ilwll on each w E F, so dominates carrier(w) for all wE F.) Note also that by Lemma 5.1 and the equivalence (1.57), the carrier projection of a set F of normal positive functionals on a JBW-algebra AI is the minimal projection p such that the image of contains F. The following definition is a special case of the notion of orthogonality in a base norm space (A 24).

U;

5.3. Definition. Positive normal linear functionals 0", algebra M are orthogonal if

We then write 0"

~ T.

T

on a JBW-

CARRIER PROJECTIONS

141

Each normal linear functional p on a JBW-algebra admits a unique decomposition p = 0" - T as a difference of positive normal functionals with 0" ~ T (Proposition 2.58). We then write p+ in place of 0" and p- in place of T, and refer to

as the orthogonal decompositwn of p. 5.4. Lemma. Posdwe normal lznear functionals 0", T on a lBWalgebra M are orthogonal zfJ thezr carrzer proJectzons are orthogonal. Proof. If 0" ~ T, let p = 0" - T. Then this is the unique orthogonal decomposition of p, so by Proposition 2.58 there is a projection p such that

Therefore U;,T = T (d. (1.56)). Thus the carrier projections of 0" and T are under p and pi respectively (by the remarks following the definition of the carrier projection (Definition 5.2)), and so are orthogonal. Conversely, suppose that p and q are the carrier projections of 0" and T, and that p ~ q. Then T(p) ::; T(q') = 0, so T(p) = 0; similarly O"(q) = O. Thus (0" -

Since

lip - qll ::;

T)(p - q) = O"(p)

+ T(q) =

110"11

+ IITII.

1, then

The opposite inequality follows from the triangle inequality, so equality holds. It follows that 0" ~ T. D Note that as a consequence, for positive normal functionals on a JBW-algebra (5.3)

0"

Indeed, if

0", T

0"

~

T

<=>

U;arrier(a) T =

0"

and

T

O.

have carrier projections p and q respectively, then ~

T

=?

P~ q

=?

U;T = U;U;T = 0,

where the last equality follows from Proposition 2.18. On the other hand,

U;T

=

0

=?

and so (5.3) follows.

U;,T = T

=?

pi 2': carrier(T) = q

=?

P ~ q,

142

5

STATE SPACES OF JORDAN ALGEBRAS

5.5. Lemma. Let p, 0"1 and 0"2 be posztwe normal functwnals on a lEW-algebra M. If p is orthogonal to 0"1 and to 0"2, then p is orthogonal to )'10"1 + ),20"2 for all posdzve scalars ),1, ),2. Proof. If p is orthogonal to of p, then by (5.3)

so p is orthogonal to

),W1

0"1

and to

+ ),20"2

0"2,

and p is the carrier projection

by (5.3). 0

5.6. Definition. If M is a lBW-algebra with normal state space K, and p is a projection in M, then we define the norm closed face Fp of K by

Fp = {O"

E

K I O"(p) = I}.

We say Fp is the face of K assocwted wlth p. A face of the form Fp we call a proJectzve face. One of the key results of this chapter will show that every norm closed face of the normal state space K of a lBW-algebra is projective (Theorem 5.32). We note that by Proposition 1.41, (5.4)

Fp = (im U;) n K = (ker U;,) n K.

Since the predual of M is positively generated (cf. Corollary 2.60), it follows that for each element a of a lBW-algebra and each projection p, we have a = 0 on Fp iff a = 0 on im Thus for a E !v!,

U;.

(5.5)

5.7. Proposition. Let a, b be posdzve elements of a lEW-algebra M. Then a -.l b lIt there eXlsts a proJectwn P such that a = 0 on Fp and b = 0 on Fp " Proof. Suppose that a -.l b. Let p = r(b). (Recall that r(b) denotes the range projection of b, cf. Definition 2.14.) Since positive elements are orthogonal iff their range projections are orthogonal (Proposition 2.16), then r(a) -.l r(b) = p, and so r(a) :::; p'. Now p = 1 on Fp implies that pi = 0 on Fp. Since r(a) :::; pi, then r(a) = 0 on Fp. By (2.6) this implies that a = 0 on Fp. Similarly since r(b) = p = 0 on F p" then b = 0 on Fp" Conversely, suppose that a and b are positive elements of M, and that there is a projection p such that a = 0 on Fp and b = 0 on F p" Then

THE LATTICE OF PROJECTIVE FACES

143

Upa = 0 (by (5.5)), so by (l.49) Up,a = a. Then by definition r(a) ::; p'. Similarly, b = 0 on F p ' implies that r(b) ::; p, and so r(a) ~ r(b). This completes the proof that a ~ b. D The lattice of projective faces Recall that a JBW-algebra Jvf can be identified as an order unit space with the space Ab(K) of bounded affine functions on K, where K is the normal state space of M (cf. Corollary 2.60). Viewing elements of M as functionals on K, we have the following result, which shows that a projection p is determined by its associated face Fp. 5.8. Proposition. If p

lS

p = inf {b E M

(5.6)

I

a projectwn m a lEW-algebra M, then

0::; b ::; 1 and b = 1 on Fp }.

Proof. To verify this, let b E M with 0 ::; b ::; 1 and b = 1 on Fp. Then 0::; 1 - b ::; 1 and 1 - b = 0 on Fp. Thus Up(l - b) = 0 (d. (5.5)), and so Up' (1 - b) = 1 - b. Therefore

which implies b 2': p. This proves (5.6). D It follows from (5.6) that the complementary face Fp ' is determined by Fp. Note also that (5.6) implies

(5.7)

carrier(Fp ) = p,

and thus p Fp is a bijection from the set of projections of a JBW-algebra M to the set of projective faces of the normal state space of M. f-)

5.9. Definition. Let M be a JBW-algebra with normal state space

K. We will write F for the set of projective faces of K, ordered by inclusion. We define the map F F' on F by (Fp)' = Fp' for each projection f-)

p, and call F' the complementary face of F.

5.10. Proposition. Let M be a lEW-algebra wlth normal state space K. The set F of proJectwe faces of K forms a complete orthomodular lattice, wdh F /\ G = F n G, and wzth OT·thocomplementatwn F F'. The map p Fp zs a lattice ortho-isomoTphzsm from the proJectwn lattzce of AI onto F. The inverse map takes a proJectwe face to ltS carner projection. f-)

f-)

Proof. The map p Fp is evidently order preserving, as is the inverse map F carrier(F). Thus P and F are order isomorphic. f-)

f-)

144

5

STATE SPACES OF JORDAN ALGEBRAS

Since the lattice of projections is a complete orthomodular lattice with orthocomplementation p I-> p' (Proposition 2.25), it follows that :F is an orthomodular lattice with orthocomplementation Fp I-> F p" Now Fp = {O" E K I O"(p') = O} and (5.1) imply that Fp n Fq = Fpl\q. Thus the greatest lower bound of projective faces in :F is their intersection. 0

Dual maps of Jordan homomorphisms 5.11. Lemma. Let M be a lEW-algebra, and let p be a proJectwn m M wdh assocwted face F = Fp. Then p lS the umque posltwe bounded affine functwnal on the normal state space K of M wdh value 1 on F and value 0 on F' = Fp' .

Proof. Recall that we can identify M with the set of bounded affine functions on K (Corollary 2.60). Let b E M+ satisfy b = 1 on F and b = 0 on F'. Then Up,b = 0, so by the complementarity of Up and Up' we have Upb = b. We next show b :S 1. If 0" E K, since 11U; I :S 1 (Proposition 1.41), then U;O" = AW for some w E F = im U; n K and A E R with 0 :S A :S 1. Then for all 0" E K, O"(b)

= O"(Upb) = (U;O")(b) = Aw(b) = A :S 1,

since b = 1 on F. Thus b :S 1 follows. Then

Since 1 - b is 0 on Fp and 1 on Fp" the same argument shows 1 - b :S p' = 1 - p, so p :S b. Thus b = p. 0 5.12. Lemma. Let A be alE-algebra wdh state space K. Then the map a I-> alK lS an lsomorphlsm from the order umt space (A, 1) onto the order umt space A(K) of w*-contmuous affine functwns on K wlth the pomtwise ordermg and norm, wlth the constant functwn 1 as dlstmgmshed order unit.

Proof. The fact that (A, 1) is a complete order unit space was established in Theorem 1.11. The rest of the lemma follows from the corresponding statement for complete order unit spaces: (A 20). 0 5.13. Lemma. If M I , M2 aTe lEW-algebms wdh nOTmal state spaces K I ,K2 , then T I-> T*IK2 lS a 1-1 cOTTespondence of positive unital 0"weakly contmuous maps T : MI ---> M2 and affine maps from K2 mto K I . Similarly, if A I ,A 2 aTe lB-algebms with state spaces K I ,K2 , then

DUAL MAPS OF JORDAN HOI\IOI\IORPHISMS

145

T f---> T* IK2 zs a 1-1 correspondence of posztwe unital maps T : Al and affine w*-contmuous maps from K2 mto K 1 .

->

A2

Proo j. Assume first that K 1, K 2 are the normal state spaces of the JBW-algebras lvh, M2 respectively. If w E K 2, then T*w = w 0 T is aweakly continuous, and thus is normal (Corollary 2.6). Thus T* maps K2 into K 1 . If Tl and T2 are bounded linear maps from Ml into M2 such that Ti and T2 agree on K 2, then Tl a and T 2a agree on all normal states of !l12 , so we must have Tl = T 2. Thus T -> T*IK2 is 1~1. Now let ¢ : K2 -> Kl be an affine map. Then ¢ extends uniquely to a linear map from the linear span of K 2 into the linear span of K 1. By Corollary 2.60, the unit ball of the preduals (M2 )*, (Md* are co(K2 U (-K2)) and co(K 1 u( -Kd) respectively, so the extension of ¢ has norm at most one. We also denote this extension by ¢, and let T = ¢* : l'vh -> M 2 . Then T is a positive unital a-weakly continuous map, and T* restricted to K2 equals ¢. This shows that T -> T* IK2 is surjective, which completes the proof of the first statement of the lemma. Assume next that K l , K2 are the state spaces of JB-algebras AI, A2 respectively. If T : Al -> A2 is a unital positive map, then T* is a w*continuous affine map from K 2 into K 1. Since states separate elements of a JB-algebra, then T -> T* IK2 is injective. To see that this map is surjective, let ¢ be a w*-continuous affine map from K2 into K l . Identify A; with A(K,) for z = 1,2 (cf. Lemma 5.12). Define T: Al -> A2 to be the map given by Ta = a 0 ¢. Then T* restricted to K2 coincides with ¢. Thus T -> T*IK2 is bijective. D

5.14. Lemma. Let Ail, !lh be lEW-algebras wzth normal state spaces K 1 , K 2 . Let ¢ : K2 -> Kl be an affine map, and T : !vh -> M2 the posztwe umtal a-weakly contmuous map such that ¢ = T*IK2' Let Fl and F2 be pTOJectwe faces of Kl and K2 respectwely, wzth assocwted pTOjectwns PI and P2. Then these are equwalent:

(i) TPI = P2, (ii) ¢~l(Fd = F2 and ¢~l(F{)

=

F~.

If ¢ zs surJectwe, then (2) and (u) are also equwalent to (iii) d;(F2)

= Fl and

¢(F~)

= F{.

PTOOj. (i) <=> (ii) Viewing elements of Ml and !viz as bounded affine functions on their normal state spaces, using Tpl = PI 0 (T*) = PI 0 ¢, and applying Lemma 5.11, we have TPI

= p2

(Tpl)~l(l)

-¢=?

= F2 and q;-l(pll(l)) = F2 and

-¢=?

d;~l(Fl)

-¢=?

=

F2

and

(Tpd~l(O)

=

d;~l(pll(O))

d;~l(F{)

=

F~.

F~

=

F~

146

5.

STATE SPACES OF JORDAN ALGEBRAS

(iii) =} (i) Assume (iii) holds, and let b = TPI' Then b ?: 0 and b = PI 0 ¢ on K 2, so by (iii) b is a positive element of M2 with value 1 on F2 and value 0 on F~. By Lemma 5.11, b = P2, so TPI = P2. (ii) =} (iii) Assume ¢ is surjective, and that (ii) holds. Then applying ¢ to each side of the equations in (ii) gives (iii). D

5.15. Lemma. Let MI and Nh be lEW-algebras wlth normal state spaces KI and K 2 , and let ¢ : K2 ---> KI be an affine map. Le~ T : MI ---> M2 be the unital posltwe a-weakly contmuous map such that T* IK2 = ¢. Then the followmg are equwalent:

(i) T lS a umtal lordan homomorphlsm from Nfl mto lvf2, (ii) ¢-l preserves complements of proJectwe faces, (iii) ¢-l as a map from the lattlce of proJectwe faces of KI mto the lattlce of projective faces of K2 preserves lattlce operatwns and complements. Proof. Note first that if p is a projection in M I , and q is the range projection of Tp', then

where the penultimate equality follows from the fact that positive elements of M2 and their range projections have the same annihilators in K2 (d. (2.6)). Thus the inverse image of a projective face is projective. (i) =} (ii) Let FI be any projective face of KI and let PI E MI be the associated projection. If (i) holds, let P2 = TPI E lvh, and let F2 be the corresponding projective face of K 2. By Lemma 5.14, ¢-I(Fd = F 2, and ¢-I(F{) = F~, so ¢-I(F{) = (¢-I(FI ))'. Thus ¢-l preserves complements. (ii) =} (i) We first show T maps projections to projections, preserving orthogonality of projections. Let PI E MI be a projection with associated face FI C K I . Let F2 = ¢-I(Fd; by the remarks in the first paragraph of this proof, F2 is a projective face of K 2. Let P2 E AI2 be the associated projection. By (ii), ¢-I(F{) = F~, so by Lemma 5.14, TPI = P2. Thus T maps projections to projections and preserves complements of projections (since T1 = 1). Now suppose P and q are orthogonal projections. Since T is a positive map, then p:S; q' implies Tp:S; T(q') = (Tq)', so Tp...L Tq. Thus T also preserves orthogonality of projections. It follows that T preserves squares of elements with finite spectral decompositions. Since T is a positive unital map and M has the order unit norm, T is norm continuous (A 15). By continuity and the spectral theorem (Theorem 2.20), T preserves squares of all elements. It follows that T is a Jordan homomorphism. (ii) ¢? (iii) The map ¢-l preserves intersections and thus lattice greatest lower bounds (d. Proposition 5.10). Since complementation is order re-

TRACES ON JB-ALGEBRAS

147

versing, if ¢-l preserves complements, it will then necessarily also preserve least upper bounds. The reverse implication (iii) implies (ii) is trivial. 0 5.16. Proposition. If M l , M2 are lEW-algebras wdh normal state spaces K l , K 2, then T f-7 T*IK2 zs a 1-1 correspondence of lordan zsomorphzsms from !vfl onto A12 and affine zsomorphzsms from K2 onto K l . Similarly, zf Al and A2 are lE-algebras wzth state spaces K l , K 2, then the map T f-7 T*IK2 zs a 1-1 correspondence of lordan zsomorphisms from Al onto A2 and affine homeomorphzsms (for the w*-topologies) from K2 onto K l .

Proof. Assume that M l , !vf2 are JBW-algebras with normal state spaces K l , K 2. Then T f-7 T*IK2 is a 1-1 correspondence of unital order isomorphisms from Ml onto M2 and affine isomorphisms from K2 onto Kl (d. Lemma 5.13). Note that T is a unital order isomorphism iff T is a Jordan isomorphism. In fact, unital order isomorphisms are Jordan isomorphisms (Theorem 2.80), and conversely, each Jordan isomorphism preserves squares, so is an order isomorphism. This completes the proof of the first statement of the proposition. The proof of the second statement is similar. 0 5.17. Corollary. Let AI be a lEW-algebra wzth normal state space K. Let F and C be pro)ectwe faces with assocwted pro)ectwns p, q respectwely. Let T be a lordan automorphzsm of M. Then T*(F) = C iff Tq=p.

Proof. If Tq = p, then T*(F) = C follows from Lemma 5.14((i) =} (iii)). Conversely, suppose T*(F) = C. Since T* is an affine automorphism of K (Proposition 5.16), it preserves complements of projective faces (see the remarks following (5.6)). Thus T*(F') = C'. Now by Lemma 5.14 ((iii)=} (i)), Tq = p. 0 Traces on JB-algebras

5.18. Lemma. Let M be a lEW-algebra, and let T be a normal state on M. The following four propertzes are equwalent.

(i) T((a 0 b) 0 c) = T(a 0 (b 0 c)) for all a, b, c EM, (ii) T(6Zoa) = T({bab}) for all a,b E M, (iii) + U;,)T = T Jar all pro)ectwns p E M, (iv) U;T = T Jar all symmetrzes s E M.

(U;

Proof. (i) =} (ii) Let a, bE M. By the definition of {bab} (cf. (1.13)),

(5.8)

T (

{bab} =

T

(2b

0

(b

0

a) - b2

0

a).

5.

148

STATE SPACES OF JORDAN ALGEBRAS

If (i) holds, then

T(2b

0

(b

0

a)) = T((2b

0

b)

0

a) = T(2b 2

0

a)

which when substituted into (5.8) gives (ii). (ii) =? (iii). Let a E M. For a projection p, taking b b = pi in (ii) gives

T((Up + Up' )(a)) = T(p 0 a)

+ T(pl 0

=

p and then

a) = T(a),

which proves (iii). (iii) ¢} (iv) By spectral theory, the symmetries s in M are the elements of the form s = 2p - 1 = p - pi, for p a projection. Now the equivalence of (iii) and (iv) follows from Us = 2(Up + Up') - I (cf. (2.25)). (iii) =? (i) Let a, b, c E M. From the definition (1.12) of the triple product we have

(5.9)

(a

0

b)

0

c - a 0 (b

0

c) = ~ ( { bac} - {acb}).

Thus to prove (i) it suffices to show (5.10)

T ( {bac} - {acb})

=

o.

By linearity of the triple product in each factor, the spectral theorem, and norm continuity of Jordan multiplication, it suffices to prove (5.10) for the case where b = p is a projection. By (iii)

T({pac} - {acp}) = T(Up + Up,)({pac} - {acp}), so we will be done if we can show (5.11) for all projections p in M and all elements a, c E M. To prove (5.11), we begin with the identity (5.12)

2{bac}

0

b = {b(a

0

c)b}

+ {b 2 ac}.

(This is easily verified for any special Jordan algebra, and then holds in all Jordan algebras by Macdonald's theorem (Theorem 1.13).) Applying (5.12) with b = p gives (5.13)

2{pac}

0

p = {p(a

0

c)p}

+ {pac}.

149

TRACES ON JB-ALGEBRAS

Interchanging the roles of a and e and subtracting gives

2p

(5.14) Let x ~(I

+ Up

0 (

{pac} - {pea})

=

{pac} - {pea}.

{pac} - {pea}. Then by (5.14), x = 2p - Up') (cf. (1.47)), then x = (I

(5.15)

+ Up

0

x.

Since pox

- Up' )x.

Applying Up to both sides of (5.15) gives Upx = 2Upx so Upx = O. Applying U; to both sides of (5.15) gives Up'x = O. This proves (5.11), which completes the proof of (i). D

5.19. Definition. Let M be a JB-algebra. A state T on A is a tracial state if T((a 0 b) 0 c) = T(a 0 (b 0 c)) for all a, b, e E A, or (when M is a JBW-algebra) if T satisfies any of the equivalent conditions of Lemma 5.1S. A tr'acial state T is fadhful if whenever a ;::: 0 and T(a) = 0, then a = O. Remark. Recall that a tracial state on a von Neumann algebra M is a state T satisfying T (ab) = T (ba) for all a, b EM. If s is a symmetry in Msa and T is a tracial state on M, then T(sas) = T(as 2 ) = T(a) so T is a tracial state on the JBW-algebra M sa' Conversely, let T be a tracial state on M = M sa and also denote by T its complex linear extension to a state on M. Then for all symmetries s, t, we have T(st) = T(t(st)t) = T(tS). Then T(pq) = T(qp) for all projections p, q EM. By linearity and the spectral theorem, T(ab) = T(ba) for all a and b in M. Thus T is a tracial state on M. 5.20. Corollary. Every tracwl state T on a lEW-algebra M satisfies

a,b;:::O

=?

T(aob);:::O.

Proof. By Lemma 5.18(ii)

and {a 1 / 2 ba 1 / 2 }

;:::

0 by Theorem 1.25. D

5.21. Lemma. A spin factor admzts a umque tracwl state. Thzs state is faithful and normal.

Proof. Write M = Rl EB N as in Definition 3.33, where N is a real Hilbert space and x 0 y = (xIY)l for x, yEN. Recall that N consists of

150

5

STATE SPACES OF JORDAN ALGEBRAS

the scalar multiples of non-zero symmetries other than ±l (Lemma 3.34). Define T : A ---7 Rl by

T(A1

+ a) =

A

for A E R and a E N.

If s is any symmetry, then Us is a Jordan automorphism (Proposition 2.34), and so Us takes multiples of symmetries to multiples of symmetries, and therefore maps N onto itself. Thus

T(Us (Al T

+ a)) =

Al

+ T(Us(a)) =

Al = T(AI

+ a).

is positive since

Thus T is a tracial state on lvI. Furthermore, T((AI + a)2) = 0 evidently implies A = 0 and a = 0, so T is faithful. We now prove uniqueness. The smallest possible dimension of N is 2, so if s is a symmetry in N there is at least one unit vector t in N orthogonal to s, and then sot = O. By the definition of the triple product (5.16)

Uts

= 2t 0 (t 0 s) - t 2 0

S

= -so

Thus for any tracial state T1,

so T1(S) = O. This holds for all symmetries in N. Since every element of N is a multiple of a symmetry, then T1 = 0 on N. Then T = T1 follows. Finally, by Proposition 3.38 every state on a spin factor is normal. 0

5.22. Proposition. Every lEW-factor M of type In (1 ::; n < (0) admds a umque traczal state. Thzs state takes the value l/n on every mimmal pTOJectwn. PTOOf. If M is of type 1 2 , then 1\1 is a spin factor by Proposition 3.37, and so there is a unique tracial state on Jvl (Lemma 5.21). Now consider the case where M is of type In with n i- 2. Then M is finite dimensional (Theorem 3.32). Let G be the group of affine automorphisms of the state space K of M. Since the unit ball of M* is co(K U -K), then G extends to a group of isometries of M*. Thus G forms an equicontinuous group of affine automorphisms of the compact convex set K. By the Kakutani fixed point theorem [108, Thm. 5.11], there is a fixed point T of G. Since for each symmetry s the dual map U; is in G, then U;T = T, so T is a tracial state.

TRACES ON JB-ALGEBRAS

151

Now we prove uniqueness. Let M be of type In with 1 ::; n < =, and let T be any tracial state on !vI. Let P and q be any two minimal projections. Recall that minimal projections and abelian projections coincide in a JBW-factor (Lemma 3.30), so P and q arc abelian. Since P and q have the same central cover (namely 1), then P and q are exchangeable by a symmetry (Lemma 3.19). It follows that T(p) = T(q). Since M is a factor of type In, by definition there is a set S of n abelian (therefore minimal) projections with sum 1. Since T takes the same value on each projection in S, it must have the value lin on each. Since all minimal projections in M are exchangeable, T takes the value lin on all minimal projections. By Lemma 3.22 any set of orthogonal non-zero projections has cardinality at most n, so every projection in !vI is a finite sum of minimal projections. By the spectral theorem T is determined by its value on minimal projections. Thus T is unique. D

5.23. Definition. Let lvi be a lBW-algebra. A positive linear map

T from !vI onto the center Z of !vI is a center-valued trace if (i) T is the identity map on Z, and (ii) T(Usx) = Tx for all x E M and all symmetries s. We say T is fmthful if x :::: 0 and Tx = 0 imply x if it is O"-weakly continuous (Proposition 2.64). Note that if A is a JBW-factor, then Z is of the form x f--+ T (x) 1 for a tracial state

= O. Recall T is normal

= R1, so a center valued trace T.

5.24. Lemma. If T zs a center-valued trace or a traczal state on a JB W-algebra .M and PI, . .. ,Pn are orthogonal pmJectzons wzth P = PI + '" + Pn, then for x E AI

Proof. We will prove the lemma for center valued traces; the proof for tracial states is essentially the same. It suffices to prove the lemma for the case n = 2. (The general case follows by induction.) Let P be any projection and let s = P - p'. Since Us = 2(Up + Up' -1) (cf. (2.25)), by the invariance of center-valued traces under the map Us, for each x E !vI We have

which implies

(5.17)

Tx = T((Up + Up' )x).

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5

STATE SPACES OF JORDAN ALGEBRAS

Let Pi and P2 be orthogonal projections in M. Applying (5.17) with Pi in place of P and UP1 +P2 x in place of x gives (5.18) Since Pi 1\ (Pi +P2) =Pi and P~I\(Pi+P2) =Pi+P2-Pi =P2 (d. (2.17)), then by Proposition 2.28, (5.19) Combining (5.18) and (5.19) gives

This completes the proof. D

5.25. Proposition. Every lEW-algebra M of type In (1 <::: n < CXJ) admzts a unique center-valued trace. ThlS trace lS falthful and normal. Proof. Let Pi, ... ,Pn be abelian exchangeable projections with sum l. Let {Sij} be symmetries such that Sij exchanges Pi and PJ for 1 <::: l, J <::: n. For each l, let M p , = {piMp;}. Define cPi : M p , ----) M by (5.20)

and : M ----) M by (5.21)

1

(x) = - ~ cPi( {p,xpd)· n

.

We will show that is a center valued trace. (For motivation, cOllsider the example where NI consists of the hermitian n x n matrices with entries in an abelian von Neumann algebra, so that each x E NI has the form x = L:i,J Zijeij· Let Pi = eii for 1 <::: l <::: n. Then cP,(PiXPi) = cPi(zi;eii) = Zii and (x) = L:i Zii.) Let Z be the center of M. It is clear that is a positive map. \lVe will now show that (NI) c Z, and that is the identity map on Z. Fix l. We first show cPi (Nip,) c Z for l = 1, ... ,n. Let 7r be an arbitrary factor representation of M, and let N10 = 7r(M) (a-weak closure). Since Pi is an abelian projection, 7r(Pi) is an abelian projection in Mo. An abelian projection in a factor is minimal (Lemma 3.30), so for x E Mpil

TRACES ON JB-ALCEBRAS

153

Fix x E M p ,. Then for each J there exists Aj E R such that

Since

the Jordan automorphism Urr(sii)U;;(~,j) takes AJ1T(Pj) to Ai1T(Pi). conclude that Aj = Ai for all J. Thus

j

j

We

j

where Ai depends on x. Therefore for all symmetries t in Iv!, and all factor representations 1T of Iv!,

Since factor representations separate points of I\I! (Lemma 4.14), Ut1Ji(X) = cPi(X). It follows that cPi(X) is central in M (Lemma 2.35), which completes the proof that cPi maps MPi into the center Z of M. Now it follows at once from the definition (5.21) of that maps Minto Z. Now let Z E Z. Then

cP,( {p,zp;})

=

2.: {SiJ {PiZP;}Sij} Z 2.: {SijPiSiJ} = Z 2.: PJ = z, j

=

j

so 1 (z) = n

2.: cPi( {PiZP;})

=

z.

i

Thus is the identity map on Z. We next show 0 Us = for all symmetries s. We first restrict to the case where M is a type In factor. Let T be the unique trace whose existence is guaranteed by Proposition 5.22. Let x E M. Then we are going to show (5.22)

(X)

= T(X)l.

Note that since the center of M is R1, by the results we have just established, (x) = Al for some scalar A. Then (5.22) will follow if we show

5.

154

STATE SPACES OF JORDAN ALGEBRAS

that T((X)) = T(X). By the definitions of cPi and , and the definition of a tracial state,

1

= -n L L T( {pixpd ) i

)

= LT( {PiXP,})· Since LiPi = 1, by Lemma 5.24 the last sum above is T(X), so T((X)) = T(X), which proves (5.22). From the definition of and (5.22), this also proves that when M is a factor the unique tracial state is given by (5.23)

The uniqueness of the trace also proves that the definition of in this case is independent of the choice of the projections {pd and the symmetries

{ Sij }. We now let M be an arbitrary JBW-algebra of type In, and will show that 0 Us = for all symmetries s. Let x be any element of M, and 1f any factor representation of Ai, and Mo = 1f(lv1). By Lemma 4.8, Mo is a type In factor. Let T be the unique trace on Mo. Then 1f(

(x) )

=

1f

(~ L

,

L {Si) {PixpdSij} ) )

By (5.23) applied to Mo (with 1f(Sij) in place of Si), 1f(x) in place of x, and 1f(Pi) in place of pd, the last expression is just T(1f(x))l, so

1f((X)) Thus if

S

=

T(1f(X))l.

is any symmetry in M, then

Since factor representations separate points, we conclude that (Usx) = (x) for all x E M and all symmetries s. Thus is a center-valued trace. All maps involved in the definition of are a-weakly continuous, so the same is true of . Thus is normal. To prove is faithful, suppose x ::::: 0

TRACES ON JB-ALGEBRAS

155

and (x) = O. Then is faithful. Finally, we prove uniqueness. Let T : lvI -+ Z be any center-valued trace. For each x E lvI, since (x) E Z, then

(x) = T((x)) =

T(~ LL{Si]{P,XpdSij }) ,

1 n

]

= - L LT( {Si] {P,XPi}Sij}) =

,

J

1 -n LLT({PiXpd) i

]

= T(L{PiXPi})'

,

By Lemma 5.24, T(L:, {pixpd) ness. 0

= Tx so

(x)

Tx, proving unique-

5.26. Lemma. If C! 2S a fa'lthful normal state on a lEW-algebra lvI of type In (1 -s: 71 < =), then there eX2sts a fmthful normal tracwl state T on M such that C! -s: nT. Proof. Let T be the unique center-valued trace on lvI, and define T on M by T(X) = C!(Tx). Then it follows from the properties of T and C! that T is a faithful normal tracial state on lvI. It remains to prove C! -s: nT. Let P be an abelian projection in lvI. Our first step is to show

Tp = ~c(p),

(5.24)

where c(p) is the central cover of p, cf. Definition 2.38. Let PI,"" Pn be abelian exchangeable projections with sum 1. Define qi = c(p )p, for i = 1, ... ,n. Then (5.25)

qi

+ ... + qn

=

c(p),

and ql, ... , qn are abelian projections (since qi -s: Pi)' and are exchangeable (since if S is any symmetry exchanging p, and Pj, then the Jordan automorphism Us fixes c(p), and so exchanges qi and q]). By the definition of a center-valued trace, T takes the same value on exchangeable projections. Thus (5.25) implies (5.26)

Tql

=

~c(p).

156

5.

STATE SPACES OF JORDAN ALGEBRAS

Since ql, ... , qn are exchangeable, then c(qd Therefore c( qd 2: qi for 1 :s; i :s; n, so

c( qd 2: ql V q2 V ... V qn = ql

=

C(q2)

+ q2 + ... + qn =

= c(p).

On the other hand, ql :s; c(p) implies c( qd :s; c(p), so c( qd = c(p). Abelian projections with the same central cover are exchangeable, so Tql = Tp, which when combined with (5.26) proves (5.24). Finally, by the definition of T, a(p)

:s;

a(c(p))

=

a(nT(p))

=

nT(p),

Thus a :s; nT on abelian projections. However, in a type I JB\V-algebra, every projection dominates a non-zero abelian projection (Lemma 3.18), so by a Zorn's lemma argument is the least upper bound of an orthogonal family of abelian projections. Since both a and T are normal, so is T, and thus a :s; nT holds on all projections. By the spectral theorem, a :s; TLT holds in general. D The following result generalizes a well known Radon-Nikodym type theorem for von Neumann algebras.

5.27. Lemma. Let M be a lEW-algebra and let a, T be two normal posztwe functionals on lvI such that a < T. Then there exists h E lvI+ with 0 :s; h :s; 1 such that a(a)

=

T(h

0

a) for all a EM.

Proof. (We follow the similar argument for von Neumann algebras in [112, Prop. 1.24.4]). For each x E M, let Tx be the map Tx(Y) = x 0 y. The map h f-> T:;T is continuous from the a-weak topology on lvI to the weak topology on /1.1* (i.e., the weak topology for the duality of lvI* and M). Since the closed unit ball lvh of M is a-weakly compact (Corollary 2.56), then the set Ko = {T:; T I h E Md is the continuous image of a a-weakly compact convex set, and so is a weakly compact convex subset of M* containing O. We are going to show that a E Ko. Let b be an arbitrary element of M such that p(b) :s; 1 for all p E Ko. We will show that a(b) :s; 1 in order to apply the bipolar theorem. Let b = b+ - b- be the orthogonal decomposition of b into the difference of orthogonal positive clements (Proposition 1.28). Let p be the range projection of b+. Then by definition of the range projection, b+ E Mp. Since b+ and b- are orthogonal, by (2.9) b- E M p" We also have Mp 0 M p' = {O} (Lemma 1.45), so

NORl\I CLOSED FACES OF JBW NORMAL STATE SPACES

157

Thus (p - p')

(5.27)

lip - p'li =

Since

0

b = b+

+ b~.

1, and by hypothesis a :S T, then T;~p'T E K a, so

By the bipolar theorem (or a direct application of the Hahn~Banach theorem), we conclude that a E K a. It remains to show that we can choose 0 :S h :S 1. Let h = h+ - h~ be the orthogonal decompositon of H, and let p = r(h+), and s = p - p'. As in the calculation above, soh = h + + h ~. Then

so T(h~)

(h~)2 :S Ilh~11 h~, then T((h~)2) = O. Now by the inequality, T(h~ 0 a) = 0 for all a E M. Thus a(a) = T(h+ 0 a), and Ilh+ II :S Ilhll :S 1, so O:S h+ :S 1. 0

= O. Since

Cauchy~Schwarz

T(h+

0

a)

=

Norm closed faces of JBW normal state spaces The main result of this section is Theorem 5.32, which shows that every norm closed face of the normal state space K of a lBW-algebra M is projective, i.e., of the form Fp = {a E K I a(p) = I} for a projection p in K. We then establish various properties of the norm closed faces of K.

5.28. Definition. If X is a subset of the normal state space of a lBW-algebra, the intersection of the faces containing X is called the face generated by X, and is denoted face(X). Similarly, the intersection of the norm closed faces containing X is called the norm closed face generated by X. If X = {a}, then we write face( a) instead of face ( {a} ), and similarly We write face (a, T) instead of face ( {a, T} ). 5.29. Lemma. Let !vI be a lEW-algebra wdh normal state space K and T a normal tracial state on !vI. Let h E M+, wzth range proJectzon satisfying r(h) = l. Define w on M by w(a) = T(h 0 a). Then T zs zn the norm closure of the face of K generated by w. Proof. Let {e.>.} be the spectral resolution for h (cf. Definition 2.21). By the definition of a spectral resolution and Theorem 2.20, for each A E R

5

158

STATE SPACES OF JORDAN ALGEBRAS

On the other hand, each e~ operator commutes with h, so by (1.62)

0< Ue ,A h < h. Combining these gives

o :::;

Ae~

:::; h.

Now for each A E R define a functional

T),(a) = T(e~ By Corollary 5.20,

T),

by

T),

0

a).

is a positive functional, and for a E M+,

AT),(a) = T(Ae~

0

a) :::; T(h

0

a) = w(a).

Thus for any A > 0, we conclude that T), is in the face of !v!: generated by w, and so IIT)'II- 1 T), is in the face of K generated by w. Finally, we show that {IIT)'II- 1 T),} converges to T in norm as A ~ O. By the definition of a spectral resolution, as A ~ 0, e~ / eS. Since eS = r(h) (cf. (2.15)), and by hypothesis, r(h) = 1, then e~ /

(5.28)

1.

By the Cauchy-Schwarz inequality (1.53), if a E M with

IT(a) - T),(a)1

=

IT(a) - T(e~

=

IT ((1 -

:::; T(l -

e~)

0

Iiall :::;

1, then

a)1 a) I

0

e~)1/2T(a2)1/2

:::; T(l - e~)1/2T(1)1/2

= T(l -

e~)1/2.

Then

and the last term goes to zero by (5.28) and normality of

T.

D

5.30. Lemma. Let M be a lEW-algebra wzth normal state space K. If T zs a fazthful normal traczal state on M, then the face of K generated by T zs norm dense m K.

Tb

E

Proof. Let F be the face of K generated by T. For b E M define M. by Tb(a) = T(b 0 a). Recall that if a, b :0:: 0, then T(a 0 b) :0:: 0

NORM CLOSED FACES OF JEW NORMAL STATE SPACES

159

(Corollary 5.20). Thus if b ~ 0, then Tb ~ 0. Note also that if b E M+, then b Ilbll1, so Tb Tllblll = IlbiIT. Thus

s:

s:

(5.29)

If b = b+ - b- is the orthogonal decomposition of b, we will prove that the orthogonal decomposition of Tb is given by Tb = Tb+ - Tb-' Let p be the range projection of b+ and pi = 1 - p. Then as in the proof of Lemma 5.27 (cf. (5.27)), we have (p - pi) 0 b = b+ + b-. Thus

The opposite inequality is clear, so it follows that Tb = Tb+ - Tb- is the orthogonal decomposition of Tb. Now let X be the linear space of all functionals Tb for b E M. Note that by the previous paragraph

Furthermore, since (Tb)+ = Tb+, then by (5.29) we have

(5.30) as long as ()+ # 0. We next show X is norm dense in M*. If a E M annihilates X, then Tb(a) = T(b 0 a) = for all b E M. In particular, T(a 2 ) = 0. Since T is faithful, then a 2 = 0, and so a = 0. Thus no non-zero element of M annihilates X, so X is norm dense in 1'v1*. Now fix w E K and choose a sequence (}n in X such that (}n --> w (in norm). Then II(}nll --> Ilwll = 1, so

°

(5.31 ) On the other hand, (}n(l)

II(}~ I -->

+ II(}~ I

-->

1.

w(l) = 1, so

(5.32) Combining (5.31) and (5.32) shows II(};;:-II --> 0, so ()~ --> W in norm. Let = 11(};t"11- 1 (};;. Then by (5.30), Wn is a sequence in F converging to w, Which completes the proof that F is norm dense in K. 0

Wn

5.31. Lemma. Let M be a lEW-algebra wdh normal state space E K, and let p be the carner proJectwn of (). Then the face generated by () zs norm dense in Fp = {w E K I w(p) = I}.

K. Let ()

Proof. We first reduce to the case where p = 1. Let Mp = {pMp}. Recall from Proposition 2.62 that we can identify the normal state space of

5

160

STATE SPACES OF JORDAN ALGEBRAS

Mp with Fp. Note that since Fp is a face of K, then face ( a) is the same whether calculated as a subset of Fp or of K. Similarly by Proposition 2.62 the norm topology on F p , viewed as the normal state space of M p , is just the norm topology inherited as a subset of K. Finally, the carrier projection of a viewed as a normal state on Mp will still be p. Therefore it suffices to prove the lemma with M replaced by M p , and we may assume hereafter that p = 1. We then have carrier( a) = 1. We now show this implies that a is faithful. If 0 ::; a E M and a(a) = 0, then a(r(a)) = 0 by equation (2.6). Then a(l - r(a)) = 1, so by the definition of the carrier projection of a, we must have 1 ::; 1 - r(a). Therefore r(a) = 0 and so a = o. Thus a is faithful as claimed. It remains to show that the face generated by a is norm dense in K. (i) First consider the case where AI has no part of type 12 or 1 3 . By Theorem 4.23, M is a JW-algebra. Let M be the universal von Neumann algebra for M (cf. Definition 4.43); we may view M as a JW-subalgebra of Msa. M acts reversibly in M (Corollary 4.30), so there is a *-antiautomorphism c1i of period 2 of M such that AI coincides with the set of self-adjoint elements of M fixed by c1i (Proposition 4.45). Observe that ~(I + c1i) will be a positive idempotent map from Msa onto M. Note that c1i preserves the Jordan product on M sa , and so is a Jordan isomorphism. Thus it is a-weakly continuous. For each normal state w on M, define = ~w 0 (I + c1i). Then is an extension of w to a normal state on M sa. We also denote by the extension of to a complex linear normal functional on M, which will then be a normal state on M. As shown above, a is faithful on M. We now show that is faithful on M. Let a E M+ and suppose a(a) = o. Then by definition a(~(a + c1i(a)) = o. Since ~(a+c1i(a)) isin M, by faithfulness of a on M, a+c1i(a) = O. Since c1i is a *-anti-isomorphism of M, it preserves order, so c1i(a) :::: o. It follows that a = c1i(a) = o. Thus is faithful on M. Let K be the normal state space of M. Since is faithful on M, its carrier projection is the identity, so Fcarrier(;) = K. Thus the norm closure of the face generated by equals K, d. (A 107). As shown above, each w E K has an extension w E K. Therefore the restriction map is an affine map from K onto K. The restriction map is norm continuous and sends face ( a) into face( a), so it follows that face( a) is dense in K. (ii) Now consider the case where M is either of type 12 or 1 3 . Let a E K be faithful. By Lemma 5.26 there is a faithful normal tracial state T on !v! such that a ::; nT where n = 2 or n = 3. By the JBW RadonNikodym type theorem (Lemma 5.27), there is a positive element h Eo M such that a(a) = T(hoa) for all a E M. We will show the range projection r(h) equals 1. Suppose q is a projection with Uqh = h. Then by (1.47)

w

w

w

w

a

a

a

a

q'

0

h

= ~(I

+ Uq, -

Uq)h

=

o.

NORM CLOSED FACES OF JBW NORMAL STATE SPACES

161

Thus a(q') = T(h 0 q') = 0. By faithfulness of a, q' = 0, so q = 1. Thus the range projection of h is 1. By Lemma 5.29, T is in the norm closure offace( a). Since a ::; nT, then a is in face( T), so face (a) = face (T). Since a is faithful and T :::: (l/n)a, then T is also faithful. By Lemma 5.30, face(a) = face(T) = K. (iii) Finally, let M be an arbitrary JBW-algebra. Write M = M2 EB M3EBMO where A12 isoftypeI2' A13 isoftypeI3, and !vIa has no summand of types I2 or I3 (cf. Lemma 3.16 and Theorem 3.23). Let a be a normal faithful state on M. Then we can write a as a convex combination of normal faithful states a2, a3, ao on !vI2, !vh, and Mo respectively. To verify this, let C2, C3, Co = 1 - C2 - C3 be the central projections such that Mi = CiM for l = 0,2,3. Let Ai = a(ci) and ai = X;lU;;a for l = 0,2,3. Then a = Aoao + A2a2 + A3a3 is the desired convex combination. If W is any normal state on M, we can similarly decompose W into a convex combination of normal states W2, W3, Wo on M 2, M 3, and Mo. By (i) and (ii), Wi E face(ai) for l = 0,2,3. Since a = Aoao + A2a2 + A3a3, each of a2, a3, ao is in face(a), so it follows that wE face(a). D Recall that a face F of K is norm exposed if there is a positive bounded affine functional a on K whose zero set equals F, cf. (A 1).

5.32. Theorem. Let M be a lEW-algebra wdh normal state space K, and let F be a norm closed face of K. If p lS the carner pro]ectzon of F, then F = Fp = {a E K I a(p) = I}. Thus every norm closed face of K is norm exposed and pro]ectwe. Proof. By the definition of the carrier projection, F C Fp. To prove the opposite inclusion, let a E Fp. For Wl and W2 in F, the carrier projection of ~(Wl +W2) dominates the carrier projections of Wl and W2. Thus the set of carrier projections of states in F is directed upward. The least upper bound of this set of projections is carrier(F) (cf. (5.2)). Therefore there is a net {we>} in F such that {carrier( we»} is an increasing net with least upper bound carrier(F). Then a(carrier(we») ----; a(carrier(F)) = 1. Thus we can choose a sequence {w;} in F such that a(carrier(wi)) ----; 1. Define an element W of F by W = 2::i 2-iwi, and observe that carrier(w) :::: carrier(wi) for each L Thus a(carrier(w)) :::: a(carrier(wi)) ----; 1. Then a(carrier(w)) = 1, so a E Fcarrier(w)' By Lemma 5.31, this implies that a is in the norm closed face generated by w, and thus is in F. Thus We have shown Fp C F, which completes the proof that F is projective. Viewing p as an affine function on K, then 1-p takes the value precisely on F, so F is norm exposed. D

°

162

5.

STATE SPACES OF JORDAN ALGEBRAS

5.33. Corollary. Let M be a lBW-algebra wIth normal state space K. Then p f-7 Fp IS an IsomorphIsm from the lattice of proJectwns of M onto the lattIce of norm closed faces of K (where Aa Fa = na Fo<). The inverse map takes a norm closed face to ItS carrIer proJectwn. Thus the norm closed faces of K form an orthomodular lattIce wIth the orthocomplementatwn F f-7 F'.

Proof, This follows from the isomorphism of the lattice of projections with the lattice of projective faces (Proposition 5.10), together with the fact that the collections of projective faces and norm closed faces coincide (by Theorem 5.32 and the observation that projective faces are norm closed). The statement Aa F" = na Fa follows from the fact that the intersection of norm closed faces is a norm closed face. 0 5.34. Corollary. Let K be the normal state space of a lBW-algebra M. Let p be a pT'OJectwn m M, and F = Fp. Then the followmg are equwalent:

(i) p IS central, (ii) F is a SplIt face, (iii) co(F U F') = K. Proof, (i) =} (ii) If p is central, then by definition p operator commutes with every element of lvI, so Up + Up' = I by Proposition 1.47. Then U; + U;, = I. If a E K, let A = a(p) = IJU;all. Then 1 - A = a(p') = IJU;,all, so

expresses a as a convex combination of states in F and F ' , cf. (5.4). Since the intersection of im U; and im U;, is {O}, K is the direct convex sum of F and F'. Thus F is a split face of K. (ii) =} (iii) follows at once from the definition of a split face (A 5). (iii) =} (i) If co(F U F') = K, then

M. = im

U; + im U;,.

Since the intersection of these subspaces is zero, the sum is a direct sum, and so U; + U;, = I. Dualizing gives Up + Up' = I. By Proposition 1.47, p operator commutes with all elements of lvI, i.e., is central. 0 5.35. Corollary. Let K be the normal state space of a lB W-algebra M. If F IS a split face of K, then F = F c , where c is ItS carrier projectIOn and is central. Furthermore, M WIll be a lEW-factor iff K has no proper split faces.

NORM CLOSED FACES OF JBW NORMAL STATE SPACES

163

Proof. A split face is norm closed (A 28), and so has the form Fe where c is its carrier projection (Theorem 5.32). By Corollary 5.34, c is central. Finally, M is a JBW-factor iff M has no non-trivial central projections, which then corresponds to K having no proper split faces. 0 5.36. Proposition. Let M be a lEW-algebra wdh normal state space

K. The map that takes split faces of K to thelr anmhdators m M gwes a 1-1 correspondence of Spllt faces and O"-weakly closed lordan zdeals. The inverse map takes a 0" -weakly closed zdeal to ltS anmhllator m K. Proof. Let F be a split face of K. By Corollary 5.34 the carrier projection c of F is central, and by (5.4), F = im U; n K. Since every normal functional is a difference of positive normal functionals (Proposition 2.52), and U; is a positive map, then im U; is positively generated, i.e., im U; = im + U; - im + U;. Thus the annihilator of F is the annihilator of imU;, which in turn is the kernel of Uc . By Proposition 2.41, the latter is exactly the O"-weakly closed ideal (1 - c)M. Conversely, if J is a O"-weakly closed Jordan ideal, then by Proposition 2.39, J = im U 1 - c for some central projection lvI. Then the annihilator of 1 in K is the annihilator of im U1 - c , which is the kernel of Ui-c. Since Uc+U1 - c = 1, the kernel of Ui-c is the image of U;. Thus the annihilator of 1 in K is im U; n K = Fe. By Corollary 5.34, Fe is a split face, which completes the proof. 0 5.37. Corollary. Let A be alE-algebra wzth state space K. The map that takes w*-closed splzt faces of K to thezr anmhzlators m A gwes a 1-1 correspondence of w*-closed splzt faces and norm closed lordan ldeals. The inverse map takes a norm closed zdeal to ltS anmhllator m K.

Proof. Let J be a norm closed Jordan ideal of A. View A as a subset of A ** and denote annihilators in the duality of A ** and A * by o. Then the O"-weak closure J of J in A** is a Jordan ideal, and by the bipolar theorem satisfies J = JOo. Since J is norm closed, we also have

J = JOo n A = J n A. Let F = JO n K. Since each element of F is O"-weakly continuous on A**, then F = (1)0 n K. By Proposition 5.36, F is a split face of K and Fo = 1. Thus

FO n A = J n A = 1. Therefore starting with a norm closed ideal J of A and taking the annihilator in K gives a split face whose annihilator in A is J again. Now suppose that F is a w*-closed split face of K. If c E A** is the carrier projection of F, then by Theorem 5.32, F = Fc. By Proposition

164

5

STATE SPACES OF JORDAN ALGEBRAS

2.62 we can identify F with the normal state space of im ue , and im U; with the predual of im Uc . By Corollary 2.60, im U; is a base norm space with distinguished base F, so by definition of a base norm space its unit ball is co( F U -F). Since F is w* -compact so is the Cartesian product F x (-F) x [0, 1], and the map (a, T, A) f--> Aa+(I-A)T is a continuous map onto co( F U -F), so it follows that the unit ball of im U; is w* -compact. By the Krein-Smulian theorem, im U; is w*-closed. Let

J

= FO n A = (im U;r n A

be the annihilator of F in A. Since FO is a Jordan ideal of A ** by Proposition 5.36, then J is a norm closed ideal in A. Since im U; is w*closed, the annihilator of J in A * will then equal im U;. The annihilator of J in K is then im U; n K = Fe = F. This completes the proof that taking annihilators gives a 1-1 correspondence of norm closed ideals of A and w*-closed split faces of K. 0 5.38. Corollary. Let A be a JB-algebra wzth state space K, let J be a norm closed Jordan zdeal of A and let 7r : A f--> AI J be the assoczated homomorphzsm. Then the dual map 7r* : (AIJ)* f--> A* zs an affine 7,80morphzsm and a w*-homeomorphzsm from the state space S(AI J) of AI J onto the w*-closed spld face F = JO n K correspondmg to J. Proof. Let a E S(AI J). For each a E J, (7r*a)(a) = a(7r(a)) = 0, so 7r*a E JO. Clearly also 7r*a E K, so 7r*a E F. Thus 7r* maps S(AIJ) into

F. Now let al and a2 be two distinct elements of S(AIJ). Then there is an a E A such that al(7r(a)) =1= a2(7r(a)). Thus 7r* is injective on S(AjJ). Next let w E F. Thus W E K, and w = on the ideal J = ker7r. Hence there exists a linear functional a on AIJ such that a(7r(a)) = w(a) for all a E A. Every positive element of AI J is a square, and so is the image of a positive element of A. Thus a E S(A), and 7r*a = w. Therefore 7r* maps S(AI J) onto F. Clearly the map 7r* is affine and w* -continuous, so it is an affine isomorphism and a w*-homeomorphism. 0

°

Recall that an atom in a JBW-algebra is a minimal projection. 5.39. Proposition. Let M be a JEW-algebra wlth normal state space K. The map p f--> Fp lS a 1-1 correspondence of atoms of M and faces of K consisting of a single extT'eme pomt. Proof. By Corollary 5.33, there is a 1-1 correspondence of atoms of M and minimal norm closed faces of K. We need to show that the latter

NORM CLOSED FACES OF JBW NORMAL STATE SPACES

165

consist exactly of the faces {a} for a an extreme point of K. Such faces are clearly minimal. Thus it remains to prove that for p an atom, Fp is a single point. This follows from the fact that Mp is one dimensional (Lemma 3.29), together with the observation that im is the predual of Mp (Proposition 2.62), and thus is also one dimensional. D

U;

5.40. Definition. Let p be an atom in a lBvV-algebra M with normal state space K. vVe will denote by p the extreme point of K satisfying

Fp

=

{p}.

Note that if p is an atom, then p is the unique normal state taking the value 1 at p. Recall that a state on a lB-algebra A is pure if it is an extreme point of the state space of A.

M

5.41. Corollary. Let M be a lEW-factor. Then M a pure normal state.

~s

of type I

~ff

adm~ts

Proof. Let K be the normal state space of M and let N be the (full) state space of l'vl. By Proposition 2.52, K is a split face of N. Since an extreme point of a face of a convex set is also an extreme point of the full convex set, a normal state is extreme in K iff it is extreme in N. By Proposition 3.44, l'vl is of type I iff it contains an atom. By Proposition 5.39, this is equivalent to K containing an extreme point, which is then also extreme in N, i.e., a pure normal state. D Recall that if a is a pure state on a C* -algebra A, and 7f (j : A --+ B(H) is the associated GNS-representation, then 7f(j(A) is irreducible (A 81). Therefore 7f(j(A)' = C1 (A 80), so by the bicommutant theorem 1i"u(A) = B(H). For each state a on a lB-algebra A, we have defined a homomorphism 1i"u : A --+ c(a)A** (Definition 4.11). We earlier showed that 7f(j(A) is a JBW-factor. Now we are able to show that this factor is of type I, as one would expect by analogy with the C*-algebra example just discussed.

5.42. Corollary. If a zs a pure state on a lE-algebra A, then 1i"u(A) = c(a)A** ~s a type I lEW-factor. Proof. By Proposition 5.39, the carrier projection of a is an atom in A**, and therefore also in c(a)A**. By Lemma 4.14, c(a)A** is a JBWfactor, which is type I by Proposition 3.44. D

5.43. Corollary. Let F and G be norm closed faces of the normal state space of a lEW-algebra M. Then these are equwalent: (i) F c G' (i. e., F and G are orthogonal in the latt~ce F). (ii) The carner projections of F and G are orthogonal

5

166

STATE SPACES OF JORDAN ALGEBRAS

(iii) Each state m F

lS

oT·thogonal to each state in C.

Furthermore, FI consists exactly of the normal states orthogonal to every element of F. Proof. The equivalence of (i) and (ii) follows at once from Corollary 5.33, since projections p and q are orthogonal precisely if p :S ql (Proposition 2.18). (ii) =} (iii) If carrier(F) ~ carrier(C), then carrier(O') ~ carrier(T) for all 0' E F, T E C. Normal states are orthogonal iff their carrier projections are orthogonal (Lemma 5.4), so each 0' E F is orthogonal to each T E C. (iii) =} (ii) If (iii) holds, then by Lemma 5.4, carrier(O') ~ carrier(T) for all 0' E F, T E C. Then taking least upper bounds in the lattice of projections we have V{carrier(O')

10'

E

F}

~

V{carrier(T)

I T

E

C}.

The left side is carrier( F) and the right side is carrier( F) (d. (5.2)), so (ii) follows. Now we will prove the final statement of the lemma. Suppose first that 0' E Fl. Since F and FI are orthogonal, then by the equivalence of (i) and (iii), 0' is orthogonal to every state in F. Conversely, suppose that 0' is orthogonal to every state in F. Then carrier(O') is orthogonal to carrier(T) for every T E F, and thus carrier(O') is orthogonal to carrier(F). Since (ii) implies (i), then Fearrier(u)

Since

0'

is in the left side, then

0'

c

Fl.

E Fl. 0

Recall that c( 0') denotes the central carrier of a normal state 0' on a JBW-algebra (Definition 4.10). 5.44. Proposition. Let 1( be the normal state space of a lBWalgebra Ivl. Let 0' E 1(. Then Fe(u) lS the smallest spilt face contammg 0'.

Proof. By Corollary 5.34, Fe(u) is a split face containing 0'. Any split face is norm closed (A 28), and so has the form Fe, where c is its carrier projection and is central by Corollary 5.34. If a split face F = Fe contains 0', then O'(c) = 1, so by definition c 2: c(O'). Then Fe(u) C Fe, and so Fe(u) is the smallest split face containing 0'. 0 5.45. Definition. Let K be the normal state space of a JBW-algebra M, and let 0' E K. We will denote the smallest split face containing 0' by

NORl\! CLOSED FACES OF JBW NOR1IAL STATE SPACES

167

Fa, and we will refer to Fa as the spilt face generated by a. (Note that by Proposition 5.44, Fa = Fe(a).) We use the same notation Fa if a is a state of a JB-algebra A. (In that case, by Corollary 2.61 we can identify the state space of A with the normal state space K of A **.)

If a is a normal state on a JBW-algebra M, we note that the central cover of the carrier projection of a equals the central carrier of a: (5.33)

c(a)

= c(carrier(a)),

which follows from the observation that a takes the value 1 on a central projection iff that projection dominates the carrier of a. 5.46. Lemma. Let M be a iE W-algebra wzth normal state space K. If a zs an extreme poznt of K, then the spilt face Fa generated by a zs a mimmal spilt face among all splzt faces of K. Proof. Let p be the carrier projection of a. Since {a} is a minimal norm closed face of K, then p is a minimal projection in Ai. Then c(p) is a minimal projection in the center of M (Lemma 3.43). By (5.33), c(a) = c(p), so c(a) is a minimal projection in the center of M. By the correspondence of split faces and central projections, it follows that Fa = Fe(a) is minimal among split faces of K. D

Recall that a state a on a JB-algebra A is a factor state if c(a) is a minimal projection in the center of A** (cf. Definition 4.13), and that this is equivalent to c(a)A** being a JBW-factor. If a is a state on a C*-algebra A, and 7fa : A ----> B(H) is the associated GNS-represenation, then 7fa(A) ~ c(a)A**, so a will be a factor state iff 7fa(A) is a von Neumann algebra factor. 5.47. Proposition. Let A be a iE-algebra, and a a state on A. Then a zs a factor state zff the splzt face Fa generated by a zs a mimmal split face of the state space of A. Proof. This follows at once from the definition of a factor state, the identification of the state space of A with the normal state space of A** (Corollary 2.61), and the correspondence of central projections in a JBWalgebra and split faces of the normal state space (Corollary 5.34). D

The following is a dual version to the analogous result in the projection lattice (Proposition 2.31).

5.

168

STATE SPACES OF JORDAN ALGEBRAS

5.48. Lemma. Let p be a proJectwn zn a lEW-algebra !v! and normal state on M. Then (5.34)

carrier(U;O')

=

0'

a

(carrier(O') V pi) 1\ p.

Proof. The proof is exactly the same as the proof of Proposition 2.31, with the positive element a there replaced by the normal statq 0', range projections replaced by carrier projections, and compressions such as Up replaced by their dual maps D

U;.

5.49. Corollary. Let p be a proJectwn zn a lEW-algebra !v! and an extreme pond of the normal state space K of !v!. Then U; 0' zs a ml1ltzple of an extreme pomt of K.

0'

Proof. Let q = carrier(U;O'), and let 11 = carrier(O'). Then by the correspondence of atoms and extreme points of K (Proposition 5.39), u is an atom. By Lemma 5.48, q

=

(carrier( 0') V pi) 1\ P

= (u

V

pi) 1\ P

= (11 V pi) - pi,

with the last equality following from (2.17). By the covering property of the projection lattice (Lemma 3.50), the last expression is either an atom or zero. If q = 0, then U;O' equals zero and thus is certainly equal to a multiple of an extreme point. If q is an atom, then 11U;0'11-1U;0' has carrier q, so is an extreme point of K. Thus again U;O' is a multiple of an extreme point, which completes the proof of the corollary. D

The Hilbert ball property Recall that the JBW -factors of type 12 are exactly the spin factors. (See Definition 3.33 and Proposition 3.37). We begin this section by describing the normal state spaces of such factors.

5.50. Definition. A Hllbert ball is the closed unit ball of a real Hilbert space of finite or infinite dimension. For convenience, we include the "zero dimensional Hilbert ball", i.e., the ball consisting of a single point. 5.51. Proposition. If !vI lS a spm factor, then every state on !vI zs normal and the state space of M zs affinely lsomorphzc to a Hilbert ball. Proof. Recall that since !v! is a spin factor, then M = N EEl Rl (vector space direct sum) where N is a real Hilbert space, with the norm given by (5.35)

Iia + Alii

=

IIal12 + IAI

THE HILBERT BALL PROPERTY

169

for a E N and A E R. (Here IIal12 denotes the Hilbert space norm.) Note that the Hilbert and JBW norms coincide on N. Recall from Proposition 3.38 that every state on M is normal. For each b in the closed unit ball of N, we define CJb : AI --4 R by

CJb(a

(5.36)

+ AI) =

(a I b)

+ A.

We will show that b f---> CJb is an affine isomorphism from the closed unit ball of the Hilbert space N onto the state space of M. Let bEN with Ilbll ::; 1. Note first that CJb(l) = 1. Furthermore, IICJbl1 = 1, because for each a E N we have

ICJb(a +

Al)1 ::;

I(a

I b)1 + IAI ::; Ilallllbll + IAI ::; Iiall + IAI = Iia + Alii·

To show CJb is positive, let 0 ::; x E M with 111- xii::; 1, so because Ihll ::; 1 we have

Ilxll ::;

1. By spectral theory,

Thus CJb (x) ;::: 0 as claimed, so CJb is a state. (Alternatively, this follows from (A 16).) If CJb, = CJb2, then for all a EN we have (a I b 1 - b2 ) = 0, so b 1 = b2 . Thus the map b f---> CJb is 1-1. Finally, we show this map is surjective. If CJ is any state, then CJ restricts to a functional on N of norm at most 1. Since the JBW and Hilbert norms coincide on N, then there exists b in the closed unit ball of N such that CJ(a) = (a I b) for all a E N. Since CJ(I) = 1, then CJ = CJb. 0 We illustrate Proposition 5.51 with a particular example. Recall that

M = (M2 (C))sa is a spin factor, cf. Example 3.36. In this case, the state space will be a 3-ball. States are given by the maps a f---> tr(da) where d is a positive matrix of trace 1. According to (A 119), an affine isomorphism from the state space of lvI onto the Euclidean ball B3 is given by the map W f---> (;31, ;32, ;33), where

For later use, suppose that p is an atom in a spin factor lvI. Then, s = p - pi is a symmetry in N, since N includes all symmetries not equal to ±1. Thus the state CJb defined in (5.36) for b = s satisfies

170

5

STATE SPACES OF JORDAN ALGEBRAS

It follows that (J"s is the unique (normal) state with value 1 at p, or in the notation of Definition 5.40, for each atom p in a spin factor, (5.37)

p

=

(J"s

where s

=p

- p'.

We are now going tQ describe the circumstances under which the face generated by a pair of extreme points in the normal state space of a JBWalgebra degenerates to a line segment. 5.52. Definition. Let (J" and T be points in a convex set K. We say and T are separated by a spld face if there is a split face F of K such that (J" E F and TEF'. (J"

Note that for any split face F of a convex set K, since the convex hull of F and F' is all of K, then each extreme point must lie either in F or in F'. Thus two extreme points not separated by a split face always lie in the same split faces. 5.53. Lemma. Let M be a JEW-algebra, and p, q dzstmct atoms zn M. Then ezther c(p) = c( q) or c(p) ~ c( q). In the former case, Mpvq zs a spm factor, and m the latter case, d zs zsomorphzc to Rl EB Rl. Proof. Since c(p) and c(q) are minimal projections in the center of M (Lemma 3.43), then either c(p) = c( q) or c(p) ~ c( q). We first consider the case c(p) = c(q). Note that by minimality of c(p) in the center of M, c(p)M is a JBW-factor. Since Mpvq

=

{(p V q)(c(p)M)(p V q)},

then Mpvq is a JBW-factor (Proposition 3.13). Define l' = P V q - p. By the covering property (Lemma 3.50), l' is an atom, and p + l' = P V q. Minimal projections are abelian, and in a factor have the same central cover (namely, the identity), so are exchangeable (Corollary 3.20). Thus p V q = p + T is the sum of two abelian exchangeable projections, so lvlpvq is by definition a type 12 factor, and therefore is a spin factor. Now consider the case c(p) ~ c( q). Then p and q are orthogonal, and qc(p) ::; c(q)c(p) = 0 implies qc(p) = O. Thus p = (p + q)c(p) is central in Mpvq (Proposition 3.13), as is q (by the same argument). Therefore by Proposition 2.41 lvlpvq is the direct sum of the algebras Mp = Up(lvlpvq) and AIq = Uq (lvlpvq). Since p and q are minimal projections, then Aip = Rp and Mq = Rq. Thus M = Rp EB Rq. 0 5.54. Lemma. Let M be a lEW-algebra wzth normal state space K. Let (J" and T be dzstmct extreme pomts of K. Then face ( (J", T) coinczdes with the line segment [(J", T] iff (J" and T aTe separated by a split face.

THE HILBERT BALL PROPERTY

171

Proof. Let F = face (cr, T). Let p and q be the carrier projections of cr T respectively. Suppose F = [cr, T]. Then F is norm closed, and so equals the minimal norm closed face containing cr and T. Then by Corollary 5.33 the carrier projection of F is p V q and F = Fpvq. Since the linear span of F is two dimensional, and F can be identified with the normal state space of Mpvq (Proposition 2.62), then Ivlpvq has dimension 2. Since spin factors have dimension at least 3, then Ivlpvq is not a spin factor. By Lemma 5.53, c(p) ..1 c(q). Then c(cr) ..1 C(T), so Fa C F~ (cf. (5.33) and Corollary 5.43). Thus cr and T are separated by the split face FT' Suppose now that F =J [cr, T]. Then the norm closed face generated by cr and T has affine dimension 2 or more, so Mpvq has dimension 3 or more. By Lemma 5.53, c(p) = c(q). Then c(cr) = C(T), so Fa = FT' Thus cr and T generate the same split face, so are not separated by any split face. (Alternatively, tllls follows from the fact that in any convex set, the face generated by two extreme points cr, T separated by a split face is the line segment [cr, T], cf. (A 29).) 0 and

5.55. Proposition. Let Ivi be a lEW-algebra, and cr, T extreme points of the normal state space K. Then face( cr, T) lS norm exposed and is affinely lsomorphlc to a Hllbert ball.

Proof. Assume first that cr =J T. Let G be the norm closed face generated by cr and T. p = carrier(cr) and q = carrier(T). Then the c:arrier of G is p V q. By Proposition 2.62 the normal state space of Alpvq is affinely isomorphic to Fpvq. By Lemma 5.53, lvlpvq is either a spin factor or isomorphic to R1 EEl Rl. In the first case, by Proposition 5.51 its normal state space is affinely isomorphic to a Hilbert ball. In the second case, the dual space of lvlpvq is two dimensional and so its state space is a line segment, which is again a Hilbert ball. Thus G is a Hilbert ball, and face(cr, T) is a face of G. Since the only proper faces of a Hilbert ball are extreme points, we must have face(cr, T) = G. Thus face(cr, T) is a Hilbert ball, and is norm exposed since G = Fpvq. Finally, if cr = T, then face(cr,T) = {cr}, which is a zero dimensional Hilbert ball. If p is the carrier projection of the norm closed face {cr}, then {cr} = Fp , so {cr} is norm exposed. 0 5.56. Corollary. Let A be a lE-algebra, and cr, T extreme pmnts of the state space K. Then face( cr, T) lS norm exposed and lS affinely isomorphlc to a Hllbert ball. In partlcular, extreme pomts of K are norm exposed.

Proof. By Corollary 2.61 we can identify the state space of A with the normal state space of the JBW-algebra A **, so the result follows from Proposition 5.55. 0

172

5

STATE SPACES OF JORDAN ALGEBRAS

In the last part of this book, we will abstract the property of state spaces described in Corollary 5.56, and call it the Hzlbert ball property. This property of JB state spaces will turn out to be a key tool in our characterization of these state spaces. Recall that if p is an atom in a JBW-algebra M, then unique normal state with value 1 at p (Definition 5.40).

p denotes

the

5.57. Corollary. (Symmetry of transition probabilities) Let p and q be atoms in a JBW-algebra M. Then

p(q) = q(p). Proo]. The result is clear if p = q, so we may assume p I q. Since in Fpvq , we can replace !v! by !v!pvq, and thus arrange that p V q = 1. Then by Lemma 5.53, M is either the direct sum of two one dimensional algebras or else is a spin factor. In the first case, there are only two distinct atoms p, q and one easily sees p(q) = 0 = q(p). Suppose instead that !v! is a spin factor, and define s = p - p' and t = q - q'. By equation (5.37)

p and q are

p( q)

=

oA ~ (t + 1)) =

~

+ ~ (t Is) =

q(p) ,

which completes the proof. D Remark. The property described in Corollary 5.57 has an interesting physical interpretation. If (Y and T are pure states, and p, q the atoms such that p = (Y and q = T, then the probability p(q) can be thought of as the "transition probability" for a system prepared in the pure state (Y = P to be found in the pure state T = q. Then Corollary 5.57 is a statement about symmetry of transition probabilities. \iVe will return to this concept in the axiomatic portion of this book. \iVe also observe that a purely geometric proof can be given for Corollary 5.57, based on the geometry of Hilbert balls. We will give such a proof in the axiomatic context later, cf. Proposition 9.14. The (Y-convex hull of extreme points

In this section we will show that every normal state on an atomic JBWalgebra can be written as a countable convex combination of an orthogonal set of extreme points, and that more generally the set of such countable convex combinations forms a split face of the normal state space of a JBWalgebra M. Recall that oeK denotes the set of extreme points of a convex set K. Recall also that a JBW-algebra is atomic if every non-zero projection

THE u-CONVEX HULL OF EXTREME POINTS

173

dominates a minimal projection, i.e., an atom, and that this is equivalent to every projection being the least upper bound of an orthogonal collection of atoms. (See the remark after Definition 3.40.) 5.58. Lemma. If Iv! ~s an atom~c lEW-algebra, then lis normal state space K ~s the norm closed convex hull of ds extreme pomts.

Proof. It suffices to show the cone generated by oeK is norm dense in the positive cone of Iv!*. By the Hahn-Banach theorem, this will follow if we show oeK determines the order on Iv!, i.e., that for a E M, (5.38)

u(a)

~

0

for all

u E oeK

=}

a

~

O.

Let a E M satisfy the left side of (5.38). Let a = a+ -a- be the orthogonal decomposition of a (Proposition l.28), and let p = r(a-). Then by (2.9) Upa+ = 0, so a- = -Upa. For each u E oeK, U;u is a multiple of an element of oeK (Corollary 5.49), so (5.39) Since a- ~ 0, by (5.39) we must have u(a-) = 0 for all u E oeK. Since M is atomic, then 1 is the least upper bound of atoms. By Corollary 5.33, the lattices of projections and norm closed faces are isomorphic, with atoms corresponding to extreme points (Proposition 5.39), so the fact that 1 is the least upper bound of atoms implies that the norm closed face of K generated by the extreme points of K is K itself. Since the set of normal states annihilated by a- is a norm closed face of K, we conclude that ais zero on K, and thus a- = O. We've then shown a = a+ - a- = a+ ~ O. This completes the proof of (5.38), and thus of the lemma itself. D Recall that a lBW-algebra A1 has a unique predual M*, i.e., a Banach space such that M is isometrically isomorphic to (M*) *, with Iv!* consisting of the normal linear functionals on Iv! (Theorem 2.55). If {Ui} is a sequence in A1* with L, Iluill < 00, then by completeness of Iv!*, Li Ui converges in norm to an element of M*. 5.59. Lemma. Let M be a lEW-algebra, P1,P2,'" orthogonal atoms, and A1, A2, ... scalars such that Li IAil < 00. Define u = Li A;j;, E M*. Then Ilull = Li IAil and the unique orthogonal decompos~tzon u = u+ -uis given by (5.40)

Proof. Define

U1

and

U2

by

174

5

Clearly Define

(5

=

(51 -

STATE SPACES OF JORDAN ALGEBRAS

(52,

and each of

(51

and

(52

is a positive linear functional

Note that for distinct 2, J, by orthogonality of p, and Pj, Pi (PJ) ::; p, (p;) = 0, so Pi(PJ) = 0. Then (5(s) = L, IA,I, and by spectral theo:y Ilsll = 1. Thus

which implies that

11(511 = Li IA,I.

which completes the proof that of (5. 0

(5

=

Furthermore,

(51 -(52

is the orthogonal decomposition

5.60. Lemma. Let AI be an atormc .fEW-algebra AI wIth normal state space K. There eXIsts a umque lmear order IsomorphIsm W from A1. mto M, such that Ilwll ::; 1 and w(p) = P for all atoms p.

Proof. Define W on lin (8eI<) by

where PI, ... ,Pn are atoms and a1, . .. ,an are scalars. vVe will show that W is well defined. By atomicity, any normal functional that annihilates all atoms will annihilate all projections, and then by the spectral theorem (Theorem 2.20) must be zero. Thus for scalars a1,' .. , an, {31, ... ,{3m and atoms PI, ... ,Pn, ql , ... , qm,

8n

m

aiPi = ~ {3J {fy

<===?

(n~ aiP, - ~m) (3J qj (u) = 0 for all atoms u.

By symmetry of transition probabilities (Corollary 5.57), this is equivalent to (5.41)

U

(t;n aiPi - ~m) {3J qJ = 0 for all atoms u.

THE u-CONVEX HULL OF EXTREME POINTS

175

As u varies over the atoms of !vi, then 11 will vary over the extreme points of K (Proposition 5.39). Thus (5.41) is equivalent to

By Lemma 5.58, this is equivalent to 2..::~1 aiPi have shown

=

2..:;~1 (3)Q). Thus we

n

111

n

111

i=1

)=1

i=1

j=1

It follows that W is well defined and I-Ion lin (oeK). Next we show Ilwll ::; 1 on lin (oeK). Let u E lin (oeK). Since W maps linear combinations of extreme points to linear combinations of atoms, by Lemma 3.53, there are orthogonal atoms PI, ... ,PH and scalars >'1," . , An such that n

w(u)

=

L:>iPi.

i=1

Since W is I-Ion lin (oeK), u = 2..::~1 A/Pi' Now by Lemma 5.59, 2..:~= 1IAi I so n

Ilw(u)11 =

=

n

IlL AiPi11 = 1<,<" max {IAil} ::; L i=1

Ilull

--

i=1

IAil = Ilull·

This shows that I WI ::; 1 on lin (oeK). By density of lin (oeK) in M* (Lemma 5.58), we can extend W to a linear map of norm 1 from 11,[* into

M. Now we show that W is bipositive. Note that by symmetry of transition probabilities,

(5.42) holds for all u and T in lin (oeK), and then by density and continuity holds for all u, T in IvI*. By definition W is positive on the convex hull of oe K , and thus on K by Lemma 5.58, so W is a positive map. If u E M* and w(u) ~ 0, then by (5.42)

for all T E oeK. Since W maps oeK onto the set of atoms in IvI, then u is positive on all atoms. By atomicity, u is positive on all projections, and so by the spectral theorem, u ~ O. Thus w- 1 is a positive map.

176

5.

STATE SPACES OF JORDAN ALGEBRAS

Finally, since lin (oeK) is norm dense in M*, and every extreme point of K has the form jJ for some atom P EM, W is determined by the requirement that 1j;(jJ) = p on all atoms p in !v!, and so is unique. 0 5.61. Theorem. Let !v! be an atomzc lEW-algebra 1mth nor-mal state space K. If rJ E K, then ther-e exzsts a (posszbly fimte) sequence of posztzve scalar-s {Ad wzth sum 1 and or-thogonal states rJi m oeK such that rJ = Li A;rJi' Pmoj. Let rJ E K, and let W be the map in Lemma 5.60. Let {e,\} be the spectral resolution for w(rJ) (cf. Theorem 2.20). Then for each A E R, e,\ = l-r-((w(rJ) - Al)+). Since w(rJ) :::: 0, then for A < 0, we have w(rJ) - Al :::: -AI = lAllI, so e,\ = 0 for A < O. By Theorem 2.20(ii), for each 0' we have (5.43) On the other hand, by Theorem 2.20(i) e a operator commutes with w(rJ), and so by (1.62) Ue;, W(rJ) ::; w(rJ). Combining this with (5.43) gives

W(rJ) :::: 0'(1 - e a ). We will show that for each 0' > 0 the projection (1- e a ) has finite dimension in the lattice of projections (cf. (A 42) and Proposition 3.51). Suppose that 0' > 0 and that PI, ... ,Pn are orthogonal atoms under 1 - en. Then

Since W is bipositive, then

Applying both sides to the identity in M, we conclude that n::; 0'~1. By atomicity, l-e a is the least upper bound of an orthogonal set of atoms, and by the result just established this set of orthogonal atoms has cardinality ::; 0'~1. Thus we conclude that 1 - e a has finite dimension for each 0' > O. This implies that there is a strictly decreasing sequence of positive scalars bd (which may be a finite sequence), such that each e~; is finite dimensional, and such that e,\ is constant for A > 0 except for jumps at the Ii. For simplicity of notation we will consider the case where the sequence {Ad is infinite, and leave to the reader the minor changes in wording needed for the case where the sequence terminates. Let I = lim; Ii. Then I> 0 would contradict finite dimensionality of e~ for I > 0, so limi Ii = O. Let qi be the jump in the spectral resolution e,\ at A = Ii, i.e., qi = e-Yi - e-Yi~E for any E with 0 < E < Ii - li+l. The

THE a-CONVEX HULL OF EXTREME POINTS

177

spectral theorem (Theorem 2.20) now gives \{t(a) = Li riqi. Each qi has finite lattice dimension (since qi :s: e-y, - e-Yi+l :s: e~'+l)' Replacing each qi by a finite sum of atoms leads to an expression \{t (a) = Li AiPi for orthogonal atoms Pi and a sequence of positive scalars Ai. Since \{t is bipositive, for each n, n

L

n

AiPi

:s: \{t (a)

=?

L AiP,

:s: a.

i=l

Applying both sides of the inequality on the right above to the identity element 1, we conclude that L~=l Ai :s: 1 for each n, and so L~l Ai :s: 1. Thus L~l AiPi converges in norm to an element of M*. Since

\{t

and

\{t

(~ AiPi) =

\{t ( a),

is bijective, we conclude that 00

a=LAiPi. i=l

Finally, applying both sides of this equation to the identity element 1 gives L~l Ai = 1. 0 5.62. Definition. The a-convex hull of a bounded set F of elements in a Banach space is the set of all sums Li A,ai where AI, A2, ... are positive scalars with sum 1 and a1,a2,'" E F. 5.63. Corollary. If "Al lS a lEW-algebra wdh normal state space K, then the a-convex hull of the extreme powts of K lS a spilt face of K. Proof. Let z be the least upper bound of the atoms in M. Recall that z is a central projection, zfvl is atomic, and z'"Al contains no atoms (Lemma 3.42). Then since every atom is under z, by the correspondence of atoms and extreme points (Proposition 5.39), every extreme point of K is in the split face F z . By Proposition 2.62, Fz is affinely isomorphic to the normal state space of zfvl, and thus equals the a-convex hull of its extreme points by Theorem 5.61. Every extreme point of a face of K is also an extreme point of K, so Fz is the a-convex hull of the extreme points of

K.D 5.64. Corollary. If A lS a iE-algebra wlth state space K, then the a-convex hull of the extreme points of K is a spilt face of K. Proof. The state space of A can be identified with the normal state space of the lBW-algebra A** by Corollary 2.61. 0

178

5.

STATE SPACES OF JORDAN ALGEBRAS

Trace class elements of atomic JBW-algebras 5.65. Definition. Let IvI be an atomic JBW-algebra. An element a E M is trace class if there are orthogonal atoms Pl,P2,'" and )'1, A2,'" E R such that

(5.44)

a = L AiPi

L

and

<

IA,I

00.

We will write M tr for the set of trace class elements, and we define the trace on M tr by

(5.45)

tr(LAiPi) ,

=

LAi. ,

To see that the tr ace is well defined on NItr , let a be defined as in (5.44), and let 0" = ~i A/Pi E M*. Let IJi be the map defined in Lemma 5.60. Then 1Ji(0") = a, and since IJi is 1-1, then 0" is the unique element of IvI* with 1Ji(0") = a. (Note for use below that this also shows 'Ij;(M*) ::) M tr .) Since ~i Ai = 0"(1), then

(5.46)

tr(a) = 0"(1) where 1Ji(0") = a,

which proves that the definition of tr is independent of the choice of the representation (5.44). Note that so far it is not obvious that NItr is a linear subspace, but we now will establish that. Soon we also will show that IvItr is an ideal, and that T satisfies T((a 0 b) 0 c) = T(a 0 (b 0 c)) for a, b, C E M tr .

5.66. Proposition. Let IvI be an atomzc lBW-algebra. Then the order zsomorphzsm IJi defined m Lemma 5.60 maps J. J* onto Nltr . Furthermore, tr zs a posztwe lmear Junctzonal on NItr , and NItr zs a Banach space wzth respect to the "trace norm" gwen by

(5.47)

Iialltr

=

tr(lal)

(where lal = a+ + a-). The map IJi zs an zsometry Jrom M* onto NItr Jor the trace norm on M tr . Proof. If 0" E M*, let 0" = 0"+ - 0"- be the orthogonal decomposition of 0". By Theorem 5.61 we can write and

0"- =

L j

{3jf'h,

TRACE CLASS ELEMENTS OF ATOMIC JBW-ALGEBRAS

179

where aI, a2, ... and (31, (32, ... are positive numbers such that 2::i ai = II(}"+II and 2:: j (3] = II(}"-II, and Pl,P2,··· are orthogonal atoms and ql, q2,··· are orthogonal atoms. Note that (5.48)

Let P = carrier( (}"+) and q = carrier( (}"-). Since ()"+ and ()"- are orthogonal, by Lemma 5.4 their carrier projections P and q are orthogonal. Since (}"+(p) = II(}"+II = 2::i ai, then the carrier projection P of ()"+ must take the value 1 at each Pi, so P ::,. Pi for each I, and similarly q ::,. q] for each J. Since P and q are orthogonal, then each Pi is orthogonal to each qj. Therefore by (5.48), \]I((}") E Mtr, so \]I maps M. into M tr . In the remarks preceding this lemma we observed that \]I(M.) contains NItr , so \]I maps M. onto AItr . Since \]I is linear, then M tr is a linear subspace of M. Since \]I is a bijection from M. onto Mtr, by (5.46) tr(a) = (\]I-l(a))(1) for each a E M tr . Since \]I is bipositive and linear, then tr is a positive linear functional on NItr . Shifting notation, we have shown above that every element ()" E M. admits a representation ()" = 2::i AiPi where PI, P2, ... are orthogonal atoms in M and 2::i 1,\1 < 00. Fix a E Mtr, let ()" = \]I-l(a), and let ()" = 2::i AS)i be such a representation of ()". Then a = \]I ((}") = 2::i AiP" and the unique orthogonal decomposition of a in IvI is given by a = a+ ~ a- where

Therefore

(5.49) so (5.50) By Lemma 5.59,

which proves that a f---> tr(lal) is a norm on NItr, and that \]I is an isometry from M. onto M tr for this norm. Since M. is a Banach space, then M tr is complete with respect to the trace norm. Finally by (5.50), (5.51 )

180

5

STATE SPACES OF JORDAN ALGEBRAS

Hence tr is a linear functional of norm at most 1 with respect to the trace norm on M tr . Since tr(p) = 1 for each atom p E lvI, the norm of tr is equal to 1. D Note that if a E M tr is represented in the form (5.44), then by (5.50) (5.52)

so the order unit norm restricted to lvItr is dominated by the trace norm.

5.67. Lemma. Let M be an atomzc lBW-algebm. If {ad lS a sequence of tmce class elements of AI wdh bounded tmce norm, and AI, A2,'" are real numbers such that Li IAil < 00, then Li Aiai converges m the tmce norm, and then also m the usual (order umt) norm, to an element a m M tr , and tr(a) = Li A,tr(ai)' Proof. Clearly the sequence of partial sums of the series Li Aiai is Cauchy for the trace norm. By Proposition 5.66, l'vItr is a Banach space for this norm, so Li Aiai converges in trace norm to an element a in lvItr . By (5.52), Li Aiai also converges to a in the (order unit) norm induced from M. By (5.51), the linear functional tr is continuous with respect to the trace norm; hence tr(a) = Li Aitr(ai). D Remark. Note that if Ai = B(H),a, then the atoms are one dimensional projections, so if an element a E lvItr is represented in the form (5.44), then there is an orthonormal set {~;} in H such that Pi is the projection onto the complex line [~il for each index l. By (5.49) lal~, = IAil~, for each l. Hence by (5.50), (5.53)

From this we conclude that the space lvItr of trace class elements as defined here is contained in the space T (H) of trace class elements in B (H) as defined in (A 86). In fact, we will see in the remarks after Theorem 5.70 that lvItr = T(H)sa. It follows from a similar argument based on (5.45) that our definition of tr on lvItr coincides with the trace inherited from

T(H)sa. In Lemma 3.41, we showed that if b is a projection in a JBW-algebra, then Ub sends atoms to multiples of atoms. The following is a generalization of that result.

TRACE CLASS ELEMENTS OF ATOlvIIC JI3W-ALGEBRAS

181

5.68. Lemma. Let M be a lEW-algebra, b any element of Ai, and m lVI. Then UbP 2S a multiple of an atom, say q, and

p any atom (5.54)

Proof. By the identity (1.15),

(5.55) Since p is an atom, then im Up = Rp by Lemma 3.29, so

for some), E R. Evaluating both sides with (1.55)) gives), = p(b 2 ), and so

p and using U;p

P (cf.

=

Substituting this into (5.55) gives (5.56)

If pW) = 0, then {bpb}2 = 0, so {bpb} = 0, and thus (5.54) holds for any projection q. If pW) i- 0, we write q = ),-l{bpb} (where), = p(b 2) as above). Then (5.54) holds, and by (5.56)

so q is a projection. It remains to show that q is an atom, which will follow if we can prove imUq = Rq (cf. Lemma 3.29). Let a E AI. By the identity (1.14) and the fact that im Up = Rp, {qaq}

= ),-2{{bpb}a{bpb}} = ),-2{b{p{bab}p}b}

E {b(Rp)b}

= Rq.

Thus im Uq = Rq, so the projection q is an atom. 0 5.69. Lemma. Let M be an atomic lEW-algebra. If a E IvI class, and b EM, then boa 2S trace class.

Proof. We have the polarization identity

(5.57)

2boa = {(I +b)a(l +b)}

~

{bab}

~a

2S

trace

5

182

STATE SPACES OF JORDAN ALGEBRAS

(cf. (2.1)). Thus in order to prove that boa is trace class, it suffices to show Uba is trace class for all b E .M. Let a = L:i AiPi as in equation (5.44). Then by Lemma 5.68 (5.58) for suitable atoms qi. Since p,(b 2 ) ::; IIb 2 1 for each l, the sequence {Pi(b 2 )q,} is bounded in trace norm, so Uba is of trace class (Lemma 5.67). 0

u E

5.70. Theorem. Let IvI be an atomlC lEW-algebra. Then Jar every IvI., there lS a umque trace class element dO' E IvI such that

(5.59)

u(b) = tr(d",

The element dO'

lS

0

w(u), where W

b) lS

Jar all bE M. the map defined m Lemma 5.60.

Proof. We are going to show that

(5.60) To see why this is of interest, note that by the polarization identity (5SI)

Applying (5.60) to (5.61) would prove the existence of dO' as stated in the theorem, since then

Let u E AI.. By Theorem 5.61, we can write u = L:i A,Pi where < (Xl and where each p, is an atom. Then by Lemma 5.68

L:i IAil

tr(Ub(w(u)))

=

tr( Ub L

,

AiPi)

= tr(LAiUbP,)

,

= tr (L AiP,(b 2 )qi),

,

where each qi is an atom. Then by Lemma 5.67

tr(Ub(W(u))) =

L

AiPi(b 2 )tr(qi) =

L i

A;]Ji(b 2 ) = u(b 2 ).

SYIVIMETRY AND ELLIPTICITY

183

This completes the proof of (5.60). Finally, to prove uniqueness, suppose that tr(d 1

0

b) = tr(d 2

for all bE M. Then taking b = d 1

-

0

b)

d 2 gives

From its definition the trace on !vltr annihilates no positive element, so d1 = d 2 . 0

Remark. We can now show that when !vI = B (H)sa, then !vltr = T(H)sa. Recall that in the remarks before Lemma 5.68, we showed that M tr c T(H)sa. Now let r E T(H)sa. Define Wr E B(H)* by wr(a) = T(ra), where T is the usual trace on B(H), cf. (A 87). By Theorem 5.70, there exists r E M tr such that tr(r 0 a) = wr(a) = T(ra) for all a E B (H). By Lemma 5.69, a 0 r E Aftr , and by the remarks following Lemma 5.67, tr and T agree on M tr . Thus T(r 0 a) = T(ra) for all a E B(H). Since T(r 0 a) = ~(T(ra) + T(ar)) = T(ra), it follows that T = r, and so T is in M tr . Thus M tr = T(H)sa. We conclude this section by noting that although we have used the term trace, we have not actually verified that tr is a tracial state. Of course, M tr is not a JB-algebra, (since it is not norm complete), so our previous Definition 5.19 of a tracial state is not applicable, nor is Lemma 5.18. Nevertheless, for each symmetry S E !vI, Us is a Jordan automorphism (Proposition 2.34), so for a = L:i AiPi as in (5.44), then Usa = L:i AiUs(Pi) and {Us (Pi)} will be a sequence of orthogonal atoms. For a E Mtfl we then have Usa E Altr and tr(Us(a)) = tr(a). A careful examination of the proof of Lemma 5.18 shows that the argument there still goes through with M tr in place of M, so that

tr((a

0

b)

0

c)

= tr(a 0 (b 0 c))

holds for all trace class a, b, c. Since we will not need this result in the sequel, we leave the details to the interested reader.

Symmetry and ellipticity In this section we will discuss two properties having to do with the existence of affine automorphisms of the state space, and thus with its degree of symmetry or homogeneity. We start with symmetry with respect to certain reflections.

184

5

STATE SPACES OF JORDAN ALGEBRAS

5.71. Definition. An affine automorphism R of a convex set K is a reflectwn if R2 = I. A convex set K is symmetnc with respect to a conve:J; subset Ko if there is a reflection R of K whose fixed point set is exactly K o. Note that if R is a reflection with fixed point set Ko, then the image of a point IJ under R is obtained by reflecting the line segment [IJ, RIJ] about its midpoint ~(IJ+RIJ) E K o. This justifies both the term "reflection" and "symmetric" above.

5.72. Proposition. If M lS a lBW-algebm wlth normal state space K, and F lS a norm closed face wzth complement F', then K lS symmetrlc wlth respect to co(F U F'). In partlcular, If p lS the carner pro]ectwn of F, then III = (Up + Up')* lS an affine pro]ectwn (ldempotent map) from K onto cor F U F'), and R = U;_p' lS a reflectwn whose fixed point set lS co(F U F').

Pmof. Recall that F = Fp = (imU;) n K. Let s = p - p', and R = U; IK. By Proposition 2.34, Us is a Jordan automorphism of M satisfying (Us )2 = I, and U;(K) c K (Proposition 5.16), so R is a reflection of K. By equation (2.25), Us = 2Up + 2Up' - I. Thus if w E K, then Rw = w iff (Up + Up' )*w = w. Then

where ,\ = IIU;wll. Since each element of co(F U F') is clearly fixed by R, the fixed point set of R equals co(F U F'). D We will refer to (Up + Up')* as the canomcal affine pro]ectwn of K onto co( F U F'). We will see in Part II of this book that this is the unique affine projection of K onto co( F U F'), and that III = U; is the unique reflection of K with fixed point set co( F U F'). (The analogous uniqueness result in the von Neumann context is given in (A 174).) We now turn to showing that certain orbits are ellipses. Recall that for each element a in a JB-algebra, the map Ta given by TaX = a 0 X is an order derivation (Lemma l.56). Thus for an element a in a JBW-algebra M, since Ta is IJ-weakly continuous, then t f-> exp(tTa)* restricts to give a one-parameter group of order automorphisms of M*. Orbits of normal states then stay inside the positive cone of M*, and if normalized the orbit will lie in K. We are going to show, for the special case where a = p - p' for a projection p, that these normalized orbits trace out half of an ellipse. Since T p_ p' = Up - Up' (cf. (1.47)), for later applications we will express our results in terms of the maps Up - Up" The following is a simple technical lemma that we will use both noW and later, and so we state it separately for easy reference.

185

SYl\IMETRY AND ELLIPTICITY

5.73. Lemma. Let a, 7 and p be lmearly mdependent elements of a vector space, let a be a real number wzth < a < 1, and set p = ~(a(1 - a))-1/2 p. For each t E R the pomt

°

(5.62)

hes on the ellzpse conszstmg of all pomts (5.63)

Thzs ellzpse has the dzameter [a,7]. When t vanes from -= to =, the parametenzed curve Wt traces out, m the dzrectzon from 7 to a, the halfellzpse between these poznts whzch contazns the poznt aa + (1 - a)7 + p (attazned for t = 0). The other half of the ellzpse zs traced out zn the same fashzon by the parameterized curve obtamed by replacmg p by -p zn

(5.62). Proof. It is easily verified that when t varies from (Xt, Yt) E R2 defined by

-= to =, the point

2(a(1 - a))1/2 Yt = aet + (1 - a)e- t

(5.64)

traces out the upper half of the circle x2 + y2 = 1 in the direction from left to right. The affine map cI> : R2 f---* M. defined by (5.65)

cI>(X, y) = ~ (a

+ 7) + x (~( a -

7))

+YP

carries this circle to an ellipse. Rewriting (5.65) in the form

cI>(X,y)

=

~(:r;+ l)a+ ~(I-X)7+YP

and substituting x = Xt, Y = Yt, we find that cI>(Xt, Yt) = Wt. Thus Wt lies on the specified ellipse for all t E R. Since cI>(I, 0) = a and cI>(-1,0) = 7, the line segment [a,7] is a diameter of this ellipse. By (5.64), (Xt, Yt) --> (±l, 0) when t --> ±=, so Wt --> 7 when t --> -= and Wt --> a when t --> =. Thus Wt traces out, in the direction from 7 to a, the half-ellipse between these points which contains the point Wo = aa + (1 - a)7 + p. In the same fashion, the parameterized curve obtained by changing the sign of p in (5.62) traces out the half-ellipse containing the "opposite" point aa + (1 - a)7 - p (i.e., the image of Wo under reflection about the diameter [a, 7]). This proves the last statement of the lemma. 0

5

186

5.74.

STATE SPACES OF JORDAN ALGEBRAS

Lemma.

Let M be a lEW-algebra wzth normal state space

K, let p be a pro]ectwn m M wzth assoczated face F = Fp and refiectwn R = U;~p' (cf. Proposztwn 5.72), and let w E K. Assume w f/c F U F ' , and define IJ = IIU;wll~lU;w, T = 11U;,wII~lUl~,w. Conszder first the orbzt {WdtER of w under the one-parameter group of order automorphzsms of M. gwen by t f-+ exp( t(U; - U;,)). Thzs orbzt zs one half of a hyperbola. Conszder also the normalzzed orbzt {WdtER of w, gwen by Wt = IlWt II ~l Wt. If w f/c co( F U F ' ), then thzs orbzt zs the (unzque) halJ-ellzpse wzth dzameter [IJ, T] whzch contams the pomt w, traced out m the dzrectwn from T to IJ, and the other half of thzs ellzpse zs the normalzzed OT'bzt of Rw. If w E co(F U F ' ), (but stzll w f/c F U F ' ), then the normalzzed orbzt of w degenerates to the line segment from T to IJ, traced out m the same dZT'ectwn.

Since w f/c F U F ' , then IIU;wll = (U;w)(l) = w(p) oF 0, and likewise 11U;,wll = W(p') oF O. Therefore the two points IJ and T of the lemma are well defined. Now we can rewrite the equation above in the form (5.66)

~ Wt = ae t IJ

+ (1 -

a ) e ~t T

+p

where a = IIU;wll and p = (1 - Up - Up' )*w. This equation shows that the orbit {wdtER is one half of a hyperbola. For each t E R, IIWtl1 = Wt(1) = ae t + (1 - a)e~t, so the normalized orbit of w is given by (5.62). Assume now w f/c co(F U F'). Then IJ, T, P are linearly independent; otherwise p would be a linear combination of the two (linearly independent) elements IJ and T, and then w = Wo would also be a linear combination of IJ E F and TEF', so w would be in im (U; + U;,) n K = co(FuF ' ). Thus we can apply Lemma 5.73, from which the description of the normalized orbit of w follows. Since U)J~p' = 2Up + 2Up' - I, the reflection R = U;~p' sends p = (1 - Up - Up' )*w to -p, and fixes IJ and T. Since U; and U;, commute with R, the orbit of Rw equals R applied to the orbit of w. Therefore the description of the normalized orbit of Rw also follows from Lemma 5.73. Finally, if w E co(F U F ' ), then p = 0 and it follows from (5.62) that Wt can be written as a convex combination Wt = AtIJ + (1 - AdT where At ----> 0 when t ----> -= and At ----> 1 when t ----> =. Thus Wt traces out the line segment from T to IJ in this case. 0 Note that the fact that the normalized orbit of the point w of Lemma 5.74 is part of an ellipse follows easily from the fact that the non-normalized

SYl\U.,1ETRY AND ELLIPTICITY

187

orbit is half of a hyperbola. Actually, the passage from the non-normalized orbit to the normalized orbit is just the projection via the origin onto the plane through p, 0', T, and since the image of the hyperbola under this projection is a conic section in the bounded set K, it must be an ellipse. Note also that if the point W of Lemma 5.74 lies in either F or F' , then it is easily verified that w is a fixed point under the one-parameter group t f-+ exp(t(U; - U;*)).

5.75. Proposition. Let !vI be a lEW-algebra, and F a norm closed face of the normal state space K. Let W = U; + U;, be the canonzcal affine proJectwn of K onto co(F U F ' ), and 0' E F, TEF'. Then w- 1 ([0', T]) has ellzptzcal cross-sectwns, z. e., the zntersectwn of thzs set wzth every plane through the lzne segment [0', T] zs an ellzptzcal dzsk (whzch may degenerate to a lzne segment). Proof. Let p be the carrier projection of F. As observed in the remarks preceding Lemma 5.73, for each w E K the normalized elliptical orbit Wt described in Lemma 5.74 stays inside K. Let Lo be the intersection of w-I([O',T]) with a plane L through [O',T]. If Lo is a line segment, then Lo is a degenerate ellipse, so we are done. Hereafter, we assume this is not the case. Since K is closed and bounded for the base norm, and all norms on the finite dimensional subspace L are equivalent (cf., e.g., [108, Thm. 1.21]), Lo = K n L is closed and bounded and thus is compact. Let W be a point on the boundary of Lo in L. Then since w(w) E [0', T], there is a scalar A with 0 < A < 1 such that u;w

+ U;,w =

AO'

+ (1

- A)T.

Applying U; to both sides, we find that 0' = IIU;wll- 1 U;w. Similarly = II U;,w 11-1 U;,w, so 0' and T have the same meaning as in Lemma 5.74. Let E(w) denote the ellipse with diameter [0', T] passing through w, together with its interior. Then each half of this ellipse is the normalized orbit of either w or the reflected point U;w (where s = p - pi) by Lemma 5.74, and so this ellipse is contained in K. Thus E(w) is contained in L o . We will show that E(w) = Lo. If WI were another element of Lo not in E(w), then the ellipse through 0', T, WI would not meet the ellipse through 0', T, W except at the points 0' and T. Therefore E(WI) would be contained in Lo and would contain w in its interior. But w is a boundary point, so this is a contradiction. We conclude that Lo = E(w) as claimed. D T

We will refer to the elliptical cross-section property described in Proposition 5.75 as ellzptzcityof K. We will define this in an abstract context later, and show that the combination of admitting a suitable spectral theory and ellipticity characterize normal state spaces of JEW-algebras. Both of the properties symmetry and ellipticity can be viewed as illustrations that there are "many" affine automorphisms of the normal state

188

5

STATE SPACES OF JORDAN ALGEBRAS

space of a JBW-algebra (or of the state space of a JB-algebra). Dually, the positive cone of every JB-algebra admits "many" Jordan automorphisms. One way to make this precise is to observe that the affine automorphisms of the interior of the positive cone of a JB-algebra A are transitive. For a more complete discussion of such homogenelty, see the discussion in the notes to Chapter 6. As discussed there, in the context of self-dual cones, another notion of homogeneity (sometimes called "facial homogeneity"), is due to Connes [36]. The definition is that exp( t( P F - PVl )) ~ 0 for all t and F, where his maps PF are projections onto faces of the associated cone, and thus are closely related to our maps Up. In the JBIV-context, positivity of the analogous maps exp( t(Up - Up ')*) is what we used above to establish ellipticity, and we will see in an abstract context that positivity of such maps is equivalent to ellipticity. Notes The introductory results of this chapter on the lattice of projective faces can be found in [10]. The characterization of Jordan homomorphisms via their dual maps on the state space (Lemma 5.15) is from [117]. The result describing alternative characterizations of traces on JB-algebras (Lemma 5.18) is due to Pedersen and St0nner [101]. The result that type In JBWalgebras have a center-valued trace (Proposition 5.25) can be found in [62]; our proof is modeled on the similar proof for type I n von Neumann algebras in [80]. For a more detailed treatment of traces on Jordan operator algebras, see [23]. The Radon~Nikodym type theorem for JBW-algebras in Lemma 5.27 generalizes a well known result for von Neumann algebras [112], and the proof is essentially the same. The result that norm closed faces of the normal state space of a JRW-algebra are projective (Theorem 5.32) is due to Iochum [68]. After quite a bit of work in the lemmas preceding Theorem 5.32, the proof eventually reduces to using the analogous result for von Neumann algebras (A 107), which is due to Effros [46] and to Prosser

[105]. In the sequel, we will only use a special case of this 1 esult: namely, that extreme points of the state space of a JB-algebra are norm exposed. This is used to show that the factor representations associated with pure states are type I (Corollary 5.42), and will play an important role in our characterization of JB-algebra and C*-algebra state spaces. In [8] and in [10] the result that extreme points are norm exposed is based on the characterization of pure states on JC-algebras due to St0rmer [126]. The general correspondence of ideals and split faces (Proposition 5.36 and Corollary 5.37) is quite analogous to the similar results for C* -algebras and von Neumann algebras, cf. (A 112) and (A 114). For C*-algebras, this result is due to Segal [115]. For JB-algebras, this result is in [65]. The Hilbert ball property for JB-algebra state spaces (Corollary 5.56) was first discussed in [10] as part of the characterization of JB state spaces

NOTES

189

(Theorem 9.38), as were the related facts that the state space of a spin factor is a Hilbert ball (Proposition 5.51), and the symmetry of transition probabilities (Corollary 5.57). The representation of normal states on an atomic JBvV-factor as a countable convex combination of orthogonal pure states (Theorem 5.61) also is in [10], and generalizes the analogous result for B(H), which can be found in [100]. The use of trace class elements to describe the normal state space of atomic JBW-algebras (Theorem 5.70) parallels the corresponding result for B (H) (A 87). The symmetry property (Proposition 5.72) first appears in [10]. As discussed at the end of this chapter, ellipticity as described in Proposition 5.75 is formally similar to the notion of Jaczal homogenedy due to Connes [36]. The geometric interpretation of ellipticity given in Proposition 5.75 appeared in [71].

6

Dynamical Correspondences

In this chapter we will discuss the notion of a "dynamical correspondence" , which is applicable in both JB and JBW contexts. This generalizes the correspondence of observables and generators of one-parameter groups of automorphisms ill quantum mechanics. It is closely related to Connes' concept of orientation [36, Definition 4.11] that he used as a key property in his characterization of the natural self-dual cones associated with von Neumann algebras (i.e., the cones of Tomita-Takesaki theory). Both can be thought of as ways to specify possible Lie structures compatible with a given Jordan structure. In the first section we introduce the general notion of a "Connes orientation", which carries over to JBW-algebras the concept Connes introduced. In the second section we introduce the notion of a "dynamical correspondence" (first defined in [12, Def. 17]). We show that a JB-algebra is the self-adjoint part of a C*-algebra iff it admits a dynamical correspondence, with a similar result for JBW-algebras, and that the dynamical correspondences are in both cases in 1-1 correspondence with "Jordan compatible" associative products (Definition 6.14). We also show that for JBW-algebras there is a 1-1 correspondence of Connes orientations and dynamical correspondences. The last section sketches a geometric interpretation of these concepts, and relates them to the concepts of orientation for C*-algebra state spaces and von Neumann algebra normal state spaces discussed in

pJ

[AS]. Connes orientations Both Connes' concept of orientation and our notion of a dynamical correspondence refer to order derivations. If A is an ordered Banach space, a bounded linear map 0 : A ---+ A is an order der'lVatwn if exp(to) :::0: 0 for all t E R, i.e., if 0 is the generator of a one-parameter group of order automorphisms of A. A bounded linear operator 0 acting on a C*-algebra A is said to be an order del'lvatwn if it leaves Asa invariant and restricts to an order derivation on Asa (A 181).

6.1. Definition. If A is a C*-algebra A, and a E A, then we define ---+ A by

(Sa : A (6.1)

DaX

= ~(ax

+ xa·).

We recall the following results from [AS].

192

6.

DYNAMICAL CORRESPONDENCES

6.2. Theorem. If A zs a C*-algebra and a E A, then 6a zs an order derzvation on A. If A zs a von Neumann algebra, then every order derzvatwn on A has the form 6a for some a EA. Proof. The first statement is (A 182), and the second is (A 183). D

6.3. Proposition. Let A be a C*-algebra. one-parameter group at = exp(t6a ) zs gwen by

If a E .4., then the

(6.2) In partzcular, zf h E Asa and a

=

h (the self-adjomt case), then

(6.3) and zf a

= zh

(the skew case), then

(6.4) Proof. Consider the left and right multiplication operators L : x f--t ax and R: x f--t xa* defined on A for a E A. Since Land R commute, exp(2t6a )(x) = exp(tL

+ tR)(x)

= exp(tL)exp(tR)(x) = etaxe ta *.

This proves (6.2), from which (6.3) and (6.4) both follow. D The orbits of the one-parameter group a; = exp(6a )* can be easily visualized in the simplest non-commutative case where A = lvI2(C). Here the state space is a Euclidean 3-ball, and the pure state space is the surface of the ball, i.e., a Euclidean 2-sphere. If a E A is self-adjoint and has two distinct eigenvalues >'1 < A2 corresponding to (unit) eigenvectors ~1,6, then the vector states W~" w6 are antipodal points on the sphere (South Pole and North Pole in Fig. 6.1). If a = zh where h E Asa (the skew case), then it can be seen from (6.4) that a; is a rotation of the ball by an angle t(A2 - Ad/2 about the diameter [W~1,W~2], d. (A 129). Thus the one-parameter group a; represents a rotational motion with rotational velocity (A2 - Ad/2 about this diameter, and the orbits on the sphere are the "parallel circles" (in planes orthogonal to [w~" w~2l). If a = h where hE Asa (the self-adjoint case), then the orbits will take us out of the state space. But this can be remedied by a normalization, i.e., by considering the parametric curves t f--t Ila;(0')11- 1 a;(0') instead of t f--t a;(O'). Now it can be seen from (6.3) that the (normalized) orbits are the "longitudinal semi-circles" on the sphere (in planes through [w~ll W~2])' cf. Lemma 5.74.

CONNES ORIENTATIONS

193

Fig. 6.1

In the example above we can easily see how the one-parameter group is determined by the geometry in the self-adjoint case, and we can also see what indeterminacy there is in the skew case. The element a E Asa determines a real valued affine function W f-7 w(a) on the state space. This function attains its minimum A1 at WI;! and its maximum A2 at w1;2. In the self-adjoint case the orbits are the longitudinal semi-circles traced out in the direction from wl;! to WE2. In the skew case the orbits are the parallel circles, but they can be traced out in two possible directions, "eastbound" and "westbound". The mere knowledge of the affine function a does not tell us which direction to choose. This would require a specific orientation of the ball (right-handed or left-handed around wl;~t:;2).

a:

If a, b are elements of a C*-algebra, then we will often make use of the identity (6.5) We are going to transfer Connes' concept of orientation [36, Definition 4.11]' originally defined in the context of certain self-dual cones, to the context of lBW-algebras. For motivation, note that if M is a von Neumann algebra and a E M sa, then

so that

Oia

essentially gives the Lie product. Thus we can express the

194

6

DYNA1VIICAL CORRESPONDENCES

ordinary left multiplication by a as

Oid from the set D( M ",) Now the idea is to axiomatize the Illap Od of order derivations into itself. Note that as it stands this map is not well defined on the set of all order derivations, since if d is not self-adjoint, then Od does not determine d. In fact, for any central skew-adjoint d the map Od will be zero. Thus we will define the Illap instead on D( M ,a) modulo the center of the set of order derivatiolls, which we now defillc. f--)

6.4. Definition. Let A be a JB-algebra. The center of the Lie algebra D(A) consists of all the order derivations 0 in D(A) that commute with all members of D(A). We will denote the center of D(A) by Z(D(A)). We review some basic facts about order derivations on JB-algebras. If

a is an element of a JB-algebra A, we let oa denote Jordan multiplication by a, i.e., oax = aox for x E A. (Note that this agrees with (6.1) when A is the self-adjoint part of a C* -algebra A.) Each such map 0(1 is an order derivation on A (Lemma 1.56). The center of a .IB-algebra A consists of those z E A such that ozoa = 0(10Z for all a E A (Definition 1.51). The space D(A) of order derivations of A is a Lie algebra with respect to the usual Lie bracket given by commutators: [01,02] = 0] 02 - 0201 (Proposition 1.59). An order derivation on A is self-adJomt if it has the form oa for some a E A. An order derivation 0 on A is skew if 0(1) = o. Every order derivation admits a unique decomposition as the sum of a selfadjoint derivation and a skew derivation (Proposition 1.60). A skew order derivation is a Jordan derivation, i.e., satisfies the Leibniz rule (6.6)

o(a

0

b)

=

o(a)

0

b + a oo(b),

and the skew order derivations are precisely the boullded lillear maps that generate one-parameter groups of Jordan automorphisms (Lemma 2.81). 6.5. Lemma. If A

(6.7)

lS

a iE-algebra, then

Z(D(A)) = {oz I Z

E

Z(A)}.

Proof. Assume first that 0 E Z(D(A)). Then [o,oal a E A. Let Z = 0(1). Then for every a E A,

o(a) = 5O a(l) = oao(l) = O(1(z) = a 0

Z

=

Z 0

o for

every

a.

Hence 0 = Oz. Also ozoa = oaoz, so z E Z(A). Assume next that Z E Z(A). By the definition of Z(A), Oz commutes with every self-adjoint order derivation oa. Therefore we only have to show

CONNES ORIENTATIONS

195

that 6z commutes with every skew derivation 6. But such a derivation is a Jordan derivation, so it follows from the Leibniz rule (6.6) that

(6.8)

66 z (x) = 6(z

0

x) = (6z)

0

x

+Z 0

(6x)

for every x E A. Since 6 is skew, 6z = 0 (Proposition 2.82), so (6.8) shows that (66z)(x) = (6 z 6)(x). Thus 66 z = 6z 6 as desired. D

6.6. Definition. Let A be a JB-algebra and 6 E D(A). Write 6 = 61 + 62 where 61 is self-adjoint and 62 is skew. Then we define 6t =61- 62. Note that if A is a JB-algebra and 6 E D(A), then (6 t )t = 6, so 6 --) 6t is linear and is an involutive map, i.e., has period 2. Note also that 6 is self-adjoint iff 6t = 6, and skew iff 6t = -6.

6.7. Definition. Let M be a lBW-algebra. We write D(M) in place of D(M)/Z(D(M)), and denote the equivalence class of an element 6 of D(M) modulo Z(D(M)) by b. Note that the involution (b)t = (bt) is well defined on D(M), for if 61 = 62, then 61 - 62 = 6z for some Z E Z(AI) (Lemma 6.5). Hence

-

-

6i - 6~ = 61 = 6z , so (6i) = (6~). 6.8. Definition. A Connes orzentatwn on a lRW-algebra IvI is a complex structure on D(M), which is compatible with Lie brackets and involution, i.e., a linear operator I on D(NI) which satisfies the requirements: (i) 12 = -1 (where 1 is the identity map), (ii) [I'll, b2 ] = [b1 , Ib2 ] = I[b1 , b2 ], and (iii) I(bt ) = -(Ib)t.

6.9. Proposition. Let M

-

be a von Neumann algebra. Then

-

I(6 x ) = 6u

for all x E M

defines a Connes onentatwn on Mba. Proof. Let IvI = M sa. We first verify that I is well defined. For this purpose, it suffices to show that if x E M and bx = 0, then bix = o. We first verify

(6.9)

ox=o

=}

xEZ(M),

196

6

DYNAMICAL CORRESPONDENCES

where Z( M) is the center of M. Write x = a + Ib with a, b E M. If Ox = 0, then Ox E Z(D(M)), so Ox = Oz for some z in the center of the JBW-algebra M (Lemma 6.5). Then z is also central in the von Neumann algebra M (Corollary 1.53). Note that the self-adjoint and skew parts of the order derivation Ox are oa and Oib respectively. Equating the selfadjoint and skew parts of the order derivations Ox and oz, we conclude that oa = Oz (so a = z), and Oib = 0 (so b is central). Thus x = a + Ib is central in M, so (6.9) is verified. Now let x E M with Sx = O. Then x is central in M, so we can write x = a + Ib with a, b self-adjoint elements of the center of M. Then Oix = Oia-b = Oia - Ob· Thus Oia = 0 and Ob E Z(D(M)), so Six = 0, which completes the proof that I is well defined. Since every order derivation on M has th~Jorm Ox for some x E M (Theorem 6.2), then the domain of I is all of D(M). Now we verify the three properties of Definition 6.8. The property (i) follows at once from the definition of I, and (ii) is an immediate consequence of the identity (6.5). To verify (iii), note that if a, b E M sa, then oa is self-adjoint and Otb is skew, so

Thus

O!

= Ox'

for all x EM.

Now it follows that I satisfies Definition 6.8(iii). D We will see later (in Corollary 6.19) that a JBW-algebra M is the selfadjoint part of a von Neumann algebra iff M admits a Connes orientation, and that such orientations are in 1-1 correspondence with W*-products on M+IM.

Dynamical correspondences in JB-algebras When a C*-algebra or a von Neumann algebra is used as an algebraic model of quantum mechanics, then it is only the self-adjoint part of the algebra that represents observables. However, the self-adjoint part of such an algebra is not closed under the given associative product, but only under the Jordan product. Therefore it has been proposed to model quantum mechanics on Jordan algebras rather than associative algebras [75], [98]. This approach is corroborated by the fact that many physically relevant properties of observables are adequately described by Jordan constructs. However, it is an important feature of quantum mechanics that the physical variables play a dual role, as observables and as generators of

DYNAMICAL CORRESPONDENCES IN JB-ALGEBRAS

197

transformation groups. The observables are random variables with a specified probability law in each state of the quantum system, while the generators determine one-parameter groups of transformations of observables (Heisenberg picture) or states (Schrodinger picture). Both aspects can be adequately dealt with in a C*-algebra (or a von Neumann algebra) A. But unlike the observables whose probability distribution is determined by the functional calculus in Asa and thus by Jordan structure, the generators cannot be expressed in terms of the Jordan product. Each self-adjoint element a = d~ of a C*-algebra defines a derivation Jih, and these derivations generate one-parameter groups as in (6.4) which describe the time development of the physical system. But generators are determined not by the symmetrized product a 0 b = ~ (ab + ba) in A sa, but instead by the antisymmetrized product a * b := ~(ab - ba) on Asa. Therefore both the Jordan product and the Lie product of a C*-algebra are needed for physics, and the decomposition

(6.10)

ab = a

0

b-

2 (a

* b)

separates the two aspects of a physical variable. It is of interest to find appropriate constructs, defined in terms of the Jordan structure of a JB-algebra (or the geometry of its state space), which makes it the self-adjoint part of a C*-algebra. One such construct is the concept of a dynam2cal correspondence (defined below), which axiomatizes the transition h f---+ bih from the self-adjoint part of a C*-algebra to the set of skew order derivations on the algebra.

6.10. Definition. A dynamzcal correspondence on a JB-algebra A is a linear map 1/J : a f---+ 1/Ja from A into the set of skew order derivations on A which satisfies the requirements (i) [1/Ju,1/Jb] = -[b a , bb] for a, bE A, (ii) 1/J"a = 0 for all a E A. A dynamical correspondence on a JB-algebra A will be called complete if it maps A onto the set of all skew order derivations on A.

6.11. Proposition. If A zs the self-adJomt part of a C*-algebra or a von Neumann algebra, then the map a f---+ 1/Ja = bia ZS a dynamzcal correspondence. In the von Neumann algebra case, thzs dynamzcal correspondence zs complete.

Proof. In both cases, property (i) above follows from the identity (6.5), and (ii) is immediate, so 1/J is a dynamical correspondence. A skew order derivation on a von Neumann algebra has the form bia for some self-adjoint element a (A 184). Thus in a von Neumann algebra the map a f---+ 1/Ja = bia is a complete dynamical correspondence. 0

6

198

DYNAMICAL CORRESPONDENCES

The skew order derivations on a JB-algebra A determine oneparameter groups of Jordan automorphisms of A (Lemma 2.81). Therefore the dual maps determine affine automorphisms of the state space of A (and also of the normal state space in the JBW case), cf. Proposition 5.16. Thus a dynamical correspondence gives the elements of A a double identity, which reflects the dual role of physical variables as observables and as generators of a one-parameter group of motions. Hencp the name "dynamical correspondence" . Since the Jordan product is commutative, there is no useful concept of "commutator" for elements in a JB-algebra, but the commutators of the associated Jordan multipliers can be used as a substitute in view of the identity [Oa,Ob] = ~O[a,bJ for elements a, b in a C*-algebra. Thus the condition (i) above is a kind of quantization requirement, relating commutators of elements to the commutators of the associated generators. Note also that the equation 1/Ja(a) = 0 is equivalent to exp(t1/Ja)(a) = a for all t E R. Thus condition (ii) says that the time evolution associated with an observable fixes that observable We will also state the definition of a dynamical correspondence in another form. In this connection we shall need the following lemma. 6.12. Lemma. Let A be a fE-algebra and let a f-7 1/Ja be a map from A mto the set of all skew order derivatIOns on A. Then for all pall'S a,b E A, (6.11)

Proof. Since 1/Ja is skew, it is a Jordan derivation. Hence for all c E A,

which can be rewritten

This gives (6.11). D Note that linearity of 1/J is not listed among the requirements in the proposition below, since it follows from the other requirements. 6.13. Proposition. Let A be a fE-algebra and let 1/J : a f-7 1/Ja be a map from A mto the set of skew order derwations of A. Then 1/J lS a dynamical correspondence lff the followmg reqmrements are satlsfied for a,b E A:

(i) [1/Ja,1/Jb] = -loa, bb], and (ii) [1fJa, bb] = [ba, 1fJb].

DYNAl\IICAL CORRESPONDENCES IN JB-ALGEBRAS

199

Proof. Assume first that 'lj; is a dynamical correspondence. Condition (i) above is satisfied as it is the same as condition (i) of the definition of a dynamical correspondence (Definition 6.10). By condition (ii) of Definition 6.10 and Lemma 6.12, for all a E A

where the last equality follows from linearity of the map a again by linearity of 'lj;, for all a, b E A,

f--->

'lj;a. Therefore,

which gives

This proves condition (ii) above. Assume next that 'lj; satisfies conditions (i) and (ii) of the proposition. Condition (i) of Definition 6.10 is trivially satisfied. It follows from condition (ii) above that for all a E A,

so ['lj;a, <Sal = 0 for all a E A. By Lemma 6.12, <S",,,a = ['lj;a, <Sa], so <S",,,a = O. Hence 'lj;aa = <S",,,a1 = 0, so condition (ii) of Definition 6.10 is also satisfied. It remains to show that 'lj; is linear. By Lemma 6.12 and condition (ii) above, for all a, b E A

Hence 'lj;ab = -'lj;ba. Since a f---> 'lj;ba is linear, it follows that a linear map from A into D(A). D

f--->

'lj;a is a

In the last paragraph of the proof above we have shown that if 'lj; is a dynamical correspondence on A, then (6.12) We shall make more use of this equation later. 6.14. Definition. Let A be a lB-algebra. A bilinear associative product * on the complexified space A + zA is a Jordan compatzble product if

200

6

(ii) (a y*

+

DYNAMICAL CORRESPONDENCES

Ib)* = a - Ib is an involution for (A for all x, yEA + LA.

* x*

+ LA, *),

i.e., (x * y)*

=

Such a product will also be called a C*-product OIl A + IA. If A is a JBWalgebra, then we call such a product a W*-product or a von Neumann product.

Remark. If A is a JB-algebra and * is a Jordan compatible product on A + LA, then we can define a norm on A + l.A by Ilxll = 11.1:*:1':111/2. With this norm, A + IA will become a C*-algebra (A 59), justifying our use of the term "C*-product". In principle this norm could depend on the associative product as well as on the norm on A, but in fact it is uniquely determined by the JB-algebra A (A 160). (This follows from Kadisoll's theorem that a Jordan isomorphism between C*-algebras is an isometry (A 159).) If A is a JBW-algebra and A + 'LA is equipped with a Jordan compatible product, then A + LA will be a C*-algebra whose self-adjoint part A is monotone complete with a separating set of normal states, so will be a von Neumann algebra (A 95), justifying the term \V*-product (or von Neumann product). We are now ready to state and prove our main theorem, which relates dynamical correspondences to associative products. 6.15. Theorem. A JE-algebra A LS (Jordan IsomorphIc to) the selfadJomt part of a C*-algebra Iff there eXIsts a dynarmcal correspondence on A, In thIS case there IS a 1-1 correspondence of C*-products on A + IA and dynamzcal cor7'espondences on A. The dynarmcal correspondence on A assoczated WIth a C* -product (a, b) f---> ab IS ( 6.13) and the C*-product on A + IA assoczated wl,th a dynaTnlc(11 correspondence IS the complex bz/mear extenS1.On of the product defined on A by

1/; on A (6.14)

ab

=

a

0

b - 11/;ab.

Proof. Assume first that A admits a dynamical correspondence 1/). By equation (6.12) we can define an anti-symmetric bilinear product (a, b) 1-7 a x b on A by writing

(6.15) Next define a bilinear map (a, b) f---> ab from the Cartesian product A x A into A + zA (considered as a real linear space) by writing (6.16)

ab=aob-z(axb).

DYNMIICAL CORRESPONDENCES IN JB-ALGEBRAS

201

This map can be uniquely extended to a bilinear product on A + ~A (considered as a complex linear space). We will show that this product IS associative. By linearity, it suffices to prove the associative law

a( cb) = (ac)b

(6.17)

for a, b, c E A. Writing out (6.17) by means of (6.16), we get

a 0 (c 0 b) - ~(a x (c 0 b)) - ~(a

0

(c x b)) - a x (c x b)

= (a 0 c) 0 b - z((a 0 c) x b) - z((a x c) 0 b) - (a x c) x b. Separating real and imaginary terms (and using the anti-symmetry of the x -product), we get two equations. The first one can be written as (6.18)

a x (b x c) - b x (a x c)

=

-a 0 (b

0

c)

+ b0

(a

0

c),

and the second one as (6.19)

a x (b

0

c) - b 0 (a x c)

= a 0 (b x c) - b x (a 0 c).

The left-hand side of (6.18) is just [1f!a,1f!b](C) and the right-hand side of (6.18) is -[b a, bb](C). Similarly the left-hand side of (6.19) is [1f!a,6b](C) and the right-hand side of (6.19) is [b a,1f!b](C). Thus these two equations follow directly from the characterization of a dynamical correspondence in Proposition 6.13. We must also show that the bilinear product on A + zA is compatible with the natural involution (a+zb)* = a-lb on A+1A, i.e., that (xy)* = y*x* for x, yEA + ~A. By linearity it suffices to show that (ab)* = ba for a, b E A. But this follows directly from the anti-symmetry of the xproduct, as

(ab) * = (a

0

b - 1 (a x b)) * = a 0 b + 1 (a x b) = boa -

1

(b x a) = ba.

We have now shown that A + ~A is an associative *-algebra. By the definition of the involution, the self-adjoint part of A A. Thus by (6.16) and the anti-symmetry of the x -product, ~ (ab

+ ba) = a 0

+ lA

is

b

for all a, b E A. Therefore the associative product in A + ~A induces the given Jordan product on A, and thus is a C*-product. Assume next that A is Jordan isomorphic to the self-adjoint part of a C*-algebra A. Extending the isomorphism from A onto Asa to a complex linear map from A + iA onto A and pulling back the associative product

6.

202

DYNAMICAL CORRESPONDENCES

in A, we get a C*-product (a,b) f--+ ab in A+2A. Defining a map 1/J from A into D(A) by equation (6.13), we get a dynamical correspondence on A (cf. the remark following Definition 6.14, and Proposition 6.11). Then we define a new product * on A + 2A to be the complex bilinear extension of (6.14). By the definition of 1/J, we have a * b = a 0 b - 21/Ja(b) = ab for a, b E A, so the new product is the same as the original. It remains to prove that if we start out from a dynamical correspondence 1/J on A and construct first the C*-product in A+iA via (6.14) as in the first part of the proof, and then from this product the dynamical correspondence given by (6.13) as above, then we will end with the dynamical correspondence with which we began. Let a, bE A. By (6.15), (6.16) and the anti-symmetry of the x -product, we find that ~i(ab - ba)

=

~(a x b - b x a)

=

a x b = 1/Ja(b),

which completes the proof. 0 Note that it follows from equation (6.13) of Theorem 6.15 that a dynamical correspondence 1/J on a JB-algebra A can be recovered from the associated C*-product on A + 2A through the equation

(6.20)

1/Ja

= Ow for all

a E A.

6.16. Corollary. A lEW-algebra M zs (lordan zsomorphzc to) the self-adJomt part of a von Neumann algebra zfJ there eX2sts a dynam2cal correspondence on M. In thzs case the constructzon m Theorem 6.15 promdes a 1-1 correspondence of W*-products on !vI + 2M and dynam2cal correspondences on M. Proof. This follows from Theorem 6.15, together with the observation in the remarks preceding Theorem 6.15 showing that if ~M is a JBWalgebra, then a Jordan compatible product on !vI + 2!vI makes M + zM into a von Neumann algebra. 0

We will now explain the relationship between Connes orientations and dynamical correspondences, and we begin with the following: 6.17. Lemma. Let I be a Cannes orientatzon on a lEW-algebra Ai. If a E !vI, then there exzsts a umque skew order denvation 6 zn I (5a ). Proof. Choose 61 E I(5a ). By Definition 6.8 (iii), since (6 a )t = Oa,

DYNAMICAL CORRESPONDENCES IN JB-ALCEBRAS

oi

Hence 5i + 51 = 0, so + 01 define 0 = 01 - ~oz = ~(01 -

203

= Oz for some z E Z(M) (Lemma 6.5). Now Then ot = -0, so 0 is skew. Also

oi).

If 0' is an arbitrary skew order derivation such that 0' E I(5a ), then 0 - 0' is both central and skew. By Lemma 6.5, a central order derivation is selfadjoint. Thus 5 - 5' is both skew and se!.f-adjoint, so 5 - 5' = o. Hence 5 is the unique skew order derivation in I(5 a ). 0 The concept of a Connes orientation is defined for JBW-algebras, while the concept of a dynamical correspondence is defined for all JB-algebras. However, in the context of JBW-algebras, they are equivalent, as we now show. In this connection we shall need the Kleinecke-Shirokov theorem, which says that if P and Q are bounded operators on a Banach space and their commutator C = [P, QJ commutes with P (or Q), then C is a quasi-nilpotent, i.e., Ilcnlll/n ---+ 0 when n ---+ 00 (cf. [63, p. 128]). (In this reference the theorem is stated for Hilbert space operators, but the proof works equally well for operators on a general Banach space.)

6.18. Theorem. If /1,1 zs a JEW-algebra, then there zs a 1-1 correspondence between Connes orzentatwns on /1,1 and dynamzcal correspondences on lvI, and any dynamzcal correspondence on M is complete. If I is a Connes onentatwn on iIi[, then the assocwted dynamzcal correspondence zs the map a f--+ 1/Ja E D( M), where Wa zs the unique skew adjomt order denvation such that (6.21)

t/Ja

E

I(5 a) for all a

E

M.

For each dynamzcal correspondence 1/J, the assocwted Connes onentatwn I is gwen by (6.22)

where 6x and Oix are defined wzth respect to the W*-pTOduct correspondmg to 1/J (as m (6.14)). Proof. Let I be a Connes orientation on !v1. Let a E M. Denote the unique skew order derivation in I(5u) by 1/Ja for each a E M. Clearly 'IjJ : a f--+ 1/Ja is a linear map from Minto D(lvI). Let a, b E M. By Definition 6.8 (i),(ii),

204

6.

DYNAMICAL CORRESPONDENCES

Thus for some z E Z(M),

Since 1/;a and 1/;b are skew, then [1/;a, 1/;b](l) = O. Also [6 a, 6b](1) = a 0 b boa = O. Hence z = 6z (1) = O. Thus [1/;a,1/;b] = -[6 a, 6b], so 1/; satisfies condition (i) of Definition 6.10. To show that 1/; also satisfies condition (ii) of Definition 6.10, we first observe that for all a E M, by Definition 6.8 (ii),

Therefore there exists z E Z(M) such that [1/;a,6 a] = 6z . Thus 6z is a commutator of two bounded linear operators on /\1, and it commutes with each of them, so it follows from the Kleinecke-Shirokov Theorem that 6z is quasi-nilpotent, i.e., lim

116:'

But 6~(1) = zn for all n. Since IlzI1 2", so

IIa 21

n-+(X)

Il

lln

=

= O.

IIal1 2 for

all a E M, then

Ilz2" I

=

hence z = O. Thus [1/;a,6 a] = O. Now it follows from Lemma 6.12 that 61jJ"a = 0, and hence also 1/;aa = 0, so we have verified condition (ii) of Definition 6.10. Thus 1/; is a dynamical correspondence. Now let 1/; be any dynamical correspondence on !'vI, and let (a, b) f--+ ab be the associated W*-product on M + ~M (Corollary 6.16). By Proposition 6.9, there is a Connes orientation I on !vI given by (6.22). With this we have from each given Connes orientation constructed a dynamical correspondence, and from each given dynamical correspondence constructed a Connes orientation. To show that these two constructions are inverses, start with a Connes orientation I on lvI. Let 1/; be the dynamical correspondence constructed from l, so that l(la) = ~a holds for all a E M. Note also that this equation implies l(~a) = l2(la) = -Ja for all a E M. From 1/;, construct the corresponding W*-product (a, b) f--+ ab on lvI +~lv[ as in Theorem 6.15, so that 1/;a = 6ia for all a E M (cf. (6.20)). Then for a, b EM,

Thus we have I(Jx ) = "SiX for all x E M + lM. By (6.22) this shows that I is the same as the Connes orientation constructed from 1/;.

COl\lPARING NOTIONS OF ORIENTATION

205

Now let 'Ij; be any dynamical correspondence on M, construct the corresponding \,y*-product (a, b) f---> ab on lYl + ~M as in Theorem 6.15, so that 'Ij;" = 6ia for all a EM. Let I be the Connes orientation defined by (6.22). Then for a E M, we have ;j;a = 5ia = 1(5a ), so 'Ij; satisfies (6.21). Thus 'Ij; coincides with the dynamical orientation constructed from the Connes orientation I. Finally, we prove that every dynamical correspondence 'Ij; on M is complete. For the W*-product on lYl + ~M associated with 'Ij;, we have 'lj;a = 6ia for all a E 1Y1 + ~M, i.e., 'Ij; is the dynamical correspondence associated with this product. Then 'Ij; is complete (cf. Proposition 6.11). 0 6.19. Corollary. Let M be a JEW-algebra. Then lYl ~s Jordan isoto the self-adJomt part of a von Neumann algebra ~ff M admzts a Connes onentatwn. There ~s a 1-1 correspondence of Connes onentatwns on M and W*-products on lvl + ~M. The Cannes onentatwn correspondmg to the W*-product (a, b) f---> ab zs gwen by morph~c

1(6 x ) = 6ix

for all x E M

+ ~M.

Proof. This follows from Theorem 6.18 and Corollary 6.16. 0 Comparing notions of orientation In [AS], two notions of orientation were defined: one for C*-algebra state spaces (A 146), and the other for von Neumann algebras and their normal state spaces, cf. (A 200) and (A 201). Here we will sketch how these are related to our current notion of a dynamical correspondence (and thus also to Connes orientations.) The relationships are easiest to see for the special example of the C*-algebra M of 2 x 2 matrices over C. In this case, the state space is affinely isomorphic to a Euclidean 3-ball (A 119). An orientation in the C* sense consists of an equivalence class of parameterizations, i.e., affine isomorphisms from the standard Euclidean 3-ball onto the state space, with two parameterizations being equivalent if they differ by composition with a rotation. An orientation in the von Neumann sense corresponds to an equivalence class of Cartesian frames for the state space, with two frames being equivalent if one can be rotated into the other. Note in each case there are two equivalence classes; these correspond to the standard multiplication on M, and to the opposite multiplication (a,b) f---> ba. Now let s be a symmetry in M. Then s can be viewed as an affine function on the state space, and will take its maximum and minimum values at antipodal points on the boundary of the 3-ba11. Let 'Ij; be the dynamical correspondence associated with the standard multiplication on M. The dual of the map 'lj;s = ,sis generates a rotation about the axis

206

G

DYNMlIICAL CORRESPONDENCES

determined by these two antipodal points. (See Figure G.l and the aCCOlllpanying remarks.) As s varies, the axis ofrotation varies, and each map 1/J, determines an orientation of the 3-ball. However, as s varies continuously, these rotations will vary continuously, and thus give the same orientation. If we replace the multiplication on M by the opposite one, the effect is to replace 1/Js by -1/Js, and so the associated rotation moves in the opposite direction. Thus in all cases, one can think of the orientatiOli as determming a direction of rotation about diameters of this 3-ball, with consistency conditions (e.g., continuity) as the axis varies. The rotations 1/Js are determined by (and determine) the dynamical correspondence a f--7 'Pa' Now we sketch how this extends to arbitrary C* and von Neumann algebras. In the case of C* -algebra orientations, each pair of equivalent pure states determine a face of the state space which is a 3-ball, and a requirement of continuity is imposed, so that the orientation of these 3- balls is required to vary continuously with the ball (A 146). In the von Neumann algebra case, there may be no pure normal states, and thus no facial 3-balls. Instead one works with "blown up 3-balls" (A 194), which also will appear later in our characterization of von Neumann algebra normal state spaces. One can describe an orientation of such blown up 3-balls, cf. (A 206), in terms of pairs (s, 1/Js), where s is a partial symmetry, andl/;s = Gis is the generator of a "generalized rotation". Thus in all contexts, the notion of orientation can be expressed in terms of maps S f--7 1/Js that are related to (generalized) rotations of (blown up) 3-balls, and then one needs to impose consistency conditions, as specified in the various definitions of orientation. Piecing these orientations together consistently is equivalent to choosing the maps 1/Js so that they can be extended to a dynamical correspondence a f--7 <Pa.

Notes The description of the order derivations on a von Neumann algebra (Theorem 6.2) is in [36]' where Connes introduced the notion of an order derivation. The proof of Theorem 6.2 quickly reduces to the result that derivations on a von Neumann algebra are inller, due to Kadison [79] and Sakai [111]. Part of the history of nonned Jordan algebras is tied to homogeneous self-d ual cones. A cone A + in a vector space A is self-dual if there exists an inner product on A such that a ;::: 0 iff (a I b) ;::: 0 for all b ;::: O. It is homogeneous if the order automorphisms are transitive on the interior of the positive cone. Note that for any lB-algebra A, if a E A is positive and invertible, then Ua l/2 is an automorphism of the interior of A + taking 1 to a, so the group of affine automorphisms of the interior of A + is transitive. Koecher [86] and Vinberg [134]' by rather different methods, showed that the finite dimensional homogeneous self-dual cones are exactly the positive cones of formally real Jordan algebras. As we remarked in the

NOTES

207

notes to Chapter 1, these are exactly the finite dimensional JB-algebras. The book of Faraut and Koninyi [52] discusses in detail finite dimensional homogeneous self-dual cones and their connections with Jordan algebras and with symmetric tube domains. The results in this chapter are related to the general problem of characterizing von I'\eumann algebras as ordered linear spaces. Sakai [109] characterized the L2 -completion (with respect to a trace) of a finite von Neumann algebra by means of self-duality of its positive cone and existence of an "absolute value", which models the left absolute value a f---t (a*a)1/2. Connes, in his order-theoretic characterization of (T-finite von Neumann algebras [36], showed that these algebras are in 1-1 correspondence with cones that are self-dual, "homogeneous", and have an "orientation". (The latter two are new concepts defined ill Connes' paper.) Later on, Bellissard and Iochum showed that the (T-finite JRW-algebras are in 1-1 correspondence with the cones that have the first two of Connes' three properties, i.e., are self-dual and homogeneous [27]. Such cones were studied in greater detail by Iochum [68], who introduced the name "facial homogeneity" for Connes' concept. As remarked at the end of Chapter 5, facial homogeneity is closely related to ellipticity. Homogeneity (in the sense of transitivity of the automorphism group on the interior of the positive cone) is equivalent to facial homogeneity in finite dimensions as shown in the paper [26] by Bellissard, Iochum, and Lima. Once one has Jordan structure, something must be added to get the C*-structure, since the Jordan product does not determine the associative multiplication. Connes' definition of orientation is one way to proceed; dynamical correspondences are another approach, closely related to Connes', and in Chapter 11 we will discuss yet a third (more geometric) approach. The results in this chapter are taken from [12] and [13]; the latter also discusses the relationship between the concepts of orientation in Connes' paper [36] and that of the authors in [10]. The notion of a "Connes orientation" (Definition 6.8) carries Connes' definition of an orientation from the natural cone of a JBW-algebra to the algebra itself. For JBW-algebras, the concept of a dynamical correspondence is closely related to that of a "Connes orientation", as illustrated by Theorem 6.18. However, the notion of a dynamical correspondence also makes sense in the broader context of a general JB-algebra. Crgin and Petersen [58] axiomatize an observable-generator duality in terms of what they call a Hamilton algebra, which is a linear space H equipped with two bilinear operations: a product T and a Lie product a, satisfying certain conditions. Emch [49] discussed "Jordan-Lie" algebras, in which both a Jordan product and a Lie product are given, satisfying certain conditions relating the two products. There is also a discussion of such algebras in the book of Landsman [91]. In each case the focus is on deformations of the algebraic structure that approach a kind of classical limit as a parameter approaches zero.

PART II Convexity and Spectral Theory

7

General Compressions

Vie will now approach the theory of state spaces from a new angle, starting with geometric axioms which are at first very general, but which will later be strengthened to characterize the state spaces of Jordan and C*-algebras. The first part of this program is to establish a satisfactory spectral theory and functional calculus. Here the guiding idea is to replace projections p by "projective units" p = PI determined by "general compressions" P defined by properties similar to those characterizing the compressions Up in (A 116) and Theorem 2.83. The first section of this chapter presents basic results on projections in cones. The second section defines general compressions. The third section contains results on projective units and projective faces. The fourth and last section gives a geometric characterization of projective faces together with some concrete examples in low dimension.

Projections in cones We will first work in a very general setting, assuming only that we have a pair X, Y of two positively generated ordered vector spaces in separating ordered duality under a bilinear form (-,.) (A 3). (An ordered vector space X is said to be positively generated if X = X+ - X+.) Unless otherwise stated we will use words like "continuous", "closed", etc. with reference to the weak topologies defined by the given duality.

7.1. Definition. If F is a subset of a convex set C C X, then the intersection of all closed supporting hyperplanes of C which contain F will be called the tangent space of C at F, and it will be denoted by TancF, or simply by Tan F when there is no need to specify C. If there is no closed supporting hyperplane of C which contains F, then TancF = X by convention. Clearly a tangent space of C is a supporting (affine) subspace, but a supporting subspace is not a tangent space in general. In Fig. 7.1 we have shown one supporting subspace which is a tangent space and one which is not.

212

7

GENERAL COlvIPRESSIONS

Fig. 7.1

By (A 1) a face F of a convex set C C X is semz-exposed if it is the intersection of C and a collection of closed supporting hyperplanes containing F. Thus F is semi-exposed iff

F = CnTancF.

(7.1)

We will use the symbol BO to denote the anmhzlator of a subset B of X in the space Y. We will now also use the symbol Be to denote the posztwe anmhzlator of B. Thus

(7.2)

Be = {y E y+ I (x, y) = 0 for all x E B}.

Similarly we use the same notation with BeY and X, Y interchanged. For each x E B C X+, the set x-1(0) = {y E Y I (x,y) = O} is a closed supporting subspace of Y+, so the positive annihilator

Be = y+ n

n

x-1(0)

xEB

is a semi-exposed face of Y+. We also make the following observation, which we state as a proposition for later reference.

7.2. Proposition. If X, Yare ordered vector spaces m separatmg ordered duaZzty and B c X+, then Bee zs the semz-exposed face generated by B m the cone X+ and Beo zs the tangent space to X+ at B. Proof The closed supporting hyperplanes of X+ are precisely the sets of the form y-l (0) for y E Y+. Thus if B c X+, then

Bee = X+ n (

n

yEBO

y-l(O))

and

B eo =

n

y-l(O).

yEBO

The intersections above are extended over all supporting hyperplanes of X+ which contain B, so the proposition follows from the definition of a semi-exposed face and of a tangent space. 0

213

PROJECTIONS IN CONES

If P : X -> X is a posdwe pro)ectwn, i.e., if P is a linear map such that P(X+) C X+ and p2 = P, then we will use the notation ker+ P = X+ n ker P and im+ P = X+n im P. Now it follows from the simple characterization (A 2) of faces in cones, and from the assumption that X is positively generated, that if P : X -> X is a positive projection, then ker+ P is a face of X+ and im P is a positively generated linear subspace of X. We will now study the tangent spaces of the cone X+ at ker+ P and im+ P, and for simplicity we will omit the subscript X+ and just write Tan (ker+ P) and Tan(im+ P). If P is a continuous positive projection on X, then we denote by P* the (continuous) dual pro)ectwn on Y, defined by (7.3)

(Px, y)

= (x, P*y) when x

E X and y E

Y.

Clearly X and Y may be interchanged in all the statements above.

7.3. Proposition. Let X, Y be posdwely generated ordered vector spaces m separatmg ordered dualdy. If P lS a contmuous positwe projection on X, then Tan(ker+ P)

(7.4)

= (ker+ pta c ker P.

Proof. By Proposition 7.2,

Since (ker P)O = imP* and im P* is positively generated, (ker Pt Hence (ker P)·o

= im+ P* - im+ P* = (ker P)· - (ker Pt.

= (ker P)OO = ker P, which completes the proof.

D

7.4. Corollary. Let X, Y be posltwely generated ordered vector spaces in separating ordered duahty. If P lS a contmuous posltwe pro)ectwn on X, then ker+ P lS a seml-exposed face of X+. Proof. Clearly (7.4) implies X+ n Tan(ker+ P) C ker+ P. The opposite inclusion is trivial, so (7.1) holds with ker+ P in place of F. Thus ker+ P is a semi-exposed face of X+. D The inclusion opposite to the one in (7.4) is not valid in general, but projections for which this is the case form an important class containing the compressions Up in the case of C*-algebras and JB-algebras (as we will show in Theorem 7.12).

7

214

GENERAL COMPRESSIONS

7.5. Definition. Let X, Y be positively generated ordered vector spaces in separating ordered duality. A continuous positive projection P on X will be called smooth if ker P is equal to the tangent space of X+ at ker+ P, i.e., if Tan(ker+ P)

(7.5)

= ker P.

Note that the non-trivial part of the equality (7.5) is the inclusion ker Pc Tan(ker+ P), which by Proposition 7.2 is equivalent to ker P

(7.6)

c

(ker+ p)eo.

We will now show that a smooth projection P is determined by ker+ P and im+ P. (This is not the case for general positive projections. A simple example of a non-smooth projection is the following. Let X = Y = R3 with the standard ordering, and let P : X ----> X be the orthogonal projection onto the (horizontal) line consisting of all points (Xl, X2, X3) such that Xl = X2 and X3 = O. Then ker P is the (vertical) plane consisting of all points such that Xl = ~X2, but the tangent space to ker+ P is only the (vertical) line consisting of all points such that Xl = X2 = 0.) 7.6. Proposition. Let X, Y be posltwely generated ordered vector spaces m separatmg ordered dualzty. If P zs a smooth projectwn on X and R zs another contmuous posztwe plOJectwn on X such that im+ R = im+ P and ker+ R = ker+ P, then R = P.

Proof.

By (7.5) and (7.4) (the latter with R in place of P), ker P

= Tan (ker+ P) = Tan(ker+ R)

C ker R.

Since im+ P = im+ R, and the sets im P and im R are both positively generated, im P = im R. From the two relations ker P C ker Rand im P = im R, we conclude that P = R. D Smoothness of P will also imply that ker+ P and im+ P dualize properly under positive annihilators. The precise meaning of this statement is explained in our next proposition. 7.7. Proposition. Let X, Y be positwely generated ordered vector spaces m separating ordered dualzty. Let P be a contmuous posztive proJectwn on X and conszder the equatwns im+ Pt

= ker+ P*,

(7.7) (7.8)

(ker+ Pt =:> im+ P*,

(7.9)

(ker+ Pt = im+ P*.

215

PROJECTIONS IN CONES

Of these, (7. 7) and (7.8) hold generally, and (7. 9) holds zff P zs smooth. Proof. Since im P is positively generated, (im+ P)O ker P*, from which (7.7) follows. Generally im P*

= (ker Pt

= (im P)O =

C (ker+ Pt,

so im+ P* C (ker+ P)·, which proves (7.8). To prove the last statement of the proposition, we assume that P is smooth. By (7.5), (ker+ P)·o = ker P. The set (ker+ Pl· is trivially contained in its bi-annihilator, so (ker+

pr

C

(ker Pt

= im P*.

Hence (ker+ Pl· C im+ P*. The opposite relation is (7.8), so the equation (7.9) is satisfied. Finally, (7.9) implies (7.5), so the equation (7.9) is only satisfied if P is smooth. 0 We saw in Corollary 7.4 that ker+ P was a semi-exposed face of X+ for every continuous positive projection P on X. The corresponding statement for im+ P is false. In fact, im+ P need not be a face of X+, and if it is a face, it need not be semi-exposed. Examples in which im+ P is not a face of X+ are easily found already in R 2 , where it suffices to consider the orthogonal projection P onto a line through the origin and some interior point of the positive cone X+. An example in which im+ P is a non-semiexposed face of the cone X+, can be obtained by ordering R 3 by a positive cone X+ generated by a convex base with a non-exposed extreme point. A standard example of a convex set with non-exposed extreme points is a square with two circular half-disks attached at opposite edges. (Such a set may be thought of as a sports stadium with a soccer field inside the running tracks. Then the non-exposed extreme points will be located at the four corner flags of the soccer field). Now the orthogonal projection P onto the line through the origin and such a non-exposed extreme point of the base will give the desired example. It turns out that smoothness of P* is a necessary and sufficient condition for im+ P to be a semi-exposed face of X+, as we will now show. 7.8. Corollary. Let X, Y be posztively generated ordered vector spaces in separatzng ordered dualzty, and let P be a contznuous positive pT'OJectwn on X. The dual prOjection P* zs smooth zff the cone im + P is a semiexposed face of X+.

Proof. By Proposition 7.2, im+ P is a semi-exposed face of X+ iff

216

7.

GENERAL COMPRESSIONS

and by (7.7) this is equivalent to im+ P

= (ker+ P*t.

But this equality is (7.9) with P* in place of P, which is satisfied iff P* is smooth. D Recall from (A 4) that two continuous positive projections P, Q on X (or on Y) are said to be complementary (and Q is said to be a complement of P and vice versa) if

(7.10) Note that since X is positively generated, (7.10) implies that PQ = QP = O. Recall also that the two projections P, Q are said to be stTOngly complementary if (7.10) holds with the plus signs removed, whIch is equivalent to

(7.11)

PQ = QP = 0,

P+Q

=

I.

Thus P admits a strong complement iff I - P ;::: O. A continuous positive projection P on X (or on Y) is to be complemented if there exists a continuous positive projection Q on X such that P, Q are complementary, in which case Q is said to be a complement of P. Also P is said to be b~complemented if there exists a continuous positive projection Q on X such that P, Q are complementary and P*, Q* are also com plementary.

Fig. 7.2

PROJECTIONS IN CONES

217

A pair of projections which are complementary, but not strongly complementary, will satisfy the first, but not the second, of the two equalities in (7.11). An example of such a pair is shown in Fig. 7.2 where X = R3 and X+ is a circular cone.

7.9. Proposition. Let X, Y be posltwely generated ordered vector spaces m separating ordered dualdy, and let P be a contmuous posltwe proJectwn on X (or on Y). If P lS complemented, then the dual projectwn P* is smooth, and If P has a smooth complement Q, then Q lS the unique complement of P.

Proof. If P has a complement Q, then the cone im+ P = ker+Q is a semi-exposed face of X+ (Corollary 7.4), so the dual projection P* must be smooth (Corollary 7.8). If P has a smooth complement Q, then an arbitrary complement R of P must satisfy ker+ R = im+ P = ker+Q and im+ R = ker+ P = im+Q. By Proposition 7.6 this implies R = Q. Thus, Q is the unique complement of

P.

D

Complementary projections are not unique in general. One can obtain an example of a positive projection with infinitely many complements by replacing the circular base of the cone in Fig. 7.2 by the convex set in Fig 7.3 which has a vertex at im+ P = ker+Q and also at im+Q = ker+ P. Then ker+Q will be a sharp edge of the cone X+, and by tilting the plane ker+Q about this edge we obtain infinitely many positive projections which are all complements of P. Similarly with P and Q interchanged. (We leave it as an exercise to picture the dual cone {y E X* I (x, y) ;::: O} with the sub cones ker+ P*, im+ P*, ker+Q*, im+Q*, and to show that in this example P, Q are complementary but not bicomplementary.)

im+Q

im+P

Fig. 7.3

The theorem below is a geometric characterization of bicomplementarity in terms of smoothness.

218

7.

GENERAL COMPRESSIONS

7.10. Theorem. Let X, Y be posztwely generated ordered vector spaces m separatmg ordered dualIty. If P and Q are two continuous POSItwe projectIOns on X, then the followmg are equwalent:

(i) P, Q are complementary smooth proJectIOns, (ii) P*, Q* are complementary smooth proJectIOns, (iii) P, Q are bIcomplementary prOJectIOns. Proof. It suffices to prove (i) ¢? (iii) since P*, Q* are bicomplementary iff P, Q are bicomplementary. (i) =} (iii) Assuming (i) and using Proposition 7.7, \ve get

Interchanging P and Q, we also get ker+Q* = im+ P*. Hence we have (iii) . (iii) =} (i) Astmming (iii) we have im+ P = ker+Q. By Corollary 7.4 ker+Q* is semi-exposed, therefore im+ P* is semi-exposed. Now it follows from Corollary 7.8 (with the roles of P and P* interchanged) that P is smooth. Similarly, Q is smooth. Hence we have (i). D

7.11. Corollary. Let X, Y be posztwely generated ordered vector spaces m separatmg ordered duahty. If P IS a blcomplemented contmuous posltwe projectIOn on X, then P has a umque complement. Proof. By Theorem 7.10, P has a smooth complement Q, and by Proposition 7.9, Q is the unique complement of P. D Our next result, concerning C*-algebras and JB-algebras, can be found in (A 116) and Theorem 2.83. But it is actually only the easy argument in the first part of the proofs of these results that is needed, so for the reader's convenience we will give this argument.

7.12. Theorem. Let X be the self-adJomt part of a C*-algebra (or von Neumann algebra) and let Y be the self-adjOInt part of ds dual (respectwely Its predual). Then the compreSSIOn Up determmed by a projectIOn p E X IS a smooth projectIOn on X, the dual map U1: IS a smooth projectIOn on Y, and the two maps Up and Uq where q = pi (= 1 - p) are blcomplementary prOjectIOns on X. The same IS true when X IS a lBalgebra (or lEW-algebra) and Y IS zts dual (respectively ItS predual). Thus m all these cases Up and Uq are blcomplementary projectwns. Proof. Let p be a projection in the C*-algebra X. We will first show that Up is a smooth projection on X by proving (7.6). Thus we must shoW that if a E ker Up, then w(a) = 0 for each wE (ker+Up)-.

PROJECTIONS IN CONES

219

Let a E ker Up, i.e., pap = 0, and W E (ker+Up)·, i.e., W ;::;. 0 and = 0 on ker+Up. Clearly the projection q = pi is in ker+Up, so w(q) = o. By (A 74) ((iii) =? (vi)), w = p. W· p. Thus w(a) = w(pap) = 0 as desired.

W

Next we observe that by (A 73) ((iii).;=;.(iv)), ker+Uq = im+Up , and likewise with p and q interchanged. Thus Up and Uq are complementary smooth projections. Thus the statement (i) of Theorem 7.10 holds when P = Up and Q = Uq . Then the statement (ii) also holds, so U; and U; are complementary smooth projections. This completes the proof in the C*-algebra case. (The same proof works in the von Neumann algebra case, now with w a normal linear functional). In the JB-algebra (and JBW-algebra) cases we can use the same argument with reference to Proposition 1.41 instead of (A 74), and Proposition 1.38 instead of (A 73). 0 We will now establish a direct sum decomposition which generalizes the Pierce decomposition of Jordan algebras [67, §2.6]' and we begin by a lemma which will also be needed later. Here we shall need the concept of a direct sum of two wedges in a linear space. Recall that a wedge W is a (not necessarily proper) cone, so it is defined by the relations W + ~V c W and AW c W for all A ;::;. 0 (cf. [AS, p. 2]). We will say a wedge W is the direct sum of two subwedges WI and W 2 iff each element of W can be uniquely decomposed as a sum of an element of WI and an element of W 2 • We will use the notation EBw for direct sums of wedges (reserving EB for direct sums of subspaces). 7.13. Lemma. Let X, Y be posdwely generated ordered vector spaces in separatmg ordered duallty. If F and G are subcones of X+ such that

(7.12)

X+

c F EBw TanG,

X+

c G EBw TanF,

and N := Tan F n Tan G, then (7.13)

Tan F

= lin F EB N, Tan G = lin G EB N, X = lin FEB lin G EB N.

Proof. Clearly lin F

+N c

Tan F

+N c

Tan F.

To prove the converse relation, we consider an element x E Tan F. Since X is positively generated, x = Xl - X2 where XI,X2 E X+. By (7.12) we also have decompositions Xi = ai + bi where ai E F and bi E Tan G for i = 1,2. Now al - a2 E lin F and b l - b2 E Tan G. But we also have

7.

220

GENERAL COMPRESSIONS

Thus Tan F = lin F + N. Similarly Tan G = lin G + N. Observe now that lin F n Tan(G) = {O}. In fact, if x E lin F n Tan(G), then x = Xl - X2 where Xl, X2 E F, and the equation X2 + X = Xl + 0 provides two decompositions of the element Xl of X+ as a sum of an element of F and an element of Tan(G), so X2 = Xl and X = O. Similarly lin G n Tan(F) = {O}. N ow also lin F n N = {O} and lin G n N = {O}, so Tan F = lin F EEl N and Tan G = lin G EEl N. Since X is positively generated, it follows from (7.12) that X = linF +Tan(G). In fact, X = lin F EEl Tan G, and then also X = lin F EEl lin G EEl N. D

7.14. Theorem. Let X, Y be posztively generated ordered vector spaces m separatmg ordered dualdy, let P, Q be a pazr of bicomplementary contmuous posdwe proJectwns on X, and set F = im+ P, G = im+Q and N = Tan F n Tan G. Then (7.12) holds, and P and Q are the proJectwns onto lin F and lin G determmed by the dzrect sum (7.14)

X = lin F EEl lin G EEl N.

Thus P + Q zs a projection wzth kernel N onto the subspace lin F EEl lin G. Proof. By Theorem 7.10, P is a smooth projection. Hence (7.15)

ker P = Tan(ker+ P) = Tan(im+Q).

Let X be an element of X+. Then we can decompose X as the sum of the element Px in the cone im+ P and the element X - Px in the subspace ker P = Tan(im+Q). If X = Xl + X2 is an arbitrary decomposition with Xl E im+ P and X2 E Tan(im+Q), then X2 E ker P, so Px = Xl. Thus the decomposition is unique. With this we have shown the first inclusion in (7.12). The second inclusion follows by interchanging P and Q. Since im P is positively generated, im P = F - F = lin F, and by (7.15) and (7.13) ker P = Tan G = lin G EEl N. Thus P is the projection onto lin F determined by the direct sum in (7.14). Similarly Q is the projection onto lin G determined by this direct sum. This completes the proof. D We will now generalize the uniqueness theorem for the conditional expectations E = Up + Up' in a von Neumann algebra (A 117) to the geometric context of bicomplementary projections in cones, which by Theorem 7.12 includes compressions in general C*-algebras, von Neumann algebras, JB-algebras and JBW-algebras.

221

PROJECTIONS IN CONES

7.15. Theorem. If X, Yare posdwely generated ordered vector spaces m separating ordered dualzty and P, Q are bzcomplementary contmuous posztwe projections on X! then the map E = P + Q zs the unzque contmuous posztive projectwn onto the subspace im (P + Q) = im PEEl im Q. Proof. Note first that since PQ = QP = 0, E is a positive projection onto im (P + Q) = im P EEl im Q and ker E = ker P n ker Q. To prove the uniqueness, we consider an arbitrary continuous positive projection Tonto im (P + Q). Thus imE = imT, so ET = T. We will prove that we also have ker EckerT, which will imply ker E = kerT and complete the proof. Observe that it suffices to prove (7.16)

ker P

c

ker PT

and

ker Q

c

ker QT,

since this will imply ker E = ker P n ker Q c ker PT n ker QT

= ker (PT + QT) = ker ET = ker T. Assume x E ker P. To show that x E ker PT, it suffices to show that (PTx, y) = 0 for all y E Y+ (since Y is positively generated). Let y E Y+ be arbitrary. Note that T* P*y E (ker+ P)·. In fact, for each a E ker+ P = im+Q C im T,

(a, T* P*y)

= (Ta, P*y) = (a, P*y) = (Pa, y) = o.

But since P is smooth, x E ker P implies that x E Tan (ker+ P) (ker+ P)·o, so we get

(PTx, y)

= (x, T* P*y) = 0

as desired. Thus ker Pc ker PT. Similarly ker Q C ker QT. With this we have proved (7.16) and have completed the proof of the theorem. D We will briefly discuss the geometric idea behind the above proof. The key to the argument is smoothness. Since P and Q are smooth projections, the cones ker+ P and ker+Q determine the subspaces ker P = Tan (ker+ P) and kerQ = Tan(ker+Q), and then also their intersection, which is shown to be the kernel not only of the projection E = P + Q but of any positive projection onto the subspace M = im (P + Q). The fact that the uniqueness of E depends on the smoothness of P and Q, can be easily seen in the example given in Fig. 7.2. Si~ce E is the projection onto M with kernel equal to the intersection of the tangent planes to the cone X* at the two opposite rays ker+ P and ker+Q, E must

222

7

GENERAL COMPRESSIONS

be the orthogonal projection onto lvI, and this is the only projection onto M which maps X+ into itself. The situation is different in the example obtained by replacing the circular cone in Fig. 7.2 by a cone with a base as in Fig. 7.3, having two antipodal vertices. Now we can find infinitely many pairs of (non-smooth) positive projections P, Q with ker+ P and ker+Q equal to the rays through these vertices (passing from one such pair to another by tilting the planes ker P and ker Q about the rays ker+ P and ker+ Q). Two distinct pairs will determine distinct intersections ker Pnker Q, and then distinct positive projections of X onto AI. Thus there is no uniqueness in this case. Definition of general compressions

We will now specialize the discussion to a pair of an order unit space

A and a base norm space V in separating order and norm duality (A 21). Note that order unit spaces and base norm spaces are positively generated, cf. (A 13) and (A 26), and so the previous results in this chapter are applicable. As in the previous section, we will study positive projections that are continuous in the weak topologies defined on A or V by the given duality. But we will no longer omit the words "weak" or "weakly" when referring to these topologies since we now also have to deal with the norm topologies. We will say that a positive projection on A or V is normalzzed if it is either zero or has norm 1, or which is equivalent, if it has norm ::::; 1. Note that if P is a normalized positive projection on A, then the dual projection P* is a normalized positive projection on V, and vice versa. Recall from (A 22) that if P is a weakly continuous positive projection on A, then (7.17)

IIP* pil

= (PI, p) for all p E V+.

We also make two observations, which we state as lemmas for later reference. 7.16. Lemma. Let A, V be an order umt space and a base norm space m separatmg order and norm dualzty. A posdwe pTOJcctwn P on A zs normalzzed zJJ PI ::::; 1.

Proof. To prove the "if" part of the lemma, we assume PI ::::; 1 and consider a E A with Iiall ::::; 1, i.e., -1 ::::; a ::::; 1, d. (A 13). Since P is positive, -PI::::; Pa ::::; PI; hence -1 <::: Pa ::::; 1, so IIPal1 ::::; 1 as desired. The "only if" part is trivial, for if IIPII ::::; 1, then IIP111 ::::; 1, which implies PI ::::; 1. D 7.17. Lemma. Let A, V be an order unit space and a base norm space m separatmg order and norm dualzty, and let K be the distinguzshed

DEFINITION OF GENERAL COMPRESSIONS

223

base of V. If P and Q are weakly contmuous posztzve proJectwns on A, then the followzng are equzvalent:

(i) PI + Ql = 1, (ii) IIP* p + Q* pll = (iii) (p*

+ Q*)(K) c

Ilpll

for all p E V+,

K.

Proof. Use the fact that

Ilpll =

(I,P! for p E V+. D

7.18. Lemma. Let A, V be an order umt space and a base norm space in separatzng order and norm dualdy, and let P, Q be two complementary weakly contmuous posdzve proJectwns on A. If one of the proJectwns P, Q is normalned, then the other one zs also normaizzed and P, Q wzll satisfy the requzrements (z), (n), (m) of Lemma 7.17.

Proof. Assume P is normalized. By Lemma 7.16, 1 - PI ::;:. 0; hence 1 - PI E ker+ P = im+Q, so

Q(1 - PI) = 1 - Pl. But PI E im+ P = ker+Q, so QPl = O. Therefore the equation above reduces to Ql = 1 - PI, which gives the desired statement (i), and then also (ii) and (iii). By statement (i), Ql .:::: 1, so by Lemma 7.16, Q is normalized. D We will now give the general definition of a concept which we have already met in the context of JB-algebras (equation (l.57)).

7.19. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality, and let P be a normalized weakly continuous positive projection on A. We will say that the dual projection P* is neutral if the following implication holds for p E V+ : (7.18)

IIP* pll = Ilpll

=}

P* p

= p.

Remark. The term "neutral" relates to physical filters which may let through one part of an incident beam and stop another part, for example, optical filters which are transparent for certain colors but not for others. The equation (7.18) represents the situation in which a beam can pass through with intensity undiminished only when the beam is unchanged, or otherwise stated, when the filter is completely neutral to the beam. 7.20. Proposition. Let A, V be an order umt space and a base norm space in separatmg order and norm dualzty, and let P be a normahzed weakly continuous positive projection on A. If P* is a neutral prOJection,

224

7

GENERAL COMPRESSIONS

then P IS smooth. If P IS a smooth prOjectIOn wIth smooth complement, then P* IS neutral. Proof. Assume first that P* is a neutral projection on V. We will show that P is a smooth projection on A by verifying (7.9). Let P E (ker+ P)-. Since 1 - PI E ker+ P, then (1 - PI, p) = O. This is equivalent to (1, p) = (1, P* p), and in turn to Ilpll = I[P* pil. Since P is neutral, we must have p = P* p. Hence (ker+ P)- C im+ P*. The opposite inclusion is generally satisfied (cf. (7.8)), so (7.9) is satisfied. Assume next that P is a smooth projection with a smooth complementary projection Q. We will show that P* is neutral by verifying (7.18). Let p E V+ with IIP*pll = Ilpll, or which is equivalent, (Pl,p) = (l,p). Since P is normalized, PI + Ql = 1 (Lemma 7.18). Hence (Ql, p) = 0, or which is equivalent, IIQ* pll = O. Thus Q* p = 0, i.e., p E ker+Q*. By Theorem 7.10, P* and Q* are complementary, so p E im+ P*, i.e., P*p = p as desired. 0

7.21. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm dualIty, and let P and Q be two normalIZed weakly contmuous posItIVe projectIOns on A. Then P and Q are blcomplementary 7,ff they are complementary and P*, Q* are both neutral. Proof. Application of Theorem 7.10 and Proposition 7.20. 0

7.22. Definition. If A, V is a pair of an order unit space and a base norm space V in separating order and norm duality, then a bicomplemented normalized weakly-continuous positive projection P on A will be called a compreSSIOn. Note that by Lemma 7.18, the complement Q of a compression P is normalized, so it is also a compression. Note also that by our previous discussion, a normalized weakly continuous positive projection P is a compression iff there is another weakly continuous positive projection Q such that anyone of the three conditions in Theorem 7.10 are satisfied, or there is a normalized weakly continuous positive projection Q such that the condition in Proposition 7.21 is satisfied.

7.23. Proposition. Let A be the self-adJomt part of a von Neumann algebra and let V be the self-adjOInt part of ItS predual, or more generally, let A be a lEW-algebra and let V be Its predual. Then A, V IS a pmr of an order umt space and a base norm space In separatmg order and norm dualIty, and the (abstract) compreSSIOns defined by Defimtion 7.22 are preCIsely the (concrete) compreSSIOns Up assocwted WIth projections pEA. Proof. In the von Neumann algebra case, the duality statement for A and V follows from (A 69), (A 94) and (A 96), and the characterization of

DEFINITION OF GENERAL COMPRESSIONS

225

the abstract compressions follows from (A 116). In the JBW-algebra case, the duality statement for A and V follows from Theorem 1.11 and Corollary 2.60, and the characterization of the abstract compressions follows from Theorem 2.83. 0 The following description of compressions will be used later in some physical remarks.

7.24. Proposition. Let A, V be a pmr of an order umt and base norm space m separatmg order and norm dualzty. Let : V -+ V and ' : V -+ V be weakly contmuous pos2twe lmear proJectwns w2th dual maps P = * : A -+ A and P' = '* : A -+ A. Then P, P' are complementary compresswns 2ff

(i) IIwll + 11'wll = Ilwll for all w E V+! (ii) and ' are neutral, (iii) ker+ Pc im+ P' and ker+ P' C im+ P. Proof. If P and P' are complementary compressions, then (i) follows from Lemma 7.18, (ii) follows from Proposition 7.21, and (iii) follows from the definition of complementary compressions. Conversely, assume (i), (ii), (iii) hold. By (i) we have IIwll S; 1 for w E K, where K is the distinguished base of V. Since the unit ball of V is co(K U -K), it follows that IIII S; 1, and similarly II'II S; 1. Since A, V are in separating order and norm duality, then IIPII = II*II S; 1, and similarly IIP'II S; 1, so P and P' are normalized. We next show that for wE

V+,

(7.19)

w = w

{=}

' w = 0.

= w, then by (i), 11'wll = 0, so 'w = 0. If 'w = 0, then IIwll = Ilwll, so by (ii), w = w, which completes the proof of (7.19). Similarly the same conclusion with and ' interchanged holds. Thus and ' are complementary projections.

If w

Finally we show that P and P' are complementary, i.e., that equality holds in (iii). Since by assumption is a projection, then for w E V+, (w) = w, so '(w) = (by (7.19), applied with w in place of w). Thus ' = 0, and similarly ' = 0. Dualizing gives P P' = P' P = 0. Therefore if P' a = a, applying P to both sides of this equality gives Pa = 0, proving that im+ P' C ker+ P. Similarly, im+ P C ker+ P'. The reverse inclusions follow from condition (iii) of this proposition, which completes the proof that P, P' are complementary compressions. 0

°

If A is an order unit space in separating order and norm duality with a base norm space V with distinguished base

(7.20)

K={PEV+lllpll=(1,p)=l},

226

7.

GENERAL COMPRESSIONS

then the map a f-> a (where a(p) = (a, p) for p E V) is an order and norm preserving isomorphism from A onto a point-separating subspace of the space V* = Ab(K) of all bounded affine functions on K, equipped with the pointwise ordering and the uniform norm (A 11). For simplicity We will identify elements a of A with their representing affine functions on K. Thus if a, b E K and F c K, then we will write a ::::: b on F instead of (a,p) ::::: (b,p) for all p E K, etc. As before we wili. use the notation for the positive part of the norm closed unit ball of A.

a

Ai

7.25. Lemma. Let A, V be an order umt space and a base norm space m separating order and norm duality. If P lS a compresswn on A, then PI lS the greatest element of whzch vamshes on ker+ p', and

Ai

(7.21)

im+ P'

= {p

E

V+ I (PI, p)

= (1, p) }.

Proof. Let P be a compression on A. If p E ker+P', then (Pl,p) = (1, P' p) = 0, so PI vanishes on ker+ P*. Consider now an arbitrary element a of Ai which vanishes on ker+ P*. Thus 0 ::; a ::; 1 and a E (ker+ P*)·. By equation (7.9) (with P' in place of P), a E im+ P. Hence a = Pa ::; PI. Thus PI is the greatest element of Ai which vanishes on ker+ p'. The equation (PI, p) = (1, p) is equivalent to IIP* pil = Ilpll, and since P is neutral, this gives (7.21). D 7.26. Proposition. Let A, V be an order unit space and a base norm space m separating order and norm dualzty, and let P be a compresswn on A. Then the two cones im + P, ker+ Pare semz-exposed faces of A +, the two cones im+ P*, ker+ P* are seml-exposed faces of V+, and each one of these four cones determmes the compresswn P. Proof. The cones in question are semi-exposed by Corollary 7.4 and the existence of a complementary positive projection. By Lemma 7.25, ker+ P* determines PI, and through (7.21) also im+ P*. By Proposition 7.6, ker+ P* and im+ P* together determine P*, and hence also P. With this, we have shown that ker+ P* determines P. When im+ P' is given, we also know the cone ker+Q* = im+ P* where Q is the complement of P. Thus it follows from what we have just proved that im+ P* determines Q, and then also its complement P (Corollary 7.11). It follows from Proposition 7.7 that im+ P determines ker+ P* and ker+ P determines im+ P*. Thus it follows from what we have shown above that each of the two cones im+ P and ker+ P determines P. D Remark. Proposition 7.26 fails if complementary compressions are replaced by bicomplementary weakly continuous positive projections. Thus

PROJECTIVE UNITS AND PROJECTIVE FACES

227

the requirement that the projections be normalized is crucial. To see this, observe that if V+ is a circular cone, for each pair of distinct faces F, G of V+ there are bicomplementary projections P, Q with im+ P* = F and im+Q* = G.

Projective units and projective faces We will continue the study of compressions defined in the context of a separating order and norm duality of an order unit space A and a base norm space V with distinguished base K. In the motivating example of a von Neumann algebra and its predual, each compression is of the form P = Up where p is a projection in the algebra, uniquely determined by the equation p = PI, cf. Proposition 7.23. Starting out from this equation, we will define a class of elements pEA which will play the same role as the projections p in the algebra, and a class of faces F c K which will play the same role as their associated norm closed faces Fp = K n im P* of the normal state space (cf. (A 110)).

7.27. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality. If P is a compression on A, then the element p = PI of is called its associated pmjectwe unit, and the face F = K n im P* is called its associated pro]ectwe face.

At

In the case of von Neumann algebras and lBW-algebras, every norm closed face of the normal state space is projective, cf. (A 109) and Theorem 5.32. This is certainly not true in general. In fact, most convex sets K encountered in elementary geometry have no projective faces at all (other than K and 0). For simplexes and smooth strictly convex sets, all faces are projective. (They are split faces in the former case, and just the extreme points in the latter.) Other examples of 3-dimensional convex sets with projective faces, are not so easy to come by. But we will soon give a geometric characterization of projective faces (Theorem 7.52), which we will use to give more examples. (Examples are pictured in Fig. 7.6 and Fig. 7.7, where F and G are complementary projective faces, and in Fig. 8.1, where all faces are projective.) We will first prove some elementary properties of projective units and projective faces.

7.28. Proposition. Let A, V be an order umt space and a base norm space m sepamtmg order and norm duaizty, and let a E A. Each projectwe unit p zs assoczated with a umque compresswn P. Also each pro]ectwe face F is associated with a unique compresswn P, and then Pa ;:::: 0 iff a ;:::: 0 on F. Proof. Assume first p = PI where P is a compression on A. By (7.21) im+ P* is determined by p, and thus by Proposition 7.26 the compression P is itself determined by p.

228

7

GENERAL COl\lPRESSIONS

Assume next F = ]{ n im P* where P is a compression on A. Then im+ P* is the cone generated by F (i.e., im+ P* consists of all AW where A E R + and w E F). Thus im + P* is determined by F, and then P is determined by F. If a ;::: 0 on F, then a;::: 0 on the cone im+ P* generated by F, hence (Pa, w) = (a, P*w) ;::: 0 for all w E K Thus a ;::: 0 on F implies Pa ;::: O. Conversely if Pa ;::: 0, then (a, w) = (a, P*w) ;::: 0 for all w E F C im+ P*. Thus Pa ;::: 0 implies a ;::: 0 on F. D

If P is a compression with associated projective unit p and associated projective face F, then it follows from the proposition above that each one of P, p, F determines the others, and we will say that they are mutually assocwted with each other. We will also denote the (unique) compression which is complementary to P by pi, and we will denote the projective unit and the projective face associated with pi by pi and F' respectively Also we will say that pi, pi, F' are the complements of P, p, F respectively. Note that by equation (i) of Lemma 7.17, pi = 1 - p, so this notation is compatible with our previous use of the prime to denote the (orthogonal) complement of projections. Note also that by (7.21), the projective faces F and F' are determined by p through the explicit formula (7.22)

F = {w E]{ I (p,w) = I},

F' = {w E]{ I (p,w) = O},

and conversely, by Lemma 7.25, that the projective unit p is determined by F or F' through the formula (7.23)

p=

V{a E At

I

a = 0 on F'} =

1\ {a E At

I

a = Ion F},

where the second equality follows from the first by replacing p by pi, F by F ' , and a by 1 - a. Thus p is the least element of At which takes the value 1 on F, and p is the greatest element of At which takes the value o on F'. Hence p is the unique element of At which takes the value 1 on F and also takes the value 0 on F'. By Definition 7.27, we have the following formula for the projective face associated with a compression: (7.24)

F = {w E ]{ I P* w = w},

F' = {w E ]{ I P* w = O}.

From this it follows that for each a E A we have (Pa, w) = 0 when wE F'. Thus

=

(a, w) when

wE F and (Pa, w)

(7.25)

Pa = a on F,

Pa = 0 on p'.

Recall from (A 2) that if a E A +, then the face generated by a in A +, denoted by face (a), consists of all b E A + such that b ::::; Aa for some

PROJECTIVE UNITS AND PROJECTIVE FACES

229

>. E R +, and that the semi-exposed face generated by a in A + is equal to {a}·· (Proposition 7.2).

7.29. Lemma. Let A, V be an order umt space and a base norm space m separatmg order and norm dualdy and conszder a compresswn P wzth

assocwted projectzve umt p. If a E A +, then the followmg are equzvalent: (i) a E im+ P, (ii) a::: Iiall, (iii) a E face(p), (iv) aE{p}··.

Proof. (i) =} (ii) If a E im+ P, then a::: IIal11 and a = Pa ::: IiallP1 = Iialip. (ii)=}(iii) and (iii)=}(iv) are trivial. (iv)=}(i) Assume (iv). By Proposition 7.26, im + P is a semi-exposed face of A +. Hence a E {p} •• C (im+ P)·· = im+ P. 0 By (A 75) and Proposition 1.40, an element of a von Neumann algebra or a JBW-algebra is an extreme point of the closed unit ball iff it is a projection. Our next proposition generalizes the "if' -part of this result. 7.30. Proposition. Let A, V be an order umt space and a base norm space m separating order and norm dualdy. If p zs a projective umt, then p is an extreme pmnt of At.

Proof. Let p be the projective unit associated with a compression P. Clearly p = PI E At. Assume (7.26)

p

= >.a + (1 - >.)b

with a, bEAt and 0 < >. < 1. Then a::: >.-lp and b::: (1- >.)-lp. Hence a, b E face(p) = im+ P, so a = Pa ::: PI = p and b = Pb ::: PI = p. By (7.26), a = b = p, so p is an extreme point of At. 0 In the proposition below (and in the sequel) we will use the standard square bracket notation for closed order intervals, so [a, b] will denote the set of all x E A such that a ::: x ::: b. 7.31. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm dualzty. If P zs a compresswn on A wzth associated proJectzve umt p and assocwted proJectzve face F, then

(i) [-p,p]=A1nimP, (ii) co(F u -F) = VI n imP*. Moreover, P( A) = im P is an order umt space wzth distinguzshed order umt p and P* (V) = im P* is a base norm space wzth dzstinguzshed base

7

230

GENERAL COMPRESSIONS

F. These two spaces are zn sepamtmg order and norm dualzty under the ordenng, norm and bzlmear form (".) relatwzzed from A and V. Proof. (i) If a E [-p,p], then it follows from Lemma 7.29 ((iii)=? (i)) that a + p E im P. Hence also a E im P, and since Iiall ::; Ilpll ::; 1, then a E At n im P. Conversely, if a E At n imP, then -1 ::; a ::; 1, so -p ::; Pa ::; p, and also Pa = a. (ii) The set at the left-hand side of (ii) is trivially contained in the set at the right. To prove the reverse containment, we consider an element w in VI n im P*. Without loss of generality, we may assume Ilwll = 1. By the definition of a base norm space (d. Definition 2.45), VI = conv(K U -K). Thus there exists (J E K, T E -K and A E [0,1] such that w = A(J + (1 A)T. Then w

= P*w = AP*(J + (1 - A)P*T.

Hence

1 = Ilwll ::; AIIP*(JII

+ (1- '\)IIP*TII,

which implies IIP*(JII = IIP*TII = 1. Thus p*(J E im+ P* n K = F, and similarly -P*T E F, so w E co(F U -F). The rest of the proposition follows from (i) and (ii) and the fact that P is a positive projection with IIPII ::; 1. 0 7.32. Proposition. Let A, V be an order umt space and a base norm space zn sepamtzng order and norm duaZzty. Let P, Q be compresswns wdh assocwted proJectwe umts p, q and assocwted proJectwe faces F, G, respectwely. Then the followzng statements are equwalent: (i) (ii) (iii) (iv)

imP C imQ, QP = P, p::; q,

Fe G, (v) PQ = P,

(vi) ker Q c ker P, (vii) im Q' C im P'. Proof. (i) =? (ii) Trivial. (ii) =? (iii) If QP = P, then p = PI = QPl ::; Ql = q. (iii) =? (iv) Clear from (7.22). (iv) =? (v) If F c G, then im+ P* C im+Q* and then also im P* C im Q*, so Q* P* = P*. Dualizing, we get PQ = P. (v) =? (vi) Trivial. (vi) =? (vii) If ker Q C ker P, then trivially ker+Q C ker+ P, so im+Q' C im+ P', and then also imQ' C imP'. (vii) =? (i) Use the implication (i) =? (vii) with Q' in place of P and P' in place of Q. 0

PROJECTIVE UNITS AND PROJECTIVE FACES

231

7.33. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality and let P, Q be compressions with associated projective units p, q and associated projective faces F, G respectively. If the condition im P C im Q (together with the equivalent conditions of Proposition 7.32) are satisfied, then we will write P ::< Q and F ::< G. Note that if P ::< Q and Q ::< P, then im P = im Q and ker P = ker Q, so P = Q. Thus the relation ::< is a partial ordering. By the equivalence of (i) and (vii) in Proposition 7.32, P ::< Q iff Q' ::< P', so P f--> P' is an order reversing map for this relation, and similarly for p f--> p' and F f--> F'.

7.34. Proposition. Let A, V be an order umt space and a base norm space m sepamtmg order and norm dualzty, and let P, Q be compressions on A wzth assocwted proJectwe umts p, q and assocwted pTOJectwe faces F, G, respectwely. Then the followmg are equwalent: (i) QP = 0, (ii) P::< Q', (iii) p::; q', (iv) Fe G', (v) PQ = O.

Proof. (i) =} (ii) If QP = 0, then im P C ker Q. Hence im+ P C ker+Q = im+Q', so P::< Q'. (ii)¢? (iii) ¢? (iv) Follows directly from Proposition 7.32. (iv)=}(v) If F c G', then it follows from the implication (iv) =} (v) of Proposition 7.32 that PQ' = P. Multiplying from the right by Q gives 0 = PQ. (v) =} (i) Use the implication (i) =} (v) with P and Q interchanged. 0 7.35. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality, and let P, Q be compressions with associated projective faces F, G and projective units p, q. If the condition PQ = 0 (or any of the equivalent conditions in Proposition 7.34) is satisfied, then we will say that P zs or·thogonal to Q and we will write P ..l Q, and we will also say that p zs orthogonal to q and write p ..l q, and that F zs orthogonal to G and write F ..l G. The notion of orthogonality of projective faces F, G depends on the duality of A and V (since G' is defined in terms of this duality). We will now show that in the important special case where A = V*, this notion corresponds to the geometric notion of antipodality defined in (A 23). Recall that two subsets F, G of a convex set are antzpodal if there is a function a E Ab(K) with 0 :s: a :s: 1 such that (7.27)

a

= 1 on

P,

a

=0

on G.

232

7

GENERAL COMPRESSIONS

7.36. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm dualdy, and assume A = V*. Let F and G be projectwe faces of K. Then F, G are orthogonal iff F, G are anttpodal. In parttcular, two extreme pomts p, 0' E K are orthogonal tff IIp-O'II = 2.

Proof. Assume first that F 1.. G, and let p, q be the projective units associated with F,G. Then p <::: q', so p+q <::: 1. By (7.22), p takes the value 1 on F. Similarly q takes the value 1 on G. Hence (7.27) is satisfied when a = p, so F, G are antipodal. Assume next that F, G are antipodal and let p, q be as above. Choose aE such that (7.27) holds. Since a = 1 on F, then a 2: p (cf. (7.23)). Similarly since I-a = 1 on G, then I-a 2: q, so a <::: q'. Hence p <::: a <::: q', so F 1.. G. The last statement follows from (A 25). D

At

7.37. Lemma. Let A, V be an order- umt space and a base nor-m space m separatmg order and norm dualtty. If P tS a compresswn and a E A +, then the following are equwalent:

(i) a = Pa + pIa, (ii) Pa <::: a. Proof. (i) =} (ii) Trivial. (ii) =} (i) Assume Pa <::: a. Since a - Pa E ker+ P = im+ P' and the two complementary projections P and pi satisfy the equation pip = 0, then a - Pa = pl(a - Pa) = pIa. Thus a = Pa

+ pia.

D

7.38. Lemma. Let A, V be an order umt space and a base norm space m separatmg order and norm dualtty. If P zs a compr-esswn and q tS a proJectwe umt such that Pq <::: q, then also Pq' <::: q'.

Proof. The compression P and its complementary compression pi satisfy the equation PI + P l l = 1 (Lemma 7.17). Therefore by Lemma 7.37, q' = 1 - q = (PI which gives Pq' ::; q'.

D

+ pi 1) -

(Pq

+ pi q) =

P q'

+ pi q' ,

PROJECTIVE UNITS AND PROJECTIVE FACES

233

7.39. Proposition. Let A, V be an order umt space and a base norm space m sepamtmg order and norm dualzty. If P and Q are compresswns wzth assoczated projective umts p and q, then the following are equwalent:

(i) PQ = QP, (ii) Pq::; q, (iii) Qp::; p. Proof. Since (i) is symmetric in P and Q, it suffices to prove the equivalence of (i) and (ii). (i) =} (ii) If PQ = QP, then Pq = PQl = QPl = Qp ::; Ql = q. (ii)=}(i) Assume now that Pq ::; q. Let a E A+ be arbitrary. Since

0::; a ::; Ilalll, then 0::; Qa ::; Iiall q. Hence 0::; PQa ::;

IlallPq ::; Iiall q,

from which it follows that (7.28)

Q' PQa = 0

for all a E A + .

Therefore PQa E ker+ Q' = im + Q, so PQa = Q PQa for all a E A +, and then for all a E A. Thus (7.29)

PQ = QPQ.

By Lemma 7.38, we also have Pq' ::; q', so we can argue as above with q' in place of q and Q' in place of Q. Therefore (7.28) holds with Q and Q' interchanged, i.e., Q PQ' a = 0 for all a E A +, and then for all a E A. Thus QPQ' = o. Dualizing, we get

Q'* P*Q* =

o.

Thus for arbitrary p E V+, we have P*Q* p E ker+Q'* = im+Q*. Therefore P*Q* p = Q* P*Q* p for all p E V+, and then for all p E V. Thus

P*Q* = Q* P*Q*. Dualizing back to A, we get (7.30)

QP=QPQ.

Combining (7.29) and (7.30) gives PQ = QP. 0

234

7

GENERAL Cm.IPRESSIONS

7.40. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality and let P, Q be compressions with associated projective units p, q and associated projective faces F, C. If the equation PQ = QP (or any of the equivalent conditions in Proposition 7.39) is satisfied, then we will say that each of the entities P, p, F is compattble with each of the entities Q, q, C. (We will also say that these entities are mutually compattble.) Note that it follows from Lemma 7.38 that if P (or either one of the associated entities p or F) is compatible with Q (or either one of q or C), then P is also compatible with the complementary compression Q' (or either one of the associated entities q' or C'). By definition, a compression P is compatible with a projective unit q iff Pq :-::: q, and this is equivalent to q = Pq + Pq' by Lemma 7.37. Thus we can extend the definition to a pair of a compression and an arbitrary element of A as follows. 7.41. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality, and let P be a compression with associated projective unit p and associated projective face F. Vie will say that P (or either one of p or F) is compattble with an element a E A if a = Pa + PIa. We will also say that a is compattble with P, and with p and F. By an easy calculation (used, e.g., in the proof of [AS, Lemma 3.39]), a projection p in a von Neumann algebra commutes with an arbitrary element a of the algebra iff a = Upa + Up' a (where p' = 1 - p). Thus the definition above generalizes not only the notion of commutation for pairs of projections in a von Neumann algebra, but also the notion of commutation for pairs consisting of a projection and an arbitrary (self-adjoint) element of the algebra. Remark. The notion of compatibility comes from quantum physics where it relates to simultaneous measurability of observables, as we will now explain in terms of the von Neumann algebra model of quantum mechanics (described in [AS, p. 121]). The (bounded) observables are represented by the self-adjoint elements of a von Neumann algebra M and the (mixed) states are represented by the elements in the normal state space K of M. The propositions, i.e., the observables that take on only the two values 0 or 1, are represented by the elements in the set P of projections. If we measure an observable a E M oa directly on a quantum system, say a beam of particles, in a state w E K, then the expected (mean) value for a is (a, w). But if we first send the beam through a measuring device for a proposition PEP, then it will be split in two parts, the one consisting of the particles appearing with the

PROJECTIVE UNITS AND PROJECTIVE FACES

235

value 1 for p, the other consisting of the particles appearing with the value o for p. Of these, the former will be in the transformed state U;w and the latter will be in the transformed state U;,w, where U; and U;, are the compressions associated with p and pi = 1 - p respectively. In this case the expected value of a (measured on the combined output from the measuring device for p) will be (a, U;w) + (a, U;,w). This value is equal to the expected value (a, w) for the unperturbed beam iff (7.31) Therefore the expected value of a on any beam is unaffected by the measurement of p iff a = Upa + Up,a. Thus a and p are simultaneously measurable iff a and Up are compatible in the sense of Definition 7.41 (applied in the von Neumann algebra context of Proposition 7.23). We will also present two other properties of quantum mechanical measurements, which are both easily obtained in the algebraic model, and will be needed in a later remark on physics. Assume pEP and w E K. If IIU;wll = 1, then (p,w) = (Up 1,w) = l. Hence by the implication (iv) =? (vi) in (A 74), (7.32)

IIU;wll =

1

=?

U;w = w,

U;

which is the defining implication for neutral projections (i.e., (7.18) with in the place of Pl. As explained above, describes the transformation of states by a measuring device for the proposition p, say a filter for a beam 0f particles. Therefore such quantum mechanical filters have the neutrality property described in the Remark after Definition 7.19. Assume next that a E A + and pEP. Now it follows from the implication (iii) =? (iv) in (A 73) that

U;

(7.33) Thus if Up' a = 0, then a = Upa + Up' a, so a and p are compatible. Therefore the positive observable a and the proposition p are simultaneously measurable in this case. What does this compatibility property mean in physical terms? Clearly Up,a = 0 is equivalent to (a, U;,w) = 0 for all w E K, and Upa = a is equivalent to (a, Ur:w) = (a, w) for all w E K. Thus the implication (7.33) says that if the positive observable a and the proposition p are such that particles which appear with the value 0 in a measurement of p never contribute anything to a subsequent measurement of a, then the expectation of a in such a measurement will have the same value (a, U;w) = (a, w) as if it were measured directly on the unperturbed beam. It is clear from Definition 7.41 that if the compression P is compatible with the element a, then pi (as well as pi or F') is also compatible with

236

7.

GENERAL COMPRESSIONS

a. Clearly, the set of elements compatible with a compression P is a linear subspace of A which contains 1. From this it follows that P is compatible with an element a of A iff P is compatible with a + Al for one (and then all) A E R. We can choose A so that a + Al 2 O. Thus the problem of showing compatibility of a compression P and an element a in A can be reduced to showing compatibility of P and an element b = a + Al in A +, and by Lemma 7.37, this can be done by showing Pb ::; ~. We will now prove some basic properties of the compatibility relation, and we begin by an observation which we state as a lemma for later reference. 7.42. Lemma. Let A, V be an order unzt space and a base norm space In separating order and norm dualdy and let P, Q be a paIr of compreSSIOns. If P ~ Q, then P and Q are compatIble. If P, Q are compatIble, then pi, Q are compatible and pi, Q' are also compatIble. If P -1 Q, then P and Q are compatIble. Proof. If P ~ Q, then PQ = P and also QP = P (Definition 7.33), so P and Q are compatible. If P, Q are compatible, then q = Pq + pi q, so pi, Q are compatible, and then pi, Q' are also compatible. If P -1 Q, then P ~ Q' implies that P is compatible with Q', and so P and Q are compatible. 0

7.43. Lemma. Let A, V be an order unzt space and a base norm space In separatmg order and norm duaizty, and let P be a compreSSIOn wIth assocwted proJectwe face F. If a IS an element of A+ such that a = 0 on F ' , then Pa = a, pia = 0 and a IS compatIble wIth P. Proof. Assume a = 0 on F ' , i.e., (a, w) = 0 for all w E F'. Since F' C im+ p l* = ker+ P*, we have a E (ker+ P*)- = im+ P (Proposition 7.7 with P* in place of P). Thus Pa = a. Applying pi gives 0 = pia, and then a = Pa + pia. Thus a is compatible with P. 0

7.44. Proposition. Let A, V be an order unzt space and a base norm space In separatIng order and norm duaizty, let P be a compreSSIOn wzth associated proJectwe face F, and let a be an arbdrary element of A. Then Pa is the unzque element compatzble wIth P whzch zs equal to a on F and vanzshes on F ' , and (P + pl)a zs the unzque element compatzble wzth P whIch IS equal to a on co(F U F'). Proof. Clearly Pa is compatible with P. If w E F, then P*w = w (7.24), so (Pa, w) = (a, P*w) = (a, w). Thus Pa = a on F. If w E F ' , then P*w = 0, so (Pa, w) = (a, P*w) = O. Thus Pa = 0 on F'. Now consider an element b compatible with P such that b = a on F and b = 0 on F'. By the definition of a projective face (Definition 7.27),

PROJECTIVE UNITS AND PROJECTIVE FACES

237

im+ P* is the cone generated by F. Similarly im+ p l * is the cone generated by F'. Therefore we also have b = a on im + P* and b = 0 on im + P'*. By the compatibility of band P, we have for each wE V+,

(b, w)

=

(Pb, w)

+ (Plb, w)

=

(b, P*w)

+ (b, pl*w)

=

(a, P*w)

=

(Pa,w).

Thus b = Pa, which proves the uniqueness. Clearly also PIa is compatible with P, and by the above PIa = a on F' and Pa = 0 on F. Therefore (P + pl)a is compatible with P and (P + pl)a = a on co(F U F'). Now consider an element c compatible with P such that c = a on co(F U F'). Then c = a on im+ P* and c = a on im+ pl*, so we can argue as above getting (P + pl)a = c, which proves uniqueness also in this case. 0 Note that by Proposition 7.44, an element a E A which is compatible with a compression P with associated projective face F is completely determined by its values on F and F'. Our next proposition gives an explicit expression for the compression associated with a projective face. 7.45. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm duahty. If F zs a proJectwe face wdh assocwted compresswn P! then for each a E A + )

Pa = (7.34) =

V{b E A + I b = 0 on F', b ~ a on F} 1\ {b E A + I b = 0 on F',

b ~ a on F}.

Proof. Assume a E A+. By (7.25) Pa = 0 on F ' , so we can choose b = Pa in (7.34). We must prove that this is the best choice, i.e., we must show that if bE A+ with b = 0 on F' and b ~ a on F (or b ~ a on F), then b ~ Pa (respectively b ~ Pa). Consider an element b of A + such that b = 0 on F' and b ~ a on F. By Lemma 7.43, b is compatible with P. Hence for each w E V+, (b, w)

=

(Pb, w)

+ (Plb, w)

=

(b, P*w)

+ (b, pl*w)

~

(a, P*w)

=

(Pa, w).

Thus b ::; Pa. (Similarly we prove that if b E A +, b = 0 on F' and b ~ a on F, then b 2: Pa.) We are done. 0 We will now specialize Theorem 7.15 to the present context, and simultaneously we will prove uniqueness under alternative conditions.

7.

238

GENERAL COMPRESSIONS

7.46. Theorem. Let A, V be an order unzt space and a base norm space in separatzng order and norm dualdy, let K be the dzstmguished base of V, and let P be a compresswn wdh assocwted proJectwe face F. The map E = P + pI is the unique weakly-contmuous proJectwn T of A onto the subspace im(P + PI) of all elements compatzble wzth P, whzch satzsfies anyone of the following three conddwns:

(i) T is posdwe, (ii) liT I ~ 1, (iii) T commutes wdh P and P'. Dually, E* zs the unique weakly contmuous proJectwn of V which maps K onto co(F U F') and leaves thzs set pomtwzse fixed. Proof. Clearly the projection T = E satisfies (i), (ii),(iii), and we will prove uniqueness under each of these conditions. (i) If T is a weakly-continuous positive projection of A onto im (P + PI), then T = E by Theorem 7.15 (with X = A, Y = V and Q = PI). (ii) Assume IITII ~ 1. Since T acts as the identity operator on the subspace im (P + PI), then Tl = 1. We will show that T is also positive. Let a E A+ and assume without loss of generality that Iiall ~ 2. Now it follows from the defining property of the order unit norm (A 13) that -1 ~ a-I ~ 1, and then in turn that Iia - 111 ~ 1. Since IITII ~ 1 and Tl = 1, then IITa-lI1 ~ 1, or which is equivalent, -1 ~ Ta-l ~ 1. Hence Ta ~ O. Thus T is positive, and the desired uniqueness follows from (i). (iii) Assume that T is a projection of A onto im (P + PI) which commutes with P and pI, and then also with E. Since imT = imE, then ET = T and T E = E, and since T commutes with E, we have T=ET=TE = E. It remains to prove the dual result on E*. Let w E K. By Lemma 7.17, E*(K) c K. By the definition of F and F ' , we have P*w = AP and P'*W = /1CJ where p E F and CJ E F' with A, /1 ~ O. Thus E*w = AP + /1CJ. Since (I,E*w) = ((P+PI)I,w) = (l,w) = 1, then A+/1 = 1. Hence E*w = AP + (1 - A)CJ E co(F U F'). Since E* leaves F and F' pointwise fixed, it also leaves co(F U F') pointwise fixed, so E*(K) = co(F U F' ). The uniqueness of E* follows from Theorem 7.15 (this time with X = V, Y = A and with P* in place of P and p l * in place of Q.) 0

7.47. Definition. Let A, V be an order unit space and a base norm space in separating order and norm duality. A compression P is said to be central if it is compatible with all a E A. An element a E A is central if it is compatible with all compressions P on A.

PROJECTIVE UNITS AND PROJECTIVE FACES

239

7.48. Proposition. Let A, V be an order umt space and a base norm space m separatmg order and norm dualzty. If P : A ---; A zs a weakly contmuous posztzve proJectzon, then the followmg are equzvalent:

(i) Pa:S: a for all a E A+, (ii) P zs strongly complemented, (iii) P is a compresszon such that pi (iv) P zs a central compresszon.

= I - P,

Proof. (i) =} (ii) Trivial, since statement (i) means that I -P is positive. (ii) =} (iii) Assume P strongly complemented and let Q = I - P 2': O. Then PI = 1 - Ql :s: 1, so P is normalized (Lemma 7.16). We also have Q* = I - P* 2': 0, so P*, Q* are also strongly complementary projections of norm :s: 1, and thus (iii) is satisfied. (iii) =} (iv) Trivial. (iv) =} (i) Clear from Lemma 7.37. 0 Recall that a convex set K is the dzrect convex sum of faces F and G if each element of K can be written uniquely as a convex combination of an element from F and one from G, in which case we write K = F EBc G. In this case, G is uniquely determined by F, and we write G = Fl. (We will see below that this is consistent with our previous meaning for Fl.) If such a face G exists, then F is a split face of K, cf. (A 5).

7.49. Proposition. Let A, V be an order umt space and a base norm space in separatzng order and norm dualzty. If P zs a central compresszon, then the assoczated proJectzve face F zs a splzt face of the dzstmguished base K, and zts complementary splzt face zs equal to the complementary projectzve face Fl. If A = V*, then the converse statement also holds, z. e., each splzt face F of K zs assoczated wzth a central compression P. Proof. The first statement of the proposition follows easily from the equation pi = 1- P. To prove the second statement, we consider an arbitrary split face F of K, and we denote the complementary split face by G. By the definition of a split face, each w E K has a unique decomposition (7.35)

W

= AO" + (1 - A)T

:s: :s:

where 0" E F, T E G and 0 A 1. It is easily verified that : W f--+ AO" is an affine map from K onto co(F U {O}). If a is an arbitrary element of the space A = V*, then the composed map a 0 is a bounded affine function on K. By elementary linear algebra, each bounded affine function on K can be uniquely extended (with preservation of norm) to a linear functional a E A, cf. (A 11). Now P : a f--+ a is a positive linear map from

7

240

GENERAL COMPRESSIONS

the space A into itself such that

(Pa, w)

=

IIPII

(a, (w))

<::: 1 and

for all a E A, wE K.

Observe also that P is weakly continuous (as the weak topology on A is given by the semi norms a f---+ l(a,w)1 where w E K). By the equation above, P* = on K, so P* is a positive projection with im-'- P* equal to the cone generated by F. It follows from (7.35) that for each wE K,

w - P*w = w - (w) = (1 -

),)T ~

0,

so P*w <::: w. Hence (Pa, w) = (a, P*w) <::: (a, w) for each a E A +. Now it follows from Proposition 7.48 that P is a central compression. By the above, K n im+ P* = F, so F is the projective face associated with P. 0

Remark. We discussed the von Neumann algebra approach to quantum mechanical measurement theory in the remark after Definition 7.4l. But the basic concepts of this theory can be defined and studied also in the general order-theoretic context of this chapter. Then the (mixed) states of a quantum system are represented by the elements in the distinguished base K of a base norm space V, the (bounded) observables are represented by the elements in the order unit space A = V*, and the expected value of the observable a E A when the system is in the state w E K is (a, w). The propositions are represented by the elements in the set P of projective units, or alternatively by the associated elements in the set F of projective faces or in the set C of compressions. We will now discuss the physical interpretation and justification of this general order-theoretic approach. The representation of states by elements in the convex set K c V and of observables by elements in the order unit space A = V* is a natural generalization of the standard algebraic model. But the representation of propositions is less obvious. Why should they be represented by projective units associated with compressions defined in terms of bicomplementarity? A quantum mechanical proposition is operationally defined by a measuring device, which we may think of as a filter for a beam of particles. We have previously explained how such a device splits the beam into two partial beams, one consisting of the particles appearing with the value 1, the other consisting of the particles appearing with the value o. If the incident beam is in the state w, then the beam of particles exiting with the value 1 will be in a transformed state WI and the beam of particles exiting with the value 0 will be in a transformed state woo In the algebraic model, where the proposition is represented by a projection p in a von Neumann algebra, the transformation of states is given by the maps U; and U*,. More precisely, WI and W2 are (up to a normalization factor) equal to and

U;w

GEOMETRY OF PROJECTIVE FACES

241

U;,w respectively (or which is equivalent, WI and W2 are in the faces F and F' associated with p and pi respectively). In the general order-theoretic context we denote the corresponding maps on the base norm space V by

II
=

Ilwll·

Now consider the dual maps P =
We will continue to work with an order unit space A and a base norm space V in separating order and norm duality. But we will now also impose the extra condition that A = V*, which is satisfied in the important special cases of von Neumann algebras and JBW-algebras, and of course also in all finite dimensional cases. Under this condition, a f--> a is an order and norm preserving isomorphism of A onto the space Ab(K) of all bounded affine

242

7.

GENERAL COMPRESSIONS

functions on K (A 11), and every bounded linear operator on V is of the form T* where T is the (weakly continuous) dual operator (defined on A = V* by (a, w) = (Ta, w) for a E A and w E V). We know that compressions occur in complementary pairs P, pi and that pi is uniquely determined by P (Corollary 7.11). We will now state the corresponding result for projective faccs in an explicit geometric form. For this purpose we associate with each face F of the distin.luished base K of V the function k F defined on K by (7.37)

kF = inf{a E Ab(K) I a = 1 on F, 0::; a::; I}.

Note that kF is a concave but not necessarily affine function for a general face F of K. Remark. The upper semi-continuous upper envelope of the characteristic function XF of a closed face F in a compact convex set K is equal to inf {a E A(K) I a 2': Ion F, 0::; a} (see, e.g., [6, p. 4] or [20, p. 22]). This expression resembles the definition of k F , but it differs from it in two respects. First, the space Ab(K) of bounded affine functions on K is replaced by the space A(K) of affine functions continuous in the given compact topology of K. But this difference is not essential from a geometric point of view. For example, in finite dimensions Ab(K) = A(K). Second, the relation 0 ::; a ::; 1 is replaced by the weaker condition 0 ::; a, and this difference is crucial. In fact, under the additional assumption a ::; 1, the function 1- a is positive and takes the value zero on F, so 1- a, and then also a, is compatible with F (Lemma 7.43). Thus this condition severely restricts the set of elements a permitted in (7.37). 7.50. Proposition. Let A, V be an order umt space and a base norm space m sepamtmg order and norm dualzty, and let K be the dzstmgU1shed base of V. Assume A = V* and let F be a proJectwe face wzth assocwted proJectwe umt p. Then p = k F, and (7.38)

F' = {w E K I k F (w) = 0 }.

Proof. By Lemma 7.25, pi (= 1 - p) is the greatest element of At which vanishes on the set K n ker+ P '* = K n im+ P* = F. Thus p is the least clement of At which is 1 on F, so p = k F . Now (7.38) follows from (7.22). D We will give a geometric characterization of pairs of complementary projective faces F, G E K. Note that when we have found conditions characterizing such a pair, then we can also decide if a single face F is projective

GEOMETRY OF PROJECTIVE FACES

243

by checking if the pair P, pi where pi is defined by (7.38) satisfies these conditions. We have already seen that two faces P, G c K are complementary projective faces corresponding to central compressions iff they are complementary split faces, i.e., iff K = PfficG (Proposition 7.49), or just K c PfficG since the opposite relation is trivial. We will now characterize general pairs of complementary projective faces by two conditions. The first of these is a modified version of the relation K c P ffic G which consists of two relations, one where P is replaced by TanK P and one where G is replaced by TanKG. The second condition is the equality TanK P n TanKG = 0. (The latter is trivial in the case of split faces, where TanK P and TanKG are the affine spans of P and G, which are disjoint since P and G are affinely independent. ) We begin by introducing some notation which will be used in the announced characterization and its proof. We will denote by H the affine hyperplane which carries the convex set K, i.e., H = {w E V I (l,w) = I}. For each face P of K we will denote by P the cone generated by P, i.e., P = {Aw I W E P, A ~ O}. If P is a face of K, then the tangent ~pace of K at P is an affine subspace of H, while the tangent space of P is a linear subspace of V. To simplify the notation, we will omit the subscripts in these two cases, writing (7.39)

TanP:= TanKP,

TanP:= Tanv+P.

7.51. Lemma. Let A, V be an order umt space and a base norm space m separatmg order and norm dualzty, assume A = V*! and let P be a non-empty subset of the dzstmguzshed base K of V. Then

(7.40)

TanP = H n TanP,

(7.41 )

Tan P = lin (Tan P).

Proof. We first prove (7.42)

TanP = peo n H.

Since peo = peo and peo = TanP (Proposition 7.2), the equality (7.40) will follow from (7.42). By definition, peo is an intersection of closed supporting hyperplanes for V+ (therefore also for K) containing F. All such hyperplanes contain Tan P, so Tan P c peo. Since H is a closed supporting hyperplane for K, then TanP c peo n H. Now suppose 0' E peonH. Let N be any closed supporting hyperplane for K containing P. Choose a E A, A E R, such that a ~ A on K, and such that N = {a E V I (a,a) = A}. Then a-AI::::: 0, and N ~ F implies a - Al = 0 on P. Then a - Al E Fe, so (a - Al,a) = O. Thus (a,a) = A,

244

7.

GENERAL COMPRESSIONS

so a E N. Hence pea n HeN, so pea n H is contained in the intersection of all such N, i.e., in Tan F. This completes the proof of (7.42), and then also of (7.40). Now we prove (7.41). By Proposition 7.2, Tan P = pea = pea ::) Tan P, so Tan P ::) lin (Tan P). Suppose a E Tan P. We consider two cases. If (l,a) = 0, then for any T E P, (l,a+T) = 1, so a+T E TanP n H = Tan P. Therefore a E Tan P - T C lin (Tan P). On the other hand, if (l,a) i= 0, then (l,a)-la E TanP nH = TanP, so a E lin (TanP). Thus Tan P C lin (Tan P), which completes the proof of (7.41). 0

7.52. Theorem. Let A, V be an order umt space and a base norm space m separatmg order and norm dualdy, assume A = V*, and let F, G be faces of the dzstmguzshed base K of V. Then F, G is a pazr of complementary pro)ectwe faces zJJ the followmg two condztions are satzsjied: (i) K

c P EBc TanG, K = 0.

C

TanP EBc G,

(ii) Tan F n Tan G

Proof. Assume first that F, G are complementary projective faces associated with the compressions P, Q. Now we apply Theorem 7.14 (noting that P and G above are denoted by P and G respectively in that theorem). Then (7.43)

V+

c

P EBw TanG,

V+

C

G EBw TanP.

We will now show (i) holds. Note that im+ P* = by smoothness of P* we have ker P*

P and

im+Q* = G, and

= Tan (ker+ P*) = Tan (im+Q*) = Tan G.

Similarly ker Q* = Tan P. Let a E K; then a = P*a

+ (a

- P*a) E im+ P*

+ ker P*

=

P EBw Tan G.

Let ,\ = I P* a II. Since I P* I :s; 1, then 0 :s; ,\ :s; 1. Let H be the hyperplane which carries K. If 0 < ,\ < I, then

0, then P*a = 0, so a E H n TanG = TanG. If,\ = I, then 1, so by neutrality P*a = a, and thus a E P. This completes the proof that (i) holds. Set for brevity N := Tan F n Tan G. Thus

If ,\

=

I!pall =

(7.44)

(P*

+ Q*) w =

0

for wEN.

GEOMETRY OF PROJECTIVE FACES

245

By Lemma 7.17, for each wE K, (1, (P*

+ Q*) w) = (PI + Q1, w) = 1.

Since H is the affine span of K, then (1, (p* + Q*) w) Now it follows from (7.44) that N n H = 0, so Tan F n Tan G

= 1 for all

wE H.

= Tan P n Tan 0 n H = N n H = O.

Thus we also have (ii). Assume next that (i) and (ii) are satisfied. By elementary linear algebra, (i) implies that lin F n lin (Tan G) = {O} (cf. [6, Prop. II.6.1]). Thus lin F n Tan 0 = {O}, and similarly lin G n Tan P = {O}. By (i) we also have V+ c P + Tan 0 and V+ c 0 + Tan P, so (7.43) follows. Thus we can apply Lemma 7.13, by which we have the equations '""-'----

~.-.-

(7.45) (7.46)

TanF

= lin FEB N, TanG = lin G EB N, V = lin F EB lin G EB N.

-

-

Let cI>, \It be the projections onto lin F and lin G determined by (7.46), i.e., im cI> = lin P, ker cI> = lin 0 EB N = Tan 0, and im \It = lin G, ker \It = lin P EB N = Tan P. By (ii) we have (7.47)

H n N = H n Tan P n Tan 0 = Tan F n Tan G = O.

This implies (7.48)

(l,T) = 0

for all TEN,

for if TEN and (1, T) = a Ie 0, then a-iT E H n N contrary to (7.47). Consider now an arbitrary wE K. By (i), (7.49) where Pi E F, Wi E Tan G, 0 ::; A ::; 1. Here (l-A)wi E Tan 0 = lin OEBN, so (1 - A)Wi = (J + T where (J E lin 0 and TEN. Setting P = APi, we can rewrite (7.49) as w

= P + (J + T,

where P E lin F, (J E lin G, TEN. Thus the first term in the decomposition of w determined by the direct sum (7.46) is of the form P = APi with

246

7

GENERAL COMPRESSIONS

Pi E F and 0 :s; A :s; 1. By the same argument, the second term is of the form a = I-lal with al E G and 0 :s; I-l :s; 1. Thus

By (7.48) (1, w) takes the form

= A + IL, so

I-l

= 1 - A. Therefore this decomposition of

w

(7.50) where Pi E F, al E G, TEN, 0 :s; A :s; 1. From this it follows that is a normalized positive projection and that im+ = F, so F = K n im+. Similarly \II is a normalized positive projection with im+\II = G, so G = K n im+\II. We will now show that , \II are complementary projections of V, i.e., that ker+\II = im+ and ker+ = im+\II, or which is the same, that ker+\II = F and ker+ = G. Clearly F c ker+ \II. To prove the converse relation, we will show K n ker+ \II c F. Assume for contradiction that there exists w E K \ F such that \IIw = o. Consider now the decomposition of w as in (7.49). Since w tI. F, we must have A < 1. Consider next the decomposition of w as in (7.50). Since \IIw = 0, we must have (1 - A)al = \IIw = o. Thus w = APi +T. Now it follows from (7.48) that (l,w) = A, which contradicts the inequality A < 1. With this we have shown ker+\II = im+. By the same argument ker+ = im+\II, so , \II are complementary projections. Observe now that ker

= TanG = Tan (im+\II) = Tan (ker+
so is a smooth projection. By the same argument, \II is a smooth projection. Since A = V*, we have = P* and \II = Q* where P, Q are weakly continuous positive projections on A. By the above, the dual projections P* = and Q* = \II are complementary smooth projections on V, so it follows from Theorem 7.10 that P, Q are bicomplernentary projections on A. Since and \II are normalized, so are P and Q. Thus P, Q is a pair of complementary compressions. We have already shown that Knim+ P* = F and K n im+Q* = G, so we are done. D In order to specify the pair A, V of the two ordered normed spaces A and V in Theorem 7.52, it suffices to specify the distinguished base K, since this convex set determines not only V, but also A = V* ~ Ab(K). We will now use this fact to give some finite dimensional examples, starting out from a bounded convex set K, which we think of as embedded as the

GEOl\IETRY OF PROJECTIVE FACES

247

base of a cone V+ in a finite dimensional linear space V, and then defining A = Ab(K). In our first example K is a simplex. Then each face is split, and therefore projective (Proposition 7.49). In this case the conditions (i), (ii) of Theorem 7.52 are trivially satisfied as remarked prior to the theorem. In our next example K is a Euclidean ball. Then all points on the surface of K are extreme points, and it is easily seen that the conditions (i), (ii) of Theorem 7.52 are satisfied by a pair of faces F = {p}, G = {O"} where p,O" are antipodal points on the surface of K. Thus each point on the surface of K is a projective face whose complementary projective face is the antipodal point on the surface of K. Fig. 7.4 shows the base norm space V and the order unit space A in the particular case where K is a two-dimensional disk. Here the norm and the ordering are given by the unit balls V1 , Al and their positive parts At· We see that the proper projective faces of K are the points on the circular boundary of K, and the associated projective units in A are the points on the circle where the boundary of the positive cone meets the boundary of the unit ball. (The associated compressions were already indicated in Fig. 7.2)

vt,

K

vt

Fig. 7.4

Readers may find it instructive to compare the example above with the similar example where the circular base is replaced by the convex set in Fig. 7.3. These two convex sets are both shown in Fig. 7.5 below. While conditions (i), (ii) are satisfied in Fig. 7.5 (a), the first of those will not hold in Fig. 7.5 (b) as is easily seen. Observe also that Fig. 7.5 illustrates the importance of smoothness for the uniqueness of the projection E* (already commented on after Theorem 7.15). In Fig. 7.5 (a) the direction of the

248

7

GENERAL COMPRESSIONS

"projection vector" (from w to E*w) is uniquely determined by the parallel tangents at F and G. But in Fig. 7.5 (b) it can be tilted up to 45° to either side with K still being projected onto co(F U G).

F..-------t G

Fig. 7.5

We will also give two slightly less trivial examples. The convex set K in Fig. 7.6 is a section of a circular cylinder, determined by two planes intersecting the cylinder in two ellipses which at a point p on the surface of the cylinder have a common tangent orthogonal to the axis. In this example, the singleton F = {p} and the opposite line segment G on the cylindrical surface will satisfy the conditions (i), (ii) of Theorem 7.52, so they are complementary projective faces.

Fig. 7.6

The convex set in Fig. 7.7 is the convex hull of a line segment F and a circular disk which is orthogonal to the segment such that the midpoint of the segment lies on the periphery of the disk. In this example the line segment F and the singleton G = {O"}, where 0" is the opposite point on the disk, will satisfy the conditions (i), (ii) of Theorem 7.52, so they are complementary projective faces.

NOTES

249

Fig. 7.7

Note, by the way, that the two examples above are closely related, as the convex sets K in Fig. 7.6 and Fig. 7.7 are in a natural way the polars of each other. (We leave the details to the interested reader.) Notes

The concept of a general compression on an order unit space A in separating duality with a base norm space V was first introduced in [7, p. 8] under the name "P-projection". The characterization of concrete compressions (i.e., the maps a f---+ pap) in order-theoretic terms as abstract compressions defined by bicomplementarity (Proposition 7.23), was first given in [7] for the von Neumann algebra case and in [9] for the JBWalgebra case. The geometric characterization of projective faces in Theorem 7.52 is new. The correspondence of split faces and strongly complementary compressions in Propositions 7.48 and 7.49 has been known for quite some time. It was used by Wils [136] in his study of the zdeal center and central decomposdwn in the general context of ordered linear spaces, which generalize the same concepts for C*-algebras (cf. Sakai's book [112]). The study of split faces is actually a central theme in much of the early theory of compact convex sets and the geometry of C*-algebra state spaces, which can be found in the book [6]. Related material published shortly after that book appeared include Vershik's paper [132]' which among other things discusses a central decomposition akin to that of Wils (although with a different approach), and the papers by Andersen [15, 16] and Vesterstnom [133] on split faces and linear extension operators for affine functions.

8

Spectral Theory

In this chapter we will study a pair A, V of an order unit space and a base norm space in separating order and norm duality under the additional assumption that each exposed face of the distinguished base K of V is projective. If A, V is such a pair, then we will say they satisfy the standing hypotheszs. This hypothesis is satisfied when A is the self-adjoint part of a von Neumann algebra and V is the self -adjoint part of its predual, and also when A is a lBW-algebra and V is its predual. (In the former case, each norm closed face, and in particular each norm exposed face, is projective by virtue of (A 107). In the latter case, the same is true by Theorem 5.32.) In the first two sections we will show that if the standing hypothesis is satisfied, then the compressions have lattice properties similar to those of the projections in von Neumann algebras and lBW-algebras. In the third section we will specialize to spaces in "spectral duality" , for which we will establish a well-behaved functional calculus. In the fourth and last section we will study some concrete examples other than the motivating examples of von Neumann algebras and lBW-algebras. Note that the two last sections of this chapter are not needed for the final results on geometric characterization of lB-algebra and C*-algebra state spaces, so they may be skipped by readers who are mainly interested in these results. Theorem 8.64 is the one result in these sections that will be used later (in the characterization of normal state spaces of lBW-algebras and von Neumann algebras.) The lattice of compressions

If A, V is a pair of an order unit space and a base norm space in separating order and norm duality, then we will denote by C, Y, P the sets of compressions, projective faces, and projective units, respectively. Recall from Chapter 7 that each of these sets has a natural ordering and complementation, and that there are natural complementation-preserving order isomorphisms mapping an element in each of these sets onto an associated element in each of the other two sets (cf. Proposition 7.32 and the explicit formulas (7.22), (7.23), (7.24) and (7.34)). 8.1. Proposition. If the standmg hypotheszs of thzs chapter is satisfied, then C, P, Fare (isomorphzc) lattzces, and the lattzce operations are giVen by the following equations for each pmr F, G E F:

(8.1)

F

1\

G = F n G,

F

V

G = (F' n G')'.

252

8

SPECTRAL THEORY

Proof. Let F, G E F and let p, q be the associated projective units. Consider the element a = ~(p' + q') E A+ and an arbitrary wE K. Then (a,w) = 0 iff (p',w) = (q',w) = 0, which is equivalent to w E FnG. Thus F n G is an exposed face, hence is projective by the standing hypothesis of this chapter. Clearly F n G is the greatest lower bound of F and G in F. Since complementation reverses order, F is a lattice with lattice operations as in (8.1). Then C and P are lattices isomorp:lic to F. 0

8.2. Lemma. Assume the standmg hypothesIs of thIs chapter. If p, q are proJectwe umts and a IS an element of A + such that a :::; p and a :::; q, then also a :::; p 1\ q (greatest lower bound m P ). Proof. Let P, Q be the compressions associated with p, q and let F, C be the projective faces associated with p, q. Since a :::; p, then a = 0 on F', and since a :::; q, then a = 0 on C'. Thus a = 0 on F' U C'. The set H = {w I (a, w) = O} is an exposed, hence projective, face. Since F' U C' c H, then also F' V C' c H. Thus a = 0 on F' V G' = (F 1\ C)'. By Lemma 7.43, a

= (P 1\ Q) a:::; (P 1\ Q) 1 =

P 1\ q.

We are done. 0 8.3. Theorem. Assume the standmg hypothesIs of thIs chapter. If P, Q are compatIble compreSSIOns, then PQ = QP = P 1\ Q. Proof. By Definition 7.40, PQ = QP and also Pq :::; q and Qp :::; p, where p, q are the projective units associated with P, Q. Define the element r E A+ by

(8.2)

r

=

Pq

=

PQ1

=

QP1

=

Qp.

Observe that r = Pq :::; PI = p, and also r = Qq :<::: Q1 = q, so r :<::: p and r :<::: q. Let s be an arbitrary element of A + such that 5 :<::: P and 5 :<::: q. Then 5 E im+ P and 5 E im+Q (Lemma 7.29). Hence 5 = Ps :<::: Pq = r. Thus r is the greatest lower bound of p and q in A + . In particular, since pl\q is a lower bound for p and q, we have r 2: pl\q. By Lemma 8.2, r :<::: pl\q, so r = pl\q, or which is equivalent, r = (P I\Q) 1. For each a EAt,

PQa :<::: PQl

= r = (P 1\ Q) 1.

253

THE LATTICE OF COl\IPRESSIONS

Hence PQa E im+(P A Q) (again by Lemma 7.29). Therefore

PQa = (P Thus PQ

=

A

Q)PQa = (P

A

Q)Qa = (P

A

Q)a.

P A Q, so we are done. D

8.4. Lemma. Assume the standzng hypothes2s of th'lS chapter. If P, Q are mutually compat'lble compresswns, both compat'lble wzth an element a E A, then P A Q and P V Q are also compat'tble wzth th'ts element.

Proof. Clearly we can assume a E A +. Since a is compatible with P and Q, then Pa::; a and Qa ::; a (Definition 7.41). Hence (P A Q) a

= PQa ::; Pa ::; a.

Thus a is compatible with P A Q (cf. Lemma 7.37). Since P, Q are compatible, then so are pi, Q' (Lemma 7.42). Also a is compatible with pi and Q', hence with pi A Q', and in turn with P V Q = (Pi A Q')'. D Recall that compressions P and Q are orthogonal if P tion 7.35).

:<

Q' (Defini-

8.5. Proposition. Assume the standzng hypothes'ts of th'ts chapter. If P, Q are orthogonal compresswns, both compat'tble w'tth an element a E A, then P V Q 'IS compat'tble wzth a, and (8.3)

(P V Q) a

= Pa + Qa.

Proof. Since P and Q are compatible with a, pi and Q' are also compatible with a. Since P, Q are orthogonal, P, Q are compatible, as are pi and Q' (Lemma 7 42). By Lemma 8.4, PVQ = (Pi AQ')' is compatible with a, so it follows from Theorem 8.3 that (8.4)

a = (P V Q) a

+ (Pi

A

Q') a = (P V Q) a

+ PIQ'a .

Since P and Q are compatible with a, pi a = a - Pa and Q' a Since P -1 Q, then Q :< pi, so PIQ = Q. Hence

plQ'a = F'(a - Qa) = pia - Qa = a - Fa - Qa. Substituting this value for FIQ' a in (8.4), gives (8.3). 0

= a - Qa.

8

254

SPECTRAL THEORY

We will occasionally write PI -+- ... -+- P n in place of PI V ... V P n if PI, ... ,Pn are mutually orthogonal compressions. Thus by definition (8.5) P

= H-+-· .. -+- P n

<¢==?

P

=

PI V ... V P" and Pi ~ P J for

2

=F ].

8.6. Lemma. Assume the standmg hypothes2s of thzs chapter. P, Q, R are mutually orthogonal compresswns, then P ~ (0 -+- R).

If

Proof. Let p, q, r be the projective units associated with P, Q, R. Since p ~ q and P ~ r, then P ::; q' and P::; r'. Thus P::; q' i\ r' = (q V r)', so P ~ (q V r). Therefore P ~ (Q V R). Since Q, R are orthogonal, we can also write P ~ (Q -+- R). 0

8.7. Proposition. Assume the standmg hypothes2s of thzs chapter. If PI"'" P n are mutually orthogonal compresszons, all compat2ble wdh a E A, then Pi -+- ... -+- P n zs compatzble wzth a, and (Pi

(8.6)

-+- ... -+- Pn) a

+ ... +

= Ha

Pna.

Proof. By Lemma 8.6, repeated use of Proposition 8.5 gives (8.6). 0

8.8. Proposition. Assume the standmg hypotheszs of th2s chapter. If PI, P2, ... ,Pn are pro]ectwe unzts, then the followmg are equwalent: n

(i) LPi::; 1, i=1

(ii) Pi ~ PJ for (iii)

n

n

i=1

i=1

V Pi = L

2

=F],

Pj'

Proof. (i) =;. (ii) If Pi + PJ ::; 1, then Pi ::; 1 - Pj = Pj, so Pi ~ PJ' (ii)=;.(iii) If Pi ~Pj for z =F], then by Proposition 8.7,

nPi = (n) V VPi 1 = L Pi 1 = L Pi . i=1

i=1

n

n

i=1

i=l

(iii) =;. (i) If (iii) holds, then trivially L~=1 Pi = V~=l Pi ::; 1.

0

L;'=l

Observe that it follows from Proposition 8.8 that Pi = P for a projection P iff V;'=l Pi = P and Pi ~ PJ for z =F ]. So if Pi, . .. ,Pn and P are the compressions associated with Pi, ... ,pn and p, then (8.7)

Pi

+ ... + Pn

= P

<¢==?

PI

-+- ... -+- P n

= P.

8.9. Lemma. Assume the standing hypothesis of this chapter. If P, Q are compressions such that Q ~ P and p, q are the associated projective

255

THE LATTICE OF COMPRESSIONS

umts, then p - q 1,S the proJectwe umt assocwted w1,th P 1\ Q' = PQ'. Moreover, R = P 1\ Q' is the umque compresswn such that P = Q R, and 1,f a E A 1,8 compatzble w1,th P and Q, then Ra = Pa - Qa.

+

Proof. Since Q :< P, then P, Q are compatible, so P, Q' are also compatible (Lemma 7.42). By Theorem 8.3, P 1\ Q' = PQ'. Hence (PI\Q')l =P(l-Ql) =p-q, so p-q=pl\q'. By (8.7), for a compression R the equation P = Q R is equivalent to p = q + r, where r is the projective unit associated with R. Therefore the unique solution to this equation is the compression R = P 1\ Q' associated with the projective unit p - q = P 1\ q'. The last statement of the lemma follows from Lemma 8.4 and Proposition 8.7. 0

+

8.10. Theorem. Assume the standzng hypotheszs of th1,s chapter. The lattzce of compresswns (as well as the 1,somorphzc lattzces of proJectwe umts and proJectwe faces) 1,S orthomodular; that zs, for each pazr P, Q E C,

(i) pI! = P, (ii) Q:< P =} P' :< Q', (iii) P 1\ P' = 0 and P V P' = I, (iv) Q:< P =} P = (P 1\ Q') V Q. Proof. (i), (ii), and P V P' = I follow directly from the corresponding statements for projective units and Proposition 8.8 (iii). Now P 1\ P' = (P' V P)' = l' = 0 follows, so (iii) holds. Finally, (iv) follows from Lemma 8.9. 0 We state the following simple observation as a lemma for later reference.

8.11. Lemma. Assume the standing hypothesis of thzs chapter. If PI, P 2 , QI, Q2 are compresswns such that PI :< Ql, P 2 :< Q2 and Ql ..1 Q2, then also PI ..1 P 2 .

8.12. Lemma. Assume the standzng hypotheszs of thzs chapter. Two compresswns P, Q are compatzble zff (8.8)

P

=

P

1\

Q

+ P 1\ Q'.

Proof. Let p, q be the projective units associated with P, Q, and observe that (by (8.7)) the equation (8.8) is equivalent to the equation

(8.9)

p

= P 1\ q + P 1\ q' .

8

256

SPECTRAL THEORY

If P, Q, and then also P, Q', are compatible, then Pi\Q = PQ and Pi\Q' = PQ' (Theorem 8.3). Thus in this case, P i\ q + p i\ q'

= (P i\ Q) 1 + (P i\ Q') 1 = P(Q + Q') 1 = PI = p,

so (8.9) is satisfied. Conversely, assume (8.9). Since pi\q::; q, then pi\q E im+Q (Lemma 7.29), so Q(pi\q) = pi\q. Since pi\q' ::; q', then pi\q' E im+Q' = ker+Q, so Q(p i\ q') = O. Now

Qp = Q(p i\ q)

+ Q(p i\ q')

= P i\ q ::; p,

so P, Q are compatible. D Note that generally (Pi\Q)..l (Pi\Q') (Lemma 8.11), so we can replace in (8.8). Also we can replace = by ::S since the opposite relation is trivial. Therefore the compatibility criterion (8.8) can be rewritten as

+ by V

P::S (P i\ Q) V (P i\ Q').

(8.10)

8.13. Theorem. Assume the standzng hypotheszs of thzs chapter. Two compresswns P, Q are compatzble zfJ there exist three compresswns R, S, T such that S ..1 T and

P = R

(8.11)

+S,

Q= R

+T.

If such a decompositwn exists, then it zs umque; zn fact R = P i\ Q, S = P i\ Q', T = Q i\ P'. Proof. Assume first that P, Q are compatible and set R = P i\ Q, S = P i\ Q', T = Q i\ pl. Clearly S ..1 T (Lemma 8.11), and (8.11) follows from Lemma 8.12. Assume next that R, S, T are three compressions such that S ..1 T and (8.11) is satisfied. Note first that since R::s P and R::s Q, then R::s Pi\Q. Note also that since R ..1 S, then R ::S Sf, and since T ..1 S, then T ::S st. Hence Q = R V T ::S Sf, so S ::S Q'. We also have S ::S P, so S::SPi\Q'. Combining the two inequalities just proven, we find that P = R V S ::S (P i\ Q) V (P i\ Q'). Hence P, Q are compatible by the criterion (8.10). Next we show that R = P i\ Q. Write r = RI, s = Sl, t = Tl. Then p = r + s. Since S ..1 Rand S ..1 T, then S ..1 Q (Lemma 8.6). Therefore Qs = QSI = O. Since r ::; q, then Qr = r (Lemma 7.29), so

r = Q(r

+ s) =

Qp = QPI = (Q i\ P)I = q i\ p,

THE LATTICE OF COMPRESSIONS

257

where the penultimate equality follows from Theorem 8.3. Finally, s

=p

- r

=

(by (8.9)), and similarly t the proof. 0

=p

- (p A q)

= P A q'

q A p'. This gives uniqueness and completes

8.14. Corollary. Assume the standmg hypotheszs of thzs chapter. If P, Q are compahble compresswns, then (8.12)

P

V

Q= P

A

Q -+- P

A

Q' -+- Q A pi = P -+- Q A P'.

Proo]. By (8.11), P V Q is the least upper bound of the compressions Q), (P A Q'), (Q A Pi). These compressions are mutually orthogonal (Lemma 8.11), so we have the first equality in (8.12). By Lemma 8.12, we also have the second equality. 0

(P

A

8.15. Corollary. Assume the standmg hypotheszs of thzs chapter. If p, q are compatzble proJectwe umts, then (8.13)

p

+q=

p V q + p A q.

Proo]. Using (8.9) (and (8.9) with p and q exchanged), we have p

+q=

p A q + p A q'

+ q A P + q A p'.

Now (8.13) follows from (8.12). 0 Recall that a Boolean algebra is a distributive orthocomplemented lattice, i.e., a lattice with a least element 0, a greatest element I and a complementation P f-> pi which satisfies the conditions (i), (ii), (iii) of Theorem 8.10 and the distributive law (8.14)

P

A

(Q

V

R) = (P

A

Q)

V

(P

A

R).

Note that for an orthocomplemented lattice (8.14) is equivalent to the "dual" distributive law, obtained by interchanging V and A. Note also that condition (iv) of Theorem 8.10 (the "orthomodular law") follows from (8.14) by replacing R by Q'. Thus Boolean algebras are a special class of orthomodular lattices. Generally we will say that a set of elements in an orthocomplemented lattice is a complemented sublattice if it contains 0 and I and is closed under the complementation P f-> pi as well as the lattice operations P, Q f-> P AQ and P, Q f-> P V Q.

8

258

SPECTRAL THEORY

8.16. Theorem. Assume the standmg hypothesis of thzs chapter. A set L of compresswns zs a Boolean algebra (under the operatwns mduced from the complemented lattzce C of all compresswns) zff zt zs a complemented sublattzce of C and each pazr P, Q E L zs compatzble. Proof. Consider first a complemented sublattice Lee which is a Boolean algebra. If P, Q E L, then by the distributive law,

P = P

A

(Q

V

Q') = (P

A

Q)

V

(P

A

Q'),

so the compatibility criterion (8.8) is satisfied. Assume next that L is a a complemented sublattice of C for which each pair P, Q E L is compatible. To prove the distributive law (8.14), we consider an arbitrary triple P, Q, R of compressions in L with associated projective units p, q, r. By Corollary 8.14 (together with Theorem 8.3 and Proposition 8.7), PA (Q V R) 1

= P(Q-tRQ') 1 = PQ1 -tPRQ'l.

Hence

s: (p A q) V (p A r). In fact, the equality sign must hold in this relation, since p A q s: P A (q V r) and also pAr s: p A (q V r). This gives (8.14) and completes the proof. 0 P A (q V r)

= (p A q)

V (p A r A q')

Recall from Proposition 7.31 that if A, V is a pair of an order unit space and a base norm space in separating order and norm duality and if Po is a compression with associated projective unit Po and associated projective face Fo, then Po(A) = im Po is an order unit space with distinguished order unit Po, Po (V) = im Po is a base norm space with distinguished base Fo, and Po(A), Po(V) are in separating order and norm duality.

8.17. Proposition. Assume the standmg hypotheszs of thzs chapter and let Po be a compresswn wzth assocwted proJectwe unit Po and associated proJectwe face Fo. Then the pmr Po(A), Po(V) wzll also satzsfy the standmg hypotheszs. For thzs pmr the compresswns are the compresswns P of A such that P .:::S Po (restncted to Po(A)}, the proJectwe umts are the proJectwe units p of A such that p Po, and the proJectwe faces arc the proJectwe faces F of K such that F C Fo. For thzs pmr, the complementary proJectwn of P .:::S Po zs P' (restrzcted to Po(A)).

s:

Proof. Let F be an exposed face of the distinguished base Fo of Po (V). This means that there exists an element of Po (A) + which vanishes

precisely on F. Otherwise stated, there exists an element a of Po(A)+ such that a = 0 on F and a > 0 on Fo \ F.

THE LATTICE OF COMPRESSIONS

259

Now define b = a + P61. We know that PM = 0 on Fa, and also that P61 = pS > 0 on K \ Fa (d. (7.25) and (7.22)). Hence b = a = 0 on F, b = a > 0 on Fa \ F, and also b > 0 on K \ Fa. Thus F is an exposed, hence projective, face of K. Let P be the compression and p be the projective unit associated with F. Since F c Fa, then P ~ Po, so P and Po, and pI and Po, are compatible, and therefore commute (Lemma 7.42). It follows that P and pI leave Po(A) invariant. The restrictions PIPo(A) and PllPo(A) are seen to be complementary compressions (with duals P* IPO' (V) and p l*IPO' (V)) whose associated projective units are p and p' 1\ Po, and whose associated projective faces are F and Fa 1\ F'. This shows that the pair Po (A), PO' (V) satisfies the standing hypothesis, and also that the compressions, projective units and projective faces for this pair of spaces are as stated in the proposition. 0

In Proposition 8.17, we observe that if A that the dual of im PO' is im Po.

= V*, then it is easy to check

8.18. Corollary. Assume the standmg hypothesIs of thIs chapter and let Po be a central compresswn. Then a compresswn P ~ Po is central for Po(A) Iff It IS central for A.

Proof. By Proposition 7.48, P is central for A (or Po(A)) iff Pa ~ a E A+ (respectively, all a E Po(A)+). is central for A, then trivially Pa ~ a for all a E Po(A)+ C A+, central for Po(A). Conversely, assume P is central for Po(A). ~ Po, then P = PPo, so for each a E A+,

for all a If P so P is Since P

Pa = P Poa

~

Poa

~

a.

Thus P is central for A. 0

Remark. We will continue the discussion of the order theoretic approach to quantum mechanical measurement theory initiated in the Remark after Proposition 7.49 in the preceding chapter, but now under the standing hypothesis of the present chapter. This hypothesis says that each exposed face, i.e., each set of the form F = {w E K I (a, w) = O} with a E A+, is a projective face. Interpreted physically, this means that for each positive observable a E A+ the question: "does a have the value 0 7" is a quantum mechanical proposition, i.e., a question which can be answered by a quantum mechanical measuring device, say a filter for a beam of particles. This is true in standard quantum mechanics, and it is natural to assume it as an axiom in the general order-theoretic approach. As explained before, the expected value of an observable a E A measured on a system in a state w E K, is (a, w). For a proposition pEP, the

8

260

SPECTRAL THEORY

expectation (p, w) is the same as the probability of the value l. Therefore the face F associated with p is the set of all states for which the particle will certainly (i.e., with probability 1) appear with the value l. Now it follows from Theorem 8.10 that when the standing hypothesis is satisfied, then the propositions form an orthomodular lattice with the same physical interpretation as in standard quantum mechanics. Thus the proposition lattice has the properties of "quantum logic", which are often used as the basic assumption in axiomatic quantum theory. (See, for example, Piron's paper [87].)

The lattice of compressions when A

= V*

In this section we will continue to study a pair of spaces A, V satisfYlllg the standing hypothesis of this chapter, but we will now also assume that A = V*. Clearly, this condition is satisfied in the motivating examples of von Neumann algebras and JBW-algebras in duality with their preduals. We know that if A = V*, then we can identify A with the space A b(]() of all bounded affine functions on K (by the map a f--+ of (A 11)). Then the ordering in A will be the same as the pointwise ordering in A b (](), and the weak topology in A (defined by the duality with V) will be the same as the topology of pointwise convergence in Ab(K). Since A = V*, the ordered vector space A is monotone complete. Thus each increasing net {aoJ bounded above has a least upper bound a E A, which is also the weak (= pointwise) limit of {aa}, and in this case we will write a", / a. Similarly each decreasing net {a",} bounded below has a greatest lower bound a E A, which is also the weak (= pointwise) limit of {an}, and in this case we will write a", ~ a. Note that if aD: / a (or a", ~ a) and P is a compression, then (by weak continuity) Pac> / Pa (respectively

a

Pa",

~

Pa).

8.19. Lemma. Assume the standmg hypothesls of thls chapteT and A = V*. If {Pc>} lS a deCTeasmg net of pTO]ectwe umts, and Pc> ~ a E A, then a lS a pTO]ectwe umt, so d lS also the gTeatest 10weT bound of {p,,} m P. SlmllaTiy, If {Pa} lS an mCTeasmg net of pTo]ective umts, and Pa / a E A, then a lS a pTO]ectwe umt, so d lS also the least uppeT bound of {Pa} m P. PTOOj. Assume P'" ~ a. Let Fa be the projective face associated with P'" for each ex, and set G = {w E K I (a,w) = O}. Now G is an exposed, hence projective, face. Let P be the compression, and p the projective unit, associated with G'. For each ex,

G => {w E K I (p""w) = O} = F~. Hence G' c Fa, so p ::; Pc" and then p ::; a. But by (7.23), p is the greatest element of At which vanishes on G, so p = a. Thus a is a

THE LATTICE OF COMPRESSIONS WHEN A

= V*

261

projective unit. The remaining statement can be proven in similar fashion, or can be derived from the first by considering the net {I - Pa}. 0

8.20. Proposition. IJ the standing hypothesis oj this chapter is satisfied and A = V*, then C, P, F are complete lattzces, and the lattzce operations are gwen by the Jollowmg equatwns Jor each Jamzly { Fa} c F: (8.15)

Moreover, zJ {PaJ zs an mcreasmg net oj compresswns compatzble wzth an element a E A, then the compresswn P = Va: P a zs also compatzble with a and Paa / Pa. Szmzlarly, zJ {Pa} is a decreasing net oj compressions compatzble wzth a E A, then P = Aa Po. is compatzble wzth a and P",a "" Pa. Proof. By Proposition 8.1, C, P, F are isomorphic lattices, so it suffices to show that F is a complete lattice with lattice operations as in (8.15). In fact, by (8.1) it suffices to show that the first equality in (8.15) is satisfied for each decreasing net {F",} in :F. Let {F",} be a decreasing net in F and let Po. be the projective unit associated with Fa for each oo. By Lemma 8.19, Pa "" P where p is the greatest lower bound of {Po.} in P. Therefore the projective face F associated with p is the greatest lower bound of {Fa} in F. For each a, F", is the set of points in K where Po. takes the value 1. Also F is the set of points in K where p takes the value 1. If w E Fa, then (Pa, w) = 1 for each a, hence also (p, w) = 1, so w E F. Conversely, if w E F, then 1 = (p,w)::; (Pa,w) for each a, so wE Fa. Thus F = naFa, and we are done. It suffices to prove the last statement of the proposition for an increasing net {Pa}. Without loss of generality we assume 0 ::; a ::; 1. We have Paa ::; a for each oo. If a < (3, then P a ::S P{3 so Paa = P{3Pa a ::; P{3a. Thus {Paa} is an increasing net bounded above by a. Since Paa ::; a, applying P to both sides gives Paa ::; Pa for each oo. Set P = Va Po. and also p = PI and Po. = Po. 1 for each oo. By Lemma 8.19, Pa / p, so p - Pa "" O. For each a, let Ra = P 1\ P~. By Lemma 8.9,

nO.

nO.

Hence

Thus Paa /

Pa.

262

8

SPECTRAL THEORY

By compatibility, Paa ::: a for each oo. Thus Pa ::: a, so P compatible with a. 0

IS

also

Recall from (A 1) that a subset F of K is a semi-exposed face if there is a family {aa} in A + such that (8.16)

F = {w E K I (aCt, w) = 0 for all oo}.

By our standing hypothesis, each exposed face of K is projective. With the additional assumption A = V*, each semi-exposed face is also projective, (and therefore is exposed, cf. (7.22)), as we will now show.

8.21. Corollary. If the standmg hypothesis of th~s chapter and A = V*, then each semi-exposed face of K zs pro]ectwe.

~s sat~sfied

Proof. Let F be a semi-exposed face of K, given as in (8.16). For each index 00, we denote by Fa the set of points in K where a", takes the value zero. Then each Fa: is an exposed, hence projective, face. By (8.16), F = F",. Now it follows from Proposition 8.20 that F is a projective face. 0

n",

8.22. Definition. Assume the standing hypothesis of this chapter and A = V*. If a E A, then the set of all compressions compatible with a and with all compressions compatible with a, will be called the C-bzcommutant of a. We will also refer to the corresponding sets of projective faces and projective units as the F-bzcommutant and P-bzcommutant respectively. The use of the term "bicommutant" is of course motivated by its use in operator algebras. It is also partly justified by the fact that in our present context, compatibility of compressions is the same as commutation.

8.23. Proposition. Assume the standmg hypothes~s of this chapter and A = V*. The C-bzcommutant of an element a E A is a complete Boolean algebra. More speczfically, it ~s closed under arbztrary (fimte or mfinite) latt~ce operatwns m C and zs a Boolean algebra under these operatwns. Proof. Let B be the C-bicommutant of a. Consider two compressions PI, P 2 E B. Each of them is compatible with a and with all compressions compatible with a. Hence they are compatible with each other. Then by Lemma 8.4, PI A P 2 and PI V P 2 are also compatible with a. Moreover, if Q is a compression compatible with a, then PI A P 2 and PI V P 2 are also compatible with Q (again by Lemma 8.4, this time with the projective unit q = Q1 in place of a). Thus B is closed under finite lattice operations induced from C. It follows by an easy application of Proposition 8.20 that B is also closed under infinite lattice operations.

THE LATTICE OF COMPRESSIONS WHEN A

= V*

263

Clearly B contains 0 and 1, and P E B implies pi E B. Thus B is a complemented sublattice of C. Since each pair of compressions in B is compatible, B is a Boolean algebra (Theorem 8.16). 0

8.24. Proposition. Assume the standzng hypothes~s of th~s chapter and A = V*. For each a E A + there ~s a least pro]ectwe unzt p such that a E face(p), and p is the unzque element of P such that

(8.17)

{w E K

I

(a,w;

= O} =

{w E K

I

(p,w;

= O}.

Proof. Let G be the set of points in K where a takes the value zero. Then G is an exposed, hence projective, face. Let p be the projective unit associated with G'. Thus G' is the set of points where p takes the value 1, and G is the set of points where p takes the value O. Now (8.17) is satisfied, and this equation determines p uniquely (by the 1-1 correspondence of projective faces and projective units). Leaving aside the trivial case a = 0, we define al = lIall-1a E By (7.23), p is the greatest element of At which vanishes on G, so al ::; p. Now a::; Iiall p, so a E face(p). Let q be a projective unit such that a E face(q). Then a ::; )..q for some ).. ::::: 0, so the projective face F of points where q takes the value zero is contained in the projective face G of points where a takes the value zero. Hence G' c F', so q' ::; pi, and therefore p::; q. Thus p is the least projective unit such that a E face(q). 0

At.

8.25. Definition. Assume the standing hypothesis of this chapter and A = V*. For each a E A+ we denote by r(a) the least projective unit p such that a E face(p). Note that by (8.17), r(a) is characterized among projective units by the following equivalence where w E K:

(8.18)

(a,w;

=0

{==}

(r(a),w;

= O.

Clearly r(a) is the customary range projection in the von Neumann algebra case, d. (A 99).

8.26. Proposition. Assume the standzng hypothes~s of th~s chapter and A = V*. If a E A+, then r(a) ~s characterized by each of the followzng statements.

(i) The compression assocwted w~th r(a) ~s the least compresswn P such that Pa = a. (ii) The complement of the compresswn associated with r(a) is the greatest compression P such that Pa = O. (iii) r(a) ~s the least projective unzt p such that a::; Iialip.

8

264

SPECTRAL THEORY

(iv) r(a) zs the greatest pro]ectwe umt contazned zn the semi-exposed face of A + generated by a. Proof. (i) and (iii) follow from Lemma 7.29, and (ii) is equivalent to f--> pi reverses order and ker+ pi = im+ P. For the proof of (iv), we recall that the semi-exposed face generated by a is equal to {a}-(Proposition 7.2). Thus a projective unit p is in this face iff;t is annihilated by each w E K that annihilates a, i.e., iff

(i) since P

{w E K I (a,w)

= O}

C {w E

K I (p,w) = O}.

By (8.17) and (7.22), the left side IS the projective face associated wIth r(a)/, and the right with pi, so this is equivalent to r(a)' ::; p'. Thus p ::; r( a), so r( a) is the greatest projective unit in the semi-exposed face generated by a. 0 By statement (iii) of the proposition above, we generally have a Note in particular that

<

Iiall r(a). (8.19)

a::; r(a)

when

0::; a::; 1.

Recall from (A 24) that two elements p,O" E V+ are said to be orthogonal, in symbols p -.l 0", if

(8.20) In particular, two elements p,O" E K are orthogonal if is, if their distance is maximal in K.

lip - 0"11

=

2, that

8.27. Theorem. Assume the standzng hypotheszs of thzs chapter and A = V*. Every w E V admits a umque decomposztion w = p - 0" such that p,O" ;::: 0 and p -.l 0", and tlm decomposztwn zs gwen by p = P*w and 0" = -pl*w for a compression P of A. Proof. By (A 26), there is a decomposition w = P-O" such that p,O" ;::: 0 and p -.l 0". We will show that this decomposition is unique and of the given form, by constructing a compression P, depending only on w, such that p = P*w and 0" = _P'*W. Assume (without loss of generality) that Ilwll = 1. Since A = V*, the closed unit ball Al = [-1,1] is w*-compact. Therefore the linear functional a f--> (a,w) attains its norm at a point x E AI. Thus -1 ::; x::; 1 and (x,w) = 1. Define a = ~(1 + x) and b = ~(1 - x). Then 0::; a ::; 1 and 0::; b ::; 1, and x = a-b. Now 1

=

(x, w)

::; (a, p)

=

(a, p) - (b, p) - (a,O")

+ (b, 0")

::;

Ilpll + 110"11

=

+ (b, 0")

Ilwll·

THE LATTICE OF COMPRESSIONS WHEN A Since

Ilw I =

= V*

265

1, the equality signs must hold. Therefore we conclude that

Ilpll

(a, p) =

(8.21)

and

(a,o-) =

o.

(Similarly (b, eJ) = IleJll and (b, p) = 0, but we shall not need these equalities. ) Let P be the compression and F the projective face associated with r (a). Leaving aside the trivial cases P = 0 and eJ = 0, we define PI = Ilpll-lp and eJl = IleJll-IeJ. We have 0 <:::: a <:::: r(a) <:::: 1 (cf. (8.19)), so by (8.21)

Thus (r(a), PI) = 1. Hence PI E F, so P E im+ P* = ker+ P'*. Therefore P* P = P and p l*p = o. By (8.21) we have (a, eJI) = 0, so it follows from (8.18) that we also have (r( a), eJI) = o. Hence eJl E F' , so eJ E im + p l* = ker+ P*. Therefore pl*eJ = eJ and p*eJ = o. Combining the equalities above, we find that P*w = p and P'*W = -eJ. The proof is complete. D

8.28. Definition. Assume the standing hypothesis of this chapter and A = V*. An element w E V is called central if (P + PI)*W = w for all compressions P. This definition is motivated by C*-algebras and von Neumann algebras. It is easily seen that a bounded linear functional w on a C*-algebra A satisfies the equation w(ab) = w(ba) for all pairs a, b E A iff its normal extension to the enveloping von Neumann algebra A ** satisfies the same requirement, and this is the case iff

(pap

+ pi api, w) =

(a, w)

for all projections pEA ** and all a E A **, i.e., iff w is central as defined above (cf. [7, Lemma 12.8 and Thm. 12.9]).

8.29. Lemma. Assume the standmg hypothesis of th7s chapter and A = V*. If w zs a central element of V, and w = p - eJ zs the orthogonal decompositwn descr7bed m Theorem 8.27, then p and eJ are also central.

Proof. For every compression P we have w = (P (8.22)

w = (P

and since (P (8.23)

+ PI)*

Ilwll

=

+ PI)* p -

(P

+ PI)*W,

so

+ PI)*eJ,

preserves norms in V+ (Lemma 7.17), then

Ilpll + IleJll

=

II(P + PI)*pll + II(P + pl)*eJll·

266

8.

SPECTRAL THEORY

By the uniqueness of orthogonal decompositions, it follows from (8.22) and (8.23) that

p=(p+pl)*p

and

0'= (p+pl)*O'.

Hence p and 0' are central. D 8.30. Proposition. If the standzng hypothes2s of thls chapter 2S sat2sfied and A = V*, then the set of central elements zn V 2S a vector lattlce Proof. The set of central elements in V is a linear subspace of V, and is positively generated by Lemma 8.29. To prove that it is a vector lattice, it suffices to show that the clement p in the orthogonal decomposition w = p - 0' is the least upper bound of wand 0 within the space of central elements. Clearly p 2: wand p 2: o. Suppose T is any central element such that T 2: wand T 2: O. Let P be a compression such that p = Pw and 0' = -plw (Theorem 8.27). Then T

=

(P

+ PI)*T 2:

P*T 2: P*w

=

p,

as desired. D By Proposition 7.30, each projective unit is an extreme point of At. In the special case of a von Neumann algebra or a JBW-algebra the converse is also true, cf. (A 75) and Proposition 1.40, but we cannot prove this in our present generality. However, we have the following partial converse. 8.31. Proposition. If the standzng hypothes2s of th2s chapter 2S satisfied and A = V*, then the pro]ectwe umts are w*-dense zn the set of extreme pmnts of At. Proof. By Milman's theorem (A 33), it suffices to prove that co P = At (w*-closure). Since the map a f--7 2a - 1 is an affine homeomorphism from At = [0,1] onto AI, we may as well show that co S = Al where S = {2p - 1 I pEP}. This will follow (by Hahn-Banach separation as in (A 30)) if we can prove that for each w E V,

(8.24)

Ilwll

= sup l(s,w)l· sES

By Theorem 8.27, for given w E V we can choose an orthogonal decomposition w = p - 0' where p,O' 2: 0, P 1- 0' and p = P*w, 0' = _pl*W for some P E C. Let p be the projective unit associated with P and set s = 2p - 1. Then s E Sand Ilwll

= Ilpll + 110'11 = (I, p) + (1,0') = (PI - P'1, w) = (8, w),

which proves (8.24).

D

267

THE LATTICE OF COMPRESSIONS WHEN A = V*

We are now going to establish an explicit formula expressing the range projection of Pa where P is a compression and a E A+, in terms of the projective unit associated with P and the range projection of a itself. This formula, proved for lBW-algebras in Proposition 2.31, is a useful technical tool. But it is also of some intrinsic interest in that it links compressions of A with the maps rpp : P ---+ P defined for each pEP by rpp (q) = (q V pi) I\p; these are the Sasaki projections of orthomodular lattice theory. 8.32. Theorem. Assume the standzng hypothes2s of th2s chapter and A = V*. If a E A+ and P zs a compresswn w2th assocwted projective umt p, then (8.25)

r(Pa)

= (r(a)

V pi) 1\ p.

Proof. Let Q be the compression associated with r(a) and let R be the compression associated with r(Pa). Thus Q V pi is the compression associated with r(a) V p'. Observe that since a E face (r(a))

= im+Q

C im (Q V Pi),

then (Q V Pi) a = a. Observe also that since pi ::S Q V pi, then pi, and hence also P, is compatible with Q V P'. Hence (Q V pl)Pa = P(Q V Pi) a = Pa.

Thus Q V pi fixes the element Pa. Therefore R ::S Q V pi (Proposition 8.26 (i)), so r(Pa) :s; r(a) V p'. Clearly P fixes Pa, so we also have r(Pa) :s; p. Thus (8.26)

r(Pa)

:s;

(r(a) V pi) 1\ p.

Since r(Pa) :s; p, then R::s P. Now R, and then also R ' , is compatible with P. Hence R' 1\ P = R'P, and since R' Pa = 0, then (R' 1\ P) a = o. Now a E ker+(R' 1\ P) = im+(R V Pi).

Thus R V pi fixes a. Therefore (by Proposition 8.26 (i)) r( a) Hence also (8.27)

r( a)

V

pi

:s; r( Pa)

V

:s;

r(Pa) V p'.

p'.

We have seen above that R:::s P, or equivalently R.l pi, so r(Pa) .l p'. Thus we can replace V by + on the right side of (8.27). Then after rearranging, r(a) V pi - pi ::; r(Pa).

268

8

SPECTRAL THEORY

By Lemma 8.9, the left side of this inequality is equal to (r(a) V pi) 1\ p. Thus (8.28)

(r(a) V pi) 1\ P ::; r(Pa).

The two inequalities (8.26) and (8.28) give (8.25). 0 For given non-zero w E V+ we will denote by ProjFace(w) the projective face generated by Wi := Ilwll~l w in K. Thus ProjFace(w) is the intersection of all F E :F which contain Wi (Proposition 8.20). We define ProjFace(O) to be the empty set.

8.33. Corollary. Assume the standing hypotheszs of thzs chapter and A

=

V*. For each w E V+ and each pEP wdh assocwted proJectwe face

F, (8.29)

ProjFace(P*w) = (ProjFace(w) V F') 1\ F.

Proof. Dualize the proof of Theorem 8.32 by replacing P by P*, a by w, r(a) by ProjFace(w) and r(Pa) by ProjFace(P*w). 0

8.34. Proposition. Assume the standmg hypotheszs of thzs chapter and A = V*. For each w E K there zs a least central proJectwn c( w) such that (c( w), w) = 1; the assocwted compresswn zs the least central compresswn P such that P*w = w, and the assocwted projective face zs the smallest spht face whzch contams w. Proof. Recall first that for w E K and PEP one has (PI, w) = 1 iff = w, cf. (7.21) and (7.24). If Ci,C2 are central projective units such that (Ci'W) = (Ci'W) = 1, then the associated compressions Pi, P 2 are also central, so P i l\P2 = P i P 2 . Hence P*w

so (Cil\c2'W) = 1. Therefore the set of central projective units c such that (c,w) = 1 is directed downward. By Lemma 8.19, this directed set has a greatest lower bound c(w) E P which is also its weak limit. Therefore, by weak continuity, (P + Pi) c(w) = c(w) for all P E C. Thus c(w) is compatible with all P E C, i.e., c(w) is central. By weak convergence (c(w),w) = 1. Clearly c(w) is the least central projective unit with this property. By the introductory remark of this proof, the compression P associated with c(w) is the least central compression such that P*w = w. By the explicit characterization of the projective face F associated with c(w) (7.24), and by Proposition 7.49, F is the smallest split face containing w. 0

THE LATTICE OF COMPRESSIONS WHEN A = V*

269

8.35. Definition. Assume the standing hypothesis of this chapter and A = V*. For each w E K the least central projective unit which takes the value 1 at w will be called the central support (or central carner) of w, and it will be denoted by c( w) (as in the proposition above). We now turn to the study of minimal projective units.

8.36. Proposition. Assume the standmg hypothes~s of th~s chapter and A = V*. If p ~s a mmimal (non-zero) proJectwe umt, then the assoc~­ ated compression P has one dzmensional range, z.e., im P = Rp, and the assocwted proJectwe face F ~s a singleton, l. e., F = {p} where p lS the only pomt of K where p takes the value 1. Moreover, the map p f---4 P ~s a 1-1 map from the set of mmzmal elements of P onto the set of exposed pomts of K.

Proof. Let p be a minimal projective unit with associated compression P and associated projective face F. Relativizing to the pair of spaces P(A), P* (V) as in Proposition 8.17 and making use of Proposition 8.31, we conclude that the positive unit ball [O,p] of P(A) = imP is the w*-closure of the projective units in P(A), which are precisely those projective units q in A such that q ::; p. By minimality of p, there are no such projective units other than q = 0 and q = 1. Thus [0, p] contains only scalar multiples of p, so im P = Rp. Since P( A) and P* (V) are in separating duality, then P*(V) is also one dimensional, so F is a singleton. The remaining statements of the proposition follow easily from the explicit formulas for F in terms of p and P, cf. (7.22) and (7.24), and from our standing hypothesis that all exposed faces are projective. 0 We will now define orthogonality in A + by a condition similar to the characterization in Theorem 8.27 of orthogonality in V+. (Like the characterization in Theorem 8.27, this condition will not be useful unless we assume our standing hypothesis, or some other hypothesis which ensures the existence of "enough" compressions.)

8.37. Lemma. Assume the standing hypothesls of thls chapter and A = V*, and let b, c E A +. If P zs a compresswn wdh assocwted projective face F, then the followmg are equwalent:

(i) Pb=b and P'c=c, (ii) P'b = 0 and Pc = 0, (iii) b = 0 on F' and c = 0 on F. Proof. (i) {c? (ii) This follows from the fact that P and pI are complementary. (ii)=}-(iii) Assume (ii). If w E F ' , then (b,w) = (P'b,w) = 0 by (7.25). Thus b = 0 on F'. Similarly c = 0 on F. (iii)=}-(i) Assume (iii). By Lemma 7.43, Pb = band F'c = c. 0

270

8

SPECTRAL THEORY

8.38. Definition. Assume the standing hypothesis of this chapter and A = V*. We will say that two elements a, bE A+ are orthogonal, and we will write a ..l b, if there exists a compression P such that anyone of the three equivalent conditions in Lemma 8.37 is satisfied. We will also say that an equality a = b - c is an orthogonal decompositwn of an element a E A if b, c E A + and b ..l c. Note that Pa ..l PIa for all a E A+ and P E C. Note also that if a = b - c is an orthogonal decomposition where Pb = b and pIC = c for a compression P, then b = Pa and c = -pia, from which it follows that a = Pa + pi a, so P is compatible with a. These facts will be used frequently and without reference in the sequel.

8.39. Proposition. Assume the standmg hypothesls of thls chapter = V*, and let a, bE A+. Then a..l b lff r(a) ..l r(b), and If a ..l b,

and A then

(8.30)

r(a

+ b) = r(a) + r(b).

Proof. Let a ..l b, say Pa = a and P'b = b for a compression P. Let Ra and Rb be the compressions associated with r(a) and r(b) respectively. Since Pa = a, we have Ra j P, and similarly Rb j P'. Then Ra ..l Rb (Lemma 8.11), so r(a) ..l r(b). Conversely, if r(a) ..l r(b), then RbRa = 0 (Proposition 7.34), so Rba = RbRaa = O. Then Rba = a and Rbb = b imply a..l b. Now assume a ..l b. We may assume without loss of generality that a, b ::; 1. Then a::; r(a) and b ::; r(b) (cf. (8.19)). By Proposition 8.8,

(8.31 )

a

+ b ::; r(a) + r(b) = r(a)

V r(b).

Thus the projective unit r(a) V r(b) majorizes a r(a

Conversely, r(a) ::; r(a

+ b)

+ b)

+ b.

Therefore

::; r(a) V r(b).

and r(b) ::; r(a

+ b),

r(a) V r(b) ::; r(a

+ b).

so

By (8.31) these two inequalities give the desired equality (8.30). D

A

8.40. Lemma. Assume the standmg hypothesls of this chapter and If a E A+ and Q is a compresswn compatible with a, then

= V*.

(8.32)

r(Qa) = Q(r(a)).

SPACES IN SPECTRAL DUALITY

271

Proof. Assume without loss that a ::; 1. We have Qa -.l Q' a, so it follows from Proposition 8.39 and compatibility that (8.33)

r(Qa)

+ r(Q'a) = T(Qa + Q'a) = T(a).

If q is the projective unit associated with Q, then r(Qa) ::; T(Q1) = q, so r(Qa) E face (q) = im+Q. Similarly T(Q' a ) E im+Q' = ker+Q. Now applying Q to (8.33) gives the desired equality (8.32). 0 8.41. Proposition. Assume the standmg hypotheszs of thzs chapteT and A = V*. If a E A +, then 1'( a) zs m the P -bzcommutant of a. Proof. If P is the compression corresponding to r(a), then Pa = a (Proposition 8.26), and so pia = 0, which gives (P + P')a = a. Thus r(a) is compatible with a. If Q is a compression compatible with a, then Qa ::; a (Lemma 7.37). Hence r(Qa) ::; T(a), and by Lemma 8.40 also Q(r(a)) ::; r(a). Thus T(a) is compatible with Q. 0

Spaces in spectral duality We will now establish a general spectral theory in the context of an order unit space A in separating order and norm duality with a base norm space V. This theory, first presented in [7] and [9], generalizes the spectral theory of von Neumann algebras and JBW-algebras. As in these cases, it may be regarded as a non-commutative measure theory which reduces to ordinary measure theory when A = LOO(fL) and V = L1(fL) for a probability measure fL (the commutative case). In the general case, the elements of A will play the role of the bounded measurable functions, and the orthomodular lattices F, P, C will play the role of the Boolean algebras of, respectively, measurable sets, characteristic functions of measurable sets and multiplication by characteristic functions of measurable sets.

8.42. Definition. Assume that A, V are a pair of an order unit and base norm space in separating order and norm duality, with A = V*. If in addition each a E A admits a least compression P such that Pa :2: a and Pa :2: 0, then we will say that A and V are in spectral duality. Generally, spectral duality will imply our standing hypothesis (that all exposed faces are projective), as we will show in Theorem 8.52. If A and V are reflexive Banach spaces, and in particular if A and V are finite dimensional, then the converse also holds, as we will show in Theorem 8.72. But in the general case, our standing hypothesis will not imply spectral duality. (A counterexample is sketched in [9, p. 508].)

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The new condition of spectral duality will ensure that there are enough elements in C (or in P) to approximate each a E A by elements 1 AiP, where PI, ... ,p" E P, and will provide a unique spectral resolution of a by the same construction as in the motivating examples. Note in this connection that in the commutative case, where the compressions are multiplication operators P : f f----+ XE . f, the requirements for P in the definition above characterize multiplication by XE, where E = a~I((O, (0)). Later on we will show that the self-adjoint part of a von Neumann algebra is in spectral duality with its predual, and likewise for a lBW-algebra (Proposition 8.76).

2::'=

8.43. Lemma. Let A, V be an order umt space and a base norm space m sepamtmg order and norm dualzty, let a E A and let P be a compresszon. Then Pa ~ a zjJ P zs compatzble wzth a and pI a :::; O. Proof. If P is compatible with a and pI a :::; 0, then trivially Pa a - pI a ~ a. Conversely, if Pa ~ a, then Pa - a E ker+ P

=

= im+ P'.

Hence pl(Pa - a) = Pa - a, so -PIa = Pa - a. From this we conclude that pI a :::; 0, and also that a = Pa + pI a, so P is compatible with a. D We state the following observation as a lemma for later reference.

8.44. Lemma. Let A, V be an order umt space and a base norm space m sepamtmg order and norm dualzty, let a E A and let P be a compresszon wzth assoczated pm]ectwe face F. Then the followmg are equwalent:

(i) Pa ~ a and Pa ~ 0, (ii) P zs compatzble wzth a and Pa ~ 0 and pIa:::; 0, (iii) F zs compatzble wzth a and a ~ 0 on F and a :::; 0 on F'. Thus A and V are m spectml dualzty zjJ there exzsts a least compresszon P compatlble wzth a such that ezther one of (z) or (zz) holds, or there exzsts a least pro]ectwe face F such that (zzz) holds. Proof. (i) <=? (ii) Clear from Lemma 8.43. (ii) <=? (iii) Clear from Proposition 7.28. D We are now going to establish that spectral duality implies the standing hypothesis of this chapter. To accomplish this, we will need to re-establish several key results with the hypothesis of spectral duality instead of the standing hypothesis. For this purpose, we now introduce a useful technical tool. For an element b ~ 0 in an order unit space A we write [0, b] for the order interval {a E A I 0 :::; a ::; b}. Recall that face(b) denotes the face generated by b in A+, i.e., the set of all a E A such that 0:::; a ::; Ab for some A E R+.

SPACES IN SPECTRAL DUALITY

273

8.45. Definition. Let A, V be in spectral duality. An element b of At has the facwl property if (8.34)

[0, b] = At n face(b).

The equivalence of (ii) and (iii) in Lemma 7.29 implies that every projective unit has the facial property. We now establish the converse, which is based on an argument of Riedel [106, 1.1(f) =? 1.1( c)].

8.46. Lemma. Assume A, V are m spectral dualzty. If bEAt has the facwl property, then b zs a pro]ectwe umt. Proof. Let bEAt have the facial property and define a = 2b - 1. Clearly -1 <::: a <::: 1. By spectral duality, there exists a compression P such that Pa 2: a and Pa 2: o. By Lemma 8.43, P is compatible with a, and then also with b. Let p = PI and pi = pl1. Since P(2b - 1) 2: 0, then 2Pb 2: p. By compatibility of P and b, 2b 2: 2Pb, so 2b 2: p. Thus by the facial property of b, we conclude that b 2: p. Since Pa 2: a, then 2b - 1 <::: P(2b - 1) <::: PI

= p.

Therefore 2b - p <::: 1. Since 2b - p = b + (b - p) 2: 0 and 2b - p <::: 2b, then 2b - p is in face(b) n [0,1]' so again by the facial property 2b - p <::: b. This implies b <::: p, so b = p, and thus b is a projective unit. D

8.47. Lemma. Assume A, V are m spectral dualzty. compatzble compresswns, then PQ = QP = P A Q.

If P, Q are

Proof. Define r = PQ1. The first two paragraphs of the proof of Theorem 8.3 apply without change to show that r is the greatest lower bound of p and q in A+. Next we show that r has the facial property. To this end we consider a E At n face(r). Since r <::: p, then a E face(p) = im+ P, so a = Pa. Since r <::: q, also a E face(q) = im+Q, so a = Qa <::: Q1 = q. From this it follows that a = Pa <::: Pq = T. Thus, r has the facial property, and by Lemma 8.46, r is a projective unit. Then by the first paragraph, T is the greatest lower bound of p and q in P. Thus T = P A q, or which is equivalent, r = (P A Q) 1. Now the final paragraph of the proof of Theorem 8.3 applies without change to show that PQ = QP = P A Q. D

Note that under the assumption of spectral duality, we do not yet know that the order isomorphic sets C, P, F are lattices. However, if P and

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Q are compatible compressions, by Lemma 8.47 the greatest lower bound P

Q exists. Furthermore, pi and Q' will be compatible, so pi A Qf exists, and then so does P V Q = (Pi A Q')'. Since orthogonal compressions are compatible (Lemma 7.42), the least upper bound of two orthogonal compressions exists. The following two results are the same as Lemma 8.4 and Proposition 8.5 respectively, except for the hypothesis being spectral duality instead of the standing hypothesis of this chapter. The proofs are the same as those of the corresponding earlier results, except that the application of Theorem 8.3 is replaced by Lemma 8.47. A

8.48. Lemma. Assume A, V are m spectral duaZzty. If P, Q are mutually compatzble compresswns, both compatzble wdh an element a E A, then P A Q and P V Q are also compatzble wzth this element. 8.49. Proposition. Assume A, V are m spectral dualzty. If P, Q are orthogonal compressions, both compatzble wdh an element a E A, then P V Q zs compatzble with a, and

(P V Q) a = Pa + Qa. Recall that if A = V*, then A is monotone complete. (See the discussion preceding Lemma 8.19). In particular, if A, V are in spectral duality, then A is monotone complete.

8.50. Lemma. Assume A, V are m spectral dualzty. If {Pn} zs a decreasmg (mcreasmg) sequence of proJectwe umts, then the element infn Pn (suPn Pn) of A zs a projectwe umt, hence zt zs the greatest lower bound (least upper bound) of {Pn} among the proJectwe units.

Proof. Assume that {Pn} is a decreasing sequence of projective units and set b = infn Pn. We will show that b has the facial property, so we consider an arbitrary a E n face(b). For every n = 1,2, ... , then a E face(b) C face(Pn), so a E im+ Pn , where Pn is the compression associated with Pn. Hence a = Pna <::: Pn1 = Pn for all n; therefore a <::: infn Pn = b, so b has the facial property. By Lemma 8.46, b is a projective unit. The case of an increasing sequence follows by taking complements. 0

At

8.51. Lemma. Assume A, V are m spectral duality, let a E A+, and let AI, A2 be real numbers such that 0 < Al < A2. If Q2 zs a compresswn compatzble wzth a, with assocwted projectwe unit Q2, such that (8.35)

275

SPACES IN SPECTRAL DUALITY

for I = 2, then there eXIsts a compreSSIOn Ql compatIble with a, with assocwted projectIVe umt q1, such that Q1 ::::: Q2, and such that (8.35) holds also for I = 1. Proof. Set b = Q2a. By Lemma 8.44 (applied to the element b - All E A), there exists a compression P compatible with b, having an associated projective unit p, such that Pb ::,. AlP and P'b <::: AlP'. We consider now the complementary compression Q = P' with associated projective unit q = p', and we observe that Q is also compatible with b, and that

(8.36) Using compatibility and (8.36), we observe that (8.37) We have Q;b = Q;Q2a = 0, and so by (8.37) Q;q' = O. Then Q2q' = q', so Q' ::::: Q2. From this it follows that Q', and then also Q, is compatible with Q2. Since Q' is compatible with b, then (8.38)

Q'a = Q'Q2a = Q'b <::: b <::: a.

Hence Q', and then also Q, is compatible with a. Define now Ql = QQ2 = Q /\ Q2, and let q1 be the projective unit associated with Q1. Since Q and Q2 are mutually compatible and both are compatible with a, then Ql = Q/\Q2 is also compatible with a (Lemma 8.48). It remains to prove that we can replace Q by Ql, b by a, and q by q1 in (8.36). Applying Q1 to both sides of the first inequality in (8.36) gives Qlb <::: Alq1' Since Qlb = Q1Q2a = Qla, this gives (8.39) which is the first of the desired inequalities. We have seen that Q' ::::: Q2, so Q' ..1 Q;, and by definition (Q /\ Q2)' = Q' V Q;. Hence by Proposition 8.49, (8.40)

Q~a

=

Q'a

Q~

+ Q~a.

By the same argument, Q~ 1 = Q'l + Q; 1, or otherwise stated, q~ = q' + q~. Note also that by (8.38) Q'a = Q'b. Combining these two facts with (8.40), (8.36), and (8.35), we get (8.41) which is the second of the two desired inequalities. With (8.39) and (8.41) we have shown that (8.35) holds also for I = 1. 0

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8.52. Theorem. If A, V are m spectral dualIty, then eveTY exposed face of K zs proJectzve. Proof. Let F be an exposed face of K and choose a E A + such that (8.42)

F = {w E K I (a,w) = O}.

Consider a sequence {An} of real numbers such that Al > A2 > ... > 0 and limn An = O. By repeated use of Lemma 8.51, we construct a decreasing sequence {Qn} of compressions compatible with a such that for each n = 1,2, ... the compression Qn and its associated projective unit qn satisfy the inequalities (8.43) Consider the compression Q = An Qn with associated projective unit q. Let G n be the projective face associated with Qn, and G the projective face associated with Q. By (8.43), for each n we have a::; An on G n , and therefore a ::; An on G = Am G m C G n . Then a ::; limn An = 0 on G. Since a:::: 0, then a = 0 on G. By (8.42), G C F. Conversely if w E F, then by (8.43) and the compatibility of Q;, and a,

so (q~,w) = 0 for n = 1,2, .... By Lemma 8.50, qn '" q, so q;, / q'. Hence (q',W) = 0, which shows that w E G. Thus G = F, which proves that F is projective. 0 Spectral duality can be defined in various other equivalent ways (among those the ones in [7] and [9]), as we proceed to show. In the next lemma (and in the sequel) we will write "a > 0 on F" when an element a E A is strictly greater than zero everywhere on a set Fe K, i.e., if (a,w) > 0 for all wE F.

8.53. Lemma. Assume the standmg hypotheszs of this chapter and A = V*. Let a E A, let P be a compression with assocwted proJectzve face F, and assume a zs compatzble wzth F and a :::: 0 on F and a ::; 0 on F'. Then G = {w E K I (Pa, w) = O} zs a proJectzve face whose complement P = G' zs compatzble wzth a and satzsfi,es the conditzons P C F, a > 0 on P and a ::; 0 on pi . Moreover, Fa '= Pa where F is the compresszo n assocwted wIth F, Proof. Observe first that Pa = 0 on F' and pia = 0 on F by (7,25), Furthermore, Pa :::: 0 and pi a ::; 0 follow from Proposition 7.28. Clearly

SPACES IN SPECTRAL DUALITY

277

G is an exposed, hence projective, face of K. Since Pa = 0 on FI, we have FI C G, and then G I c F. If wE G, then (a,w)

If wE G I

C

= (Pa,w) + (pla,w) = (pla,w)

<::: O.

= (Pa, w) + (PI a, w) = (Pa, w)

::::>

F, then (a, w)

O.

Here the inequality must be valid, otherwise w E G. Thus a <::: 0 on G and a> 0 on G I • Let Q be the compression associated with G. We have Pa = 0 on G, so QPa = 0 and QI Pa = Pa (Lemma 7.43). Since G I C F, we have pI a = 0 on G I. Therefore also QI pI a = 0 and Q pI a = pI a. Hence Qa

+ Qla =

Q(Pa

+ pIa) + QI(Pa + pIa) =

pIa

+ Pa =

a.

Thus Q, and then also G and G I , are compatible with a. Now has all the properties announced in the lemma. Finally, the compression F = QI associated with F satisfies so the equality QI Pa = Pa gives Fa = F Pa = Pa. D

F=

GI

F ::S

P,

8.54. Lemma. Assume that A, V are a pair of an order unit and base norm space m separatmg order and norm dualzty, wzth A = V*. Then the followmg are equwalent:

(i) A and V are m spectral dualzty. (ii) For each a E A there exzsts a umque pTOJectwe face F compatzble wzth a such that a > 0 on F and a <::: 0 on Fl. If these condztwns are safzsJied, then the face F m (u) zs also the least pTOJectwe face compatzble wzth a such that a ::::> 0 on F and a <::: 0 on Fl. PTOOf. (i) =? (ii) Assume A and V are in spectral duality, and let a E A. By Theorem 8.52, every norm exposed face is projective. By Lemma 8.44 there exists a least projective face F compatible with a such that a ::::> 0 ~n F and a <::: 0 on Fl. By Lemma 8.53, there exists a projective face Fe F with the same properties as F, but now with the strict inequality a> 0 on F. By minimality, F = F. Thus a> 0 on F. To prove uniqueness, we consider an arbitrary projective face FI compatible with a such that a > 0 on FI and a <::: 0 on F{. By minimality, Fe Fl. Thus we can apply Theorem 8.10 (iv) (the orthomodular law), by which FI = F V (FI 1\ F') = F V (FI n F').

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SPECTRAL THEORY

w

w;

Here we must have Fl n F' = 0. For if E Fl n F ' , then (a, > 0 since w E Fl and (a, w; ::; 0 since w E F ' , a contradiction. Thus Fl = F as desired. (ii) =} (i) We first show that exposed faces of K are projective. Fix a E A+, and let G = {w I (a,w; = O}. Let F be as in (ii); we will show

G=F'. Since a 2 0 on K and a ::; 0 on F ' , then a = 0 on F ' , so F' c G. Let w E G, and let P be the compression associated with F. Then P and a are compatible, so (8.44)

0= (a, w; = (Pa

+ pi a, w; =

(a, P*w;

+ (a, P'*w;.

Note that P*w and P'*W are multiples of elements of F and F' respectively. Since a = 0 on F ' , then (a, pl*w; = o. Thus by (8.44), 0= (a, P*w;. Since a > 0 on F, we must have P*w = 0, and so w E F'. Thus G = F ' , and G is projective. Now let a E A be arbitrary, and let F be the projective face with the properties in (ii). Let Fl be an arbitrary projective face compatible with a such that a 2 0 on Fl and a ::; 0 on F 1 . Consider once more the projective face F C Fl which satisfies the same requirements as F l , but now with the strict inequality a > 0 on F (cf. Lemm~ 8.53). Since F is unique under these requirements, we must have F = F C Fl. Thus F is the least projective face compatible with a such that a 2 0 on F and a ::; 0 on F'. By Lemma 8.44, the proof is complete. D 8.55. Theorem. Assume the standmg hypothesis of thlS chapter and A = V*. A and V aTe m spectral duallty lff each a E A has a umque orthogonal decomposltwn.

Proof. Assume A and V are in spectral duality. Let a E A and let

F be the unique projective face compatible with a such that a > 0 on F and a::; 0 on F' (Lemma 8.54). Then the compression P associated with F is compatible with a, and satisfies the inequalities Pa :;,. 0 and pi a ::; 0 (Lemma 8.44). Thus we have the orthogonal decomposition a = b - c where b = Pa and c = -pia. To prove uniqueness, we consider an arbitrary orthogonal decomposition a = b1 - Cl with b1 = PI a and Cl = - P{ a for a compression Pr compatible with a. Let Fl be the projective face associated with Pr. Since PI a = bl :;,. 0 and P{ a = -Cl ::; 0, we have a :;,. 0 on Fl and a ::; 0 on Fl (Lemma 8.44). By Lemma 8.53, there is a projective face F c F{ compatible with a such that a > 0 on F and a ::; 0 on F'. Moreover, Fa = PIa where F is the compression associated with F. But F was the unique projective face compatible with a such that a > 0 on F and a ::; 0 on F ' , so F = F and F = P. Thus

br

=

PIa = Fa = Pa = b,

Cl

= bi

-

a = b - a = c,

SPACES IN SPECTRAL DUALITY

279

so a = b - c is the unique orthogonal decomposition of a. Assume next that a is an element of A with a unique orthogonal decomposition, say b = Pa and c = -pia for a compression P compatible with a, and let F be the projective face associated with P. Again we consider the projective face F of Lemma 8.53. Thus a is compatible with F, and a > 0 on F and a ::::: 0 on Fl. Recall also that F = G I where G = {w E K I (Pa,w) = O}. We will prove that A and V are in spectral duality by showing that F is contained in an arbitrary projective face FI compatible with a such that a :;:, 0 on FI and a ::::: 0 on FI (Lemma 8.44). If PI is the compression associated with F I , then we have PIa:;:' 0 and P{a ::::: 0 (Lemma 8.44), so a = PI a - (- P{ a) is an orthogonal decomposition. By uniqueness, PIa = band -P{a = c. We have b = 0 on F{ (7.25), so F{ c G. Hence F = G I C F I , which completes the proof. 0

8.56. Definition. Assume A and V are in spectral duality. For each a E A we will write a+ = band a- = c where a = b - c is the unique orthogonal decomposition of a. 8.57. Proposition. Assume A and V are m spectral dualzty, and let a EA. Then the projective face F assoczated wzth the pro]ectzve umt r(a+) is the umque projective face F compatzble wzth a such that a > 0 on F and a ::::: 0 on F I , and zt zs also the least pro]ectzve face F compatzble with a such that a :;:, 0 on F and a ::::: 0 on Fl. The compresszon P associated with r(a+) zs the least compresszon such that Pa :;:, a and Pa :;:, O. Proof. Let H be the projective face associated with r(a+). By (8.18) (and (7.22)), HI = {w E K I (a+,w) = O}. Therefore we will show that H = F where F is the unique projective face compatible with a such that a> 0 on F and a::::: 0 on Fl. Let P be the compression associated with F. Then Pa :;:, 0, pia::::: 0 and a = Pa - (_pia) (Lemma 8.44). Thus a+ = Pa and a- = -pia (by uniqueness of orthogonal decompositions). Since a+ = Pa, the set G of Lemma 8.53 is equal to HI, so H = G I = F where F has the same properties as those that determine F. Thus H = F. With this we have proved the first characterization of r(a+) in terms of its associated projective face F. The second characterization of F follows from the first by Lemma 8.54. Then the characterization of r(a+) in terms of its associated compression P follows by the equivalence of (iii) and (i) in Lemma 8.44. 0 In a von Neumann algebra the spectral projections e).. of a self-adjoint element a are explicitly given as the complements of the range projections r((a - .>..1)+) (and similarly for a JBW-algebra, d. (A 99) and Theorem 2.20. This will now be generalized to the context of an order unit space

280

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SPECTRAL THEORY

A = V* in spectral duality with a base norm space V. For simplicity of notation we will write r)..(a) := r((a - A1)+).

(8.45 )

Note that by Proposition 8.57, T)..(a) is compatible with a - Al and then also with a, and the projective face F).. associated with T).. (a) is the unique projective face compatible with a such that (8.46)

a

> A on F)..,

a::;

A on

F~.

It also follows from Proposition 8.57 that the compression R).. associated with r)..(a) is the least compression compatible with a such that (8.47) Clearly r)..(a) = r(a) if a 2': 0 and A = O. In this case r)..(a) is compatible, not only with a, but also with all compressions compatible with a (Proposition 8.41). It turns out that the same is true for arbitrary a E A and A E R. This is an important technical result, which we now proceed to prove (Proposition 8.62).

8.58. Lemma. Assume the standmg hypotheszs of this chapter and A = V*. Let F, C be a pair of projectwe faces both compatzble with a E A and let A E R. If F ..1 C and a > A on F U C, then a > A on F V C.

Proof. Let p, q be the projective units and P, Q be the compressions associated with F, C. Let wE F V C. By Proposition 8.8 (and (7.22)), (p,w)

(8.48)

+ (q,w) =

(pVq,w) = 1.

Set ~ = (p,w), and note that by (8.48), (q,w) = 1 -~. Note also that (p, w) = 1 implies w E F and that (q, w) = 1 implies w E C. In these two cases there is nothing to prove, so we will assume 0 < ~ < 1. Since (1, P*w) = (p, w) = ~ and (1, Q*w) = (q, w) = 1 - ~, the two positive linear functionals p = ~-lp*w and (J = (1 - ~)-lQ*w are states. Moreover, p E F and (J E C (7.24). Since w E FvC, then (PVQ)*w = w. Hence by Proposition 8.5,

(a,w) = ((PVQ)a,w) = (Pa,w) + (Qa,w) = (a, P*w) + (a, Q*w) = ~ (a, p) + (1 -~) (a, We are done. 0

(J)

> A.

SPACES IN SPECTRAL DUALITY

281

8.59. Lemma. Assume A and V are m spectral duallty. If a and A > 0, then r.da) :s: r(a).

E

A+

Proof. We know that r(a) is compatible with all compressions compatible with a (Proposition 8.41). Therefore r(a) is compatible with the compression associated with r).,(a), or which is equivalent, the projective face F associated with r(a) is compatible with the projective face F).,(a) associated with r).,(a). Hence F' is compatible with F)." so by Lemma 8.12,

F'

=

(F'

A

F).,)

V

(F'

A

F{).

We will show that F' A F)., = 0. If w E F' A F)., = F' n F)." then (a,w) = 0 since wE F, and (a,w) > A > 0 since w E F)., (cf. (8.46)), a contradiction. Thus F' = F' A F{ c F{, so F)., c F. D

8.60. Lemma. Assume A and V are m spectral dualzty. Let a, b E A+ and let A> O. If a -.l b, then r).,(a) -.l r).,(b) and (8.49)

Proof. Since a -.l b, then also r(a) -.l r(b) (Proposition 8.39). By Lemma 8.59, r).,(a) :s: T(a) and r).,(b) :s: T(b), so we have T).,(a) -.l r).,(b). Note that by Proposition 8.8, this implies (8.50) Let F and G be the projective faces associated with r(a) and r(b) respectively. Also let F)., and G)., be the projective faces associated with T).,(a) and r).,(b) respectively. Then F)., is compatible with a and G)., is compatible with b (Proposition 8.57). Since r(a) and r(b) are orthogonal and T).,(a) :s: r(a), then T).,(a) -.l r(b). If R)., is the compression associated with r).,(a), then 0 :s: R).,b :s: IlbIIR).,r(b) = 0, so R)., is compatible with b (Lemma 7.37). Thus F)., is compatible with b, and similarly G)., is compatible with a. Hence F)., and G)., are compatible with both a and b, and therefore also with a + b. Since F)., -.l G)." then F)., V G)., is also compatible with a + b (Proposition 8.5). By (8.46) a > A on F)., and b > A on G)." so a + b > A on F)., u G).,. By Lemma 8.58, also a + b > A on F)., V G).,. We next prove (8.51 )

a:::; Ar(a) on F{

and

b:::; AT(b) on

G~.

To prove the first ofthese inequalities, let R)., be the compression associated with r).,(a), and Ro the compression associated with r(a). Since r).,(a) :::;

8.

282

SPECTRAL THEORY

r(a) (Lemma 8.59), then R>.. and Ro are compatible, and then also and Ro. Then by (8.47)

R~

Thus R~(Ar(a) ~ a) 2' 0, which gives the first inequality in (8.51); the second follows in a similar manner. Since l' (a) -.l 1'( b), then l' (a) + l' (b) ::; 1. Then

a

+ b ::;

A(r(a)

+ r(b))

::; A on

F~

n G~ = (F>..

V

G>..)'.

With this we have shown that (8.46) is satisfied with a + b in place of a and F>.. V G>.. in place of F>... Thus F>.. V G>.. is the projective face associated with 1'>.. (a + b). Hence

1'>.. (a

+ b) = 1'>.. (a)

V

r>..(b).

By (8.50) this gives (8.49). 0

8.61. Lemma. Assume A and V are m spectral dualzty. If a and Q lS a compresswn compatzble wzth a, then for each A > 0,

E

A+

r>..(Qa) = Q(r>..(a)).

(8.52)

Proof. We have Qa -.l Q'a, so it follows from Lemma 8.60 (and compatibility) that (8.53)

r>..(Qa) + 1'>.. (Q'a) = r>..(Qa + Q'a) = T>..(a).

By Lemma 8.59, T>..(Qa) ::; T(Qa) ::; q. Hence r>..(Qa) E face (q) = im+Q. Similarly r>..(Q'a) E im+Q' = ker+Q. Now applying Q to (8.53) gives (8.52). 0

8.62. Proposition. Assume A and V are m spectral dualdy. a E A, then each r>..(a) lS m the P-bzcommutant of a.

If

Proof. Let Q be a compression compatible with a. Assume first a 2' 0. If A < 0, then T>..(a) = 1, so r>..(a) is compatible with Q. If A = 0, then r>..(a) = r(a), so it follows from Proposition 8.41 that Q and T>..(a) are compatible in this case also. Assume now A > 0. By Lemmas 8.61 and 8.60, Q(r>..(a))

+ Q'(T)..(a))

=

1').. (Qa)

+ TA(Q'a)

=

Thus Q and r)..(a) are compatible in this case.

r)..(Qa

+ Q'a)

=

1').. (a).

283

SPACES IN SPECTRAL DUALITY

In the general case we define b = a + Ilalll and 11 = A + Iiali. Then bE A+ and Q is compatible with b. By the above, Q is also compatible with rll(b). Let F).. be the projective face associated with r)..(a) and let Gil be the projective face associated with rll(b). Now Gil is compatible with b, and then also with a. For W E K, (b, w) > 11 iff (a, w) > A, so it follows from the criterion (8.46) that Gil = F)... Thus rll(b) = r)..(a), so Q and r)..(a) are compatible. 0 8.63. Corollary. Assume A and V are m spectral dualzty. Let a E A and A E R, and let F).. be the pro)ectwe face assoczated with r).. (a). If F is a pro)ectwe face compatzble wzth a such that a ;::: 11 on F where 11 > A, then Fe F)...

Proof. By Proposition 8.62, F is compatible with F)... Hence (by Lemma 8.12) F = (F 1\ F)..) V (F 1\ F~). But F 1\ F~ = F n F~ = 0 since a ;::: 11 on F and a A on F~. Thus F = (F 1\ F)..) c F)... 0

:s:

We are now ready to prove our spectral theorem which generalizes the corresponding theorem for von Neumann algebras (A 99) and JBWalgebras (Theorem 2.20). As in these cases, we define the norm 111'11 of a finite increasing sequence 1= {Ao, AI, ... , An} of real numbers by

(8.54) 8.64. Theorem. Assume A and V are m spectral dualzty, and let a EA. Then there zs a umque famzly {e).. hER of projective umts wzth assoczated compresswns U).. such that

(i) e).. zs compatzble wzth a for each A E R, (ii) U)..a:S: Ae).. and U~a ;::: Ae~ for each A E R, (iii) e).. = 0 for A < -ilall, and e).. = 1 for A> (iv) e)..:S: ell for A < 11, (v) 1\11 >).. ell = e).. for each A E R.

Iiall,

The famzly {e)..} zs gwen bye).. = 1 - r((a - Al)+), each e).. zs m the P-bzcommutant of a, and the Rzemann sums n

(8.55)

S-y

=

L Ai(e)..; -

e)..i_l)

,=1

converge

m norm

to a when

iii I

---->

o.

Proof. (i)-(v) Define e).. = 1 - r((a - A1)+) and let U).. be the compression associated with e)... By Proposition 8.62 each e).. is in the Pbicommutant of a. With the notation from (8.45) and (8.47) we have e).. = r)..(a)' and U).. = R~, which gives (i) and (ii).

284

8

SPECTRAL THEORY

If A < -ilall, then a e,\ = T,\(a)' = O. Similarly,

Al > 0 on K, so T,\(a) = T(a - AI) = 1 alld A> Iiall implies e,\ = 1. This proves (iii). Let F,\ be the projective face associated with T,\ (a) for each A E R. Then assume A < IL. By Corollary 8.63, F I , C F,\. Hence T I , :::: T,\ which gives e,\ :::: ell and proves (iv). We will prove (v) by establishing the corresponding equality for the projective faces F~ associated with the projective units e,\ = T,\ (a)" i.e., Ap >,\ F~ = F~. Choose an arbitrary A E R and consider the decreasillg net {F~} p>'\. By Proposition 8.20, the projective face G

=

1\ F~ = p>'\

n

F:l

11 >'\

is compatible with a. Since F(, :J F~ for each IL > A, then G :J F~. "Ve must show that G = F~. Since G' C F,\, then a> A on G' by (8.4G). Assume now w E G. For each IL> A we have w E F~, so (a, w) :::: 1". Hence (a, w) :::: A. Thus a:::: A on G. With this we have shown that the criterion (8.4G) is satisfied with G' in place of F,\. Thus G' = F,\, so G = F~ as desired. To prove uniqueness, we consider an arbitrary family {e,\ hER of projective units with associated compressions U,\ such that (i)-(v) are satisfied. We will show that for a given A E R the projective face H,\ associated with e,\ is equal to the projective face F~ associated with T,\(a)'. We know from (8.47) that the compression R,\ associated with T,\(a) is the least compression compatible with a such that

By (ii), this condition is satisfied with U~ in place of R,\. Therefore R,\ ::S U~. Hence H,\ C F~. By (i) each Hil is compatible with a, and by (ii) Hil :J Hv when IL> 1/. Then by (v), (8.56)

H,\ =

1\ Hll = 11>'\

n

HI"

Il>'\

For each IL, we have U;la = a on H;l' cf. (7.25). By (ii), U;,a ~ lLe~, so a ~ IL on H;l' If IL > A, we can apply Corollary 8.63 with the projective face H~ in place of F. This gives H;1. C F,\ for each IL > A, so F~ c HJ1' Thus by (8.56), F~ c H,\, so H,\ = F~ as desired. To prove convergence of the Riemann sums, we choose for given E > o a finite increasing sequence {AO, AI,"" An} of real numbers such that AO < -ilall, An> Iiall and Ai - Ai-1 < E for z= 1, ... , n. By (iv), e" ::; e p when A < IL, so in this case e" -1 e~ and e p - e" = e p 1\ e~ (Lemma 8.9). If A < IL, we also have (Up 1\ U~) a = Upa - U"a

SPACES IN SPECTRAL DUALITY

285

(Lemma 8.9). By (ii), U/1a::; /-LeI' and U~a:::: Ae~, so applying U~ to the first inequality and UJ1 to the second gives

Therefore

Now define Pi = e;." ~ e;",_l for l = 1, ... , n. Pi = U;.,; 1\ U~'_l associated with Pi satisfies

Note that 2:.,Pi gives

Thus

= 1 by (iii). Since

2:.:'=1 Pia = a

n

n

i=l

i=l

Ils'Y ~ all < E

Then the compression

(Proposition 8.7), this

and the proof is complete. 0

8.65. Corollary. If A and V are m spectral duality, then each a E A can be approxlmated m norm by lmear comlnnatwns AiP, of mutually orthogonal proJectwns Pi m the P-blcommutant of a.

2:.:'=1

Proof. Clear from Theorem 8.64. 0 Note that if A and V are in spectral duality, then the first of the two requirements for the C-bicommutant (compatibility with a) is redundant in Definition 8.22. In fact, if a compression is compatible with all compressions compatible with a, then it is compatible with the projective units e;., of a, and by an easy application of the spectral theorem (Theorem 8.64) also with a itself. (Similarly for the P-bicommutant and the F-bicommutant.)

8.66. Definition. If A and V are in spectral duality and a E A, then the family {e;.,} ;"ER of Theorem 8.64 will be called the spectral resolutwn of a, and the projective units e;., in this family will be called the spectral units of a. If we define Riemann-Stieltjes integrals with respect to {e;.,} ;"ER as the norm limit of approximating Riemann sums in the usual way, then we can restate the last result in Theorem 8.64 as

(8.57)

a= JAde;.,.

8.

286

SPECTRAL THEORY

Similarly the integral with respect to {e A } AER of an arbitrary continuous function f defined on the interval [-Ilall, Iiall] is given by (8.58) where the (norm) limit exists by the same argument as in the last part of the proof of Theorem 8.64. More generally we can define the integral with respect to {eAhER of a bounded Borel function f defined on this interval as the weak mtegral in A = V', determined by the equation

(J

(8.59)

f(>-.) de A , ¢ ) =

J

f(>-.) d(e A , ¢)

for all ¢ E V+,

where d(e A , ¢) denotes the measure associated with the increasing, right continuous function >-. f---+ (e A , ¢). Using this notion of integral, we define the spectral functzonal calculus for an element a E A. This is the map which assigns to each bounded Borel function f on [-Ilall, Iiall] the element f(a) E A given by (8.60)

f(a)

=

J

f(>-.) de A ,

where {e A hER is the spectral resolution of a.

8.67. Proposition. If A and V are m spectral duahty and a E A, then the spectral functional calculus for a sat7sfies the followmg reqwrements (where f,g and all fn are bounded Borel functzons on [-Ilall, I all] and Q, f3 are real numbers):

(i)

Ilf(a)ll::; Ilflloo.

(ii) (Qf

+

(iii) f::; g

f3g)

=

=}

f (a) ::; g (a) .

Qf(a)

+

f3g( a).

(iv) If Un} 7S a bounded sequence and fn f(a) m the w*-topology of A = V'.

f---+

f pomtw7se, then fn(a)

f---+

Proof. (i) We know from (A 11) that

Ilf(a)11

=

sup IU(a),w)1 ::; sup wEK

wEK

JIf('\)1

d(eA,w).

For each state w, the function ,\ f---+ (e A, w) is increasing and right continuous with values ranging from 0 to 1, so d(eA'w) denotes a probability measure in the integral above. TherEfore Ilf(a)11 ::; Ilflloo. (ii), (iii) Trivial. (iv) Easy application of the monotone convergence theorem. 0

287

SPACES IN SPECTRAL DUALITY

Remark. The functional calculus described in Proposition 8.67 has a natural physical interpretation. If a is an observable and f is a bounded Borel function, then f(a) is the observable determined by measuring a and then applying the fUllction f to the result. Now define pos('\) = max(,\,O) and neg('\) = - min('\, 0), and let a+ = pos(a) and a- = neg(a). Then a = a+ -a-, and a+ and a- are orthogonal since a+ Iialip and aIlall(l-p), where p = X(o.oo)(a). Thus the existence of orthogonal decompositions, which will be a key assumption in our characterization of state spaces of C*-algebras (Theorem 1l.59), is well justified on physical grounds. Uniqueness of such decompositions is perhaps less evident, but we will see that uniqueness is automatic in the context in which we will be using it. By virtue of Theorem 8.55, this also provides another physical justification for the assumption of spectral duality.

s:

s:

We will now give a characterization of spectral duality which involves the concept of the F-bicommutant.

8.68. Lemma. Assume the standmg hypothesIs of thIS chapter and A = V*. For each a E A there IS a greatest compression Q m the Cbicommutant of a such that Qa 0; equwalently there IS a largest proJective face C m the F-bIcommutant of a such that as: 0 on C.

s:

Proof. Let Q be the set of compressions Q in the C-bicommutant of a such that Qa O. Recall that the C-bicommutant of a is closed under the lattice operations of C, and is a Boolean algebra (Proposition 8.23). Thus if Ql, Q2 E Q, then Q1 V Q2 is in the C-bicommutant of a, and by the distributive law for Boolean algebras, (Q1 VQ2) = (Q1 +Q~ /\Q2)' By Proposition 8.5

s:

Hence Ql V Q2 E Q. Thus Q is directed upwards, so we can organize Q to an increasing net {Q",} such that Q", / Q := Va Qa. By Proposition 8.20, Qaa / Qa, so Qa O. By Proposition 8.23, Q is in the Q-bicommutant of a. Thus Q is in Q, and in fact is the greatest element of Q. This proves the result for compressions. The result for projective faces follows by Proposition 7.28. 0

s:

A

8.69. Theorem. Assume the standmg hypotheszs of thzs chapter and Then the following are equwalent:

= V*.

(i) A and V are in spectral duality. (ii) For each a E A there is a proJectwe face C m the F-bIcommutant of a such that a ::; 0 on C and a > 0 on C/. (iii) For each a E A the largest proJectwe face C m the F-bIcommutant of a such that a ::; 0 on C satIsfies the mequality a > 0 on C'.

288

8

SPECTRAL THEORY

Proof. (i) =} (ii) If A and V are in spectral duality and a E A, then the projective face G associated with the spectral unit eo = r(a+)' has the properties in (ii) by virtue of Proposition 8.62 and equation (8.46). (ii) =} (iii) Let Go be a projective face in the F-bicommutant of a such that a:"::: 0 on Go and a > 0 on GS. If G is the largest projective face in the F-bicommutant of A such that a :"::: 0 on G (cf. Lemma 8.68), then Go c G. Hence G' c GS, so we also have a> 0 on G' . (iii) =} (ii) Trivial. (ii) =} (i) Suppose F is a projective face compatible with a such that a> 0 on F and a:"::: 0 on F'. Then FlnG' = 0 and GnF = 0. Since G is in the F -bicommutant of a, then G and F are compatible. By (8.8)

G = (G /\ F) so G

C

V

(G /\ F') = G /\ F',

V

(F' /\ G' ) = F' /\ G,

F'. We also have

F' = (F' /\ G)

so F' c G. Therefore F' = G, and so F = G' , which proves condition (ii) of Lemma 8.54. Thus A and V are in spectral duality. 0 Note that statement (iii) in Theorem 8.69 differs from all the preceding characterizations of spectral duality in that it imposes conditions on a specific face G, whereas the other ones just demand the existence of a projective face or compression with certain properties without telling how it can be found. Therefore it is easier to apply statement (iii) when proving spectral duality in special cases, as we will do in Theorem 8.72. In this theorem V is supposed to have a weakly compact base, and we shall need the following elementary result on compact convex sets. 8.70. Lemma. Let K be a compact convex set (m some locally convex space), and let A(K) be the order umt space of all contmuous affine functwns on K. If ¢ ~s a state on A(K), (~.e., ¢ E A(K)*, ¢ ~ 0 and ¢(1) = I), then there zs a pomt w E K such that ¢(a) = a(w) for all a E A(K).

Proof. Recall that a linear functional1/; on the order unit space A(K) is positive iff 1/;(1) = 111/;11, d. (A 16). Therefore the linear functional ¢ on A(K) can be extended with preservation of the properties ¢ ~ 0 and ¢(1) = 1 to a linear functional on C(K) by the Hahn-Banach theorem. This functional on C(K) can be represented by a probability measure !L on K (Riesz representation theorem), and this measure has a barycenter

SPACES IN SPECTRAL DUALITY

289

wE K (cf. [6, Prop. 1.2.1]) such that

cjJ(a)

=

J

adp.

=

a(w)

for all a E A(K). 0 A subset of a Banach space is weakly compact if it is compact in the weak topology determined by the dual space. Thus the distinguished base K of a base norm space V is weakly compact iff it is compact in the weak topology determined by the order unit space V*. By elementary linear algebra, V* is isomorphic to the order unit space Ab(K) of all bounded affine functions on K, cf. (A 11). Hence K is weakly compact iff it is compact in the weak topology determined by Ab(K), i.e., in the weakest topology for which all bounded affine functions are continuous. Thus if K is weakly compact, then A(K) = Ab(K). Recall also that a Banach space is rejiexzve if it is canonically isomorphic to its bidual. Thus a base norm space V with distinguished base K is reflexive iff it is isomorphic to the space V** ~ Ab(K)* under the map w f---> w (where w(a) = a(w) for all a E Ab(K)), or equivalently, iff the map w f---> W from V into V** is surjective. 8.71. Lemma. A base norm space V IS rejiexzve Iff Its dIstmgUIshed base K IS weakly compact. Proof. Assume K is weakly compact. By Lemma 8.70, each state cjJ on A(K) = AdK) is of the form cjJ = w for some w E K. The dual of the order unit space Ab(K) is a base norm space (A 19), so the state space of Ab (K) spans Ab (K) * = V**. Therefore the map w f---> W maps V onto V**, so V is reflexive. Conversely, if V is reflexive, then the closed unit ball V1 of V is weakly compact (Banach-Alaoglu), so the weakly closed subset K of V1 is also weakly compact. 0

We will now specialize the discussion to dual pairs (A, V) for which A and V are reflexive spaces with A = V*. Clearly this is true in all finite dimensional cases. 8.72. Theorem. If the dual paIr (A, V) satIsfies the standmg hypothesis of thIS chapter with A = V*, and A and V are rejiexzve, then A and V are m spectral dualIty. Proof. For given a E A we will show that the largest projective face G in the F-bicommutant of a such that a ::::; 0 on G satisfies the inequality

8.

290

SPECTRAL THEORY

a > 0 on C'. Assume for contradiction that (a, w) :::: 0 for some w

E

C'

and define

f3= inf(a,w)::::O.

(8.61)

wEG'

a

a

(8.62)

H

Let = a - f31. Then ~ 0 on C'. Let Q be the compression associated with C and observe that Q'a ~ 0 (Proposition 7.28). Then define

= {w

E

K

I

(Q'a,w)

= O}.

Since V is reflexive, the distinguished base K, and then also the set C' c K, is weakly compact. Hence there is an Wo E C' such that (a, wo) = f3. Therefore

(Q'a,wo)

= (a,Q'*wo) = (a,wo) = (a,wo) - f3 = o.

Thus Wo E H, so H n C' -I- 0. We will show that H n C' is in the F-bicommutant of a. Observe first that H' is the projective face associated with r(Q'a) (8.18). It follows from Proposition 8.41 that H' is in the F-bicommutant of Q'a. Therefore the compression R associated with H is in the C-bicommutant of Q'a. Clearly Q is compatible with Q'a, so R is compatible with Q and then also with Q'. Since Rand Q' are mutually compatible and both are compatible with Q'a, then R 1\ Q' is compatible with Q'a (Lemma 8.4). Since Q'a = Q'a - f3Q'l, and R 1\ Q' ::::: Q' is compatible with Q'l, then R 1\ Q' is compatible with Q' a. Since Rand Q' are compatible, R 1\ Q' = RQ', so (R 1\ Q')Qa = O. Observe now that Qa :::: 0 since a :::: 0 on C (Proposition 7.28), and note that a compression which annihilates a positive or negative element of A is compatible with this element (Lemma 7.37). Therefore R 1\ Q' is compatible with Qa. Thus R 1\ Q' is compatible with both Q' a and Qa, so is compatible with a = Qa + Q' a. Let P be a compression compatible with a. Then P and also P' are compatible with the compression Q' in the C-bicommutant of a, so

P(Q'a)

+ P'(Q'a)

=

Q'(Pa

+ P'a)

=

Q'a.

Thus P is compatible with Q' a, and then is also compatible with Q'a = Q' a - f3Q'l. Therefore P is compatible with the compression R in the C-bicommutant of Q'a. Hence

RQ'P

= RPQ' = PRQ'.

Thus the compression R 1\ Q' = RQ' is compatible with P. With this we have shown that R 1\ Q' is in the C-bicommutant of a. Now H 1\ C' is in the F-bicommutant of a, as claimed.

SPECTRAL CONVEX SETS

291

Consider the projective face G := G V (H 1\ G') and note that G properly contains G since H 1\ G' = H n G' -=I- 0. The projective faces G and H 1\ G' are both in the F-bicommutant of a, and therefore G is also in the F-bicommutant of a (Proposition 8.23). We have a::; 0 on G, so Qa ::; 0 (Proposition 7.28). Since a ::; ii, then Q'a::; Q'ii. Since Q'ii = 0 on H, then Q'a ::; 0 on H, and so we also have RQ'a;:; o. Now by Proposition 8.5 the compression Q RQ' associated with G satisfies the inequality

+

(Q

+RQ') a

=

Qa

+ RQ'a::; o.

Hence a ::; 0 on G. This contradicts the maximality of G. By Theorem 8.69, the proof is complete. D

Remark. Theorem 8.72 is not true without the condition that V be reflexive, by which K is weakly compact. A counterexample is given in [7, Prop. 6.11]. We will just sketch the idea leaving the details to the interested reader. In the example, K is a smooth and strictly convex set, and the space A = V* has elements not attaining their maximum and minimum on K. (Actually, K is affinely isomorphic to the set of sequences that are eventually zero in the closed unit ball of the Banach space l2 of square summable sequences.) The standing hypothesis of this chapter is trivially satisfied, as the only faces are the extreme points and each of them is a projective face (whose complement is the unique antipodal point). But no omch face is compatible with an element a E A that does not attain its maximum and minimum, so the only projective faces compatible with this element are K and 0. Thus by Lemma 8.44 (iii), A and V are not in spectral duality. In this example K is not norm complete. But spectral duality can not be rescued just by demanding norm completeness. In fact, one can construct a similar example where K is a norm complete (but not weakly compact) smooth and strictly convex set, as explained in [9, §2]. Spectral convex sets In our definition of spectral duality for a pair of spaces A, V we assumed A = V*. Then the pair A, V is completely determined by the distinguished base K of V, as we will show in Proposition 8.73 below. In this section we will shift the focus from the pair of ordered normed spaces A, V to the geometry of K, and we will give various examples of convex sets which in this way determine spaces in spectral duality. The proof of Proposition 8.73 involves elementary properties of order unit and base norm spaces, presented in [AS, Chpt. 1]. For the readers convenience we will briefly review what is needed.

292

8

SPECTRAL THEORY

If K is a convex set for which the bounded affine functions separate points, then the set Ab(K) of all such functions is a norm complete order unit space (whose distinguished order unit is the constant unit function), the dual space Ab(K)* is a norm complete base norm space (whose distinguished base is the "state space" consisting of all linear functionals ¢ such that ¢ ;::: 0, or equivalently II¢II = ¢(1)), and these two spaces are in separating order and norm duality. Observe that Ab(K) is also in separating duality with the subspace of Ab(K)* which consists of all alscrete linear functionals, i.e., functionals of the form L::'=l AiWi with Wl, ... ,Wn E K. (Here, and in the sequel, W denotes the evaluation functional at a point w, i.e., w(a) = a(w) for all a E Ab(K).) Recall also that an lsomorphlsm of order unit spaces, or base norm spaces, is a bijective map that preserves all relevant structure (i.e., linear structure, ordering and norm). 8.73. Proposition. Let K be the dlstmgmshed base of a base norm space V, and set A = V* (so A lS a norm complete order umt space m separatmg order and norm duality wlth V). Then the dual pmr A, V lS canomcally lsomorphlc to the pmr conslstmg of the space Ab(K) and the space of all dlscrete lmear functwnals on Ab(K).

Proof. Each element (8.63)

W

W=Ct(T~(3T

in the base norm space V can be written as where CT,TEKanda,(3ER+.

This representation is of course not unique, but by elementary linear algebra each ao E Ab(K) has a well defined extension to a linear functional a E V* given by the equation

(a,w)

=

aao(CT)

~

(3ao(T),

d. (A 11). Since V* = A and the spaces A and V are in separating order and norm duality, the extension map 1> : ao f---> a is an isomorphism of the order unit space Ab(K) onto the order unit space A. By linearity, the dual map 1>* : V --+ Ab(K)* is given by 1>*w = aCT ~ (37 when W is represented as in (8.63). Thus 1>*w is a discrete linear functional on Ab(K). Clearly each discrete linear functional ¢ = L:~=l AiWi with Wi E K and Ai E R for l = 1, ... , n can be expressed as a difference ¢ = aCT ~ (37 where CT, T E K and a, (3 E R +. Therefore 1>* is an isomorphism of the base norm space V onto the base norm space of all discrete linear functionals on Ab(K). 0 8.74. Definition. A convex set K will be called spectral if it is affinely isomorphic to the distinguished base of a complete base norm space V in spectral duality with the order unit space A = V*.

SPECTRAL CONVEX SETS

293

By Proposition 8.73, this definition is intrinsic in that the spaces A and V are determined by the convex set K. Note however, that A and V will be given in all of our concrete examples, so the identification of these spaces in Proposition 8.73 will not be needed for this purpose. The requirement that K be affinely isomorphic to the distinguished base of a base norm space is equivalent to the assumption that the bounded affine functions on K separate points of K, together with a weak kind of compactness requirement, namely, that co(K U -K) be radially compact when K sits on a hyperplane missing the origin, cf. [6]. In particular, any compact convex subset of a locally convex space will satisfy this requirement. We begin with the motivating examples already mentioned in connection with the definition of spectral duality.

8.75. Proposition. If /-L zs a measure on a O"-field

X, and K zs the set of all 1

E £1(1<) such that f

01 subsets

2': 0 and

of a set

J 1 d/-L

=

1,

then K zs the dzstznguzshed base 01 the base nann space Ll (/-L) whzch zs zn spectral dualzty wzth the order unzt space L 00 (Ii), so K zs a spectral can vex set.

1 = f+ - 1- is the usual decomposition of positive and negative parts, then 11111 = 111+11 + 111-11, so L 1 (/-L) is a base norm space. Each measurable set E c X determines a face FE of K which consists of all 1 E K such that f = 0 on X \ E. This face is easily seen to be a projective face associated with the projective unit p = XE and with the compression P : 9 f-+ XE . 9 for 9 E LOO(/-L). Clearly also, each 1 ELI (/-L) admits a least compression P such that P f 2': 1 and P 1 2': 0, namely the compression associated with the face FE where E = 1- 1 ((0, <Xl)). Thus P satisfies the requirements in the definition of spectral duality. 0 Proof. If fEU (/-L) and

1 into

8.76. Proposition. If IvI zs a lEW-algebra wzth normal state space K, then K zs the dzstznguzshed base of the predual M. which zs a base norm space zn spectral dualzty wzth the order unzt space M, so K zs a spectral convex set. The same zs true when we speczalzze to the self-adJoznt part of a von Neumann algebra. Proof. M. is a base norm space with lary 2.60. M is in spectral duality with that in the von Neumann algebra case no needed for the proof of Proposition 2.19.)

distinguished base K by CorolM. by Proposition 2.19. (Note knowledge of Jordan algebras is 0

From now on we will concentrate on compact convex sets K. These lIlay initially be given as subsets of an arbitrary locally convex (Hausdorff) space. But this ambient vector space will play no role in our discussion since

294

8.

SPECTRAL THEORY

we will always pass to the regular zmbedding of K as the distinguished base of a complete base norm space V, as explained in [6, Thm. II.2.4]. We will briefly describe this imbedding. As usual we denote by A(K) the order unit space consisting of all affine functions on K that are continuous in the given compact topology. Then the canonical map w f--> W is an affine isomorphism from K into the dual space V = A(K)*. In fact, this map is a homeomorphism from the topology on K to the w*topology on A(K)*, since the latter is pulled back to a Hausdorff topology which must be identical with the given compact topology on K. The space V = A(K)* is a complete base norm space for the natural ordering and norm (A 19), and the map w f--> W carries K onto the distinguished base of V, which consists of all states on A(K) (Lemma 8.70). Now the homeomorphic affine isomorphism of K onto the distinguished base of the complete base norm space V = A(K)* is the regular imbedding referred to above. When a compact convex set K is regularly imbedded in V = A(K)*, then V is in separating order and norm duality with both A(K) and Ab(K), and we will now specify some terminology to distinguish between concepts defined by these two dualities. The duality with A(K) determines the w*-topology on V = A(K)*. On K this topology coincides with the given compact topology, which we will henceforth refer to as the w*-topology on K. In particular, we will say that a face F of K is w *-closed if it is closed in this topology, and we will say that F is w*-exposed if it is exposed by a w*-continuous function a E A(K)+ (i.e., if F = {w E K I (a,w) = O}). The duality with Ab(K) ~ V* determines a (stronger) weak topology on V, which determines the same continuous linear functionals and closed convex sets as the norm topology. We will say that a face F of K is norm closed if it is closed in this topology, and we will say that F is norm exposed if it is exposed by a norm continuous function a E Ab(K)+. Note that this is all standard terminology when K is the state space of a C*-algebra A with enveloping von Neumann algebra A ** (then A(K) ~ Asa and Ab(K) ~ (A **)sa, cf. (A 70) and (A 102)), or when K is the state space of a JB-algebra A with enveloping JBW- algebra A** (then A(K) ~ A and Ab(K) ~ A** (Lemma 5.12 and Corollary 2.50)).

It is not hard to show that each finite dimensional simplex is spectral. But it is also true that a general Choquet simplex is spectral, as we now proceed to show. Recall first that a convex compact set K (regularly embedded in V = A(K)*) is a Choquet simplex if V is a vector lattice with the ordering defined by the positive cone V+ = cone(K), cf. (A 8) or [6, Ch. II, §3]. In fact, in this definition, rather than assuming that K is regularly imbedded, we may just assume that K is a compact convex subset of a locally convex space V, such that the linear span of K equals V and the affine span of K does not contain the origin, cf. [85].

SPECTRAL CONVEX SETS

295

8.77. Lemma. If K zs a Choquet szmplex, then the space Ab(K) zs a vector lattzce. Proof. if

71, 72,

(J

The proof makes use of the Riesz decomposition property, i.e.,

2: 0, then

and can be found, e.g., in [84, Prop. 23.9].) 0 Note that the space A(K) of a Choquet simplex is not a vector lattice in general. Nevertheless it enjoys the Riesz decomposition property. In fact, a compact convex set K is a Choquet simplex iff A(K) enjoys the Riesz decomposition property (or the equivalent Riesz interpolation property, cf., e.g., [6, Cor. 11.3.11]). The following is the well known theorem of Stone-Kakutani-KreinYosida.

8.78. Theorem. If a complete order umt space A zs a vector lattzce, then A ~ GR(X) for some compact Hausdorff space x. Proof. We sketch the proof. Let X be the set of lattice homomorphisms from A into R that take the value 1 at the distinguished order unit. Then X is compact and Hausdorff (when equipped with the w*topology), and coincides with the set of pure states on A. The evaluation map is then an isomorphism from A into GR(X), which is surjective by the lattice version of the Stone-Weierstrass theorem. The details can be found in [6, Cor. II.l.11] or [84, Thm. 24.6]. 0 8.79. Corollary. If K zs a Choquet szmplex, then Ab(K) ~ CR(X) for a compact Hausdorff space X, and K zs affinely isomorphzc to the normal state space of the lEW-algebra GR(X), and to the normal state space of the abelian von Neumann algebra Cc(X). Proof. Assume K is regularly imbedded in V = A(K)*. The first statement follows from Lemma 8.77 and Theorem 8.78. Thus Ab(K) ~ CR(X) can be equipped with a product making it a lBW-algebra with predual V. Therefore by the uniqueness of the predual (Theorem 2.55), K is affinely isomorphic to the normal state space of GR(X). Alternatively, Cc(X) will be a C*-algebra with a separating set of normal states, so will be an abelian von Neumann algebra (A 95), and the self-adjoint part of the unique predual will be V (Proposition 2.59), so K will be affinely isomorphic to the normal state space of Cc(X). 0

296

8

SPECTRAL THEORY

In the proof above, (A 95) refers back to a theorem abstractly characterizing von Neumann algebras among C*-algebras (due to Kadison and to Sakai). The proof is non-trivial, and only a reference to the proof of the hard part is given in [AS, Thm. 2.93]. But what is needed here is only the commutative case, for which much less difficult arguments suffice (see [45, Cor. 5.2]). 8.80. Proposition. EveTY Choquet szmplex K zs (' spectral convex set fOT whzch eveTY nOTm closed face zs a splzt face, and thus zs nOTm exposed Pmoj. By Corollary 8.79, K is affinely isomorphic to the normal state space of the JBW-algebra CR(X) and to the normal state space of the abelian von Neumann algebra Cc(X), so is spectral by Proposition 8.76. By Theorem 5.32 or (A 109), every norm closed face F of K is a projective face associated with a projection p, and is then exposed by pi = 1-p. Every projection p in CR(X) or Cc(X) is central, so the associated compression F will be a split face, cf. Corollary 5.34 or (A 112). 0

Remark. If K is a compact spectral convex set regularly imbedded in V = A(K)*, then every a E A ~ Ab(K) has a unique orthogonal decomposition a = a+ - a- (Theorem 8.55). But it is not generally true that a+, a- E A(K) for all a E A(K). This is the case iff all spectral units e).. (Definition 8.66) of continuous affine functions are upper semicontinuous [9, Prop. 2.4]. A compact spectral convex set K with this property was called stmngly spectml in [7]. There it was shown that K is strongly spectral iff A(K) is closed under functional calculus by continuous functions [7, Thm. 10.6]. The state space of every C*-algebra is a strongly spectral convex set [7, Thm. 11.6]' and the state space of every JB-algebra is also a strongly spectral convex set [9, Cor. 3.2]. But it is not true that every Choquet simplex is strongly spectral. In fact, a Choquet simplex K is a strongly spectral convex set iff the extreme boundary oeK is closed, i.e., iff K is a Bauer simplex [7, Prop. 10.9]. We will now define a general concept of "trace" in the context of spectral convex sets, and then show that such traces form a Choquet simplex.

8.81. Definition. Let K be a spectral convex set and assume K is imbedded as the distinguished base of a base norm space V in duality with A = V*. Then we will say that w E K is a tmczal state if it is central in V, i.e., if (P + PI)*W = w for all compressions P on A. Clearly this notion generalizes the notion of a tracial state for C*algebras (cf. the remarks after Definition 8.28).

Remark. By definition each tracial state is "central" in the sense of Definition 8.28. In the special case of the C*-algebra M 2 (C) it is also

SPECTRAL CONVEX SETS

297

central in the usual geometric sense, since the state space of this algebra is a 3-dimensional Euclidean ball and the center of this ball is seen to be the unique tracial state (A 119). Actually, this statement can be generalized, as we will now explain. If we define a generalIzed dwmeter of a compact spectral convex set K to be a subset of the form co(F U F') where F is a projective face, then it follows easily from Definition 8.81 that the set of all tracial states in K is the intersection of all generalized diameters. Note that such generalized diameters of the normal state space of a von Neumann algebra were studied (but not named) in (A 174), and also that they are closely related to the generallzed axes (A 185), which were used in the definition of generalIzed rotatwns (A 186) and 3-frames (A 190). 8.82. Proposition. If K lS a compact convex set for which every norm exposed face lS pTO]ectwe, then the set of tmcwl states m K lS either empty or a Choquet slmplex.

Pmoj. Let K be imbedded as the distinguished base of a base norm space as in Definition 8.81, and assume that there exists at least one tracial state in K. Then the set of all central elements is a vector lattice (Proposition 8.30). The set of central elements is the linear span of the set of tracial states (Lemma 8.29). Hence K is a Choquet simplex. 0 The proposition above gives the following alternative definition of a Choquet simplex which is perhaps more geometrically appealing than the original lattice definition, as it immediately shows that ordinary finite dimensional simplexes are Choquet simplexes. 8.83. Proposition. A compact convex set K lS a Choquet slmplex iff every norm closed face lS spilt.

Prooj. If K is a Choquet simplex, then every norm closed face is split (Proposition 8.80). On the other hand, if every norm closed face of K is split, since every split face is projective (Proposition 7.49), then every norm closed face of K is projective. For every compression P, the associated norm closed face is then split, so all compressions P are strongly complemented (Proposition 7.49), i.e., P + pI = I. Thus in this case every P E K is a tracial state. By Proposition 8.82, K is a Choquet simplex. 0 Remark. It follows from Proposition 8.82 that the set of tracial states of a C*-algebra (or more generally, a lB-algebra) is a Choquet simplex. There is also a remarkable converse result, proved independently by Blackadar [30] and Goodearl [57], by which every metrizable Choquet simplex is the trace space of a C*-algebra A. (See the notes at the end of this chapter for more references on this topic.)

298

8

SPECTRAL THEORY

We will now specialize to weakly compact convex sets. Recall from the discussion in the previous section that such a set is compact in the weak topology determined by the space Ab(K) of all bounded affine functions on K, which is equal to the space A(K) of all continuous affine functions on K. In this case there is no difference between w* -closed and norm closed faces of K, or between w*-exposed and norm exposed faces of K. Therefore we will use the terms "closed face" and "exposed face" without further specification. Similarly we will use the term "taLgent space" and the notation Tan ( F) for a face F of K without further specification (d. Definition 7.1). We say that a weakly compact (and in particular a finite dimensional) convex set K is stnctly convex if the only proper faces (i.e., non-empty faces =F K) are the extreme points, and we say that K is smooth at an extreme point w E K if Tan(w) is a hyperplane (or otherwise stated, if w is contained in a unique closed supporting hyperplane). We will also say a strictly convex weakly compact convex set K is smooth if every extreme point of K is smooth.

8.84. Proposition. If K zs a stnctly convex and smooth weakly compact convex set, then K zs spectral. Proof. As usual we assume K is regularly imbedded as the distinguished base of the base norm space V = A(K)* which is in separating duality with the order unit space A = V* ~ Ab(K). Then V is a reflexive space (Lemma 8.71), so we also have V ~ V** ~ A *. Thus we can apply Theorem 8.72, by which it suffices to show that the dual pair (A, V) satisfies the standing hypothesis of this chapter, i.e., that each exposed face of K is projective. By strict convexity, each face of K is of the form F = {w}, where w is an extreme point, and by smoothness its tangent space H = Tan(w) is a hyperplane, i.e.,

H = {p E aff(K) I (h,p) = O} where hE A+, aff(K) is the affine span of K, and H n K = {w}. By weak compactness, h attains its maximum value at a non-empty face G of K. (We may assume this maximum value is l.) By strict convexity, G = {Wi} where Wi is an extreme point of K, in fact, by smoothness Wi is the unique antipodal point of w. Let p be an arbitrary point of K. By smoothness of K, the tangent space of K at w' is the set of points in the affine span of K where h takes on the value l. Then there is a unique scalar A such that (h, Aw+(I-A)p) = 1, so K is contained in the direct convex sum of F = {w} and the tangent space of K at G = {w'}. A similar result holds with the roles of wand w' interchanged. Thus the conditions (i) and (ii) of Theorem 7.52 are satisfied,

SPECTRAL CONVEX SETS

299

so F and G are complementary projective faces of K. Thus each face of K is projective, and we are done. D 8.85. Corollary. The closed umt ball of a Hllbert space, and m partzcular any fimte dzmenswnal Euclzdean ball, zs a spectral convex set. Also the closed unit ball of LP({l) zs a spectral convex set for each measure {l and each exponent p such that 1 < p < 00. Proof. The closed unit ball of LP({l) is weakly compact, and is smooth and strictly convex (cf. [87, p. 351]). Now the result is clear from Proposition 8.84. D

If VI and V2 are base norm spaces with distinguished bases KI and K2 respectively, then it is easily seen that their direct sum V = VI EB V2 is a base norm space with distinguished base K = KI EBc K2 (direct convex sum), and that the direct sum A = Al EBA2 of the order unit spaces A = Vt and A2 = V2* is an order unit space isomorphic to V*, in separating order and norm duality with V under the bilinear form (8.64) Observe that the faces of K are the sets FI EBcF2 where FI and F2 are faces of KI and K2 (possibly improper, i.e., Fi = 0 or Fi = K, for z = 1,2). In fact, each face of K is of the form

(8.65)

F

= FI EBc F2 where F, = F n Ki for z = 1,2.

Assume for z = 1,2 that Fi is a projective face of K i , with associated projective unit Pi and associated compression Pi, and let Fi, Pi be the complementary projective face, projective unit, and compression, respectively. Then the direct sum PI EB P2 is seen to be a compression of Al EB A2 with complementary compression P{ EB P~. Its associated projective unit is PI EB P2 with complementary projective unit p~ EB p~, and its associated projective face is FI EBc F2 with complementary projective unit F{ EBe F~. Thus if Fi is a projective face of Ki for z = 1,2, then FI EBe F2 is a projective face of K = KI EBe K 2, and its complementary projective face is F{ EBc F~. From this we draw the following conclusion.

p;,

8.86. Proposition. The dzrect convex sum K = KI EBe K2 of two spectral convex sets KI and K2 zs a spectral convex set. A subset of K is a proJectwe face zjJ zt zs of the form FI (Be F2 where Fi zs a proJectwe face of Ki (posszbly improper) for z = 1,2, and then zts complementary projective face 7S given by

(8.66) Proof. Clear from the argument above. 0

300

8.

SPECTRAL THEORY

Proposition 8.86 provides many new spectral convex sets. For example, in dimension 3, in addition to Euclidean balls and tetrahedrons (3sim plexes), there are also circular cones (direct convex sums of a circular disk and a single point). Such direct convex sums are perhaps not so interesting since their facial structure is completely described by the facial structure of their direct summands by virtue of Proposition 8.86. But there exist other examples of spectral convex sets already in finite dimensions. Some examples are given in our next theorem.

8.87. Theorem. If

~s the set of all pomts

R

(8.67)

where m, n = 0,1,2, ... (if m = 0 or n = 0, then there are no y -terms orz -teTms) wh~ch sat~sfy the mequal~ty

(8.68)

together- wzth the

mequallt~es m

(8.69)

then K

0::::;

~s

x::::; 1,

n

'~ " y2t < x2 ,

L z; ::::; (1 -

;=1

)=1

x)2,

a non-decomposable spectml convex set.

Proof. K is constructed from the direct convex sum K = B EBc C of a Euclidean m-ball B and a Euclidean n-ball C by "blowing up" each line segment between a point in B and a point in C to an elliptical disk in a plane orthogonal to the affine span of Band C. This construction involves some rather lengthy convexity arguments which are not needed in the sequel. Therefore we have relegated the details of the proof to the appendix of this chapter. 0 If m = n = 0, then (8.68) reduces to

(8.70)

t2

::::;

x(l - x),

which defines a circular disk in R 2 (obtained by "blowing up" the line segment between the points (0,0) and (0, 1) ). If m = 1 and n = 0, then (8.68) reduces to (8.71)

SPECTRAL CONVEX SETS

301

This inequality determines a convex set in R3 which is shown in Fig. 8.1.

Fig. 8.1

Note that the convex set defined by (8.71) is constructed from the direct convex sum of the origin (0,0,0) and the line segment between (1,1,0) and (1, -1,0)) by blowing up each line segment [(0,0,0), (0, Yo, 0)] with Iyol ::; 1 to the elliptical disk consisting of all points (x, y, t) where y = XYo and t satisfies the inequality

where 0 =

VI - Y5'

The longer axis of the ellipse is the line segment

[(O,O,O),(O,Yo,O)] in the x,y-plane and the shorter axis is a vertical line segment of length 0, so the ellipse degenerates to a line segment iff Yo = ± 1.

In the present case the convex set R looks like a "triangular pillow" . The faces are the three edges and the three vertices of the triangle together with all boundary points on the curved surface off the triangle. The affine tangent spaces at the edges are the vertical planes which they determine; the affine tangent spaces at the vertices are the vertical lines which they determine; and the affine tangent spaces at the (smooth) points on the curved surface are the ordinary tangent planes at these points. An edge of the triangle and its opposite vertex are seen to satisfy the requirements for the faces F and G of Theorem 7.52, so they are complementary projective faces. Similarly, a boundary point on the curved surface and its antipodal boundary point (with parallel tangent plane) are complementary projective faces. Thus all faces are projective, and by Theorem 8.72 that is all that is needed to show this convex set is spectral. We will also describe in some detail the two four dimensional spectral convex sets obtained from Theorem 8.87. If m = 1 and n = 1, then (8.67) reduces to (8.72)

1'his inequality determines a convex set in R4 which is constructed in the same way as above from the direct convex sum of the two line segments

302

8

SPECTRAL THEORY

{x,y,z,t) I x = y = t = 0, Izl ::; I} and {x,y,z,t) I x = 1, Iyl ::; 1,z = t = O}, so a tetrahedron in R3 will play the same role as the triangle in the example above. If m = 2 and n = 0, then (8.67) reduces to (8.73) This inequality determines a convex set in R4 which is constructed in the same way from the direct convex sum of the circular disk {(:r:, Yi , Y2, t) I x = 1, yi + y~ ::; 1, t = O} and the single point (0,0,0,0), so a circular cone in R 3 will play the same role as the triangle or the tetrahedron in the examples above. (The spectral convex set obtained for Tn = and n = 2 is isomorphic to the one in this example under the map x f-> 1 - x, y, f-> Zi, Zi f-> Yi with z = 1,2). It is possible to construct many other examples of non-decomposable spectral convex sets by generalizing the construction in the proof of Theorem 8.87 in various ways, but a systematic study of such examples is beyond the scope of this book.

°

Remark. We will now continue the discussion of the order theoretic approach to quantum mechanics under the assumption that A and V are in spectral duality, and we begin by explaining how this assumption is motivated by physics. Assume A = V*, and recall that by Lemma 8.54, A and V are in spectral duality iff for all a E A there exists a unique projective face F compatible with a such that (8.74)

a

> 0 on F

and

a::; 0

OIl

F'.

Note that since the projective unit p associated with F is compatible with a, the proposition pEP and the general observable a E A are simultaneously measurable. We know that a measuring device for the proposition p will split a beam of particles in a state w into two partial beams, one consisting of particles that have the value 1 on p and are in a transformed state Wi E F, and one consisting of particles that have the value 0 and are in a transformed state Wo E F'. By (8.74) the value of the observable a (which is unaffected by measurement of p) is strictly greater than zero on the first partial beam and less than or equal to zero on the second. Therefore the proposition p is the question that asks if the observable a is strictly positive. Thus, the spectral axiom means that for each observable a E A, the question: "does a have a value > O?" is a quantum mechanical proposition, i.e., a question that can be answered by a quantum mechanical measuring device. By Theorem 8.52, the spectral axiom implies the standing hypothesis of this chapter. Thus if A and V are in spectral duality, then the propositions

APPENDIX PROOF OF THEOREM 8 87

303

form a complete orthomodular lattice as in standard quantum mechanics (Theorem 8.10 and Proposition 8.20). But the standing hypothesis is not enough to give all aspects of quantum mechanical measurement theory. An important ingredient that is missing is a mathematical procedure to determine the entire probability distribution (not only the expectation) of an observable a E A. Thus what we would like to have is a method to compute the probability that the value of an observable is in a given Borel set E c R when the observable is measured on a system in a given state w E K. We will show that this can be achieved if we assume that A and V are in spectral duality. Spectral duality of A and V provides a spectral resolution {e)..} )..ER of each a E A (Theorem 8.64 and Definition 8.66). This spectral resolution provides a functional calculus for a, i.e., a map f f-+ f(a) from the space B(R) of bounded Borel functions into A satisfying the conditions (i)-(iv) of Proposition 8.67. As explained in the remark after Proposition 8.67, the element f(a) of A represents the observable operationally defined by a measurement of the given observable a followed by evaluation of the function f on the recorded result. Thus we can for each Borel set B c R compute the observable XE(a) E A, which has the value 1 when the value of a is in E and the value 0 when the value of a is in R \ E. This observable is a proposition, more specifically, it is the question that asks if the value of a is in E. The expected value (XE(a), w) is the same as the probability that the proposition XE(a) has the value 1, and therdore that the observable a has a value in E, when the system is in the state w. In this way, the functional calculus gives the desired probability distribution of the observable in every state of the quantum system. Note, however, that although all aspects of quantum mechanical measurement theory which depend on spectral theory and functional calculus can be adequately described in the context of spectral duality, there are also features which require additional assumptions. This is the case, for example, for the pure state properties (among those the symmetry of transition probabilities) which will be introduced and studied in the next chapter.

Appendix. Proof of Theorem 8.87 In this appendix S will denote a finite dimensional real linear space, i.e., S ~ R k for a natural number k. If N is an affine subspace of S, then the vertzcal subspace over N is the affine subspace N x R of S x R ~ R k+1 which consists of all pairs (w, A) with wEN and A E R. An affine subspace M of S x R is said to be vertlcal if it is the vertical subspace OVer the subspace N = AI n S of S. Note that in order to prove that an affine subspace !vI of S x R is vertical, it suffices to show that M contains a single vertical line {w} x R. (This observation is based on the simple fact that if an affine subspace contains three of the vertices of a parallelogram, then it also contains the fourth vertex.)

8.

304

SPECTRAL THEORY

8.88. Lem~a. Let K be a compact convex set m a fimte d!menswnal space S and let K be a compact convex set m S x R such that K n S = K. Assume that FI and F2 are complementary proJectwe faces of K and that the followmg condztzons are satzsfied:

(i) (w, t)

K

zmplzes wE K. (ii) FI and F2 are faces of K as well as of K. (iii) The affine tangent spaces of K at FI and F2 are the vertzcal subspaces over the affine tangent spaces of K at FI and F2 respectwely. E

Then FI and F2 are complementary proJectwe faces of K. Proof. Let NI and N2 be the affine tangent spaces of FI and F2 respectively. By Theorem 7.52, (8.75) We must show that also

K (8.76)

C

FI EBc (N2 x R),

Kc

F2 EBc (NI x R),

(NI x R) n (N2 x R)

= 0.

To prove the first inclusion in (8.76), we consider an arbitrary point E K and we will show that it can be uniquely expressed as a convex combination of a point in FI and a point in N2 x R. By condition (i), W E K, and by (8.75) the point w can be uniquely expressed as a convex combination

(w, t)

(8.77) where WI E F I , W2 E N 2 , and 0 ::; A ::; 1. Here we can assume A < 1; otherwise the point (w, 0) is in the face FI of K and there is nothing to prove. Assuming A < 1, we define t2 = (1 ~ A)~lt. Then

and by the uniqueness of the convex combination (8.77), this expression of (w, t) as a convex combination of a point in FI and a point in N2 x R is also unique. The second inclusion in (8.76) is proved in the same way, and the last equality in (8.76) follows trivially from the last equality in (8.75). 0 8.89. Lemma. Let K be a compact convex set in a finite dimenswnal R be a compact convex set in S x R such that R n S = K.

space S and let

APPENDIX PROOF OF THEOREM 8 87

305

Assume that FI and F2 are complementar'Y proJectwe faces of K and that the followmg conddwns are satzsjied:

(i) (w, t) (ii) (w, t)

K

lmphes wE K. and w E F) or w E F2 zmplzes t = O. (iii) There zs a lme L through two pomts Wl E Fl and W2 E F2 such that the plane set D = K n (L x R) has vertzcal tangents at F) and F 2 . E

E K

Then F) and F2 are complementary pT'OJectwe faces of K.

wE

Proof. Vie will first prove statement (ii) of Lemma 8.88. Fl and

Assume

K for z = 1,2 and 0 < A < l. By condition (i) above, Since W = AUl + (1 - A)U2 E F l , then Ul, U2 E Fl. By condition (ii) above, t; = 0 so that (u;, t;) E F, x {O} for z = 1,2. Thus Fl is a face of K. Similarly F2 is a face of K. We will now prove statement (iii) of Lemma 8.88. Let AI be any supporting hyperplane of K at Fl' Either the hyperplane AI contains the plane L x R or it intersects it in an affine subspace which will be a supporting affine subspace of the convex set D in L x R, so it must contain the vertical tangent {Wl} x R of D at Wl. In either case, by the observation preceding Lemma 8.88, AI will be a vertical subspace of S x R. Observe that the vertical hyperplane IvI in S x R supports K at Fl iff it is of the form IvI = H x R where H is a hyperplane in S that supports K at Fl. (If ¢ is a defining function for H, then (w, t) f-> ¢( w) is a defining function for H x R.) The intersection of all such hyperplanes H x R is equal to N) x R (since Nl is the intersection of all hyperplanes H in S that supports K at F)). Therefore the tangent space for K at Fl is equal to N) x R. By the same argument, the tangent space for K at F2 is equal to N2 x R. This completes the proof. 0 where

(11;, t i ) E

Ul, U2 E K.

Consider a direct convex sum K = B EElc C of two Euclidean balls, say that Band C are the closed unit balls of Rm and Rn respectively, and admit also the case where morn is zero so that B or C is an "improper ball" consisting of a single point. Let X = R, Y = Rm, Z = Rn and consider the Euclidean space S = X x Y x Z ~ Rm+n+l. Assume (without loss) that K is imbedded in S by the affine isomorphism which identifies the point b E B with the point (1, b, 0) E S and the point c E C with the point (0,0, c) E S. Then the point A b EEl (1 - A) c E K with b E B, C E C and A ::; 1 will be identified with the point (x, y, z) E S where x = A E X, Y = AbE Y, Z = (1 - A) c.

° ::;

8

306

SPECTRAL THEORY

Thus we can represent K as the set of all points

w = (x,y,z) E 5,

(8.78) where 5 (8.79)

= X x Y x Z with X = R, Y = Rm, Z = Rn and

0:S x :S

1,

IIYII:S x,

Ilzll:S 1 -

x.

8.90. Lemma. Let K be as above. Then all pTOper faces of KaTe proJectwe, and they are of the followzng types where 'U and v are boundar'y pmnts of Band C respectwely:

(i) B or C (balls), (ii) [u,v] = {u} EBc {v} (lzne segments), (iii) B EBc {v} or {u} EBc C (caps of cones), (iv) {u} or {v} (szngletons). Band C are complementary spht faces. If u ' zs the antzpodal poznt of u on the boundary of the ball B and Vi zs the antipodal poznt of v on the boundary of the ball C, then the lzne segments [u, v] and [U', Vi] aTe complementary pTOJectwe faces, the czrcular cone B EBc {v} and the szngleton {Vi} are complementary pTOJectwe faces, szmzlarly the czrcular cone {u} EBc C and the szngleton {u ' } are complementary pTOJectwe faces. Proof. Clear from Proposition 8.86. 0

Note that in Lemma 8.90, if B is a single point, then only (i) and the first part of (iii) need be considered, and similarly if C is a single point, then only (i) and the second part of (iii) apply. The theorem below is Theorem 8.87 restated in the terminology of this appendix. 8.91. Theorem. If K zs the dzrect convex sum of two Euclzdean balls Band C represented as a subset of the Euclzdean space 5 = X x Y x Z where X = R, Y = RTn, Z = Rn as in (8.78) and (8.79), then the set K C 5 x R conszstzng of all poznts (w, t) with w = (x, y, z) E K, t E R and

(8.80) zs a non-decomposable spectral convex set. Proof. Let f be the positive function defined on K by the equation f(w) = t where w = (x, y, z) and (8.81 )

APPENDIX. PROOF OF THEOREM 8.87

307

Now K consists of all points (w, t) E S x R such that wE K, t E Rand

It

(8.82)

I

S;

f(w).

Clearly f is a continuous function, so K i~ compact. We will show that is a concave function, which implies that K is convex. Consider the two functions

(8.83)

g(x, y) = /x 2 -

Ily112,

h(x, z) = /(1 - x)2

f

-llzI12,

where 9 is defined on the set of all points (x, y) E X x Y such that o S; x S; 1 and IIYII S; x, and where h is defined on the set of all points (x,z) E XxZ such that 0 S; x S; 1 and Ilzll S; I-x. Observe that for u 20 the equation u = g(x, y) is equivalent to u 2 + IIyl12 = x 2, which defines the boundary of a convex cone in X x Y x R (with top-point (0,0,0) and with base {(1,y,u) I u 2 + IIyl12 S; I} that is a Euclidean ball). The subgraph of 9 (i.e., the set of all points (x, y, t) such that 0 S; t S; g(x, y)) is the "upper half' of this cone (i.e., the intersection with the upper half-space consisting of all points (x, y, t) E X x Y x R such that t 2 0). Thus the subgraph of 9 is convex. Therefore 9 is concave. Similarly, the subgraph of h is the upper half of a convex cone in X x Y x R) (with top-point (1,0,0) ), from which it follows that h is concave. Note that f(x,y,z) = /g(x,y)h(y,z). From this and the concavity of 9 and h it follows that f is concave, as we will now show. Let W1 = (Xl, Y1, zd and W2 = (X2' Y2, Z2) be two distinct points of K, let 0 < ), < 1, and define Wo = (xo, Yo, zo) by

(8.84) For simplicity of notation, set fi = f(Wi), gi = g(Xi' Yi) and hi = h(yi' Zi) for 2 = 0,1,2. By the Cauchy~Schwarz inequality and the concavity of 9 and h, we have

)'h + (1 - )')12 =yf):i; J):h; + /(1 (8.85)

S; /),gl S;

+ (1 -

,;go ~ =

),)g2 /(1 - )')h2

),)g2 /)'h 1 + (1 - )')h2 fo.

Thus f is concave on K, so K is convex. We will now show that f is strictly concave in the interior of the set K c S. By the definition of K and f (i.e., by (8.79) and (8.81)), the boundary of K consists of all points W E K such that f(w) = o. Thus f(w) > 0 for w in the interior of K. Assume for contradiction that Wo is an interior point of K for which f is not strictly convex. Then Wo can be written as a convex combination

8

308

SPECTRAL THEORY

(8.84) with 0 < A < 1 and WI =f. W2 for which the equality signs hold in (8.85). Moreover, we can and will choose this convex combination such that WI and W2 are interior points of K. Then for z = 0,1,2 the variables fi, gi, hi are all non-zero. Since the last ::; relation in (8.85) holds with equality, (8.86) From the first of these equalities, it follows that the line segment between the points ((xl,yd,gl) and ((X2,Y2),g2) in (X x Y) x R lies on the boundary of the (conical) subgraph of g, hence on a ray from the point (0,0,0). By the second equality in (8.86), the line segment between the points ((Xl, zd, hd and ((X2' Z2), h2) in (X x Z) x R lies on the boundary of the (conical) subgraph of h, hence on a ray from the point (1,0,0). Thus there exist two scalar factors a, {3 such that

(8.87)

X2= OOX l, 1 - X2

Y2=OOYl,

= {3(1 - xd,

Z2

g2=oogl,

= {3zl'

h2 = {3h l .

Since the first::; relation in (8.85) holds with equality and this relation follows from Cauchy-Schwarz, the two R 2 -vectors (JAgl, J(l - A)g2) and (JAh l , J(l- A)h2) are proportional. Thus hl/g l = h2/g2, so g2/g) = h2/ hI. Therefore there exists a scalar factor " such that g2 = "gl, h2 = "hI. Combining with (8.87), we find that a = {3 = " and that

Adding these two equations, we conclude that a = {3 = " = 1 so that the two points WI = (Xl, Yl, zd and W2 = (X2' Y2, Z2) coincide, contrary to assumption. The interior of the set K consists of all points (w, t) E K x R for which the inequality (8.80) is strict, that is, the ones for which It I < f(w). Therefore the boundary of K consists of all (w, t) E K x R for which f(w) = Itl, and the boundary points with w in the interior of K are those for which It I = f(w) > O. Since f is strictly concave in the_interior of K, all such boundary points are seen to be extreme points of K. We will now determine the faces of K. By the above, they include all singletons {(w,t)} on the boundary of K for which w is an interior point of K. We will show that the only other faces of K are the subsets that are proper faces of K. Let F be a proper face of K (i.e., F =f. K). To prove F is a face of K, let (w,O) be an arbitrary point of F and assume

APPENDIX PROOF OF THEOREM 8 87

309

where 0 < A < 1 and (wi,td E K for l = 1,2. Then W = AWl +(1-A)w2' so wI and W2 are in F. Since F is a proper face, it contains no interior point of K. Hence Iti I -s: f(wd = 0 for l = 1,2. Thus Itll = It21 = 0, so that (WI, td and (W2' t 2 ) are in F. This shows F is a face of i<. Clearly every face of K which is conta~.-ned in K must be a face of K, so it only remains to show that a face of K that is not contained in K is a singleton. But this is clear, for otherwise such a face would contain an open line segment disjoint from K, and for a point (w, t) of this segment we would have It I > 0, so that W would~be an interior point of K and then (w, t) would be an extreme point of K, contrary to the fact that all these points are located on an open line segment in K. ~ ~With this we have shown that the faces of K are those extreme points of K that are not in K together with the faces of K. We will show K is spectral by proving that all these faces are projective (Theorem 8.72). The non-trivial faces of K are the ones described in Lemma 8.90. We will show they are all projective in K by verifying the assumptions in Lemma 8.89. Conditions (i) and (ii) are easily verified, so we only have to construct lines L with the property demanded in condition (iii) for pairs F I , F2 of projective faces of K. First assume FI = Band F2 = C. Let L be the line through the centers WI = (1,0,0) and W2 = (0,0,0) of the two balls Band C. Thus L consists of the points w).. = (A, 0, 0) with A E R, and Lx R consists of the points (w)..,t) with (A,t) ER2. By (8.80), the convex set D=Kn(LxR) consists of all points (w).., t) for which t 4 -s: A2(1 - A)2. Thus D is the circular disk defilled by the inequality (8.88) Clearly, condition (iii) of Lemma 8.89 is satisfied in this case. Next assume that FI = [u, v] and F2 = [u' , Vi] where u and v are boundary points of Band C respectively, and where u ' and Vi are the antipodal boundary points. Now let L be the line through the mid-points WI = ~(u + v) and W2 = ~(U' + Vi) of the line segments [u, v] and [u' , Vi]. Therefore u = (I, b, 0) and u ' = (I, -b, 0) where b E Band Ilbll = l. Similarly v = (0,0, c) and Vi = (0,0, -e) where eE C and Ilell = l. Hence WI = (~, ~b, ~c) and W2 = (~, -~b, -~e). L consists of all points (8.89)

W)..

=

AWl + (1 - A)W2

and with the above values for W)..

=

WI

with A E R,

and W2,

(~, (A - ~)b, (A - ~)e).

Substituting x = ~, Y = (A - ~)b, Z = (A - ~)e into (8.80), we get the same inequality (8.88) as in the preceding case. Thus D is a circular disk

8

310

SPECTRAL THEORY

and condition (iii) of Lemma 8.89 is satisfied also in this case. Finally assume Fi = B EBc {v} and F2 = {Vi} where v is a boundary point of C and Vi is the antipodal boundary point. (The case where Fi = {u} EBc C and F2 = {u' } is similar.) Let 11 be the center of Band let L be the line through the points Wi = ~ (u + v) and W2 = Vi. Now u = (1,0,0), v = (0,0, e) and Vi = (0,0, -e) where e E C and Ilell = 1. Hence Wi = (~, 0, ~e) and W2 = (0,0, -e). The points of L are expressed in terms of Wi and W2 as in (8.89). Therefore we find that W).=(~A,O, ~A-1).

Substituting x

=

~ A, Y

= 0,

Z

=

~ A -1 into (8.80), we get the inequality

(8.90) This inequality defines an oval disk (of degree 4) with vertical tangents for A = 0,1. Thus D has vertical tangents at Wi and W2, so condition (iii) of Lemma 8.89 is satisfied also in this case. Now we consider extreme points of X. Clearly f is cOEtinuously differentiable in the interior of X, from which it follows that X has a unique tangent hyperplane at each boundary point (w, t) with W in the interior of X, and this hyperplane is equal to Tan {( w, t)} (as defined in Definition 7.1). Clearly (w, t) has a unique antipodal point (Wi, t') (also with Wi in the interior of X) defined by the property that the tangent space of K at (Wi, t') is parallel to the tangent space at (w, t). (By the results of the previous paragraph, this antipodal point is not in X.) Now it follows by an elementary argument in plane geometry that each point in X can be expressed in a unique way as a convex combination of the point (w, t) and a point in Tan{(w ' , t')} (cf. the proof of Proposition 8.84). Thus the conditions (i) and (ii) of Theorem 7.52 are satisfied by the pair o~ faces F = {(w, t)} and G = {(Wi, t')}. Th~refore all extreme points of X that are not in X are projective faces of X. The proof is complete. D It can be shown that the set D = K n (L x R) defined by the line L between arbitrary interior points Wi and W2 of Band C is an elliptical disk (which degenerates to a line segment if Wi or W2 are boundary points.) Thus K is obtained by "blowing up" the line segments between points in Band C to elliptical disks in an extra dimension. If m = n = 0, then K is a circular disk, so it is the state space of the spin factor Nh(R)sa. In all other cases K is different from the state space of a lB-algebra. (The inequality (8.90) violates the elliptical cross-section property of Proposition 5.75.) The convex sets K in Theorem 8.91 are mutually non-isomorphic except for the trivial isomorphism between sets with transposed pairs of parameters m, nand n, m (under the map x ........ 1 - x, Y ........ z, z ........ Y).

NOTES

311

Notes The results in this chapter on the lattice of compressions, and on spectral duality, were established either in [7] or [9], with a few exceptions noted below. Loosely speaking, spectral duality requires that there be "enough" projective faces or compressions. The concept of spectral duality in [7] was defined in terms of projective faces by statement (ii) of our present Theorem 8.69, which involves projective faces in the bicommutant. The alternate characterization of spectral duality in statement (ii) of Lemma 8.54 was given in [7, Thm. 7.5], and was used to prove the existence of unique orthogonal decompositions (Theorem 8.55) in [9, Thm. 2.2]. Our present definition of spectral duality in terms of compressions (Definition 8.42) does not involve the bicommutant, but we have to work harder to prove the key result that norm exposed faces are projective (Theorem 8.52). An important first step in the argument that leads up to this theorem is Lemma 8.46, which was originally proved by Riedel [106] in a slightly different context (with his "fundamental units" in place of our projective units). Note that in [7] there is also a concept of "weak spectral duality", which is shown to imply existence, but not, a prwn, uniqueness, of spectral resolutions. It is still an open question if weak spectral duality implies spectral duality (or, which can be seen to be equivalent, if the minimality requirement is redundant in Definition 8.42). The proof given here that Choquet simplexes are spectral (Proposition 8.80) reduces to the fact that normal state spaces of a JBW-algebra or a von Neumann algebra are spectral. A more direct lattice-theoretic proof is given in [7]. The "exotic" examples in Theorem 8.87 are presented here for the first time. What was known about the relationship of split faces to Choquet simplexes evolved over several years. It was shown by Alfsen [5] that every w*-closed face of a Choquet simplex is a split face (where for a regularly imbedded compact convex set we refer to the given compact topology as the w*-topology). Then Asimow and Ellis [19] showed that every norm closed face of a Choquet simplex is split. The fact that a compact convex set is a Choquet simplex iff every norm closed face is split (Proposition 8.83) is proved in [9], and the result that a compact convex set is a Choquet simplex iff every w*-closed face is split was shown by Ellis [51]. The fact that the traces of a C*-algebra form a Choquet simplex is due to Thoma [127]. The converse result by which every metrizable Choquet simplex is the trace simplex of a C*-algebra, was found by Blackadar [30] and Goodearl [57], who independently proved that for every metrizable Choquet simplex K there is a simple, unital C*-algebra whose trace simplex is affinely homeomorphic to K. This result can also be obtained from the characterization of dimension groups among all ordered abelian groups, which was found slightly later by Effros, Handelman and Shen [47]. Note that the trace simplex has recently found application as an impor-

312

8

SPECTRAL THEORY

tant invariant for C*-algebras, and that a series of interesting new results on the interplay between the theory of infinite dimensional simplexes and C*-algebras have been found in connection with the Elliott program for a K-theoretic classification of various kinds of C*-algebras. (See [107]' and in particular the references to R0rdam, Thomsen and Villadsen in that book, and the book [92].) Many authors have studied spectral theory from an order-theoretic point of view, but we will just mention a few papers that relate fairly closely to the approach in this book. Some of those are the papers by Abbati and Mania [1], and by Bonnet [31]' which are based on axiom systems similar to (but not identical with) that in the paper by Riedel [106] mentioned above. Another relevant reference is the paper by C. M. Edwards [41], which extends spectral duality from order unit spaces to the wider class of G M -spaces (not necessarily with unit). There is also a rich literature on quantum mechanics modeled on ordered vector spaces, and again we will only give a few references that are directly related to the material in this book. The "operational approach" to axiomatic quantum mechanics (consistent with the point of view discussed in the remark after Proposition 7.49) goes back at least as far as Davies and Lewis [39], who proposed analyzing the space of observables and the convex set of states, expressed as an order unit and a base norm space in duality. In this context, it is natural to axiomatize filters, and numerous authors have taken that approach. We will just point out the papers of Mielnik [97]' Gunson [59]' Araki [17], and Guz [60]. The attempt to axiomatize quantum mechanics by assuming a linear space of observables equipped with a suitable functional calculus was the basis of Segal's paper [114], and is also present in von Neumann's paper [98]. In both cases one is led to define a candidate for a Jordan product given by squares: a 0 b = -j-((a + b)2 - (a - b)2). However, distributivity of such a product is not automatic, so additional axioms must be assumed to guarantee it. This is exactly what is needed to make the transition from spectral theory to Jordan algebras, and is the topic of the next chapter.

PART III State Space Characterizations

9

Characterization of Jordan Algebra State Spaces

In this chapter we will give a characterization of the state spaces of JB-algebras and of normal state spaces of JBW-algebras. As a preliminary step, we will characterize the normal state spaces of JBW-factors of type I. (This includes as a special case a characterization of the normal state space of B(H).) Our standing hypothesis for this chapter will be that A, V is a pair of an order unit space and a complete base norm space in separating order and norm duality with A = V*, with the additional assumption that each exposed face of the distinguished base K of V is projective. (This is the same as the standing hypothesis of the last chapter, except for the additional requirements that V be complete and that A = V*.) By Proposition 8.76, the normal state space of a JBW-algebra A is spectral, i.e., A is in spectral duality with its predual V = A*. Thus by Theorem 8.52, such a pair A, V satisfies the standing hypothesis of this chapter. If instead B is a JB-algebra with state space K, then K can be identified with the normal state space of the JBW-algebra B** (Corollary 2.61), so again K is spectral and thus satisfies the standing hypothesis for the duality of V = B* and A = B**. We will add to the standing hypothesis two equivalent sets of axioms. One set (the "pure state properties") is of particular physical interest, while the other set is geometric in nature. We will use these axioms to characterize the normal state spaces of JBW-factors of type I, and then use this result to achieve the characterization of state spaces of JB-algebras. The axioms just discussed involve the extreme points of the state space, and so are not relevant in the more general situation of normal state spaces of arbitrary JBW-algebras, where there may be no extreme points. The ellipticity property (cf. Proposition 5.75 and the remarks following) will be used to characterize these normal state spaces, and we will also give a characterization in which ellipticity is replaced by a more physically relevant assumption. The pure state properties 9.1. Definition. If V is a base norm space with distinguished base ----+ V a positive map, we say T preserves extreme rays if T maps each extreme point of K to a multiple of an extreme point.

K and T : V

316

9

CHARACTERIZATION OF JORDAN STATE SPACES

Recall that under the standing hypothesis of this chapter, the sets C,

P, :F of compressions, projective units, and projective faces form isomorphic lattices. Each is a complete orthomodular lattice (Theorem 8.10 and Proposition 8.20). We now briefly review some lattice terminology. The least element of a complete lattice is called zero. A projective unit that is minimal among the non-zero projective units is called an atom. The same term is used for any element of a complete lattice that is minimal among the non-zero elements of the lattice, cf. (A 41). A complete lattice is atomIC if each non-zero element is the least upper bound of atoms. In this section, we will sayan element in a complete lattice is fimte if it is the least upper bound of a finite set of atoms. Note that this lattice-theoretic concept may differ from the notion of finiteness for projections in von Neumann algebras (A 166), as defined in terms of Murray-von Neumann equivalence, if the projection lattice is not atomic. Recall that under the standing hypothesis, for each atom u in A there is a unique extreme point u of K such that U(u) = 1 (Proposition 8.36).

9.2. Definition. Assume the standing hypothesis of this chapter. Then K has the pure state properiles if (i) every extreme point of K is norm exposed, (ii) for every compression P, the map P* preserves extreme rays, and (iii) (11, v) = (v, u) for all atoms u, v in A. 9.3. Proposition. The normal state space of a lEW-algebra and the state space of a lE-algebra have the pure state propertzes.

Proof. Since the state space of a lB-algebra can be identified with the normal state space of its bid ual (Corollary 2.61), it suffices to establish the pure state properties for normal state spaces of lBW-algebras. By Theorem 5.32 every norm closed face is projective, so (i) holds. By Corollary 5.49, each compression on a lBW-algebra preserves extreme rays of the normal state space. Finally, (iii) follows from Corollary 5.57. 0

Remark. If we interpret K as the set of states of a physical system, we can view each rJ E V+ as representing a beam of particles with intensity IlrJll. We further interpret compressions P as representing physical filters. (See the remark after Proposition 7.49.) If each norm exposed face is projective, then (i) says that for each pure state there is a filter that prepares that state. The property (ii) says that filters transform pure states to pure states. Finally, (iii) is a statement of "symmetry of transition probabilities". See the remarks after Corollary 5.57, and [AS, remarks following Lemma 4.10 and Cor. 6.43], for further discussion of this interpretation.

THE PURE STATE PROPERTIES

317

In a general finite dimensional compact convex set, there may be faces that are not exposed. However, we have the following result, due to Minkowksi, cf. [123, Thm. 3.4.11]. 9.4. Theorem. If F zs a proper face of a non-empty convex set K m R n, then there zs a proper exposed face G of K contaznzng F. Proof. By working in the affine span of K, we may assume without loss of generality that the affine span of K is Rn. The interior of K is non-empty (e.g., it contains the arithmetic mean of any finite number of points in K whose affine span equals the affine span of K), and is convex and dense in K. Since F is a proper face, then F does not meet the interior of K. By Hahn-Banach separation, there is an affine function a with a > 0 on the interior of K and a :s; 0 on F. Since the interior of K is dense in K, then a = 0 on F. Then G = a- 1 (0) n K is the desired proper exposed face. 0

In the current context, we can say more. The following result shows that the first pure state property is automatic in finite dimensions. (This result will not be needed in the sequel.) 9.5. Proposition. Let K be a jinzte dzmenszonal compact convex set m whzch every exposed face zs pro]ectzve. Then every face F of K is exposed, and therefore pro]ectzve. Proof. Let G be the intersection of the exposed faces of K that contain

F. By finite dimensionality, G is itself an exposed face of K. (An intersection of any family of hyperplanes will equal the intersection of some finite subfamily, and if a1, ... ,am are positive affine functions on K such that i a;I(O) = F, then F is exposed by al + ... +a m .) Suppose F i- G. By Minkowksi's theorem (Theorem 9.4) applied to F c G, there is an affine function a on G, not the zero function, such that a = 0 on F, and a :::- 0 on G. Let P be the compression associated with G, so that G = im P* nK. Note that G is a base for the cone im+ P*, and so we can extend a to a linear function on lin G = im+ P*. Let b = a 0 P*. Then b = 0 on F, and b :::- 0 on K. Thus H = {O" E K I (b,O") = D} is an exposed face of K containing F and properly contained in G, contrary to the definition of G. We conclude that F = G. 0

n

The second of the pure state properties is closely related to the covering property for lattices, cf. (A 43), which we now review for the reader's convenience. 9.6. Definition. Let p and q be elements in a complete lattice L. We say q covers P if p < q and there is no element strictly between p

318

9

CHARACTERIZATION OF JORDAN STATE SPACES

and q. We say L has the covermg property if for all pEL and all atoms 'U

E

L,

(9.1)

p Vu

= p or p Vu covers p.

We say L has the fimte covermg property if (9.1) holds for all finite p in L and all atoms u E L. Recall that by Lemma 8.9, (p V u) 1\ pi = (p Vu) - p for each p, u E P. Thus P has the covering property iff for each atom u E P and each pE P,

(9.2)

(p V u)

1\

pi is an atom or zero.

We will use the following result to relate the covering property to the second of the pure state properties.

9.7. Proposition. Assume the standmg hypotheszs of thzs chapter, and let P be a compresswn wzth assocwted proJectwe umt p. Assume also that each extreme pomt of K zs norm exposed. The followmg are equwalent:

(i) P* preserves extreme rays. (ii) For each atom u E P, (u V pi) 1\ P zs ezther an atom or zero. (iii) P maps atoms to multzples of atoms. Proof. Let P be the projective face associated with p. Let u be an atom, and ()' the associated extreme point of K. By Theorem 8.32, r(Pu)

(9.3)

= (pi

V u) 1\ p,

and by Corollary 8.33, (9.4)

ProjFace(P*(),)

= (Pi

V {()'}) 1\ P,

where ProjFace(P*(),) denotes the least projective face containing IIP*()'11- 1 P*()', or the empty set if P*()' = O. (i) =? (ii) By (i), P*(), is a multiple of an extreme point, and thus of an exposed point of K. Therefore ProjFace(P*(),) is an atom in the lattice :F, or else the empty set. Now (ii) follows from (9.4) and the isomorphism of the lattices P and :F. (ii) =? (iii) By (ii) and (9.3), r(Pu) is zero or an atom. If Pu = 0, then Pu is certainly a multiple of an atom. Otherwise q = r(Pu) is an atom. Let Q be the associated compression. By Proposition 8.36, im Q = Rq. By Lemma 7.29, Pu E face(q) = im+Q c Rq, which proves (iii). (iii) =? (i) By (iii), Pu is a multiple of an atom, so r(Pu) is an atom or zero. Thus by (9.3), (pi V u) I\p is an atom or zero. By the isomorphism

THE HILBERT BALL PROPERTY

319

of 'P and F, (F'v {(j}) /\ F is an atom in F or equals the empty set. By (9.4) the same is true of ProjFace(p*(j). Since minimal projective faces are sets consisting of a single extreme point (Proposition 8.36), then p*(j is a multiple of an extreme point. D 9.8. Corollary. Assume the standmg hypothesIs of tlm chapter', and assume that eveTY extTeme pomt of K lS nOTm exposed. Then the second of the pUTe state pmpertles lS equwalent to the coveTmg property JOT the lattlces C, 'P, F. Pmoj. Let 71 be an atom in 'P and let p E 'P. By (9.2), the covering property for 'P is equivalent to (ii) in Proposition 9.7 holding for all p E 'P and all atoms 71. The corollary now follows from the equivalence of (i) and (ii) in Proposition 9.7. D

The Hilbert ball property Recall that a HdbeTt ball is a convex set affinely isomorphic to the closed unit ball of a finite or infinite dimensional real Hilbert space. 9.9. Definition. A convex set K has the HllbeTt ball propeTty if the face generated by each pair of extreme points of K is a norm exposed Hilbert ball. Note that in Definition 9.9 we allow the pair of extreme points to coincide, so the Hilbert ball property implies that each extreme point of K is norm exposed. Note also that by the standing hypothesis of this chapter, the Hilbert ball property implies that the face generated by each pair of extreme points of K is projective.

9.10. Proposition. State spaces of lE-algebras and C*-algebras, and normal state spaces of lEW and von Neumann algebras, have the Hllbert ball propeTty. Prooj. By Proposition 5.55 and Corollary 5.56, normal state spaces of JBW-algebras and state spaces of JB-algebras have the Hilbert ball property. This implies as a special case the stated results for C*-algebra state spaces and von Neumann algebra normal state spaces, but the fact that the face generated by two distinct pure states of a C*-algebra is a Hilbert ball (of dimension 1 or 3) is also stated directly in (A 143). Since norm closed faces of the normal state space of a von Neumann algebra are norm exposed (A 108), and the state space of a C*-algebra can be identified with the normal state space of the enveloping von Neumann algebra (A 101), it follows that the face generated by a pair of pure states in a C*-algebra state space is norm exposed. This completes the proof that C*-algebra state spaces have the Hilbert ball property. 0

320

9.

CHARACTERIZATION OF JORDAN STATE SPACES

We will now derive some consequences of the Hilbert ball property. As remarked above, the first of the pure state properties follows from the definition of the Hilbert ball property. Recall that the second pure state property is equivalent to the covering property (Corollary 9.8). We now show that the Hilbert ball property implies the finite covering property.

9.11. Proposition. Assume the standmg hypotheszs of thzs chapter. If K has the Hzlber-t ball pmper-ty, then the latt?ces C, P, F have the fimte covenng pmper-ty.

Pmof. Assume K has the Hilbert ball property. We first establish the implication for atoms 1L, v and projective units q, q<

(9.5)

U

Vv

q is an atom or zero.

=}

If 1L = v, then 1L V v = 'U is an atom, so by definition q must be an atom or zero. If 1L i v, the projective face corresponding to 1L Vv is generated by the pair of distinct extreme points associated with u and v, so by assumption is a Hilbert ball. The projection q corresponds to a projective face properly contained in this Hilbert ball. The only proper faces of a Hilbert ball are the extreme points, so q must be an atom or zero. Now let u be any atom in P and p any finite projective unit. By (9.1) we must show u V p covers or equals p, or equivalently,

(u

(9.6)

V p) - p

is an atom or zero.

We will prove (9.6) by induction on the lattice dimension of p. (Recall that the lattice dimension of p is the smallest number of atoms whose l.u.b. is p, cf. (A 42).) If dim(p) = 1, then p is an atom, so (9.6) follows from (9.5) (with p in place of v and (u V p) - p in place of q). Let dim(p) = n :;,. 2, and assume (9.6) holds for all smaller dimensions of p. Choose atoms 1LI, ... ,Un such that p = U I V ... Vu". By definition of dimension, the clement q = UI V ... V Un-I satisfies dim(q) < nand q < p (since dim(p) = n). The map x f---+ :r - q is order preserving with order preserving inverse, and thus induces a lattice isomorphism from the lattice interval [q,l] onto the interval [O,q'j. Since

p V 11. = (q V u) V (q Vu,,), then

(p

V

11.) - q =

(( q V

11.) - q)

V (( q V un) -

q) .

By our induction hypothesis, since dim(q) < n, both terms on the right are atoms or zero, so we conclude that dim((p

V

11.) - q) ::;: 2.

THE HILBERT BALL PROPERTY

321

Note that p> q implies

(p V u) - p < (p V u) - q. By (9.5), the only elements under an element of dimension 2 are atoms or zero, so this finishes the proof that (p V u) - p is an atom or zero. Thus we have shown that the finite covering property follows from the Hilbert ball property. D Remark. In the proof above, the only property of the Hilbert ball that is used is that it has no proper faces other than singletons. Thus the finite covering property would be implied by the weaker assumption that the face generated by each pair of extreme points is norm exposed and strictly convex.

If an additional geometric assumption is added to the Hilbert ball property, we will see in Proposition 9.13 below that the second pure state property follows. Recall from Definition 5.71 that a convex set K is symmetllc wzth respect to a convex subset Ko if there exists a refiectwn of K whose set of fixed points is precisely Ko. 9.12. Lemma. Assume the standmg hypotheszs of thzs chapter. If F is a pro)ectwe face of K and P zs the correspondmg compression on A, then the followmg are eqwvalent:

(i) K zs symmetllc wzth respect to co(F U F ' ), (ii) 2P + 2P' - I :::: o. If these eqwvalent condztwns hold, then (2P + 2P' - 1)* zs the umque refiectwn of K wzth co(F U F') as zts set of fixed pomts. Pmoj. Let T = 2P+2P' -I. If (ii) holds, then T is a positive map such that T2 = I, and T1 = 1. It follows that T* is an affine automorphism of K of period 2. For IJ E K we have T*IJ

= IJ

¢===}

(P

+ PI)*IJ = IJ

¢===}

IJ E co(F U F').

(The last equivalence follows from Theorem 7.46). Thus T* is a reflection of K with fixed point set co(F U F ' ), so by definition (i) holds. Conversely, suppose (i) holds, and let R be a reflection of K with fixed point set co(F U F'). Then !(R + I) is an affine projection of K onto co(F U F'). By Theorem 7.46, there is just one affine projection of K onto co(FUF'), namely (p+p/)*, so we must have !(R+1) = (p+pl)*. Thus R = (2P+2P' -I)*, and (ii) follows. This also establishes uniqueness of the reflection that has co(F U F') as its set of fixed points. 0

322

9

CHARACTERIZATION OF JORDAN STATE SPACES

Remark. By Proposition 5.72, the normal state space K of a JBWalgebra (and the state space of a JB-algebra) is symmetric with respect to co( F u F') for every norm closed face F. The same holds for von Neumann algebra normal state spaces, since the self-adjoint part of a von Neumann algebra is also a JBW-algebra. In the von Neumann context, let p = PI be the projection associated with the face F, let s be the symmetry 2p - 1, and let Us be the map a f----o> sas. Then Us = 2Up + 2Up ' - I, so from Lemma 9.12 it follows that U; is the reflection whose fixed point set is co(F U F'). The following result gives one circumstance in which the Hilbert ball property implies the second of the pure state properties, but will not be needed in the sequel.

9.13. Proposition. Assume the standmg hypotheszs of thzs chapter. If K has the Hzlbert ball property and zs symmetrzc wzth respect to co(F U F') for every proJectwe face F, then P* preserves extreme rays for every compresswn P. Proof. Let 0' be an extreme point of K, and let P be a compression. We will show that P*O' is a multiple of an extreme point of K. If P*O' = 0' or P*O' = 0 this is trivial, and so we assume neither occurs. By bicomplementarity of P and P', it follows that neither P'*O' = 0' nor P'*O' = 0 occurs. Let A = IIP*O'II. Note that

IIP'*O'II

= (1, P'*O') = (1 - PI, 0') = 1 - IIP*O'II = 1 - A.

By neutrality of P* (Proposition 7.21) and our assumptions, 0 Define 0'1

= A-I P*O' and

0'2

< A<

l.

= (1 - A)-l P'*O'.

Note that 0'1 and 0'2 are positive and of norm 1, hence are in K. Now define T = (2P + 2P' -1)*0'. By Lemma 9.12, (2P + 2P' - 1)* is an affine automorphism of K of period 2, and so T is an extreme point of K. Now we have

It follows that 0'1 and 0'2 are in the face of K generated by 0' and T. Furthermore, 0'1 and 0'2 are in the disjoint faces F and F', where F is the projective face associated with P. Therefore the line segment [0'1,0'2] cannot be extended beyond 0'1 or 0'2 and remain in K. (Otherwise, one of 0'1, 0'2 would be in the face generated by the other, contradicting F n F' = 0.) Hence both are on the boundary of the Hilbert ball face(O', T). For a

323

ATOMICITY

Hilbert ball, boundary points are extreme points. Thus 0"1 is an extreme point of this face, and then also of K. Hence P*O" = AO"l is a multiple of an extreme point. 0 Now we show that the third pure state property (symmetry of transition probabilities) follows from the Hilbert ball property. 9.14. Proposition. Assume the standmg hypothesIs of thzs chapter. If K has the HIlbert ball property, then for every pazr of atoms u, v E P,

(u,7J) = (v,ii).

(9.7)

Proof. By hypothesis F = face(ii,7J) is a norm exposed face affinely isomorphic to a Hilbert ball. Being norm exposed, the face F is also projective, therefore of the form F = K n im Q* for a compression Q. Since ii,7J E F, we can pass to the corresponding projective units and obtain u,v :::; Ql. Now by Proposition 8.17, u and v will be projective units for the relativized duality of (imQ,Q1) and (imQ*,F). Thus the restriction ofu and v to F will be affine functions with values in [0,1]; the former with the value 1 at ii and 0 at its antipodal point, and the latter with the value 1 at and 0 at its antipodal point. Denoting by w the center of the Hilbert ball F, and by the angle between the vectors ii - wand 7J - w, one easily computes that (ii, v) = (7J, u) = ~ + ~ cos e. 0

v

e

Atomicity Most of our effort in this chapter will be spent to characterize the normal state spaces of atomic JBW-algebras (i.e., those for which the lattice of projections is atomic). These are precisely the JBW-algebras that are the direct sum of type I factors (Proposition 3.45), so this also will lead to a characterization of normal state spaces of type I factors. In this section, we give a geometric property that is closely related to atomicity. Recall that the O"-convex hull of a set X in a Banach space consists of all (necessarily norm convergent) countable sums Li AiXi where A1, A2, ... are non-negative real numbers such that Li Ai = 1, and Xl, X2, . .. are points in X. By Theorem 5.61, the normal state space of an atomic JBWalgebra is the O"-convex hull of its extreme points. By virtue of the next result, this property for JBW-algebras is equivalent to atomicity, and will playa role in our state space characterizations. 9.15. Proposition. Assume the standmg hypothesIs of thzs chapter. If every extreme point of K is norm exposed and the O"-convex hull of the extreme points of K equals K, then the lattices C, P, and F aTe atomIc. Proof. Let G be a non-empty projective face in F. We will show that G contains an extreme point. Let w E G. By assumption, we can find a

324

9.

CHARACTERIZATION OF JORDAN STATE SPACES

sequence of extreme points W1,W2, ... and non-negative scalars A1, A2, ... with sum 1 such that W = Li AiWi' We may assume without loss of generality that A1 is not zero. Then CXJ

(9.8)

W = A1W1

+ (1

- Ad((l - Ad- 1

L

A;Wi)'

i=2

Since C is a face of K, this implies that W1 E C. By assumption, the extreme point Wj is exposed, so {W1} is an atom in F dominated by C. It follows that in the isomorphic lattice P, every non-zero projective unit dominates an atom. Let p be any projective unit, and let q be the least upper bound of the atoms dominated by p. If q -I p, then p - q must dominate an atom, and this atom is then dominated by both q and 1 - q, contrary to q /\ q' = O. vVe conclude that p = q, so every projective unit is a least upper bound of atoms. Thus P and the isomorphic lattices F and C are atomic. 0 Recall that by Lemma 3.42 the least upper bound of the atoms in a lBW-algebra !vI is a central projection z such that zM is atomic and (1 - z)M contains no atoms. We will now show that given the standing hypothesis of this chapter, the covering property implies a similar splitting into atomic and non-atomic parts.

9.16. Proposition. Assume the standmg hypothesls of thls chapter. Assume also that each extreme pomt of K lS norm exposed, and that the lattlces C, P, F have the covenng property. Let z be the least upper bound of the atoms in P. Then z lS central, every non-zero projectwn under z domznates an atom, and z' = (1 - z) domlnates no atom.

Proof. We first establish the following implication involving the projective unit r(a) associated with an element a E A+, and arbitrary compressions P and Q: (9.9)

Pa E imQ

=}

P(r(a)) E imQ.

Indeed, if Pa E im Q, then Q' Pa = 0, so a is zero on P*Q'*a for all a E K. By (8.18), a and r(a) annihilate the same elements of K, so r(a) is zero on P*Q'*a for all a E K. Thus Q' Pr(a) = 0, which implies Pr(a) E imQ. Now let z be the least upper bound of the atoms in A, which exists since the lattice of projective units is complete (Proposition 8.20). We will prove that z is central. Let Q be the compression corresponding to z; note that im Q contains all atoms. Let P be any compression, and Ul, U2, . . . , Un atoms. By the covering property (9.2) and Proposition 9.7

THE TYPE I FACTOR CASE ((ii)

=}

325

(iii)), PU1, ... PUn are multiples of atoms. Therefore

P(U1

+ ... + un)

E im Q.

By (9.9)

P(r(U1 Since Ui = r(ui):::: r(u1 + ... 7'(U1 + ... + un). Therefore

+ ... + Un)) + un)

for

2

E

imQ.

= 1, ... ,n, then U1 v··· VU n

::::

Since im+Q is a face of A+, then P(U1 v··· Vu n ) E imQ. By Lemma 7.29, P( U1 V ... V un) :::: z. The family of all 1. u. b. 's of finite sets of atoms is directed upwards with l.u.b. equal to z. By Lemma 8.19, z is also the weak limit of this directed set. By weak continuity of compressions, we conclude that pz :::: z. By Proposition 7.39, P is compatible with z. Since P was an arbitrary compression, then z is central. Let p be any non-zero projective unit such that p :::: z, and let P be the corresponding compression. We will show p dominates an atom. We are going to show that there exists an atom U such that Pu I O. Suppose instead that Pu = 0 for each atom u. Then pi U = U, so U :::: p'. Since this holds for all atoms u, and the least upper bound of the set of atoms is z, it follows that z :::: pi and so p:::: Zl, which contradicts p:::: z. Thus there exists an atom U such that Pu I O. By the covering property and Proposition 9.7, Pu = AV for some atom v and some A I O. Then v E im P, so by Lemma 7.29, v :::: p, and we are done. Finally, if U is any atom and u :::: Zl, then U :::: z 1\ Zl = 0, a contradiction, so Zl dominates no atoms. 0

The type I factor case In this section we will give two characterizations of the normal state spaces of JBW-factors of type I, and more generally of atomic JBWalgebras. One characterization (Theorem 9.33) will rely on the pure state properties, while the other (Theorem 9.34) will rely on the Hilbert ball property and the property that K is the (J'-convex hull of its extreme points. Our approach is to work with the following set of properties, which will be shown to be implied by either of the aforementioned sets of axioms, and to imply both of the said characterizations. In this section we will assume that the following conditions are satisfied. (i) A, V satisfy the standing hypothesis of this chapter. (ii) The lattices C, P, F are atomic.

326

9.

CHARACTERIZATION OF JORDAN STATE SPACES

(iii) The lattices C, P, :F have the finite covering property. (iv) Every extreme point of K is norm exposed. (v) (u, v) = (v, u) for each pair of atoms u, v E A. We will refer to (i) - (v) as the workmg hypotheses of thzs sectwn.

9.17. Proposition. Assume that the pazr A, V satzsfy the workmg hypotheses of thzs sectwn, and let P be a compresswn. Then the pmI' im P, im P* also sahsfy the workmg hypotheses of thzs sectwn.

Proof. The properties (i) and (iv) follow from Proposition 8.17, and the remarks following that proposition. By the same result, the lattice of compressions for the pair im P, im P* can be identified with the compressions dominated by P, and (ii) and (iii) follow. In order to establish (v) for this duality, we first show that for atoms u and compressions P we have the equivalence

(9.10)

u E im P

-¢==}

u E im P* .

To see this, note that u E im P is equivalent to u ::::: PI (Lemma 7.29). By the order preserving correspondence of projective units and projective faces (Proposition 7.32), u::::: PI is equivalent to {u} c (imP*)nK, which proves (9.10). Now let u and v be atoms under PI. Then is in im P*, so is the unique element of K n im P* with the value 1 on the atom u. Thus (v) follows for the duality of im P, im P*, which completes the proof that the working hypotheses of this section hold for the pair im P, im P*. 0

u

We observe that we have proven slightly more than was stated in the proposition. If A, V satisfy the standing hypothesis of this chapter, and if P is any compression, then the pair im P, im P* also satisfies the standing hypothesis of this chapter. Furthermore, if anyone of the properties (ii) (v) holds for A, V, then it holds for the pair im P, im P*. The finite covering property assures that there is a well-behaved notion of dimension for elements of the lattices C, P, :F. In an atomic lattice L, the dzmenswn of a finite element p is the least number of atoms whose least upper bound is p (A 42). 'Ne will make frequent use of the following result.

9.18. Proposition. Assume the standmg hypothesls of thls chapter, and that P lS atomic and has the fimte covenng proper·ty. Let p be a fimte element of P. Then p can be expressed as a sum of atoms. In fact, p = PI + ... + Pk for each maxlmal set of orthogonal atoms PI, ... , Pk under P and the cardmality of any set of atoms with sum P is dim(p). Furthermore, every element q ::; P lS fimte, and If q ::; P with q #- p, then dim(q) < dim(p).

THE TYPE I FACTOR CASE

327

Proof. The lattice P is a complete atomic orthomodular lattice with the finite covering property, so all but the last strict inequality follows from (A 44). So suppose now that q ::; p with q i- p. Then each of q and p - q can be written as finite sums of dim( q) and dim(p - q) atoms respectively, and so p can be expressed as the sum of dim(q) + dim(p - q) atoms. It follows that dim(p) = dim(q) + dim(p - q) > dim(q). D We say a compression P is cofimte if its complement pI is finite (and we define cofimte similarly in the lattices P and F). 9.19. Lemma. Assume A, V satzsfy the workmg hypotheses of thzs sectwn. Then each fimte or cofimte compresswn P maps atoms to multiples of atoms and P' maps extreme rays to extreme rays.

Proof. Let P be a cofinite compression and p = PI. By the working hypotheses, P has the finite covering property. Thus for each atom u E P, p/Vu covers or equals pi, and so by Lemma 8.9, (p'Vu) I\p = (p/Vu) _p' is an atom or zero. By Proposition 9.7, P* maps extreme points to multiples of extreme points and P maps atoms to multiples of atoms. Now suppose that P is finite. Let q = p V u, where u is an atom. By definition q is finite. By Proposition 9.17, the working hypotheses also hold for the duality of im Q and im Q*. Since Q is finite, then so are all compressions below Q (Proposition 9.18.) In particular, the complementary compression for P viewed as a compression on im Q will be finite, so P will be cofinite. It follows from the first part of this proof that Pu will be a multiple of an atom. Thus P maps atoms to multiples of atoms, and so P* maps extreme rays to extreme rays (Proposition 9.7). D We remind the reader of our convention that the weak topology on A refers to that given by the duality of A and V. (In the context of JBWalgebras or von Neumann algebras, this would be the a-weak topology.) Since we are always working in the context where A = V*, this coincides with the w*-topology. In particular, the unit ball of A is weakly compact. 9.20. Lemma. ASS1lme the workmg hypotheses of thzs section. If a E V and (u, a) ~ 0 for all atoms u, then a ~ o.

Proof. If p is any projective unit, then atomicity of the lattice P implies that each maximal orthogonal collection of atoms under p has l.u.b. equal to p. The finite sums from this collection form an increasing net of projective units with l.u.b. p. Thus p is the weak limit of finite SUms of atoms (Lemma 8.19), and so (p, a) ~ o. By Proposition 8.31, the projective units are weakly dense in the set of extreme points of the order interval [0,1]. Since [0,1] is convex and weakly compact, by the Krein-Milman theorem, a ;0:: 0 on [0,1]' and so a ;0:: 0 on A +. 0

328

9.

CHARACTERIZATION OF JORDAN STATE SPACES

9.21. Definition. Assume the working hypotheses of this section. Then Af denotes the linear span of the finite projective units in A. Recall that if G is a convex set, oeG denotes the set of its extreme points.

9.22. Lemma. Assume the workzng hypotheses of Af and V aTe in sepamtzng order and norm dualdy.

th~s

sectzon. Then

Proof. Af and V are in separating ordered duality (cf. (A 3)) by Lemma 9.20. To complete the proof we must show that the norm of an element of V is equal to its supremum on the unit ball of A f , d. (A 21). Let 0' E V and assume 0' restricted to Af has norm 1. Since the projective units are weakly dense in oe [0,1]' and a I-> 2a - 1 is an affine homeomorphism of [0,1] onto [-1,1]' then {2p - 1 I pEP} is weakly dense in oe[-I,I]. As in the proof of Lemma 9.20, for each pEP we can choose a net Po: of finite projective units converging weakly to p and a net qo: of finite projective units cOllverging to 1 - p. Then 1 ::;> (Po: - qo:, 0') for each a, and (Po: - q"" 0') ----> (2p - 1,0'). Hence (2p - 1, 0') ~ 1. By weak compactness of [-1,1] and the Krein-Milman theorem, we conclude that 110'11 ~ 1. 0

9.23. Lemma. Assume the workzng hypotheses of th~s sectzon. Then there ex~sts a umque posdive lznear map ¢ : Af ----> V such that ¢( u) = U fOT all atoms u. The equatzon (alb) = (a, ¢(b)) defines a symmetnc bzlznear form on Af such that (9.11 )

(ulv)

=

(u, v)

for all pairs of atoms u, v. Furthermore, each fimte or cofinite compresszon P maps Af znto itself and sat~sfies the symmetry conditzon

(9.12)

(Palb)

=

(aIPb)

for all a, bE A f . Proof. First let

Ul,""

Un

be atoms and

)'1, ... ,

An

scalars such that

~~=l Aiui ::;> 0. By (v) of the working hypotheses,

for each atom v. Hence ~~=l AJii 2: 0 by Lemma 9.20. In particular, if L~=l AiUi = 0, then L~=l AiUi = O. Thus there is a well defined positive

THE TYPE I FACTOR CASE

329

linear map ¢ : Ar --) V given by (9.13)

Bilinearity of the form (alb) = (a,¢(b» is trivial. So is (9.11), and symmetry of the bilinear form follows from (9.13) and (v) of the working hypotheses. Now assume that P is a finite or cofinite compression. By Lemma 9.19, P maps atoms to multiples of atoms, hence P maps Ar into A r . Let u and v be atoms. In order to prove that P is symmetric with respect to the bilinear form on A r , we start by showing that (9.14)

((1-P)uIPv) =0.

Write Pv = AW, where W is an atom. We may assume A # 0, since otherwise (9.14) is clear. Since wE imP, then by (9.10), iii E imP*. This together with the definition of the bilinear form on Ar gives ((I - P)u I Pv) = A((1- P)u I w) = A((I - P)u, iii) = A(U, (1- P)*iii) = 0,

which proves (9.14). From (9.14) we conclude that (u I Pv) = (Pu I Pv) for all atoms u, v. Exchanging the roles of u and v, and using (Pu I Pv) = (Pv I Pu), we get (Pulv)

= (PuIPu) = (uIPv)

for all atoms u, v. Now (9.12) follows by linearity, since by definition Ar is the linear span of atoms. D 9.24. Proposition. Assume the workmg hypotheses of thls sectwn. Let P be a compresswn, and let ¢ be the map defined m Lemma 9.23. If P is fimte or cofimte , then (9.15)

¢(Pa)

= P*(¢(a») for all a

EAr.

If P lS fimte and Q zs any compr-esswn such that Q ~ P, then

(9.16)

(Q

+ Q')*¢(P1)

=

¢(P1).

Proof. Assume that P is finite or cofinite. By (9.12) and the definition of the bilinear form (-I,), for all a, bEAr, (b, ¢(Pa) = (b I Pal = (Pb I a) = (Pb, ¢(a) = (b, P*¢(a).

330

9.

CHARACTERIZATION OF JORDAN STATE SPACES

Since Af and V are in separating duality (Lemma 9.22), then ¢(Pa) P*¢(a) follows. Assume now that P is finite and Q is any compression such that Q ~ P. By Lemma 7.42, P and Q are compatible. Since P is finite and Q ~ P, then Q is finite (Proposition 9.18). Applying (9.15) and compatibility of Q with PI, we get

(Q

+ Q')*¢(P1) = ¢((Q + Q')(P1)) = ¢(P1),

which proves (9.16). 0 Note that (9.16) says that ¢(P1) is a tracial state of imP for the duality of im P and im P*, cf. Definition 8.81. Recall that (J, T E K are defined to be orthogonal if II(J - Til = 2 (cf. (8.20)).

9.25. Lemma. Assume the workzng hypotheses of tiLlS sectwn. Let u, v be atoms. Then u, v are orthogonal lff U, v are orthogonal.

PTOOj. If u ..1 v, then (v, u) :s: (1 - u, u) = 0, so (v, u) = 0, and similarly (u, v) = 0. Then (u - v, U - v) = 2. Since Ilu - vii :s: 1, then Ilu - vii = 2, so U, are orthogonal. Conversely, suppose that ..1 Define w = then this must coincide with the unique orthogonal decomposition of w (cf. Theorem 8.27). Thus by the same result, there exists a compression P such that P*U = and p'*v = v. Then by (9.10), Pu = u and P'v = v, so u and v are orthogonal. 0

v

u v.

u - v;

u

9.26. Lemma. Assume the workzng hypotheses of thls sectwn. Let P be a fimte compression. Then every (J E im P* can be WTltten as a fimte lznear combmatwn (J = 2.:~=1 Ai(Ji of orthogonal poznts (JI, ... ,Un E (OeK) n imP*.

PTOOj. By working in the duality of im P and im P* (cf. Proposition 8.17), we may assume that PI is equal to the order unit 1, (i.e., that P is the identity map), so that 1 is finite. We will use induction on the lattice-theoretic dimension of P. If dim(P) = 1, then P is a minimal non-zero compression, so by Proposition 8.36 the corresponding projective face consists of a single extreme point, and thus the proposition holds in this case. Now assume that the result is true for dim(P) < n. Let dim(P) = 71, and let (J E im P* be arbitrary. By assumption 1 is finite, and thus is a finite sum of orthogonal atoms (Proposition 9.18). Define T = ¢(1). By Lemma 9.25, T is a finite sum of orthogonal extreme points of K. By Proposition 9.24, for all compressions Q we also have (9.17)

(Q

+ Q')*T

= T.

THE TYPE I FACTOR CASE

331

If a IS a multiple of T, then we are done. If not, then there is a scalar A E R such that neither a :::: AT nor a ;::: AT. Thus for that scalar A, if we define w = a - AT, then w is neither positive nor negative. By Theorem 8.27, a can be written as a difference of positive elements w = w+ - w- in such a way that there is a compression Q such that w+ = Q*w, w- = -Q'*w. Then (Q + Q')*w = w. Thus by equation (9.17) (Q

+ Q')*a =

(Q

+ Q')*(w + AT) =

(w

+ AT) = a.

Now let al = Q*a and a2 = Q'*a; note that a = al +a2. Since w L- 0 and Q*w = w+ ;::: 0, then we cannot have Q = P (since P is the identity map), and similarly Q' I:- P. By Proposition 9.18, the lattice dimensions of Q and of Q' are strictly smaller than that of P. By our induction hypothesis, al is a finite linear combination of orthogonal extreme points of K in im Q*, which we can write in the form Ul, ... ,Uk for atoms U1, ... , Uk in imQ (cf. (9.10)). Similarly we can write a2 as a finite linear combination of orthogonal elements ih, ... , vm in im Q'* n oeK, where VI, ... , Vm are atoms in im Q'. Then each Ui is dominated by Q1 and each Vj is dominated by Q'l, so Ui 1.. Vj for 1 :::: z :::: k and 1 :::: J :::: m. By Lemma 9.25, the corresponding extreme points of K are orthogonal. Substituting these decompositions into a = al + a2 gives the desired decomposition of a. 0

9.27. Corollary. Assume the workmg hypotheses of thls sectwn. Then every a E Ar can be wrztten as a finde lmear combmatwn a = 2::7=1 Aiui of orthogonal atoms Ul,··· ,Un·

a = ¢(a)

Proof. We first show that the map ¢ : Ar -> V is one-to-one. Let 2::7=1 Aiui be any finite linear combination of atoms 'Ul, ... , Un. If = 0, then for all atoms v in A,

Let P be the finite compression corresponding to Ul V ... V Un. Then a annihilates every E oeK in im P*, so by Lemma 9.26, a annihilates all of im P*. Since a E im P, the separating duality of im P and im P* implies that a = o. Finally, applying Lemma 9.26 to ¢(a) shows that we can write ¢(a) = 2:::: 1 f3Wi for orthogonal points al, ... , am E oeK. If we choose atoms vI, ... , Vm such that Vi = ai for each l, then ¢(a) = ¢(L::: 1 f3iVi) and so a = L:::l f3iVi is our desired representation. 0

v

332

9

CHARACTERIZATION OF JORDAN STATE SPACES

9.28. Definition. Assume the working hypotheses of this section. For each atom u = Pe (with P a compression), and each bE A we define (9.18) Note that by (1.47), u

*b=

u

0

b when A is a JBW-algebra.

9.29. Lemma. Assume the workzng hypotheses of thl.s secb'Jn. Let u, v be atoms zn A. Then'

(i) u * v = v * u. (ii) If u -.l v and bE A, then u (iii) If u -.l v, then u (iv) U H l = U.

* v = O.

* (v * b) = v * (u * b).

Proof. (i) Let P, Q be the compressions such that PI = u and Q1 = u V v. On im Q the complement of P is the restriction of P' (Proposition 8.17), and thus u * v is the same whether calculated in A or in im Q. Hence without loss of generality we may assume that uVv = 1. If u = v then (i) is obvious, so we may also assume that u # v. Then dim(l) = dim(uVv) = 2. Now dim(l - u) = 1, so 1 - u is also an atom. Since u is an atom, then the image of P is one dimensional (Proposi.. tion 8.36), and thus im P consists of all multiples of u and im P* consists of all multiples of (cf. (9.10)). Therefore for each atom w,

u

PW=AU for some scalar A. Applying u to both sides and using P*u (w, u) = A, and so (9.19) Since u'

Pw

= (w,u)u = (ulw)u.

= 1 - u is also an atom, we have P'w

= (w, ;;')u' = (u'lw)u'.

By (9.18) we can compute u

u

*v =

*v

as follows.

+ (ulv)u - (u'lv)u') = ~ (v + (ulv)u - (u'lv)(l - u)) = ~ (v + (ulv)u + (u'lv)u - (u'lv)l) = ~ (v + (llv)u - (u'lv)l). ~(v

= u gives

THE TYPE I FACTOR CASE

Since (1Iv)

333

= (1, v) = 1, then u *v

= ~(v

+ u-

(1 - (ulv))I).

Since this last expression is symmetric in u and v, (i) follows. (ii), (iii) Now assume u ~ v, and let P, Q be the compressions corresponding to u, v. Then P, Q, pi, Q' commute (Lemma 7.42), so (ii) follows. Also, v :"::: u' implies that Pv = 0, and then P'v = v, so (iii) follows from the definition (9.18). (iv) follows at once from (9.18). 0 In the proof of several results below we shall also need the following elementary result on order unit spaces: If Ao is a linear subspace of an order unit space A and 0 is commutative bilinear product on Ao with values in A such that

(9.20)

-1:":::a:":::l

=}

O:":::aoa:":::1

for a E A o , then

Iia 0 bll :": : Ilallllbll

(9.21)

for all pairs a, bE Ao (A 50). This can be readily seen from (9.20) and the identity

9.30. Proposition. Assume the workmg hypotheses of thls sectwn. Let Ao be the norm closure of Af + Rl. Then there lS a umque product a 0 b on Ao such that Ao lS alB-algebra wlth ldentlty 1 and such that

uov=u*v. Proof. We first define the product on Af by

(9.22)

a 0

b=

L

cxi{3jUi

*vJ

i,j

where a = 2.:i CXiUi and b = 2.: j {3jVj with Ul, ... , Un and Vi, ... , Vm atoms. From Definition 9.28 it follows that the right side of (9.22) is independent of the representation of b as a linear combination of atoms. From Lemma 9.29 (i) it follows that it is also independent of the representation of a, and so the product is well defined on A f . There is clearly a unique extension of this product to a commutative bilinear product on Af + R1 such that 1 acts as the identity. (If it happens that 1 is in A f , then from the definition (9.18) it is easy to check that 1 already acts as the identity on Ad

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CHARACTERIZATION OF JORDAN STATE SPACES

Next we show that this product on Af

+ Rl

satisfies

(9.23) By Corollary 9.27, each element b in Af is a finite linear combination of orthogonal projective units, say VI, .. . , V". Then the same is true of each element b + Al with A E R, since b + Al is a linear combination of the orthogonal projective units VI, ... , V"' q where q = 1 - U1 - ... - Un. It is straightforward to check that q2 = q, and that q * Uj = 0 fo~ J = 1, ... , n. Thus for a E Af + Rl we can find real scalars AI, . .. ,An and A such that

Now by applying the compressions associated with U1, ... , Un, q to an element a E Af + Rl such that -1 S; a S; 1, we conclude that AI, . .. , An and A are between -1 and 1, and thus that (9.23) holds. A similar calculation shows that all groupings a * (a * ... * (a * a) * ... ) of n factors of a give the same product, namely

and thus the product is power associative. Now by the remarks preceding this proposition, (9.23) implies that (9.21) holds in Af + Rl. Hence there is a unique extension of the product o to the norm closure Ao of Af + Rl. By continuity, the resulting product on Ao is commutative, power associative, squares are positive, the order unit 1 acts as an identity, and (9.21) holds. Then by definition Ao is a commutative order unit algebra (cf. Definition l.9), and thus Ao is a JB-algebra by Theorem 2.49. D 9.31. Lemma. Assume the workmg hypotheses of thIs sectIOn. For each a E Ao and each a E V there IS a umque element a 0 a E V such that (9.24)

(b, a 0 a) = (a

0

b, a)

for all b E Ao.

Proof. Suppose first that a is an atom, and let P be the compression such that a = Pl. Then by definition of the Jordan product on A o , for bE A o , a

0

b = ~(I

+P

- P')b.

Then

(aob,a) = (~(I+P-P')b,a) = (b,~(I+P-P')*a),

335

THE TYPE I FACTOR CASE

and so (9.24) holds with a 00"= ~(I + P - P')'O". We can take 100" to be 0", and so by linearity an element a 0 0" satisfying (9.24) exists for all a in Af + Rl. For an arbitrary element a of A o , choose a sequence {an} in Af + Rl converging in norm to a. By the norm property (9.21), {an 0 O"} is a Cauchy sequence in V. By completeness of V, this sequence converges in norm to an element of V; we define that element to be a 0 0". It is then evident that (9.24) holds. Since Af and V are in separating duality, the element a 0 0" satisfying (9.24) is unique. D

9.32. Lemma. Assume the workmg hypotheses of thzs sectwn. Then A can be equzpped wzth a product that makes zt an atomic ]B W-algebra wlth predual V (wzth the gwen order and norm on A and V ). Proof. By Lemma 9.22, Af and V are in separating norm duality, as are A and V by hypothesis. Thus Ao and V are also in separating norm duality, so we can imbed V isometrically into AD. We therefore identify V with a subspace of AD. By Corollary 2.50, AD' is a lBW-algebra, with a separately w*-continuous product that extends the product on Ao. By w*-density of Ao in AD', together with (9.24), we conclude that for each 0" E V, (9.25)

(b, a

0 0")

= (a 0 b, 0") for all a E Ao and all bEAD'.

Now let] be the annihilator of V in AD'. By (9.25), a 0] c ] for all (= O"-weakly closed), it follows that ao] C ] for all a E AD', i.e., ] is a O"-weakly closed ideal of AD'. By Proposition 2.39 there is a central projection c E Ao' such that ] = im Uc = cAD*. Since V is complete, it is a norm closed subspace of AD' so by the bipolar theorem, V is the annihilator of ] = im Uc in AD. The annihilator of im Uc is ker U;, which coincides with im Ui-c. By Propositions 2.9 and 2.62, Ui_c(A D) is the predual of the lBW-algebra im U 1 - c , and thus we have the isometric isomorphism a E Ao. Since] is w*-closed

Hence A can be equipped with a product making it a lBW-algebra with predual V. The order on an order unit space is determined by the norm and the order unit (since for an element a with norm 1, a 2: iff Ill-all :s; 1), so the order on A as a lBW-algebra coincides with that inherited as the dual of the base norm space V, and thus matches the given order on A. By uniqueness of the predual of a lBW-algebra (Theorem 2.55), K will be affinely isomorphic to the normal state space of A. By assumption the lattice F of norm exposed faces of K is atomic, and therefore so is the projection lattice of A, i.e., A is an atomic lBW-algebra. D

°

336

9

CHARACTERIZATION OF JORDAN STATE SPACES

Remark. For the interested reader, we sketch the structure of the proof above in a familiar context, without providing details. In the case where A = B (H),," then Af consists of the finite rank operators, and Ao will be ]{ + R1 where ]{ is the space of (self-adjoint) compact operators Then Ao = V EEl Rwo, where Wo is the unique state on ]{ + R1 that annihilates ]{, and Ao* ~ B(H)"" + Rco, where Co is a central projection and an atom. Thus essentially we have defined the Jordan product on the finite rank operators and then recovered the Jordan product on B(H)s" as the bidual. 9.33. Theorem. Let ]{ be the base of a complete base nOTm space. Then ]{ zs affinely zsomoTphzc to the nOTmal state space of an atormc lEW-algebm ziJ all of the followmg hold.

(i) EveTY nOTm exposed face of ]{ zs pTO)ectzve. (ii) EveTY non-empty spilt face of ]{ contams an extTeme pomt. (iii) ]{ has the pUTe state pTOpeTtzes (cf. Defimtwn 9.2). FUTtheTmoTe, ]{ wzll be affinely zsomoTphzc to the nOTmal state space of a lEW-factoT of type I zff ]{ satzsfies the thTee condztwns above and m addztwn]{ has no pTOpeT splztface (Le., one otheT than]{ OT 0). PTOOj. Assume first that ]{ is affinely isomorphic to the normal state space of an atomic JBW-factor lvI. By Theorem 5.32 every norm closed face of ]{ is projective. so (i) holds. By definition of an atomic JBVv-algebra, the lattice of projections of AI is atomic. By Proposition 5.39, atoms in the projection lattice correspond to projective faces consisting of a single extreme point, so every projective face of ]{ contains an extreme point. By Proposition 7.49 every split face of ]{ is a projective face. (Alternatively, every split face is norm closed (A 28), and therefore projective (Theorem 5.32)). Thus (ii) holds. By Proposition 9.3, the normal state space of AI has the pure state properties, so (iii) holds. Now suppose that ]{ is the distinguished base of a complete base norm space V, and satisfies (i), (ii), and (iii). Let A = V*. We are going to show that the working hypotheses of this section hold. The pair A, V are in separating order and norm duality (A 27). Then (i) implies that the standing hypothesis for this chapter holds. The properties (iv) and (v) of the working hypotheses of this section are part of the pure state properties. The covering property follows from the second of the pure state properties (Corollary 9.8). It remains to show that the lattices C, P, :F are atomic. Let z be the least upper bound of the atoms in P. By the covering property and Proposition 9.16, z is central, every projection under z dominates an atom, and 1 - z dominates no atom. We will show z = 1, which will then imply that the lattices P, C, :F are atomic.

THE TYPE I FACTOR CASE

337

The projective face F corresponding to z is a split face by Proposition 7.49. Let F' be the complementary split face. If F' is not empty, then by (ii) it contains an extreme point a, which is norm exposed by the first of the pure state properties. Then z' = 1 ~ z must dominate the atom corresponding to a (cf. Proposition 8.36), which contradicts the fact that z' dominates no atom. Thus F' must be the empty set, and F must be all of K, so z = 1, which completes the proof of atomicity. Therefore the working hypotheses of this section are satisfied. By Lemma 9.32, A = V* can be equipped with a product making it a JBWalgebra. By the uniqueness of the predual (Theorem 2.55), K is affinely isomorphic to the normal state space of A. Finally, suppose that K satisfies (i), (ii), (iii), and has no proper split faces. Then as established in the first part of this proof, K is the normal state space of a JBW-algebra IvI. Since K has no proper split faces, then M is a JBW-factor (Corollary 5.35). Since K contains extreme points, then M is a JBW-factor of type 1 (Corollary 5.41). Conversely, if K is the normal state space of a JBW-factor AI of type I, then M is atomic, so (i), (ii), (iii) hold, and since M is a factor, then K has no proper split face (Corollary 5.35). This completes the proof of the last statement of the theorem. D

9.34. Theorem. Let K be the base of a complete base norm space. Then K zs affinely zsomorphzc to the normal state space of an atomzc lEW-algebm iff all of the followmg hold.

(i) Every norm exposed face of K zs proJectwe. (ii) The a-convex hull of the extreme pomts of K equals K. (iii) K has the HIlbert ball property. Furthermore, K wzll be affinely zsomorphzc to the normal state space of a lEW-factor of type I zff K satIsfies the three condItIOns above and m addition K has no proper spilt face. Proof. Assume first that K is affinely isomorphic to the normal state space of an atomic JBW-factor M. By Theorem 5.32 every norm closed face of K is projective, so (i) holds. By Theorem 5.61, (ii) holds. By Proposition 9.10, (iii) holds. Now suppose that K is the distinguished base of a complete base norm space V, and satisfies (i), (ii), and (iii). Let A = V*. We are going to show that the working hypotheses of this section hold. The pair A, V are in separating order and norm duality (A 27). Every extreme point of K is norm exposed by the Hilbert ball property. Property (ii) implies that the lattices C, P, :F are atomic (Proposition 9.15). The Hilbert ball property implies the finite covering property (Proposition 9.11). The symmetry of transition probabilities, i.e., (v) of the working hypotheses of this section, also follows from the Hilbert ball property (Proposition 9.14).

338

9

CHARACTERIZATION OF JORDAN STATE SPACES

Thus the working hypotheses of this section are satisfied. The remainder of the proof of the current theorem is the same as the last two paragraphs of the proof of Theorem 9.33. D

Remark. Instead of assuming in Theorems 9.33 and 9.34 that every norm exposed face of K is projective, we could instead assume the stronger property that K is a spectral convex set, d. Definition 8.74 and Theorem 8.52. Recall that if C5, T are extreme points of the normal state space of a JBW-algebra, the face they generate is a norm exposed Hilbert ball (Proposition 5.55). \Ve will see that the affine dimension of this ball distinguishes the normal state spaces of type I JBW-factors. Recall from Theorem 3.39 that type I JBW-factors are either R, spin factors, H 3 (O), or the bounded self-adjoint operators on a real, complex, or quaterniollic Hilbert space.

9.35. Definition. A type I JBW-factor is said to be roeal (resp. complex resp. quaternwmc) if it is isomorphic to B(H)sa for some real (resp. complex resp. quaternionic) Hilbert space H. 9.36. Proposition. Let C5, T be dzstmct extreme pamts of the normal state space K of a lEW-factor M of type 1.

(i) If If If If If

(ii) (iii) (iv) (v)

M M M M M

real, then dzm face(C5, T) = 2. complex, then dzm face(C5, T) = 3. quaternwmc, then dzm face(C5, T) = 5. a spm factor, then dIm face(C5, T) = dzm M = H 3 (O), then dzm face(C5,T) = 9. zs zs zs zs

~

1.

Proof. Let p and T be the carrier projections of C5 and T. Then by the isomorphism of the lattice of norm closed faces and the lattice of projections (Corollary 5.33), p and T are atoms (i.e., minimal projections), and the carrier of face (C5, T) is pVT. Let q = p V T ~ p. Then p and q are orthogonal atoms (Lemma 3.50), with p V q = pVT. Thus face(C5, T) can be identified with the normal state space of lvIp +q • The affine dimension of face(C5, T) is then one less than the linear dimension of lvIp +q . By the classification of type I JB\V-factors, lvI is isomorphic to a spin factor, to H 3 (O), or to B(H)sa for a real, complex, or quaternionic Hilbert space. If lvI is a spin factor, then !vI is of type 12 , so any pair of orthogonal projections must add to 1 (Lemma 3.22). Thus in this case lvIp +q = lvI, aud hence (iv) holds. So it remains to consider the case where !vI is H 3 (O), or B (H)sa for a real, complex, or quaternionic Hilbert space. In each case, there are orthogonal minimal projections p and q such that !vI~+~ p q is isomorphic to 1\,f2(R)sa, lvI2 (Ck" M 2 (H)sa, or M 2 (O)sa' The results stated in the proposition will follow if we show that p + q is equivalent to p + q, so that M p + q is isomorphic to M;+q'

CHARACTERIZATION OF STATE SPACES OF JB-ALCEBRAS

339

Since p and p are minimal projections in a JBW-factor, they are exchangeable by a symmetry s (Lemma 3.19). Then Usq is orthogonal to UsP = p, as is (j, so both are minimal projections in M 1 _ p' which is a JBW-factor (Proposition 3.13). Thus there is a (1 ~ p)-symmetry to that exchanges Usq and (j. Then t = P+ to is a symmetry that also exchanges Usq and (j (since Jordan multiplication by p on M 1 _ p is the zero operator), while Ut fixes p. Thus UtUs takes p to p and q to (j, so takes p + q to p + (j. This completes the proof. D

Characterization of state spaces of JB-algebras We will continue to assume the standing hypothesis of this chapter, namely, that A is an order unit space and V a complete base norm space with distinguished base K, such that every norm exposed face of K is projective. Recall from Proposition 8.34 and Definition 8.35 that for each a E K there is a smallest central projective unit c(a) (called the central carT"'ler of a) such that (c( a), a) = 1. The corresponding face FeCa) is the smallest split face containing a. We will write Fa in place of FeCa).

9.37. Lemma. Assume the standmg hypothes7s of th7s chapter. If is an extreme pomt of K, then Fa contams no proper spht face.

(J

Proof. Suppose that G is a non-empty split face of K properly contained in Fa. Since K is the direct convex sum of G and the complementary split face G' , and a is an extreme point of K, then (J must be in either G or G' . By minimality of Fa, a cannot be contained in G, so a E G' . Then by (A 7), G' n Fa is a split face of K containing a and properly contained in Fa (since it contains no point of G). This violates the fact that Fa is the smallest split face containing (J. Thus we conclude that no such split face G of K exists. Finally, a split face of Fa would also be a split face of K, as can be verified directly from the definition of a split face (A 5), or from Corollary 8.18. Thus Fa contains no proper split face. D Recall that positive elements al, a2 of A are orthogonal if there exists a projective face F of K such that al = 0 on F and a2 = 0 on F' (Definition 8.38). Recall that under the standing hypotheses of this chapter, every element a E A admits a unique decomposition as a difference of orthogonal positive elements iff A, V are in spectral duality (Theorem 8.55). In the rest of this section K will be a compact convex set. Without loss of generality, we may assume that K is regularly imbedded as the base of the base norm space V = A(K)*, where A(K) denotes the continuous affine functions on K. (See the description of this imbedding following

340

9

CHARACTERIZATION OF JORDAN STATE SPACES

Proposition 8.76.) We let A = Ab(K) = V* be the space of bounded affine functions on K (A 11). Then A and V are in separating order and norm duality (A 27). Note that A(K) C Ab(K), and so notions such as orthogonality of positive elements of A(K) are to be interpreted with respect to the duality of A = Ab(K) and V. One of the key properties that we will use to characterize JB-state spaces is that each a E A(K) admits a decomposition as a difference of positive orthogonal elements of A(K). (For a discussion of a physical interpretation of this property, see the remarks after Proposition 8.6:.) Note that we will not require uniqueness of this decomposition, but we will require that the orthogonal elements be in A(K) and not just in A = Ab(K). Combining this property with the properties previously discussed, we can now characterize the state spaces of JB-algebras. 9.38. Theorem. A compact convex set K zs affinely homeomorphzc to the state space of a lB-algebm (wdh the w*-topology) ztf K satzsjies the conditwns:

(i) Every a E A(K) admds a decomposdwn a A(K)+ and b.l c.

=

b - c wzth b, c E

(ii) Every norm exposed face of K zs pTOJectwe. (iii) K has the pure state propertzes.

Here the condition (m) can be replaced by the alternatwe conddwn (iii ') The a-convex hull of the extreme pmnts of K zs a spld face of K and K has the Hilbert ball pTOperty.

Proof. Let B be a JB-algebra with state space K, and let V = B*. We will show K satisfies the properties above. Recall that we can identify K with the normal state space of the JBW-algebra B** (Corollary 2.61), and so we work in the duality of V and A = V* = B**. (i) By Proposition 1.28, each element of A admits a unique decomposition as the difference of orthogonal positive elements. Those elements will also be orthogonal as elements of the JBW-algebra A **. By Proposition 2.16, positive elements a, b of the JBW-algebra B** arc orthogonal iff their range projections are orthogonal. Viewing a, b as elements of A = Ab(K), this is the same as the notion of orthogonality with respect to the duality of A and V, d. Definition 8.38 and Proposition 8.39. Thus (i) holds. (ii) Norm closed faces of the normal state space of a JBW-algebra are projective (Theorem 5.32), so (ii) follows. This also follows more simply from the fact that positive elements of the JBW-algebra B** and their range projections have the same annihilators in the normal state space, d. (2.6). (iii) follows from Proposition 9.3. (iii ') follows from Corollary 5.63 and Corollary 5.56.

CHARACTERIZATION OF STATE SPACES OF JB-ALCEBRAS

341

Conversely, assume that (i), (ii), and either (iii) or (iii ') hold. For each extreme point (J" of K, let Pe(a) denote the compression associated with C((J") , and let Aa = im Pe(a) and Va = im Pe'(a)" By Proposition 8.17 and Proposition 8.34, we can identify Aa with the space of all bounded affine functions on Fa (the minimal split face generated by (J"). Fix an extreme point (J" of K. By Lemma 9.37, Fa contains no proper split faces. We are going to show that Fa satisfies the conditions (i), (ii), and (iii) of Theorem 9.33 or of Theorem 9.34, and thus that Aa can be equipped with a product making it a type I JBW-factor. By (ii) of the current theorem, A, V satisfy the standing hypothesis of this chapter. Thus by Proposition 8.17, we can identify Aa with the dual of the complete base norm space Va (with distinguished base Fa), and every norm exposed face of Fa is projective for the duality of Aa and Va. The projective faces for this restricted duality will be precisely the projective faces of K that are contained in Fer. Assume that the pure state properties hold for K. Then by the remarks following Proposition 9.17, the first and third of the pure state properties will also hold for Fu' By the second pure state property and Corollary 9.8, the covering property holds for the lattice of projective faces of K, and therefore also for the lattice of projective faces of Fa. Therefore again by Corollary 9.8, the second of the pure state properties holds for Fa. Since Fu contains the extreme point (J", and has no proper split faces, then we have shown that Fu satisfies the hypotheses of Theorem 9.33, and so we conclude that Aa admits a product making it a JBW-factor of type I, with the given order and norm. Now assume alternatively that (iii ') holds. If w, T are extreme points of Fa, then the face G generated by w, T in K is the same as in Fa, so the fact that G is norm exposed and a Hilbert ball in K implies the same in Fa. Thus the Hilbert ball property holds in Fa. Now let G be the split face of K that is the (J"-convex hull of the extreme points of K. Then G n Fa is a split face of K that contains (J", so by minimality of Fa, the split face G must contain Fu. Thus if W is any element of Fa, then W is a (J"-convex combination of extreme points of K. Each of those extreme points of K must in fact be in the face Fu (cf. (9.8)), so must be extreme points of Fa. Thus Fa is the (J"-convex hull of its extreme points. Hence (iii ') holds for Fu. Now by Theorem 9.34, Aer can be equipped with a product making it a JBW-factor of type I, with Fu as its normal state space. Thus we've shown that (i), (ii), and (iii) or (iii ') imply that each Fu is the normal state space of a type I JBW-factor. For each extreme point (J" of K, define Tru to be the restriction map which sends a E A(K) to alFu E Ab(Fu). We have identified Aa with Ab(Fu ), so Trer maps A into Au. Define B = EBerE&,K Au (cf. Definition 2.42) where oeK denotes the set of extreme points of K. Then B is a JB-algebra with pointwise operations and the supremum norm. By the Krein~Milman theorem, the map Tr that takes a to EBuE&e K TrO"(a) is an isometric order isomorphism of A(K) into B.

342

9

CHARACTERIZATION OF JORDAN STATE SPACES

Next, we are going to show that 7r(A(K)) is a lB-subalgebra of B. We first will show that 7r(A(K)) is closed under the map x I-> x+ where x = x+ - x- is the unique orthogonal decomposition in the lB-algebra B (cf. Proposition 1.28). For that purpose, let a E A(K) and let a = b-c be a decomposition of the type described in (i). By the definition of orthogonality, there is a projective face C such that b = 0 on C and c = 0 on C'. For each (J' E 8 e K, by (A 6) the faces F(TnC and F(TnC' are complementary split faces of Fa, and thus are complementary projective faces of Fa by Proposition 7.49. Thus band c restricted to Fa are orthJgonal elements of the order unit space AT' As discussed at the start of this proof, b and c restricted to Fa will also be orthogonal as elements of Aa viewed as a lB'-V-algebra. Thus by the uniqueness of the orthogonal decomposition 7r(a) = 7r(a)+ - 7r(a)- in the lB-algebra A a , we have 7r a (b) = 7r a (a)+ for all (J' E 8 e K. Hence

and in particular 1T(A) is closed in B under the map x I-> x+. Now let X be the spectrum of 1T( a) (cf. Definition 1.18). We will show f (1T( a)) E 1T(A) for each f E CR(X). Let Co denote the set of all f E CR(X) such that f(1T(a)) E 1T(A), where f acts on 1T(a) by the continuous functional calculus (Definition 1.20). Since A(K) is complete, and 1T is an isometry, then 1T(A(K)) is complete. Therefore Co is a linear subspace of CR(X) closed under the map f I-> f+, and so is a vector sublattice of CR(X). Since f I-> f(1T(a)) is an isometry on CR(X) (Corollary 1.19), then Co is a norm closed vector sublattice of CR(X) containing the constants and the identity function. Hence by the lattice version of the Stone-Weierstrass theorem (A 36), Co = CR(X). In particular, Co contains the squaring map, so 1T(A(K)) is a lB-subalgebra of B. Thus 1T is a unital order isomorphism from A(K) onto a lB-algebra, and so its dual gives an affine homeomorphism of the state space of a lB-algebra onto K. 0 Remark. Instead of assuming (i) and (ii) in Theorem 9.38, we could assume the stronger property that K is a strongly spectral convex set. (See the remark after Proposition 8.80.) Recall that a lC-algebra is a lB-algebra which is isomorphic to a norm closed 10rdan subalgebra of B (H)sa (Definition 1.7). 9.39. Corollary. A compact convex set K lS affinely homeomorphlc to the w*-compact state space of a le-algebra iff it satisfies, not only the conditlOns of Theorem 9.38, but also the additional requirement that every mmimal Spllt face of dimenslOn 26 lS a Hllbert ball.

CHARACTERIZING NORMAL STATE SPACES OF JBW ALGEBRAS

343

Proof. By Lemma 4.17 and Corollary 4.20, a JB-algebra A is a JCalgebra iff there is no closed Jordan ideal J in A such that AI J is isomorphic to the exceptional Jordan factor H3(O). Recall from Corollary 5.38 that for every closed ideal J of A there is a natural affine isomorphism of the state space of AI J onto the split face F = JO n K corresponding to J. Thus it remains to show that H3(O) is the only JB-algebra whose state space is a 26-dimensional convex set which has no proper split face and is not a Hilbert ball. Counting (real) parameters in self-adjoint 3 x 3-matrices over the octonions, we find that H3(O) is a 27-dimensional vector space (over R). Therefore the state space of H3(O) is a 26-dimensional convex set. Since H3(O) is a factor, it contains no non-trivial central projection. Therefore its state space contains no proper split face. The state space of H3(O) is not a Hilbert ball since it has proper faces that are not singletons, e.g., the projective faces associated with projections p such that pH3(O)P is isomorphic to H2(O). If B is any JB-algebra whose state space is a 26-dimensional convex set which has no proper split face and is not a Hilbert ball, then B is a JBWfactor (Corollary 5.35), and is of type I (since it is finite dimensional). We conclude from Theorem 3.39 that the only type I factors with 26dimensional state space are the 27-dimensional spin factor and H3(O). The state space of a spin factor is a Hilbert ball (Proposition 5.51), so B = H3(O). We are done. 0 The state spaces of JC-algebras can also be characterized among JBstate spaces by a condition on the dimensions of the Hilbert balls generated by pairs of pure states: cf. [10, Prop. 7.6].

Characterization of normal state spaces of JEW algebras In this section we will always be working with a spectral convex set K which is then the distinguished base of a base norm space V, in spectral duality with A = V* (cf. Definition 8.74). Note then that A = Ab(K) (Proposition 8.73). By Theorem 8.52, every norm exposed face of K will be projective, so the standing hypothesis of the previous chapter is satisfied.

9.40. Lemma. Let K be a spectral convex set, and A = Ab(K). Let {e>.} be a spectral resolutwn for a E A, and let E = {e>'2 - e>., I A1 < A2}. Then:

(i) Each pazr e, f in E is compatzble. (ii) E zs closed under the map (e, 1) f---> e !\ f. (iii) Each element b E lin E can be expressed as a linear combination of orthogonal elements of E.

344

9.

CHARACTERIZATION OF JORDAN STATE SPACES

Proof. Let Eo be a finite subset of E, say Eo

=

{e/3, - e",; II'S

2

'S n},

where ai < Pi for each L Let Al, ... , A2n be the set {al' ... ,an, Pl, ... ,Pn} arranged in increasing order. Let F = {e'\,+l - e,\, 11 'S 2 'S 2n - I}. The projective units in F are orthogonal, and each member of Eo is a sum of elements of F. The items (i), (ii), (iii) follow. 0

9.41. Lemma. Let K be a spectr'al convex set, and A = Ab(K). Let {e,\} be a spectral resolutwn for a E A, let E = {e'\2 - e,\, I Al < A2}, and let M be the norm closure of the lmear span of E. Then there 2S a compact Hausdorff space X and an 2sometnc order 2somorph2sm from CR(X) onto M such that the mduced product * on M sat2sfies e * f = e 1\ f for all e,f E P. Proof. Let Mo

= lin E. Define * on Mo by a*b=

L aJ3 ei J

1\

fj

t,J

where a = 2:i aiei and b = 2: j pjfj with el,···, en, il,··., fm E E. We will show this product is well defined. By Lemma 9.40 all elements of E are compatible. Thus if Pe ; is the compression associated with ei for 2 = 1, ... ,n, then by Theorem 8.3,

for 1 'S

2,

J 'S n. Hence

Thus the definition of a * b is independent of the representation of b, and similarly of the representation of a. Since the operation 1\ is commutative and associative, the same is true of * on 1\110 . Clearly * is bilinear. Next we show that for a E 1\110, (9.26) By Lemma 9.40, we can write a as a linear combination of orthogonal elements el, ... , en of E, say a = Li Aiei. Since -1 ::; a ::; 1, then IAil ::; 1 for all i (as can be seen by applying suitable compressions to both sides of the equation a = Li Aiei)' Then a*a = Li Arei, so 0::; a*a ::; 1, which proves (9.26).

CHARACTERIZING NORMAL STATE SPACES OF JBW ALGEBRAS

Thus (9.20) holds for a E M o, and holds, i.e., for a, bEMa,

*

345

is commutative, so (9.21) also

Iia * bll ::; Iiall Ilbll·

(9.27)

(See also the argument after (9.21).) Now we can extend * to 11'1 (the norm closure of Mo) by continuity. The resulting product on M still satisfies (9.27) and is commutative and associative. By norm continuity of the map * and the fact that A + is norm closed, it follows that a*a 2: 0 for all a E M. It is clear from the definition of the product that the order unit acts as an identity for M. Thus M is a commutative, associative order unit algebra, and so AI ~ CR(X) follows from (A 48). D

9.42. Definition. Let K be a spectral convex set, and A = Ab(K). Let e E P and let P e be the corresponding compression. Then we define Te : A ---> A by

I,

Note that if e and so

I

are compatible projective units, then (Pe+P:)1

=

Thus for compatible e, I we have (9.28) Recall from (1.47) that if A is a JB-algebra, then Te is the Jordan multiplication operator a I--> e 0 a. Thus in this case Tel = Tie = eo I for every pair of projections e, I. For a general spectral convex set K, the equation Tel = Tie may fail for a pair of projective units e, f. But we will now show that if it holds, then the map (e, f) I--> Tel = Tie from P x P into A extends to a bilinear map from A x A into A which organizes A to a JBW-algebra with normal state space affinely isomorphic to K. This result provides our first characterization of normal state spaces of JBW-algebras. Note that this construction of a bilinear product in A is similar to that in Proposition 9.30 with atoms replaced by general projective units. However, in the proof of Proposition 9.30 we made use of the fact that every linear combination of atoms can be written as a linear combination of orthogonal atoms (Corollary 9.27), which is not true if atoms are replaced by arbitrary projective units. Thus we take a different tack, using the spectral theorem (Theorem 8.64) and Lemma 9.41.

346

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CHARACTERIZATION OF JORDAN STATE SPACES

9.43. Theorem. A spectral convex set K is the normal state space of a lEW-algebra iff (9.29) for all paIrs of proJectwe umts e, f E A = Ab(K). More specl.ficaliy, If thIs condztwn IS satIsfied, then there IS a umque bIlmear product 0 on A whIch satIsfies the norm conditwn (9.21) for a, bE A and the equatwn eo e = e for e E P, and thIs product organizes A to the umque lEW-algebra whose normal state space IS affinely IsomorphIc to K. Proof. Clearly (9.29) is a necessary condition that K be the normal state space JBW-algebra since Te is the Jordan multiplication operator a f-> e 0 b in this case. To prove sufficiency, we assume (9.29). Define (e, J) = Tef = Tfe for each pair e, f E P. Let a = Li ociei for ei E P, OCi E R where 1 ::; i ::; 71, and b = LJ3 j f) for fj E P, Pj E R where 1 ::; J ::; m. Then

(2:OCiTei) (b) = i

2: OC,p) (ei' f)) = (2: PjTfJ ) (a). i,]

J

Thus there is a commutative bilinear product 0 on the space Ao := lin P with values in A given by the following definition, which is independent of the representation of a and b:

(9.30) Clearly, this product is norm continuous in each variable separately. We next show that the implication (9.20) holds in Ao. Fix a E Ao. Define E, lvI, and the product * as in Lemma 9.41. By Lemma 9.40, each pair e, fEE is compatible, so by (9.28) e0 f

= Te f = e /\ f = e * f.

Thus 0 and the product * agree on linE. Let {an} be a sequence in linE converging to a. (Such a sequence exists by the spectral theorem (Theorem 8.64)). By separate norm continuity of 0 on Ao and norm continuity of * on M ~ CR(X), (9.31 )

a

0

a

= lim lim an n Tn

0

am

= lim lim an * am = a * a. n Tn

Now in M ~ CR(X) the implication (9.20) holds, so it follows from (9.31) that it also holds in Ao. Thus (9.32)

CHARACTERIZING NORMAL STATE SPACES OF .JBW ALGEBRAS

347

for a E Ao. Now (9.21) also holds, i.e.,

Iia 0 bll :::: Ilallllbll

(9.33)

for all pairs a, bE Ao. By (9.33) and the spectral theorem (Theorem 8.64), there is a unique continuous extension of 0 to a product on A. By continuity, the extended product is still bilinear and commutative, and satisfies (9.32) and (9.33). To verify that A is a JB-algebra for this product, by Theorem 1.11 we need only show that A satisfies the Jordan identity

a2

(9.34)

0

(b

0

a) = (a 2

0

b)

0

a.

So suppose that el, ... , en are orthogonal projective units and AI, ... , An scalars such that a = Li Aiei' Then for each ~,J, the compressions Pei , P:" Pe), P:) associated with ei, ej commute (Lemma 7.42), so Tei and Tej commute. From the definition (9.30) together with commutativity, we have (ei 0 b) 0 ej = ei 0 (b 0 ej), so

',]

i,j

We have then shown that the Jordan identity (9.34) holds for a E Ao with finite spectral decomposition and all b E Ao. By continuity and the spectral theorem (Theorem 8.64), the Jordan identity holds for all a, bE A, By Theorem 1.11, the order from the JB-algebra structure (given by the cone of squares) coincides with the given order from the order unit space

A. The spectral convex set K is the distinguished base of a complete base norm space V (Definition 8.74) whose dual space is the order unit space A = Ab(K) (Proposition 8.73). Thus A is a dual space, hence a JBW-algebra. The JBW-algebra A has a unique predual space A., which consists of all normal linear functionals on A (Theorem 2.55). Therefore the Banach space V is isometrically isomorphic to the space of all normal linear functionals on A. Since V as well as A. are in separating order duality with A, these two base norm spaces are also order isomorphic. Therefore the distinguished bases of these two base norm spaces are affinely isomorphic. Thus K is affinely isomorphic to the normal state space of A. The uniqueness of the product 0 follows from the fact that two bilinear products on A which satisfy the norm condition (9.21) and the equation eo e = e for e E P, must agree on P and by linearity on Ao, and then by norm continuity and the spectral theorem also on A. If A is any J~W -algebra whose normal state space is affinely isomorphic to K, then A is isomorphic to A by Proposition 5.16. D

348

9

CHARACTERIZATION OF JORDAN STATE SPACES

The corollary below gives another characterization of normal state spaces of JBW-algebras. It is of interest mainly in showing that the functional calculus for affine functions on a spectral convex set provides a natural candidate for a Jordan product, which will give a JBvV-algebra exactly under the simple but unappealing requirement that this product be bilinear. Let a E A and let E, lvI, and * be defined as in Lemma 9.41. vVe observe for use below that the proof above showed that the products * and o agree on lin E, and thus by norm continuity of each ~roduct, agree 011

M. 9.44. Corollary. Let K be a spectral convex set, and A = Ab(K). Define a product

0

on A by

(9.35) where the squares are gwen by the functwnal calculus on A defined by equatwn (8.60). A wzth thzs product and the order umt norm zs a JBWalgebra zff thzs product zs bzlznear. Proof. Suppose that this product is bilinear, and let a E A. Then squares for this product by definition coincide with those given by the functional calculus, so by Proposition 8.67,

(9.36) Furthermore, with the notation of Lemma 9.41, a*a = aoa (see the remarks preceding this corollary), so in particular a 0 a is again in AI. Then we have (a 0 a) 0 a = (a * a) * a, and similarly for any product of factors of a grouped in any fashion. Since AI ~ CR(X) is power associative, the same follows for the product o. Thus A is a complete order unit space with a commutative, power associative product for which 1 is an identity, and for which (9.36) holds. By (A 51), A is a commutative order unit algebra with positive cone A2 (A 47), and thus by Theorem 2.49, A is a lB-algebra. Since A is monotone complete and admits a separating set of normal states, then A is a lBW-algebra. 0 The corollary below shows that the condition Tef = Tje in Theorem 9.43 can be restated in terms of the general version of the "closeness operator" (9.37)

c( e, 1) = e f e

+ e' J' e'

associated with a pair of projections e, f in a von Neumann algebra. This operator, which was first defined in B(H) by Chandler Davis [40], is a useful tool in the theory of algebras generated by a pair of two non-commuting

CHARACTERIZING NORl\lAL STATE SPACES OF JBW ALGEBRAS

349

projections, sometimes referred to as "non-commutative trigonometry". (If e and f are projections onto two lines in C 2 with an angle a, then c(e, 1) = (cos 2 a)I. The element c(e, 1) has an interesting physical intepretation, discussed in the remark after Corollary 9.46. For more information on this topic, see the discussion in [AS, Chpt. 6] and also the references in the notes to that chapter.) The compression Pe associated with a projection e in a von Neumann algebra is the map a f-+ eae, and the compression P~ associated with the complementary projection e' = 1 - e is the map a f-+ e' ae'. Therefore we define c(e, 1) for a pair e, f of two projective units in an order unit space A in spectral duality with a base norm space V by

(9.38)

c(e,1)

= Pef +

P~f'.

9.45. Corollary. A spectr'al convex set K of a lEW-algebra zJJ

(9.39)

c(e,1)

IS

the normal state space

= c(f, e)

for all pazrs of pro)ectwe umts e, f E A := Ab(K). Proof. Let e, f be a pair of projective units in A. Using the equality P~f'

=

P~(1 -

1) = 1 - e -

P~f,

we find that Tef

= ~(f + Pef - P~1) = ~(e + f -1 + Pef + P~f').

Subtracting the corresponding equation with e and find that

Thus the equation Tef

f

interchanged, we

= Tje is equivalent to the equation

(9.40)

which is the same as c(e,1) Theorem 9.43. 0

=

c(f, e).

Now the corollary follows from

The above characterization of the normal state space of a JBW-algebra Was first proved in [9, Cor. 3.7] under the condition (9.40), written as the commutator relation [Pe, PI] I = [Pj, P:]l, and with the Jordan product defined as in (9.35).

350

9

CHARACTERIZATION OF JORDAN STATE SPACES

The symmetry condition c( e, 1) = c(j, e) for projective units in a spectral convex set is related to symmetry of transition probabilities (the third of the pure state properties). In fact, if e and f are minimal projections in a von Neumann algebra or JBW-algebra M, computing in the subalgebra Mevf gives

c(e, 1)

= (e,l) (e V 1) = (j,e) (e V 1).

Of course, by Corollary 9.45 and Corollary 5.57, the symmetry condition c(e,1) = c(j, e) for projective units in a spectral convex set K implies that Ab(K) can be equipped with a product making it a JBW-algebra with normal state space K, and so in particular that K has the property of symmetry of transition probabilities.

9.46. Corollary. A compact spectral convex set K zs the state space of a iE-algebra zJJ zt satzsfies the conddwns

(i) c(e, 1) = c(j, e) for all pairs e, f E P, (ii) a 2 E A(K) for all a E A(K). Proof. Clearly these are necessary conditions that K be the state space of a JB-algebra (which must be isomorphic to A(K) equipped with the product 0 of (9.35)). They are also sufficient, for if (i) is satisfied, then Ab(K) is a JBW-algebra with the product 0 of (9.35); and if (ii) is also satisfied, then A(K) is closed under this product which will organize A(K) to a JB-algebra with state space affinely isomorphic and homeomorphic to

K.o Remark. The "closeness operator" c( e, 1) associated with projections in a von Neumann algebra has a physical interpretation discussed in the remark following [AS, Cor. 6.43]. We will show that the element c(e,1) of A+ defined by (9.38) for a pair of projective units in the space A = Ab(K) of a spectral convex set K can be interpreted in a similar way. Then we assume that e and f represent propositions (or questions) and that the compressions Pe and Pf represent measuring devices for these propositions, as explained in the remark after Proposition 7.49. Thus each of these measuring devices records if particles in a beam appear with the value 1 or the value 0 (the two possible values for the proposition). What happens if the beam from one apparatus is sent through the other one? Clearly, there are four possible outcomes with probabilities p(l,l), p(l, 0), p(O, 1), p(O, 0) respectively. (Here the first variable of the function p(., .) refers to the first apparatus and the second variable refers to the second apparatus). We will now determine these probabilities. If a beam of particles in a state w E K is sent into the apparatus Pe , then the particles that are recorded with the value 1 will leave the apparatus in a new state represented by P;w E V+. More specifically,

CHARACTERIZING NORMAL STATE SPACES OF .JBW ALGEBRAS

351

they will be in the state IlPe*wll- 1 Pe*w E K. Here IIP;wll is the relative intensity of the partial beam consisting of all particles that emerge with the value 1. Otherwise stated, IIP*wll is the probability that a particle in the state w is recorded with the value 1 by the apparatus Pe . Assume now that the beam from the apparatus Pe is sent into the apparatus Pf . Then the states are transformed once more in the same fashion as in the first apparatus. Therefore the particles that are recorded with the value 1 by both apparata will emerge in a state represented by Pj Pe*w, and the partial beam of such particles will have the relative intensity IIPjPe*wll. Thus the probability that a particle is recorded with the value 1 by both apparata is

In the same way we find that p(l,O) = (Pe!',W),

p(O, 1) = (P:f,w),

p(O,O) = (p:!"w).

Note that in general these probabilities will depend on the order in which the measurements are performed. However, it follows from the above that

p(l,l)+p(O,O) =

(Pef+P~!',w)

= (c(e,f),w).

Thus (c( e, f), w) is the probability that two consecutive measurements on particles in the state w by the apparata Pe and Pf in that order coincide (both with the value 1 or both with the value 0). Therefore the equation c(e, f) = c(j, e) says that for all w this probability is the same when the order of the measurements is reversed. (Thus we might refer to this equation as "the symmetry of coincidence probabilities".) We now give an alternate characterization of normal state spaces that is more geometric in nature, replacing the condition (9.29) by ellipticity. We have previously discussed this concept in the context of normal state spaces of lBW-algebras (d. Proposition 5.75). We now define the corresponding concept for spectral convex sets and show that it characterizes normal state spaces of lBW-algebras.

9.47. Definition. Let K be a spectral convex set. Let P and pi be complementary projective faces of K, with corresponding compressions P and P'. Let I}i = (P + PI)* be the canonical projection of K onto co(P U Pi), let a E F, T E pi, and let [a, T] be the line segment with endpoints a, T. Then we say I}i-l ([a, T]) has ellzptzcal cross-sectwns if the intersection of this set with every plane through the line segment [a, T] is an ellipse together with its interior. We say K is elhptzc if such cross-sections are elliptical for all F, F ' , a E F, T E P'.

352

9.

CHARACTERIZATION OF JORDAN STATE SPACES

By Proposition 5.75, the normal state spaces of JBW-algebras are elliptic. Now let K be a spectral convex set, represented as the distinguished base of a complete base norm space V. Let F and FI be complementary projective faces of K, with corresponding compressions P and pi, and fix (J E F, TEFl, and W E K \ co(F U FI). Consider the orbit Wt of W under the action of the one-parameter group t f--+ exp(t(P - Pi)) *, and let w; = Ilwtll-1Wt denote the normalized orbit. Then the proof of Lemma 5.74 (with the concrete compression Up replaced by the abstract compression P) shows that the normalized orbit of any such W is the unique half ellipse that passes through wand has (J and T as endpoints. If the cross-sections described in Definition 9.47 are ellipses, then each such normalized orbit will stay inside K, or equivalently, t f--+ exp(t(P - Pi))' will be a oneparameter group of order automorphisms of V. Thus we have for t E R, PEe, (9.41)

K is elliptic

=}

exp(t(P - Pi))' ~ O.

This is equivalent to requiring exp(t(P - Pi)) ~ 0 for all t E R. This says that P - pi is an order derivation of the order unit space A = Ab(K), cf. (A 66). Note that in the case of a JBW-algebra, ~(I + P - PI)b = (PI) 0 b, so the fact that P - pi is an order derivation follows from the fact that Jordan multiplication by any element of a JB-algebra is an order derivation (Lemma 1.56.) For use below, we recall that if A is any complete order unit space and 15 : A -+ A is a bounded linear map, then by (A 67), 15 will be an order derivation iff for a E A + , (J E (A') + , (9.42)

(a, (J) = 0

=}

(ba, (J) = o.

We will also need the fact that the set of order derivations is closed under commutators (A 68).

9.48. Theorem. A conve1; set K zs affinely isomorphzc to the normal state space of a lEW-algebra iff K zs spectral and ellzptzc.

Proof. We have already shown that the normal state space of a JBWalgebra is spectral and elliptic (Proposition 5.75 and Proposition 8.76). Conversely, assume that K is spectral and elliptic, and let A = Ab(K). We are going to prove that K satisfies the condition (9.29), which by Theorem 9.43 will complete the proof. Let P and Q be arbitrary compressions. We will show that (9.43)

[P - PI,Q - Q/]1 = 0,

CHARACTERIZING NORMAL STATE SPACES OF JBW ALGEBRAS

353

which is evidently equivalent to [Te, T f ]1 = 0, and then to (9.29), where e and f are the projective units associated with P and Q. By (9.41) the operators Dl := P - P' and D2 := Q - Q' are order derivations. Consider also the positive operators El := P + P' and E2 := Q + Q'. Note that (9.44) For an arbitrary a E A+ and (J E K we have (Qa, Q'*(J) = 0, so we can use (9.42) with Qa in the place of a and Q' (J in the place of (J to conclude that (D1Qa,Q'*(J) = O. Hence Q'D1Q = O. Similarly QD1Q' = O. An immediate consequence is

(Q - Q')(P - P')(Q - Q') = (Q

+ Q')(P -

P')(Q

+ Q')

(both sides being equal to QD1Q + Q'D1Q'). Clearly the same equality holds with P in the place of Q and D2 in the place of D l . Thus (9.45 ) We next establish (9.46) To verify this, we expand the left side using (9.44):

[Dl' D2]21

= =

D1D2D1D21 D1D2D1D21

+ D2D1D2Dll- D1D~Dll + D2D1D2Dd - D1E2Dd -

D2DiD21 D2E1D21.

Clearly Ell = 1 and E21 = 1, so in the last two terms above we can replace 1 by E21 and Ell (in this order), getting

By (9.45) this gives (9.46). From (9.46) we have (9.47) By the fact that the set of order derivations is closed under commutators, [Dl' D 2 ] is an order derivation. Therefore the left side of (9.47) is positive for all t. From this it follows that [Dl' D2]1 = 0, which proves (9.43) and completes the proof. D

354

9

CHARACTERIZATION OF JORDAN STATE SPACES

Notes The geometric characterization of the normal state spaces of type I JBW-factors (in Theorem 9.33) was first given in [10]' and it was then followed by the characterization of the state spaces of JB-algebras (Theorem 9.38) in the same paper. (Note that the condition in Theorem 9.38 that requires the (j-convex hull of the extreme points to be a split face may be redundant, as there are no known examples to the contrary.) In a paper proposing axioms for quantum mechanics, Gunson l59] showed under various assumptions that the linear span of the atoms formed a Jordan algebra, and the proof of Proposition 9.30 is related to his. The characterization of normal state spaces of general JBW'-algebras (Theorem 9.48) is due to Iochum and Shultz [71]. The Hilbert ball property and the pure state properties were introduced in [10]' and the ellipticity property in [71]. Note that the equivalent condition on the right side of (9.41) formally is very similar to the notion of (facial) homogeneity in self-dual cones due to Connes [32]. (See the discussion following Proposition 5.75.) The characterization of normal state spaces of JB\V-algebras and state spaces of JB-algebras by means of the condition (9.39) was first given (in a slightly different form) in [9], where there is also a brief discussioll of the physical interpretation. For a thorough discussion of an approach to axiomatic quantum mechanics via JBW-algebras, based on Corollary 9.45, see the papers of Kummer [88, 89] and Guz [60]. In the paper [21], the authors Ayupov, Iochum and Yadgorov characterize the state spaces of finite dimensional JB-algebras by three properties. The first one, called "projectivity", is the same as the "standing hypothesis" of Chapter 8 (exposed faces being projective), and the two others are "symmetry" and "modularity". In [22] the same authors show that a sclfdual cone in a real Hilbert space is facially homogeneous iff it is symmetric and modular. TI:ansition probabilities have long played a central role in quantum mechanics. For example, in [135] \Vigner shows that a bijection of the set of pure normal states of B (H) onto itself that preserves transition probabilities is implemented by a unitary or a conjugate linear isometry, and thus induces a *-isomorphism or *-anti-isomorphism of B(H). Araki has given a careful justification of JB-algebras as a model for quantum mechanics in finite dimensions [17]. In this paper he discusses the role of symmetry of transition probabilities (cf. (9.7)), which is one of his axioms, and he also assumes that filters send pure states to multiples of pure states. In the finite dimensional context of Araki's paper, his axioms force the filters to be compressions. The assumption that filters should send pure states to multiples of pure states is also in the paper of Pool [104], and "pure operations" (those that send pure states to pure states) were discussed in Haag and Kastler's influential paper [61]. The lattice-theoretic covering property (Definition 9.6),

NOTES

355

which is equivalent to compressions sending pure states to multiples of pure states (Corollary 9.8), plays a key role in the "quantum logic" approach to axioms for quantum mechanics, e.g., in the work of Piron [102]. Other relevant references for axiomatic foundations of quantum mechanics related to our presentation here can be found in the books of Busch, Grabowski, and Lahti [35], Emch [49]' Landsman [91]' Ludwig [94]' and Upmeier [130]. Finally, we mention briefly JB*-triples. These generalize JB-algebras, and axiomatize the triple product {a,b,c} f--t ~(abc+cba). Recall that JB-algebras are in 1-1 correspondence with symmetric tube domains. Bya theorem of Kaup [83], JB*-triples are in 1-1 correspondence with bounded symmetric domains. Thus they are of considerable geometric interest. There is a Gelfand-Naimark type theorem for JB*-triples, due to Friedman and Russo [54]. Friedman and Russo also showed that most of the geometric properties of JB-algebra state spaces in Theorem 9.38 generalize to preduals of JBW*-triples [53], and they gave a characterization of the preduals of atomic JBW*-triples [55], which generalizes Theorem 9.33. These results are quite remarkable, since they generalize order-theoretic concepts and results to a context where there is no order, requiring totally different proofs.

10

Characterization of Normal State Spaces of von Neumann Algebras

vVe hegin this chapter hy characterizing the normal state space of the von Neumann algebra B(H). Our starting point is Theorem 9.34 which characterizes normal state spaces of JBW-factors of type I by geometric axioms, among those the Hilbert ball property by which the face generated by each pair of extreme points is a Hilbert ball. In the case of B (H) these balls will be 3-dimensional (A 120), and this "3-ball property" is the single additional property we need to characterize the normal state space of B(H) (Theorem 10.2). Since general von Neumann algebras may admit no pure normal states, assumptions involving pure states are not useful to characterize their normal state spaces. Our starting point for general von Neumann algebras is Theorem 9.48 which characterizes normal state spaces of general JBWalgebras, and we show that a "generalized 3-ball property" that involves pairs of norm closed faces rather than pairs of extreme points (Definition 10.20) is the single additional property needed to characterize the normal state spaces of general von Neumann algebras (Theorem 10.25).

The normal state space of B (H) 10.1. Definition. A 3-ball is a convex set affinely isomorphic to the standard 3-ball

Note that a 3-ball is a special case of a Hilhert ball.

10.2. Theorem. Let K be the base of a complete base norm space. Then K 7S affinely 7somorphic to the normal state space of B (H) for a complex Hz/bert space H 7ft all of the followmg hold.

(i) Every norm exposed face of K 7S pro)ectwe. (ii) The cy-convex hull of the extreme pomts of K equals K. (iii) The face generated by every pozr of extreme pomts of K 7S a 3-ball and 7S norm exposed. Proof. Assume first that K is affinely isomorphic to the normal state space of B(H). Since B(H)sa is an atomic lBW-algebra, then properties (i) and (ii) follow from Theorem 9.34, as does the Hilbert hall property. If

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iJ, T are extreme points of K, then the face they generate is norm exposed by the Hilbert ball property. The fact that the face generated by iJ, T is a 3-ball is (A 120). Now assume K satisfies (i), (ii), (iii). By Theorem 9.34, K is affinely isomorphic to the normal state space of an atomic JBW-algebra IvI. Suppose that K contains a proper splIt face F. Since AI is atomic, then F and F' contain minimal projective faces, which are then singleton extreme points (Propositions 8.36 or 5.39). Let iJ E F and TEF' be such extreme points. Then the face of K generated by iJ and T is the line segment joining these two points (Lemma 5.54 or (A 29)), which contradicts (iii). Thus K contains no proper split face, and so by Theorem 9.34, K is affinely isomorphic to the normal state space of a type I JBW-factor M. By (iii), M is isomorphic to 13 (H)sa for a complex Hilbert space H (Proposition 9.36). D

3-frames, Cartesian triples, and blown up 3-balls Let (iJ, T) be a pair of antipodal points in the standard Euclidean 3ball B3. Then there is a unique reflection (i.e., period 2 automorphism) of B3 whose set of fixed points is the axis determined by (iJ, T) (i.e., the line segment with endpoints iJ, T). Two such axes are orthogonal iff the reflection associated with each axis reverses the other axis, i.e., exchanges the corresponding pair of antipodal points of that axis. If we have three orthogonal axes, then the product of the associated reflections will be the identity map. This is characteristic of dimension 3: for higher dimensional Hilbert balls the product of any such triple of reflections will invert any fourth axis orthogonal to the first three, and thus cannot be the identity map. These observations motivate our generalization of 3-balls. We are going to define generalued axes, 3-frames, and blown up 3-balls. (See the definitions (A 185), (A 190), and (A 194) for a discussion of these concepts for normal state spaces of von Neumann algebras.) Our context in the rest of this section will be a JBW-algebra IvI and its normal state space K. Recall that there is a 1-1 correspondence of projections in AI and norm closed faces in K given by p f--t Fl' = {iJ E K I iJ(p) = I} (Corollary 5.33). The faces Fl' and F~ = Fl" (where pi = I-p) are said to be complementary.

10.3. Definition. If K is the normal state space of a JB\V-algebra, an ordered pair (F, F') of complementary norm closed faces is called a generallzed aXlS of K. Observe that if p is a projection and pi = 1 - p, then s = p - pi is a symmetry (and every symmetry arises in this way). If F = Fl" we will call (F, F') the generalzzed axis assocwted with the symmetry s = p - pi .

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Recall that a reflection of a convex set is an affine automorphism of period 2, and that the map U; (dual to Us : x f--+ {sxs} ) acts as a reflection of K with fixed point set co(F U F') (Proposition 5.72). We now show that this reflection is unique, and will then refer to it as the refiection about the generallzed aXlS (F, F'). This uniqueness result in the von Neumann alge bra context is (A 174).

10.4. Proposition. If p lS a proJectwn m a lEW-algebra !vI wlth assocwted pTOJectzve face F, and s = p - pi where pi = 1 - p, then U; lS the umque refiectwn of the nOTmal state space K wlth fixed pomt set co(F U F'). Proof. Let R be any reflection of K with fixed point set co(F U F'). Then ~(I +R) is an affine projection of K onto co(FUF ' ). Since Up, Up' are complementary compressions (Theorem 2.83) with associated projective faces F, F' (cf. (5.4)), then (Up + Up')* is the unique affine projection of K onto co(F U F') (Theorem 7.46). Thus ~(I + R) = (Up + Up')*' so R = (2(Up + Up') - 1)*. Then R = follows from (2.25). 0

u.:

If F is a projective face of K with complementary face F ' , and an affine automorphism ¢ of K exchanges F and F ' , then we will say that ¢ reveTses the generalued aXlS (F, F'). \Ve can use this to define a notion of orthogonality for generalized axes.

10.5. Definition. If !vI is a lBW-algebra with normal state space K, a pair of generalized axes is orthogonal if the reflection about the first generalized axis reverses the second generalized axis. By the following result, the reflection about the second generalized axis will also reverse the first, so orthogonality is a symmetric relation as one would hope.

10.6. Lemma. Let p and q be pTOJectwns of a lEW-algebra M with complementary pToJectwns pi and q'. Let F and G be the norm closed faces associated wlth p and q, and let RF and Rc be the refiectwns of K about the generahzed axes (F, F') and (G, G ' ). The following are equivalent.

(i) The Tefiectwn RF reverses the generallzed aXlS (G, G ' ). (ii) The Tefiection Rc reverses the generallzed aXlS (F, F ' ). (iii) (p - pi) 0 (q - q') = O. Pmof. Let s = p - pi and t = q - q'. Then sand t are symmetries and RF = U; and Rc = Ut (Proposition 10.4). By Corollary 5.17, (i) is equivalent to

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(i ') Us exchanges q and q' while (ii) is equivalent to (ii ') Ut exchanges p and p'. Each of (i') and (ii') is equivalent to (iii) by Lemma 3.6. 0 We say that symmetries s, t are Jordan orthogonal if sot = o. Thus orthogonality of generalized axes is equivalent to Jordan orthogonality of the associated symmetries. The following definition abstracts the key properties of the set of three orthogonal axes of B:3.

10.7. Definition. Let K be the normal state space of a JBvV-algebra M. A 3-fmme is an ordered triple of mutually orthogonal generalized axes of K such that the product of the associated reflections of K is the identity map. Recall that if F is the projective face associated with a projection p, then F can be identified with the normal state space of the JBVIsubalgebra im Up, d. Proposition 2.62. Therefore we can apply the notions of generalized axes and 3-frames to F as well as to K. Recall that a Cartesian triple of symmetries is an ordered triple (r, s, t) of Jordan orthogonal symmetries such that UrUsUt is the identity map (Definition 4.50). We now show that Cartesian triples are the algebraic objects corresponding to 3-frames. 10.B. Definition. If ct = (r, s, t) is a Cartesian triple, then denotes the ordered triple of generalized axes associated with r, s, t.

ct.

10.9. Lemma. Let AJ be a JEW-algebm, The map ct f--+ ct. lS a 1-1 correspondence of Carteszan trzples of symmetrzes and 3-fmmes of K. Proof. Let ct = (r, s, t). By Lemma 10.6, the orthogonality of the generalized axes of ct. is equivalent to Jordan orthogonality of r, s, t. The refiections about the three generalized axes are U;, U; ,Ut' respectively. Then the product of the three refiections is the identity iff U;U;Ut = Id, which in turn is equivalent to UtUsUr = I d. By the remarks after Definition 4.50, if (r, s, t) is a Cartesian triple of symmetries, then the maps Un US) Ut commute, so this is equivalent to (1', S, t) being a Cartesian triple. 0 From thc proof above and Proposition 10.4, it follows that the reflections associated with the generalized axcs in a 3-frame commute (and therefore their product in any order is the identity).

Remark. By Lemma 10.9, the existence of 3-frames in K is equivalent to the existence of Cartesian triples in M. By Definition 10.5 and

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361

Lemma 10.6, a necessary condition for this is the existence of a projection p E !vI such that p and pi can be exchanged by a symmetry (as in (ii ') of the proof of Lemma 10.6). (In the case of von Neumann algebras, the above condition is sufficient as well as necessary for the existence of 3-frames in K, cf. (A 193) where this condition is referred to as "halvability of the identity element".) Note that the above exchangeability condition is satisfied in the Jordan matrix algebra Mn(C)sa iff n is an even number. Therefore the state space K of Mn(C)sa has no 3-frame when n is odd. But K has closed faces with 3-frames for all n > 1, namely all faces that are isomorphic to state spaces of subalgebras of the form im Up ~ !vh(C)sa where k = rank(p) is even. We are now ready to define our generalization of 3-balls.

10.10. Definition. Let M be a JBW-algebra with normal state space K. A face F of K (possibly F = K) is a blown up 3-ball (of K) if it is a norm closed face admitting a 3-frame. We will see later that this implies that the facial structure and automorphism group of F are quite similar to those of the standard 3-ball

B3. 10.11. Example. In the special case of M = !vh(C)sa, we can identify the state space of !vI with the positive matrices of trace 1, and thus with B3 as follows (see (A 119)):

(10.1) Let 0O. be the ordered triple of pairs of singleton faces of B3 located along each of the three coordinate axes. Each of these pairs is a "generalized axis" in our current terminology. We will refer to 0O. as the standard 3-fmme of B3 . The reflections corresponding to each generalized axis will be exactly the reflections of K ~ B3 in each of the coordinate axes. The associated 3-frame is

(10.2) which are the Pauli spin matrices. If a JBW-algebra M possesses a Cartesian triple of symmetries, then M is isomorphic to the self-adjoint part of a von Neumann algebra M (Theorem 4.56). Furthermore, this von Neumann algebra M will contain

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a set of 2 x 2 matrix units, so that M "" lvh(C) I8i B for a von Neumann algebra B. Using the correspondence of 3-frames and Cartesian triples, we can characterize normal state spaces of such von Neumann algebras.

10.12. Proposition. If the normal state space K of a lEW-algebra M contams a 3-frame, wzth assocwted Carteswn tnple of symmetnes (r, s, t), then M = M + ~M can be eqUIpped w~th a umque von Neumann algebra product compat~ble w~th the gwen lordan product and satzsfymg ~rst = 1. For th~s product, the complex lmear span of 1, T, s, t ~s a *subalgebra *-~somorph~c to M2 (C)! and thus M "" M2 (C) I8i B fOT a von Neumann algebra B. Proof. The proposition follows from the correspondence of 3-frames and Cartesian triples (Lemma 10.9) and Theorem 4.56. 0 The following result generalizes the fact that the state space of lvI2 (C) is a 3-ball, cf. (A 119).

10.13. Corollary. The normal state space K of a lEW-algebra M is a blown up 3-ball ~ff M ~s ~somoTph~c to the self-adJomt pad of a von Neumann algebra M of the fOTm M "" Jvh(C) I8i B for a von Neumann algebra B. Proof. This follows at once from Proposition 10.12 and the definition of a blown up 3-ball. 0 We will now present some elementary concepts and facts that will be used in the proof of our next result (Proposition 10.15). This result is not needed in the sequel, but provides a further illustration of the appropriateness of the term "blown up 3-ball".

10.14. Definition. If K is the normal state space of a JBW-algebra, then Auto(K) is the (norm-continuous) path component of the identity in the group of affine automorphisms of K. (Here we extend affine automorphisms of K to V = lin K and give them the norm topology as bounded linear maps on V.) Note that affine automorphisms of the standard 3-ball B3 extend uniquely to orthogonal transformations of R 3. The latter have determinant ±1. Those with determinant +1 are precisely the rotations, while those with determinant -1 are the reversals (i.e., compositions of a rotation and a reflection in the origin.) Rotations are evidently connected by a path to the identity, while reversals are not (by continuity of the determinant map.) Thus Auto(B3) can be identified with the rotation group SO(3). By (A 131), the rotations of B3 are precisely the duals of the

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363

maps x f---> 'uxu* for unit aries u in IvI2 (C), while the reversals arise as the duals of the maps x f---> vx*v- l , where v is conjugate unitary. Thus rotations of B3 are the duals of *-automorphisms of M 2 (C), while reversals correspond to *-anti-automorphisms (A 124). Recall that if K is the normal state space of a JBW-algebra, then the projective faces and norm closed faces of K coincide, and the lattice of norm closed faces of K is an orthomodular lattice (Corollary 5.33). An ortho-lsomorphlsm from one orthomodular lattice into another is an injective map that preserves complements and (finite) least upper bounds and greatest lower bounds. Let NIl and 11:12 be JBW-algebras with normal state spaces K I , K 2 . Recall that the map T f---> T* is a 1-1 correspondence of positive unital (Jweakly continuous maps T: M] --+ IvI2 and affine maps from K2 into KI (Lemma 5.13). If T is also a Jordan homomorphism, and p is a projection in MI with associated projective face Fp , then by Lemma 5.14,

(10.3) and if T is a Jordan isomorphism from NIl onto NI2 , and q is a projection in M 2 , then

(10.4) We will make use of these results several times in the next proof, in a context where T will be a unital *-homomorphism between von Neumann algebras. Recall that for each closed face F of the normal state space of a JBWalgebra, there is a reflection RF with fixed point set co(FUF'), d. Proposition 10.4.

10.15. Proposition. Let K be the normal state space of a lBWalgebra IvI. If K lS a blown up 3-ball, then there ],s an affine map ¢ from K onto B3 such that

(i)

an ortho-lsomorphlsm from the lattlce of faces of B3 mto the lattzce of norm closed faces of K, (ii) there lS an lsomorphzsm (J --+ (j of the rotation group Auto(B3) mto the group of affine automorphlsms of K WhlCh takes RF to Rq,-l(F) for each face F of B 3 , and WhlCh satzsfies (J 0 ¢ = ¢ 0 (j. ¢-l

lS

Proof. Since K is a blown up 3-ball, there is a von Neumann algebra B such that M is isomorphic to the self-adjoint part of M = Ivh (C) ® B (Corollary 10.13). Identify M and M sa, and identify K with the normal state space of M. Define T : M 2 (C) --+ M by Tx = x ® l. We identify the state space of M 2 (C) with B 3 , and let ¢ : K --+ B3 be T* restricted to K.

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To see that ¢ is surjective, it suffices to show ¢(K) contains every pure state on M2(C), Recall that there is a 1-1 correspondence of pure states on lvh(C) and minimal projections in M 2(C) (e.g., Proposition 5.39). Let p be a minimal projection in lvh(C) and w the unique state with w(p) = 1. If T is any normal state on M with T(p ® 1) = 1, then ¢(T)(p) = T(Tp) = T(p ® 1) = 1, so ¢(T) = w. Thus ¢ is surjective. Since ¢ is affine and norm continuous, then ¢-l maps faces of B3 to norm closed faces of K. For faces F and G of B 3 ,

Thus ¢-l preserves greatest lower bounds. Since x f---> x ® 1 is a unital *-homomorphism from lvI2(C) into lvI2(C) ® B, then ¢-l preserves complements (Lemma 5.15). Since (F!\ G)I = FI V G I for projective faces F, G, then ¢-l also preserves least upper bounds, and thus is an orthoisomorphism from the lattice of faces of B3 into the lattice of norm closed faces of K. This proves (i). For each !J E Auto(B3) there is a unique *-automorphism S of .M2(C) such that !J = S*. (See the discussion preceding this proposition.) Let I d be the identity automorphism of B. Then S ® I d is a *-automorphism of M, so (S ® Id)* is in the space Aut(K) of affine automorphisms of K. Observe that !J f---> S is an anti-isomorphism from the group Aut o(B 3) onto the group of all *-automorphisms of M2(C), Define 7r : Auto( B 3 ) -+ Aut(K) by 7r(!J)

= (S®Id)*,

where !J = S*. Then 7r is a group isomorphism into Aut(K). Let F be an arbitrary face of B 3 , with associated projection p. Now we show that 7r(RF) = R¢-l(F)' Then RF is the reflection U;, where s = p - pi (Proposition 10.4). Therefore

Note that s ® 1

= P ® 1 - pi ® 1. Thus 7r(RF) is the reflection Rc where

G is the face of K associated with the projection p ® 1

= Tp. By (10.3), G = FTp = (T*)-l(Fp) = ¢-l(F), so 7r(RF) = R¢-l(F)' Next we show !J 0 ¢ = ¢ 0 7r(!J) for each !J E Auto(B3). Write !J = S*, with S a *-automorphism of lvI2(C), For x E M 2 (C), we have

T(S(x)) = S(x) ® 1 = (S ® Id)(x ® 1) = (S ® Id)(Tx) ,

THE GENERALIZED 3-BALL PROPERTY

so To S

365

= (S 181 Id) 0 T. Dualizing gives U

Setting (j

0 9 = 9 0 (S 181 I d) * = 90 7r( u).

= 7r(u) proves (ii) and completes the proof of the proposition.

0

Remark. By Proposition 10.15 a blown up 3-ball can be mapped onto the standard 3-ball B3 by an affine map which "pulls back" the face lattice of B3 into the face lattice of K and "lifts" the rotation group of B3 into the group of all affine automorphisms of K. This shows that a blown up 3-ball possesses much of the affine structure of the standard 3-ball, and is another reason for our choice of that name. Note that it is not the case that each affine surjection 9 : K ----) B3 which admits a group isomorphism from Auto(B3) into Aut(K) with these properties will admit a group isomorphism from Aut(B 3) with the same properties. \Ve will show this by an example where 9 is as in the proof above and B is chosen to be a von Neumann factor not *-anti-isomorphic to itself, (e.g., [37]). Then it is straightforward to check that the von Neumann algebra M 2 (C) 181 B is also a von Neumann factor. As in the proof above, define T: lvh(C) ----) M by Tx = x 1811, and recall that 9 is the dual map from K onto B 3 , and has the properties described in (ii). Now let 1j; be the transpose map on Jl.h (C). Then 1j;* is an affine automorphism of B 3 , but since 1j; is a *-anti-automorphism of B 3 , then U = 1j;* is in Aut( B 3 ) but is not in Auto(B3). Now suppose that there is a lift (j E Aut(K), i.e., 9 0 (j = U 0 9. Let 1.{t : M ----) M be the Jordan isomorphism such that 1.{t* = (j (Proposition 5.16). By Kadison's theorem, cf. (A 158), since M is a factor, 1.{t is either a *-isomorphism or an *-anti-isomorphism. Since 90 (j = U 0 9, then T* 0 1.{t* = 1j;* 0 T*. Therefore 1.{t 0 T = To 1j;, so for all x E lvh(C) we have 1.{t(x 181 1) = 1j;(x) 181 1. Hence 1.{t restricts to a *-anti-isomorphism on the subalgebra lVh(C) 181 C1. Since 1.{t is either a *-isomorphism or a *-anti-isomorphism on M, the latter must hold. Since 1.{t leaves invariant the *-subalgebra lvI2 (C) 181 C1, it must leave invariant the relative commutant of this subalgebra, namely 1181 B. Then 1.{t induces a *-anti-isomorphism of B, a contradiction. Thus in Proposition 10.15, Auto(B3) cannot be replaced by Aut(B 3 ). It follows that in the definition of global 3-balls in [71], Aut( B 3 ) should be replaced by Auto (B 3 ). With this change in definition, the results and proofs in [71] hold without other change. The generalized 3-ball property

In this section we will define the key property needed to insure that a JBW-algebra normal state space is in fact a von Neumann algebra normal state space. We start by establishing some useful properties concerning equivalence of orthogonal projections. These properties generalize known

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properties of projections in von Neumann algebras, cf. (A 175) for Lemma 10.16, and (A 179) and (A 195) for PropositIOn 10.17.

10.16. Lemma. If p and q aTe oTthogonal, equwalent PToJcctwns m a JB W-algebm M, then p and q aTe exchanged by a symmetTY.

Pmoj. Suppose first that A1. has no 12 or 13 direct summand. By Theorem 4.23, we may assume IvI is represented as a concrete JW-subalgebra of B(H)sa, and is reversible (Corollary 4.301. By assumption there are symmetries 81, ... ,8" with 8182'" SnP8 n '" 81 = q. Let v = 8182 ... 8n p. Then vv* = q and v*v = p, so v is a partial isometry with initial projection p and final projection q. Let 8 = V + v*. Then 8 is a (p + q)symmetry in B(H)sa that exchanges p and q. By reversibility of AI, 8 = 8182 ... 8n P + p8 n ... 8281 E M. By Lemma 3.5, p and q are exchangeable by a symmetry. It remains to establish the lemma's conclusion when p and q are orthogonal, equivalent projections in a JBW-algebra ]1.1 of type 12 or 1 3 . We claim that in this case p and q are abelian projections, and so are exchangeable (Corollary 3.20). Since factor representations separate the points of M (Lemma 4.14), it suffices to show that 7r(p) and 7r(q) are abelian projections for every factor representation 7r. Let 7r be a factor representation of ]1.1., and define N = 7r(lvf) (a-weak closure). Here N will be a type 12 or 1:3 JBW-factor (Lemma 4.8). Then 7r(p) and 7r(q) are exchangeable orthogonal projections in N. If they were not zero and not minimal projections, then each would dominate at least two non-zero orthogonal projections, so N would contain at least four orthogonal non-zero projections, contrary to the properties of factors of type h or 13 (Lemma 3.22). Thus 7r(p) and 7r(q) must be either minimal projections or zero, and so in either case are abelian. This completes the proof that p and q are abelian, and thus are exchangeable by a symmetry. D A key step in the next proof is to show that if p and q are projections with lip - qll < 1, then p and q can be exchanged by a symmetry. (See (A 178) for the corresponding von Neumann algebra result.) We will see that this follows from the fact that for any projections p and q, the projections p - p 1\ q' and q - q 1\ pi can always be exchanged by a symmetry. For von Neumann algebras, a proof of the latter fact can be found in (A 177). For JBW-algebras, note that

= P 1\ (pi V q) = T(Upq), q - (q 1\ pi) = q 1\ (q' V p) = T(Uqp),

p - (p

1\

q/)

where the last equality in each equation follows from Proposition 2.3l. There is a symmetry 8 that exchanges Upq and Uqp (Lemma 3.46). Since Us is a Jordan automorphism, it will also exchange the range projections of Upq and Uqp, and therefore will exchange p - P 1\ q' and q - q 1\ p'.

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10.17. Proposition. If p and q are orthogonal projectIOns m a JBWalgebra AI, then p and q are equwalent Iff there IS a (norm continuous) path of projectIOns from p to q. Proof. Suppose first that p and q are equivalent, orthogonal projections. By Lemma 10.16, p and q are exchanged by a symmetry. If e = p+q, by Lemma 3.5 we can choose an e-symmetry s that exchanges p and q, i.e., Us(p) = q. Then Us(p - q) = q - p, so by the equivalence of (iii) and (iv) in Lemma 3.6 (applied to Me), we have so (p - q) = o. Now let St = (cos 1ft )(p - q) + (sin 1ft)s for 0 S; t S; 1. Then for each t, the element St is an e-symmetry, and So = P - q, while SI = q - p. Thus there is a path of e-symmetries from p - q to q - p. Then Pt = ~ (St + e) will be a path of projections from p to q. Conversely, suppose that p and q are projections such that there exists a path of projections from p to q. Then we can find a finite sequence of projectionll> Po = p, PI, ... ,Pn = q such that Ilpi - Pi+lll < 1 for 0 S; i S; n - 1. We will be done if we show that there is a symmetry Si such that Us,Pi = PHI for 0 S; I S; n - 1. For simplicity of notation we may assume without loss of generality that n = 1, so that lip - qll < 1. We are going to show that p 1\ q' = q 1\ p' = O. Since p 1\ q' S; q', then p 1\ q' is orthogonal to q, so (p 1\ q') 0 q = o. Thus by the lB-algebra norm requirement (1.6),

lip 1\ q'll = II(p 1\ q') 0

(p -

q)11

S;

II(p 1\ q')llllp - qll

S;

lip - qll < 1.

Since the projection p 1\ q' must have norm 1 or 0, then p 1\ q' = o. By a similar argument, q 1\ p' = o. As observed before the statement of this proposition, the projections p-pl\q' and q-ql\p' can always be exchanged, so we conclude that p and q can be exchanged by a symmetry whenever lip - qll < 1. This completes the proof of the proposition. 0

10.18. Corollary. If F and G m·e orthogonal norm closed faces of the normal state space of a JB W-algebra !vI, wIth associated projectIOns p and q, the followmg are eqUIValent.

(i) p and q (ii) p and q (iii) There IS (iv) There IS that RH

are equwalent. are exchangeable by a symmetry. a (norm contmuous) path of prOjectIOns from p to q. a norm closed face H WIth assocwted reflectIOn RH such exchanges F and G.

Proof. The equivalence of (i) and (ii) is Lemma 10.16. The equivalence of (i) and (iii) is Proposition 10.17. A symmetry s exchanges p and q iff U; exchanges F and G (Corollary 5.17). Since the set of maps U; for s a symmetry is the same as the set of reflections RH for H a projective face (Proposition 10.4), this establishes the equivalence of (ii) and (iv). 0

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VON NEUMANN ALGEBRA NORMAL STATE SPACES

In a von Neumann algebra, projections can be exchanged by a finite product of symmetries iff the projections are unitarily equivalent (A 179), so the statements above correspond to unitary equivalence of F and G. For von Neumann algebras, the statement (iii) above is equivalent to there being a (norm continuous) path from F to G in the Hausdorff metric (A 196). 10.19. Definition. Let M be a JBW-algebra with normal state space K. Projective faces F, G of K are equwalent if their associated projections are equivalent. Note that if F and G are orthogonal, then F and G will be equivalent iff any of the conditions (i), (ii), (iii), or (iv) of Corollary 10.18 hold. If K is the normal state space of a JBW-algebra, we topologize the norm closed faces of K by using the 1-1 correspondence of norm closed faces and projections and carrying over the norm topology from the set of projections. Then by Corollary 10.18 orthogonal norm closed faces are equivalent iff there is a norm continuous path from one face to the other. We are now ready for the key definition. By (A 120), the face generated by any two extreme points of the normal state space of B (H) is a 3- ball, and by Theorem 10.2 and Theorem 9.34, this property distinguishes the normal state space of B(H) among the normal state spaces of JBW-factors of type I. More generally, if two pure states on a C* -algebra are unitarily equivalent, then the face they generate in the state space will be a 3ball (A 143). The following definition can be thought of as generalizing this property to the context of general norm closed faces, and is useful in situations where there may be no extreme points. Recall that norm closed faces F, G are orthogonal if they are orthogonal in the orthomodular lattice F, i.e., if F c G' , or equivalently, if the associated projections are orthogonal.

10.20. Definition. Let K be the normal state space of a JBWalgebra M. Then K has the generalzzed 3-ball property if every norm closed face E = F V G generated by a pair F, G of orthogonal, equivalent projective faces is a blown up 3-ball. 10.21. Lemma. Let M be a lEW-algebra wzth normal state space K wzth the generalzzed 3-ball property, let p and q be orthogonal proJectwns exchanged by a symmetry, and let e = p+q, and r = p-q. Then there zs an e-symmetry s that exchanges p and q, and for each such partwl symmetry s there zs a partwl symmetry t such that (r, s, t) zs a Carteswn trzple of e-symmetries. Proof. Let F, G be the projective faces associated with p, q respectively. By Corollary 5.17, since p and q are exchanged by a symmetry w,

THE GENERALIZED 3-BALL PROPERTY

then

F

V

369

U,7J exchanges G and F, so F and G are equivalent. We can identify

G with the normal state space of !'vie (Proposition 2.62). By the gen-

eralized 3-ball property, F V G is a blown up 3-ball, and so by Corollary 10.13, lvIe is Jordan isomorphic to the self-adjoint part of a von Neumann algebra M. We identify Me with Msa. Let l' = P ~ q. Since p and q are exchanged by a symmetry, by Lemma 3.5 there is an e-symmetry s exchanging p and q, so by Lemma 3.6, ros = O. Then 1', s anticommute in M. Define t = zrs E Msa = Me. It is straightforward to verify that (1', s, t) is a Cartesian triple of e-symmetries, cf. (A 192). D

10.22. Lemma. If M zs a lEW-algebra whose normal state space

K has the generalzzed 3-ball property, and !'vil zs any dzrect summand of M, then the normal state space Kl of lvh also has the generalzzed 3-ball property. Proof. Let F and G be orthogonal equivalent norm closed faces of K l . If p and q are the projections in !'vh corresponding to F and G, then p and q are orthogonal and equivalent in M l , and then also in lvI (e.g., by Lemma 3.5). Thus F and G are equivalent in K, so F V G is a blown up 3-ba11. (Note that the latter property is the same whether F V G is viewed as a projective face of K or of K l , cf. Definition 10.10). D

10.23. Proposition. If M zs a von Neumann algebra wdh normal state space K, then K has the generalzzed 3-ball property. Proof. Let F and G be orthogonal equivalent projective faces of K with associated projections p and q. Let e = p + q. There is a symmetry that exchanges p and q (Lemma 10.16), and then by Lemma 3.5 there is an e-symmetry s that exchanges p and q. By Lemma 3.6 the e-symmetries l' = P ~ q and s satisfy l' 0 S = O. By direct calculation or (A 192), if we define t = zrs, then (1', s, t) is a Cartesian triple of e-symmetries. It follows that the face E = F V G associated with e is a blown up 3-ball (Lemma 10.9). Thus K has the generalized 3-ball property. D

10.24. Lemma. If a lEW-algebra .M has the generalzzed 3-ball property, then lvI zs a lW-algebra, and zs reverszble m every representatwn as a concrete lW-algebra. Proof. We first show that lvI is a JW-algebra. Write M = Mo EEl M3 where M3 is the type 13 summand of lvI. Then Mo is a JW-algebra (Theorem 4.23). Let Pl, P2, P3 be equivalent abelian projections in Nh with sum the identity of M 3. By Corollary 3.20, Pl and P2 can be exchanged by a single symmetry. By Lemma 10.21 there exists a Cartesian triple (1', s, t) of (Pl + P2)-symmetries with l' = Pl ~ P2.

370

10

VON NEUl\IANN ALGEBRA NORMAL STATE SPACES

Now let 7r be a factor representation of M 3 , and define N = 7r(M3) (aweak closure). Since P1, P2, P3 are exchangeable and have sum the identity (of M 3 ), then their images are exchangeable with sum 1. In particular, none of the images can be zero, and so (7r (r), 7r ( S ), 7r (t)) is a non-zero Cartesian triple of partial symmetries in N. By Lemma 4.52, H3(O) contains no such triple, so N cannot be the exceptional .1BW-factor. By Corollary 4.20, M3 is a .1W-algebra, and thus so is M. Now assume that M is represented as a concrete .1W-algebra, and let !vh be the 12 direct summand of M. By Corollary 4.30, M is reversible iff M2 is reversible. By definition of type 1 2, the identity of M2 is the sum of equivalent abelian projections P and q, which can be exchanged by a symmetry. By the generalized 3-ball property and Lemma 10.21, AI contains a Cartesian triple of symmetries. Thus by Lemma 4.53, 11/[2 is reversible, and hence AI is reversible. D

10.25. Theorem. A convex set K zs affinely zsomorphzc to the normal state space of a von Neumann algebra zff K IS spectral, ellzptIc, and has the generalned 3-ball property.

Proof. The necessity of the first two of these conditions follows from Theorem 9.48, and the third follows from Proposition 10.23. Conversely, assume that K is spectral, elliptic, and has the generalized 3-ball property. From Theorem 9.48, K is affinely isomorphic to the normal state space of a .1BW-algebra M. Identify K with the normal state space of M. We will show that Ai is isomorphic to the self-adjoint part of a von Neumann algebra. By Lemma 10.24, we may assume that !v[ is represented as a reversible concrete .1W-subalgebra of B (H),a' Let Ro(J'vl) denote the real *-algebra generated by AI; by reversibility, Ro(AI)sa = AI. Recall that by Lemma 10.22 the normal state space of each direct summand of M also has the generalized 3-ball property. If M is of type I 1, then M ~ CR(X) for a compact Hausdorff space X, so Iv[ ~ Msa where M = Cc(X). Since AI is monotone complete and has a separating set of normal states, then M is a von Neumann algebra (A 95). If Mis of type Ik for 2 -s: k -s: 00, or has no type I part, then by Propositions 3.24 and 3.17, the identity is the sum of a finite set Pl, P2, ... ,pn of n exchangeable projections, with n 2 2. For z = 1,2, ... ,n let Si be a symmetry that exchanges P1 and p,. Define

By Lemma 4.28 the family {eij} is a system of matrix units for Ro(AI), and eij + eji = SiP1Sj + SJP1Si E M (by reversibility, or by (1.12)). Now S = e12 + e21 exchanges ell and e22, and is a (ell + e22)-symmetry. By the

THE GENERALIZED 3-I3ALL PROPERTY

371

generalized 3-ball property and Lemma 10.21, there is a partial symmetry t such that (ell ~ e22, e12 + e21, t) is a Cartesian triple of partial symmetries. Thus we can apply Lemma 4.57 to find J central in Ro(M) such that J* = ~J and J2 = ~l. Now by Lemma 4.55, M is isomorphic to the self-adjoint part of a von Neumann algebra. Finally we consider the case where !vI is an arbitrary JBW-algebra. Recall that !vI can be written as the direct sum of JBW-algebras of type I and a JBW-subalgebra with no summand of type I (Lemma 3.16), and the former can in turn be written as the direct sum of JBW-algebras of type Ik for 1 k CXJ (Theorem 3.23). We have shown that each of these summands is isomorphic to the self-adjoint part of a von Neumann algebra, so it follows that the same is true of !vI. This completes the proof of the theorem. 0

s: s:

Remark. If K is affinely isomorphic to the normal state space of a von Neumann algebra M, then K determines the Jordan product on M. In fact, the evaluation map is a bijection from M ,a onto the bounded affine functions on K (A 97), and the functional calculus for such bounded affine functions was defined in Proposition 8.67. Then the Jordan product on Msa can be recovered from ~(ab+ba) = i((a+b)2~(a~b)2). However, the associative product on M is not determined by the affine structure of K. For example, the transpose map on !vh(C) is a *-anti-automorphism, although its dual map is an affine automorphism of the state space. Thus some additional structure on K is needed to determine the associative product on M. One such structure is that of an onentatwn of K (A 201). We recall: 10.26. Theorem. If M lS a von Neumann algebra, then there lS 1-1 correspondence between Jordan compatlble assocwtwe products in M and global orlentatwns of the normal state space K of M. Proof. [AS, Thm. 7.103]. 0 For a discussion of the geometric interpretation of orientations on the normal state space of a von Neumann algebra, see the remarks at the end of Chapter 6. State spaces of C* and von Neumann algebras can also be characterized among state spaces of Jordan algebras (JB and JBW) by the existence of a dynamical correspondence (cf. Definition 6.10), which also determines the associative product. This is explained in the proposition below, which is essentially a reformulation of information from Theorem 6.15 and Corollary 6.16, restated in terms of state spaces instead of algebras. 10.27. Proposition. The state space K of a fE-algebra A lS affinely homeomorphic to the state space of a C*-algebra A iff there is a dynamical

372

10

VON NEUMANN ALGEBRA NORMAL STATE SPACES

correspondence on A. The nor'mal state space J( of a lEW-algebra M zs affinely zsomorphzc to the state space of a von Neumann algebra M ziJ there zs a dynamzcal correspondence on A-1. In these cases, the dynamzcal correspondence determznes a unzque C*-product on A + zA (respectzvely, W*-product on M + zM ). Proof.

J(

is affinely homeomorphic to the state space of a C*-algebra

A iff A is isomorphic (as a JB-algebra) to the self-adjoint part of A (Proposition 5.16). Furthermore, this 0CCurS iff A admits a dynamical correspondence, and dynamical correspondences are in 1-1 correspondence with C*-products on A + zA (Theorem 6.15). The JBW-algebra result is proven in the same way by means of Corollary 6.16. 0 Recall from Theorem 6.18 that there is a 1-1 correspondence between dynamical correspondences and Connes orientations (Definition 6.8) on a JBW-algebra M. Thus we also have the following characterization.

10.28. Proposition. The normal state space

J(

of a lEW-algebra M

zs affinely zsomorphzc to the normal state space of a lEW-algebra M ziJ there zs a Connes orzentatzon on M. In thzs case such a Connes orzentatzon

determines a unzque W* product on M

+ zA1.

Proof. This follows from Proposition 10.27 and Theorem 6.18. 0 Note that a Connes orientation holds more information than the generalized 3-ball property, since it provides a uniquely determined von Neumann algebra whereas the W*-product constructed from the generalized 3-ball property is not unique. (It depends on the choice of symmetries used to construct matrix units in the proof of Theorem 10.25.) Since a Connes orientation determines the associative product when the Jordan product is known, then in this respect it serves the same purpose as the concept of orientation defined for a von Neumann algebra (or its normal state space) in (A 200) and (A 201). Otherwise the two concepts of orientation have little in common. The concept of a Connes orientation is algebraic and global in nature, while the concept of orientation from [AS] is geometric and locally defined in terms of facial blown up 3-balls (A 194). The relationship between these concepts of orientation was discussed in greater detail in the last section of Chapter 6. A dynamical correspondence provides a physically meaningful way to single out C*-algebra state spaces among those of lB-algebras. In the next chapter we will show how one can characterize state spaces of general C*algebras by a set of geometric axioms involving a concept of "orient ability" defined in terms of facial 3-balls as in (A 146), but for convex sets more general than state spaces of C*-algebras.

NOTES

373

Notes The geometric characterization of the normal state space of B (H) (Theorem 10.2) follows easily from the characterization of normal state spaces of type I JBW-factors given in Theorem 9.34, which first appeared in [10]. The characterization of von Neumann algebra state spaces (Theorem 10.25) is due to Iochum and Shultz [62]. It was actually preceded by the characterization of state spaces of C* -algebras (Theorems 11.58 and 11.59 in the next chapter), which appeared in [11]. Theorem 10.26 was announced in [13]' and the proof first appeared in [AS]. In the C* -algebra theorem in Chapter 11, we will see that the Hilbert ball property used in the characterization of JB-algebra state spaces is replaced by the 3-ball property. However, this property is of no use in characterizing normal state spaces of von Neumann algebras, since there may be no extreme points for the normal state space, and thus no facial 3-balls. Therefore, the 3-ball property has been replaced by the concepts of a "blown up 3-ball" and of the "generalized 3-ball property" (Definitions 10.10 and 10.20), which were first given in [71]. There the blown up 3-balls were called "global 3-balls", and they were defined, not by the existence of a 3-frame as in this book, but by the existence of an affine surjection onto a bona fide 3-ball which pulls back the lattice of faces and the group of affine automorphisms in an appropriate way, as in Proposition 10.15.

11

Characterization State Spaces

of

C*-algebra

In this chapter we will reach our final goal: a characterization of the compact convex sets that are state spaces of C*-algebras. (We will assume our C*-algebras have an identity, but our characterization can easily be adapted for non-unital algebras). We will start with our previous characterization of state spaces of JB-algebras (Theorem 9.38). Then we will add two additional properties that characterize C*-state spaces among state spaces of JB-algebras. A key axiom for state spaces of JB-algebras is the Hilbert ball property, by which the face generated by any two pure states is a norm exposed Hilbert ball (Definition 9.9). These balls may be of any finite or infinite dimension (as can be seen from Proposition 5.51). But in the case of C*-algebras, they have dimension 0, 1 or 3, with the last occurring iff the two states are distinct and unitarily equivalent, or which is the same, iff they generate the same split face, cf. (A 142) and (A 143). This is the 3-ball property (Definition 11.17), which will be a key axiom for our characterization of state spaces of C* -algebras. We know (from Corollary 5.42) that each pure state p on a JB-algebra A determines a unital Jordan homomorphism 1fp : A ----+ !v! where M is a JBW-factor of type I and 1fp(A) is o--weakly dense in M. Briefly stated: 1fp is a dense type I factor representatwn. The dense type I factor representations separate the points of A (Lemma 4.14), and each of them is Jordan equivalent (Definition 11.4) to 1fp for a pure state p on A (Lemma 11.16). We show in this chapter that the 3- ball property for the state space K of a JB-algebra A is equivalent to A being of complex type, i.e., that for each dense type I factor representation 1f : A ----+ !v! we have M ~ B (H)sa with H a complex Hilbert space (Theorem 11.19). Thus if the state space K of A has the 3- ball property, then we can associate to each pure state P E K a concrete representation of A on a complex Hilbert space. This concrete representation is irreducible (Lemma 11.3), but it is not generally unique (up to unitary equivalence). By associating an irreducible concrete representation to each pure state as described above, and by forming the direct sum of all these representations, we can obtain a faithful representation 1f : A ----+ B (H)sa on a complex Hilbert space. But this does not solve our characterization problem, since the lB-algebra 1f(A) is not in general equal to the self-adjoint part of the C*-algebra it generates in B (H). However, after imposing a

376

11

C*-ALGEBRA STATE SPACES

second key axiom on K, we can choose concrete representations associated with the pure states in such a way that this is in fact the case. This second axiom involves the concept of a "global orientation" , which is a direct generalization of the same notion for the state space of a C*algebra (A 146). Thus a global orzentatwn of the state space K of a JBalgebra is a continuous choice of orientation of all those faces of K that are 3-balls, and K is said to be oT'zentable if it admits such a global orientation (Definition 11.50). Together, the 3-ball property and orientability will characterize state spaces of C* -algebras arlOng all state spaces of JB-algebras. Our proof of this result involves the non-trivial fact that a JB-algebra of complex type is a universally reversible JC-algebra (Theorem 11.26). A universally reversible JC-algebra is imbedded in its universal C*-algebra in a particularly nice fashion (Corollary 4.42), and we use this fact to show that in our case each pure state is associated with no more than two inequivalent concrete representations. More specifically, if the state space K of a JB-algebra A has the 3-ball property, then each pure state p on A is the restriction to A of just two pure states a and T on the universal C*-algebra U of A (with a = T iff P is "abelian"). Then the GNS-representations of U associated with a and T restrict to a pair of conjugate irreducible representations of A, and those are (up to unitary equivalence) the only irreducible concrete representations associated with the pure state p on A (Propositions 11.32 and 11.36). Now the problem is to select for each pure state p on A either the one or the other of the two pure states a and T on U in such a way that the direct sum of the corresponding irreducible concrete representations maps A onto the self-adjoint part of a C*-algebra. We show that if K is orientable, then this is possible; in fact, each global orientation of K will determine such a selection, so in this case A is isomorphic to the self-adjoint part of a C*-algebra. Here the orient ability condition is necessary since the state space of every C* -algebra is orientable (A 148), and is irredundant since there are examples of JB-algebras of complex type with non-orientable state space (Proposition 11.51). Thus a JB-algebra is isomorphic to the self-adjoint part of a C* -algebra iff its state space has the 3-ball property and is orient able (Theorem 11.58). Combining this result with the characterization of state spaces of JBalgebras (Theorem 9.38), we obtain a complete geometric characterization of the compact convex sets that are state spaces of C*-algebras (Theorem 11.59). By a short argument this also gives the 1-1 correspondence of Jordan compatible C*-products and global orientations (Corollary 11.60), which was proven without use of Jordan algebra theory in (A 156). States and representations of JB-algebras

Recall that a factor representation of a lB-algebra A is a unital homomorphism from A into a lBW-algebra such that the a-weak closure of 1T(A)

STATES AND REPRESENTATIONS OF JB-ALGEBRAS

377

is a JBW-factor (Definition 4.7). A factor representation is fmthful if it is 1-1. Since there are JB-algebras that admit no non-zero homomorphisms into B(H) (e.g., the exceptional algebra H 3 (O)), we need a substitute for the notion of irreducibility of a representation. For a C* -algebra A, a representation 7T : A -+ B (H) is irreducible iff the image is a-weakly dense, cf. (A 80) and (A 88). That motivates singling out the following class of representations.

11.1. Definition. Let A be a JB-algebra and M a JBW-algebra. A factor representation 7T : A -+ !vI is dense if the a-weak closure of 7T(A) equals M. Recall from Proposition 2.68 that the a-weak and a-strong closures of convex sets in a JBW-algebra coincide, so we could use either topology in the definition above. In general, for subspaces X of a JBW-algebra, X will denote the a-weak closure.

11.2. Definition. A concrete representation of a JB-algebra A is a Jordan homomorphism 7T from A into B(H)sa for a complex Hilbert space H. Such a representation is zrreduczble if there are no non-trivial closed subspaces invariant under 7T(A). 11.3. Lemma. Let A be a JB-algebra. Then every dense concrete representatwn 7r: A -+ B (H)sa zs lrreduclble. Proof. If p is a projection in B (H)sa such that the closed subspace pH is invariant under 7r(A), then p7r(a)p = 7r(a)p for all a in A. Then taking adjoints gives 7r(a)p = p7r(a) , and by density of 7r(A) in B(H)sa, p must be central in B(H), and thus equal to 0 or 1. Thus 7r(A) has no non-trivial closed invariant subspaces. 0 Note that the converse to Lemma 11.3 is not true. For example, the JB-algebra of real n x n symmetric matrices acts irreducibly on C n (since the complex algebra generated includes all matrix units and thus all of Mn(C)), and yet is not dense in Mn(C)sa. Recall that representations 7rl, 7r2 of a C* -algebra A are said to be quasz-equivalent if there is a *-isomorphism from 7rl (A) onto 7r2 (A) carrying 7rl to 7r2 (A 136). Now we define the analogous Jordan notion.

11.4. Definition. Let A be a JB-algebra, Ml and lvh JBW-algebras, and let 7Tl : A -+ Ml and 7r2 : A -+ M2 be homomorphisms. Then 7rl and 7T2 are Jordan equivalent if there exists a Jordan isomorphism from 7rl(A) onto 7r2(A) such that (7rl(a)) = 7r2(a) for all a E A.

378

11.

C*-ALGEBRA STATE SPACES

Recall that if A is a JB-algebra and M a JBW-algebra, then each homomorphism 11' : A ---> M admits a unique extension to a normal homomorphism n: A** ---> AI (Theorem 2.65). Then n is a-weakly continuous (Proposition 2.64), so its kernel is a a-weakly closed ideal of A **. Thus by Proposition 2.39, the kernel has the form cA ** for some central projection c in A**.

11.5. Definition. Let A be a JB-algebra, M a JBW-algebra, and 11' : A ---> AI a homomorphism. The central cover of 11' is the central projection c(11') in A** such that kern = (1 - c(11'))A**.

n,

Note that 1 - c( 11') is the largest central projection in A ** killed by and thus c(11') is the smallest central projection c such that n(c) = 1. Most of the remaining results in this section are close analogs of similar results for C*-algebras, d. [AS, Chpt. 5].

11.6. Lemma. Let A be alE-algebra, M a lEW-algebra, 11' : A ---> IvI a homomorphzsm, and n : A ** ---> /vI the umque normal extension. Then n restrzcted to c(11')A** zs a *-zsomorphzsm onto 11'(A) eM.

Proof. By Proposition 2.67, n(A**) = 11'(A). Since (kern) nC(11')A** = (1 - c(11'))A** n c(11')A** = {O}, then n is faithful on c(11')A**. For each element a E A** we have n(a) = n(c(11')a + (1 - c(11'))a) = n(c(11')a), so n maps c(11')A** onto n(A**) = 11'(A). Thus n restricted to c(11')A** is a *-isomorphism onto 11'(A). 0 11.7. Proposition. Let A be a lE-algebra and 11'1 and 11'2 homomorphisms of A mto lEW-algebras. Then 11'1 and 11'2 are lordan equwalent zff c(11'd = C(11'2)'

Proof. Let AI, = 11'i(A) for z of M1 onto IvI2 such that

= 1,2. Let

be a Jordan isomorphism

(11.1) for all a in A. Every Jordan isomorphism between JBW-algebras is normal, and therefore a-weakly continuous (Proposition 2.64). By a-weak continuity of n1 and n2, and a-weak density of A in A **, the equality (11.1) extends to a E A** if we replace 11'; by ni for z = 1,2. It follows that ker n1 = ker n2, and so by definition c( 11'd = c( 11'2)' Conversely, if c(11'd = C(11'2)' then kern1 = kern2, and so we can define : n1(A**) ---> n2(A**) by

(n1(x)) = n2(x).

STATES AND REPRESENTATIONS OF JB-ALGEBRAS

379

The agreement of the kernels is exactly what is needed to make well defined and a *-isomorphism. By Proposition 2.67, 7ri(A**) = Jri(A) = Mi for ~ = 1,2. Thus is a Jordan isomorphism from M1 onto ]vh carrying Jr1 to Jr2. 0 Proposition 11. 7 can be thought of as generalizing the fact that representations of a C*-algebra are quasi-equivalent iff their central covers coincide (A 137).

11.8. Lemma. Let A, B be lB-algebras and Jr : A -> B a lordan from A onto B. If 1 ~s the kernel of Jr and F ~s the anmh~lator of 1 m the state space of A, then Jr* ~s an affine homeomorphism from the state space of B onto the splzt face F. homomorph~sm

Proof. This follows from Corollary 5.38. 0

Recall that if A is a JB-algebra, each state on A has a unique extension to a normal state on A**, so the restriction map is an affine isomorphism from the normal state space of A ** onto the state space of A (Corollary 2.61). As usual we identify the state space of A and the normal state space of A **. Thus if p is a projection in A **, Fp denotes the set of normal states on A ** (or equivalently, states on A) with value 1 on p.

11.9. Lemma. Let A be alE-algebra wdh state space K and let A -> M be a dense factor representatzon mto a lE W-algebra M. Then Jr* ~s an affine ~somorphism from the normal state space of M onto Fe(IT) c K.

Jr :

Proof. Let 7r : A** -> ]vI be the normal extension of Jr. For each normal state a on M, a 0 7r is the unique normal extension of a 0 Jr = Jr* a. Thus to prove the lemma we need to show that a f---> a 0 7r is an affine isomorphism from the normal state space of AI onto Fe(IT). Since the annihilator of ker 7r in K is

((1 - c(Jr))A**r n K

= (im U 1- c (IT)r n K = (ker U~-e(IT)) n K =

Fe(IT)

(cf. equation (5.4)), the lemma follows by an argument similar to that in the proof of Corollary 5.38. 0 11.10. Proposition. Let A be a lE-algebra, and let Jr1 : A -> M1 and Jr2 : A -> M2 be dense factor representatzons. Let K1 and K2 be the normal state spaces of M1 and M2 respectwely. Then Jr1 and Jr2 are Jordan equwalent iff Jri(Kd = Jr2(K2). Proof. We have Jri(KI) = Jr 2(K 2 ) iff Fe(IT,) i.e., iff C(Jrl) = C(Jr2). This in turn holds iff Jr1 and (Proposition 11. 7). 0

= Jr2

F e(1I"2) (Lemma 11.9), are Jordan equivalent

380

11.

C*-ALGEBRA STATE SPACES

For our purposes, we will be most interested in concrete dense representations. In this case, Jordan equivalence is closely related to unitary equivalence as we shall now see.

11.11. Definition. Let A be a JB-algebra. Then concrete representations 7ri : A ----) B(Hd and 7r2 : A ----) B(H2 ) are umtarzly equwalent (respectively, conJugate) if there is a complex linear (respectively, conjugate linear) isometry u: Hi ----) H2 such that

for all a E A.

11.12. Proposition. Let 7ri : A ----) B(Hdsa and 7r2 : A ----) B(H2)sa be concrete representatwns of a JB-algebm A. If 7ri and 7r2 are umtan.ly equwalent or conJugate, then they are Jordan equwalent. If 7ri and 7r2 are Jordan equwalent and dense, then they are ezther umtarzly equwalent or conJugate, with both posszbziztzes occurrmg zff the representatwns are one dzmenswnal. Proof. Suppose that 7ri and 7r2 are either unitarily equivalent or conjugate. Let u be as in Definition 11.11. Define : B(Hd ----) B(H2) by (x) = ux'u- i if u is complex linear, and by (x) = ux*u- i if u is conjugate linear. In the former case, is a *-isomorphism, and in the latter case it is a *-anti-isomorphism (A 124). Thus in either case is a Jordan isomorphism of B(Hdsa onto B(H2 )sa. A Jordan isomorphism is normal, and thus and -i are cr-weakly continuous (Proposition 264). Hence maps the cr-weak closure of 7ri(A) onto the cr-weak closure of 7r2(A). It follows that concrete representations that are either unitarily equivalent or conjugate are also Jordan equivalent. Let 7ri : A ----) B(Hdsa and 7r2 : A ----) B(H2)sa be dense and Jordan equivalent. Let be a Jordan isomorphism from B(Hdsa onto B(H2 )sa carrying 7ri to 7r2. Extend to a complex linear Jordan isomorphism from B(Hi ) onto B(H2), By Kadison's decomposition theorem for Jordan isomorphisms of von Neumann algebras (A 158), is either a *-isomorphism or a *-anti-isomorphism. Thus by (A 126) there is an isometry v : Hi ----) H2 which is either complex linear or conjugate linear such that

Therefore

and so

7rl

and

7r2

are either unitarily equivalent or conjugate.

STATES AND REPRESENTATIONS OF JB-ALGEBRAS

381

Finally, if 1f1 and 1f2 are unitarily equivalent and conjugate, then there is a *-isomorphism and a *-anti-isomorphism IlJ from B(H 1) onto B(H2) such that

(1f1(a)) = 1lJ(1f1(a)) = 1f2(a)

for all a E A

(A 124). Then 1lJ- 1 0 fixes 1f1(A). By density of 1f1(A) in B(Hdsa and complex linearity, 1lJ-1 0 is the identity on B(H1)' Since 1lJ- 1 0 is a *-anti-isomorphism, this implies that B(Hd is abelian. Thus H1 and H2 must be one dimensional. Conversely, suppose 1f1 and 71'2 are Jordan equivalent, and that H1 and H2 are one dimensional. Let : B(Hdsa ---> B(H2)sa be a Jordan isomorphism carrying 1f1 to 71'2' Then it is straightforward to check that any complex linear or conjugate linear isometry from H1 onto H2 implements , so 71'1 and 71'2 are both unitarily equivalent and conjugate. D Let CJ be a state on a JB-algebra A. Recall that the central carrier c(CJ) is the least central projection with value 1 on CJ (Definition 4.10), and that 71'" : A ---> c(CJ)A** is defined by 71',,(a) = c(CJ)a (Definition 4.11).

11.13. Lemma. Let CJ be a state on a fB-algebra A. Then c(71',,) equals c( CJ).

Proof. The unique normal extension of 71'" to A** (cf. Theorem 2.65) is given by 1i'" (a) = c( CJ)a for a E A **. The kernel of this extension is (1 - c(CJ))A**, which must by definition equal (1 - c(71',,))A**, and so c(71',,) = c(CJ) follows. D 11.14. Corollary. If A zs a fE-algebra wzth state space K, and CJ and T are states on A, then the followmg are equwalent.

(i)

71'"

and 71'T are fordan equwalent.

(ii) c(CJ) = C(T). (iii) The spilt faces of K generated by CJ and T coinczde.

Proof. By Proposition 11.7, 71'" and 71'T are Jordan equivalent iff c(1f,,) = C(1fT)' By Lemma 11.13, this is equivalent to (ii). By Proposition 5.44, the minimal split faces generated by CJ and Tare F e (,,) and Fe(T) respectively. By the correspondence of projections and norm closed faces (Corollary 5.33), Fe(,,) = Fe(T) iff c(CJ) = C(T), which implies the equivalence of (ii) and (iii). D

Remark. If CJ is a state on a C*-algebra A, temporarily denote the associated GNS representation (A 63) by 1/;" : A ---> B(H). Identify (A**)sa with (Asa)** (cf. Lemma 2.76). Then the central cover of the Jordan representation 1/;O'IAsa is c(CJ) (cf. (A 135)). Define 71'0' : Asa ---> (Asa)** by 1fO'(a) = c(CJ)a. The central cover of 1f0' is c(CJ) (Lemma 11.13), so by Proposition 11.7, 1/;0'1 Asa is Jordan equivalent to 71'0"

11

382

C*-ALGEBRA STATE SPACES

11.15. Definition. Let A be a JB-algebra, and M a JBW-algebra. A homomorphism 7r : A --> M is a type I factor representation if the 0'- weak closure of 7r(A) is a type I JBW-factor. Recall that if~ a pure state, then 7r a is a dense type I factor representation, i.e., 7r a (A) is a type I JBW-factor (Corollary 5.42). In fact, up to Jordan equivalence all dense type I factor representations arise in this way, as we now show. Recall that Fa denotes the smallest split face containing 0'. Lemma. Let A be a lE-algebra with state space K, and M a dense type I factor representatwn mto a lEW-algebra /vI. Then Fe(K) contains at least one pure state, and for every such pure state 0' m Fe(K)' 7r zs lordan equwalent to 7ra, and Fe(K) = Fa.

11.16.

7r :

A

-->

Proof. By Lemma 11.9, 7r* is an affine isomorphism from the normal state space of M onto Fe(K)' Since M is of type I, by Corollary 5.41 the normal state space of M contains an extreme point, and therefore so does Fe(K) . Let 0' be an extreme point of Fe(K)' Since M is a factor, it contains no non-trivial central projections, and thus its normal state space contains no proper split face (Corollary 5.35). Hence the split face Fa = Fe(a) generated by 0' must equal Fe(K) , so c(O') = c(7r). Since c(O') = c(7r a ) (Lemma 11.13), then c(7r) = c(7r a ). Therefore 7r is Jordan equivalent to 7r a (Proposition 11. 7). 0

Reversibility of JB-algebras of complex type We now define one of the key properties that distinguishes C*-algebra state spaces. Recall that a 3-ball is a convex set affinely isomorphic to the closed unit ball B3 of R3 (Definition 10.1). In the case of C*-algebras, the faces generated by distinct pure states are either line segments (iff the states are unitarily inequivalent), or 3-balls (A 143). We now abstract this property. Recall that for the state space of a JB-algebra, the face generated by any pair of pure states is a Hilbert ball (Corollary 5.56), but that this Hilbert ball might be of any dimension (cf. Proposition 5.51). If 0' and 7' are distinct pure states, the split faces Fa and FT generated by 0' and 7' will be minimal split faces (Lemma 5.46). These must either coincide or be disjoint, since the intersection of split faces is also a split face (A 7). The face generated by 0' and 7' will degenerate to a line segment iff Fa and FT are disjoint (Lemma 5.54), or equivalently, iff 7r0' and 7rT are not Jordan equivalent (Corollary 11.14).

REVERSIBILITY OF JB-ALGEBRAS OF COMPLEX TYPE

383

11.17. Definition. If K is a convex set, then K has the 3-ball property if the face generated by every pair of extreme points of K is norm exposed, and is either a point, a line segment or a 3-ball. Note that in Definition 11.17 we allow the two extreme points to coincide, in which case the generated face will be a single norm exposed extreme point. Note also that the convex sets with the 3-ball property are a subclass of the convex sets with the Hilbert ball property (for which every extreme point is norm exposed, cf. Definition 9.9). Of course, the 3-ball property will be useful only when there are many extreme points, e.g., when K is compact, or K is the normal state space of an atomic JBW-algebra (cf. Lemma 5.58).

11.18. Definition. A JB-algebra is of complex type if for every dense type I factor representation 7f : A ---+ A1 we have M ~ B (H)sa for a complex Hilbert space H. 11.19. Theorem. A iE-algebra A lS of complex type lff ltS state space has the 3-ball property. Proof. Suppose first that A is of complex type, and let 0' and T be pure states. If 0', T are separated by a split face, then face ( 0', T) is the line segment [0', T], cf. Lemma 5.54 or (A 29). Now suppose 0' and T are not separated by any split face, and let F = FO' = FT' Let 7f (J : A ---+ c( O')A ** be multiplication by c( 0'). Then 7f (J is a type I factor representation (Corollary 5.42). Since A is of complex type, then c(O')A** ~ B(H)sa. The normal state space of c(O')A** is F = FC((J) = F(J (Proposition 2.62). Hence we conclude that F is affinely isomorphic to the normal state space of B (H)sa. The latter has the 3- ball property (A 120), and thus so does F. The face of K generated by T and 0' is also a face of F, and so we conclude that it is a 3-ball. Thus we have shown that K has the 3-ball property. Conversely, assume K has the 3-ball property, and let M be a JBWalgebra, and 7f : A ---+ M a dense type I factor representation. Then M = 7f(A) is a type I JBW-factor. By Lemma 11.9, 7f* is an affine isomorphism from the normal state space of M onto FC(7r)' The normal state space of M is a split face of K, and thus inherits the 3-ball property from K. Let 0' and T be distinct extreme points of the normal state space of M. Since M is a factor, its normal state space has no proper split faces (Corollary 5.35). Then 0' and T cannot be separated by a split face, so face(O', T) is not a line segment (Lemma 5.54). By the 3-ball property, face(O', T) is a 3-ball. Thus M is a type I JBW-factor for which the face of the normal state space generated by each pair of distinct pure normal states is a 3-ball. Therefore M is isomorphic to B (H)sa for some complex Hilbert space H (Proposition 9.36). Hence A is a JB-algebra of complex type. 0

11.

384

11.20.

C*-ALGEBRA STATE SPACES

Corollary.

The self-adJomt part of a C*-algebra IS a JE-

algebra of complex type.

Proof. This follows from Theorem 11.19 and the fact that the state space of a C*-algebra has the 3-ball property, cf. (A 143) and (A 108). 0 The following simple observation will be used frequently.

11.21. Lemma. Every JEW-algebra M of fype Jordan orthogonal symmetrIes.

h

contams a pazr of

Proof. By the definition of type I2 (Definition 3.21), there are exchangeable projections P1 and P2 in M with sum 1 (Corollary 3.20). Let 8 be a symmetry that exchanges P1 and P2. Then by Lemma 3.6, sand the symmetry r = P1 - P2 are Jordan orthogonal. 0 Our next lemma involves Jordan polynomials in several variables. Such a polynomial is a sum of products, just like a standard polynomial. But by non-associativity, parentheses are needed to indicate how these products are to be multiplied out. Therefore each standard polynomial corresponds to a whole class of Jordan polynomials (obtained by inserting parentheses). Clearly all those take the same values as the given standard polynomial on every associative subalgebra, and then in particular on the center.

11.22. Lemma. Let S be any spm factor and let Sl and 82 be any two Jordan orthogonal symmetrIes m S. For each natural number k we can find a Jordan polynomzal P k m k + 2 varzables, with P k mdependent of the chozces 81, 82, such that k elements a1,"" ak of S are 1m early dependent Iff (11.2)

Proof. Recall that S = R1 EB N (vector space direct sum), where N is a real Hilbert space and the Jordan product in S is defined by (11.3)

(AI

+ x)

0

(p,1

+ y) =

(Ap,

+

(xly))l

+

(AY

+ p,x)

for x, YEN, and that N consists of all scalar multiples of symmetries in S other than ±1 (Lemma 3.34). We extend the inner product from N to all of S by setting (11.4)

(AI

+ x I p,1 + Y)

= Ap,

+

(xly)

for x, YEN, so that S becomes the Hilbert space direct sum of R1 and N. By the Gram criterion for inner product spaces, k elements a1, .. . ,ak

REVERSIBILITY OF JB-ALCEBRAS OF COMPLEX TYPE

385

of 5 are linearly dependent iff' det( (ai la J )) = O. We will now use this criterion to construct Pk . Observe first that if ai = Ail + x, with Xi E N and Ai E R for I = 1, ... , k, then by (11.4) (11.5) Furthermore, for any a = Al by means of (11.3) gives

+x

with x E N and A E R, multiplying out

(11.6) and then also (11. 7) Now it follows from (11.5), (11.6) and (11.7) that there is a Jordan polynomial Q such that (11.8) Choose a Jordan polynomial f in k 2 variables which takes the same values as the determinant of order k on the center Rl of 5, i.e.,

for all scalars a'j with 1 ~ I, J ~ k. (See the remark in the paragraph preceding this lemma). Substitute Q(sl,s2,ai,a J ) for the (IJ)th variable of f for 1 ~ I, J ~ k, and then denote the resulting Jordan polynomial by Pk . By (11.8),

Recall that the JB norm and the Hilbert space norm on a spin factor are equivalent (cf. equation (3.10)), so closed subspaces for the two norms coincide. Observe for use in the next proof that if 5 is a spin factor and B is a norm closed subspace of dimension 3 or more, containing the identity 1, then B is also a spin factor for the inherited product and norm. In fact, if 5 = Rl E8 N where N is a real Hilbert space, then B is seen to be the spin factor R1 E8 (B n N).

386

11

C*-ALGEBRA STATE SPACES

11.23. Lemma. Let A be a lB-algebra of complex type, and c the central proJectzon such that cA ** IS the type 12 summand of A **. Then every factor representatzon of cA ** IS onto a spzn factor of dzmenszon at most 4.

Proof. Let 1f be a factor representation of cA **, and let S = 1f( cA **) Note that S is of type I2 (Lemma 4.8), and thus is a spin factor (Proposition 3.37). We must show S has dimension 4 or less. Vie will first show 1f(cA) has dimensi0n 4 or less. Suppose 1f(cA) has dimension 4 or more. The homomorphic image 1f(cA) ofthe JB-algebra cA is a norm closed unital sub algebra of the spin factor S (Proposition l.35), so it is itself a spin factor. (See the observation preceding this lemma.) Therefore a f----> 1f( ca) is a dense type I factor representation mapping A onto the spin factor 1f(cA). Since A is of complex type, 1f(cA) ~ 6(H)Sd for a complex Hilbert space H. But 6(H)sa is a spin factor iff H is two dimensional (Theorem 3.39). Hence 1f(cA) is four dimensional. We have now shown that 1f( cA) is of dimension 4 or less. It remains to prove the same result for 1f(cA**) in place of 1f(cA). Since cA** is of type I 2 , it contains a pair r, s of Jordan orthogonal c-symmetries (Lemma 1l.21). Then since 1f is unital, 1f(r) and 1f(s) are seen to be Jordan orthogonal symmetries in S. Let al,"" a5 E A. Since the dimension of 1f(cA) is 4 or less, then 1f(ad, ... , 1f(a5) will be linearly dependent in S, so by Lemma 1l.22, with P5 as described in Lemma 11.22, (1l.9) Since 1f is an arbitrary factor representation of cA ** and the factor representations of a JB-algebra separate points (Lemma 4.14), then (1l.1O) By the Kaplansky density theorem for JB-algebras (Proposition 2.69), the unit ball of A is a-strongly dense in the unit ball of A **, so by a-strong continuity of Jordan multiplication on bounded sets (Proposition 2.4), the equality (11.10) will hold not only for arbitrary al,'" ,a5 in cA, but also for all al,"" a5 in cA **. Therefore (1l.9) will hold for every factor representation 1f of cA ** and all aI, ... ,a5 in cA **. By Lemma 11.22, 1f(ad, ... ,1f(a5) are linearly dependent, so the dimension of 1f(cA**) is at most 4. We are done. 0 Recall that an element I of a JBW-algebra M is central iff UsI = I for all symmetries s in M (Lemma 2.35). We need the following slightly stronger result for JBW-algebras of type h.

REVERSIBILITY OF JB-ALCEBRAS OF COMPLEX TYPE

387

11.24. Lemma. Let M be a JEW-algebra of type h, and let r, S be a pazr of Jordan orthogonal symmetrzes zn M. If x E M satisfies Urx = Usx = x, then x zs central zn lVI.

Proof. Let 7T be a factor representation of M, and S = 7T(M). We will show 7T(X) is central in S. By Lemma 4.8, S is of type 1 2 , and thus is a spin factor. Recall that S = Rl EB N where N consists of all scalar multiples of symmetries in S other than ±1 (Lemma 3.34). Since 7T is unital, 7T(r) and 7T(S) are Jordan orthogonal symmetries. Now 7T(r) 07T(S) = 0 implies that neither 7T(r) nor 7T(S) can equal ±1, and so in particular 7T(r) E N. Thus we can write 7T(X) = Al

(11.11)

+ Q7T(r) + w,

where Q, A E R, and wEN is orthogonal (and then Jordan orthogonal) to 7T(r). By Lemma 3.6, Un(r)W = -w. By hypothesis, x = Urx = {rxr}, and since 7T preserves Jordan products, then 7T(X) = Un(r)7T(X). Substituting (11.11) into this last equality gives Al

+ Q7T(r) + W =

Al

+ Q7T(r)

- w.

Thus w = 0, so 7T(X) E R1 + R7T(r). Similarly 7T(X) E Rl + R7T(S), so 7T(X) E R1. Thus 7T(X) is central for all factor representations of lVI, and therefore x is central in M. 0 We will now prove a technical result that will playa key role in showing that JB-algebras of complex type are universally reversible, cf. Definition 4.32. We begin by listing a few elementary general facts. By common usage, 0 is often omitted when multiplying by a central element in a Jordan algebra. (See the remarks following Proposition 1.52). Let q be a central projection in a JBW-algebra lVI. If x is a general element of lV!, then (11.12)

qx = qox = {qxq} = Uqx.

(cf. (1.65)). Furthermore, multiplication by q is a unital Jordan homomorphism from M onto the JBW-algebra Mq (cf. (1.66)). Note also that if rand S are orthogonal symmetries in a JB-algebra, then Ur and Us will commute. In fact, Urs = -s (Lemma 3.6), so by the identity (1.16) (11.13) and then multiplying both sides by Ur gives UrUs = UsUr . Recall that every JBW-algebra of type 12 contains a pair of Jordan orthogonal symmetries (Lemma 11. 21).

11

388

C*-ALGEBRA STATE SPACES

11.25. Lemma. Let M be a lEW-algebra of type 12 , and let r, s be a pa2r of lordan orthogonal symmetnes m M. If M admlts a homomorph2sm onto the four dlmenswnal spm factor, then M con tams a central projectwn q and an element t such that qr, qs, tare lordan orthogonal q -symmetnes. Proof. Let 7r : M --) S be a representation onto a four dimensional spin factor S. Recall that S = Rl EB N, where N is the span of the symmetries other than ±1, and N is a three dimensional Hilbert space such that a 0 b = (alb)1 for a, bEN. We begin by constructing an element y E M, Jordan orthogonal to rand s. Note that 7r(r) and 7r(s) are not equal to ±1 (since 7r(r) and 7r(s) are Jordan orthogonal symmetries), and so 7r(r) and 7r(s) are in N. Let v E N be a non-zero vector orthogonal to 7r( r) and 7r( s), and choose x E M such that 7r(x) = v. By the definition of the Jordan product on S, 7r (x) = v is Jordan orthogonal to 7r (r) and 7r (s ) . Note then by Lemma 3.6, U 1T (r)7r(x) = ~7r(x) and U7C(S)7r(X) = ~7r(x). Thus

(11.14)

(1

~ U7C (r))(1 ~ U7C(S))7r(x)

=

47r(x)

i- O.

Define (11.15) By (11.14), Y is not zero. Since U; = Id, then Ury = ~y. Now (11.15), together with the fact that Ur and Us commute (see the remarks preceding this lemma), implies that UsY = ~y. Then Ury = ~Y and Usy = ~Y imply that Y is Jordan orthogonal to rand s (Lemma 3.6). Thus we have succeeded in finding an element of M Jordan orthogonal to rand s. Next we construct a non-zero central projection q, and a q-symmetry t E M Jordan orthogonal to rand s. Define

where r(a) denotes the range projection of a positive element a (cf. Definition 2.14), and Y = y+ ~ y- is the unique orthogonal decomposition of y (Proposition 1.28). By Proposition 2.16, r(y+) and r(y-) are orthogonal, and since y i- 0, they are not both zero. Hence q is a non-zero projection, and t is a q-symmetry. Since U r and Us are Jordan automorphisms of M (Proposition 2.34), they preserve orthogonal decompositions. The unique orthogonal decomposition of ~y is y- ~ y+, and as shown above Ur exchanges y with ~y, so it follows that Ur exchanges y+ and y-, and then also exchanges their range projections. Hence Urt = ~t, and similarly Ust = ~t. Thus t is a q-symmetry, Jordan orthogonal to rand s.

REVERSIBILITY OF JB-ALGEBRAS OF COMPLEX TYPE

389

We finally show that q is central. Since Ur and Us exchange r(y+) and r(y-), then by the definition of q, (11.16) Thus q is central (Lemma 11.24), and so a f-* qa is a unital Jordan homomorphism from M onto the JBW-algebra M q . A unital Jordan homomorphism preserves Jordan orthogonality, so qr, qs, qt = t are Jordan orthogonal. A unital homomorphism also takes symmetries to symmetries, so qr, qs, qt = t are the desired Jordan orthogonal q-symmetries. 0 The following theorem is a key property of JB-algebras of complex type that will playa crucial role in what follows. For use in the proof below, we review the notion of a Cartesian triple (Definition 4.50). If e is a projection in a JBW-algebra lvI, a Cartesian tnple of e-symmetnes is a triple (r, s, t) such that (i) ros = sot = tor = 0, (ii) r2 = s2 = t 2 = e, and (iii) UrUsUt is the identity on Me. Any JBW-algebra admitting a Cartesian triple of symmetries is a JW-algebra, and is reversible in any faithful representation as a concrete JW-algebra (Lemma 4.53). The condition UrUsUt = I in the definition of a Cartesian triple of symmetries is not redundant. (A counterexample is given in the remark after [AS, Thm. 7.29]). But note for use in the next proof that it is redundant in case M is a four dimensional spin factor. Indeed, if M = Rl EB N where N is a three dimensional Hilbert space, then by using the general fact that Urs = -s for any pair r, s of Jordan orthogonal symmetries (Lemma 3.6), and the fact that a given Jordan orthogonal triple r, s, t of symmetries is a basis for N, one readily computes that UrUsUtx = x for x E N, so that UrUsUta = a for every a = Al + x E N.

11.26. Theorem. Every iE-algebra A of complex type sally reversible iC-algebra.

lS

a umver-

Proof. We first show that A is a JC-algebra. For each pure state a of the state space of A, the map 7r a : A ---> c( a) A * * is a dense type I factor representation (Corollary 5.42), and such representations separate elements of A by Lemma 4.14. Since A is of complex type, then by definition c(a)A** must be isomorphic to B(H)sa for some complex Hilbert space H. The direct sum of such representations then provides a faithful representation of A as a JC-algebra. The bidual of a lC-algebra is a JW-algebra, so A** is a JW-algebra (Proposition 2.77). We will show that A** is reversible in any faithful representation as a concrete lW-algebra on a Hilbert space H. By Lemma 4.33, this implies that A is universally reversible, which will complete the proof. Assume A** C B(H)sa. Let c be the central projection such that cA ** is the b summand of A **. By Corollary 4.30 it suffices to prove that

390

11

C*-ALGEBRA STATE SPACES

cA ** is reversible. We are going to show cA ** contains a Cartesian triple of symmetries, and thus is reversible (Lemma 4.53), which will complete the proof. Let {(rc;, Sc;, tc;)} be a maximal collection of Jordan orthogonal triples of partial symmetries in cA** such that the projections Cn = T~ = s; = are orthogonal and central. Let TO = Lc; r c;, So = La Sa, to = Lc; t a , Co = La Ce;. (For each of these sums, the terms live on orthogonal subspaces of H, so the sums converge strongly. Therefore the sums also converge a-weakly in B(H)sa, and then in M as well.) Then ro, So, to are Jordan orthogonal co-symmetries. We will show that Co = c, and then show that ro, So, to form a Cartesian triple. Let w = C - Co and suppose that w is not zero. Since wA ** is a direct summand of cA **, it is also of type 1 2 . We now show that wA ** admits a homomorphism onto the four dimensional spin factor. Suppose not, and let 7r be any factor representation of wA **. Since wA ** is of type 12, then 5 = 7r(wA**) (a-weak closure) is an 12 factor (Lemma 4.8), i.e., 7r( wa) is a spin factor representation of cA", a spin factor. Then a so by Lemma 11.23 the dimension of 5 is 4 or less. We are assuming the dimension of 5 is not 4, so the dimension is 3 or less. Since A is of complex type, it has no spin factor representations onto the three dimensional spin factor. Hence 7r(wA) is of dimension 2 or less. Then 7r(wA) is spanned by the identity and at most one other element, so must be associative. Thus 7r( wA) is associative for all factor representations 7r of wA **. Since factor representations of wA ** separate points, we conclude that wA is associative. Then wA ** must also be associative (by a-weak density of wA in wA** and separate a-weak continuity of Jordan multiplication). This contradicts wA** being of type 1 2 . Thus wA** admits at least one representation 7r onto a four dimensional spin factor. Let r, S be a pair of Jordan orthogonal w-symmetries in wA** (cf. Lemma 11.21). Since wA** admits a homomorphism onto the four dimensional spin factor, wA** contains a non-zero central projection q and an element t such that qr·, qs, t are Jordan orthogonal q-symmetries (Lemma 11.25). This contradicts the maximality of the collection {(ra, Sa, ta)}, and thus proves that w = o. Hence Co = c. We finally show that TO, So, to form a Cartesian triple. All that remains is to show that Uro Us[) Uto is the identity on cA **. We prove this by considering factor representations of cA **. By Lemma 11.23, an arbitrary factor representation 7r maps cA ** onto a spin factor 5 of dimension 4 or less. Thus 5 = R1 EEl N where N is a real Hilbert space of dimension 3 or less. In fact, N must be of dimension 3, since 7r(ro), 7r(so), 7r(to) are Jordan orthogonal symmetries in 5, hence orthogonal vectors in N. Therefore 5 is four dimensional. Now it follows from the observation in the paragraph before the theorem that (7r(ro), 7r(so), 7r(to)) will be a Cartesian

t;

f--)

REVERSIBILITY OF JB-ALGEBRAS OF COlVIPLEX TYPE

391

triple in S. Therefore for all x E cA **,

Hence UTa Usa Uto = I d on cA ** . This completes the proof that cA** admits a Cartesian triple. We are done. 0 We have worked hard to prove reversibility of JB-algebras of complex type. The following corollary is the key consequence of reversibility that we will use in this chapter. Recall (Definition 4.37) that with each JBalgebra there is associated its universal C*-algebra C~(A), and a canonical homomorphism from A into C~ (A) such that the image of A generates C~ (A). If A is a JC-algebra, this homomorphism is 1-1, and we identify A with its image under this imbedding. There is also a canonical *-antiautomorphism

11.27. Corollary. Let A be a lB-algebra of complex type, and C~(A) the umversal C*-algebra wdh canomcal *-antz-automorphzsm
A is universally reversible by Theorem 11.26, and thus A

C~(A);a by Corollary 4.42. 0

We will now develop some properties of representations of JB-algebras of complex type. Recall that a dense concrete representation of a JBalgebra is irreducible (Lemma 11.3). We have already observed that the converse is not true in general, but now we show that the converse holds for JB-algebras of complex type.

11.28. Proposition. Let A be a lB-algebm of complex type. If A ---> B (H)sa zs an zrreduczble concrete representatwn, then 7f zs dense, i.e., the (J"-weak closure of 7f(A) equals B(H)sa.

7f :

Proof. Let M = 7f(A) ((J"-weak closure). We will show first that M is a JW-factor, then that it is reversible, and finally that it is of type I. Since A is of complex type, then it will follow that M COo' B(K)sa for some Hilbert space K. Since M is irreducible, Proposition 4.59 will imply that

M

=

B(H)sa.

If there were a non-trivial central projection c in M, then cH would be a closed invariant subspace for M, contradicting the irreducibility of 7f(A). Thus M is a JW-factor. We next show that M is reversible in every representation as a concrete JW-algebra. Recall that a concrete JW-algebra is reversible iff its 12 summand is reversible (Corollary 4.30). Since M is a factor, we need

392

11.

C*-ALGEBRA STATE SPACES

only deal with the case where !vI is of type 1 2 , i.e., a spin factor. In this case M is a type I factor, and A is of complex type, so we must have M ~ B(K)sa for some Hilbert space K. The self-adjoint part of a C*algebra is reversible in every concrete representation (Proposition 4.34), so M is reversible in every concrete representation. This completes the proof that the 12 summand of M is reversible in every concrete representation, and thus that AI is reversible in every faithful representation as a concrete JW-algebra. \Ale note for use below that in particular !vI is reversible in its universal von Neumann algebra W~ (M) (Definition 4.43). Finally we show that !vI is a type I factor by showing that !vI contains a minimal projection. We start by showing that W,7 (AI) contains a minimal projection. Let M be the von Neumann algebra generated by !vI in B (H). Then M is irreducible, and so its commutant is C1 (A 80). Thus, by the bicommutant theorem (A 88), M = B(H). By the defining property of W~ (M), the inclusion of M in B (H)sa can be lifted to a normal *homomorphism 71' from W~ (M) onto B (H). Then the kernel of 71' is a O'-weakly closed ideal of W~(!vf), and so there is a central projection c in W~(M) such that the kernel of 71' is (1 - c)W~(M) (A 105). Thus 71' is a *-isomorphism from cW~(M) onto B(H). It follows that cW~(M) contains a minimal projection p. Then p also is minimal in HT~ (!vI). Now we show that AI contains a minimal projection. Let be the canonical *-anti-isomorphism of W~(M). Since M is reversible in W~(M), by Proposition 4.45 the set of self-adjoint fixed points of is M. Let q = p V (p) (the least upper bound in the projection lattice of W,7 (M) ). Since is a *-anti-isomorphism, it is also a Jordan isomorphism and an order isomorphism, and thus is a lattice isomorphism on the projection lattice. Since is of period 2, (q) = (p) V p = q. Thus q is in M. Since p and (p) are minimal projections ("atoms"), in the lattice of projections of W~(M), the lattice-theoretic dimension of q = p V (p) is at most 2, cf. (A 42) (applied to the atomic part of W~(M), cf. Lemma 3.42). If q is an atom in !vI we are done. If not, let v be any non-zero projection in !vI with v < q. By Proposition 3.51, v is a finite projection in W~(M) with dim(v) < 2. Thus v is an atom in W~(M), and therefore also is a minimal projection in AI. This completes the proof that AI is a type I factor. As remarked at the start of this proof, this shows that M = B (H)sa and thus that 71' is dense. D Next we study the relationship of pure states and irreducible representations for JB-algebras. Recall that if K is any convex set, then 8e K denotes the set of extreme points of K. Recall also that if 0' is a pure state of a C*-algebra A and 71'0' the associated GNS representation (A 63), then there is a unit vector ~ such that O'(a) = (71'0'(a)~ I ~) for all a E A. Such a vector ~ is said to be a representmg vector for 0'.

THE STATE SPACE OF THE UNIVERSAL C*-ALGEBRA

393

11.29. Definition. Let A be a lB-algebra of complex type, and a a pure state on A. An irreducible concrete representation 7r : A -+ B (H)sa is assocwted wzth a if there is a unit vector ~ in H such that

(11.17)

a(a)

=

(7r(a)~ I~)

for all a EA.

Then we also say that a is assocwted wzth 7r. By Proposition 11.28, 7r(A) is a-weakly dense in B (H)sa, and so the vector ~ is uniquely determined up to multiplication by a scalar of modulus 1. We will say the vector ~ represents a with respect to 7r. Recall that we denote the vector state on B (H) associated with a vector ~ by wI;. 11.30. Lemma. Let A be a iE-algebra of complex type, 7r : A -+ B (H)sa an zrreducible concrete representation. The pure states associated wzth 7r are preczsely the states of the form WI; 0 7r for umt vectors ~ E H. Furthermore, 7r is associated wzth a pure state a iff c( 7r) = c( a), or equwalently, zfJ a zs an extreme powt of Fe(K)' In that case, 7r* zs an affine zsomorphzsm from the normal state space of B (H)sa onto the split face Fu generated by a.

Proof. By definition, 7r is associated with a iff there is a vector state such that a = 7r* wd = WI; 0 7r). By Proposition 11.28, 7r is a dense type I factor representation, and so 7r* is an affine isomorphism of the normal state space of B(H)sa onto Fe(K) (Lemma 11.9). Since the pure normal states on B (H) are the vector states (A 118), the image of the set of vector states under 7r* is precisely the set of extreme points of Fe(K); these are then the pure states with which 7r is associated. In particular, there is at least one pure state associated with 7r. Since B(H)sa is a factor, its normal state space has no non-trivial split faces (Corollary 5.35). Thus the split face generated by any state in Fe(K) is Fe(K)' Since the split face generated by a is Fe(u) (Proposition 5.44), then a pure state a is in Fe(K) (and therefore is associated with 7r) iff c( a) = c( 7r). 0

w~

The state space of the universal C*-algebra of a JB-algebra In this section, A will be a lB-algebra of complex type, with state space

K. We will let U denote the universal C*-algebra for A (cf. Definition 4.37), and K will denote the state space of U. q, will be the canonical *-anti-automorphism of U (Proposition 4.40), and r : K -+ K will be the restriction map. Recall that q, has period 2, and that the set of self-adjoint fixed points of q, is A (Corollary 11.27).

394

11.

C*-ALGEBRA STATE SPACES

11.31. Proposition. Let A be a iE-algebra of complex type, and let U, K, K, r be as defined above. Then each pure state of U rest7"lcts to a pure state of A, and for each pure state C5 in K, r is a bi)ectzon from the spl~t face generated by C5 onto the sp12t face generated by r( (5). Proof. Let C5 be a pure state on U, and let 7r : U ---+ B (H) be the eNS-representation of U associated with C5. Then 7r is an irreducible representation of U (A 81). By the definition of U, the C*-algebra generated by A is U, so 7r(A) is irreducible. Since A is of complex type, irreducibility of 7r(A) implies that 7r : A ---+ B (H),a is dense (Proposition 11.28). Thus 7r* is a bijection of the normal state space N of B (H) onto a split face of K (Lemma 11.9), and (7rIA)* is a bijection from N onto a split face of K. Since (7r IA) * = r 0 7r*, then r is an affine isomorphism from 7r*(N) onto (7rIA)*(N). By the definition of the eNS-representation, C5 = 7r*wl; for a representing vector ( Hence C5 is in 7r*(N) and r(C5) is in (7rIA)*(N). An affine isomorphism of a split face of K onto a split face of K will take minimal split faces to minimal split faces and extreme points to extreme points, and so the proposition follows. 0

We now show that for a JB-algebra of complex type, a pure state is associated with two irreducible representations which are not unitarily equivalent (unless the representations are one dimensional), but rather are conjugate (cf. Definition 11.11).

11.32. Proposition. Let A be a iE-algebra of complex type, and let r, q, and U be as defined at the begmmng of th~s sectzon. If C5 ~s a pur'e state of U and 7ra ~s the associated GNS representation of U, then the representatzons 7r a IA and 7r.p* a IA are conjugate ~rreduc~ble representatzons of A assocwted wzth r(C5) = r(q,*C5). Every ~rreduc~ble representatzon of A assocwted w~th r(C5) ~s umta7"lly equwalent to 7r a iA or to 7r.p*aIA, with both poss~bd~tles occur7"lng ~ff they are one d~mensional. Proof. Since q, is a *-anti-isomorphism, it is a unital order automorphism, so q,* is an affine automorphism of K (A 100). Since C5 is a pure state on U, then q,*C5 is also a pure state on U. Thus the eNS representations 7ra and 7r.p*a are irreducible representations of U (A 81). Since A generates U, then 7ra and 7r.p*a are also irreducible when restricted to A. It remains to show that 7ra and 7r.p*a restricted to A are conjugate. We will do this by constructing an auxiliary representation 1j; of U conjugate to 7ra , and then showing 1j; is unitarily equivalent to 7r.p*a as a representation of U (and then also when restricted to A). Let ~ be a cyclic vector for 7r
(11.18)

THE STATE SPACE OF THE UNIVERSAL C*-ALCEBRA

395

for all a E U. Recall that a conjugation of H is a conjugate linear isometry ) such that )2 = 1 (A 121). Extend {O to an orthonormal basis of H, and let) be the associated conjugation of H fixing that basis, cf. (A 123). Let a f----> at be the associated transpose map (A 125), i.e.,

at = )a*)

(11.19)

for all a E B(H).

Define 1/J : U --+ B (H) by 1/J (a) = 7T a ( (a)) t. Since and the transpose map are *-anti-isomorphisms, then 1/J is a *-homomorphism. Note that for a E A, since fixes each element of A, we have

Thus by the definition of conjugacy, 1/J restricted to A is conjugate to 7T a restricted to A. It remains to show that 1/J and 7Tq,*a are unitarily equivalent. For a E Usa, by the definition (11.19), and the identity ()7]1 1)7]2) = (7]1 17]2) for 7]1,7]2 E H (A 122), we get

(11.20)

(1/J(a)f,

I f,)

= ( 7Ta((a))tf, =

If,) =

(J7T a ((a));f, I f,)

(7T a ((a));f, I )f,) = (7T a ((a))f, I f,) = (*O')(a).

If we can show that f, is a cyclic vector for 1/J, then the unitary equivalence of 1/J and 7Tq,*a will follow from (11.20) and (A 77). Since 1fa(U) is irreducible, so is 1/J( U) = )7T a( ( U)) *) = )7T a( U)). (If Ho were a proper closed subspace invariant for 1/J(U), then )Ho would be a proper closed invariant subspace for 7Ta(U) = )1/J(U);.) It follows that f, is a cyclic vector for 1/J. Thus 1/J is unitarily equivalent to 7Tq,*a as a representation of U. Since 1/JIA is conjugate to 7T a , this completes the proof that 7T a iA and 7Tq,*aIA are conjugate. By (11.18) 7T a iA is associated with r(O'). Since fixes A, then r(*O') = r(O'). Next, let 7T be any irreducible representation of A associated with r(O'). Then by Lemma 11.30, 7T and 7T a iA have the same central covers, and so are Jordan equivalent (Proposition 11. 7). Since 7T and 7T a are dense (Proposition 11.28), then 7T and 7T a iA are unitarily equivalent or conjugate (Proposition 11.12). In the latter case, 7Tq,*aIA and 7T are both conjugate to 7T a lA, and thus are unitarily equivalent to each other. Recall that conjugate dense representations are Jordan equivalent, and are also unitarily equivalent iff they are one dimensional (Proposition 11.12). Density of 1fa and 7Tq,*a follows from Proposition 11.28. Hence the last statement of the proposition holds. 0 11.33. Corollary. If A is a iE-algebra of complex type, then every pure state on A is associated with at least one irreducible representation.

11

396

C*-ALGEI3RA STATE SPACES

Proof. Let U be the universal enveloping C*-algebra for A, and let a be a pure state on A. The set of states on U that extend a form a w*closed face of the state space of U. Thus by the Krein-Milman theorem, there is a pure state extension a. Let 7r be the GNS representation of U associated with a. By Proposition 11.32, 7rIA is an irreducible representation of A associated with a. D 11.34. Definition. Let A be a JB-algebra. A state a

IS

abelzan if

7r(T(A) is associative. For motivation, let A be a C*-algebra, and let 7r(T be the GNS representation associated with a state a. Then 7r IT (A) is abelian iff 7r (T (A )sa is an associative JB-algebra (Corollary 1.50). As remarked after Corollary 11.14, 7r(T1 Asa is Jordan equivalent to the Jordan homomorphism described in Definition 4.11, so 7r a (Asa) ~ c(a)A sa . Thus a is an abelian state on Asa iff 7r a (A) is abelian. (See also (A 155).) 11.35. Lemma. Let a be a pure state on a iE-algebra A. Then the followmg are equwalent:

(i) a is abelzan, (ii) 7r a (A) 1.S one dzmenswnal, (iii) {a} zs a splzt face of the state space of A. Proof. (i) {=? (ii) If a is abelian, then 7r a (A) = c(a)A C c(a)A** is associative, and by density so is c(a)A**. Then each projection in c(a)A** operator commutes with all elements of c( a)A ** (Proposition 1.47), i.e., is central. Since a is a pure state, then a is a factor state, i.e., c(a)A** is a factor (Lemma 4.14). Therefore, the only central projections are 0 and c(a). Since we have shown every projection is central, then c(a)A** has no non-trivial projections. By the spectral theorem (Theorem 2.20), c(a)A** must consist of multiples of the identity c(a), which proves (ii). Conversely, if (ii) holds, then 7r a (A) is associative, so a is abelian. (ii) =? (iii) If 7r a (A) is one dimensional, since 7r a (A) = c(a)A is dense in c(a)A**, then c(a)A** is one dimensional. Hence the normal state space of c( a)A ** is a single point. By Proposition 2.62, the normal state space of c(a)A** can be identified with Fe(a), which equals Fa (Proposition 5.44). It follows that Fa is a single point, and thus must equal {a}. (iii) =? (ii) If {a} is a split face, then Fe(a) = Fa = {a}. Hence the normal state space of c( a)A ** is a single point, so c( a)A ** is one dimensional. Now (ii) follows. D 11.36. Proposition. Let A be a iE-algebra of complex type, let U, cI> and r be as defined at the beginning of this section, and let a be a pure state on U. Then

THE STATE SPACE OF THE UNIVERSAL C*-ALGEBRA

397

(i) <1>*a = a ~ff a ~s abelzan. (ii) If a ~s not abelzan, then the spl~t faces generated by a and <1>* a are d~sJomt. (iii) The only pure states of U that restnct to r(a) on A are a and <1>*a, and the inverse ~mage of r(a) cons~sts of the lme segment wdh endpoints a, <1>*a. Proof. (i) Assume <1>*a = a. The eNS representations "Tra and "Trq,*a of U when restricted to A are conjugate irreducible representations (Proposition 11.32). Since <1>* a = a, these representations coincide, and by Proposition 11.32, "Tra(A) must be one dimensional. Since A generates U, then "Tr a (U) is one dimensional, so a is an abelian state on U. Conversely, suppose that a is abelian, so that {a} is a split face of K (Lemma 11.35). Let ("Tra, Ha,~a) be the associated eNS-representation of U on the Hilbert space H a , with the cyclic vector ~a. Then the normal state space of B(Ha) is affinely isomorphic to Fe(a) = {a} (A 138), and so Ha is one dimensional. Thus for each a E U, the vector "Tra(a)~a is a multiple of ~a. Since a(a) = ("Tra(a)~a I ~a), one has

This holds for all a E U, but we will make use of it for a E A. Similarly, since <1> * is an affine automorphism of K, then {<1>* a} = <1>* ({ a}) is a split face. For simplicity of notation, set <1>* 0" = T. If ("TrTlHTl~T) is the eNS-representation of U associated with T, then for a E A,

where the last equality uses the fact that <1>(a) = a for a E A. It is easily verified that A~ A~T is a unitary intertwining these representations of A, and so "Tra and "TrT are unitarily equivalent as representations of A. Since A generates U, then they are also unitarily equivalent as representations of U. Therefore a and T generate the same (singleton) split face (A 142), and so T = a. (ii) Suppose that the split faces generated by 0" and T are not disjoint. Since 0" and T are pure, they generate minimal split faces (Lemma 5.46). The intersection of two split faces is a split face (A 7), and so minimal split faces must either coincide or be disjoint. Thus the split faces generated by 0" and T coincide. Recall that r(O") = r(T) (Proposition 11.32), and that r is bijective on the split face generated by 0" (Proposition 11.31). Thus We conclude that T = 0". By (i) this implies that 0" is abelian, and now (ii) follows. (iii) Suppose w is a pure state of U such that r(w) = r(O"). Let "Trw be the associated GNS representation of U. Then "Trw IA is an irreducible

398

11.

C*-ALCEBRA STATE SPACES

representation associated with 1'(w) = 1'(0-) (Proposition 11.32), and so is unitarily equivalent to either 7r u IA or to 7rT IA. Since A generates U, the unitary that carries 7rw iA to either 7r u iA or to 7r T IA will also carry trw to either 7r u or 7rT' Therefore the split face generated by w coincides with the split face generated by one of 0- or 1 (A 142). Since l' is bijective on each of these split faces (Proposition 11.31), and 1'(w) = 1'(0-) = 1'(1), we must have w = 0- or w = I. Finally, let F = 1'-1(1'(0-)). Since 1'(0-) is a pure state of A, and l' : K --> K is surjective (by a Hal:n-Banach argument), then F is a non-empty w*-closed face of K. By the Krein-IvIilman theorem F is the w* -convex hull of its extreme points. The latter are pure states of K that restrict to 0-, and so by the last paragraph, the only possibilities are 0- and I. It follows that F is the line segment with these points as endpoints. D

If K is any convex set, we let oe,O K denote the extreme points of K that generate singleton split faces. If K is the state space of U as above, then oe,O K will be the set of abelian pure states of U. 11.37. Corollary. Let A be a lB-algebra of complex type, and let 1', K and K be as defined at the begmmng of th~s sectwn. The restnctwn map l' : K --> K maps oe,O K b~]ectzvely onto oe,oK and maps oe K \ oe,O K two-to-one onto oeK \ oe,oK. Proof. By Proposition 11.31, r(oeK) C oeK, and for 0- E oeK, {o-} is a split face iff {r(o-)} is a split face. By the Krein-Milman theorem, each 0- E oeK has an extension (j E oe K. Thus l' maps oe,O K onto oe,oK and oe K \ oe,O K onto oeK \ oe,oK. The last statement of the corollary follows from Proposition 1l.36 (iii). D

Remark. The abelian states on U form a w* -closed split face of K (A 154). Thus oe,O K is the set of extreme points of this split face, and by the Krein-Milman theorem, the w* -closed convex hull of oe.O K is the set of all abelian states. 11.38. Lemma. If A ~s a lB-algebra of complex type, and 1', K, K and <1> are defined as at the begmmng of th~s sectwn, then the fixed poznt set of <1>* m K ~s mapped b'l]ectwely onto K by 1'. Proof. Let 0- E K. Since <1> fixes A, then 1'( * 0-) = 1'(0-). Let K 1 denote the set of fixed points of * . Note that ~ (0- + * 0-) is in K 1, and has image 1'(0-) = 1'( * 0-). Thus l' maps the fixed point set of <1>* onto 1'( K) = K. To show that l' is injective, let 0- and w be states in K 1 such that 1'(0-) = 1'(W), and let x E: U Sd' Note that ~(x + <1>x) is fixed by <1>, and thus is in A (Corollary 11.27). Then 1'(0-) = 1'(W) implies that 0- and

THE STATE SPACE OF THE UNIVERSAL C*-ALCEBRA w

agree on ~ (x

+ 1>x).

399

By assumption, u and ware fixed by 1>*, so

u(x) = ~(u + 1>*u)(x) = u(~(x + 1>x)) = w(~(x + 1>x)) = w(x). Thus u

=

w, which completes the proof that r is injective on K

1.

0

11.39. Lemma. Let A be a iE-algebra of complex type, let rand K be as defined at the begmnmg of thls sectwn, and let F be a w*-closed spllt face of K. If r lS one-to-one on oeF, then r is one-to-one on F.

Proof. Let G = 1>* (F). Since 1>* is of period 2, then 1>* (G) = F, so that 1>* exchanges F and G. Since 1>* is an affine isomorphism, it preserves complementation of split faces, so 1>* also exchanges FI and G I . We first show that F n G is pointwise fixed by 1>*. Let u be a pure state in F n G. Then u E G implies that 1>*u is a pure state in F. Thus u and 1>* u are pure states in F that restrict to the same state on A, so by our hypothesis 1>*u = u. Hence 1>* fixes each pure state of the w*-closed split face F n G. By w*-continuity of 1>* and the Krein-Milman theorem, 1>* fixes every state in F n G. Now let u and w be states in F such that r(u) = r(w). Then since r(u) = r(1)*u) and r(w) = r(1)*w), we have r(~(u

+ 1>*u))

=

r(~(w

+ 1>*u)

=

~(w

+ 1>*w)).

By Lemma 11.38, ~(u

(11.21)

+ 1>*w).

Since F is a face, by the definition of a split face of K, F is the direct convex sum of F n G and F n G I . Write (11.22)

u

= SU1 + (1 - S)U2 and w = tW1 + (1 - t)W2

where U1, W1 E F n G, and U2, W2 E F n G I , and s, t are scalars with 0:::; s, t :::; 1. Substituting (11.22) into (11.21) gives

HSU1 + (1 - S)U2 + S1>*U1 + (1 - S)1>*U2) =

Htw1 + (1 - t)W2 + t1>*W1 + (1 - t)1>*W2).

Since 1>* exchanges F with G and FI with G I , the four terms in each sum above are in F n G, F n G I , G n F, G n FI respectively. Since 1>* fixes F n G pointwise, then

400

11

C*-ALGEBRA STATE SPACES

By affine independence of F n G, F n G' , G nF', cf. (A 6) or [6, Prop. II.6.6]' we conclude that for 0 < s < 1, then s = t, al = WI, a2 = W2, and so a = w. We leave it to the reader to show that a = s = 0 or s = 1. Thus r is one-to-one on F. D

W

also follows for

Let A be a JB-algebra, A + zA its complexification. Recall from Definition 6.14 that an associative product * on A + zA is iordan compatzble if the associated Jordan product coincides with the original one on A (i.e., a * b + b * a = ab + ba for a, b E A), and If the map a + zb f--+ a - zb for a, b E A is an involution for the product *. We refer to such a product as a C*-product on A + zA.

11.40. Lemma. If A zs a iE-algebra and A+zA admzts a C*-product jjxjj = jjx*xjjl/2 zs a norm on A + zA, and makes A + zA into a C*-algebra.

*, then

Proof. A JB-algebra is also an order unit algebra (Lemma 1.10), so the lemma follows from (A 59). D Thus if A is a JB-algebra and A + zA admits a C*-product, then A is isomorphic to the self-adjoint part of a C*-algebra. Conversely, every isomorphism of A with the self-adjoint part of a C*-algebra induces a C*product on A + zA, as we now will see. In what follows, if A is a JB-algebra and 1f is a Jordan isomorphism onto the self-adjoint part of a C*-algebra, then we will also denote by 1f the complex linear extension of 1f to A + zA.

11.41. Proposition. Let 1f : A -> Asa be a iordan zsomorphzsm from a iE-algebra A onto the self-adJomt part of a C*-algebra A. Then there zs a umque C*-product on A + zA makmg 1f : A + zA -> A into a *-zsomorphzsm. Two such *-zsomorphzsms 1f1 : A + zA -> A I and 1f2 : A + zA -> A2 mduce the same C*-product on A + zA zff 1f2 01f1 1 zs a *-zsomorphzsm from Al onto A 2 . Proof. The product (x,y) f--+ 1f- I (1f(x)1f(Y)) on A + zA is easily seen to be a C* -product making 1f : A + zA -> A into a *-isomorphism. In fact, it is the only such product, for if * is any product on A + zA such that 1f(x * y)

= 1f(x)1f(Y)

for X, yEA + zA, then applying 1f- 1 to both sides gives x * y = 1f- I (1f(x),1f(Y))· For the second statement of the proposition, define = 1f2 01f 11 and let \jJ be the identity map from A + zA equipped with the C*-product determined by 1f1 onto A + zA equipped with the C*-product determined

THE STATE SPACE OF THE UNIVERSAL C*-ALGEBRA

401

If these two products coincide, then it follows from the equality W 07r 1 1 that is a *-isomorphism. Conversely, if is a *isomorphism, then it follows from the equality W = 7r21 0 0 7rl that the identity map W is a *-isomorphism, so that the two products coincide. 0 by

7r2.

= 7r2 0

Thus for a JB-algebra A, specifying a C*-product on A + zA is equivalent to giving a Jordan isomorphism of A onto the self-adjoint part of a C* -algebra. We next give a necessary and sufficient condition for a JBalgebra of complex type to admit such a C* -product, in terms of the state space K of the universal C*-algebra.

11.42. Proposition. Let A be a iE-algebra of complex type, and let U, K, K and r be as defined at the begmmng of thzs sectwn. Then A zs zsomorphzc to the self-adJomt part of a C*-algebra zff there exists a w*-closed splzt face F of K such that r maps F bZJectwely onto K. Furthermore, there zs a 1-1 correspondence of such splzt faces F and C*products on A + tA.

Proof. Let F 0 be the set of w* -closed split faces F of K that are mapped bijectively by r onto K. We will establish the 1-1 correspondence of C*-products and split faces F in F o. Fix a C*-product on A + tA. This product organizes A + tA to a C*-algebra, which we will denote by A for short. Let L be the imbedding of the JB-algebra A into its universal C*-algebra U, and let 1/J be the unique lift of the inclusion 7r : A -> A + tA to a *-homomorphism from U onto A. Thus 1/J 0 L = 7r. Let J be the kernel of 1/J. Then 1/J* is an affine isomorphism from the state space S (A) of A onto the annihilator F of J in K, which is a w*-closed split face of K (A 133). On the other hand, 7r* is an affine isomorphism from S(A) onto K (Proposition 5.16). Dualizing 1/J 0 L = 7r gives i* o1/J* = 7r*. But i* = r, so r 0 1/J* = 7r*. Since 1/J* is an affine isomorphism from S( A) onto F and 7r* is an affine isomorphism from S (A) onto K, we conclude from this that r is an affine isomorphism from F onto K. Thus to each C*-product we have associated a split face in Fo. Suppose that two C*-products lead to the same split face FE F 0, and let 7rJ : A -> A + zA for J = 1,2 be the inclusions of A into the C* -algebra A + tA equipped with each product. Then the kernels of the associated *-homomorphisms 1/Jl : U -> A + tA and 1/J2 : U -> A + tA must coincide, since both are the annihilators of F in U (cf. (A 114)). Hence there is a *-isomorphism from A + zA with the first product, to A + zA with the second product, such that 01/Jl = 1/J2. By definition of 1/Jl and 1/J2, we have 7rl = 1/Jl 0 Land 7r2 = 1/J2 0 i. Thus 0 7rl = 7r2, so = 7r2 07rll. By Proposition 11.41, the two C*-products must coincide. Thus we have shown that the map from C*-products into F 0 is injective. To prove surjectivity, let F E F 0 and let J be the annihilator of F in U. Recall that J is a closed ideal in U and that F is the annihilator

11

402

C*-ALGEBRA STATE SPACES

of J m K (A 114). Let 'ljJ : U --> U / J be the quotient map, and define = 1j; 0 i : A --> U / J. We will show that 1i' maps A bijectively onto (U / J)sa. Recall that ;j;* is an affine isomorphism from the state space of U / J onto F (A 133). Since!' E F 0, then T is an affine isomorphism from F onto K, so 1i'* = TO 'ljJ* is an affine isomorphism from the state space of U / J onto K. It follows that 1i' is a Jordan isomorphism from A onto (U / J)sa (Proposition 5.16). Give A + zA the C*-product indpced by 1i' (cf. Proposition 11.41). Then 1i' : A --> U / J extends to a *-isomorphism 1i' : A + zA --> U / J Define ¢ = (1i') -1 : U / J --> A + zA, and 'ljJ = ¢ 0 ;j; : U --> A + zA. Let 7f : A --> A + zA be the inclusion map.

1i'

Then 'ljJ is a *-homomorphism, and for all a E A, ('ljJ

0

t)(a) = ¢(;j;(i(a))) = ¢(1i'(a)) = a = 7f(a),

~ 'ljJ 0 i = 7f. Since ¢ is a *-isomorphism, the kernels of 'ljJ = ¢ o;j; and 'ljJ coincide, and thus both equal F. Hence the split face associated wIth the given product on A + zA is F, which completes the proof of a 1-1 correspondence of C*-products and split faces in F o. 0 Lemma 1l.39 and Proposition 1l.42 will playa key role in the proof of our characterization of C* -algebra state spaces.

Orient ability The 3-ball property is not sufficient to guarantee that a JB-algebra state space is a C*-algebra state space, since there are JB-algebras of complex type that are not the self-adjoint part of C*-algebras, as we will see in Proposition 11.5l. An extra condition is needed ("orientability"), which we will now discuss. The notion of orientation of 3-balls and of C*-algebra state spaces was introduced in [AS, Chpt. 5], where it was shown that orientations of the state space of a C*-algebra are in 1-1 correspondence with C*-products

ORIENTABILITY

403

(A 156). C*-algebra state spaces are always orientable (A 148), and we will see that this property (extended to JB-algebra state spaces) is exactly what singles out C*-algebra state spaces among all JB-algebra state spaces satisfying the 3-ball property. We briefly review the relevant concepts, and extend them to our current context. Recall that B3 denotes the closed unit ball of R 3, and that a 3-ball is a convex set affinely isomorphic to B3.

11.43. Definition. Let K be a convex set. A facial 3-ball is a 3-ball that is a face of K. Our primary interest will be in the facial 3-balls of the state space of a JB-algebra of complex type. However, we will first review some elementary notions concerning general 3-balls, d. (A 128).

11.44. Definition. If B is a 3-ball, a parametenzatwn of B is an affine isomorphism from B3 onto B. Each parameterization eP of a 3-ball B is determined by an orthogonal frame (i.e, an ordered triple of orthogonal axes), given by the image under eP of the usual x, y, z coordinate axes of B3. If ePl and eP2 are parameterizations of a 3-ball B, then eP;;l 0 ePl is an affine isomorphism of B 3 , which extends uniquely to an orthogonal transformation of R3.

11.45. Definition. If B is a 3-ball, an oT'ientatwn of B is an equivalence class of parameterizations of B, where two such parameterizations ePl and eP2 are considered equivalent if det (eP;;l 0 ePd = l. If eP : B3 ---+ B is a parameterization, we denote by [eP] the associated orientation of B. An orientation of B corresponds to an equivalence class of orthogonal frames, with two frames being equivalent if there is a rotation that moves one into the other. Note that each 3-ball admits exactly two orientations, which we will refer to as "opposite" to each other.

11.46. Definition. If Bl and B2 are 3-balls equipped with orientations [ePl] and [eP2] respectively, and 1jJ : Bl ---+ B2 is an affine isomorphism, we say 1j; preserves onentatwn if ['l,6 0 ePl] = [eP2], and reverses onentation if the orientation [1jJ 0 ePl] is the opposite of [eP2]' We now outline how we will proceed. In the special case of the JBalgebra A = M 2 (C)sa, the state space is a 3-ball. The affine structure of the state space determines the Jordan product on A uniquely. There are two possible C*-products on A + ~A compatible with this Jordan product, namely, the usual multiplication on M 2 (C) and the opposite one. This choice of products corresponds to a choice of one of the two possible orientations on the state space (A 132).

404

11.

C*-ALGEBRA STATE SPACES

More generally, if K is the state space of a JB-algebra A of complex type, then a choice of orientation for each facial 3-ball may be thought of as determining a possible associative multiplication locally (i.e., on the affine functions on that 3-ball). It isn't necessarily the case that these choices of local multiplications can be extended to give an associative multiplication for A. We will see that this is possible iff the orientations of the facial 3-balls can be chosen in a continuous fashion. To make sense of a continuous choice of orientation, we need to define topologies on the set of facial 3-balls and on the set of oriented facial 3balls. We will then have a Z2 bundle with the two possible orientations (or oriented balls) sitting over each facial 3-ball. A continuous choice of orientation then will be a continuous cross-section of this bundle. The precise definition of this concept will be given for convex sets with the 3-ball property. Since each face of a convex set with the 3-ball property also has the 3-ball property, this definition will apply not only to state spaces of JB-algebras of complex type, but also to all faces of such state spaces.

11.47. Definition. Let K be a compact convex set with the 3-ball property in a locally convex linear space. Param(K) denotes the set of all affine isomorphisms from the standard 3-ball B3 onto facial 3-balls of K. We equip Param(K) with the topology of pointwise convergence of maps from B3 into K. Note that 0(3) and SO(3) acting on B3 induce continuous actions on Param(K) by composition. Two maps ePl, eP2 in Param(K) differ by an element of 0(3) iff their range is the same facial 3-ball B, and differ by an element of SO(3) iff they induce the same orientation on B. Thus we are led to the following definitions.

11.48. Definition. Let K be a compact convex set with the 3-ball property in a locally convex linear space. We call 03 I( = Param(K) / SO(3) the space of on.ented facwl 3-balls of K. We equip it with the quotient topology. If eP E Param( K), then we denote its equivalence class by [eP]. (Note that [eP] is an orientation of the 3-ball eP(B 3 ).) We call 3 I( = Param(K) / SO(3) the space of facwl 3-balls, equipped with the quotient topology from the map eP f---+ eP( B 3 ). When there is no danger of confusion we will write 03 for 03I( and 3 for 3 I(. Note that the definition above generalizes the corresponding one for C*-algebra state spaces (A 144). But the proposition below, which involves the canonical map from 03 to 3 that takes an oriented facial ball to its underlying (non-oriented) facial ball, is only a partial generalization of the corresponding result for C*-algebra state spaces (A 145) and (A 148), since in the C*-algebra case, this canonical map defines a trivial bundle, and not merely a locally trivial one.

o RIENTABILITY

405

11.49. Proposition. Let K be a compact convex set with the 3-ball property m a locally convex lmear space V. The spaces 0 B K and B K are Hausdorff, the canonical map from OB K onto B K ~s contmuous and open, and OB K -> B K ~s a locally trivial prmc~pal Z2 bundle.

Proof. By definition of the quotient topologies, the quotient maps from Param(K) onto Param(K)/50(3) and onto Param(K)/0(3) are continuous. Since these are quotients of a Hausdorff space by the action of compact groups, both are Hausdorff [32, Props. III.4.1.2 and III.4.2.3]). Since the relevant equivalence relations on Param(K) are given by continuous group actions, the quotient maps are open. Since the quotient maps Param(K) -> OB and Param(K) -> B are continuous and open, it is straightforward to check that the canonical map OB -> B is continuous and open. Now let cPo be any element of 0(3) that has determinant -1 and whose square is the identity. Then the continuous action of the group {cPo, I} on Param(K) induces a continuous Z2 action on OB = Param(K)/50(3) that exchanges the two elements in each fiber. To finish the proof we need to show that for each 3-ball B in B there is an open neighborhood W containing B such that the bundle OB -+ B restricted to W is isomorphic to the trivial bundle Z2 x W -+ ~V. This is equivalent to showing that for each cP E Param(K) there is an open neighborhood 5 of [cP] that meets each fiber of the bundle OB -+ B just once. Let A(B 3 , K) denote the set of affine maps from B3 into K, with the topology of pointwise convergence. We may assume without loss of generality that is not in the affine span of K. (Otherwise translate K so that this is true). We claim that the set of injective maps is open. To see this, let cPo E A(B 3 ,K) be injective. Let {ai, ~ = 1,2,3,4} be affinely independent elements of B3. Then {cPO(ai), ~ = 1,2,3,4} are affinely independent, and since 0 is not in the affine span of K, then these vectors in V are also linearly independent. Thus we can pick continuous linear functionals al,a2,a3,a4 such that ai(cPo(aj)) = 6iJ (the Kronecker delta). Then the determinant of the matrix (ai (cPo (a j ) )) will be 1. If cP is near enough to cPo then the determinant of the matrix (ai(cP(aj))) will be nonzero, and so {cP(ai), ~ = 1,2,3,4} must be linearly independent. It follows that the affine span of {cP(ai), ~ = 1,2,3,4} must be three dimensional, and so cP must be injective. Note further that for each E > 0 the set of cP such that 16ij - (ai (cP( a J ))) 1< E is a convex open neighborhood of cPo consisting of injective maps for E sufficiently small. Now choose a convex open neighborhood No of cPo in A( B3 , K) consisting of injective maps, and let N be the intersection of this neighborhood with Param(K). Let S = {[cP] cP EN}. Then 5 is an open subset of OB K , which we claim meets each fiber of our bundle at most once. Suppose that cPl and cP2 are in N and [cPt] and [cP2] are in the same fiber. Then cP1 and cP2 map B3 onto the same facial 3-ball B. For each t E [0,1] define

°

1

406 'Yt : B3

11 --->

C*-ALGEBRA STATE SPACES

K by 'Yt

= t¢l + (1 - t)¢2.

Then each 'Yt is in the convex neighborhood No, and so must be an injective map of B3 into B. For each t E [0, 1], ¢:;1 0 'Yt is an injective affine map from B3 into B 3 , taking 0 to 0, which then admits a unique extension to an invertible linear map from R3 into R3. Thus t f-> det(¢:;l 0 'Yt) is a continuous map from [0,1] into R \ {O}. Since its value at t = 0 is 1, its value at t = 1 cannot be -1, and so ¢1 and ¢2 must determine the same orientation. We have shown the bundle VB ---> B over W = {¢( B 3 ) I ¢ E N} is isomorphic to the trivial Z2 bundle Z2 x W ---> W, and thus is a locally trivial Z2 principal bundle. 0 Let 1r : VB ---> B be the bundle map that takes each oriented 3-ball to the underlying facial 3-ball. To specify an orientation for each facial 3-ball, we can give a map 8 : B ---> VB such that 1r(8(F)) = F for all FEB. Such a map is determined by its range, which will be a subset of VB that meets each fiber exactly once, i.e., includes exactly one of the two oriented 3-balls sitting over each facial 3-ball. We refer to both the map 8 and its range as cross-sectwns. If 8 is a continuous crosssection, then its range is a closed cross-section, since VB is Hausdorff and 8(B) = {[¢] I 8(1r([¢])) = [¢]}. Conversely, suppose X C VB is a closed cross-section. Then the complementary set Y of oriented 3-balls is also closed (since it is the image of X under the Z2 action of the bundle). We refer to Y as the opposzte cross-sectwn to X. Then X and its opposite cross-section Yare both open and closed. If 8 : B ---> VB is the crosssection such that 8(B) = X, then 8 will be continuous, and there is a 1-1 correspondence between closed cross-sections and continuous crosssections. Note that a closed cross-section X provides a trivialization of the bundle: the bundle is isomorphic to the trivial bundle X x Z2 ---> X. Clearly each trivialization of the bundle gives a closed cross-section.

11.50. Definition. Let K be a convex set with the 3-ball property in a locally convex space. Then K is orzentable if the bundle VB K ---> B K is trivial. A continuous cross-section of this bundle is called a global orzentation, or simply an orzentatwn, of K. Note that the above definition is a direct generalization of the corresponding definition for C*-algebra state spaces (A 146). Characterization of C*-algebra state spaces among JB-algebra state spaces We now pause to give an example showing that orient ability is not automatic for state spaces of JB-algebras of complex type.

C*-ALGEBRA STATE SPACES AMONG JB-ALGEBRA STATE SPACES 407

11.51. Proposition. Let T be the umt clrcle and let A consist of all contmuous functions f from T mto M 2 (C)sa such that f( ->.) = f(>.)t for all >. E T, where f(>.)t denotes the transpose of f(>.). Then A lS a iE-algebra of complex type whose state space is not onentable.

Proof. A is a norm closed Jordan sub algebra of the self-adjoint part of the C*-algebra C of all continuous functions from T into M 2 (C), and thus is a JB-algebra. Let a be a pure state of A. \Ve will show that the split face of the state space K of A generated by a is a 3-ball. The set of states of C that restrict to a form a w* -closed face of the state space of C, and so by the Krein-Milman theorem there is a pure state 0- of C that restricts to a. Let 7r be the GNS representation and ~ the cyclic vector associated with 0-. Since 0- is pure, then 7r is an irreducible representation of C (A 81). We will now verify that 7r is unitarily equivalent to evaluation at some point in T. Let Co be the subalgebra of C consisting of all f E C whose values are scalars (i.e., are in C1 where 1 is the identity matrix) at each point in T. Then 7r(Co) is central in 7r(C). Since 7r is irreducible, 7r(C) is a-weakly dense in B(H) (A 141), so 7r(Co) is also central in B(H). Thus 7r(Co) must consist of scalar multiples of the identity. Therefore 7r is a multiplicative linear functional on Co. By a well known theorem, every multiplicative linear functional on the commutative C*-algebra of all continuous complex valued functions on a compact set is evaluation at a point. Therefore there exists >'0 E T such that 7r(J) = f(>.o) for all f E Co. Write M2 for M2(C)' For 1 :S l,] :S 2 let mil E C be the constant function whose value at each point of T is the standard matrix unit ei) of lVh. Then every element of C admits a unique representation in the form Li) fi]mi] where each fij E Co. Thus

7r(

2.:= ];jmij ) ']

=

L

fij(>'O)7r(mi])'

']

Then 7r( mij) are 2 x 2 matrix units and generate the image 7r( C) of C, so 7r(C) must be isomorphic to M 2 . Identify 7r(C) with lvh. Choose a unitary v E 11/[2 such that V7r( mij )v* = e,] for all l,]. Then v7rv* is exactly evaluation at >'0. Thus 7r is unitarily equivalent to evaluation at some point in T, and we assume hereafter that 7r is given by evaluation at such a point >'0. We claim that 7r(A) = M 2 (C)sa' Choose a continuous function f : T ~ R so that f(>.o) = 1 and f( ->'0) = O. Given m E (M 2 )sa, the map a : >. f-> f(>.)m + f( _>.)m t is in A, and satisfies 7r(a) = m. Thus 7r(A) = M 2 (C)sa' Therefore (7rIA)* is an affine isomorphism of the state space of M2 (i.e., the 3-ball B 3 ) onto a split face F of K (Lemma 1l.8). Since a(a) = 0-( a) = (7r( a)e I e) for a E A, this split face contains (J. Thus the split face

408

11

C*-ALGEBRA STATE SPACES

generated by each pure state a is a 3-ba11. If T is another pure state of A, then either T E F, in which case face(T,a) = F is a 3-ball, or else T and a are separated by the split face F, in which case face(T, a) is the line segment [a, T], cf. Lemma 5.54 or (A 29). It follows that the state space K of A has the 3-ball property, and so A is of complex type (Theorem 11.19). Next we are going to construct a path between two distinct oriented facial 3-balls of A in the same fiber of OB --> B. We will see that this implies that K is not orientable. We identify the state space S(M2 ) with B 3 , and thus view an affine isomorphism from the state space of M2 onto a 3-ball of K as an element of Param(K). For each s E [0,1]' define a homomorphism 'irs : A --> M2 by 'irs(f) = f(e i7rs ). Then 'ir; will be an affine isomorphism from S(M2) = B3 onto a split face of K, which will also be a facial 3-ball of K. Thus 'ir; is a parameterization of a facial 3-ball of K; as usual we denote the associated oriented 3-ball by ['ir;]. Thus S f--+ 'ir; is a continuous map from [0,1] into Param(K), and so S f--+ ['ir;] is a continuous path in OB. If 1/; : M2 --> M2 denotes the transpose map, then by the definition of A, 'irl (f) = 1/;( 'ira (f)) for all f E A. Thus 'ira 0 1/;* = 'iri. It follows that the ranges of the orientations 'ira and 'iri are the same facial 3-ba11. Furthermore, since the dual of the transpose map, extended from S(M2 ) = B3 to R 3 , has determinant -1 (A 130), then the orientations given by 'ira and 'iri are different, so ['ira] and ['iril are the two distinct elements of a single fiber of OB --> B. However, if K were orient able , then the bundle of oriented 3-balls would be trivial, and so ['ira] and ['iril would lie in different connected components of OB. This contradicts the fact that we have just found a path in OB from ['ira] to ['iri], so K is not orientable. 0

Remark. In the example above, K is e,ctually affinely isomorphic to a C*-algebra state space, but by Proposition 11.51 the affine isomorphism cannot be chosen to be a homeomorphism, since the state space of a C*algebra is orientable (A 148). We sketch how to construct such an affine isomorphism. Let D be the C*-algebra of continuous functions f from T into M 2 (C) such that f( -A) = f(A) for all A E T. Let J be an ideal of D consisting of functions vanishing at 1, and similarly J the Jordan ideal of A consisting of functions vanishing at 1. For each f E Jsa or f E J, we associate the map S f--+ f(e i7rs ) from [0,1l into M 2 (C)sa. In this fashion, Jsa and J both can be identified with the Jordan algebra of all continuous functions from [0,1] into M 2 (C)sa vanishing at 0 and 1. Let Fv and FA be the annihilators of Jsa and J in the respective state spaces of D and A. Since J and J are the kernels of homomorphisms onto M2 (C) and M 2(C)sa respectively, it follows that Fv and FA are affinely isomorphic to 3-balls. Their complementary split faces F-fy and F;" are then affinely isomorphic to the state space of .:J and J respectively, i.e., the set of positive linear functionals of norm 1 on .:J or J respectively.

C*-ALGEBRA STATE SPACES A:tvIONG JB-ALGEBRA STATE SPACES 409

In particular, since Jsa ~ 1, then F-b and F~ are affinely isomorphic. This isomorphism, combined with the affine isomorphism of Fv and FA, extends uniquely to an affine isomorphism of the state spaces of D and A. We do not know if the state space of an arbitrary JB-algebra A of complex type admits a (possibly discontinuous) affine isomorphism onto the state space of a C* -algebra; nor do we know if A ** is isomorphic to the self-adjoint part of a von Neumann algebra for each JB-algebra A of complex type.

If K is the state space of a JB-algebra of complex type and if F is a split face of K, then the subspace Param(F) denotes the subset of Param(K) consisting of maps with range in F, with the relative topology from Param( K). B F denotes the set of facial 3-balls contained in F with the quotient topology from Param(F). We define VB F in an analogous way. Since Param(F) is saturated with respect to the action of 0(3) and SO(3), and the actions of 0(3) and SO(3) induce open quotient maps, B F has the topology inherited from B K, and VB F has the topology inherited from VBK ([32, I.5.2, Prop. 4]). If N is the normal state space of B (H), we can view N as a split face of the state space of B (H) (A 113). Then B N is path connected (A 152), and more generally we have the following result for state spaces of JB-algebras of complex type. 11.52. Lemma. If A is a lB-algebra of complex type, and F the spilt face generated by a pure state, then the space B F of faczal 3-balls m F zs path connected. Proof. Let a be a pure state on A, and let 71" be an irreducible concrete representation of A on a Hilbert space H associated with a (cf. Corollary 11.33). Then 71"* is an affine isomorphism from the normal state space N of B(H) onto the split face F generated by a (Lemma 11.30). Furthermore, 71"* is continuous for the weak topology on N given by the duality of B (H) and B (H) * and the w* -topology on F. It follows that 71"* induces a continuous surjective map from Param(N) to Param(F), and then also between the quotient spaces B Nand B F. Since B N is path connected (A 152), then B F is the continuous image of a path connected set, and therefore is itself path connected. 0 11.53. Definition. Let Kl and K2 be arbitrary convex sets, with each facial 3-ball in Kl and K2 given an orientation. Let ¢ : Kl -> K2 be an affine isomorphism of Kl onto a face of K 2. Then ¢ preserves onentation if ¢ carries the given orientation of each facial 3-ball F of Kl onto the given orientation of ¢(F), and reverses onentatwn if ¢ carries the given orientation of each facial 3-ball F of Kl onto the opposite of the given orientation of ¢(F).

410

11

C*-ALCEBRA STATE SPACES

Let A be a JB-algebra of complex type. We continue the notation of the previous section: U is the universal C*-algebra for A, is the canonical *-anti-automorphism of U, K is the state space of U, K is the state space of A, and r : K -> K is the restriction map.

11.54. Lemma. Let A be a iE-algebra of complex type, and let U, , K and r be as defined above. Then the map rP f--> r 0 ¢ zs a contmuous map from Param(K) onto Param(K) , and mduces contmuous maps [rPl f--> [r 0 rPl from OS K to K, and B f--> r(B) from S K onto SK

os

Proof. Recall that r maps oeK onto oeK (Corollary 11.37), and for each pure state a on U, r maps the minimal split face Fa bijectivelyonto the minimal split face Fr(a) (Proposition 11.31). Therefore r maps facial 3-balls of K bijectively onto facial 3- balls of K, and every facial 3- ball of K is the image of a facial 3-ball of K. Thus the map rP f--> r 0 ¢ is a continuous map from Param( K) onto Param( K). This map commutes with the action of 0(3) and 50(3), and hence induces a continuous map [¢l f--> [r 0 rPl from OS K to OSK, and a continuous map B f--> r(B) from SK to SK. D We now make use of the fact that the state space of any C*-algebra (in particular, that of U) has a global orientation induced by the algebra (A 148), i.e., an orientation assigned to each facial 3-ball in such a way as to give a continuous cross-section to the bundle OSK -> SK. This orientation is defined in (A 147), and a key property of this orientation, which will be used below, is that the dual * of a *-anti-isomorphism of U reverses the orientation of K (A 157).

11.55. Lemma. Let A be a iE-algebra of complex type whose state space K zs onentable, let U, , K and r be as defined before Lemma 11.54, and let a global onentatwn of K be gwen. Then for each pure state a on A, there IS a umque pure state (j of U such that the restnctwn map r zs an orzentatwn preservmg affine zsomorphzsm from the spld face F(j onto the splzt face Fa, and orzentatwn reversmg from Fip'(j onto Fa.

Proof. Let w be any extension of a to a pure state on U. Then r is an affine isomorphism from Fw onto Fa (Proposition 11.31). Equip each facial 3-ball in K with the orientation induced by U (A 148). Let [rPol and [¢ll be two of the resulting oriented 3-balls in Fw. By Lemma 11.52 there is a path of facial 3-balls from rPo (B 3 ) to rPl (B 3 ). Composing with the orientation induced by U gives a path of oriented 3-balls [rPtl from [rPol to [rPd· By continuity of the map [rPl f--> [r 0 rPl (Lemma 11.54), [r 0 rPtl is a continuous path in OB K. Then the path must lie either in the closed

C*-ALGEBRA STATE SPACES AMONG JB-ALGEBRA STATE SPACES 411

cross-section given by the specified orientation of K, or in its opposite, since these cross-sections are both open and closed. Thus [r 0 *w, and reverses orientation on the other. Since wand
(11.24) and then, for b E A, (11.25) This shows that a representing vector for aa is obtained by applying the representing operator 7r a (a) for a to the representing vector ~a for a, and then normalizing to get a unit vector. In this way, the action a f----> aa of a on states corresponds to the action of 7r a (a) on their representing vectors.

11.

412

C*-ALGEBRA STATE SPACES

11.56. Lemma. Let A be a C*-algebra, a an element of A, and U a pure state on A such that u(a*a) i- O. Then U a zs also a pure state. The two pure states U a and U are umtarzly equwalent, and are distinct zff

(11.26) m whzch case the face generated by

U

and

Ua

zs a facwl 3-ball B(u, u a ).

Proof. By (11.25), we have U a = Wry 07r(7. Since vector states are pure normal states on B(H) (A 118), and 7r; is an affine isomorphism from the normal state space of B(H) onto the split face F(7 generated by U (A 138), then U a = Wry 07r(7 is a pure state in F(7' Since the split face F(7 is minimal among all split faces (not only those containing u) (A 140), the split face generated by U a must be all of F(7' By (A 142), U a and U are unitarily equivalent. Since 7r = 7r (7 is irreducible, 7r (7 (A) is U -weakly dense in B (H) (A 141). Therefore U a and U will be distinct iff the representing vectors 'r) and ~(7 are linearly independent. This will be the case iff strict inequality holds in the Schwarz inequality, i.e., by (11.25) and (11.24), iff

Thus U a and U are distinct iff (11.26) is satisfied. We know that the face generated by a pair of distinct pure states of a C*-algebra is a 3-ball (A 143). Therefore the face generated by U and U a is a 3-ball when the equality (11.26) is satisfied. 0 By the lemma above, each a E A determines a map U f---+ B(u, u a ) from pure states to facial 3-balls whose domain is the w*-open set of all pure states for which (11.26) is satisfied. Observe for use in the theorem below that if U is any non-abelian pure state, then there exists a E A such that U is in the domain of the map U f---+ B(u, u a ). In fact, in this case 7r(7 is not one dimensional, so there exists a E A such that 7r (7 (a )~(7 and ~(7 are linearly independent, and then strict inequality will hold in the Schwarz inequality, so by (11.24)

11.57.

u

Lemma.

If A

zs a C*-algebm and a E A, the map

B(u, u a ) zs contmuous from the w*-topology of ds domam (defined by (11.26)) to the topology of the space B of facial 3-balls. f---+

Proof. Cf. (A 151). 0 Now we can prove the main result of this section.

C*-ALGEBRA STATE SPACES Al\IONG JB-ALGEBRA STATE SPACES 413

11.58. Theorem. Let A be a iE-algebra wzth state space K. Then

A zs zsomorphzc to the selJ-adJoznt part oj a C*-algebra iff K has the 3-ball property and zs orzentable. Proof. The necessity of the latter two conditions follows from (A 108), (A 143) and (A 148). Suppose conversely that K has the 3-ball property and is orientable. Then A is of complex type (Theorem 11.19). Fix a global orientation of K. Let U be the universal enveloping C*-algebra of A, and J( the state space of U with the global orientation induced by the C*-algebra U. We are going to construct a w*-closed split face F of J(, such that the restriction map r is a bijection from F onto K. By Proposition 11.42, that will complete the proof. For each a E oeK, let ¢(a) be the unique pure state in J( such that the restriction map r is an orientation preserving affine isomorphism from F¢(O") onto FO" (cf. Lemma 11.55). Let F be the w*-closed convex hull of {¢(a) I a E OeK}. We are going to show that F is the w*-closed convex hull of the collection {F¢(a) I a E oeK} of split faces, and thus is itself a split face (A 115). The w*-closed convex hull of this collection evidently contains F, so to prove equality it suffices to show that F¢(a) C F for each a E oeK. Fix a E oeK. If w is a pure state in F¢(a) , then r(w) is a pure state on A (Proposition 11.31). Since pure states generate minimal split faces, Fw = F¢(a). By definition ¢ maps F¢(O") onto Fa, so in particular r(w) E Fa. It follows that Fr(w) = Fa. Thus r is an orientation preserving affine isomorphism from Fw onto Fr(w), so by the definition of ¢, ¢(r(w)) = w, which implies that w E F. Thus oeF¢(a) c F. The split face F¢(a) is the a-convex hull of its extreme points (A 139). Therefore F¢(a) C F, which completes the proof that F is a split face of J(. Now we will show that oeF = {¢(a) I a E oeK}. Since r(¢(a)) = a, this will show that the restriction map r is a bijection from oeF onto oeK, and thus that r is a bijection from F onto K (Lemma 11.39). By Proposition 11.42, this will imply that A is the self-adjoint part of a C*algebra and complete the proof. Suppose for contradiction that w E oeF and w tJ. ¢( oeK). If w is abelian, then r(w) is abelian and w = ¢(r(w)) (Corollary 11.37), contrary to our supposition. Thus we may assume w is not abelian. By Proposition 11.36, the pure state r(w) of K has exactly two pure state extensions, namely wand *w where is the canonical *-anti-automorphism of U. The pure state extension ¢( r( w)) cannot be equal to w since w tJ. ¢( oeK). Therefore ¢(r(w)) = *w. Recall from Lemma 11.55 that the restriction map r is orientation preserving on one of the two split faces Fw and Fq,.w and orientation reversing on the other. Since * w = ¢( r( w)), then r is orientation preserving on Fq,.w. Hence r is orientation reversing on Fw. By Milman's theorem (A 33), oeF is contained in the w*-closure of ¢( oeK), so there is a net {P"Y} in ¢( oeK) such that P"Y --> w (in the

414

11

C*-ALGEBRA STATE SPACES

w*-topology). As observed above, there exists a E U such that W IS m the domain of the map p f---+ B(p, Pa). Since this domain is a w*-open subset of oeK, it contains P"( for all " large enough. By Lemma 11.57, B(p,,(, (P"()a) f---+ B(w,wa)) in the topology of B/C. Write B"( in place of B(Ii"(, (Ii"()a) and Bo in place of B(w, wa). Then B"( --t Bo implies that r(B"() --t r(Bo) (Lemma 11.54). Therefore, by continuity of the given orientation on K, the orientation of r( B"() converges to the orientation of r(Bo). On the other hand, the map induced by r on the space of oriented 3-balls is continuous (Lemma 11.55), and r preserves orientation on B"( and reverses it on Bo (as shown above), so the orientation of r(B"() must converge to the opposite of the orientation on r(Bo). This contradiction completes the proof that oeF equals {Ii I a E oeK}. We are done. D We record for use below some additional facts established in the proof above. Let F be the w*-closed convex hull of {¢(a) I a E oeK}, where ¢( a) is the state Ii defined in Lemma 11.55. Then F is a split face, every pure state in F has the form ¢( a) for some a E oeK, and the restriction map is an orientation preserving affine isomorphism from the split face F onto K.

Characterization of C*-algebra state spaces among convex sets Our main result now follows by combining previous results.

11.59. Theorem. A compact convex set K 2S affinely homeomorph2c to the state space of a C*-algebra (with the w*-topology) 2ff K sat2sfies the followmg properties:

(i) Every norm exposed face of K 2S pro)ectwe. (ii) Every a E A(K) admds a decompos2twn a = b - c wdh b, c E A(K)+ and b..l c. (iii) The a-convex hull of the extreme pomts of K 2S a splzt face of K, and K has the 3-ball property. (iv) K 2S orzentable. Proof. This follows by combining our characterization of the state spaces of lB-algebras (Theorem 9.38), and our characterization of those lB-algebra state spaces that are state spaces of C*-algebras (Theorem 11.58). D Remark. In (iii) above, we could replace the assumption that the a-convex hull of the extreme points of K is a split face of K by the assumption that for each compression P, the dual map P* sends pure states to multiples of pure states. The latter is one of the three pure

C*-ALGEBRA STATE SPACES AI\IONG CONVEX SETS

415

state properties, cf. Definition 9.2. The other two pure state properties follow from the 3-ball property, cf. Proposition 9.14, and now the proof of Theorem 11.59 still is valid, if we use the part of Theorem 9.38 that involves the pure state properties. According to (A 156), there is a 1-1 correspondence of Jordan compatible C*-products on a C*-algebra and orientations of its state space. The following result essentially says the same thing, adapted to our present context, but the results in this chapter allow us to give a much shorter proof than for (A 156) in [AS]. 11.60. Corollary. If A ~s a iE-algebra whose state space K has the 3-ball property and ~s onentable, then C*-products on A EO ~A are m 1-1 correspondence w~th global onentatwns of K.

Proof. There is a 1-1 correspondence between C*-products on A EO iA and those w* -closed split faces F of the state space K of the universal C*algebra of A for which r maps F bijectively onto K (Proposition 11.42). Thus what remains is to establish a 1-1 correspondence of such split faces of K and global orientations of K. Suppose first that we are given a global orientation e of K. As in the proof of Theorem 11.58 we equip K with the global orientation induced by the C*-algebra U, and assign to each pure state 0" E K the unique pure state ¢(O") E K for which the restriction map r is orientation preserving from the orientation of Fcp(u) to the orientation e on K. Let Fe be the w*-closed convex hull of {¢(O") I 0" E oeK}. Then it follows from the remarks following Theorem 11.58, that Fe is a split face, and that r is an orientation preserving affine isomorphism from Fe onto K. Hence the orientation e of K can be recovered from the orientation of K restricted to Fe, so the map e f-> Fe is 1-1 from the set of global orientations of K into the set of w* -closed split faces F of K for which r maps F bijectively onto K. To prove surjectivity, we suppose that we are given a w*-closed split face F of K which is mapped bijectively by r onto K. The global orientation on K restricts to give a global orientation of F. Since r : F f-> K is an affine homeomorphism, there is a unique global orientation e of K such that r : F --> K is orientation preserving. For each w E oeF, the restriction map r is an orientation preserving affine isomorphism on the generated split face Fw c F. Thus w has the defining property of the pure state ¢(r(w)), so w = ¢(r(w)). Hence oeF c {¢(O") I 0" E oeK}. On the other hand, if 0" E oeK, and w E oeF is the unique pure state such that r(w) = 0", then w = ¢(r(w) = ¢(O"), so {¢(O") I 0" E oeK} c oeF, and thus equality holds. Since Fe is by definition the w*-closed convex hull of {¢(O") 10" E oeK}, we conclude by the Krein-Milman theorem that F = Fe, as desired. Thus the map e f-> Fe is a 1-1 correspondence from

416

11.

C*-ALGEBRA STATE SPACES

global orientations of K to w*-closed split faces F mapped bijectively onto K by r. This completes the proof of the 1-1 correspondence of C*-products on A and global orientations of K. D The above characterization of C*-algebra state spaces was obtained by adding new assumptions to those characterizing JB-algebra state spaces. We did the same thing in Proposition 10.27, where we added the assumption about the existence of a dynamical correspondence. However, the two sets of extra assumptions are quite different in character. The one in Theorem 11.59 is "geometric and local", whereas the one in Proposition 10.27 is "algebraic and global". The former proceeds from local data of geometric nature (a continuous choice of orientations of facial 3-balls) from which a C*-product is constructed, while the latter postulates the existence of a globally defined map a f-> 1/Ja from elements to skew order derivations, from which the Lie part of the C*-product is obtained by an explicit formula (equation (6.14)). Thus, the starting point is closer to the final goal in the latter than in the former of the two characterizations. But the characterization in terms of dynamical correspondences is nevertheless of considerable interest, both because it explains the relationship with Connes' approach to the characterization of O"-finite von Neumann algebras, and because of its physical interpretation, which gives the elements of A the "double identity" of observables and generators of one-parameter groups (explained in Chapter 6). Note, however, that the conditions of Theorem 11.59 also relate to physics. We know that a Euclidean 3-ball is (affinely isomorphic to) the state space of ]l;h(C), which models a two-level quantum system (A 119). Thus the key role that is played by 3-balls in Theorem 11.59 reflects the fact that properties that distinguish quantum theory from classical physics are present already in the simplest non-classical quantum systems, the two-level systems, and that multilevel systems do not introduce any new features in this respect. The state space of a C*-algebra has at least two different global orientations. One is the orientation induced by the given C*-product (A 147), and the other is the opposite orientation, obtained by reversing the orientation of each facial 3-ba11. But there may be many other global orientations. For example, if we have a general finite dimensional C*-algebra A = L.~ Mj(C), then each summand M](C) has just two possible associative Jordan compatible multiplications, each associated with one of the two orientations of the finite dimensional split face corresponding to that summand. In each such split face, there are just two ways to choose a continuous orientation, and so the orientation for each such split face is determined by the orientation of a single facial 3-ball. There are n connected components of the space of facial 3-balls, each consisting of those 3-balls contained in a single minimal split face, and there are 2 n possible global orientations.

NOTES

417

Finally we observe that if the global orientation of a C*-algebra state space is changed to the opposite orientation, then the associated dynamical correspondence a f---> Wa (determined by the associated Lie product) will be changed to the opposite dynamical correspondence a f---> -Wa, and then the generated one-parameter group exp (tWa) will be changed to the oneparameter group exp (-tWa), which has opposite sign for the time parameter t. In this sense, changing to the opposite orientation can be thought of as time reversal.

Notes The results in this chapter first appeared in [11]. These results can be thought of as saying that the state space together with an orientation is a dual object for C*-algebras. Instead of the state space, some papers focus on the set of pure states as a dual object, e.g., [118]' and the papers of Brown [34] and Landsman [90].

Appendix

Below are the results from State spaces of operator algebras: baszc theory, orzentatwns and C*-products, referenced as [AS] below, that have been referred to in the current book. Many of these are well known results; see [AS] for attribution. A 1. A face F of a convex set K c X is said to be semz-exposed if there exists a collection 1{ of closed supporting hyperplanes of K such that F = K n nHEH H, and it is said to be exposed if 1{ can be chosen to consist of a single hyperplane. Thus, F is exposed iff there exists a y E Y and an a E R such that (x,y) = a for all x E F and (x,y) > a for all x E K \ F. A point x E K is a semz-exposed poznt if {x} is a semi-exposed face of K, and it is an exposed pomt if {x} is an exposed face of K. The intersection of all closed supporting hyperplanes containing a given set B c K will meet K in the smallest semi-exposed face containing B, called the semz-exposed face generated by B. [AS, Def. 1.1] A 2. A non-empty subset of a linear space X is a cone if C + C c C and )"'C c C for all scalars)... 2:: o. A cone C is proper if C n (-C) = {O}. Let C be a proper cone C of a linear space eX, and order X by x -::; Y if y - x E C. If F is a non-empty subset of C other than {O}, then F is a face of C iff F is a subcone of C (i.e., F + F c F and )"'F c F for all )... 2:: 0) for which the following implication holds:

It follows that if x E C, then facec(x) = {y E C I y -::;)...x

for some

)... E R+}.

[AS, p. 3] A 3. Two ordered vector spaces X and Yare in separatmg ordered dualzty if they are in separating duality and the following statements hold for x E X and y E Y:

x 2:: 0

¢=}

(x, y) 2:: 0 all y 2:: 0,

2:: 0

¢=}

(x, y)

y

[AS, Def. 1.2]

2:: 0 all x 2:: o.

420

APPENDIX

A 4. We will say that two positive projections P, Q on an ordered vector space X are complementary (and also that Q is a complement of P and vice versa) if ker+Q

= im+ P,

ker+ P

= im+Q.

We will say that P, Q are complementary m the strong sense if ker Q = im P,

ker P

= im Q.

[AS, Def. 1.3]

A 5. We say that a convex set K c X is the free convex sum of two convex subsets F and G, and we write K = FfficG, if K = co(FuG) and F, G are affinely independent. Observe that if K = F ffic G, then the two sets F and G must be faces of K. We say that a face F of K is a spilt face if there exists another face G such that K = F ffic G. In this case G is unique; we call it the complementary spilt face of F, and we will use the notation F' = G. More specifically, F' consists of all points x E K whose generated face in K is disjoint from F, in symbols F' = {x E K I face K (x) n F = 0}. [AS, Def. 1.4] A 6. If F and G are split faces of a convex set K, then every x E K can be decomposed as x = ~~,j=l a'jXi] where aij;::: 0 for 2,J = 1,2 and Xll E FnG, x12 E FnG', X21 E F'nG, X22 E F'nG'. This decomposition is unique in that every aij is uniquely determined and every Xij with nonvanishing coefficient aij is uniquely determined. [AS, Lemma 1.5] A 7. If F and G are split faces of a convex set K, then F n G and co (F U G) are also split faces and

(F n G)'

=

co(F'

U

G').

[AS, Prop. 1.6]

A 8. Let K be a compact convex set regularly embedded in a locally convex vector space X. K is said to be a Choquet simplex if X is a lattice with the ordering defined by the cone X+ generated by K. [AS, Def. 1.8] A 9. An ordered normed vector space V with a generating cone V+ is said to be a base norm space if V+ has a base K located on a hyperplane

421

APPENDIX

H (0 ~ H) such that the closed unit ball of V is co(Ku -K). The convex set K is called the d2stmguzshed base of V. [AS, Def. 1.10] A 10. If V is an ordered vector space with a generating cone V+, and if V+ has a base K located on a hyperplane H (0 ~ H) such that B = co(K U -K) is radwlly compact, then the Mmkowski Iunctwnal

Ilpll = inf{a > 0 I p E aB} is a norm on V making V a base norm space with distinguished base K. [AS, p. 9] A 11. If V is a base norm space with distinguished base K, then the restriction map I f---> 11K is an order and norm preserving isomorphism of V* onto the space Ab(K) of all real valued bounded affine functions on K equipped with pointwise ordering and supremum norm. [AS, Prop. 1.11]

A 12. A positive element e of an ordered vector space A is said to be an order umt if for all a E A there exists A ::::: 0 such that -Ae

<::::

a

<::::

Ae.

The order unit is called Arch2medean if for all a E A there exists A ::::: 0 such that

na

<::::

e for n

=

1, 2, . . .

=}

a

<::::

O.

[AS, Def. 1.12] A 13. An ordered linear space A with an Archimedean order unit e admits a norm

(1)

Iiall = infp > 0 I -Ae <:::: a <::::

Ae}

which satisfies the following relation for all a E A:

-ilalle <:::: a

<::::

Iialle.

[AS, p. 10] A 14. An ordered normed linear space A is said to be an order unit space if the norm can be obtained as in equation (1) from an Archimedean

APPENDIX

422

order unit, which is called the dzstzngmshed order unzt and will be denoted by 1. [AS, Def. 1.13] A 15. If A is an order unit spa:;e, then every positive linear map T: A ----+ A is norm continuous with norm IITII = IITlll. [AS, Lemma 1.15] A 16. A linear functional p on an order unit space A is positive iff it is bounded with Ilpll = p(I). [AS, Lemma 1.16] A 17. A linear functional p on an order unit space A is called a state if it is positive and p(l) = 1. The set K of all states on A is called the state space of A. An extreme point of the state space is called a pure state. [AS, Def. 1.17] A 18. If a is an element of an order unit space A with state space

K, then a E A+

{==}

p(a) 2: 0 for all p E K,

and Iiall = sup{lp(a)llp E

K}.

[AS, Lemma 1.18] A 19. The dual of an order unit space A is a base norm space and the dual of a base norm space V is an order unit space. More specifically, the distinguished base of A * is the state space of A, and the distinguished order unit of V* is the unit functional on V. [AS, Thm. 1.19] A 20. Let A be an order unit space and define for each a E A the function a on the state space K by writing a(p) = p( a) for all p E K. Then a f-+ is an order and norm preserving isomorphism of A onto a dense subspace of the space A(K) of all real valued w*-continuous affine functions on the w * -compact convex set K. If A is complete in the order unit norm, then the range of the map a f-+ will be all of A(K). [AS, Thm. 1.20]

a

a

A 21. An order unit space A and a base norm space V are said to be in separating order and norm dualzty if they are in separating ordered duality and if for a E A and p E V,

Ilall= sup l(a,a)1 11""119

[AS, Def. 1.21]

and

Ilpll= sup l(b,p)l· Ilbll:Sl

APPENDIX

423

A 22. If an order unit space A and a base norm space V are in separating order and norm duality, then (1, p) = Ilpll for all p E V+, the norm of V is additive on V+, and the distinguished base K is located on the hyperplane H = {p E V I (1, p) = I}. Moreover, if T : A --> A is a continuous positive linear map, then the equality (Tl, p) = IIT* pil holds for an arbitrary p E V+. [AS, Lemma 1.22] A 23. Two subsets F and G of a convex set K in a vector space X are said to be antzpodal if they are located on parallel supporting hyperplanes on opposite sides of K, i.e., if there exists a linear functional h on X with values in [0,1] such that h(p) = 1 for p E F and h(p) = for pEG. If we have two antipodal singletons F = {p}, G = {O"}, we will also say that the two points p and 0" are antzpodal. [AS, Def. 1.23]

°

A 24. Two points p and 0" in the positive cone V+ of a base norm space V are said to be orthogonal (written p ..L 0") if

lip - 0"11 = Ilpll + 110"11· In particular, two points p and 0" in the distinguished base K are orthogonal if lip - 0"11 = 2, i.e., if the distance between p and 0" is maximal. [AS, Def. 1.24] A 25. Let V be a base norm space, and let p,O" E V+ with p i= 0, 0" i= 0. Then p..L 0" iff Ilpll-1p and 110"11- 10" are antipodal points of the distinguished base K. [AS, Prop. 1.25]

°

A 26. Each point w i= in a base norm space V can be decomposed as a diflerence of two orthogonal positive components, i.e., there exists p,O" E V+ such that w

=

p -

0"

and

Ilwll = Ilpll + 110"11·

[AS, Prop. 1.26]

A 27. If V is a base norm space with dual space A = V*, then A and V are in separating order and norm duality. [AS, Cor. 1.27] A 28. Each split face F of the distinguished base K of a base norm space is norm closed. [AS, Prop. 1.29] A 29. Let K be the base of a base norm space V, and let 0" and T be extreme points of K. If there is a split face F such that 0" E F and

424

APPENDIX

rt- F, then the face of K generated by [AS, Prop. 1.30]

T

IJ

and

T

is the line segment

[IJ, T].

A 30. (Hahn-Banach separatiun) Let X be a locally convex space and let B c X be closed, convex and balanced (i.e., x E Band IAI :::; 1 implies AX E B). If Xo E X and Xo rt- B, then there exists an w E X* such that Iw(x)1 :::; 1 for all x E Band w(xo) > 1. [AS, Prop. 1.31] A 31. (Bipolar theorem) Let X and Y be vector spaces over R or C in separating duality under a bilinear form ( , ). If M is a weakly closed subspace of X with the annihilator MO in Y and if MOO is the annihilator of MO in X, then MOO = M. [AS, Thm. 1.35] A 32. If Y is a norm closed subspace of a Banach space X and ¢ : X --> XjY is the quotient map, then the dual map ¢* is an isometric isomorphism from (XjY)* onto the annihilator of Y in X*. [AS, Prop. 1.36] A 33. (Milman's theorem) If E is a closed subset of a compact convex set K in a locally convex vector space, then the only extreme points in co(E) are points in E. Thus K = co(E) only if oeK c E. [AS, Thm. 1.37]

A 34. (Krein-Smulian theorem) A convex set in the dual X* of a Banach space X is w* -closed iff its intersection with each positive multiple of the closed unit ball in X* is w* -closed. [AS, Thm. 1.38] A 35. In this book we will use the word algebra to denote a real or complex vector space with a bilinear product (not necessarily associative or commutative). An algebra A will be called normed if it is equipped with a norm such that for a, bE A,

(2)

Ilabll :::; Ilallllbll·

As usual, an associative normed algebra which is (norm) complete, is called a Banach algebra. [AS, Def. 1.42]

A 36. (Lattice version of the Stone-Weierstrass theorem) Let X be a compact Hausdorff space and let A be a closed linear subspace of CR(X) which is closed under the lattice operations and contains the unit function (taking the value 1 at all points of X). Then A = CR(X) iff A separates the points of X. [AS, Thm. 1.43] A 37. An ordered vector space X is said to be monotone complete (monotone IJ -complete) if every increasing net (sequence) in X which is

425

APPENDIX

bounded above has a least upper bound in X. (Then also every decreasing net (sequence) in X which is bounded below has a greatest lower bound in x.) [AS, p. 3]

A 38. If X is a compact Hausdorff space and GR(X) is monotone complete, then for each a E GR(X), E = {s E X I a(s) > O} is simultaneously closed and open in X. Furthermore, (i) a?: 0 on E, a ::::; 0 on X\E, and E is the smallest closed subset of X such that a::::; 0 on X \ E, (ii) E is the smallest closed subset of X for which XEa+ = a+ (pointwise product), (iii) XE is the supremum in GR(X) of an increasing sequence in

face(a+). If a ?: 0, then we write r(a) for XE defined as above. [AS, Lemma 1.55] A 39. Let X be a compact Hausdorff space such that GR(X) is monotone complete and let a E GR(X). Then there is a unique family {e>J .\ER of projections in GR(X) such that

(i) (ii) (iii) (iv)

e.\a::::; Ae.\ and e~a

e.\

?:

= 0 for A < -ilall

e.\::::; e,", for

Ae~

for all A E R, and e.\ = 1 for A>

Iiall,

A < /1,

1\,",>.\ e,l = e.\ for all A E

R.

The family {e.\} is given by e.\ = 1 - r((a - A1)+). For each increasing finite sequence 1= {Ao, AI,"" An} with AO < -ilall and An > Iiall, define Ihll = max(Ai - Ai-I! and s, = L~=l Ai(e.\, - e'\,_l)' Then lim

1h11~0

lis, - all = O.

[AS, Thm. 1.58] A lattice L with least element 0 and greatest element 1 is

A 40.

orthomodular if there is a map p

f----+

pi, called the orthocomplementatwn,

that satisfies

(i) pI!

= p,

(ii) p::::; q implies p' ?: q', (iii) p V p' = 1 and p A p' = 0, (iv) If p ::; q, then q = p V (q A p').

If p ::::; q' in an orthomodular lattice L, then we say p is orthogonal to q and write p ~ q. If p ~ q, then we sometimes write p + q in place of p V q. [AS, Def. 1.60] A 41. Let L be a complete lattice. A non-zero element p of L is an atom if each element q ::; p either equals p or is the zero element. L is

426

APPENDIX

atomzc if every non-zero element is the least upper bound of atoms, and an element P of L is finzte if p is the least upper bound of a finite set of atoms. [AS, Def. 1.61] A 42. Let p be a finite element in a complete atomic lattice. The minimum number of atoms whose least upper bound is p is called the dzmension of p and is denoted dim(p). [AS, Def. 1.62]

A 43. Let p and q be elements in a complete lattice L. We say q covers p if p < q and there is no element strictly between p and q. We say L has the covermg property if for all pEL and all atoms U E L,

(3)

pV

U

= P or

PV

U

covers p.

We say L has the finzte covermg property if (3) holds for all finite p in L and all atoms U E L. [AS, Def. 1.63] A 44. Let p be a finite element in a complete atomic orthomodular lattice L with the finite covering property. Then p can be expressed as a sum of atoms. In fact, p = PI + ... + Pk for each maximal set of orthogonal atoms PI, ... ,Pk under p, and the cardinality of any set of atoms with sum p is dim(p). Furthermore, every element q ::; p is finite with dim(q) < dim(p). [AS, Prop. 1.66]

A 45. An algebra is said to be power assoczatzve if parentheses can be inserted freely in products with identical factors. Thus the n-th power an of an element a is well defined in this case. We will write An for the set of all n-th powers of elements in a power associative algebra A. [AS, Def. 1.70] A 46. An order unit space A which is also a power associative complete normed algebra (for the order unit norm) and satisfies the following requirements: (i) the distinguished order unit 1 is a multiplicative identity, (ii) a 2 E A+ for each a E A, will be called an order unzt algebra. [AS, Def. 1.72]

A 47. If A is an order unit algebra, then A+

=

A2. [AS, Lemma 1.73]

A 48. If A is an order unit algebra, then the following are equivalent: (i) A is associative and commutative, (ii) abEA+ for each pair a,bEA+,

427

APPENDIX

(iii) A ~ CR(X) for a compact Hausdorff space X. [AS, Thm. 1.74]

A 49. The Jordan product in an associative (real or complex) algebra A is given by

a 0 b = ~ (ab

+ ba)

for a, bE A .

[AS, Def. 1.76]

A 50. Let Ao be a linear subspace of an order unit space A, and suppose that a, b f-7 ab is a bilinear map from Ao x AD into A such that for a, bEAD (and with the standard notation a 2 = aa),

(i) ab

=

ba,

(ii) -1::; a ::; 1

=}

0::; a 2

::;

1.

Then for all a, bEAD,

Ilabll ::; Iiall Ilbll· [AS, Lemma 1.79]

A 51. Suppose A is a complete order unit space which is a power associative commutative algebra where the distinguished order unit 1 acts as an identity. Then A is an order unit algebra iff the following implication holds for a E A:

[AS, Lemma 1.80]

A 52. Suppose A is a real Banach space which is equipped with a power associative and commutative bilinear product with identity element 1. Then A is an order unit algebra with positive cone consisting of all squares, distinguished order unit 1 and the given norm, iff for a, b E A,

(i) (ii) (iii)

Ilabll::; Ilallllbll, IIa 2 1 = Ila11 2 , Ila 2 11::; IIa 2 + b2 11.

Moreover, the ordering of A is uniquely determined by the norm and the identity element 1. [AS, Thm. 1.81] A 53. Let A be a commutdtive order unit algebra that is the dual of a base norm space V such that multiplication in A is separately w*continuous. Then for each a E A and each E: > 0 there are orthogonal

APPENDIX

428

projections P1, ... ,Pn in the w*-closed sub algebra W(a, 1) generated by a and 1 and scalars A1,"" An such that n

,=1 [AS, Thm. 1.84]

A 54. For each pair of elements x, y in a unital *-algebra, sp(yx) \

{O} = sp(xy) \ {O} .

[AS, Lemma 1.90]

A 55. If a is a continuous complex valued function on a compact Hausdorff space X, then the spectrum of a relative to the Banach algebra Cc(X) is equal to the range of a, i.e., sp(a) = {a(s) I SEX}. Moreover, the spectrum of a relative to Cc(X) is the same as the spectrum of a relative to the norm closed *-subalgebra C(a,l) generated by a and the unit function 1. The same statements hold with CR(X) in the place of Cc(X). [AS, Lemma 1.91] A 56. If A is a unital complex *-algebra and the real vector space Asa is an order unit algebra with distinguished order unit equal to the multiplicative identity and with the Jordan product induced from A, then A+ = A2 = {x*x I x E A}. [AS, Lemma 1.92]

A

=

A 57.

A C*-algebra is a complex Banach *-algebra A such that for all x E A. Unless otherwise stated, we will assume that C*-algebras mentioned have an identity. [AS, Def. 1.93]

Ilx*xll = I xl1 2

A 58. If A is a unital C*-algebra, then its self-adjoint part A = Asa is an order unit algebra for the Jordan product and the norm induced from A, and with distinguished order unit equal to the multiplicative identity and positive cone A+ = A2 = {x*x I x E A}. [AS, Thm. 1.95]

A 59. If A is a unital complex *-algebra and Aba is a complete order unit algebra for the Jordan product induced from A with distinguished order unit equal to the multiplicative identity, then the positive cone consists of all elements x* x where x E A, and A is a C* -algebra for a unique norm which restricts to the order unit norm on Asa. [AS, Thm. 1.96], where "order unit space" should read "order unit algebra". A 60. An element of a C*-algebra A is said to be positive if it is of the form x* x for some x E A, and we will denote the set of all positive

APPENDIX

429

elements in A by A +. A linear functional p on a C*-algebra A with identity element 1 is called a state if it is positive on positive elements and p(l) = l. The set of all states on A is called the state space of A, and it will be denoted by S( A), or just by K when there is no need to specify A, and the extreme points of K are called pure states. [AS, Def. l.97] A 61. A representatwn of a *-algebra A is a *-homomorphism 7r : l3 (H) where H is a real or complex Hilbert space. If we want to specify H, then we say that 7r is a representatwn of A on H. If 7r is injective, then we say that 7r is fa~thful. If ~ E H and the linear subspace 7r(A)~ = {7r(x)~ I x E A} is dense in H, then we say that ~ is a cychc vector (or a generatmg vector) for 7r. A representation with a cyclic vector is called a cycl~c representatwn. [AS, Def. l.98]

A

->

A 62. Suppose A is a (real or complex) unital *-algebra whose selfadjoint part A = Aba is a complete order unit space with distinguished order unit equal to the multiplicative identity and with the positive cone A+ = {x*x I x E A}. For each state p on A there is a Hilbert space H p, a cyclic unit vector ~p E Hp and a representation 7r p of A on Hp such that p(x) = (7rp(x)~p I ~p) for all x E A. Moreover, II7rp(x)11 ::; Ilx*xII1/2 for all x E A. [AS, Prop. l.100] A 63. If A is a C*-algebra and p is a state on A, then the representation described above is the GNS-representatwn associated with p. We will denote it by (7r p, H p, ~p), or just by 7rp when there is no need to specify Hp or ~p. Thus

[AS, Def. 1.101] A 64. (Gelfand-Naimark Theorem) Each unital C*-algebra A is isometrically *-isomorphic to a C*-subalgebra of l3(H) for a complex Hilbert space H. One such isomorphism is the direct sum of all GNSrepresentations associated with states on A. [AS, Thm. l.102] A 65. Suppose A is a real *-algebra with identity 1 whose self-adjoint part Asa is a complete order unit space with the distinguished order unit 1 and the positive cone A + = {x* x I x E A}. Then there is a representation 7r of A on a (complex) Hilbert space H which is an isometry on Asa. [AS, Prop. l.103] A 66. Let A be an ordered Banach space (i.e., a real Banach space ordered by a positive cone A + ). A bounded linear operator 8 on A is an order derivation if et8 is a positive operator on A, i.e., if et8 (A+) C A+

APPENDIX

430

for all t E R, or equivalently, if {e tO } is a one-parameter group of order automorphisms. We will write D(A) for the set of all order derivations on

A. [AS, Def. 1.104] A 67. Let A be a complete order unit space, and 0 a bounded linear map from A into A. The following are equivalent: (i) 0 is an order derivation. (ii) If x E A+, 0::::; u E A* and u(x)

= 0, then u(ox) = O.

[AS, Prop. 1.108] A 68. The set D(A) of order derivations of a complete order unit space A is a real linear space closed under Lie brackets [01,02] = 0102 - 0201. [AS, Prop. 1.114] A 69. Let A be a (unital) C*-algebra with state space K. Its selfadjoint part Asa is a complete order unit space under the norm induced from A, the ordering determined by the positive cone A + = A 2 = {x* x I x E A}, and with the distinguished order unit equal to the multiplicative identity 1. [AS, Prop. 2.2] A 70. Let A be a C*-algebra with state space K, and define for each on K by fL(p) = p(a) for p E K. Then the map a f--? is an order and norm preserving linear isomorphism of Asa onto the space A(K) of all continuous affine functions on K. [AS, Prop. 2.3]

a E Asa the function

a

a

A 71. A linear functional p on a C*-algebra A is positive iff it is bounded with Ilpll = p(I). [AS, Prop. 2.11]

A 72. Let A and E be two possibly non-unital C*-algebras. If is a *-homomorphism from A into E, then (A) is a C*-subalgebra of E. [AS, Prop. 2.17] A 73. Let p be a projection in a C*-algebra A and set q = p'. If a E A is self-adjoint, then the following two statements are equivalent:

(i) aq = 0 (by taking adjoints also qa = 0), (ii) ap = a (by taking adjoints also pa = a), and if a ~ 0, then they are also equivalent to each of the following four statements: (iii) (iv) (v) (vi)

qaq = 0, pap = a,

a::::; Iiallp, a E face(p).

[AS, Lemma 2.20]

APPENDIX

wE

431

A 74. Let p be a projection in a C* -algebra A and set q = p'. If A* is self-adjoint, then the following four statements are equivalent:

(i) q. w (ii) p. w

= 0 (by taking adjoints also w· q = 0), = w (by taking adjoints also w . p = w),

and if w ;::: 0, then they are also equivalent to each of the following four statements, (iii) w(q) (iv) w(p)

= 0, = Ilwll,

(v) q·w·q=O, (vi) p. w . p = w. [AS, Lemma 2.22] A 75. Let A be a C * -algebra. Then the extreme points of precisely the projections in .A. [AS, Prop. 2.23]

Ai

are

A 76. If {a,} is an increasing net bounded above in B(H)sa, then {a,} has a least upper bound a E B (H)sa, and a is also a strong (and weak) limit of {a,}. Similarly for a decreasing net and its greatest lower bound. [AS, Prop. 2.29] A 77. Let 7rl be a representation of a C*-algebra A on a Hilbert space Hl with cyclic vector~. Another representation 7r2 of A on a Hilbert space H2 is unitarily equivalent to 7rl iff it has a cyclic vector 6 such that

and then the equivalence can be achieved by a unitary that u6 = 6. [AS, Prop. 2.31]

U E

B(Hl' H 2 ) such

A 78. A representation 7r of a C*-algebra A on a Hilbert space H is said to be zrreducible if {O} and H are the only closed subspaces invariant under 7r(A). [AS, Def. 2.34] A 79. For each subset 5 of B(H) for a Hilbert space H, the set of operators in B (H) that commute with all operators in 5 is called the commutant of 5 and is denoted by 5'. [AS, Def. 2.35] A 80. A representation 7r of a C*-algebra A on a Hilbert space H is irreducible iff it has trivial commutant, i.e., iff 7r(A)' = Cl. [AS, Lemma 2.39]

A 81. The GNS-representation associated with a state p on a C*algebra is irreducible iff p is a pure state. [AS, Cor. 2.41]

432

APPEND~X

A 82. The irreducible representations of a C*-algebra A separate the points of A. [AS, Cor. 2.43] A 83. If A is a C* -algebra with state space K, then EB pEK 7r P (the direct sum of all GNS-representations) is called the umversal representatwn of A. [AS, Def. 2.45] A 84. If A is a *-subalgebra of B(H), then the weak, respectively stT'Ong, (operator) topology of A is the locally convex topology determined by the semi-norms a f---> l(a~I7])I, respectively a f---> Ila~ll, where ~,7] E H; the a-weak topology is determined by the semi-norms

where {~;}and {7];} are two sequences in H such that I::l II~i112 < 00 and I::l II7]i 112 < 00, and the a-stT'Ong topology is determined by the semi-norms

where {~;} is a sequence in H such that I:: l II~d2 < 00. Note that aa ---> a a-strongly iff (aa -a)*(aa -a) ---> 0 a-weakly. [AS, Def. 2.27 and Def. 2.64]

A 85. For each positive operator a E B(H) we define the trace of a as the sum tr(a) = I:,(a~'YI~'Y) (with values in [0,00]), where {~'Y} is any orthonormal basis in H. For arbitrary a E B(H) we define Ilalll = tr(lal)· [AS, Def. 2.51] A 86. An operator a E B (H) is of trace class if Ilalll < 00. The set of trace class operators is denoted by T(H) (or by L 1 (H)). By Lemma 2.53, T(H) is a two-sided ideal in B(H) and Ilalll is a norm on this ideal; we call it the trace norm. [AS, Def. 2.54] A 87. If r is a trace class operator, we define a linear functional on B (H) by Wr (a) = tr( aT'). The map r f---> Wr is an isometric order isomorphism of the ordered Banach space T(H) of trace class operators onto the subspace of B (H)* which consists of all a-weakly continuous linear functionals. [AS, Thm. 2.68]

Wr

A 88. If A is a unital *-subalgebra of B(H), then the bicommutant A" is the closure of A in any of the weak, strong, a-weak, or a-strong

topologies. [AS, Cor. 2.78]

APPENDIX

433

A 89. A unital *-subalgebra M of B(H) is called a concrete von Neumann algebra if it is weakly closed. A C*-algebra which admits a faithful representation as a concrete von Neumann algebra is called an abstract von Neumann algebra. [AS, Def. 2.80] A 90. A positive linear said to be normal if ¢( a) = least upper bound a in A. said to be normal if it is a functionals on A. [AS, Def.

functional (or state) ¢ on a C* -algebra A is lim-y ¢( a-y) for each increasing net {a-y} with More generally, a linear functional on A is linear combination of normal positive linear 2.82]

A 91. Let M be a von Neumann algebra acting on a Hilbert space H. The O"-weakly continuous linear functionals on M form a norm closed subspace of M'. [AS, Lemma 2.85] A 92. If M is a von Neumann algebra acting on a Hilbert space, then the O"-weakly (and O"-strongly) continuous linear functionals on M are precisely the normal linear functionals on M. [AS, Cor. 2.87] A 93. The O"-weak topology on a von Neumann algebra M is determined by the semi norms x f--+ Iw(x)1 where w is a normal state on M, and the 0" -strong topology on M is determined by the seminorms x f--+ w(x'x)1/2 where w is a normal state on M. [AS, Cor. 2.90] A 94. If M is a von Neumann algebra, then the map W : M f--+ (M.)* defined by (Wa)(¢) = ¢(a) for a E M and ¢ E M. is a surjective isometric isomorphism and a homeomorphism from the O"-weak topology on M to the w*-topology on (M.)*. Moreover, Wa ::,. 0 iff a ::,. o. [AS, Thm. 2.92] A 95. A C*-algebra A is a von Neumann algebra iff it satisfies either one of the following two conditions: (i) Asa is monotone complete and the normal states separate the points of A. (ii) There is a Banach space B such that A is (isometrically isomorphic to) the dual of B. Moreover, if condition (ii) is satisfied, then B is unique, in fact it is (up to an isometry) the Banach space of normal linear functionals on A. [AS, Thm. 2.93] A 96. The self-adjoint part (M.)sa of the predual of a von Neumann algebra M is a base norm space whose distinguished base is the normal state space K of M. [AS, Cor. 2.96] A 97. Let M be a von Neumann algebra with normal state space K, and define for each a E Msa the function a on K by ii(p) = p( a)

434

APPENDIX

a

for p E K. Then the map a f--+ is an order and norm preserving linear isomorphism of M,a onto the space Ab(K) of all bounded affine functions on K. [AS, Cor. 2.97]

A 98. The set of projections in a von Neumann algebra is an orthomodular lattice (cf. (A 40)) with p 1\ q = pq and p V q = p + q - pq for a pair of commuting projections p, q. [AS, Thm. 2.104] A 99. To each self-adjoint element a in a von Neumann algebra M there exists a unique resolution of the identity {e)J .\ER such that (i) e.\a::::: >.e.\ and (1 - e.\)a ;::: >.(1 - e.\) for all >. E R, (ii) e.\ = 0 for>. < -ilall and e.\ = 1 for>. > Iiall, (iii) e.\ commutes with a for all >. E R. The projections (iv) a =

e.\

are given by e.\

= 1 - T((a - >'1)+), and

J >.de.\.

Here for x E M +, T(X) denotes the least projection p such that pxp = x, and is called the mnge pTOJectwn of x. [AS, Def. 2.107 and Thm. 2.110] A 100. Let Al and A2 be C*-algebras. Then T f--+ T* is a 1-1 correspondence of unital order isomorphisms T from Al onto A 2 , and affine homeomorphisms from the state space of A2 onto the state space of A 1 . Similarly, if M 1 and M 2 are von Neumann algebras, then there is a 1-1 correspondence of unital order isomorphisms from M 1 onto lvh and affine isomorphisms from the normal state space of M 2 onto the normal state space of M 1 . [AS, Cor. 2.122]

A 101. If A is a C*-algebra with the universal representation 7r (cf. (A 83)), then the von Neumann algebra A = 7r(A) (weak closure) is called the envelopmg von Neumann algebm of A. A is isomorphic (as a Banach space) to the bid ual A ** of A, and the normal state space of A is affinely isomorphic to the state space of A. By common usage, we will denote the enveloping von Neumann algebra by A **. Thus we identify A with A ** equipped with the induced involution and product. [AS, Def. 2.123, Cor. 2.126 and Cor. 2.127] A 102. Let A be a C*-algebra with state space K and enveloping von Neumann algebra A** (cf. (A 101)). Then (A**)sa is isomorphic (as an order unit space) to the space Ab(K) of all bounded affine functions on K under the map a f--+ (with a(p) = p(a) for p E K) which is the unique O'-weakly continuous extension of the corresponding isomorphism of Asa onto A(K) (cf. (A 20)). [AS, Prop. 2.128]

a

A 103. A unital *-homomorphism ¢ : A

-+

M from a C*-algebra

A into a von Neumann algebra M has a unique extension to a normal *-homomorphism ¢: A ** -> M. [AS, Thm. 2.129]

APPENDIX

435

A 104. If u is a normal state on a von Neumann algebra M, then there is a smallest central projection e E M such that u( e) = 1. This projection e is called the central car'r1,er of u, and is denoted e(u). [AS, Prop. 2.130 and Def. 2.131]

A 105. Let J be a u-weakly closed two-sided ideal in a von Neumann algebra M. Then there is a unique central projection e such that J = eM; in fact, e is the unique two-sided identity of J. [AS, Cor. 3.17]

t,

A 106. If M is a von Neumann algebra and F is a subset of M then the least projection p E M such that w(p) = Ilwll (or equivalently w(p') = 0) for all w E F is called the car'r1,er pmJectwn of F and is denoted by carrier (F). [AS, Def. 3.20] A 107. Let F be a face of the normal state space K of a von Neumann algebra M and set p = carrier(F). Then the norm closure F of F consists of all u E K such that u(p) = 1. [AS, Prop. 3.30] A 108. A norm closed face F of the normal state space of a von Neumann algebra is norm exposed. [AS, Prop. 3.34] A 109. Let M be a von Neumann algebra with normal state space K, and denote by :F the set of all norm closed faces of K, by P the set of all projections in M, and by 3 the set of all u-weakly closed left ideals in M, each equipped with the natural ordering. Then there are an order preserving bijection : p f---> F from P to :F and an order reversing bijection ~ : p f---> J from P to 3, and hence also an order reversing bijection e = ~ 0 -l from :F to 3. The maps , ~,e and their inverses are explicitly given by the equations (i) F

= {u E K I u(p) = I}, p = carrier(F),

(ii) J={aEMlap=O},p=r(J)', (iii) J F

= {a E M I u(a*a) = 0 all u E F}, = {u E K I u(a*a) = 0 all a E J}.

Here in (ii) r(J) denotes the right identity of J. [AS, Thm. 3.35] A 110. If p is a projection in a von Neumann algebra M with normal state space K, the associated face Fp = {u E K I u(p) = I} satisfies

Fp = {u

E

Kip· u . p = u}.

[AS, equation (3.14)] A 111. If p is a projection in a von Neumann algebra M, then the following are equivalent:

436

APPENDIX

(i) p is central, (ii) a = pap + pi apl for all p EM, (iii) W = p. W . P + pl. W . pi for all wE M *.

[AS, Lemma 3.39] A 112. Let p be a projection in a von Neumann algebra M, let J be the associated a-weakly closed left ideal in M, and let F be the associated norm closed face of the normal state space K of M. Then the following are equivalent: (i) p is a central projection, (ii) J is a two-sided ideal, (iii) F is a split face.

If these conditions are satisfied, then the complementary split face of F is the norm closed face FI associated with pl. [AS, Prop. 3.40] A 113. The normal state space of a von Neumann algebra M is a split face of the state space K of M. [AS, Cor. 3.42] A 114. Let A be a C*-algebra with state space K. Then the canonical 1-1 correspondence between closed two-sided ideals J in A and w*-closed split faces F in K maps the ideal J to its annihilator ;0 n K in K and the face F to its annihilator FO in A. [AS, Cor. 3.63] A 115. If {Fo:} is a collection of split faces in the state space K of a C*-algebra A, then the w*-closed convex hull co(Uo: Fo:) is a split face. [AS, Prop. 3.77] A 116. Let M be a von Neumann algebra and let P be a positive, a -weakly continuous, normalized projection on M sa' There exists a projection p E M such that P = Up iff P is bicomplemented; in this case p is unique: p = PI, and the complement Q of P is also unique: Q = Uq (where q = 1 - p). [AS, Thm. 3.81] A 117. The conditional expectation E = Up + Uq associated with a projection p with complement q = pi in a von Neumann algebra M is the unique normal positive projection of M onto the relative commutant {py = pMp + qMq. [AS, Thm. 3.85] A 118. The extreme points of the normal state space K of B (H) are the vector states wT) (with TJ a unit vector in H), and each a E K is an infinite convex combination of vector states, i.e., a = 2::::: 1AiWT)i (norm convergent sum in B(H)*) where 2:::::1 Ai = 1 and where Ai ? 0 and IITJil1 = 1 for ~ = 1,2, .... [AS, Prop. 4.1] A 119. For each Hilbert space H the normal state space K of B(H) can be inscribed in a Hilbert ball; in fact, K can be inscribed in a ball

APPENDIX

437

defined by the norm of the space HS(H) of Hilbert-Schmidt operators by the map Wr f-> r. If dimH = n < 00, then K is a convex set of dimension n 2 - 1 which can be inscribed in a Euclidean ball of the same dimension. If n = 2, then K is affinely isomorphic to the full Euclidean ball B3; in fact, the positive trace-one matrix representing a state W E K relative to an orthonormal basis {6,6} can be written in the form

where

W

f->

((31, (32,(h) is an affine isomorphism of K onto B3. [AS, Thm.

4.4]

A 120. The face generated by two distinct extreme points of the normal state space of B(H) is a Euclidean 3-ball. [AS, Cor. 4.8]

A 121. A map v from a Hilbert space Hi into a Hilbert space H2 is said to be conjugate linear if it is additive and v( ,\~) = '\v~ for all ,\ E C and ~ E Hi. A conjugate linear isometry from Hi into H2 is called a conjugate umtary. If j is a conjugate unitary map from the Hilbert space H to itself and J2 = 1, then J is said to be a conJugatwn of H. [AS, Def. 4.21] A 122. If v is a conjugate linear isometry from a Hilbert space Hi into a Hilbert space H 2, then (v~IV7)) = (ei7)) for all pairs ~,7) E Hi. [AS, Lemma 4.22]

A 123. Every orthonormal basis a conjugation, namely

{~;}

in a Hilbert space H determines

We will refer to this as the conjugatwn assoczated wzth the orthonormal basls {~;}. [AS, equation (4.13)]

A 124. Let Hi and H2 be Hilbert spaces and let Kl and K2 be the normal state spaces of B (Hd and B (H 2) respectively. If u is a unitary from Hi to H 2, then the map 1> : a f-> Adu(a) is a *-isomorphism from B(Hd onto B(H2) and 1>* is an affine isomorphism from K2 onto K 1 . If v is a conjugate unitary from Hi onto H 2, then the map W : a f-> Adv(a*) is a *-anti-isomorphism from B(Hd onto B(H2) and w* is also an affine isomorphism from K2 onto K 1 . [AS, Prop. 4.24] A 125. Let H be a complex Hilbert space, and {l;aJ an orthonormal basis. Define the map a f-> at on B (H) by at = ja* j where j is the

438

APPENDIX

conjugation associated with the orthonormal basis. The map a f-7 at is called the transpose map with respect to the orthonormal basis {~,,}. [AS, Def. 4.25] A 126. Let HI and H2 be Hilbert spaces. A map from B(Hd onto B(H2 ) is a *-isomorphism iff it is implemented by a unitary and is a *anti-isomorphism iff it is implemented by a conjugate unitary, cf. (A 124). [AS, Thm. 4.27] A 127. The convex set B3 is the closed unit ball of R3. A convex set (in a linear space) is called a 3-ball if there is an affine isomorphism of B3 onto the set. A parametenzatzon of a 3-ball F is an affine isomorphism ¢ from B3 onto F. [AS, p. 193] A 128. The relation ¢1 rv ¢2 mod 50(3) divides the set of all parametrizations of a 3-ball B into two equivalence classes, each of which is called an onentatzon of B. We refer to each of the two orientations of a 3-ball as the opposite of the other. We denote the orientation associated with a parametrization ¢ by [¢]. An affine isomorphism 'ljJ : BI --> B2 between two 3-balls oriented by parametrizations ¢I and ¢2, is said to be onentatzon preservmg if [¢2] = ['ljJ 0 ¢d, and it is said to be orzentatzon reversmg if ['ljJ 0 ¢d is the opposite of [¢2]. [AS, Def. 4.28] A 129. Let a be a non-scalar self-adjoint operator on a two dimensional Hilbert space H, consider the w*-continuous affine function on the state space K of B(H) (defined by a(w) = w(a)), and assume that attains its maximum J.l at the point (J E K and its minimum l/ at the point T E K. Then the unitary u = e ia determines a rotation Ad: with the angle J.l - l/ about the axis c;:;. [AS, Thm. 4.30]

a

a

A 130. If {6, 6} is an orthonormal basis in a two dimensional Hilbert space H, then the dual of the transpose map with respect to this basis is a reflection about the plane orthogonal to the axis W7)~7)2 where

Thus the dual of the transpose map reverses orientation. [AS, Lemma 4.33] A 131. Let K be the state space of B (H) for a two dimensional Hilbert space H, so that K is a 3-ba11. Then the rotations of K are precisely the maps * where is implemented by a unitary operator u, i.e., : a f-7 uau* = Adu(a), and the reversals of K (i.e., the orientation reversing maps, which are the compositions of a reflection with respect to any diametral plane, and a rotation), are precisely the maps * where

439

APPENDIX

is implemented by a conjugate unitary operator v, i.e.,
f->

va*v- I

=

A 132. Let H be a two dimensional Hilbert space and let K be the state space of B(H). If
A 133. Let A, B be C*-algebras and 7r a *-homomorphism from A onto B. If J is the kernel of 7r and F = JO n K is the annihilator of J in the state space K of A, then 7r* is an affine homeomorphism from the state space of B onto the split face F. In particular, if (5 E K and 7r 0- is the associated GNS representation, then

where Fo- is the split face generated by Prop. 5.3]

(5,

and F 0- is its w*-closure. [AS,

A 134. Let A be a C* -algebra and 7r : A --+ B (H) a representation. The central cover of 7r is the central projection c( 7r) in A ** such that ker 7r = (1 - c( 7r)) A **, where 7r : A ** --+ B (H) is the normal extension of 7r, cf. (A 103). [AS, Def. 5.6] A 135. If (5 is any state on a C*-algebra, and 7r0- the associated GNS representation, then the central cover c( 7r 0-) and the central carrier c( (5) are equal. [AS, equation (5.6)] A 136. Two representations 7r1 and 7r2 of a C*-algebra A on Hilbert spaces HI and H2 are quasz-eqmvalent if there is a *-isomorphism
A 137. Let 7r1 and 7r2 be representations of a C*-algebra A. Then the following are equivalent: (i) 7rl and 7r2 are quasi-equivalent. (ii) The normal extensions 7rl and 7r2 to A ** have the same kernel. (iii) c(7rd = C(7r2)' [AS, Prop. 5.10]

440

APPENDIX

A 138. If A is a C*-algebra and 1f : A --> 13(H) is an irreducible representation, then 1f* is an affine isomorphism of the normal state space of 13(H) onto FC(7r)' In particular, if (J is a pure state on A, then 1f; is an affine isomorphism of the normal state space of 13 (HOI) onto Fa, where Fa is the split face generated by (J. [AS, Cor. 5.16]

A 139. If (J is a pure state on a C* -algebra A, then the split face Fa is the (J-convex hull of its extreme points. [AS, Cor. 5.17] A 140. Let A be a C*-algebra with state space K. The split face Fa generated by each pure state (J properly contains no non-empty split face of K. [AS, Cor. 5.18]

A 141. If 1f is a representation of a C*-algebra A on a Hilbert space H, then the following are equivalent: (i) 1f is irreducible. (ii) 1f( A)' = Cl. (iii) 1f(A)" = 13(H). (iv) 1f(A) is weakly (or (J-weakly) dense in 13(H). (v) The normal extension 7r of 1f maps A * * onto 13 (H) . [AS, Prop. 5.15]

A 142. Let (J and T be pure states on a C*-algebra A with state space K. The following are equivalent: (i) (ii) (iii) (iv) (v) (vi)

is unitarily equivalent to T. 1fa is unitarily equivalent to 1fT , 1fa is quasi-equivalent to 1f T , (J and T generate the same split face of K. c( (J) = c( T) . There is a unit vector T) in HOI such that T = (J

w1)

01f a .

[AS, Thm. 5.19] A 143. Let (J, T be distinct pure states on a C*-algebra A. If the GNS-representations 1fa and 1fT are unitarily equivalent, then the face generated by (J and T is a 3-ball. If these representations are not unitarily equivalent, then the face they generate is the line segment [(J, T]. [AS, Thm. 5.36]

A 144. Let K be the state space of a C*-algebra. Param(K) denotes the set of all parametrizations of facial 3-balls of K, cf. (A 127). vVe equip Param(K) with the topology of pointwise convergence of maps from B3 into the space K with the w*-topology. We call OB K = Param(K)/SO(3) the space of oriented facial 3-balls of K. We equip it with the quotient

APPENDIX

topology. We let quotient topology ¢(B 3 ). Thus 8 K identify these two

441

8

K denote the set of facial 3-balls, equipped with the from the map of Param (K) onto 8 K given by ¢ f-+ is homeomorphic to Param(K)JO(3) and we will often spaces. [AS, Def. 5.38 and Def. 5.39]

A 145. Let K be the state space of a C*-algebra. The spaces 08K and 8 K are Hausdorff, the canonical map from 08 K onto 8 K is continuous and open, and 08 K --) 8 K is a Z2 bundle. [AS, Prop. 5.40] A 146. Let K be the state space of a C*-algebra. A continuous crosssection of the bundle 08 K --) 8 K (of oriented facial 3- balls over facial 3-balls) is called a global onentatwn, or simply an orientatwn, of K. [AS, Def. 5.41] A 147. Let A be a C*-algebra with state space K, and F a facial 3-ball of K with carrier projection p. Let Jr be any *-isomorphism from pA**p onto lv12 . The orzentatwn on F mduced by A is the (equivalence class) of the map Jr* from B3 onto F, i.e., the orientation determined by the parameterization Jr*. [AS, Def. 5.44] A 148. The state space K of a C*-algebra A is orientable. Specifically, the orientation of each facial 3-ball induced by A gives a global orientation of K. [AS, Thm. 5.54] A 149. Let A be a C*-algebra, a E A, and let such that (J"(a*a) i= O. Then we define a state (J"a by

(J"

be any state on A

[AS, Def. 5.57]

A 150. Let A be a C*-algebra, a an element of A, and (J" any pure state on A such that (J"(a*a) i= O. Then (J"a is a pure state, (J"a and (J" generate the same split face, and their associated GNS-representations are unitarily equivalent. [AS, Lemma 5.58] A 151. Let A be a C*-algebra with state space K and let a E A. Then the map (J" f-+ B((J",(J"a) is continuous from its domain into the space of facial 3-balls 8 K. [AS, Lemma 5.59]

A 152. If H is a complex Hilbert space, and N the normal state space of B(H), then the space B N of facial 3-balls is path connected. [AS, Lemma 5.60J A 153. The commutator zdeal of a C*-algebra A is the (norm closed) ideal generated by all commutators of elements in A. It is denoted [A, A J. [AS, Def. 5.62J

APPENDIX

442

A 154. The annihilator Fa = [A, A]a n K of the commutator ideal of a C*-algebra A in the state space K is a w*-closed split face such that (i) Fa consists of all a E K with 1fa(A) abelian, (ii) oeFO consists of all a E oeK with 1fa(A) one dimensional, or equivalently, with Fa = {a}. [AS, Prop. 5.63]

A 155. Let A be a C*-algebra with state space K. We will denote by Fa the split face that is the annihilator of the commutator ideal, i.e., Fa = K n [A, A] a. We will say that a state a is abelian if 1fa( A) is abelian. Thus Fa consists of all abelian states. [AS, Def. 5.64] A 156. If A is a C*-algebra with state space K, then there is a 1-1 correspondence between (i) Jordan compatible associative products on A, (ii) global orientations of K, (iii) ordered pairs (Fl' F2) ofw*-closed split faces with CO(Fl uF2) = K and Fl n F2 = Fa (where Fa is defined as in (A 155), (iv) direct sum decompositions [A, A] = J 1 EEl J 2 • [AS, Thm. 5.73]

A 157. If : A ----) B is a Jordan isomorphism between C*-algebras, then is a *-isomorphism iff * preserves orientation, and is a *-antiisomorphism iff * reverses orientation. [AS, Thm. 5.71] A 158. Let be a Jordan isomorphism of a von Neumann algebra

M onto a von Neumann algebra N. Then there exists a central projection e such that restricted to eM is a *-isomorphism onto (e)N, and restricted to (1 - e)M is a *-anti-isomorphism onto (1 - (c))N. [AS, Cor. 5.76]

A 159. Every Jordan isomorphism of a C*-algebra A onto a C*algebra B is an isometry. [AS, Cor. 5.77] A 160. If

* is a Jordan compatible associative product on a C*-algebra

A, then the norm associated with * coincides with the original norm, so that Ai is a C*-algebra when equipped with the product * and the original involution and norm. [AS, Cor. 5.78]

A 161. Two projections e and f in a von Neumann algebra M are said to be equivalent (in the sense of Murray and von Neumann) if there exists a partial isometry v E M with initial projection e and final

APPENDIX

projection f, i.e., if there exists v E M [AS, Def. 6.2]

443

such that v*v

= e and vv* = f.

A 162. Let e and f be projections in a von Neumann algebra M. We write e ::S f if e is equivalent to a subprojection of f, and we write e --< f if e is equivalent to a subprojection of f but not to f itself. [AS, Def. 6.6]

A 163. Let e and f be two projections in a von Neumann algebra M with decompositions e = L:a e a and f = L:a fa where e a and fa are projections in M. If e a rv fa for all 0:, then e rv f. In fact, if e a = v~va and fa = vav~ for all 0: and v = L:a Va, then e = v*v and f = vv*. Also, veav* = fa for all 0:. [AS, Lemma 6.7] A 164. If e and f are projections in a von Neumann algebra M, then there exists a central projection c E M (with complementary projection c' = 1 - c) such that ce ::S cf

and

c' f ::S c' e.

[AS, Thm. 6.11] A 165. A projection e in a von Neumann algebra M is said to be an abelwn pro]ectwn if eM e is an abelian subalgebra of M. [AS, Def. 6.18] A 166. A projection e in a von Neumann algebra M is said to be infimte (relative to M) if it has a non-zero subprojection eo -=I- e such that eo rv e. Otherwise, e is said to be fimte (relative to M). If e is infinite and ce is also infinite for each central projection c such that ce -=I- 0, then e is said to be properly infimte. The von Neumann algebra M is said to be finite or properly mfimte if the identity element 1 is finite, respectively, properly infinite. [AS, Def. 6.12] A 167. Let M be a von Neumann algebra. M is said to be of type I if it has an abelian projection e such that c( e) = 1, and is said to be of type In (where n is a cardinal number) if 1 is the sum of n equivalent abelian projections. M is said to be of type I I if it has no non-zero abelian projection but has a finite projection e such that c(e) = 1, (and then is said to be of type III if 1 is a finite projection, and of type 1100 if 1 is a properly infinite projection). M is said to be of type I I I if it has no non-zero finite projections. [AS, Def. 6.22]

A 168. If M is a von Neumann algebra, then there is a cardinal number no and unique orthogonal central projections Pn (for each n ::; no), CI, coo, and q, summing to 1, and such that Pn M is of type In (or zero)

444

APPENDIX

for all n, clM is of type III (or zero), cooM is of type 1100 (or zero), and q M is of type II I (or zero). [AS, Thm. 6.23] A 169. Let {eij} be a self-adjoint system of n x n matrix units in a unital *-algebra M and let T be the subalgebra of M consisting of those elements commuting with all eij' For each t E M and each pair ~,J, the sum tij = Lk ekitejk is in T, and the map ¢ : t f-> [till is a *-isomorphism of M onto Mn(T). Moreover, eiiMeii is *-isomorphic to T for each ~, so M is also *-isomorphic to Mn(eiiMeii) for each L [AS, Lemma 6.26]

A 170. If M is a von Neumann algebra of type In (n < 00) with center Z and e is an abelian projection with central cover 1, then M is *-isomorphic to both Mn(Z) and Mn(eM e). [AS, Thlll. 6.27] A 171. We say two projections p and q in a C*-algebra are umtar~ly equwalent, and we write p "'u q, if there is a unitary v in the algebra such that vpv* = q. [AS, Def. 6.28] A 172. An element s in a von Neumann algebra M is a symmetry if 8* = 8 and 8 2 = 1, and is a partwl symmetry (or e -symmetry) if s* = s and s2 = e for a projection e EM. We say two projections p and q in M are exchanged by a symmetry, and we write p "'s q, if there is a symmetry 8 E M such that 8p8 = q. [AS, pp. 251-2] A 173. Each e-symmetry 8 in a von Neumann algebra M can be uniquely decomposed as a difference s = p - q of two orthogonal projections, namely p = ~ (e + 8) and q = ~ (e - 8). This is called the canomcal decomposdwn of 8. Conversely, each pair p, q of two orthogonal projections determines an e-symmetry 8 = P - q with e = p + q. Moreover, if 8 is an e-symmetry, then 8 = e8e = e8 = se, and s is a symmetry in the von Neumann subalgebra eMe. [AS, Lemma 6.33] A 174. Let 8 be a symmetry with canonical decomposition 8 = P - q (where q = pi) in a von Neumann algebra M with normal state space K, and let F and G be the norm closed faces in K associated with p and q respectively. Then the set of fixed points of Us is equal to the relative commutant {s y of 8 in M and also to the range space im E = pMp+qMq of the conditional expectation E = Up + Uq, and Us is the unique normal order preserving linear map of period 2 whose set of fixed points is equal to im E. Furthermore, F and G are antipodal and affinely independent faces of K, the restriction of the map E* to K is the unique affine projection (idempotent map) of K onto co(F U G) and the reflection U; determined by s is the unique affine automorphism of K of period 2 whose set of fixed points is equal to co(F U G). [AS, Thm. 6.36]

APPENDIX

445

A 175. Let e and f be two orthogonal projections in a von Neumann algebra M. Then e and f are unitarily equivalent iff e and f are exchanged by a symmetry iff e and f are Murray-von Neumann equivalent. [AS, Prop. 6.38] A 176. To each pair of projections p, q in a von Neumann algebra we assign the element c(p, q) = pqp + p' q' p' (which is a "generalized squared cosine", or "closeness operator" in the terminology of Chandler Davis). [AS, Def. 6.40 and Lemma 6.41] A 177. If p, q is a pair of projections in a von Neumann algebra M, then p - p 1\ q' and q - q 1\ p' can be exchanged by a partial symmetry. [AS, Prop. 6.49] A 178. Let p and q be distinct projections in a von Neumann algebra lip - qll < 1, then p and q can be exchanged by a unitary u E C*(p,q,l) which is the product of two symmetries in M and satisfies the inequality

M. If

111 - ull < hllp - qll· [AS, Thm. 6.54] A 179. Two unitarily equivalent projections p, q in a von Neumann algebra M can be exchanged by a finite product of symmetries. [AS, Prop. 6.56] A 180. If p and q are projections in a von Neumann algebra M, then there is a central symmetry c E M such that cP.:Sscq

and

c'q.:Ssc'p,

i.e., cp can be exchanged with a subprojection of cq by a symmetry, and likewise for c'q and c'p. [AS, Thm. 6.60] A 181. A bounded linear operator <5 acting on a C*-algebra A is said to be an order denvatwn if it leaves Asa invariant and restricts to an order derivation on A,a, or which is equivalent, if it is *-preserving (i.e., <5(x*) = <5(x)* for all x E A) and exp(t<5)(x) E A + for all x E A + and all t E R. [AS, Def. 6.61]

<5 a

:

A 182. If a is an element of a C* -algebra A, then the operator A --> A defined by 6a x = ~(ax + xa*) is an order derivation, and the

APPENDIX

446

associated one-parameter group is given by

[AS, Prop. 6.65] A 183. A bounded linear operator 0 on a von Neumann algebra M is an order derivation iff 0 = Om for some m EM; in particular 0 is an order derivation such that 0(1) = 0 iff 0 = Oia for some a E Msa. [AS, Thm. 6.68] A 184. If 0 is a linear operator on a von Neumann algebra M, then the following are equivalent: (i) 0 is an order derivation with 0(1) = 0, (ii) 0 = Oia for some a E M sa, (iii) 0 is a *-derivation (i.e., o(x*) = o(x)*, and o(xy) = o(x)y+xo(y) for all x, y EM), (iv) 0 is a *-preserving Jordan derivation (i.e., o(x*) = o(x)*, and o(x 0 y) = o(x) 0 y + x 0 o(y) for all x, y EM). [AS, Cor. 6.69] A 185. If p and q are complementary projections in a von Neumann algebra M (i.e., p+q = 1), then the associated norm closed faces Fp and Fq of the normal state space K (cf. (A 110)) are said to be complementary. We will say that an ordered pair (Fp, Fq) of complementary faces is a genemhzed axzs of K, and we will call (Fq, Fp) the opposite genemlzzed axzs of (Fp, Fq). [AS, Def. 6.77] A 186. Let (F, G) be a generalized axis of the normal state space K of a von Neumann algebra M. A one-parameter group of affine automorphisms of K will be called a genemhzed mtatwn of K about (F, G) if the set of fixed points is co(F U G) and the orbit of each w tf. co(F U G) is (affinely isomorphic to) a circle with center in co(F U G). [AS, Def. 6.78] A 187. A partial symmetry s in a von Neumann algebra M will be called balanced if it has a canonical decomposition s = p - q where p ~ q. [AS, Def. 7.4] A 188. Let rand s be e-symmetries for a given projection e in a von Neumann algebra and let l' = P - q be the canonical decomposition of r (cf. (A 173)). Then the following are equivalent: (i) r 0 s = 0, (ii) Usr = -1', (iii) UsP = q.

[AS, Prop. 7.5]

APPENDIX

447

A 189. An e-symmetry r in a von Neumann algebra M is balanced (cf. (A 187)) iff there is another e-symmetry s E M which is Jordan orthogonal to r. [AS, Cor. 7.6] A 190. A 3-frame in the normal state space of a von Neumann algebra is an ordered triple of generalized axes whose associated reflections satisfy the following two requirements: (i) The reflection about each of the three generalized axes reverses the direction of the other two generalized axes. (ii) The product of all three reflections in any order is the identity map. [AS, Def. 7.27] A 191. A Carteswn tnple of symmetr~es (r, s, t) in a von Neumann algebra M satisfies the two requirements:

(i) r 0 s = sot = tor = 0, (ii) UrUsUt = I (where I is the identity operator and Usx := sxs for all x E M and likewise for rand t). The Cartesian triples in M are in 1-1 correspondence with the 3-frames in the normal state space of M (cf. (A 190)) under the map which assigns to s the generalized axis (Fp, Fq) determined by the canonical decomposition s = p - q (cf. (A 173)) and likewise for rand t. If e is a projection in M, then a Carteswn tnple of e-symmetnes is a Cartesian triple of symmetries in the von Neumann algebra Me, and a Carteswn tnple is a Cartesian triple of e-symmetries for any projection e in M. [AS, Prop. 7.28 and Def. 7.29] A 192. If (r, s) is a Jordan orthogonal pair of symmetries in a von Neumann algebra M (i.e., if r 0 s = 0), then this pair can be extended to a Cartesian triple of symmetries, and the possible extensions are those triples (r, s, t) where t = lZrs for a central symmetry z EM. [AS, Prop. 7.31] A 193. There exists a Cartesian triple of symmetries in a von Neumann algebra M iff the identity element 1 is halvable, i.e., is the sum of two equivalent projections. [AS, Cor. 7.33] A 194. Let K be the normal state space of a von Neumann algebra. If K has a 3-frame, then K will be called a blown-up 3-ball. If a norm closed face F of K has a 3-frame, then F will be called a facwl blown-up 3-ball. [AS, Def. 7.37] A 195. Two projections p and q in a von Neumann algebra Mare homotopic iff they are unitarily equivalent. [AS, Prop. 7.49]

448

APPENDIX

A 196. If p and q are projections in a von Neumann algebra M with normal state space K, then

so the map e f---+ Fe is a homeomorphism from the set P of projections in M to the set :F of norm closed faces in K (with the Hausdorff metric). [AS, Lemma 7.54] A 197. Let M be a von Neumann algebra with normal state space K and assume that M has halvable identity, or which is equivalent, that K is a blown-up 3-ball. An onentation of M is a homotopy class of Cartesian triples of symmetries in M, or which is the same, a unitary equivalence class of such triples. Similarly, an olzentatwn of K is a homotopy class of 3-frames in K, or which is the same, a unitary equivalence class of such frames. [AS, Def. 7.67] A 198. We will denote the set of all halvable projections in a von Neumann algebra M by X, and we will assume that this set is equipped with the norm topology. We also will assume that the set T of all Cartesian triples in M is equipped with the topology inherited from M x M x M, where M has the norm topology. We will denote the set of all local orientations of M, i.e., the set of all equivalence classes [0:] of Cartesian triples in M, by 0, and we will equip this set with the quotient topology induced from T. Clearly, the set 0 of all local orientations of M is a bundle over X with the bundle map 7f which assigns the projection e to each orientation of a local subalgebra eM e. Thus 7f([0:]) = e" for each 10:] E O. [AS, Def. 7.70] A 199. Let e1 and e2 be halvable projections with e1 ::; e2 in a von Neumann algebra M, and let [0:1] and [0:2] be orientations of e1 Mel and e2 M e2 respectively. We will say that [0:1] is a ,est,zctwn of [0:2] and that [0:2] is an extension of [o:d, and we will write [0:1] « [0:2], if the representing Cartesian triples 0:1 = (11, Sl, td and 0:2 = (12, S2, t 2 ) can be chosen such that

(i) (ii)

'1« '2, 1/J",

=

1/J"2

on e1Me1· [AS, Def. 7.81]

A 200. A cross-section e : X f---+ 0 of the bundle 0 of local orientations of a von Neumann algebra M is called conszstent if e( ed « e( e2) for all pairs e1, e2 E X with e1 ::; e2, and a consistent continuous cross-section of 0 is called a global onentation of M. [AS, Def. 7.85]

A 201. A cross-section e : 8 f---+ 08 of the bundle 08 of oriented facial blown-up 3-balls of the normal state space K of a von Neumann

APPENDIX

449

algebra M is called cons7stent if B(BI) « B(B 2) for all pairs B I , B2 E B with BI C B 2 , and a consistent continuous cross-section of OB is called a global onentatwn of K. [AS, Def. 7.93]

A 202. A one-parameter group {UT }TER of linear operators on a real or complex vector space X will be called rotatwnal if the orbits are either fixed points or (affinely isomorphic to) circles traced out with common minimal period 27r. [AS, Def. 1.109]

A 203. An order derivation 8 of an ordered Banach space A will be called a rotatwnal derivation if 83 + 8 = 0, or, which is equivalent, if the associated one-parameter group {e Tb } TER is rotational. [AS, Def. 1.112] A 204. If 8 is a *-derivation of a von Neumann algebra M, then the following are equivalent: (i) 8 is a rotational derivation, (ii) sp(8) C {-7, 0, 7} and the spectral points are eigenvalues whose eigenspaces span M, (iii) 8 = 8is for a symmetry S EM. [AS, Thm. 6.76]

A 205. If B is a global orientation of a von Neumann algebra M, then we can assign to each balanced e-symmetry r (where e is halvable) a unique rotational derivation 'ljJ(r) of the local subalgebra eM e such that 'ljJ(r) = 'ljJexleMe for each Cartesian e-triple a such that the first entry of a is r and [a] = B(e). [AS, Lemma 7.95] A 206. If B is a global orientation of a von Neumann algebra M, then the map 'ljJ from balanced e-symmetries in M to rotational derivations of eM e defined above, satisfies the following conditions: (i) For each r, ker'ljJ(r) is equal to the relative commutant {ry n

eMe. (ii) If a unitary v E M carries rl to r2, then it carries 'ljJ(rd to 'ljJ(r2), i.e., Adv'ljJ(rl)Ad~1 = 'ljJ(r2) on e2Me2. (iii) If rl «r2, then 'ljJ(rd = 'ljJ(r2) on elMel (where el = rn. Conversely, each map 'ljJ which assigns rotational derivations on eM e to balanced e-symmetries in such a way that the conditions above are satisfied, arises in this way from a unique global orientation B. [AS, Thm. 7.96] A 207. If M dence between

is a von Neumann algebra, then there is 1-1 correspon-

(i) central symmetries in M which are 1 on the abelian part of M, (ii) Jordan compatible associative products in M,

450

APPENDIX

(iii) global orientations of M, (iv) global orientations of the normal state space K of M. The orientation associated with the given product on M assigns to each halvable projection e the equivalence class of the Cartesian triple (r, s, t) determined by e = 2 r st, and this orientation is called the natural global onentatwn associated with M. [AS, Thm. 7.103]

A 208. Let PI be the largest central abelian projection in a von Neumann algebra M. We will say a Cartesian e-triple in M is full iff c(e) = 1 - Pl. [AS, p. 338] A 209. Let M 1 and M 2 be von Neumann algebras with natural global orientations 8 1 and 82 respectively. Let : M 1 -> M 2 be a Jordan isomorphism, and let ex = (r, s, t) be a full Cartesian e-triple such that 2rst = e. Then the following are equivalent: (i) (ii) (iii) (iv)

is a *-isomorphism. is orientation preserving. (r st) = (r ) (s) (t). carries Bl(e) to 82 ((e)).

Similarly, the following are equivalent: (v) (vi) (vii) (viii)

is a *-anti-isomorphism. reverses orientation.

(rst) = (t)(s)(r) (= -(r)(s)(t)). carries 8 l (e) to the opposite of 82 ((e)).

[AS, Cor. 7.108]

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Index

A(K),422 A*,24 A+,8 A** ,60 At, 226 Ab(K), 421 Ap,25 a+, a-, 15 E, E K , 404 B((J",(J"a)' 412

B 3 ,358 B-,212 BO,212

b", / b, 37 C,251

C,90 C(a,1),8 C:(A), 121 c(e, f), 348 c(p), 56

c( 7r), 378 c( (J"), 109 D(A),32 8d ,30 ~t , 195 8, 195

e)..,46 F,251 F V G, 251 F 1\ G, 251 Fp, 142 Fa, 167

H,89 H n (B),80 K,393 K,393 M*,62

M tr , 178 0,89

OE, OE K , 404 P,251 Param(K), 404 , 122, 393 P..l Q, 231 P::s Q, 231 P*,75 p,269 a..l b, 15 F ..1 G, 231 p..l q, 425 p ..1 (J", 264 7ra,109 'if, 68 F', 143 pI, 75 p', 22 p '" q, 79 p "'s q, 444 p "'u q, 444 p V q, 48 p 1\ q, 48 R,90 R(A), 113 Ro(A), 113 R F ,359 T,393 T(a), 43 T)..(a),280 sp(a), 11 (J"a,411 TanF,211 TancF,211 Ta ,26 tr, 178 U,393 Ua ,9 W(a, 1),41 W';(M), 124

INDEX

460

Z(A),28 Z(D(A)), 194 [a, b], 23 [0", T], 170 [¢], 403 (".), 211 EB,56 EB e , 239 EB w , 219 E9a M a , 57 Oe, 172 {abc}, 8

abelian projection full, 450 in JBW-algebra, 85-86 in von Neumann algebra, 443 abelian state on C*-algebra, 442 on JB-algebra, 396 adjoin an identity, 34 algebra, 424 alternative algebra, 88 annihilator, 212 positive, 212 antipodal, 231, 423 approximate identity, 16 associated projection, face, compression for JBW-algebra, 142 associated projective unit, projective face, compression, 228 atom in JBW-algebra, 89, 164 in lattice, 96, 425 minimal projective unit, 316 atomic JBW-algebra, 96-101 atomic lattice, 316, 426 balanced partial symmetry, 446 Banach algebra, 424 Banach dual space, 62, 64, 433 Banach-power-associative algebra, 6

base

distinguished, see distinguished base base norm space, 58, 67, 420 bicomplemented positive projection on JBW-algebra, 75 on ordered linear space, 216222 bidual of C*-algebra, 434 of JB-algebra, 58-62 bipolar theorem, 424 blown up 3-ball, 358-365 blown-up 3-ball, 447 Boolean algebra, 257, 262 C*-algebra, 428 C*-product, 200, 372, 400-402 C-bicommutant, 262 canonical *-anti-automorphism of universal C*-algebra, 122127, 391 of von Neumann algebra, 125 canonical affine projection in ordered linear space context, 238 of JBW normal state space, 184 of von Neumann normal state space, 444 canonical decomposition of partial symmetry, 444 carrier projection, 140 in JBW-algebra, 161 in von Neumann algebra, 435 Cartesian triple in JBW-algebra, 127-137 of e-symmetries, 128, 389 of partial symmetries, 128 of symmetries, 128 Cartesian triple in von Neumann algebra, 447 of e-symmetries, 447 of symmetries, 447 center

INDEX

of a JBW-algebra, 55-58 of C* -algebra, 29 of JB-algebra, 28 of JBW-algebra, 84 of order derivations, 194 center-valued trace, 151 faithful, 151 normal, 151 central compression, 238 element in JB-algebra, 28 element of base norm space, 265 element of order unit space, 238 central carrier in JBW-algebra, 109 in ordered linear space context, 269, 339 in von Neumann algebra, 435 central cover of Jordan homomorphism, 378-382 of projection, 56 of representation of C*algebra, 439 central decomposition, 249 central support, see central carrier Choquet simplex, 294-297, 311, 420 closeness operator, 348-351, 445 cofinite, 327 commutant, 431 commutator ideal, 441 comparison of projections in JBW-algebra, 83-84 in von Neumann algebras, 443, 445 comparison theorem for JBW-algeba, 84 for von Neumann algebra, 443 compatible in ordered linear space context, 234-238 in orthomodular lattice con-

461 text, 52, 255-258 complement of compression, 228 of positive projection, 216, 420 of projective face, 228, 242 of projective unit, 228 com plementary face in JBW normal state space, 143 in the strong sense, see strongly complementary positive projection on ordered linear space, 75, 216, 420 projective face, 358, 446 split face, 420 complemented sublattice, 257 complete lattice, 49 compression on JB-algebra, 22 on JBW-algebra, 75, 218 on ordered linear space, 224 on von Neumann algebra, 218 concrete representation of JBalgebra, 108, 377 conditional expectation, 220, 436 cone, 419 proper, 419 conjugate Jordan representations, 380 conjugate linear, 380, 437 conjugate unitary, 437 conjugation, 437 associated with orthonormal basis, 437 Connes orientation, 191-196, 203 covering property, 101, 318, 319, 426 in JBW-algebras, 99 covers, 99, 317, 426 cross-section, 406 consistent, 448, 449 opposite, 406 cyclic representation, 429 cyclic vector, 429

462 dense factor representation, 377380 type I, 375, 382 dimension in JBW-algebra projection lattice, 99 of element in lattice, 326, 426 direct sum of algebras, 56 of convex sets, 239, 420 of wedges, 219 discrete linear functional, 292 distinguished base, 58, 421 distinguished order unit, 6 distributivity, 312 dynamical correspondence, 196207 complete, 197 e-symmetry in JBW-algebra, 80 in von Neumann algebra, 444 elliptic, 351 elli ptical cross-sections, 351 ellipticity, 187, 370 enveloping von Neumann algebra of C*-algebra, 434 equivalent, see also Murrayvon Neumann equivalent, unitarily equivalent projections, 79-83, 367 projective faces, 367, 368 equivalent projections, 101 exceptional ideal, 108, 111 exceptional Jordan algebra, 3, 107 exchangeable projections, 79-84 exchanged by a symmetry, 79-84, 367, 444 exposed, 317 exposed face, 419 exposed point, 419 F-bicommutant, 262 face, 23 face(X),157 facial 3-ball, 403

INDEX

facial blown-up 3-ball, 447 facial homogeneity, 188, 189, 207 facial property, 229, 273 factor representation, 108, 109 type I, see type I factor representation factor state, 109, 167 faithful representation, 377, 429 filter, 223, 235, 240, 259, 312, 316, 354 finite element of lattice, 316, 426 projection in atomic JBWalgebra, 99 projection in von Neumann algebra, 316, 443 von Neumann algebra, 443 finite covering property, 318, 320, 426 formally real Jordan algebra, 34 free convex sum, see direct sum of convex sets function systems, 6 functional calculus Borel, 286 continuous, 12 spectral, 286 fundamental units, 311 Gelfand-Naimark theorem, 429 Gelfand-Naimark type theorem for JB-algebras, 111 generalized 3-ball property, 365373 generalized axis, 297, 358, 446 associated with a symmetry, 358 opposite, 446 reverse, 359 generalized diameter, 297 generalized rotation, 297, 446 generating vector, 429 global 3-balls, 373 global orientation, 406 of C* state space, 441

463

INDEX

of normal state space of von Neumann algebra, 371, 449 of state space of JB-algebra, 376, 415 of von Neumann algebra, 448 natural, 450 GM-spaces, 312 GNS-representation, 429 halvability, 361 halvable projection, 447 hereditary Jordan subalgebra, 54 Hilbert ball, 168, 319 Hilbert ball property, 168-172, 319, 340 homogeneous cone, 188 facially, see facial homogeneity self-dual, 206 ideal center, 249 ideal of a Jordan algebra, 16 infinite projection in von Neumann algebra, 443 inverse, 10 Jordan, 10 irreducible concrete JW-algebra, 136 irreducible representation of C*-algebra, 431 of JB-algebra, 377 isomorphism of ordered normed spaces, 292 JB*-algebra,35 JB*-triples, 355 JB-algebra, 5 of complex type, 375, 383 JB-subalgebra, 25 JBW-algebra, 37 type I, see type I JBW-algebra type In, 86 type l oo , 86 JBW-factor, 87, 165 of type I, see type I JBWfactor JBW-subalgebra, 41 JC-algebra, 111, 112

acting on H, 6 concrete, 6 Jordan algebra, 3 exceptional, see exceptional Jordan algebra formally real, see formally real Jordan algebra special, see special Jordan algebra Jordan automorphism, 73 Jordan comparison theorem, see comparison theorem Jordan compatible product, 133, 199,400 Jordan coordinatization theorem, 80,88 Jordan derivation, 31, 74 Jordan equivalent homomorphisms, 377-382 Jordan orthogonal, 92, 360 Jordan product, 3, 5, 427 Jordan representation associated with pure state, 393 JW-algebra, 70-72, 103-108, 112 concrete, 70 JW-factor representation, 108 Kaplansky density theorem for JB-algebras, 69 Krein-Smulian theorem, 424 lattice, 425-426 of compressions, projective faces, projective units in ordered linear space context, 251-271 of projections and compressions in JBW-algebra, 48-53 of projections in von Neumann algebra, 434 Leibniz rule, 31, 194 linearization of an identity, 4 linearized Jordan axiom, 27 Macdonald's theorem, 9

464 matrix units (self-adjoint system), 114 measurement, 25, 235, 240, 259 Milman's theorem, 424 minimal projection, 89 Minkowski functional, 421 modular pair, 99 monotone (J -complete, 424 monotone closed, 63 monotone complete, 37, 424 Murray-von Neumann equivalent, 79, 442 neutral positive projection, 25, 223-225 non-commutative trigonometry , 349 norm closed face, 161, 294, 435 norm closed face generated by X, 157 norm exposed face, 161,294,435 norm exposed point, 188, 316 normal linear functional on JBW-algebra, 37, 63-64 on ordered Banach space, 37 on von Neumann algebra, 37, 433 normal positive map, 68 normal state space of B(H), 357 of JBW-algebra, 37, 343 of von Neumann algebra, 353, 357 normalized positive projection, 75, 222 normed algebra, 7, 424 observables, 6, 191, 196, 198, 234, 240, 259, 287, 302, 312, 416 octonions, 102 operational approach, 312 operator commute, 26-30 order automorphisms of JB-algebra, 73-74 order derivation on C*-algebra, 191-193,445

INDEX

on JB-algebra, 30-32 on ordered Banach space, 191, 429 self-adjoint, 31, 194 skew, 31, 194 skew-adjoint, see skew order derivation order unit, 421 Archimedean, 421 distinguished, see distinguished order unit, 422 order unit algebra, 7, 61, 426-428 order unit space, 6-7, 421-423 orientable, 376, 406 orientation Connes, see Connes orientation extension of, 448 global, see global orientation of 3-ball, 403, 438 opposite, 403, 438 of blown-up 3-ball, 448 of C* state space, 441 of convex set with 3-ball property, 406 of von Neumann algebra, 448 if identity is halvable, 448 of von Neumann normal state space, 371 on facial 3-ball induced by C*-algebra, 441 opposite, 416 restriction of, 448 orientation preserving, 403, 409, 438, 439, 450 orientation reversing, 403, 409, 438, 439 ortho-isomorphism, 363 orthocomplementation, 49, 143, 162, 425 orthogonal compressions, 231 elements of base norm space, 67, 264, 423 elements of JB-algebra, 15, 44

INDEX

elements of order unit space, 270, 339 elements of orthomodular lattice, 50, 425 generalized axes, 359 normal linear functionals on JBW-algebra, 140 projective faces, 231, 368 projective units, 231 orthogonal decomposition in base norm space, 67 in JB-algebra, 16 in order unit space, 270 of normal functionals on JBW-algebra, 65, 141 orthomodular lattice, 49, 77, 162, 255, 425 orthomodular law, 257 P-bicommutant, 262 P-projection, 249 parameterization, 403, 438 partial symmetry in JBW-algebra, 80 in von Neumann algebra, 444 Pauli spin matrices, 92, 361 physics, 6, 25, 76, 172, 196, 198, 223, 234, 235, 240, 241, 259, 287, 315, 316, 340, 350, 354, 372, 416 positive element of a C*-algebra, 428 power associative, 4, 5, 32, 426 predual of Banach space, 62 of JBW-algebra, 62-70 of von Neumann algebra, 433 preserves extreme rays, 168, 315, 318, 319, 322 preserves orientation, see orientation preserving projection dual, 213 element of JB-algebra, 22 neutral, see neutral positive

465 projection normalized, see normalized positive projection on ordered linear space, 213 smooth, see smooth positive projection projections exchanged by a symmetry, see exchanged by a symmetry projective face of convex set, 227-249 of normal state space of JBW-algebra, 142, 161 projective geometry, 102 projective unit, 227-241 properly infinite projection in von Neumann algebra, 443 properly infinite von Neumann algebra, 443 proposition, 234, 240, 259, 302, 350 pure state on C*-algebra, 429 on JB-algebra, 109, 165 on order unit space, 422 pure state properties, 316, 340, 415 pure states associated with Jordan representation, 393 purely exceptional Jordan ideal, 110 quadratic ideals, 77 quantum logic, 102, 260, 355 quantum mechanics, 6, 34, 76, 191, 196, 234, 259, 302, 354 quasi-equivalent representations, 377, 439 quotient of JB-algebra, 16-22 radially compact, 293, 421 range projection in JBW-algebra, 43 in order unit space, 263 in von Neumann algebra, 434 reflection about generalized axis, 359

466 of convex set, 184, 321 reflexive, 289 regular imbedding, 294 representation of a *-algebra, 429 representing vector, 392, 393 resolution of the identity, see spectral resolution reverses orientation, see orientation reversing reversible Jordan subalgebra of l3(H) , 113-120, 129, 389 Riesz decomposition property, 295 rotational derivation, 449 one-parameter group, 449 Sasaki projections, 267 self-dual cone, 206 semi-exposed face, 212, 262, 419 generated by set of points, 419 semi-exposed point, 419 semimodular lattice, 99 separated by a split face, 170 separating order and norm duality, 45, 422 separating ordered duality, 419 Shirshov-Cohn theorem, 9 iT-convex hull, 177, 323, 340, 414 iT -strong topology on *-subalgebra of l3(H) , 432 on JBW-algebra, 38 iT-weak topology on *-subalgebra of l3(H) , 432 on JBW-algebra, 38 smooth at an extreme point, 298 convex set, 298 positive projection, 214 space of facial 3-balls, 404 space of oriented facial 3-balls, 404, 440 special Jordan algebra, 3, 107 spectral convex set, 291-312, 352, see also spectral duality spectral duality, 271-291, see also

INDEX

spectral convex set weak, 311 spectral projection, 47 spectral resolution, 46-48, 285, 434 spectral theorem for JBW-algebras, 46 for monotone complete

C R (X),425 for spaces in spectral duality, 283 spectral theory, 8 spectral units, 285 spectrum of an element, 11 spin factor, 91-95, 103, 168 split face, 420 generated by a normal state, 167 standard 3-frame of B3 , 361 standing hypothesis, 251 *-anti-isomorphism, 354 state on C*-algebra, 429 on JB-algebra, 24 on order unit space, 422 state space of C*-algebra, 375-417, 429 of JB-algebra, 24, 315-343 of order unit space, 422 Stone-Weierstrass theorem (lattice version), 424 strictly convex, 298 strong operator topology, 432 strongly complementary, 216, 420 strongly connected, 82, 88 strongly spectral convex set, 296, 342 support projection, see carrier projection symmetric tube domains, 35 symmetric with respect to convex subset, 184, 321 symmetry element in JB-algebra, 55 element in von Neumann algebra, 444

INDEX

of convex set, 184, 322 of normal state space of JBW-algebra, 184 symmetry of transition probabilities, 172, 316, 323 tangent space, 211-220, 244 3-ball, 357, 382, 403, 438 standard, 357 3-ball property, 357, 375, 383, 414 3-frame, 297, 358, 360, 447 trace B(H) context, 432 atomic JBW-algebra context, 178 trace class B(H) context, 432 atomic JBW-algebra context, 178 trace norm B(H) context, 432 atomic JBW-algebra context, 178 trace simplex, 311 tracial state faithful, 149 on JB-algebra, 149 on JBW-algebra, 147-157 on order unit space, 296 transition probability, 172, 354, see also symmetry of transition probabilities transpose map with respect to orthonormal basis, 438 triangular pillow, 301 type I factor representation, 165, 382 type I JBW-algebra, 85-87 type I JBW-factor, 87-96, 98, 165 complex type, 338 quaternionic type, 338

467

real type, 338 unital order automorphisms of JB-algebra, 73 unitarily equivalent Jordan representations, 380 projections in C*-algebra, 444 universal C* -algebra for JBalgebra, 120-124, 127 universal representation of C*algebra, 432 universal von Neumann algebra for JBW-algebra, 124-127 universally reversible, 118, 119, 124, 126 vertical, 303 vertical subspace, 303 von Neumann algebra abstract, 433 concrete, 433 type 1,443 type 11,443 type II I, 443 type Ih, 443 type I 100 , 443 type In, 443 von Neumann product, see W*product w* -closed face, 294 w* -exposed face, 294 W*-product, 200, 202, 205, 372 w*-topology on state space, 294 weak integral, 286 weak operator topology, 432 weak topology defined by spaces in duality, 222 weakly compact, 289 wedge, 219 working hypotheses of this section, 326